Optimal Control Of Hybrid Electric Vehicles Based On .

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Optimal Control of Hybrid Electric Vehicles Basedon Pontryagin’s Minimum PrincipleNamwook Kim, Sukwon Cha, Huei PengAbstract - A number of strategies for the power management ofHEVs (Hybrid Electric Vehicles) are proposed in the literature. Akey challenge is to achieve near-optimality while keeping themethodology simple. The Pontryagin’s minimum principle (PMP)is suggested as a viable real-time strategy. In this paper, theglobal optimality of the principle under reasonable assumptionsis described from a mathematical viewpoint. Instantaneousoptimal control with an appropriate equivalent parameter forbattery usage is shown to be possibly a global optimal solutionunder the assumption that the internal resistance and opencircuit voltage of a battery are independent of the state-of-charge(SOC). This paper also demonstrates that the optimality of theEquivalent Consumption Minimization Strategy (ECMS) resultsfrom the close relation of ECMS to the optimal-control-theoreticconcept of PMP. In static simulation for a power-split hybridvehicle, the fuel economy of the hybrid vehicle using the controlalgorithm proposed in this paper is found to be very close –typically within 1% – to the fuel economy through global optimalcontrol that is based on DP (Dynamic Programming).Index Terms—road vehicle control, cost optimal control, fueloptimal control, dynamic programming, Pontryagin maximumprinciple.I. INTRODUCTIONThe optimal control of HEVs (Hybrid Electric Vehicles) isan important topic not only because it is useful for powermanagement control but also indispensible for the optimaldesign of HEVs. Different vehicle systems can be compared toeach other only when the controllers guarantee the optimalityfor each deployed system. Technically, we can obtain optimalcontrol trajectories if the whole driving-cycle information isgiven prior, and if we have determinate performance indexes,such as fuel consumption, exhaust emission, or accelerationperformance. Under those circumstances, the DynamicProgramming (DP) approach guarantees the global optimalresults and had been investigated in several prior publications[1], [2], [3]. The results obtained through DP are unbeatablebut, unfortunately, cannot be implemented directly. Instead, apost-processing step is required by using rule extraction, e.g.,through Neural Networks, which approximates the results ofthe optimal control pattern. Even with this post-processingstep, these strategies cannot cover all driving conditions.Hence, the real-time controller based on DP is effective onlyfor the driving cycle that is used for rule extraction. To remedythis problem, stochastic dynamic programming and drivingpattern detection with multiple driving cycles had beensuggested as possible solutions [4], [5]. Another approachbased on optimal control theory which, basically, realizes theminimization of the Hamiltonian has been applied to theoptimal control problem for HEVs in [6], [7]. The EquivalentConsumption Minimization Strategy (ECMS), which reallybegan from the heuristic concept that electric energy could beequivalent to fuel usage, was introduced [8]. Real-timeapplications of ECMS were suggested in [9], [10]. As ageneral case of the Euler-Lagrange equation, Pontryagin’sminimum principle (PMP) was also introduced as an optimalcontrol solution [11], [12], [13], wherein the Hamiltonian isconsidered as a mathematical function. In this paper, we showthat the Hamiltonian can be calculated from numerical modelsand further, prove that the control concept based on PMP canbe a global optimal solution under reasonable assumptions.The optimal control based on PMP is simple enough to beimplemented in real-time applications because it is based oninstantaneous optimization. Assuming that the cost function tobe optimized involves only fuel consumption, the controlconcept minimizes the Hamiltonian, which is defined as: ( SOC , P )H m fc ( Pbat ) p SOC(1)batwhere ṁfc is the rate of fuel consumption, p is an adjustmentvariable, which is called ‘costate’ in PMP, and SȮC is a timederivative of SOC (the state-of-charge).II. GENERAL APPROACHES FOR THE OPTIMAL CONTROLOF HEVSAs stated above, assuming that minimum fuel consumption isthe goal of optimal control, the problem of HEVs can bedefined as (2), in which the engine speed, Se, and the enginetorque, Te, can be used to determine the fuel consumption. min J t f L S , T , t dt t0 ( e e ) subject to : SOC ( t0 ) SOC t f (2) f ( SOC , S , T )SOCee SOCmin SOC SOCmax Smin Se Smax Tmin Te Tmax {}( )

where L(Se, Te) is the rate of fuel consumption of the engine.SOC is determined by a battery model, which will bedescribed in (8). Further, Te and Se are restricted by operatingconstraints such as the maximum possible engine speed or themaximum possible engine torque given the impact ofconstraints on components, such as the maximum motor speed,maximum torque, or maximum battery power. This optimalcontrol problem can be solved from optimal controltechniques, which are described in the next section.A. Optimal Control TheoryTo solve a deterministic optimal control problem, which isdefined as (2), there are two representative approaches. One isthe Hamilton-Jacobi-Bellman (HJB) approach, which is basedon Bellman’s principle of optimality, and the other istrajectory optimization, which originates from the Calculus ofVariation.Fuel consumption (g)Field of optimal . 1. A field of the optimal cost. The field is a family of optimal fuelconsumptions. The starting point is SOC 0.6 and t 0. Contrary to this figure,in general, a field of cost-to-go is widely used in optimal control problemsbecause it is more useful for dealing with state equations.As a kind of numerical method for the HJB equation,Dynamic Programming (DP) solves a field of optimal controlthat is based on the principle of optimality; the field is afamily of optimal fuel consumptions, as shown in Fig. 1. Onthe other hand, Pontryagin’s minimum principle (PMP), whichis a general case of the Euler-Lagrange equation in theCalculus of Variation, considers the optimality of a singletrajectory. (See Fig. 2.)Optimal trajectory solved by PMP0.95Optimal(PMP)Trajectory 1Trajectory 2Trajectory 2001400tme (s)Fig. 2. The trajectory derived from PMP. The trajectory is superior to onlyneighboring trajectories (i.e., it is only locally optimal), which means thetrajectory from PMP could be inferior to Trajectory 3, which is not adjacent tothe PMP-derived trajectory.In general, the DP approach guarantees the global optimalsolution by obtaining all possible optimal trajectories from thefield of optimal control. On the other hand, PMP, as onemethod of trajectory optimization, yields us necessary – butnot sufficient – conditions that the absolute (i.e., globaloptimal) trajectory must satisfy. Hence, there could be asuperior solution that is distant from the local optimaltrajectory that is obtained from PMP. (See Fig. 2.)B. DP vs. PMPAs has been stated above, the trajectory derived from PMPmight not be a global optimal solution. Therefore, the controlbased on PMP can be considered as inferior to the (globallyoptimal) control based on DP. On the other hand, DP requiresmore computing time than PMP because DP solves allpossible optimal controls to fill the optimal field [11]. SinceDP is a numerical representation of the HJB equation, DPneeds a similar computation load as the HJB equation, whichsolves a partial differential equation (PDE), whereas PMPsolves just nonlinear second-order differential equations. Thedrawback of DP with regard to the computational loadbecomes compounded due to the ‘curse of dimensionality,’ i.e.,when the state variables increase in number, the computationalload of the PDE exponentially increases in accordance withthe increase in the dimension of the optimal field. However, inPMP, the number of nonlinear second-order differentialequations linearly increases with the dimension. In conclusion,we can say that the control based on PMP can reduce thecomputational time for getting an optimal trajectory but itcould be a local optimal solution, not a global solution ingeneral problems.C. Sufficient Conditions for Global Optimality of PMPIn specific cases, the optimal control based on PMP can be aglobal optimal control. For example, it is well-known that theoptimal control based on the Euler-Lagrange equation can be aglobal optimal control in a linear system [15]. In general, thefollowing three approaches are effective to establish that thenecessary conditions from PMP become sufficient for theglobal optimal control: 1) the optimal trajectory obtained fromPMP is a unique trajectory that satisfies the necessary andboundary conditions; 2) some geometrical properties of theoptimal field provide the possibility of optimality verification;and 3) as a general statement of the second approach, theabsolute optimality is, mathematically, proved by clearpropositions [21]. If one of these three approaches isapplicable to the optimal control problem of HEVs, we canreplace DP with PMP, which can save on time to yield optimalresults and also guarantee global optimality.

III. APPLICATION OF PONTRYAGIN’S MINIMUMPRINCIPLEIn the optimal control problem of HEVs, Pontryagin’sminimum principle (PMP) produces a boundary value problemwithin second-order nonlinear differential equations. In thissection, we introduce a static model of a target vehicle anddescribe the techniques for obtaining the Hamiltonian from themodel and deciding on the optimal control from theHamiltonian, which is a local process of getting the optimaltrajectory through PMP.A. Vehicle ModelIn this paper, we assume that the hybrid vehicle uses a powersplit hybrid, viz., the Toyota Hybrid System (THS), whichintegrates two motors/generators and an engine through aplanetary gear set (see Fig. 3). In the static model of thepower-split system, we have two independent control variables,the torque of the engine and the speed of the engine, when therequested output speed and torque are given, which means allother variables can be fixed by these two control variables.and Sreq are the speeds of the engine, MG1, and MG2, and therequested output speed, respectively. The required power ofthe battery can then be calculated as:Pbat ηc1k Tmg1 Smg1 ηc 2k Tmg 2 Smg 2(5)where the efficiencies of MG1 and MG2, ηc1 and ηc2, areobtained based on motor efficiency maps of each MG, whichinclude the motor and inverter losses, and 1, recuperatingk (6)motoring 1,We can also use a fuel consumption rate map to obtain ṁfc.m fc L (Teng , S eng )(7)Finally, the derivative of SOC, SȮC can be calculated fromthe battery power and the current SOC. Considering that theequivalent open-circuit voltage and internal resistance arefunctions of SOC, SȮC is a function of SOC and Pbat, whichcan be expressed as [2].2 1 Vbat Vbat 4 Rbat Pbat(8)SOCQbat2 RbatIn conclusion, ṁfc in (7) and SȮC in (8) depend on Teng, Seng,and SOC, when the requested output condition is given.B. Confined Optimal Operating Line (C-OOL)Fig. 4. The lever diagram for the lever analogy of the THS system by which theforce and momentum equilibria are simply obtained.The torque and speed relations among the power resourcesare well-known under static conditions [2]. We can obtain theoperating torques and speeds of the MG1 and MG2, which arefunctions of the engine torque and the engine speed, when therequested output torque and speed are already calculated fromthe driving cycle. Tmg1 1 1/ ζ Treq 01(3) T 1 R R Teng mg 2 (1 R ) (())Optimal points for C-OOL11010090Engine Torque(Nm)Fig. 3. A schematic diagram of a power-split hybrid system considering thefinal differential gear ratio.Prior to solving the time horizon optimal problem of HEV,we introduce an inner-loop optimal process to get ConfinedOptimal Operating Line (C-OOL), which is the family of thebest engine operating points confined to specific output torqueand speed. This optimal process makes the Hamiltonian as afunction of only one control variable, Pbat. To get theHamiltonian, first, we calculate ṁfc and Pbat which arefunctions of the control variables, Teng and Seng described in (7)and (9) from the static model (3)-(5) in every second.Pbat h(Teng , Seng )(9)The optimal problem to minimize the fuel consumptionsubject to a specific battery power, Pbat, is defined as:min m fc L Teng , Seng (10)s. t. h Teng, Seng Pbat 0807060504030C-OOLMinimum fuel consumption pointsOperating line according to Pbat S mg 1 R 1 R ζ S req Equivalent fuel consumption line20(4) S SEngine max torque1010 eng mg 2 10001500200025003000where Teng, Tmg1, Tmg2, and Treq are the torques of the engine,Engine speed(rpm)MG1, and MG2, and the requested torque of output,Fig. 5. An example of C-OOL when Treq 100Nm and Sreq 100rad/s. Dottedrespectively. Further, R and ζ are the gear ratio of the planetary lines are feasible operating lines for specificvalues of Pbat. The resolution ofgear set and the final gear ratio, respectively. Seng, Smg1, Smg2, Pbat in the figure is 1.5kW whereas it is 0.05kW in our simulation.

The optimized control variables, T*eng and S*eng, in the problemcan be decided by choosing a minimum point of fuelconsumption on each feasible engine operating line as per Pbat(see the dotted-line in Fig. 5). Finally, the instantaneousoptimal fuel consumption rate is a function of the optimizedcontrol variables, T*eng and S*eng, subject to the specified Pbat.( h (T**m fc L Teng, SengPbat*eng,S*eng))(11)(12)In light of the time-horizon optimal control, this optimalprocess reduces the dimension of the control variable fromTeng and Seng to only Pbat, whereby we do not consider theinferior engine operating points when solving the problem inthe time-horizon plane. Now, from (11) and (12), we obtainthe fuel consumption, ṁfc, which is a function of Pbat, asshown in Fig. 6.Confined-optimal fuel consumption rateHybrid drivingPure electric driving2.10.5The state equation and the costate equation can beexpressed as (16) and (17), respectively. HSOC f ( SOC , Pbat )(16) p p f(17) ( SOC ) ( SOC )For optimality, another condition in (18) should beconsidered to determine the optimal control variable, Pbat, atevery time step.*H ( Pbat, SOC * , p* , t ) H ( Pbat , SOC * , p* , t )(18)Finally, when both the final time and the final state are fixed,as in the optimal control problem of HEVs, the boundarycondition of the final state is added for the sake of optimality.SOC ( t0 ) SOC t f(19)p H( )0-20-15-10-50510Pbat (kW)Fig. 6. Instantaneous optimal fuel consumption rate line in the domain set,{h(Treq, Sreq), L(Treq, Sreq) [Treq, Sreq] [T *req , S *req ]}, when Treq 100Nm andSreq 100rad/s.Now, the fuel consumption rate, ṁfc, can be determined byPbat, which could be decided by a supervisory algorithm. Then,the engine operating point and all other operating points ofpower resources are calculated based on C-OOL. The optimalfuel consumption rate line in Fig. 6 can be interpreted as akind of Pareto frontier for all the engine operating points [22].The physical interpretation of the line in Fig. 6 is clear: giventhat the engine always operates at the best point, less fuelconsumption is needed when more battery power is used, andvice versa. In general, the requested output torque and speedvary over time; hence, we can assert that in the time-horizonplane, the fuel consumption rate, ṁfc, is a function of just Pbatand t, as in (13).m fc g ( Pbat , t )(13)Additionally, the pure electric driving point shown in Fig. 6is the operating point at which the battery supplies all theenergy needed to drive the vehicle while the engine does notoperate or, if appropriate, operates at an optimal speed with nofuel consumption but with engine drag.C. Necessary Conditions from PMPFrom the assistance of the inner-loop optimal process, onlyPbat is the control variable that decides all the operating pointsThese conditions, (16) (19), are necessary and boundaryconditions that the optimal trajectory must satisfy [19].D. Optimal Control using the HamiltonianThe Hamiltonian is obtained from Eq. (13), Eq. (8), and thecostates. The optimal control is determined by the necessarycondition in (18). Fig. 7 shows three examples of theHamiltonian and the associated optimal controls. We canchoose the optimal control, Pbat, which minimizes theHamiltonian if the costate is given beforehand, though theoptimal costate is obtained over the entire time-horizon time tosatisfy the boundary condition in (19).Optimal control on Hamiltonianp: -280.50p: -330.02p: -379.501.11Hamiltonianmfc (g/s)1.5in the time-horizon plane of the optimal control problem. Thecontrol variable, Pbat, decides the fuel consumption rate andthe engine operating point in the C-OOL. All the other systemvariables, such as the motor speed, torque, and transmissionstatus, are fixed from the engine operating point. To obtain theoptimal Pbat, the performance index can be defined as:tfmin J g ( Pbat ( t ) , t ) dt (14)t0 In (14), g is the best fuel consumption rate function in (13),which is a function of Pbat and t. From PMP, the Hamiltonianis defined as (15), where p is the costate.H g ( Pbat ) p f ( SOC , Pbat )(15)0.90.80.7-25-20-15-10-5051015Pbat (kW)Fig. 7. Three examples of the Hamiltonian and the associated optimal controlsfor varying costates when Treq 100Nm, Treq 100Nm, Sreq 100rad/s, and SOC

is 0.6.If we have an appropriate costate, this instantaneousoptimal control could be an optimal solution in the timehorizon control problem and the Hamiltonian function wouldnot need to be an explicit function. In Fig. 7, the higher costatemakes the controller choose the higher P *bat as an optimalcontrol, which lowers SOC. Additionally, the Hamiltonian inFig. 7 is an almost-convex function and the constraint on thestate is presented by a linear summation of SOC. In that case,mathematically, the optimization problem possibly possessesan appropriate costate, which implies that we can find theappropriate costate to satisfy the boundary condition; this is asituation of strong duality in optimization [23]. As shown inFig. 7, the control concept is based on instantaneousminimization but the control can be optimal only when anappropriate costate is given.IV. CHARACTERISTICS OF THE HEV PROBLEMIn view of optimal control, the HEV problem has specialcharacteristics. The fuel consumption rate in Fig. 6 is not afunction of the state variable, SOC, and SȮC in (16) is notonly independent of time but also highly depends on Pbatrather than SOC. These characteristics influence the propertiesof the solution from PMP. In this section, we describe thespecialties of the optimal control problem of HEVs withregard to the above characteristics. We also introduce severaltechniques to apply PMP in general problems of HEVs.A. Constant CostateThe costate in (15) originates from Lagrange multipliers forthe incorporation of dynamic constraints, which, occasionally,possess physical meanings. In the HEV problem, the costatecan be interpreted as a ‘weight’ coefficient of the timederivative of SOC, by which the second term in (15) can beinterpreted as an equivalent fuel consumption. On the otherhand, the costate in our problem can be considered as aconstant under some assumptions about the battery. In general,the SOC range of the battery usage is limited between 0.2 and0.9 but in charge-sustaining problems, the battery mainlyoperates in a narrower range, e.g., from 0.5 to 0.7; hence, thevoltage and the resistance may not vary so much in the range.In that case, the costate stays near the initial value because,mathematically, the absolute value of f/ (SOC) in (17) is verysmall when compared to the absolute value of the costate forthe entire driving cycle. If the resistance and the voltage areconstants or depend on only Pbat, we can assume that SȮCdepends on only Pbat and not on SOC. Then, the costateexpression, i.e., (17), vanishes in our optimal control problem,which is reasonable in the primary range of usage of SOC. f ( SOC, P ) f ( P )SOC(20)batbatNow, the costate can be considered as a constant.p p f 0, p constant ( SOC )(21)Owing to the constant costate, we do not only reduce thecomplexity of the optimal control based on PMP but alsodiscover an interesting feature of the optimal control problemof HEVs.B. Condition of Global OptimalityThe idea of assuming the costate as constant is suggested inseveral prior studies to simplify the computations insimulation; the assumption has been described as anintuitively reasonable assumption [6], [10], [14]. However, theessential point of the assumption is that the optimal controlbased on PMP can become a global optimal control under theassumption. To prove this proposition, we have to return to thefirst approach of section II.C. As stated in that section, if anoptimal trajectory that satisfies the necessary conditions ofPMP and the boundary conditions is unique, the optimaltrajectory should be considered as a global optimal trajectory.In our problem, we can consider two distinct trajectories thathave different costates but satisfy all the necessary conditionsand the boundary condition in (16) (19) shown in Fig. 9.Under the assumptions of the battery, the Hamiltonian can bepresented as:H g ( Pbat ) p f ( Pbat )(22)Battrery characteristicsΩ disΩ chaVoutPrimary usage2702500.52300.4210Voltage (V)Resistance ( Ω)0.6Fig. 9. The two optimal SOC trajectories that satisfy the necessary conditionsand the boundary condition.1900.317000.20.4SOC0.60.81Fig. 8. Plot of the battery’s open-circuit voltage and resistance (used in Prius04).From the necessary condition in Eq. (18), the optimalcontrol variable for SOC*1 , P*bat,1, satisfies the condition in Eq.(23).()(***H Pbat,1 , p1 , t H Pbat , p1 , t)(23)This condition should be satisfied for all admissible valuesof Pbat shown in Fig. 7, including P*bat,2, i.e., either()()****H Pbat,1 , p1 , t H Pbat ,2 , p1 , t ,or(24)

()()()()******g Pbat,1 , t p1 f Pbat ,1 g Pbat ,2 , t p1 f Pbat ,2 ,(25)We can apply the same argument for SOC*2 , which can beexpressed as:()()()()******g Pbat,2 , t p2 f Pbat ,2 g Pbat ,1 , t p2 f Pbat ,1 .(26)Another inequality is obtained by summing the above twoinequalities; this can be expressed as:(p*1){ () (** p2* f Pbat,1 f Pbat ,2)} 0 .(27)Now, we can replace the state equation, f(Pbat), with SȮC.(p*1)() * SOC * 0. p2* SOC12(28)The condition in Eq. (28) indicates that the two existingoptimal trajectories, which have the same initial and final SOCvalues, are impossible because the sign of (SȮC*1 SȮC*2 ) doesnot change for all t under different constant costates. Finally,the supposition that there are two different optimal trajectoriesthat satisfy both the necessary and the boundary conditions isrefuted. Additionally, from Eq. (28), we can derive theproposition that the SOC trajectory that has a higher costatealways either increases at a slower rate or decreases at a fasterrate than the trajectory with a lower costate under theassumptions of the battery, which makes sense in light of theobservations on the optimal control of the Hamiltonian insection III.D.C. State Variable ConstraintsThe constraints on the control variable, Pbat, can be applied tothe optimal problem when calculating the Hamiltonian fromthe static model. The constraints on the state variable, such asmaximum or minimum limits on SOC, however, are notintroduced above. If needed, we can apply a state variableinequality constraint by adding a new imaginary state variable,SOCC, and augmenting the Hamiltonian with the adjustedterms. To apply the constraint, we have to define the stateequation of SOCC as in [19].2 SOCC ( SOC - SOCmin ) U ( SOCmin - SOC ) ( SOCmax - SOC ) U ( SOC - SOCmax )2(29)In (29), U is a unit Heaviside step function and SOCmin andSOCmax are minimum and maximum limits on SOC, i.e., 0.2and 0.9. If we set both SOCC(t0) and SOCC(tf) to zero, theimaginary state variable, SOCC, becomes zero for the entiredriving cycle because SȮCC can never be nonzero when thestate does not violate the constraint on SOC for the entiredriving cycle. The Hamiltonian, including the augmented termabout SOCC, is defined as: p SOC H m fc p SOC(30)CCFrom PMP, the costate equation corresponding to the newstate can be expressed as:p C H 0(31) ( SOCC )The state equation and the costate equation, (29) and (31),should be added to the original necessary conditions of PMP.This additional state is trivial in the optimal control problemfor a charge-sustaining HEV because there is little likelihoodthat its optimal SOC trajectory violates the SOC limitation.This consideration, however, is effective in the optimal controlproblem for plug-in hybrid vehicles.D. Cost FunctionIn the PMP algorithm, the Hamiltonian can be modified if wehave to consider new components of cost, such as emissions orthe estimation of drivability, or new states, such as thetemperature. The performance index of the problem with thesenew costs and states can be defined as:(( ) )J h x tf ,tf tft0( m fc) g new dt(32)In (32), gnew is a new cost function, x is a new state, and h isa cost function for the final states. The modified Hamiltoniancan be expressed as: pT ( t ) x ( t )H m fc g new p SOC(33)newnewIn (33), pnew is a new costate vector. The necessaryconditions of PMP, (16) (18), can be also applied for the newHamiltonian. In general, if the final state of the new states,x(tf), is not fixed, the new boundary condition, (34), should beadded instead of the state boundary condition for the fixedboundary [19]. h (34) x new p new t 0fIf the following two conditions are satisfied, this addition ofnew costs does not affect the global optimality of PMP, whichmeans that the uniqueness that we established in section IV.Bis still applicable. 1) The new cost is not a function of the statevariables, SOC or xnew; therefore, the cost can be directlycalculated from the control variable, Pbat, i.e., the new cost isdetermined by engine operating points, as is the case with thefuel consumption. 2) There is no state except SOC, or, if thereis such a state, the state equations are not functions of the statevariables at all. In this case, we can apply the same argumentto the new Hamiltonian, for which we can consider twodistinct trajectory vectors, x1 and x2, which satisfy thenecessary condition in Eq. (18). Now, we have a similar resultas with Eqs. (23) (28). The result is expressed as:(p*1)() p*2 x 1* x *2 0(35)*In (35), p is a constant costate vector that includes p and pnewand x* is a state vector that includes SOC and the new state,xnew. When x*1 (t0) x*2 (t0), there is no possibility that x*1 (tf) x*2(tf) except that(p*1)() p*2 x 1* x *2 0, t(36)*1*1The inequality in (35) is an equality, i.e., (p p ) is anorthogonal vector of (ẋ *1 ẋ *2 ), only when the minimumHamiltonian for x1 is coincident with the minimumHamiltonian for x2 for all time-points, which is an unusualsituation. Therefore, we can conclude that x*1 (t) x*2 (t) for all tis the only solution if the final trajectory vector, x*1 (tf), equalsx *2 (tf) when x *1 (t0) x *2 (t0). This proposition states that the

optimal trajectory x* is still unique even though new costfunctions may be added. In general, if there are additionalcosts or states, such as emission costs or the temperature,either the cost can be a function of the new states or the statesmight be coupled to each other. Then, the optimal controlbased on PMP cannot guarantee global optimal control thoughit remains the case the control based on PMP is locally optimal.E. ECMS and PMPThe equivalent consumption minimization strategy (ECMS)was introduced as an optimal control idea. Several controlstrategies based on ECMS, such as Adaptive-ECMS andTelemetric-ECMS, were suggested as real-time optimalcontrol concepts in [8], [9], [10]. m fc scha Pbat , Pbat 0 min EFC (37) m fc sdis Pbat , Pbat 0 The performance index of ECMS in (37) is generally calledthe equivalent fuel consumption (EFC) though the forms ofapplication of the idea vary slightly. ECMS was reallydeveloped from a heuristic concept that current battery usagewould be compensated for in the future; so, there are twodifferent coefficients, scha and sdis, for the charging anddischarging statuses, which influence the electric-energybalance. From the similarity between the Hamiltonian andEFC, ECMS was described as being fundamentally linked tothe Euler-Lagrange equation [14]. It is natural to view ECMSfrom the concept of PMP because PMP states that theHamiltonian does not have to be an analytic or explicitfunction. From the comparison between the Hamiltonian andEFC, scha Pbat , Pbat 0p SOC(38) sdis Pbat , Pbat 0When the equivalent factors in ECMS are optimized for theHEV problem, these factors, scha and sdis, can be linked to theoptimal costate of PMP, as in Eq. (38).4x 10Relation between SOC and Pbat-3SOC 0.2SOC 0.3SOC 0.5SOC 0.9321SOC0-1-2-3equivalent coefficients of ECMS, scha and sdis, are optimizedfor the entire driving cycle to satisfy the energy balance, theboundary condition, SOC(t0) SOC(tf), is satisfied. These twocoefficients are linked to the charging and discharging slopesin Fig. 10. By the use of two parameters rather than one, theEFC can be brought closer to the Hamiltonian. If more distinctand equivalent parameters are used, such as s1, s2, s3, s4, and s5,as in Fig. 10, the closer is the fuel consumption of ECMS tothe optimal value. On the other hand, in the optimal controlbased on ECMS, if the correlation between Pbat and SȮC istoo complicated to be represented by these two slopes, ECMSmight not show

optimal control, dynamic programming, Pontryagin maximum principle. I. INTRODUCTION he optimal control of HEVs (Hybrid Electric Vehicles) is an important topic not only because it is useful for power-management control but also indispensible for the optimal des

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