research article global optimal energy management strategy

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 708261 11 pageshttpdxdoiorg1011552013708261

Research ArticleGlobal Optimal Energy Management StrategyResearch for a Plug-In Series-Parallel Hybrid ElectricBus by Using Dynamic Programming

Hongwen He Henglu Tang and Ximing Wang

National Engineering Laboratory for Electric Vehicles Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Hongwen He hwhebitbiteducn

Received 3 August 2013 Revised 5 October 2013 Accepted 5 October 2013

Academic Editor Hui Zhang

Copyright copy 2013 Hongwen He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Energy management strategy influences the power performance and fuel economy of plug-in hybrid electric vehicles greatly Toexplore the fuel-saving potential of a plug-in hybrid electric bus (PHEB) this paper searched the global optimal energymanagementstrategy using dynamic programming (DP) algorithm Firstly the simplified backward model of the PHEB was built which isnecessary for DP algorithm Then the torque and speed of engine and the torque of motor were selected as the control variablesand the battery state of charge (SOC) was selected as the state variables The DP solution procedure was listed and the way waspresented to find all possible control variables at every state of each stage in detail Finally the appropriate SOC increment isdetermined after quantizing the state variables and then the optimal control of long driving distance of a specific driving cycle isreplaced with the optimal control of one driving cycle which reduces the computational time significantly and keeps the precisionat the same time The simulation results show that the fuel economy of the PEHB with the optimal energy management strategyis improved by 537 compared with that of the conventional bus which can be a benchmark for the assessment of other controlstrategies

1 Introduction

In recent years the problems of energy shortage and envi-ronmental pollution have greatly promoted the develop-ment of electric vehicles (EVs) Among the EVs the pureelectric vehicles (PEVs) run with zero emissions and renew-able electricity but their disadvantages such as the shortoperation range high battery price and long battery chargingtime have limited the userrsquos acceptability The hybrid electricvehicles (HEVs) have longer operation range and higherperformance than PEVs but the electricity that keeps thebattery state of charge (SOC) in a narrowwindow is still fromthe onboard fossil fuel [1 2]While the plug-in hybrid electricvehicle (PHEVs) with larger battery capacity can run a longpure electric mileage and make full use of the cheap powerfrom gird hence it is more competitive than EVs and chargesustainable HEVs [3]

The energy management strategy is one of the key factorsthat influence the fuel economy and power performance ofthe PHEVs In the PHEVs in order to make full use of

the electricity energy stored in batteries it is preferred that thebattery energy drops to its minimumwhen the vehicle arrivesat the destinationTherefore the energymanagement strategybecomes more complicated than that of the HEVs Similarto HEVs the energy management strategies in PHEVs canbe usually classified into two categories rule-based con-trol strategies and optimization-based control strategies [4]The main idea of rule-based control strategies is to makeeach component work in efficient area individually [5ndash7]The reference [2] put forward a PHEV rule-based controlstrategy after considering the all-electric range and chargedepletion range operations The reference [5] proposed aPEHV rule-based energy management strategy by using theADVISOR The rule-based control strategies are simple andeasy to implement but they cannot safeguard the systematicoptimization and cannot fully exploit the advantages ofPHEVs

The optimization-based control strategies include globaloptimization and real-time optimization The real-time opti-mization realizes a local optimum step by step real timely

2 Mathematical Problems in Engineering

and loses the potential to get a global optimumThe adaptivecontrol is a good example of real-time optimization andthe Hinfin control theory is powerful in adaptive control [8ndash11] The global optimization finds an optimal solution forthe whole process which is suitable for energy managementissues of the PHEB with regular driving cycle For exampledynamic programming (DP) algorithm which is effectiveto solve the constrained and nonlinear optimization prob-lems is selected to realize a global optimization of energymanagement for HEVs [12] The reference [12] studied anoptimal energy management of a parallel HEV with theknown driving cycle using DP algorithm The reference [13]built the driving cycle model using traffic information withthe help of intelligent transportation systems and utilizedthe DP algorithm to study the global energy managementoptimization of a parallel plug-in hybrid electric sport utilityvehicle (SUV) The reference [14] presented a way on how toimplement the DP algorithm in the optimization of HEVsand carried out a global optimization with Toyota Priusas an example The reference [15] determined an optimalenergy management law for a two-clutch single-shaft parallelHEV by using optimization software named KOALA Bycomparisons the DP algorithm has been proved to bepowerful and effective in the global optimization of controlstrategies in HEVs In this paper a global optimization ofthe energy management strategy for a plug-in series-par-allel hybrid bus (PHEB) is explored and the battery energystate control is specially considered and discussedThe PHEBmodel was built in Section 2 and the global optimizationproblem with DP algorithm was put forward in Section 3The DP numerical computation method was discussedand put forward in Section 4 and the simulation resultswere given in Section 5 Section 6 gives the main conclu-sions

2 Plug-In Series-Parallel Hybrid ElectricBus Modeling

21 Plug-In Series-Parallel Hybrid Electric Bus ConfigurationFigure 1 shows a schematic view of the PHEB powertrainwhich includes a diesel engine an integrated starter generator(ISG) motor and a main drive motor The ISG is connectedto engine through a torsion damper There is an on-off modeclutch between the ISGmotor and the main drive motorThePHEB works in parallel mode or engine-only mode when themode clutch is in ldquoONrdquo condition and the diesel engine ISGmotor and main drive motor drive the wheel mechanicallyWhile the PHEBworks in its series mode or all-electric modewhen the mode clutch is in ldquoOFFrdquo condition the main drivemotor drives the wheels directly and the diesel engine drivesthe ISG motor to generate electricity or not according to thebattery state of charge The main specific parameters of thePHEB are listed in Table 1

22 Plug-In Series-Parallel Hybrid Electric Bus SimulationModels There are two modeling methods in PHEV simu-lations One is forward modeling which is more accurate

Diesel engine

ISG motor Model clutch

Main drivemotor Final drive

Figure 1 The plug-in series-parallel hybrid electric bus powertrainconfiguration

Table 1 Main specific parameters of the plug-in series-parallelhybrid electric bus powertrain

Diesel engineMaximum power (kW) 147Maximum torque (Nm) 730

ISG motorMaximum power (kW) 55Maximum torque (Nm) 500

Drive motorMaximum power (kW) 166Maximum torque (Nm) 2080

BatteryCapacity (Ah) 60Voltage (V) 580

Final drive ratio 617Curb weight (kg) 12500Gross weight (kg) 18000Air resistance coefficient 055Frontal area (m2) 66Tire dynamic radius (mm) 473

but with heavier computational burden and is always usedto test the vehicle dynamic performance and drivability Theother one is the backward modeling which is calculated withfixed time steps ignoring the dynamics of the powertraincomponents and usually is used to evaluate the vehicle fueleconomy [16] Since the global optimization is based on thefixed driving cycle and the DP problem is solved backwardfrom the terminal of the driving cycle we built the facing-backward simulation models as follows

The diesel engine is modeled as a 3-dimension look-up table where the inputs are the engine torque and speedand the output is the fuel consumption rate as shown inFigure 2The fuel consumption efficiencymap is based on theexperimental data

The ISG motor is modeled as a 3-dimension look-uptable based on the experimental efficiency map as shown in

Mathematical Problems in Engineering 3

800700600500400300200100

1000 1500 2000 2500

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

Te-max

BSFC (g(kWmiddoth))

Figure 2The fuel consumption efficiency map of the diesel engine

ISG speed (rmin)

ISG

torq

ue (N

m)

100 400 700 1000 1300 1600 1900 2200 2500 2800minus500minus400minus300minus200minus100

0100200300400500

Efficiency ()TISG-max

Figure 3 The ISG efficiency map

Figure 3 Due to the limit of battery power the output torqueof ISG 119879ISG is described as follows

119879ISG

=

min (119879ISG req 119879ISG dis max (119899ISG)

119879ISG bat dis max (119899ISG SOC)) 119879ISG req ge 0

max (119879ISG req 119879ISG chg max (119899ISG)

119879ISG bat chg max (119899ISG SOC)) 119879ISG req lt 0

(1)

where 119879ISG req is the required ISG torque 119899ISG is the ISGspeed SOC is the battery state of charge (SOC) 119879ISG dis maxand 119879ISG chg max are the maximum ISG torque when driv-ing and generating respectively and 119879ISG bat chg max and119879ISG bat dis max are the torque limits due to battery currentlimits in the charging and discharging modes which arefunctions of ISG speed and torque

The main drive motor is modeled as a 3-dimension look-up table based on the experimental efficiency map as shown

Efficiency ()Tm-max

300 600 900 1200 1500 1800 2100 2400Motor speed (rmin)

200016001200

800400

0minus400minus800

minus1200

Mot

or to

rque

(Nm

)

Figure 4 The efficiency map of the main drive motor

in Figure 4 Due to the limit of battery power the outputtorque of the main drive motor 119879

119898is as follows

119879119898=

min (119879119898 req 119879119898 dis max (119899119898)

119879119898 bat dis max (119899119898 SOC)) 119879

119898 req ge 0

max (119879119898 req 119879119898 chg max (119899119898)

119879119898 bat chg max (119899119898 SOC)) 119879

119898 req lt 0

(2)

where 119879119898 req is the required torque of the main drive motor

119899119898

is the speed of the main drive motor 119879119898 dis max and

119879119898 chg max are the maximum torque when driving and

regenerative braking respectively and 119879119898 bat chg max and

119879119898 bat dis max are the torque limits due to battery current limits

when charging and discharging respectively which are thefunctions of the main drive motor speed and torque

The static equivalent circuit battery model described in[12] is usedThemodel inputs are the speed and torque of ISGand main drive motor the model output is the battery SOCwhich is calculated by (3)

SOC (119896 + 1) = SOC (119896)

minus (119881oc minus (1198812

oc minus 4 (119877int + 119877119905)

times (119879119898sdot 119899119898sdot 120578minus sgn(119879

119898)

119898

+ 119879ISG sdot 119899ISG sdot 120578minus sgn(119879ISG)ISG ))

12

)

times (2 (119877int + 119877119905) sdot 119876119887)minus1

(3)

where 119877int is the internal resistance 119881oc is the open-circuitvoltage 119877int and 119881oc are the function of SOC 119876

119887is the

maximum battery capacity 119877119905is the terminal resistance

120578119898

and 120578ISG are the efficiencies of the main drive motorand ISG accordingly and 119896 denotes the calculation step indiscretization way


700 1000 1300 1600 1900 2200 2500

800700600500400300200100

0

Torq

ue (N

m)

P-out (kW) Opt-line

Speed (rmin)

TISG-min

Te-maxBSFC (g(kWmiddoth))

Figure 5 Engine-ISG fuel consumption map

The mode clutch works in three conditions disengagedwhere clutch = 0 engaged where clutch = 1 and half-engaged where 0 lt clutch lt 1 The transition duration isvery small less than the time interval of the sample pointof the driving cycle so only two working conditions areconsidered in the dynamic optimization If clutch = 1 thetorque of engine and ISGmotor can be delivered to final drivecompletely If clutch = 0 only themain drivemotor drives thevehicle

When the PHEB works in series hybrid mode takingthe efficiency of engine and ISG into consideration theminimum fuel consumption curve can be found as shownin Figure 5 For each required 119875ISG only the points on theminimum fuel consumption line are considered

The time interval is set to 1 second Assuming that thetorques of the engine main drive motor and ISG remainconstant in one time interval we can calculate the vehicledynamics as

VV (119896 + 1) = VV (119896)

+

1

120575119872

(

119879req (119896) sdot 119894119900 sdot 120578

119903119889

minus

VV (119896)1003816100381610038161003816VV (119896)

1003816100381610038161003816

(119865119891+ 119865119886(VV (119896))) )

(4)

where 119879req is the total required torque as the input of thefinal drive 120578 is the mechanical powertrain efficiency 119894

119900is the

final drive ratio VV is the vehicle speed 119903119889 is the dynamic tireradius 120575119872 is the effective mass of the vehicle and 119865

119886and

119865119891are aerodynamic drag force and rolling resistance force

respectivelyAccording to (4) we can get the 119879req as the driving

cycle is known in advance For the PHEB powertrain therelationship between 119879req and the powertrain componentscan be expressed as

119879req (119896) = 119879119890 (119896) + 119879ISG (119896) + 119879119898 (119896) +119879119887(119896)

119894119900

(5)

where 119879119887is the hydraulic brake torque and 119879

119890is the engine

torque

3 Global Optimal Energy ManagementStrategy Modeling

For PHEB DP aims to find the control of each stage tominimize the cost function over the whole driving cyclesThecontrol variables and state variables are determined beforeDP problem is formulated The state variables includingvehicle speed VV and SOC reflect the operating state of thesystem As the driving cycle and the vehicle speed VV inevery stage are known SOC is chosen as the state variableThere are many control variables in the PHEB such as enginetorque 119879

119890 engine speed 119899

119890 motor torque 119879

119898 ISG torque

119879ISG and hydraulic brake torque 119879119887 but only three of them

are independent Here the 119899119890 119879119890 and 119879

119898are chosen as the

independent control variablesIn the discrete-time format the PHEB system can be ex-

pressed as

119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896)) (6)

where 119909(119896) and 119906(119896) are state vector and control vectorrespectively

For PHEB the price of electricity is very low comparedwith that of diesel when used to drive the same distance [13]so we focus our research on minimizing the fuel consump-tion The cost function is built as follows

119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896)) =

119873minus1

sum

119896=0

fuel (119896) (7)

where 119873 is the duration of the driving cycle and 119871 is theinstantaneous cost fuel denotes the diesel consumption

If the SOC drops below the lower limit the battery willnot supply electricity to the main drive motor To avoid thecondition that the engine cannot supply the required torqueanother cost function should be considered besides (7) thenthe cost function rewrites as follows

119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0

[fuel (119896) + 120572(119879req (119896) minus 119879119890 (119896)

minus 119879ISG (119896) minus 119879119898 (119896) minus119879119887(119896)

119894119900

)

2

]

(8)

where 120572 is a positive weighting factorConstrains (5) and (9) are necessary to ensure a smooth

operation of the engine ISG main drive motor and batteries


during the optimization Consider

119899119890-min le 119899119890 (119896) le 119899119890-max

119879119890-min (119899119890 (119896)) le 119879119890 (119896) le 119879119890-max (119899119890 (119896))

119879ISG-min (119899ISG (119896) SOC (119896)) le 119879ISG (119896)

le 119879ISG-max (119899ISG (119896) SOC (119896))

119879119898-min (119899119898 (119896) SOC (119896)) le 119879

119898(119896)

le 119879119898-max (119899119898 (119896) SOC (119896))

SOCmin le SOC (119896) le SOCmax

119899119898(119896) = 119899

119890(119896) = 119899ISG (119896) if clutch = 1

119899119890(119896) = 119899ISG (119896) if clutch = 0

119879119890(119896) + 119879ISG (119896) = 0 if clutch = 0

(9)

4 A Numerical Computation forthe DP Problem

Based onBellmanrsquos Principle ofOptimization the global opti-mization problem can be solved by dealing with a sequenceof subproblems of optimization backward from the terminalof the driving cycle [17 18] Then the DP problem can bedescribed by the recursive equation (10)-(11)The subproblemfor (119873 minus 1) step is

119869lowast

119873minus1(119909 (119873 minus 1)) = min

119906(119873minus1)

[119871 (119909 (119873 minus 1) 119906 (119873 minus 1))] (10)

For step 119896 (0 le 119896 lt 119873 minus 1) the subproblem is

119869lowast

119896(119909 (119896)) = min

119906(119896)

[119871 (119909 (119896) 119906 (119896)) + 119869lowast

119896+1(119909 (119896 + 1))] (11)

where 119869lowast119896(119909(119896)) is the optimal cost-to-go function at state119909(119896)

from stage 119896 to the end of the driving cycle and 119909(119896+1) is thestate in stage 119896+1 after the control 119906(119896) is applied to state 119909(119896)at stage 119896 according to (6)

41 Solving the DP Problem Backward The recursive equa-tion (10)-(11) is solved backward and quantization and inter-polation are needed to solve the equation The continuousstate SOC is discretized into finite grids first and the numberof discretized state 119878 is

119878 =

(SOCmax minus SOCmin)

120575SOC (12)

where 120575SOC is the increment of the discretized SOC andSOCmax and SOCmin are upper and lower constrains of SOC

Then find all possible control solutions at every state ofeach stageThe function 119869lowast

119896(119909(119896)) at every grid points of SOC

is evaluated and 119869lowast119896+1(119909(119896 + 1)) is evaluated by interpolation

if the calculated value of admissible SOC119896+1

in (3) does notfall exactly on grid points The way of interpolation is shownin [13] The procedure of solving the DP problem backward

Yes

Yes

Yes

Yes

No

No

No

No

k = k minus 1

j lt C

i lt Si = i + 1

j = j+ 1

Find the minimum cost J at stage k state i and therelevant optimum control u then save them in

Calculate the cost from stage k to N

Number of stage N

Number of state S

Calculate the required speed and torque at the

k ge 1

J(x(i) u(j) k) = L(x(i) u(j) k) +J(f(x(i) u(j) k) k + 1)

Calculate the cost from stage k + 1 toN J(f(x(i) u(j) k) k + 1) calculate the cost

from stage k to k + 1 at state x(i) with the controlu(j) (x(i) u(j) k)

k = N minus 1

Calculate all the possible controls u(j)

(j = 1simsimC C is the number of controls)j = 1

Calculate the new state f(x(i) u(j) k) at the stagek + 1 when control u(j) is applied to x(i) at stage k

Calculate the cost Jat stage N minus 1

J = L(x(i) u(j))

At the state i i = 1

At the stage N minus 1 k = N minus 1

stage k nreq(k) and Treq(k)

matrixes COSTmin(ki) and (ik) separately

COSTminUopt

Uopt

L

Figure 6 The flow chart of solving the DP problem backward

is shown in Figure 6 where the required speed 119899req is thesame as driving cycle and required torque is determined byinversely solving vehicle dynamic model as shown in (4)

411 To Find All Possible Control Solutions It is very impor-tant to find all possible control solutions in the procedureof solving the DP problems backward The possible controlsolutions are the possible combination of the discrete torqueof the components in each state of the driving cycle whichmeets the torque need of the vehicle The number of thecontrol solutions influences the accuracy of the optimizationgreatly and the way to search for all the control solutionsinfluences the computational burden significantly To get acompromise here we find the possible working modes of thePHEB first and then we find all possible control solutions inevery mode finally we get all the control solutions at everygrid point of SOC

The PHEB works in many modes such as engine-onlymode battery electric (EV) mode engine-ISG parallel mode


u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k

Figure 7 Schematic diagram of state transformation with controlvariables

engine-motor parallel mode and series mode The ISG isused as a starter and generator and will not drive the busdirectlyOnlywhen the torque required by the vehicle exceedsthe maximum torque that can be provided by the maindrive motor and engine together the ISG will provide theremaining torque

According to (6) the state variables in the 119909(119896 + 1) mayexceed the range of SOC as shown in Figure 7 where thestate at stage 119896 + 1 exceeds the range of SOC with the controlvariables 119906

1(1) and 119906

119899(3) To avoid this situation the control

variables should be limited We divide SOC into three areasSOCmax ge SOC ge SOChigh SOCmin le SOC le SOClow andSOClow lt SOC lt SOChighThe initial SOC and terminal SOCare usually in the area SOClow lt SOC lt SOChigh

(A) When the SOC is higher than high limit SOChigh themotor drives the bus without regenerative brakingOnly when the torque required by the vehicle exceedsthe maximum torque that can be provided by drivemotor ISGwill supply positive torque to drive the busIf more driving power is required the engine comesto work to supply the remaining torque

(B) When the SOC drops below the low limit SOClowthe battery will not supply electric energy any moreAccording to the required torque and required speedareas of final drive shown inFigure 8 the PHEBworksin different modes as listed in Table 2 The blue linein Figure 8 represents the maximum output torque ofthe motor with the power supplied by engine ISGwhen PHEB works in series mode

(C) If SOClow lt SOC lt SOChigh the torque that can besupplied by the powertrain components is shown inFigure 9 The possible working modes are shown inTable 3

If the PHEB works in series mode discretize the min-imum fuel consumption curve into finite points and theengineISG works on these points If the PHEB works inengine-motor parallel mode such as the PHEV which works

0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)

Figure 8 The required speed and torque of the final drive whenSOC lt SOClow

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min

Figure 9 The required speed and torque of final drive whenSOClow lt SOC lt SOChigh

on area 3 in Figure 9 the flow chart to find all possiblecontrols is shown in Figure 10(a) If the PHEB works inthe same mode on areas 4 5 and 6 in Figure 9 the initialcondition of 119879

119890is set to be 119879req If the PHEB works in engine-

ISG parallel mode the way to find the possible controls isshown in Figure 10(b)

42 To Find the Optimal Control Path Forward The optimalcontrols at every state point of every stage are obtainedby solving the DP problem backward if the initial SOC isspecified the optimal control path will be found forwardThe interpolation is also needed to find the optimal controlpath as shown in Figure 11 If the optimal control at stage 119896is 119906119896 the optimal control 119906

119896+1at stage 119896 + 1 is got through

interpolation between the controls 119906119896+1(119894) and 119906

119896+1(119894 + 1)

which are the optimal controls at state grid points 119909(119894) and119909(119894 + 1) respectively at stage 119896 + 1

5 Simulation Results

For the PHEB it is reasonable to make full use of the batteryenergy Considering the health and the efficiency of thebattery the low level the high level and the initial SOCs were


ne = nreq Te = Te-maxTISG = 0 j = 0

Tm = Treq minus Te

j = j + 1

Te = Te minus 120575Te

Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No

u(j) = (ne Te Tm TISG)

(a) For engine-motor parallel mode

ne = nreq Te = TreqTm = 0 j = 0

TISG = Treq minus Te

j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u


(b) For engine-ISG parallel mode

Figure 10 The way to find possible control solutions

Table 2 Working modes when SOC lt SOClow

The required speed andtorque areas of the finaldrive in Figure 8

PHEB possible working mode Remarks

1 Series mode EngineISG works in maximum power point2 Engine-only mode Engine only drives the PHEB

3 Engine-ISG parallel mode Engine drives the PHEB and the ISG generates as much electricenergy as possible to charge the battery

x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2

Figure 11 The schematic diagram of interpolation

selected to be 03 08 and 06 respectively The driving cycleis the Chinese typical urban drive cycle (CTUDC) as shownin Figure 12 The total distance of one CTUDC is 5897 kmand the duration of one CTUDC is 1314 seconds

The battery capacity of PHEB is much higher than thatof HEVs and the PHEB drives with the mode of one day

0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)

Figure 12 The profile for one CTUDC driving cycle

one charge so the PHEB would drive for many consecutivedriving cycles To show the energy distribution between theengine ISG and main drive motor the model is simulatedwith the input of 15 consecutive CTUDC cycles The incre-ment of discretized SOC 120575SOC is selected to be 0001 theincrement of engine torque 120575119879

119890is selected to be 5Nm the

weighting factor 120572 is selected to be 100 and the PHEB weightof simulation is set to be the gross weight

Figure 13 shows the simulation result of SOC underthe DP optimal control for 15 consecutive CTUDC cycles


Table 3 Possible PHEB modes when SOClow lt SOC lt SOChigh

Required torque andspeed areas of final drive PHEB possible working modes Remarks

1 EV mode series mode If the motor cannot supply sufficient torque the ISGmotor will provide the remaining torque

2 Engine-ISG-motor parallel modeThe required torque of final drive exceeds the maximumtorque provided by engine and motor together and ISG

motor provides remaining torque3 Engine-motor parallel mode None

4 EV mode series mode engine-motor parallelmode None

5 engine-ISG parallel mode engine-motor parallelmode engine-only mode None

6 Engine-only mode EV mode engine-ISG parallelmode engine-motor parallel mode series mode None

06

05

04

03

SOC

0 05 1 15 2times104

Time (s)

Figure 13 The SOC simulation result for 15 CTUDC cycles

The SOC decreased to 0314 when the bus reached thedestination Because of the regenerative braking at the endof the cycle the terminal SOC is a little higher than the lowlevel but it is still very close to the low level so the bus canmake full use of the battery energy with the optimal control

The SOC decreases evenly from the initial SOC to thelow level of the SOC When the optimal control was appliedto the 15 consecutive CTUDC cycles the SOC reductionwould be 002 for one cycle on average To relieve the heavycomputational burden the optimal control for one CTUDCcycle can be used as the optimal control for 15 consecutiveCTUDC driving cycles through restricting the initial SOCand terminal SOCThe initial SOC and desired terminal SOCare selected to be 05 and 048 respectively To ensure that theSOC at final time is the desired value an additional terminalconstrain on SOC needs to be imposed and the cost functionwould be

119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0

[fuel (119896) + 120572(119879req (119896) minus 119879119890 (119896) minus 119879ISG (119896)

minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)

Simulation resultCTUDC

Figure 14 Velocity comparison between simulation and CTUDCcycle

SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046

Figure 15 The SOC simulation result for one CTUDC cycle

where 120573 is positive weighting factor and SOC119891is the desired

SOC at the end of driving cycleThe increment of SOC 120575SOC is selected to be 0001 the

increment of engine torque 120575119879119890is selected to be 5Nm and

the weighting factors 120572 and 120573 are selected to be 100 and 1 times107 respectively The difference between simulation result ofvelocity and desired vehicle velocity is very small as shown inFigure 14 Figure 15 shows the simulation result of SOC withthe optimal control for one driving cycle and the terminalSOC turned out to be 04804 which is very close to thedesired value


Table 4 The simulation results for different driving cycles

Number of driving cycle 15 1Distance (km) 8846 5897Initial SOC 06 05Terminal SOC 0314 04804Electric energy consumption (kWh) 995 0682Fuel consumption (L) 1744 1174Fuel consumption per 100 km (L100 km) 1972 1990Computation time (s) 502208 26432

The simulation results for 15 consecutive driving cyclesand one driving cycle are shown in Table 4 The fuel con-sumption per 100 km of 15 driving cycles is 1972 L andthe fuel consumption per 100 km of one driving cycle withrestricted terminal SOC is 1990 L The fuel consumptionper 100 km increased by 091 while the computation timedecreased by 947 Considering that the internal resistanceof the battery is a function of SOC the optimal control ofone driving cycle with restricted terminal SOC is appliedto the simulation for 15 consecutive driving cycles and thesimulation result is shown in Table 5

With the optimal control for one driving cycle the fuelconsumption per 100 km increased by 091 and the electricenergy consumption increased by 32 but the terminal SOCis still higher than the low level The computation time ofsolving DP problem to find the optimal control decreasedsignificantly so it is feasible to find the optimal controls forconsecutive driving cycles by solving the DP problem for onedriving cycle with restricted initial SOC and terminal SOC

The state increment 120575SOC in (12) also influences the accu-racy of the optimization If the 120575SOC is smaller the quantizedsearch area will be larger hence the computational burdenwill be heavier To study the tradeoff between accuracy of theoptimization and computation time 120575SOC is selected to be00005 0001 and 0005 respectively The SOC simulationresults are shown in Figure 16 The terminal SOC droppedto 04802 04804 and 04806 respectively which are veryclose to the desired valueWhen 120575SOC is selected to be 0001the curve of SOC is very similar to the curve when 120575SOC isselected to be 00005

The fuel consumption and computation time results aresummarized in Table 6 Compared with the results when120575SOC is 00005 the fuel consumption increased by 061and the computation time decreased by 539 when 120575SOCis 0001 while the fuel consumption increased by 703 andthe computation time decreased by 910 when 120575SOC is0005 Considering the tradeoff between fuel consumptionand computation time it is feasible to set 120575SOC to be 0001And the simulation results in this case are shown in Figure 17

The output torque of the PHEB components is shown inFigure 17 It can be seen that the ISG seldom works as a gen-erator to charge the battery and most of the negative poweris from regenerative braking If the ISG works as a generatorthere are engine efficiency losses ISG efficiency losses maindrive motor efficiency losses and battery efficiency losseshence the system is ineffective so in most cases the optimal

0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005

Figure 16 The SOC simulation results with different 120575SOCs

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0

0minus100minus200minus300

Te

(Nm

)T

m(N

m)

TIS

G(N

m)

Figure 17 The simulation results with DP optimal control when120575SOC is 0001

0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e

Figure 18 The PHEB working modes under optimal control strat-egy

control strategy based on DP avoids the situation when theISG works as a generator At the end of the cycle to forcethe terminal SOC to be the desired value the ISG suppliednegative power to charge the battery

The working modes of the PHEB is shown in Figure 18where mode = 0means that the bus stops and no powertraincomponents is working mode = minus1 means regenerative


1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor

Figure 19 Working areas of the components

Table 5 Simulation results for different control strategies

Control strategy Number ofdriving cycle Initial SOC Terminal SOC Electric energy

consumption (kWh) Fuel consumption (L)Fuel consumption

per 100 km(L100 km)

DP optimal controlfor 15 consecutivecycles

15 06 0314 995 1744 1972

DP optimal controlfor one cycle withrestricted initialand terminal SOCs

15 06 0305 1027 17603 1990

Table 6 Simulation result of fuel consumption and computationtime

120575SOC 00005 0001 0005Fuel consumption per 100 km (L100 km) 1978 1990 2117Computation time (s) 57334 26432 5174

braking mode mode = 1means EV mode mode = 2meansengine-only mode mode = 3 means parallel mode andmode = 4 means series mode It can be seen that the systemdoes not work in series mode in case of the low efficiency

When the PHEBworks in full load condition and the loadrate of engine is high the engine can work in high efficiencyarea without need of the load regulation of ISG so theISG seldom supplies negative torque The working points ofengine andmain drivemotor are shown in Figure 19 It showsthat the engine works in high efficiency areas in most casesThe main drive motor drives the bus alone when the vehiclespeed is lowThe engine works if the vehicle required speed ishigh and themain drive motor supplies the remaining torqueto maintain the engine working in high efficiency The ISGseldom works under the optimal control based on DP butit does not mean that the ISG is useless because the energycontrol strategy of the PHEB is based on rules in reality thedriving distance is not a fixed value and the ISG is neededto charge the battery to maintain the SOC level There is

no standard PHEB fuel consumption for rule-based energymanagement control strategy and the PHEB may consumemore fuel than traditional bus if the control rule parameterswere not properly designed So we made a comparison ofthe fuel consumption between the PHEB with optimizedcontrol strategy and the prototype conventional diesel busThe results show that the experimented fuel consumptionof the prototype conventional diesel bus is 43 L100 km inCTUDC driving cycle and the fuel consumption of theoptimized PHEB is 1990 L100 km with additional electricityconsumption of 1161 kWh100 km the fuel consumptiondecreased by 537 The optimal control based on DP canimprove fuel economy significantly It should be noted thatthe fuel consumption results given by the optimal controlbased on DP are maximum potential gains and they cannotbe reached in a real vehicle because the entire driving cycleis known in advance and neither comfort constraints norhighly dynamic phenomena are taken into account [15]

6 Conclusions

It is very complicated to determine the energy manage-ment strategy for a series-parallel PHEB and the dynamicprogramming is a powerful tool to get global optimizationresults The backward simulation model of the series-parallelPHEB was built Then to explore the potential of fueleconomy the dynamic programming algorithm is utilized to


realize an optimal control on a known-in-advance drivingcycle The procedure of DP for the series-parallel powertraintopology is introduced in detail An appropriate methodis proposed to improve the computational efficiency whichcan reduce the computation burden greatly and keep theprecision of DP

The simulation results show that with the global optimalcontrol the battery SOC can reach its lower limit at theend of the cycle which means that the bus can make fullof the battery energy Meanwhile the ISG seldom works ingeneration mode under given cycle and SOC interval whichavoids the inefficient situation It is proved that the optimalcontrol based onDP can reduce the fuel consumption greatly

The drawback of optimal control based on DP is thatthe driving cycle should be known in advance and thecomputational burden is still very heavy so it is difficult tobe applied in a real vehicle In the further study a near-optimal control law will be extracted according to the globaloptimization results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National High Tech-nology Research and Development Program of China(2011AA11A228 2012AA111603 and 2011AA11A290) in partthe International Cooperation Research Program of ChineseMinistry of Science and Technology (2011DFB70020) in partand the Program for New Century Excellent Talents inUniversity (NCET-11-0785) in part The authors would alsolike to thank the reviewers for their corrections and helpfulsuggestions

References

[1] M Ehsani Y Gao and A Emadi Modern Electric HybridElectric and Fuel Cell Vehicles Fundamentals Theory andDesign CRC Press New York NY USA 2009

[2] Y Gao and M Ehsani ldquoDesign and control methodology ofplug-in hybrid electric vehiclesrdquo IEEE Transactions on Indus-trial Electronics vol 57 no 2 pp 633ndash640 2010

[3] CH Stephan and J Sullivan ldquoEnvironmental and energy impli-cations of plug-in hybrid-electric vehiclesrdquo Environmental Sci-ence and Technology vol 42 no 4 pp 1185ndash1190 2008

[4] S G Wirasingha and A Emadi ldquoClassification and review ofcontrol strategies for plug-in hybrid electric vehiclesrdquo IEEETransactions on Vehicular Technology vol 60 no 1 pp 111ndash1222011

[5] H Banvait S Anwar and Y Chen ldquoA rule-based energy man-agement strategy for plugin hybrid electric vehicle (PHEV)rdquo inProceedings of the American Control Conference (ACC rsquo09) pp3938ndash3943 St Louis Mo USA June 2009

[6] N J Schouten M A Salman and N A Kheir ldquoFuzzy logiccontrol for parallel hybrid vehiclesrdquo IEEE Transactions onControl Systems Technology vol 10 no 3 pp 460ndash468 2002

[7] P Pisu andG Rizzoni ldquoA comparative study of supervisory con-trol strategies for hybrid electric vehiclesrdquo IEEE Transactions onControl Systems Technology vol 15 no 3 pp 506ndash518 2007

[8] H Zhang Y Shi and M Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[9] H Zhang Y Shi and A S Mehr ldquoRobust 119867infinPID con-

trol for multivariable networked control systems with distur-bancenoise attenuationrdquo International Journal of Robust andNonlinear Control vol 22 no 2 pp 183ndash204 2012

[10] H Zhang Y Shi and A S Mehr ldquoRobust weighted119867infinfiltering

for networked systems with intermittent measurements of mul-tiple sensorsrdquo International Journal of Adaptive Control and Sig-nal Processing vol 25 no 4 pp 313ndash330 2011

[11] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

[12] C-C Lin H Peng J W Grizzle and J-M Kang ldquoPowermanagement strategy for a parallel hybrid electric truckrdquo IEEETransactions on Control Systems Technology vol 11 no 6 pp839ndash849 2003

[13] QGong Y Li andZ-R Peng ldquoTrip-based optimal powerman-agement of plug-in hybrid electric vehiclesrdquo IEEE Transactionson Vehicular Technology vol 57 no 6 pp 3393ndash3401 2008

[14] WWang and SM Lukic ldquoDynamic programming technique inhybrid electric vehicle optimizationrdquo in Proceedings of the IEEEInternational Electric Vehicle Conference pp 4ndash8 GreenvilleSC USA March 2012

[15] J Scordia M Desbois-Renaudin R Trigui B Jeanneret FBadin and C Plasse ldquoGlobal optimisation of energy manage-ment laws in hybrid vehicles using dynamic programmingrdquoInternational Journal of Vehicle Design vol 39 no 4 pp 349ndash367 2005

[16] W Zheng Reseach on power train parameter matching methodand control strategy for hybrid electric vehicle [PhD thesis]Harbin Institute of Technology Harbin China 2010

[17] Y Zou S J Hou and E L Han ldquoDynamic programming-basedenergy management strategy optimization for hybrid electriccommercial vehiclerdquo Automotive Engineering vol 34 no 8 pp663ndash668 2012

[18] R BellmanDynamic Programming PrincetonUniversity PressPrinceton NJ USA 1957

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and loses the potential to get a global optimumThe adaptivecontrol is a good example of real-time optimization andthe Hinfin control theory is powerful in adaptive control [8ndash11] The global optimization finds an optimal solution forthe whole process which is suitable for energy managementissues of the PHEB with regular driving cycle For exampledynamic programming (DP) algorithm which is effectiveto solve the constrained and nonlinear optimization prob-lems is selected to realize a global optimization of energymanagement for HEVs [12] The reference [12] studied anoptimal energy management of a parallel HEV with theknown driving cycle using DP algorithm The reference [13]built the driving cycle model using traffic information withthe help of intelligent transportation systems and utilizedthe DP algorithm to study the global energy managementoptimization of a parallel plug-in hybrid electric sport utilityvehicle (SUV) The reference [14] presented a way on how toimplement the DP algorithm in the optimization of HEVsand carried out a global optimization with Toyota Priusas an example The reference [15] determined an optimalenergy management law for a two-clutch single-shaft parallelHEV by using optimization software named KOALA Bycomparisons the DP algorithm has been proved to bepowerful and effective in the global optimization of controlstrategies in HEVs In this paper a global optimization ofthe energy management strategy for a plug-in series-par-allel hybrid bus (PHEB) is explored and the battery energystate control is specially considered and discussedThe PHEBmodel was built in Section 2 and the global optimizationproblem with DP algorithm was put forward in Section 3The DP numerical computation method was discussedand put forward in Section 4 and the simulation resultswere given in Section 5 Section 6 gives the main conclu-sions

2 Plug-In Series-Parallel Hybrid ElectricBus Modeling

21 Plug-In Series-Parallel Hybrid Electric Bus ConfigurationFigure 1 shows a schematic view of the PHEB powertrainwhich includes a diesel engine an integrated starter generator(ISG) motor and a main drive motor The ISG is connectedto engine through a torsion damper There is an on-off modeclutch between the ISGmotor and the main drive motorThePHEB works in parallel mode or engine-only mode when themode clutch is in ldquoONrdquo condition and the diesel engine ISGmotor and main drive motor drive the wheel mechanicallyWhile the PHEBworks in its series mode or all-electric modewhen the mode clutch is in ldquoOFFrdquo condition the main drivemotor drives the wheels directly and the diesel engine drivesthe ISG motor to generate electricity or not according to thebattery state of charge The main specific parameters of thePHEB are listed in Table 1

22 Plug-In Series-Parallel Hybrid Electric Bus SimulationModels There are two modeling methods in PHEV simu-lations One is forward modeling which is more accurate

Diesel engine

ISG motor Model clutch

Main drivemotor Final drive

Figure 1 The plug-in series-parallel hybrid electric bus powertrainconfiguration

Table 1 Main specific parameters of the plug-in series-parallelhybrid electric bus powertrain

Diesel engineMaximum power (kW) 147Maximum torque (Nm) 730

ISG motorMaximum power (kW) 55Maximum torque (Nm) 500

Drive motorMaximum power (kW) 166Maximum torque (Nm) 2080

BatteryCapacity (Ah) 60Voltage (V) 580

Final drive ratio 617Curb weight (kg) 12500Gross weight (kg) 18000Air resistance coefficient 055Frontal area (m2) 66Tire dynamic radius (mm) 473

but with heavier computational burden and is always usedto test the vehicle dynamic performance and drivability Theother one is the backward modeling which is calculated withfixed time steps ignoring the dynamics of the powertraincomponents and usually is used to evaluate the vehicle fueleconomy [16] Since the global optimization is based on thefixed driving cycle and the DP problem is solved backwardfrom the terminal of the driving cycle we built the facing-backward simulation models as follows

The diesel engine is modeled as a 3-dimension look-up table where the inputs are the engine torque and speedand the output is the fuel consumption rate as shown inFigure 2The fuel consumption efficiencymap is based on theexperimental data

The ISG motor is modeled as a 3-dimension look-uptable based on the experimental efficiency map as shown in


800700600500400300200100

1000 1500 2000 2500

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

Te-max

BSFC (g(kWmiddoth))


ISG speed (rmin)

ISG

torq

ue (N

m)


0100200300400500




119879ISG

=





(1)



Efficiency ()Tm-max

300 600 900 1200 1500 1800 2100 2400Motor speed (rmin)

200016001200

800400

0minus400minus800

minus1200

Mot

or to

rque

(Nm

)



119898is as follows

119879119898=

min (119879119898 req 119879119898 dis max (119899119898)

119879119898 bat dis max (119899119898 SOC)) 119879

119898 req ge 0

max (119879119898 req 119879119898 chg max (119899119898)

119879119898 bat chg max (119899119898 SOC)) 119879

119898 req lt 0

(2)


119899119898







SOC (119896 + 1) = SOC (119896)


oc minus 4 (119877int + 119877119905)


119898)

119898


12

)


(3)


119887is the


120578119898



700 1000 1300 1600 1900 2200 2500

800700600500400300200100

0

Torq

ue (N

m)

P-out (kW) Opt-line

Speed (rmin)

TISG-min






VV (119896 + 1) = VV (119896)

+

1

120575119872

(

119879req (119896) sdot 119894119900 sdot 120578

119903119889

minus

VV (119896)1003816100381610038161003816VV (119896)

1003816100381610038161003816

(119865119891+ 119865119886(VV (119896))) )

(4)


119900is the


119886and




119879req (119896) = 119879119890 (119896) + 119879ISG (119896) + 119879119898 (119896) +119879119887(119896)

119894119900

(5)


119890is the engine

torque





119898 ISG torque





pressed as

119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896)) (6)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896)) =

119873minus1

sum

119896=0

fuel (119896) (7)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0

[fuel (119896) + 120572(119879req (119896) minus 119879119890 (119896)


119894119900

)

2

]

(8)





119899119890-min le 119899119890 (119896) le 119899119890-max

119879119890-min (119899119890 (119896)) le 119879119890 (119896) le 119879119890-max (119899119890 (119896))


le 119879ISG-max (119899ISG (119896) SOC (119896))

119879119898-min (119899119898 (119896) SOC (119896)) le 119879

119898(119896)

le 119879119898-max (119899119898 (119896) SOC (119896))


119899119898(119896) = 119899

119890(119896) = 119899ISG (119896) if clutch = 1

119899119890(119896) = 119899ISG (119896) if clutch = 0

119879119890(119896) + 119879ISG (119896) = 0 if clutch = 0

(9)



119869lowast

119873minus1(119909 (119873 minus 1)) = min

119906(119873minus1)

[119871 (119909 (119873 minus 1) 119906 (119873 minus 1))] (10)


119869lowast

119896(119909 (119896)) = min

119906(119896)

[119871 (119909 (119896) 119906 (119896)) + 119869lowast

119896+1(119909 (119896 + 1))] (11)




119878 =


120575SOC (12)







Yes

Yes

Yes

Yes

No

No

No

No

k = k minus 1

j lt C

i lt Si = i + 1

j = j+ 1



Number of stage N

Number of state S


k ge 1




k = N minus 1





J = L(x(i) u(j))





COSTminUopt

Uopt

L






u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k




1(1) and 119906







0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min








119896+1(119894 + 1)






Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










800700600500400300200100

1000 1500 2000 2500

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

Te-max

BSFC (g(kWmiddoth))


ISG speed (rmin)

ISG

torq

ue (N

m)


0100200300400500




119879ISG

=





(1)



Efficiency ()Tm-max

300 600 900 1200 1500 1800 2100 2400Motor speed (rmin)

200016001200

800400

0minus400minus800

minus1200

Mot

or to

rque

(Nm

)



119898is as follows

119879119898=

min (119879119898 req 119879119898 dis max (119899119898)

119879119898 bat dis max (119899119898 SOC)) 119879

119898 req ge 0

max (119879119898 req 119879119898 chg max (119899119898)

119879119898 bat chg max (119899119898 SOC)) 119879

119898 req lt 0

(2)


119899119898







SOC (119896 + 1) = SOC (119896)


oc minus 4 (119877int + 119877119905)


119898)

119898


12

)


(3)


119887is the


120578119898



700 1000 1300 1600 1900 2200 2500

800700600500400300200100

0

Torq

ue (N

m)

P-out (kW) Opt-line

Speed (rmin)

TISG-min






VV (119896 + 1) = VV (119896)

+

1

120575119872

(

119879req (119896) sdot 119894119900 sdot 120578

119903119889

minus

VV (119896)1003816100381610038161003816VV (119896)

1003816100381610038161003816

(119865119891+ 119865119886(VV (119896))) )

(4)


119900is the


119886and




119879req (119896) = 119879119890 (119896) + 119879ISG (119896) + 119879119898 (119896) +119879119887(119896)

119894119900

(5)


119890is the engine

torque





119898 ISG torque





pressed as

119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896)) (6)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896)) =

119873minus1

sum

119896=0

fuel (119896) (7)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0

[fuel (119896) + 120572(119879req (119896) minus 119879119890 (119896)


119894119900

)

2

]

(8)





119899119890-min le 119899119890 (119896) le 119899119890-max

119879119890-min (119899119890 (119896)) le 119879119890 (119896) le 119879119890-max (119899119890 (119896))


le 119879ISG-max (119899ISG (119896) SOC (119896))

119879119898-min (119899119898 (119896) SOC (119896)) le 119879

119898(119896)

le 119879119898-max (119899119898 (119896) SOC (119896))


119899119898(119896) = 119899

119890(119896) = 119899ISG (119896) if clutch = 1

119899119890(119896) = 119899ISG (119896) if clutch = 0

119879119890(119896) + 119879ISG (119896) = 0 if clutch = 0

(9)



119869lowast

119873minus1(119909 (119873 minus 1)) = min

119906(119873minus1)

[119871 (119909 (119873 minus 1) 119906 (119873 minus 1))] (10)


119869lowast

119896(119909 (119896)) = min

119906(119896)

[119871 (119909 (119896) 119906 (119896)) + 119869lowast

119896+1(119909 (119896 + 1))] (11)




119878 =


120575SOC (12)







Yes

Yes

Yes

Yes

No

No

No

No

k = k minus 1

j lt C

i lt Si = i + 1

j = j+ 1



Number of stage N

Number of state S


k ge 1




k = N minus 1





J = L(x(i) u(j))





COSTminUopt

Uopt

L






u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k




1(1) and 119906







0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min








119896+1(119894 + 1)






Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










700 1000 1300 1600 1900 2200 2500

800700600500400300200100

0

Torq

ue (N

m)

P-out (kW) Opt-line

Speed (rmin)

TISG-min






VV (119896 + 1) = VV (119896)

+

1

120575119872

(

119879req (119896) sdot 119894119900 sdot 120578

119903119889

minus

VV (119896)1003816100381610038161003816VV (119896)

1003816100381610038161003816

(119865119891+ 119865119886(VV (119896))) )

(4)


119900is the


119886and




119879req (119896) = 119879119890 (119896) + 119879ISG (119896) + 119879119898 (119896) +119879119887(119896)

119894119900

(5)


119890is the engine

torque





119898 ISG torque





pressed as

119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896)) (6)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896)) =

119873minus1

sum

119896=0

fuel (119896) (7)



119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0

[fuel (119896) + 120572(119879req (119896) minus 119879119890 (119896)


119894119900

)

2

]

(8)





119899119890-min le 119899119890 (119896) le 119899119890-max

119879119890-min (119899119890 (119896)) le 119879119890 (119896) le 119879119890-max (119899119890 (119896))


le 119879ISG-max (119899ISG (119896) SOC (119896))

119879119898-min (119899119898 (119896) SOC (119896)) le 119879

119898(119896)

le 119879119898-max (119899119898 (119896) SOC (119896))


119899119898(119896) = 119899

119890(119896) = 119899ISG (119896) if clutch = 1

119899119890(119896) = 119899ISG (119896) if clutch = 0

119879119890(119896) + 119879ISG (119896) = 0 if clutch = 0

(9)



119869lowast

119873minus1(119909 (119873 minus 1)) = min

119906(119873minus1)

[119871 (119909 (119873 minus 1) 119906 (119873 minus 1))] (10)


119869lowast

119896(119909 (119896)) = min

119906(119896)

[119871 (119909 (119896) 119906 (119896)) + 119869lowast

119896+1(119909 (119896 + 1))] (11)




119878 =


120575SOC (12)







Yes

Yes

Yes

Yes

No

No

No

No

k = k minus 1

j lt C

i lt Si = i + 1

j = j+ 1



Number of stage N

Number of state S


k ge 1




k = N minus 1





J = L(x(i) u(j))





COSTminUopt

Uopt

L






u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k




1(1) and 119906







0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min








119896+1(119894 + 1)






Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899119890-min le 119899119890 (119896) le 119899119890-max

119879119890-min (119899119890 (119896)) le 119879119890 (119896) le 119879119890-max (119899119890 (119896))


le 119879ISG-max (119899ISG (119896) SOC (119896))

119879119898-min (119899119898 (119896) SOC (119896)) le 119879

119898(119896)

le 119879119898-max (119899119898 (119896) SOC (119896))


119899119898(119896) = 119899

119890(119896) = 119899ISG (119896) if clutch = 1

119899119890(119896) = 119899ISG (119896) if clutch = 0

119879119890(119896) + 119879ISG (119896) = 0 if clutch = 0

(9)



119869lowast

119873minus1(119909 (119873 minus 1)) = min

119906(119873minus1)

[119871 (119909 (119873 minus 1) 119906 (119873 minus 1))] (10)


119869lowast

119896(119909 (119896)) = min

119906(119896)

[119871 (119909 (119896) 119906 (119896)) + 119869lowast

119896+1(119909 (119896 + 1))] (11)




119878 =


120575SOC (12)







Yes

Yes

Yes

Yes

No

No

No

No

k = k minus 1

j lt C

i lt Si = i + 1

j = j+ 1



Number of stage N

Number of state S


k ge 1




k = N minus 1





J = L(x(i) u(j))





COSTminUopt

Uopt

L






u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k




1(1) and 119906







0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min








119896+1(119894 + 1)






Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










u1(1)

u1(2)

un(1)

un(2)

un(3)

xn

x1

k + 1k




1(1) and 119906







0

1

2

3500 1000 1500 2000 2500 3000

3000

2500

2000

1500

1000

500

0

T-APU-motor-max

Torq

ue (N

m)

Te-max

Speed (rmin)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Torq

ue (N

m)

4 3

5

1 2

6

Te-max

Speed (rmin)

Te-max + Tm-max

Tm-max

ne-min








119896+1(119894 + 1)






Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Tm = Treq minus Te

j = j + 1


Te le Te-min

Tm ge Tm-max

u

Yes

Yes

No

No





j = j + 1

Te = Te + 120575Te

Te ge Te-max

TISG le TISG-min

Yes

Yes

No

No

u









x1 x1 x1

xn xn xn

uk

uk+1(i)

uk+1(i + 1)

uk+1

k k + 1 k + 2




0 200 400 600 800 1000 1200 1400

80

60

40

20

0

Velo

city

(km

h)

Time (s)















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















06

05

04

03

SOC

0 05 1 15 2times104

Time (s)




119869 =

119873minus1

sum

119896=0

119871 (119909 (119896) 119906 (119896))

=

119873minus1

sum

119896=0


minus 119879119898(119896) minus

119879119887(119896)

119894119900

)

2

]

+ 120573(SOC (119873) minus SOC119891)

2

(13)

0 200 400 600 800 1000 1200 1400Time (s)

60

40

20

0

Velo

city

(km

h)



SOC

0 200 400 600 800 1000 1200 1400Time (s)

05

049

048

047

046














0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















0 200 400 600 800 1000 1200 1400Time (s)

SOC

05

049

048

047

046

120575SOC = 00005120575SOC = 0001120575SOC = 0005


0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400

0 200 400 600 800 1000 1200 1400Time (s)

1000

1000

minus1000

500

0

0


Te

(Nm

)T

m(N

m)

TIS

G(N

m)


0 200 400 600 800 1000 1200 1400Time (s)

4

3

2

1

0

minus1

Mod

e





1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1000 1500 2000 2500100200300400500600700800

Te-max

Engi

ne to

rque

(Nm

)

Engine speed (rmin)

BSFC (g(kWmiddoth))

(a) engine

200016001200

800400

0minus400minus800

minus12000 300 600 900 1200 1500 1800 2100 2400

Mot

or to

rque

(Nm

)

Tm-max

Efficiency ()

Motor speed (rmin)

(b) motor





per 100 km(L100 km)


15 06 0314 995 1744 1972


15 06 0305 1027 17603 1990






6 Conclusions








Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article global optimal energy management strategy

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