This paper presents a study of the energy-efficient operation of all-electric vehicles leveraging route information, such as road grade, to adjust the velocity trajectory. First, Pontryagin's maximum principle (PMP) is applied to derive necessary conditions and to determine the possible operating modes. The analysis shows that only five modes are required to achieve minimum energy consumption: full propulsion, cruising, coasting, full regeneration, and full regeneration with conventional braking. Then, the minimum energy consumption problem is reformulated and solved in the distance domain using dynamic programming to find the optimal speed profiles. Various simulation results are shown for a lightweight autonomous military vehicle. The sensitivity of energy consumption to regenerative-braking power limits and trip time is investigated. These studies provide important information that can be used in designing component size and scheduling operation to achieve the desired vehicle range.

## Introduction

With the technologies and advancements made in the field of vehicle electrification, the search is not only to find the next technological breakthrough but also to make the most out of what is presently available. Today's vehicles already have the sensors, software, and capabilities to predict the estimated time of arrival and travel distance based on a driver's chosen destination and real-time traffic data. Utilizing such data can provide the necessary information to operate the vehicle in an energy-efficient manner. The improved energy efficiency of autonomous driving systems has been studied by researchers and engineers for the past three decades [1], addressing the challenges that lie in decision-making, path planning, and trajectory generation [2–6]. Traditionally, speed profile optimization has been extensively studied in application to railway vehicles because of their operating environment: a given route, known road grade, and speed limits [7–11].

Minimizing energy consumption is still a huge challenge for ground vehicles [12]. Powertrain technology has been in continuous improvement; nonetheless, drivers have not adapted their driving style to improve efficiency. Eco-driving or energy-efficient driving is the term given to the idea of determining the speed trajectory that minimizes the vehicle energy consumption under final time and distance constraints. This problem has been addressed as an optimal control problem (OCP) with related work done by researchers [3–5,7–23], the solutions of which can provide a vehicle speed profile that decreases the amount of energy consumed over the trip.

In solving the trajectory optimization problem, numerical approaches such as genetic algorithm [7,14,16], ant colony algorithm [16], sequential quadratic programming [13], nonlinear programming [24], and dynamic programming (DP) [15–17,23,25] have been used for nonlinear characteristics, such as varying speed limits, resistive forces, and signal-phase timing. Among these, DP has been widely and extensively used because it can find a global optimum even for nonlinear systems with nonlinear constraints. In Ref. [23], authors applied DP to determine optimal torque distribution between front and rear in-wheel motors with preview terrain information. However, these methods typically require substantial computation time due to discretization of states and control inputs. In Refs. [24] and [25], the authors propose approaches to improve computational efficiency through approximated vehicle operation and problem reformulation, respectively. Pontryagin's maximum principle (PMP) can be used to analyze the system and to decrease the computational cost by reducing the number of possible control actions to the set, which achieves the optimal solution. Standard results for the minimization of energy consumption at wheels for conventional vehicles consist of only four modes: full propulsion, cruising, coasting, and full braking [5].

The optimal control modes of all-electric vehicles may not necessarily be the same as those of conventional vehicles. This paper extends the basic idea of applying PMP to all-electric vehicles and synthesizes the results of the PMP analysis into DP formulation. When applying PMP, we deliberately introduce three control inputs: motoring, regenerating, and friction braking, unlike approaches in Refs. [8] and [10]. For the given vehicle model, namely, point-mass longitudinal dynamics with invariant motor efficiency, five modes are required to achieve energy-efficient operation: full propulsion, cruising, coasting, full regeneration, and full regeneration with conventional braking. Based on this result, the minimum energy consumption problem is reformulated and solved by DP for the case of a lightweight military vehicle. In particular, two different driving environments including nonhilly and rugged roads are studied. The simulation results demonstrate the effectiveness of eco-driving in regards to energy consumption and driving time, which could be useful in planning mission trips.

Furthermore, this paper investigates the sensitivity of performance such as net energy consumption and journey time to regenerative braking power capability, typically not considered in speed optimization problems. As regenerative braking capability changes, for example, because of the size of batteries or thermal management system malfunction, the optimal trajectory may change. Thus, the optimal trajectory planning accounting for regenerative braking power would provide beneficial information in both design and control aspects. Moreover, efficient use of control modes has a significant impact on the overall energy consumption and vehicle performance. The importance of the operational environment also plays an important role in the mode changes that occur. Additionally, the constraints of deceleration, acceleration, speed, and distance limit play an essential role in the selection of control modes. The impact of battery sizing and regenerative power impact on the change of control modes are reflected in this paper.

This paper is organized as follows: Sec. 2 details modeling of a ground vehicle and analysis of the optimal control problem. The reformulation and implementation in Dynamic Programming are described in Sec. 3. In Sec. 4, case studies in application of a lightweight military vehicle, trajectory optimization results obtained by DP, and the impact of battery power capability are discussed in detail. Finally, conclusions and directions for future work are presented in Sec. 5.

## Minimum Energy Consumption Problem

In this section, the minimum energy consumption problem is presented and analytically solved by applying Pontryagin's maximum principle (PMP). PMP is a powerful computational and analytical tool used to solve optimal control problems. PMP provides a set of necessary conditions that an optimal control must satisfy while maximizing (or minimizing) the Hamiltonian function. Despite the fact that PMP generally yields necessary conditions, it is still useful and effective: (1) to find a small subset of control actions and (2) to numerically compute optimal control trajectory significantly fast compared to DP. For a certain class of optimal control problems, such as power management of hybrid electric vehicles, solutions obtained from PMP and DP are hardly different, which makes PMP widely used in automotive applications [22,26,27].

For simplicity, the following assumptions are made: (1) the efficiency of the system is invariant, that is, motoring and generating efficiency of the electric motor is fixed; (2) the vehicle moves forward only; and (3) road grade and velocity limit are distance-dependent and known a priori. It is noted that although the efficiency of an electric motor is generally nonlinear [23], the efficiency is relatively constant above certain force or torque levels covering nominal operating range, which makes the first assumption reasonable.

### Model Description.

where *s* and *v* represent distance and velocity, respectively, while *M* is vehicle mass and *t* is time. The variables *A*, *B*, and *C* are coefficients used to determine resistance forces by rolling and aerodynamic drag, and the grade angle *θ* is a function of distance, i.e., *θ* = *θ*(*s*). The variables *F _{p}* and

*F*denote control inputs regarding propulsion and braking, respectively. Hereafter, $*\xaf$ and $*\xaf$ are used to refer to the maximum and minimum values of parameter *.

_{b}*b*) is rewritten as

where *F _{m}*,

*F*, and

_{g}*F*represent motoring, generating, and friction braking forces, respectively. This choice is deliberate to clearly distinguish operational modes of the vehicle in the OCP analysis discussed later.

_{fb}*A*,

*B*, and

*C*are used to compute the resisting forces; rolling and aerodynamics force, with the resisting force being speed-dependent, while the grade force depends on the angle,

*θ*, such that

where *t _{f}* and

*S*are operational time and distance traveled, and these are bounded, meaning that the vehicle stops at time

_{f}*t*after traveling a given distance,

_{f}*S*. The vehicle is assumed to move forward only, i.e., $x2(t)>0,t\u2208(t0,tf)$.

_{f}### Optimal Control Modes.

where $Peelec$ is the electrical power by the motor; and *η*_{1} and *η*_{2} denote motoring and generating efficiency of the electrical machine, respectively.

where $\theta \u2032=(\u2202\theta (x1)/\u2202x1)$.

*u*, in order to determine the driving modes. Then Eq. (6) becomes

Based on the values of the switching functions, $(x2/\eta 1)+p2,\u2009x2\eta 2+p2$, and *p*_{2}, the control inputs to minimize $H$, and the fact that each have three options of being greater than, less than, or equal to 0, 27 different modes are found possible. It should be noted that not all of the modes are feasible due to the assumptions made. In other words, infeasible modes can be first ruled out by investigating the assumptions. First, efficiency of the electric motor must be greater than 0 and less than 1; i.e., $0<\eta i<1,i\u2208{1,2}$. Second, the electric vehicle moves forward only, i.e., *x*_{2} > 0. From these two conditions, it is found that $(x2/\eta 1)>\eta 2x2$. For instance, when $p2\u22650$, the conditions $(x2/\eta 1)+p2<0$ and $x2\eta 2+p2<0$ are infeasible because they violate *x*_{2} > 0. The conditions $(x2/\eta 1)+p2<0$ and $x2\eta 2+p2=0$ cannot occur simultaneously since they violate $(x2/\eta 1)>\eta 2x2$. For the same reason, the conditions $x2\eta 2+p2\u22650$ and $(x2/\eta 1)+p2<0$ cannot occur simultaneously as well.

*u*

_{1},

*u*

_{2}, and

*u*

_{3}are undefined because the optimal control mode is not uniquely determined from the given condition. The existence of the optimal control mode can be analyzed by checking the necessary condition, known as Kelley's condition [28]. For a single control problem, the control

*u*for a singular-arc can be determined by differentiating $\xi =(\u2202H/\u2202u)$ with respect to time

*t*until

*u*appears explicitly, and then Kelley's condition must be satisfied. That is,

where *l* is an integer variable.

*e*),

*u*

_{1}is undefined. Consider $\xi 1=(\u2202H/\u2202u1)=(x2/\eta 1)+p2=0$. Its first and second time derivatives are given by

*f*), the solution can be found to be

The cases (9*e*) and (9*f*) are both cruising mode as the same amount of input is applied to the system in order to keep it at constant speed; (9*e*) applies the input in propulsion while (9*f*) applies it in braking.

*g*),

*u*

_{3}is undefined. Consider $\xi 3=(\u2202H/\u2202u3)=p2=0$. Then, its first and second time derivatives are expressed as following:

Since $p2=0,\u2009p\u02d91=0$ from Eq. (7*a*). The solution would never arrive to a situation in which *u*_{3} (friction braking) would stand alone and have a distinct solution; therefore, this mode is also infeasible.

Finally, the possible control modes are summarized in Table 1. Note that the number of control modes is five and that two different braking modes are used in an electrified vehicle unlike a conventional vehicle without regenerative braking capability.

### Constraints on Propulsion and Braking.

Propulsion and regenerative-braking of the electrified vehicle are limited by the maximum power from electric components such as electric motor(s), a fuel cell and a battery. As a result, the influence of these limits or constraints on vehicle operation need to be clearly understood to properly formulate the control problem.

*v*−

*f*relationship can be obtained from the motor torque,

*τ*, the tire radius,

*r*, and the final drive ratio

*i*

_{a}*ω*, from the following relation:

*ϕ*

_{1}and

*ϕ*

_{2}are velocity-dependent upper and lower limits of the force. However, the motor power is constrained by power/energy sources. For instance, when fuel cells and batteries are used, motoring power is limited by the maximum power from both fuel cells and batteries, whereas charging power is limited by batteries only; in this case, the shape of force limits is no longer symmetric. The functions

*ϕ*

_{1}and

*ϕ*

_{2}are influenced by the size of powertrain components, i.e.,

where *P _{b}*,

*P*and

_{fc,}*P*denote battery power, fuel-cell power, and motor power, respectively. Note that the switching functions are the same as those described in Sec. 2.2; therefore, $u\xaf1$ and $u\xaf2$ become

_{m}*ϕ*

_{1}(

*x*

_{2}) and

*ϕ*

_{2}(

*x*

_{2}), respectively. The influence of propulsion and braking power on the optimal trajectory will be presented and discussed in Sec. 4.

## Trajectory Optimization in Spatial Domain

The formulation of an optimal control problem is complicated and has implications for the obtained solution. By solving the same problem in different ways, we can determine more straightforward and less computationally intensive methods. Specific factors impact the optimization of energy consumption in an electric vehicle. These factors include the constraints on the speed, acceleration, time, distance, and braking force. Additionally, the problem can be formulated in the discrete or continuous time domain, or even translated into the distance domain.

where *η* is the motor efficiency and the mode-dependent variable *q* is defined such that *q *=* *1 for regenerative-braking and *q* = −1 for propulsion. Note that the equations of motion are expressed in terms of *s* and hence $x\u2032$ is used instead of $x\u02d9$ to represent a derivative with respect to distance rather than time. i.e., $x=[x1,x2]T=[t,v]T$.

To solve the optimal control problem, dynamic programming (DP) is used in this study. DP is a powerful numerical technique to determine optimal trajectories explicitly using the Bellman's optimality principle while searching from the final state backward in time. It is well known that DP suffers from computational burden when the number of state and control variables increase, i.e., *curse of dimensionality*. Nonetheless, DP is still useful to find a global optimum even for nonlinear systems with constraints when the number of variables are small enough [29].

To perform optimization using DP, the equation of motion needs to be expressed in a discrete domain. Discretization inherently introduces numeric errors, which degrades the accuracy of the found solution. To avoid these numerical errors, a careful implementation is important [30]; in this study, discretization steps for time, velocity, and distance are chosen as provided in Table 2.

For each of the five control modes in (18), a set of equations are used to compute the corresponding acceleration, velocity, and force for a given distance of Δ*s* as follows:

In the equations above, the larger magnitude of the resisting force, *f _{r}*, or grade force,

*f*, in addition to whether the grade is positive or negative, determines whether the force applied to the vehicle is a propulsion force or braking force. Thus, the force in Cruising mode can be either positive or negative depending on the grade and its dominance over the other resisting forces. Note that mode 4, Full braking mode, involves friction braking in addition to regenerative braking. It is important to clarify that the benefit of this mode is reflected in its use when braking over the regenerative braking limit is needed. This mode would allow the vehicle to respect the speed and time constraints; that is, the vehicle would be allowed to operate at higher speed because rapid deceleration is possible. Additionally, this mode enables the system designer to choose an energy storage device with a smaller capacity that might otherwise limit regenerative braking capability.

_{g}*electrical*forces, denoted by $feelec$, the motor and generator efficiencies are included, such that:

*electrical*energy consumption is computed as:

where *N _{f}* is the total number of finite horizon in a distance space and Δ

*s*is the distance discretization as given in Table 2. Note that Δ

*s*= 30 m is used after careful simulation proved that the quality of optimal solutions could be kept while computation load is decreased. It can be found that the same discretization length is considered in Ref. [23].

## Case Study

In this work, a light-weight military ground robot is considered as a target vehicle. As the needs of military vehicles rapidly increase, energy-efficient maneuvering becomes one of the most important requirements among robust performance, mobility, and ability to support a variety of electrical loads. Because of the last requirement, electrified powertrains have been receiving attention in all branches of the military (ships, airplanes, and combat vehicles) for their potential strategic benefits [31]. It should be noted that, although the application considered in this work is to a military ground vehicle, the presented work and approach can be applied to any electrified vehicles.

The army ground vehicle programs use various drive cycles including time, speed, and grade, for testing and validation of new vehicle systems and models. These cycles have traditionally been characterized by run speed, number of stops, and terrain profile. For the sake of powertrain analysis, there have been a number of additional metrics proposed for characterization of such drive cycles in the context of fuel economy evaluation. The drive cycles for ground vehicles focus on running at a constant speed over varied terrain for practical reasons [32]. Thus, two different drive conditions are studied: relatively flat and hilly roads as shown in Fig. 2. Specifically, these two drive cycles are considered as the baseline operations when comparing the performance of the optimized speed profile obtained from DP; that is, the total energy consumption and trip time traversing these two cycles are computed from a given set of time, speed, and grade information.

### Nonhilly Environment—Convoy Cycle.

The Convoy Cycle, shown in Fig. 2, has some deviation in speed and a small variation in grade as shown in Fig. 2. From the cycle and parameters in Table 3, the resisting force, *f _{r}*, the grade force,

*f*, and the acceleration required to operate the robot can be computed. The total electrical energy consumed from the Convoy cycle was calculated to be 1.06 MJ, which represents the baseline value for a nonhilly operation. The top subplot of Fig. 3 displays the results of total energy consumption for different trip durations, which clearly shows a tradeoff between energy consumption and trip time. Particularly, when the trip time is set to be similar to that of the baseline, the total electrical energy consumed to traverse the optimal speed profile obtained by DP is 21% less, i.e., 0.84 MJ. When the amount of energy consumed is set to be similar to that of the baseline Convoy cycle, a 14% reduction in trip time is achieved.

_{g}Figure 4 shows the speed profiles over two different trip times in addition to the baseline Convoy cycle. With the speed limit set at 23 m/s and with a nonhilly road, the vehicle does not show frequent mode changes, but a stable cruising operation that allows the vehicle to stop within the set trip time. Additionally, for the same distance but with a longer trip time, the vehicle tends to decrease its speed, which obviously reduces the energy consumption. The propulsion and cruising that the vehicle executes are due to the little deviation in grade that allows the vehicle to increase its speed at downhills without enduring additional energy loss. The regenerative braking is used at the end of the trip to stop in accordance with the final velocity constraint.

### Rugged Hilly Environment—Churchville B Cycle.

Unlike the Convoy cycle, the Churchville B cycle provides constant velocity throughout the trip. However, this cycle has a more discernible grade change that highlights the performance benefits of velocity optimization over a more irregular course or terrain that has steeper grades. The total electrical energy consumed from the Churchville B cycle was calculated to be 0.41 MJ, which represents the baseline value for the rugged-hilly operation. Using trip time similar to the baseline, the solution to the OCP shows a 24% reduction in energy consumption, i.e., 0.31 MJ. Moreover, it is observed from Fig. 3 that a 24% reduction in total trip time can be achieved when consuming the same amount of electrical energy as the baseline. The results in Fig. 5 display the speed profile for Churchville B cycle over two different trip times compared to the baseline. The vehicle exhibits several mode changes as it drives over an uphill and downhill distinctively; in particular, coasting and cruising modes are effectively utilized to minimize energy consumption. As the trip time decreases, the vehicle tends to make better use of increased speed at downhills and uses the propulsion mode more often. In contrast, for a longer trip time, the vehicle tends to use more braking that could benefit in regenerating energy while respecting the time constraint.

### Impact of Hybridization—Battery Power Capability.

Note that motor power is set at 30 kW, which can be interpreted such that all power sources including fuel cells and batteries can provide sufficient power to operate the motor over the considered driving cycles. In this study, it is assumed that the battery energy or state-of-charge constraints do not become active throughout the trip. That is, any regenerative braking events following full propulsion do not recuperate more than the energy used for vehicle propulsion. This assumption allows us to avoid adding the battery state-of-charge dynamics. The inclusion of energy constraint to address long downhill operation will be addressed in our future work.

As shown in Fig. 6, the Convoy cycle reflects no difference in energy consumption for different regenerative power capability ranging between 10 and 25 kW. When it comes to the Churchville B cycle, the results show that as the regenerative power capability increases, energy consumption is decreased; however, as the regenerative power capability reaches 20 kW, there is no further improvement. In other words, the sizing parameter has a saturation point after which increasing the regenerative power capability adds no value to the vehicle. As shown in Fig. 7, increasing the regenerative power from 10 kW to 15 kW improves the energy consumption by as much as 35%. It can be observed that as the trip time set increases, the impact of resizing the regenerative power decreases to as low as 6%. Moreover, increasing the regenerative power from 15 kW to 20 kW has a lower benefit on the energy savings. The maximum energy savings is as high as 6% and drops to as low as 2%. These results reflect on the idea of the reduced impact of increasing the regenerative power.

The vehicle has capability constraints set on it through the power limit and acceleration limit. For instance, due to the deceleration limit set on the vehicle, the regenerative power required from the vehicle would not exceed 25 kW. As a result, increasing the regenerative power of the vehicle would not have a significant impact. Referring to Fig. 8, it can be observed that during braking, a majority of the points fall in the low power zones. Furthermore, as the regenerative power is increased from 15 kW to 20 kW, only 5 points fall outside the 15 kW domain while 1 point only falls in the 25 kW domain. Hence, there is no significant impact on the minimization of energy consumption with an increase in regenerative power beyond 20 kW.

These results highlight the importance of operational environment in design and control. The hilly Churchville B cycle allows the vehicle to use coasting and regenerative braking more often at downhills. The use of the regenerative braking mode requires higher regenerative power capabilities of the motor. On the other hand, the Convoy cycle has minimal grade change and requires the regenerative braking mode only at the end of the cycle to stop the vehicle within the time constraint. As a result, there is no significant impact of battery size on energy consumption. Furthermore, increasing the regenerative power capability reflected greater flexibility in decreasing the driving time for a trip. With an increased regenerative braking capability, the vehicle has more flexibility to increase its speed while sticking to the final distance set.

### Impact of Control Modes.

Statistical information about control modes over the two driving cycles for different time constraint is shown in Fig. 9. It is observed that the hilly Churchville B terrain exhibits more use of coasting. Driving downhill, the vehicle can use the grade force to its benefit for as long as the vehicle is maintaining the speed limit. In the nonhilly Convoy cycle, the vehicle is focused more on cruising mode. The straight path for the vehicle allows the vehicle to keep the balance of overcoming the resisting forces through cruising. Second, increasing trip time increases the use of coasting which is an efficient mode in minimizing energy consumption. Having more time flexibility, the vehicle tends to save energy by exerting no additional forces. Nonetheless, in the Churchville B cycle, the coasting mode is not inevitably increasing with increased trip time. The reason comes down to the importance of the operational environment that plays a significant role in the mode changes that occur. Notably, the constraints of deceleration, acceleration, speed, and distance limit play an important role in selecting the optimal control modes. Finally, it is evident that as the regenerative power capability increases, the use of the full propulsion and regenerative braking modes decreases. The reason for the decrease in using these modes comes down to the increase in the use of coasting, which leads to higher averaged velocity.

One of the main advantages of the PMP analysis performed in this paper is the optimization approach using only five modes. The use of full braking mode is advantageous in many ways. First, when the trip time is a constraint, the full braking mode allows the vehicle more flexibility in applying the maximum speed for a longer duration while arriving at the destination with zero velocity within the time set. Although there is a deceleration limit set on the vehicle that might cause the full braking mode not to be beneficial in this case, including the mode is still the proper way to compute a speed profile. Second, the full braking mode allows operation with smaller batteries, at times when regenerative braking capability is insufficient. As a result, this would allow more freedom to downsize the battery to compensate for the size of another energy source that could be used to hybridize the vehicle and provide an even higher overall efficiency.

## Conclusion

In this paper, the approach of synthesizing PMP analysis and Dynamic Programming has proved effective in computing the desired speed profile for energy minimization. The minimum energy consumption problem was formulated and analyzed by PMP. The analysis showed that only five modes are required to achieve the minimum energy consumption, which could significantly reduce computation time for DP. Then, the OCP was reformulated in the distance domain and solved by DP. For the considered vehicle, a light-weight military electrified vehicle, the simulation results demonstrate a tradeoff between energy consumption and trip time for both flat and hilly roads. The optimal cycle uses 20% less energy for the same trip duration or could reduce the time by 14% with the same energy consumption.

The impact of battery size, which dominates the regenerative braking capability, on energy consumption and control modes was also investigated. The benefit from increasing battery size, or regenerative power capability, depends on the operational environment. That is, for the nonhilly terrain, no significant reduction in energy consumption is observed regardless of the increase in regenerative power capability. On the other hand, for the hilly terrain, it is beneficial for the considered vehicle if larger batteries are installed; yet, no greater than 20 kW.

Future work includes further investigation of the influence of vehicle design on the optimal speed profile in the presence of a nonlinear motor efficiency map. Moreover, co-optimization of speed profile and power management problems will be conducted to minimize total fuel consumption. The co-optimization problem will include battery state-of-charge dynamics and their constraints, in addition to nonlinear efficiency of the fuel cell system.

## Acknowledgment

The authors would like to acknowledge the technical and financial support of the Automotive Research Center (ARC) in accordance with Cooperative Agreement No. W56HZV-14-2-0001. U.S. Army Combat Capabilities Development Command (CCDC) Ground Vehicle System Center (GVSC) in Warren, MI and the Advanced Vehicle Power Technology Alliance (AVPTA) between the Department of Energy (DOE) and the Department of the Army (DA). UNCLASSIFIED: Distribution Statement A. Approved for public release; distribution is unlimited OPSEC#1991.

## Funding Data

Automotive Research Center (W56HZV-14-2-0001: Funder ID: 10.13039/100008192).

U.S. Army Tank Automotive Research, Development and Engineering Center (Funder ID: 10.13039/100009922).