Optimization of power management in an hybrid electric vehicle using dynamic programming

Mathematics and Computers in Simulation 73 (2006) 244–254

Optimization of power management in an hybrid electricvehicle using dynamic programming

Laura V. Perez ∗,1, Guillermo R. Bossio 1, Diego Moitre, Guillermo O. Garcıa 1

Facultad de Ingenierıa, Universidad Nacional de Rıo Cuarto, Cordoba, Republica Argentina

Available online 27 July 2006

Abstract

Hybrid electric vehicles are those powered from two different sources. Typically, they are equipped with an internal combustionengine, and also with an electrical storage system, such as a bank of batteries or ultra-capacitors. While braking, these vehiclesmay convert kinetic energy to electrical energy and send it back to the electrical storage system (regenerative braking). The wholevehicle system may be abstracted to one consisting of two energy sources, one of them rechargeable and the other consumable, thatfeed or receive energy from an energy consumer. A centralized control strategy is required to define the instantaneous power flowsamong these three main components. In this work, we derive the power split between the two sources such that fuel consumption isminimized, while the vehicle performs a given velocity cycle. Bounds on the power flows from both sources are considered. Thereis also a constraint of an integral nature that arises from the fact that the energy of the electrical storage system must remain betweenproper limits, in order to avoid physical damage. The problem is posed as a finite horizon dynamical optimization problem withconstraints and solved by a dynamic programming (DP) approach. The hybrid electrical vehicle being developed in the Universityof Rıo Cuarto, Argentina is taken as the case study.© 2006 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Cost optimal control; Dynamic programming; Road vehicle control

1. Introduction

Hybrid electric vehicles (HEVs) are those whose architecture includes two or more energy sources and associatedenergy converters. Typically, they are equipped with an internal combustion engine with an associated fuel tank (FT)and an electric motor with an associated electrical storage system (ESS), such as a bank of batteries or ultra-capacitors.These vehicles take advantage of the higher efficiency of electrical traction, while overcoming its main disadvantage:its low range (a consequence of the low energy density of current ESSs). The addition of a fuel converter compensatesfor this fact, since fuel has an energy density two or three order of magnitude larger. HEVs are expected to be lesspolluting, have lower fuel consumption and a range similar to that of conventional vehicles.

The storage elements and energy converters can be arranged following different topologies. Fig. 1 shows the “series”configuration. In this case, the electric motor moves the wheels and is fed by the ESS. The engine, which is in turn fed bythe chemical storage system, i.e., the fuel tank, drives an electrical generator. This generator provides electrical energy

∗ Corresponding author. Tel.: +54 358 4676255; fax: +54 358 4676246.E-mail addresses: [email protected] (L.V. Perez), [email protected] (G.R. Bossio), [email protected] (D. Moitre),

[email protected] (G.O. Garcıa).1 With Universidad Nacional de Rıo Cuarto and also with Consejo Nacional de Investigaciones Cientıficas y Tecnicas.

0378-4754/$32.00 © 2006 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2006.06.016

mailto:[email protected]




dx.doi.org/10.1016/j.matcom.2006.06.016

L.V. Perez et al. / Mathematics and Computers in Simulation 73 (2006) 244–254 245

Fig. 1. Scheme of a series HEV.

to the traction motor when the driver power demand exceeds that delivered by the batteries. If the power generated bythe engine–generator system exceeds what is needed, then it is diverted to recharge the ESS.

Hybrid power-trains share with electrical ones the additional advantage of regenerative braking. This means thatelectric motors, when used for traction, are able to become generators during braking. Hence, the vehicle kinetic energy,which otherwise is burnt in the brake drums in the form of heat, can be converted into electrical energy and sent backto the batteries. For that reason, Fig. 1 shows double arrows from the wheels to the storage system.

An electronic power manager determines at each instant, the amount and direction of power flow in each one ofthe paths. This higher-level control is known as “supervisory control”. At a lower level, power electronic devices willcontrol each one of the power converters of the system to satisfy the supervisory controller commands.

As has been said, there may be other topologies. The addition of power from the two sources may be mechanicalinstead of electrical, etc., but regardless of the particular configuration, the coordination between them and the systemoperational and physical limitations force trade-offs. So, to make full use of the benefits of HEV, optimizing thesupervisory control strategy is necessary. Optimization will depend on the objective pursued, on the control actionselected and on the constraints considered.

Our research group is developing an HEV series aimed for city work. It is in city work where the advantages ofHEVs are most noticeable because of the sequence of acceleration and deceleration intervals. This work is a first steptowards the definition of an optimal strategy for the supervisory control of the vehicle. The control objective is tominimize fuel consumption while the vehicle performs a scheduled velocity cycle.

Clearly, these control problems include combinations of linear and non-linear, discrete and continuous, algebraicand dynamical systems. Hence, most of the reported optimal supervisory control strategies use either intelligent controltechniques, such as rule-based, fuzzy logic and neural networks, or optimal control approaches, such as those in Delpratet al. [5,6] and Steinmauer and del Re [10]. Within the same approach, Zaremba et al. [11], Brahma et al. [2,3] andSciaretta et al. [9] use a discrete approach and a DP algorithm. In this work, we follow Brahma et al. [2,3] in solvingour supervisory control problem. The drawback in this approach is that the treatment of constraints of integral formcomplicates the computations. This kind of constraint appears because of the need to preserve the batteries fromdepletion or overcharge, and/or if a “charge sustaining” operation of the vehicle is imposed. Brahma et al. proposeheuristic methods based on penalization terms added to the objective function. We apply their approach to our caseand propose an alternative method based on checking the constraints as the algorithm proceeds. The method is alsosuboptimal but strictly ensures the safe operation of the ESS.

The paper is organized as follows. Section 2 gives to the statement of the problem; Section 3 summarizes the DPapproach to solve this dynamic optimization problem; Section 4 shows some results; finally, Section 5 contains ourconclusions and intended future work.

2. Statement of the problem

To define the supervisory control, a control objective, a control action (or independent variable), a state variable (ordependent variable) and system constraints have to be established. These elements may interchange roles and hence

246 L.V. Perez et al. / Mathematics and Computers in Simulation 73 (2006) 244–254

the problem of defining the supervisory control strategy may be posed in different ways. Two remarks are in order: (a)even if there may be different objectives to be pursued, such as minimizing emissions, maximizing, driveability, etc.,usually the primary objective is fuel economy and (b) because of regenerative braking, it seems preferable to performacceleration and deceleration using mainly the electrical path, since it is reversible. If there were no losses, all theenergy wasted during acceleration could be recovered during braking. As losses do exist, the engine should be usedonly to compensate for them. Nevertheless, current batteries not only have a low power density, as has been said, butalso must not be depleted nor overcharged at any time because they get irreversibly damaged. So, a trade-off solutionhas to be found.

2.1. Abstracted model

In order to consider the supervisory control problem, it suffices to consider an abstracted scheme for the system[2,3,8]. In this model, only the net power flow in each path is considered. We shall indicate by PFT(t) the power flowat time t from the fuel tank/engine/generator path and PESS(t) the power flow at time t from the ESS (Fig. 2).

The following convention is also established: a positive power flow means power flowing away from the ESS. Inconsequence, during regenerative braking a negative flow will take place in the electrical path. Accordingly, the powerflow from the engine cannot be negative, as it cannot absorb any power.

In this approach, the required velocity profile is considered a given function. The required power can be computedfrom it using a model of the vehicle longitudinal dynamics [7] and hence it is also considered a known function thatwill be denoted by Preq(t).

A balance equation can naturally be established, since the sum of power from both sources has to be equal to therequired power at all times:

PFT(t) + PESS(t) = Preq(t). (1.1)

Regarding the net energy consumed from each source in a time interval, we must take into account that not allthe power delivered by the source can be actually used to supply the demand, because in every energy conversionprocess there are losses. Hence, only a portion of the power delivered will reach the summing junction in Fig. 2. Inour abstracted model, this portion will be represented by two functions ηFT and ηESS that take values between 0 and1, and will be referred to as the efficiencies associated with each path. Indeed, efficiencies depend on the power flows,so we have for each t

0 ≤ ηFT(PFT(t)) ≤ 1, (1.2)

0 ≤ ηESS(PESS(t)) ≤ 1. (1.3)

Fig. 2. Abstracted scheme for an HEV.


We consider them as known functions that in practice are experimentally determined. Hence, the net energy consumedfrom the FT at time t can be computed as follows:

EFT(t) =∫ t

0

PFT(s)

ηFT(PFT(s))ds. (1.4)

Considering that the denominator is less than 1 it is clear that the total energy consumed is indeed greater than theintegral of the power delivered.

To compute the net energy consumed from the ESS it has to be considered that during acceleration (PESS > 0) theeffect of losses is represented by dividing by ηESS as in the previous case, but during regenerative braking, because oflosses, only a portion of the kinetic energy coming from the wheels will reach the ESS, so the inverse situation has tobe represented. Let us define

f (PESS(t)) =

⎧⎪⎨⎪⎩

ηESS(PESS)PESS if PESS < 0

PESS

ηESS(PESS)if PESS ≥ 0.

(1.5)

Then, the net consumed energy from the ESS at time t is

EESS(t) =∫ t

0f (PESS(s)) ds. (1.6)

2.2. Control objective

The control objective was taken as minimizing fuel consumption in a time interval [0,T], T known. That is, minimizingthe net energy consumed from the chemical storage system (fuel tank) in the interval, which is given by (1.4) witht = T.

2.3. Control action and state variable

Our purpose is to determine for each t in [0,T] the values of PESS and PFT that minimize the objective. Using thebalance Eq. (1.1), we can eliminate one of these functions. Hence, the problem can be posed in two alternative forms.Either PFT or PESS can be taken as the control action or independent variable, over which the minimization will beperformed. The remaining one is considered the state or dependent variable and is obtained from the first.

Physically, this implies that either the supervisory control will be exerted by the engine acceleration command or bythe ESS power controller. The alternative device should provide the remaining necessary power to satisfy the demandedpower Preq.

2.4. Constraints

Clearly, power flows are physically limited, hence

0 ≤ PFT(t) ≤ PFTmax ∀t (1.7)

and

PESSmin ≤ PESS(t) ≤ PESSmax ∀t. (1.8)

In addition, if the ESS is, as usual, a bank of batteries, it has to be protected from depletion and from overcharge.This implies that the net consumed energy from the ESS has to be maintained between proper limits at each instant t.Hence,

EESSmin ≤ EESS(t) ≤ EESSmax ∀t. (1.9)

Let us take EESS as the state variable. The statement of the two problems can be summarized as follows.


Problem 1. Find PFT(t) that minimizes

V [PFT(t)] =∫ T

0

PFT(t)

ηFT(PFT(t))dt (1.10)

subject to

EESS(t) = f (Preq − PFT(t)) ∀t ∈ [0, T ] (state equation)

EESS(0) = 0(1.11)

0 ≤ PFT(t) ≤ PFTmax ∀t ∈ [0, T ] (1.12)

PESSmin ≤ Preq − PFT(t) ≤ PESSmax ∀t ∈ [0, T ] (1.13)

EESSmin ≤ EESS(t) ≤ EESSmax ∀t ∈ [0, T ] (1.14)

where Preq(t), ηFT and f are given functions.

Problem 2. Find PESS(t) that minimizes

V [PESS(t)] =∫ T

0

Preq(t) − PESS(t)

ηFT(Preq − PESS(t))dt (1.15)

subject to

EESS(t) = f (PESS(t)) ∀t ∈ [0, T ] (state equation)

EESS(0) = 0(1.16)

0 ≤ Preq(t) − PESS(t) ≤ PFTmax ∀t ∈ [0, T ] (1.17)

PESSmin ≤ PESS(t) ≤ PESSmax ∀t ∈ [0, T ] (1.18)

EESSmin ≤ EESS(t) ≤ EESSmax (1.19)

where Preq(t), ηFT and f are given functions.

Some remarks are worth mentioning

a) The state equation, Eq. (1.11), is an indefinite integral rather than a differential equation, since its right-hand sidedoes not depend on the state variable but only on the control input:

f (Preq(t) − PFT(t)) =

⎧⎪⎨⎪⎩

Preq(t) − PFT(t)

ηESS(Preq(t) − PFT(t))if PFT(t) < Preq(t)

ηESS(Preq(t) − PFT(t))(Preq(t) − PFT(t)) if PFT(t) ≥ Preq(t).

(1.20)

The same occurs with (1.16) for Problem 2.b) The dependence of efficiencies on the power flows adds non-linearities to the state equation.c) There is no feedback: the solution is an open loop control.


3. DP solution

Because of (a) and (b) above, using calculus of variations or Pontryagin optimal control theory to solve theseproblems is not straightforward [4]. In addition, the many constraints to be satisfied imply including many Lagrangemultipliers and using the Karush–Kuhn–Tucker conditions. To manage these difficulties more easily, we choose adiscrete approach and the DP solution from Brahma et al. [2,3]. From the point of view of the search of the DP solutionboth problems are almost symmetric. So, we will only refer to Problem 1 henceforth.

We considered the network in Fig. 3. Along the horizontal axis, the interval [0,T] is divided in N stages of length�t. The possible discrete values for the control function searched appear along the vertical axis, equally spacedbetween 0 and PFTmax . Thus, each node ik of the network represents the ith choice for the control at step kth, whichwill be denoted PFTik

.All possible connections between nodes of successive stages have been considered. This implies that it is possible

to change the power delivered by the engine/generator to any other possible value in a time equal to the time step.Nevertheless, for each particular engine this will probably have to be limited according to the particular time constantof the engine. Let us call Sk the set of feasible nodes of stage k.

Control sequences are represented by paths from the initial state to terminal states. Let us denote them by

π = {PFTiπ (1),1 , PFTiπ (2),2 , . . . , PFTiπ (N),N

}, (1.21)

where PFTiπ (k)k is the value of PFT at the node iπ(k) selected by the particular sequence π at stage k. The cost of eachcontrol sequence is the cost of the corresponding path or, in other words, the value V(π) of the discretized objectivefunctional for the particular sequence.

Each control sequence, named policies in the DP jargon, is the result of a decision since at each stage there may beseveral possible nodes where the path can lead to. Each intermediate segment between nodes from successive stages iscalled an arc and it has an associated cost of transition. Let us denote by ak

ij the cost of transition from node i of stagek to node j of stage k + 1 (k = 1, 2, . . ., N, i ∈ Sk, j ∈ Sk+1).

For this problem, transition costs are computed as follows:

akij = 1

2

(PFTi,k

ηFTi,k

+ PFTj,k+1

ηFTj,k+1

)�t. (1.22)

In consequence, the total cost of a control sequence will be the sum of the transition costs of the particular arcs thatdefine it. The optimal policy will be that of minimum total cost. In this sense, the problem is equivalent to a shortestpath algorithm. The optimal cost will be the value of the objective functional for the optimal control sequence.

To find the solution, Bellman’s optimality principle is used [1]. Let

π∗ ={

P∗FTiπ (1),1

, P∗FTiπ (2),2

, . . . , P∗FTiπ (N),N

}. (1.23)

Fig. 3. Network considered in the supervisory control problem, exemplified for the case of four time stages.


be the optimal policy. Let us define the “tail subproblem” from the node i of the stage k which consists of finding theoptimal solution from node i of stage k to the terminal node. Consider the policy

π∗ ={

P∗FTiπ (k),k

, P∗FTiπ (k+1),k+1

, . . . , P∗FTiπ (N),N

}(1.24)

The optimality principle states that the policy π∗ is optimal for the tail subproblem. In other words, if the solutionto the whole problem happens to pass by node i of stage k, then the solution of the tail subproblem coincides with itfrom this point to the terminal node. This leads to the following recursion:

Initial step: Solve all the tail subproblems for the last stage k = N − 1Generic induction step: Solve all the tail subproblems of length N − k, using the solutions of the tail subproblems

of length less than N − k, i.e., solve all tail subproblems of stage k, by using the solutions of the tail subproblems ofstage k + 1. In this case, this results in the following recursive algorithm:

DP algorithm

VN (i) = aNit , i ∈ SN

Vk(i) = minj ∈ Sk+1

[akij + Vk+1(j)], i ∈ Sk, k = 0, . . . , N − 1,

(1.25)

where aNit is the arc-cost from node i of stage N to a fictitious terminal node t and Vk(i) is the minimum cost from node

i of stage k to the terminal node. This algorithm is known as the backward algorithm. A similar forward algorithm canalso be used.

In our particular implementation, transition costs are computed as the search for the optimal path proceeds. For eachnode, it is checked if the constraint (1.13) is satisfied. If not, the path from this node to the terminal node is discardedfrom the network. Thus, the minimum in (1.25) is taken on the subset S′

k ⊂ Sk of all nodes of stage k that satisfy (1.13).The treatment of the constraint (1.14) is more difficult, because it is of integral form. Its value depends not only on

the values of Preq and PFT at each stage k, but also on their values in the rest of the nodes of each particular policy.Different approaches can be found in the bibliography to manage this type of constraint. In ref. [2], a penalization

term is introduced in the objective functional. This implies modifying arc-costs computation to

akij = 1

2

(PFT i,k

ηFT i,k

+ PFT j,k+1

ηFT j,k+1

)�t + α

∣∣∣∣12(

Preq(k�t) − PFT i,k

ηESS i,k

+ Preq((k + 1)�t) − PFT j,k+1

ηESS j,k+1

)�t

∣∣∣∣(1.26)

The parameter α is chosen by trial and error, in such a way that for the demanded cycle, the resulting consumedenergy profile satisfies (1.14) for each t.

To ensure strictly that this constraint will not be violated, we instead implemented the following alternative strategy.The result given by this algorithm is also suboptimal, but the order of complexity is not increased.

The net consumed energy from the ESS is backward accumulated at the same time that the energy consumed Vi(k)from the FT is computed. For each tail subproblem considered as the algorithm progresses, the constraint is checked. Ifit is violated, the path corresponding to this tail subproblem is discarded from the network. This leads to the algorithmthat follows. Let Qi(k) be the net consumed energy corresponding to the consumption Vi(k) and let

bkij = 1

2

(Preq(k�t) − PFTi,k

ηESSi,k

+ Preq((k + 1)�t) − PFTj,k+1

ηESSj,k+1

)�t (1.27)

be the net consumed energy transition costs.Modified DP algorithm

VN (i) = aNit , i ∈ SN

QN (i) = Q0, i ∈ SN


For k = N−1, . . ., 0For i ∈ Sk

For j ∈ Sk+1’compute ak

ij , bkij

compute Q1kij = −bk

ij + Qk+1(i)end forFind S′′

k+1 = {j ∈ S′k+1 such that Q1k

ij ensures that (1.14) is satisfied}Vk(i) = min

j ∈ S′′k+1

[akij + Vk+1(j)]

save j* that realizes the minimum in Index (i,k)Qk(i) = −bk

ij∗+Qk+1(i)end for

end for

c Compute solution sequence

u(0) = index of initial PFT;For k = 1, . . ., N

u(k) = Index(u(k − 1),k − 1)P∗

FT(k) = PFT(u(k))end for

4. Simulation results

The above method has been applied to the case of the experimental HEV that is being developed in our group. Thisprototype is at present powered in a purely electric form by a bank of batteries and is equipped by a 32 kW electricmotor for traction. In previous work, the longitudinal dynamics of this prototype has been modeled and validatedthrough road tests [7]. This model allows computation of the power demanded by the vehicle to follow a velocity cycle,including mechanical losses and losses from the electrical motor. So it has been linked to the DP algorithm to providethe values for Preq.

A hypothetical fuel converter system capable of delivering a maximum power of 40 kW has been considered. Itsefficiency function ηFT, was taken from ref. [3]. The discretization step for the power from this system was �PFT = 1 kW,since, as also found in refs. [2,3], it was observed that the results did not change substantially for smaller values.

The electric storage system is a bank of 20 batteries Yuasa-Exide EV-5, with nominal charge equal to 197 Ah each.The minimum charge for this system to work properly is 20% of the nominal charge. That means that the bank has towork with a charge value in the interval [Qmin, Qnom], with Qmin = 0.2Qnom. Actually, the bounds for the net consumedenergy EESSmin and EESSmax are not constant, since they depend on how far is the initial charge Q0 from the limits ofthe interval. They also depend on the ESS voltage, which for now was considered constantly equal to its nominal value(Unom = 120 V). Hence, they are computed at the beginning, using

EESSmin = (Q0 − Qnom) × Unom

EESSmax = (Q0 − Qmin) × Unom.(1.28)

The top graphs in Figs. 4 and 5 show the power profiles obtained using the proposed algorithm for an elementaryvelocity cycle that consists of 10 s at constant acceleration until reaching a speed of 30 km/h, 10 s at this speed and 10 sat constant deceleration until completely stopping. Obviously, if the ESS has enough energy to perform the cycle andcompensate for the losses, the solution of minimum consumption will be PFT ≡ 0. Hence, in order to show results ofinterest, low initial energy contents within the ESS were considered and then a hypothetical low value for Qnom wasused. The initial charge is indicated bottom-left in the figures as a percentage of Qnom.

The dashed line is the corresponding demanded power. The full line is the algorithm output, i.e., the optimal PFTprofile. The dotted line is the complementary PESS profile. After the 8th stage approximately, the dashed line coincideswith the dotted one and so, cannot be seen. Negative values of the PESS profile means batteries recharging during


Fig. 4. Optimal power split profiles and net consumed energy from the ESS for Qnom = 0.92 Ah.

braking. The profile obtained indicates that to ensure that the ESS is operated in a safe region during the whole cycle,the engine will have to provide part of the required power in the first stages. Then, it can be turned off and the ESSwill perform the rest of the cycle by itself.

Similar results are obtained if the total initial energy in the ESS is even lower, as in Fig. 5. In this case, the contributionof the ESS is smaller than before, while that from the engine has increased, in order to keep the net consumed energyfrom the ESS between the prescribed limits.

The bottom graphs in Figs. 4 and 5 are the corresponding net consumed energy from the ESS profiles. It can beseen that they move within the bounds.

Results for the European Normalized Cycle [5,7] are presented in Fig. 6. In this example, the demanded power mayexceed the maximum power of the ESS. As the initial charge of the ESS is sufficiently high, the solution indicates thatthe engine has only to be used when the demanded power is greater than the bound PESSmax = 30 kW.

Fig. 5. Optimal power split profiles and net consumed energy from the ESS for Qnom = 0.92 Ah.


Fig. 6. Optimal power split profiles and net consumed energy from the ESS for the European Normalized Cycle and for Q0 equal to 80% of thenominal charge of the bank of batteries of the prototype.

Table 1Consumption from both sources corresponding to the case in Fig. 4

Fuel consumption (kWh) Energy cons. from the ESS (kWh)

Case Q0 = 60% (proposed algorithm) 0.0645 0.0221Case α = 1.97 (using penalization term) 0.0661 0.0237

Table 2Consumption from both sources corresponding to the case in Fig. 5

Fuel consumption (kWh) Energy cons. from the ESS (kWh)

Case Q0 = 50% (proposed algorithm) 0.0751 0.0166Case α = 2 (using penalization term) 0.0755 0.0129

In Tables 1 and 2, results concerning optimal consumption are shown. In order to compare the two methods used, itis also necessary to take into account the corresponding values for the consumed energy from the ESS. So, for the caseof the algorithm that uses the penalization term, we choose a value α such that the resulting energy consumption fromthe ESS is approximately the same as for using the other algorithm. It can be seen that the minima for fuel consumptionare very similar for both methods.

5. Conclusions

We have implemented an algorithm to find the split between the two sources using a priori knowledge of the drivingcycle. Solutions are suboptimal because of the treatment of the constraint on the consumed energy from the ESS.Results seem to be dependent on the backward direction chosen in the implementation of the algorithm. Moreover, inthe backward algorithm, the computation to check the integral constraint (1.14) has to be initialized without a prioriknowledge of the final value of the net consumed energy EESS. Once it reached stage 0, the whole profile can beshifted to satisfy the initial condition EESS(0) = 0. However, to ensure that the shifted profile will also satisfy (1.14), itis necessary to force the profile actually computed into a narrower band. This problem would not appear at all in theforward algorithm, so we are now working on this. The treatment of this integral constraint within this DP approachis still a subject of research [6,9].


On the other hand, the algorithm has some advantages: (a) it overcomes switching and non-linearities of stateequation; (b) it manages easily those constraints that are local in time; (c) it may apply to other systems of interestthat include two sources of energy, such as electronic devices, stand-alone wind or solar stations, power networksincluding both hydro electrical and nuclear power stations, etc.; (d) it may be extended to include stochastic featuresin the system (stochastic DP).

References

[1] D. Bertsekas, Lecture slides on dynamic programming, based on lectures given at MIT, Cambridge, MA, 2002.[2] A. Brahma, Y. Guezennec, G. Rizzoni, Optimal energy management in series hybrid vehicle, in: Proceedings of the American Control

Conference, 2000.[3] A. Brahma, Y. Guezennec, G. Rizzoni, Dynamic optimization of mechanical/electric power flow in parallel hybrid electric vehicles, OSU,

Mech. Eng. Dep. and Center for Automotive Research, 2000, preprint.[4] A. Chiang, Elements of Dynamic Optimization, McGraw-Hill, 1992.[5] S. Delprat, T.M. Guerra, G. Paganelli, J. Lauber, M. Delhom, Control strategy optimization for an hybrid parallel powertrain, in: Proceedings

of the American Control Conference, 2001, pp. 1315–1320.[6] S. Delprat, J. Lauber, T.M. Guerra, J. Rimaux, Control of a parallel hybrid powertrain: optimal control, IEEE Trans. Veh. Technol. 53 (May

(3)) (2004) 872–881.[7] L. Perez, E. Oliva, C. Careglio, R. Leidhold, C. Falco, G. Garcıa, Simulacion de un Vehıculo Electrico Experimental, in: Anales del XVIII

Congr. Arg. de Control Autom., AADECA, 2002.[8] G. Rizzoni, L. Guzzella, B. Baumann, Unified modeling of HEV drive trains, IEEE/ASME Trans. Mech. 4 (September (3)) (1999).[9] A. Sciaretta, M. Back, L. Guzzella, Optimal control of parallel hybrid electric vehicles, IEEE Trans. Control Syst. Technol. 12 (May (3)) (2004).

[10] G. Steinmauer, L. del Re, Optimal control of dual power sources, in: Proceedings IEEE International Conference on Control Applications,2001.

[11] A.T. Zaremba, R.M. Stuntz, A. Borichev, Control strategy development for HEV in Matlab/Simulink using constrained optimization, 2002,preprint.

Optimization of power management in an hybrid electric vehicle using dynamic programming

Documents

Transcript of Optimization of power management in an hybrid electric vehicle using dynamic programming