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This paper presents a novel method to solve the energy management problem for hybrid electric vehicles (HEVs) with engine start and gearshift costs. The method is based on a combination of deterministic dynamic programming (DP) and convex optimization. As demonstrated in a case study, the method yields globally optimal results while returning the solution in much less time than the conventional DP method. In addition, the proposed method handles state constraints, which allows for the application to scenarios where the battery state of charge (SOC) reaches its boundaries.

Hybrid electric vehicles (HEVs) are one option to reduce the CO_{2} emissions of passenger light-duty vehicles. Such hybrid vehicles consist of at least two power sources, typically an internal combustion engine and one or more electric motors, as well as an energy buffer, typically a battery. The control of the power flows among these devices is commonly referred to as the energy management or supervisory control [

However, in all those convex optimization approaches, costs for switching the engine on/off state or changing the gears are not considered. As reported for instance in [

Therefore, this paper presents a method to solve the energy management problem for a parallel HEV taking such engine start and gearshift costs into account by using a combination of convex optimization and DP. In a case study, the method is shown to converge to the globally optimal solution while still being computationally efficient. Moreover, the proposed method can handle state constraints. Although not presented in this paper, the algorithm can potentially be applied to an efficient sizing of hybrid powertrains or to the on-board control of hybrid vehicles in the form of receding horizon control.

The paper is structured as follows: Section 2 details the vehicle model used,; Section 3 explains the mathematical problem; Section 4 presents the novel method; Section 5 compares the novel method to DP; Section 6 discusses the novel method; and Section 7 concludes on the ideas shown in this paper.

The vehicle under investigation is a pre-transmission parallel HEV passenger car of the executive class.

In order to apply the concepts presented in Section 4 below, the generic, nonlinear vehicle model has to be approximated by a convex model, see [

A convex description of the model requires the component models to be expressed as a convex function of the free optimization variables which are used by the convex solver. These free optimization variables typically comprise more free variables than are actually required to solve the problem using different methods such as, for instance, DP. Examples of such free optimization variables are the torques of the motor and the engine, the electrical power consumption of the motor, the fuel consumption, the battery current and the battery state of charge (SOC). Inherently discrete variables, such as the engine on/off decision or the gear decision, are not convex, and thus, they cannot be included in a convex description. However, for given trajectories of the non-convex variables, the remaining model can be convex.

In this paper, the energy management of the vehicle is assumed to have three decision variables, namely the engine on/off, the gear, and the torque split between the engine and the motor. Since the engine on/off and the gear decision are inherently discrete, these variables are assumed to be known prior to solving a convex optimization problem. The only convex decision variable is the torque split. Therefore, the power flows from the road load up to the gearbox input, where the torque split is decided, can be expressed by any nonlinear function. Only the power flows upstream of the gearbox input have to be described in a convex form. In the following, the approximations needed to obtain a suitable model are explained in detail. All vehicle parameters used in the model are listed in

Assuming that a quasi-static driving cycle is represented by its velocity _{w}_{w}

The variable _{w}_{air} the air density, _{d}_{r}_{υ}_{g}_{r}_{r}

The speed _{g}_{g}_{g}_{g}_{,0}, _{g}_{,1}, _{g}_{,1} being the parameters to model the speed-dependent gearbox losses which account for the increased friction at higher gearbox input speeds [

Moreover, losses occurring during gearshifts are assumed to consume _{g}

The gear _{g}

These shift losses in

The torque split between the electric motor and the internal combustion engine is determined by the motor torque _{ts}_{m}_{e}

This equation is convex in any of the variables involved.

The electric motor, with a nominal power of 40 kW, is directly coupled to the input shaft of the gearbox such that the motor speed _{m}_{g}

This description is convex in the variable _{m}

The operation of the motor is limited by its minimum and maximum torque limitations, as well as by its maximum speed:

The model fit for the electric motor compared to the underlying measurement data is shown in

The mass flow _{f}_{e}_{e}

At all times, the engine is only allowed to operate within its limits defined by a maximum torque, minimum speed and a maximum speed condition,

The model fit of the engine compared to measurement data is shown in

The engine on/off decision is determined via the variable _{e}_{g}_{e}

These start costs represent the fuel-equivalent costs to start the engine. For example, these costs can be calculated based on an electrical energy consumption to crank up the engine to its start speed and a fuel consumption to synchronize the engine speed with the gearbox input speed, see for instance [

The battery is assumed to be of the LiFePO_{4} type [_{oc}_{i}

In the convex case, the discrete-time equation of the dynamics for the battery SOC reads:
_{b}_{aux}_{b}_{b}

The battery current and the battery SOC are constrained to:

In summary, the control variables

The problem to be solved is the energy management problem formulated as an optimal control problem as follows:
_{0} with _{0} being the initial state of charge.

The energy management problem can be solved using DP. However, to account for the engine start and gearshift losses, three state variables are required,

A more efficient approach to solve the problem was proposed in [

In this paper, these drawbacks are avoided by extending the ideas presented in [

As shown in [

By further defining:

_{ts}_{g}_{e}

Therefore, in this paper, the difference to the approach described in [

Minimize:

Note that the gearshift and engine on/off sequence must be defined prior to solving _{g}_{e}_{g}

The problem in

As illustrated in _{in}, and the output of the DP consists of the gearshift strategy _{g}_{e}_{in} stands for _{in}(_{out}_{e}_{g}_{out} is calculated by solving the convex optimization problem in _{g}_{e}

Assuming that the optimal value of _{in} = ^{⋆} is known, the optimal values of
^{⋆} stands for optimal, and ≡ stands for identity of the trajectory. On the other hand, solving the convex problem for the optimal values of
_{out} ≡ ^{⋆} [_{in} ≡ ^{⋆}, then _{out} ≡ _{in} ≡ ^{⋆}. As a consequence, the identity “_{out} ≡ _{in}” is a necessary condition for the optimality of a solution.

To show that this condition is also a sufficient condition, the condition “if and only if _{out} ≡ _{in} ⇒ _{out} ≡ ^{⋆}” has to be shown. To do so, two further scenarios need to be considered for the case if _{in} ≠ ^{⋆}: in the first scenario, it may happen that solving _{in} ≠ ^{⋆}_{out} ≡ ^{⋆}, and therefore _{out} ≠ _{in}.

In the second scenario, solving _{out} ≠ ^{⋆} and also _{out} ≠ _{in} as explained in the following:

Assume that the values of _{in} are lower than those of the optimal solution ^{⋆}. This means that electric energy is considered to be cheap in the optimization problem in _{ts}, u_{g}, u_{e}

Then, at the next step of the sequential optimization, the optimal power split is calculated via convex optimization for the values of _{e}_{g}_{out} to be higher than those of _{in}. Therefore, if _{in} ≠ ^{⋆}_{out} ≠ s_{in}. Finally, it is concluded that if and only if _{out} ≡ _{in} holds, then _{out} ≡ ^{⋆}.

Based on these arguments, an iterative algorithm can be constructed which starts with an initial guess of _{in} and converges towards a solution which fulfills the condition _{in}≡ _{out}. In the following, this iterative algorithm is referred to as the DP-C method.

The iteration counter is initialized by

In this step, based on

As a result, new values of

For the given values of

To check whether the DP-C method has converged, the trajectories of

If the value of _{TOL}_{TOL}

In practice, it is more convenient to define a convergence tolerance Δ_{TOL}_{g}_{e}_{TOL}^{−5} L/100 km. However, this method of checking the convergence is not equivalent to the one described in the first paragraph above. Therefore, checking whether

Performing Steps 2 and 3 does not necessarily yield the optimal solution since the optimal equivalence factor is unknown. Initializing the equivalence factor _{in} with a too high value will result in a too low value for _{out} (and ^{⋆} lies between _{in} and _{out}. To ensure convergence towards ^{⋆}, damping is introduced by including an update law for the equivalence factor
^{⋆} for

By using the tolerance criterion introduced in Step 4 (Δ_{TOL}

For the generation of the optimization results shown below, the indicator for the fuel consumption in _{f}_{l}

To assess the performance of the DP-C method, it is compared to a pure DP implementation of the convex vehicle model described in Section 2. For the following case studies, unless stated otherwise, the minimum and maximum values of the SOC are assumed to be 20% and 80%, respectively. The SOC state of the DP method is discretized with a resolution of 1% SOC.

Both the DP-C and the DP method are applied on three well-known driving cycles, namely the New European Driving Cycle (NEDC), the Federal Test Procedure (FTP), and the Common Artemis Driving Cycle (CADC). In a further case, referred to as “CADC bounded”, the DP-C and the DP method are both applied to the CADC with tighter SOC constraints,

As

Comparing the computational burden associated with the DP-C and the DP methods, the DP-C significantly outperforms the DP method. As

Summarizing the findings described in this section, the DP-C method yields more accurate results in much less time than a pure DP implementation.

To assess the performance of the DP-C method on the nonlinear vehicle model, the control strategy obtained with the DP-C method is applied to the nonlinear model. The result obtained is then compared to the result of a pure DP implementation including the nonlinear model. In this case study, the only differences of the nonlinear model compared to the convex model are the use of the original consumption maps instead of second-order polynomials in

To increase the accuracy of the DP-C method on the nonlinear vehicle model, the DP-C algorithm is modified at Step 2 (“DP for _{g}_{e}

As

In terms of computation time, the DP-C method is more than 90% faster than the DP method in the cases where the full range of the SOC is discretized (NEDC, FTP, CADC), and it is 75% faster in the case with the reduced SOC discretization (CADC bounded).

In conclusion, even for the nonlinear model, the DP-C method is able to yield close-to-optimal results in much less time than the pure DP method.

In the case of the convex model, the above values for the fuel consumption calculated by DP are always worse than those obtained with the DP-C method. The accuracy of the DP solutions could generally be improved by increasing the resolution of the discretization of the SOC state. However, the improvements in accuracy are very small compared to the additional computational burden required.

A substantial advantage of the DP-C method is its ability to yield globally optimal results for the convex vehicle model much faster than the DP method. In the case of a nonlinear model, the performance of the DP-C method is very close to that of the DP method. Moreover, the equivalence factor related to the battery state is obtained directly from the convex solver without any error-prone calculation via the optimal cost-to-go from DP. Furthermore, more accurate results can be obtained with the DP-C method than with the DP method because no discretization of the SOC state is necessary. In addition, as opposed to the method presented in [

The only drawback of the DP-C method are the modeling errors introduced by the convex modeling of the powertrain. However, in the case of the HEV in this paper, the errors are small.

This paper presents a method to calculate the globally optimal energy management strategy for a parallel HEV on a given driving cycle taking into account penalties to avoid frequent engine start and/or gearshift events. The proposed method combines DP and convex optimization in an iterative scheme. The algorithm converges to the globally optimal solution after a few iterations. The optimal gearshift and engine on/off strategy is evaluated by DP while convex optimization is used to determine the optimal power split strategy. The proposed method delivers globally optimal results with respect to the convex vehicle model, even in the presence of state constraints. When compared to the basic DP algorithm, the proposed method results in a substantial reduction of the evaluation time.

For the convex vehicle model, the proposed method delivers a higher precision than DP (0.1%–0.2% lower fuel consumption) while incurring significantly less computational effort (75%–98% less). The higher precision is due to the fact that convex optimization does not require a discretization of the continuous control and state variables, which inherently introduces interpolation errors. To evaluate the magnitude of the error that is introduced by using convex model approximations, the strategy that was optimized for the convex model is applied to the nonlinear model. Compared to the true globally optimal solution obtained by applying DP on the nonlinear model, the proposed method results in a slight deterioration of precision only (up to 0.3% increased fuel consumption).

The proposed method can be extended to other vehicle topologies and different formulations of the energy management problem. For example, the battery state of health or the engine temperature could be considered by including further continuous state variables. Additional discrete state variables, such as for example the clutch state, as well as additional continuous and discrete control variables, such as for example the desired clutch state, the motor speed or torque of a series-parallel hybrid vehicle, could be included as well.

The authors express their sincere gratitude to Daimler AG for having supported this project.

The authors declare no conflicts of interest.

Hybrid electric vehicle (HEV) architecture considered in this paper: pre-transmission parallel HEV.

Fit of the convex motor model.

Fit of the convex engine model.

Comparison of the nonlinear and the approximated model for the open-circuit voltage of the battery.

Sequential application of dynamic programming (DP) and convex optimization.

Iteration scheme of the DP-C algorithm.

Comparison of the results obtained with DP and the DP-C algorithm on the New European Driving Cycle ((NEDC) driving cycle.

Comparison of the results obtained with DP and the DP-C algorithm on the Federal Test Procedure (FTP) driving cycle.

Comparison of the results obtained with DP and the DP-C algorithm on the Common Artemis Driving Cycle (CADC) driving cycle.

Comparison of the results obtained with DP and the DP-C algorithm on the CADC driving cycle with limited SOC range.

Parameters of the pre-transmission parallel HEV executive class passenger car. SOC: state of charge.

Wheel radius | _{w} |
0.32 m |

Air density | _{air} |
1.24 kg/m^{3} |

Effective frontal area | _{d} |
0.60 m^{2} |

Rolling friction coef. | _{r} |
0.012 |

Gravit. constant | _{g} |
9.81 m/s^{2} |

Total vehicle mass | _{υ} |
1800 kg |

Rot. equiv. mass (gear-dep.) | _{r} |
[129, 84, 72, 61, 55, 52, 51] kg |

Gear ratios | _{g} |
[10.8, 7.1, 4.7, 3.4, 2.5, 2.0, 1.8] |

Gearbox efficiency param. | _{g}_{,0} |
0.95 |

Gearbox efficiency param. | _{g}_{,1} |
0.02 1/(rad/s) |

Gearbox efficiency param. | _{g}_{,1} |
400 rad/s |

Nominal motor power | 40 kW | |

Max. motor speed | _{m,max} |
628 rad/s |

Nominal engine power | 150 kW | |

Min. engine speed | _{e,min} |
105 rad/s |

Max. engine speed | _{e,max} |
628 rad/s |

Max. battery capacity | _{0} |
7.64 A h |

Open circuit voltage | _{oc} |
263 V |

Battery int. resistance | _{i} |
0.24 Ω |

Min. battery current | _{b,min} |
−200 A |

Max. battery current | _{b,max} |
200 A |

Min. SOC | _{min} |
0.20 |

Max. SOC | _{max} |
0.80 |

Aux. power demand | _{aux} |
400 W |

Results with the convex vehicle model.

NEDC | FC [L/100 km] | 4.405 | 4.399 | (−0.1%) |

ICE starts [#] | 7 | 7 | (+0.0%) | |

Gearshifts [#] | 86 | 86 | (+0.0%) | |

CPU time [s] | 791 | 6 | (−99.2%) | |

| ||||

FTP | FC [L/100 km] | 4.288 | 4.279 | (−0.2%) |

ICE starts [#] | 28 | 28 | (+0.0%) | |

Gearshifts [#] | 180 | 182 | (+1.1%) | |

CPU time [s] | 1287 | 14 | (−98.9%) | |

| ||||

CADC | FC [L/100 km] | 5.635 | 5.627 | (−0.1%) |

ICE starts [#] | 78 | 75 | (−3.8%) | |

Gearshifts [#] | 316 | 310 | (−1.9%) | |

CPU time [s] | 2363 | 105 | (−95.5%) | |

| ||||

CADC (bounded) | FC [L/100 km] | 5.657 | 5.643 | (−0.2%) |

ICE starts [#] | 90 | 85 | (−5.6%) | |

Gearshifts [#] | 322 | 310 | (−3.7%) | |

CPU time [s] | 527 | 122 | (−76.9%) |

Results with the nonlinear vehicle model.

NEDC | FC [L/100 km] | 4.453 | 4.449 | (−0.1%) |

ICE starts [#] | 7 | 7 | (+0.0%) | |

Gearshifts [#] | 56 | 54 | (−3.6%) | |

CPU time [s] | 684 | 18 | (−97.4%) | |

| ||||

FTP | FC [L/100 km] | 4.317 | 4.270 | (−1.1%) |

ICE starts [#] | 28 | 28 | (+0.0%) | |

Gearshifts [#] | 138 | 138 | (+0.0%) | |

CPU time [s] | 1111 | 17 | (−98.5%) | |

| ||||

CADC | FC [L/100 km] | 5.625 | 5.642 | (+0.3%) |

ICE starts [#] | 84 | 74 | (−11.9%) | |

Gearshifts [#] | 354 | 350 | (−1.1%) | |

CPU time [s] | 1959 | 155 | (−92.1%) | |

| ||||

CADC (bounded) | FC [L/100 km] | 5.639 | 5.654 | (+0.3%) |

ICE starts [#] | 93 | 88 | (−5.4%) | |

Gearshifts [#] | 366 | 394 | (+7.7%) | |

CPU time [s] | 434 | 107 | (−75.5%) |