Drive Mode Optimization and Path Planning for Plug-in Hybrid Electric Vehicles

Drive Mode Optimization and Path Planning for Plug-in Hybrid Electric Vehicles

Chi-Kin Chau, Khaled Elbassioni and Chien-Ming Tseng C.-K. Chau, K. Elbassioni, and C.-M. Tseng are with the Department of EECS at Masdar Institute of Science and Technology, UAE (e-mail: {ckchau, kelbassioni, ctseng} paper appears in IEEE Transactions on Intelligent Transportation Systems (DOI:10.1109/TITS.2017.2691606).

Drive modes are driver-selectable pre-set configurations of powertrain and certain vehicle parameters. Plug-in hybrid electric vehicles (PHEVs) typically feature special options of drive modes that can affect the hybrid energy source management system, for example, electric vehicle (EV) mode (that draws fully on battery) and charge sustaining (CS) mode (that utilizes internal combustion engine to charge battery while propelling the vehicle). This paper studies an optimization problem to enable the driver to select the appropriate drive modes for fuel minimization. We develop optimization algorithms that optimize the decisions of drive modes based on trip information, and integrated with path planning to find an optimal path, considering intermediate filling and charging stations. We further provide an online algorithm that is based on the revealed trip information. We evaluate our algorithms empirically on a Chevrolet Volt, which shows significant fuel savings.

Energy-efficient transportation, plug-in hybrid electric vehicles, fuel optimization

I Introduction

Modern vehicles are provided with a plethora of configuration options. Drive modes are a set of pre-set profiles of configurations of powertrain and other vehicle parameters that are selectable by drivers during driving. For example, Sport mode maximizes the engine performance by allowing larger horsepower, whereas ECO mode suppresses the vehicle performance by constraining acceleration and throttle response.

In this paper, we consider the drive modes specifically in plug-in hybrid electric vehicles (PHEVs). PHEVs are equipped with rechargeable batteries and electrical machines (which double as electric motors and generators), as well as conventional internal combustion engines. PHEVs are benefited by the convenience of fuel refilling and cheap electrical energy. In addition to regenerative braking and electric-motor-assisted engine stop/start, PHEVs can harness the diversity of energy efficiency of electric motors and combustion engines by optimizing the hybrid energy sources.

There are special drive modes in PHEVs that can affect the hybrid energy source management system. For example, Electric Vehicle (EV) mode allows the PHEV to draw solely on battery without relying on internal combustion engine, and Charge Sustaining (CS) mode utilizes internal combustion engine to charge battery and propel the PHEV simultaneously.

Many prior results [1, 2, 3] in energy management consider the internal optimization processes of PHEVs, which assume complete controls of all system components of a PHEV (see Sec. III). As a departure from prior work, this paper focuses on a driver-centric approach to let the driver to select the appropriate drive modes while driving.

In this paper, we consider general optimization problems of drive modes for fuel minimization in the following settings:

  • Route-based Drive Mode Optimization: Given the trip information for a particular route (e.g., a forecast of vehicle speed profile), we find an optimal solution of drive mode decisions for each segment of the trip.

  • Online Drive Mode Optimization: The decisions of drive modes are made in an online fashion, based on only the revealed trip information over time.

  • Integrated Drive Mode Optimization and Path Planning: Given the source and destination of a trip, we find an optimal path with a solution of drive mode decisions, taking into account various fuel prices at intermediate filling stations and the availability of battery charging.

In Sec. IV, we formulate the preceding problem by an integer programming problem, which captures several practical aspects of PHEVs (e.g., multi-mode transmission, and vehicle speed dependency in combustion engine management). In Sec. V, we devise effective algorithms for solving these drive mode optimization problems. We also provide a fast approximation algorithm that can scale with large problem sizes. To demonstrate the practical value of our results, we evaluate our algorithms empirically on a Chevrolet Volt in Sec. VIII. Validated by real-world data, we observe that our system can provide a significant improvement in fuel savings. We also discuss several practical issues in Sec. IX.

The future vehicle platforms will be likely to support third-party applications. Thus, our system can be loaded as a software application on the vehicle for automatic drive mode decisions, even without explicitly relying on drivers’ inputs.

Ii Background

The powertrain mechanics of PHEVs can be classified as series hybrids, parallel hybrids, and series/parallel hybrids. In series hybrids, the internal combustion engine is always connected to the generator to charge the battery. The drivetrain is only powered directly by electric motor. Once the state-of-charge (SoC) of battery becomes low, the internal combustion engine will start to charge the battery. In parallel hybrids, the combustion engine and electric motors can operate in tandem to power the drivetrain. There is a power split system to combine the parallel power sources, and a possible clutch to enable the combustion engine to charge battery and propel the vehicle simultaneously. In series/parallel hybrids, a combination of planetary gear trains allows flexible power split between the combustion engine and a number of electric motors.

Despite the differences of powertrain mechanics, the internal operations of PHEVs are often transparent to drivers. There are automatic systems to manage the transmission gear, powertrain and hybrid energy sources. Given the steering and pedal control status, the automatic system controls the torque, rpm of combustion engine111In practical PHEV systems, the rpm of combustion engine is normally related to the vehicle speed, even for series hybrids in which the combustion engine is not directly connected to the drivetrain. Apparently, there is a safety hazard for the combustion engine operating in a high speed, when the vehicle is stationary. Also, production PHEVs often use a variable number of electric motors/generators conditional on the vehicle speed. In high speed, more than one electric motor may be used., transmission gears, output power of battery, etc., to match the load of drivetrain accordingly [1, 2]. Although the low-level mechanics are not controllable by drivers, typical vehicles are usually customizable by setting certain high-level drive modes. We survey the available drive modes in several production PHEV models that can affect the behavior of hybrid energy management system in Table I.

 Vehicle Model Drive Mode       Description
Chevrolet Volt (model 2011-2015) Normal Draw only on battery until SoC drops to 22%. Then, use combustion engine to charge battery and propel vehicle.
Mountain Same as Normal mode, but draw only on battery until SoC drops to 45%.
Hold Use combustion engine to maintain the current SoC.
Toyota Prius Plug-in (model 2012-2015) Normal Use both combustion engine and electric motor to propel vehicle.
EV Always draw on battery, if there is sufficient SoC and within EV mode speed limit.
Ford Fusion Energi (model 2014-2015) EV Now Draw on battery entirely, if there is sufficient SoC.
EV Auto Use both combustion engine and electric motor, depending on vehicle speed.
EV Later Conserve battery by mostly using combustion engine, which also charge battery.
TABLE I: Examples of drive modes in production PHEVs.

Since there are a variety of model-specific drive modes, this paper aims to present a study as general as possible and to be extensible to future models. Rather than considering model-specific drive modes, we consider four generic drive modes: (1) Electric Vehicle (EV) mode that draws solely on battery, (2) Combustion Engine (CE) mode that relies solely on internal combustion engine, (3) Charge Sustaining (CS) mode that utilizes internal combustion engine to charge battery and propel the vehicle, and (4) Aggregate Power (AP) mode that combines both electric motor and internal combustion engine to propel the vehicle. In Sec. IV, we formulate a general optimization problem of generic drive mode decisions. We remark that our problem is sufficiently general to capture a variety of existing PHEV models. We will discuss the mapping from the model-specific drive modes to generic drive modes, and validate our model for Chevrolet Volt empirically.

Iii Related Work

There is a body of work about optimizing energy management systems for PHEVs. For example, [1] uses heuristic control strategy to optimize energy consumption for given torque and speed. A similar concept relying on rule-based management policies has been presented in [2]. [4] considers continuous-time optimization control of hybrid energy sources. Some studies focus on sub-optimal solutions that can be computed faster than dynamic programming [5][6]. These prior results often assume complete controls of internal energy management system in PHEVs, and are mostly based on simulations. Our work considers limited control by only selecting the drive modes available in the PHEVs, and we evaluate the results on a real-world PHEV. The models of PHEVs and experimental validations were studied in [7, 8].

This papers utilizes the estimated information of vehicles. Participatory sensing can provide data with good geographic penetrations [9]. Our previous work employs participatory sensing for distance-to-empty prediction [10, 11, 12].

There are prior papers about online energy management strategies for PHEVs. For example, [3] utilizes nonlinear optimization for parallel and serial HEVs. However, this paper considers competitive online algorithms [13] that can provide proven worst-case guarantees from the offline optimal solutions. The path planning problem considering various gas prices at filling stations has been studied in[14]. However, this paper considers a more general path planning problem with energy management strategies for PHEVs. This paper extends substantially the preliminary results in [15]. We now provide approximation solution to path planning and extend the online algorithm to consider four generic drive modes.

Iv Model and Problem Formulation

Notation  Definition
The state-of-charge of the battery at time .
The fuel tank level at time .
The power of drivetrain of PHEV at time .
The power from battery to electric motor at time .
The power from generator to charge the battery at time .
, The charging and discharging efficiency coefficients at time .
The output power from combustion engine at time .
A function maps the output power to the required amount of fuel.
The per-unit cost of , .
The power from engine to charge the battery at time .
The charging efficiency coefficient by combustion engine at time .
The maximum available power to charge battery
from combustion engine at time .
The maximum portion of power contributed by electric motor at time .
TABLE II: Key Notations

Our goal is to develop a systematic study for drive mode decisions, based on a generic vehicle model characterized by parameters using measurements or standard vehicle information. We consider a semi-blackbox model of PHEV that is abstracted away from the underlying vehicle control systems. A table of key notations is given in Table II.

This paper considers a discrete-time setting from time slot to , where the inputs within one time slot are assumed to be quasi-static. Let be the fuel tank level and be the SoC of the PHEV at time . and are the initial fuel tank level and SoC respectively.

Define the driving profile be , where is the vehicle speed and is the gradient of road at time . The driving profile can be obtained by prediction using historic data, or crowd-sourced data collection [16, 9, 10, 11, 12]. Note that is non-negative for all . We assume that the energy consumption of PHEV is solely characterized by the driving profile, for example, under moderate weather and traffic conditions. Let the acceleration be . The load of drivetrain of a generic vehicle [17, 18] is given by


where is the vehicle weight, is the gravitational constant, is the density of air, is the frontal area of the vehicle, is aerodynamic drag coefficient of the vehicle, is the rolling friction coefficient, and is the default load (e.g., due to air-conditioning). These parameters can be obtained from standard vehicle information or simple measurement.

Note that can be positive or negative (possibly due to negative ). Let and . represents the power captured by regenerative breaking.

In the following subsections, we describe the four generic drive modes (EV, CE, CS, AP modes), as illustrated in Fig. 1. These generic drive modes provide abstract representations of the model-specific drive modes in PHEVs.

Iv-a Electric Vehicle (EV) Mode

In EV mode (also called charge depleting mode), the PHEV is only powered by battery, which is also charged by regenerative braking when decelerating or stopping. Let be the allowable range of SoC to operate in EV mode. Let be the power from battery to electric motor (when ), and be the power from generator to charge the battery (when ). If , then the SoC is given by


subject to , and . Parameters and denote the charging and discharging efficiency coefficients. Note that and are time-variable222The efficiency coefficients are often assumed time-invariant in previous work (e.g., [1, 2, 3])., because there may be a variable number of generators/motors to be utilized in the PHEV, depending on the driving profile (as observed in production PHEVs such as Chevrolet Volt).

Note that regenerative braking incurs no fuel cost. Hence, Eqn. (2) and the constraints are equivalent to setting


Iv-B Combustion Engine (CE) Mode

In CE mode, the PHEV is only powered by internal combustion engine. Let the output power from combustion engine at time be . The fuel tank level is given by


subject to . is an increasing convex function that maps the output power to the required amount of fuel.

We also allow the battery to be charged by regenerative braking, if possible. The SoC is given by


subject to and . This implies that .

Iv-C Charge Sustaining (CS) Mode

In CS mode, the internal combustion engine is used to propel the vehicle and charge the battery simultaneously. Let be the power from engine to charge the battery at time . The fuel tank level is given by


subject to . The SoC is given by


subject to , , and . denotes the charging efficiency coefficient by combustion engine. In this paper, we allow possibly different generators used by regenerative braking and combustion engine (as observed in Chevrolet Volt). is the maximum available power to charge battery from combustion engine at time .

Fig. 1: System models of four generic drive modes.

Note that practical control system in PHEVs (e.g., Chevrolet Volt) may set the rpm of combustion engine related to vehicle speed. captures the limitation of available power from combustion engine depending on vehicle speed at time .

We assume that the energy management system in PHEV attempts to charge battery up to whenever possible. Then, Eqns. (6)-(7) and the constraints are equivalent to setting

Iv-D Aggregate Power (AP) Mode

In AP mode, the PHEV is propelled by internal combustion engine and electric motor (that is powered by battery) together. The SoC is given by


subject to , and . is the maximum portion of power contributed by electric motor to drivetrain. In this paper, we allow a variable number of generators/motors to be utilized in the PHEV, conditional on the driving profile, which can induce a variable portion of power split by electric motor and combustion engine over time. The fuel tank level is given by


subject to . We assume that the energy management system in PHEV attempts to use electric motor to power the drivetrain by whenever possible. Then, Eqns. (8)-(9) and the constraints are equivalent to setting

Note that our model is abstracted away from the underlying mechanics, like automatic transmission, powertrain control, etc. But the parameters are sufficiently general to capture the effects of the underlying mechanics.

Iv-E Mapping to Generic Drive Modes

We next discuss the mapping from model-specific drive modes to generic drive modes. Chevrolet Volt has four drive modes: Normal, Sport, Mountain and Hold. We consider EV and CS modes only, and SoC lies in the ranges . To trigger EV mode, one can enter Normal mode. To trigger CS mode, one can enter Mountain mode. If SoC is higher than , only EV mode is allowed. If SoC is lower than , only CS mode is allowed.

subject to (10) (11) (12) (13) (14) (15) (16) (17) (18)
subject to (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)

Toyota Prius Plug-in provides four drive modes: ECO, Normal, Power and EV. EV mode is present. Normal mode may be mapped to CS or CE modes, whereas Power mode may be mapped to AP or CE modes. Ford Fusion Energi offers three drive modes: EV Now, EV Auto and EV Later. EV mode is present. EV Auto mode may be mapped to AP mode, whereas EV later mode may be mapped to CS mode.

The mapping of model-specific drive modes to generic drive modes can be validated by empirical studies. In this paper, we validate our model particularly for Chevrolet Volt in Sec. VIII.

Iv-F Drive Mode Optimization Problem

Let , , , be the binary decision variables indicating if EV, CE, CS, and AP modes are enabled at time , respectively. Let . We formulate a fuel minimization problem of drive mode decisions by integer programming problem (DMOP), given a driving profile , initial SoC and fuel tank level . The objective of DMOP is to minimize the total cost: .

Note that DMOP does not always have a feasible solution, for example, when there is insufficient fuel. Also, if a drive mode is not present, we can disable a certain drive mode in DMOP by adding an additional zero constraint to the respective drive mode. For example, to disable AP mode, we set for all . In particular, we denote by DMOP that has only EV and CS modes, without CE and AP modes, which can model series hybrid Chevrolet Volt.

  Algorithm 1.   1: 2:for each  do 3:        4:       if  then 5:              6:                      7:if  then 8:       return 9:else 10:       return INFEASIBLE     Algorithm 2.   1:if  then 2:        3:       for each  do 4:              5:              6:             if  then 7:                     8:                     9:                                          10:       return 11:else 12:       return     Algorithm 3.   1: 2:, 3: 4:if  and  then 5:       , , Use EV mode 6:else if  and or not ( and  then 7:       , , Use AP mode 8:else if  and or not  then 9:       , , Use CS mode 10:else if  then 11:       , , Use CE mode 12:return  

V Offline Solution

V-a Exact Solution by Dynamic Programming

This section provides offline solutions to solve DMOP. Our exact solution is based on dynamic programming.

Consider a sub-problem at time , given the previous SoC and intended current SoC :

subject to  Cons. (11)-(18)

Note that given a fixed drive mode , can be solved straightforwardly. Hence, can be solved by selecting the minimum-cost solution among the four drive modes. Let be the minimum-cost solution, denoted by . If there is no feasible solution, returns infinite cost . Assuming the range is divided into a set of discrete levels, such that we can enumerate each possible level. A dynamic programming approach is presented in Algorithm . A similar approach can be applied to DMOP with absent drive modes (e.g., DMOP) by restricting each to only the available drive modes.

Theorem 1

Algorithm provides an optimal solution for DMOP with pseudo-polynomial running time333The running time of a pseudo-polynomial-time algorithm depends polynomially on the size of input in unary representation, whereas the running time of a polynomial-time algorithm depends polynomially on the size of input in binary representation.. The proof can be found in the Appendix.

V-B Approximation Solution by Convex Relaxation

DMOP is a non-convex problem, even if we relax the integrality Cons. (18), because of Cons. (14)-(16). Thus, we define a convexified problem (cDMOP) in the following way:

  1. We replace Cons. (14)-(16) by simple upper bounds in Cons. (23)-(25). Cons. (22) can ensure in cDMOP are not larger than those in DMOP.

  2. We add a constraint () to ensure the load to be entirely satisfied by electric motor in EV mode.

  3. We relax the decision variables to be fractions between 0 and 1 in Cons. (27).

By rounding the largest fractional variable among up to one and other smaller variables down to zero (subject to the respective feasibility constrain of each mode), this provides a fast approximation solution to solve DMOP.

Vi Online Solution

We present an online algorithm (Algorithm ) for DMOP that does not require the future driving profile, but only the current vehicle state. Let the inputs of DMOP be . DMOP can be solved optimally, when all inputs are given in advance. However, is revealed gradually over time in practice, which requires decision-making without future information.

An algorithm is called online, if the decision at the current time only depends on the information before or at the current time slot (i.e., ). Given input , let be the cost of algorithm , and be the cost of an offline optimal solution (that may rely on an oracle to obtain all future inputs). In competitive analysis for online algorithms [13], competitive ratio is a metric defined by the worst-case ratio between the cost of the online algorithm and that of an offline optimal solution, namely,


Algorithm selects the drive mode following priority: EV AP CS CE, based on the respective conditions. Define normalized cost by the fuel cost over the amount of used fuel (i.e., in CS mode and in AP mode). First, EV mode is selected, if it is present and there is sufficient SoC. Else, AP mode is selected, if it is present and the normalized cost is less than threshold or CS and CE modes are not present. Else, CS mode is selected, if it is present and the normalized cost is less than or CE mode is not present. Otherwise, CE mode is selected. The basic idea of is to make conservative decision at each time step, such that the incurred energy consumption is within a certain bounded range from the offline optimal solution.

Define the per-unit cost by . Note that is strictly increasing convex and is an increasing function. Suppose for some constants and . We assume can be estimated in advance for a specific trip444First, set . Then are updated to be the maximum and minimum observed so far at each time . If is relatively large, the estimated will converge to the true values.. Let Next, we determine proper and with a good competitive ratio.

Theorem 2

We consider the initial SoC and we require the final SoC to be . Assuming for all , let the thresholds in Algorithm be


where , then the competitive ratio of for solving DMOP is


Vii Integrated Path Planning

We consider an integrated optimization problem with both path planning and drive mode decisions for the PHEV:

  • There are multiple paths between source and destination.

  • There are possible intermediate nodes in each path to provide fuel refilling or battery charging.

A road network is represented by a connected directed graph that connects from the source to the destination . For each edge , may be a stop, such that the PHEV can receive refilling of fuel at price per unit, or battery charging at most unit at price per unit. Let be the set of paths connecting and . We label the edges in each by and write . Let be the number of time slots required for traveling . Let be the initial fuel tank level at time , when traveling . Let the fuel tank capacity be .

The path planning problem with drive mode decisions is formulated as an integer programming problem (PPDM).


In PPDM, we find the optimal path from to and the corresponding drive modes , given the initial fuel level () and SoC ().

Vii-a Exact Solution by Dynamic Programming

Vii-A1 Uniform Cost Case

First, we consider the uniform case with identical fuel price ( and for all ). For a path , by Eqn. (33), we obtain


By Eqn. (32) and Eqn. (34), we obtain


Thus, we can rewrite PPDM as follows:

subject  to Eqns. (45) and (35)-(43)

To solve uPPDM, we construct a weighted directed graph as follows (see Fig. 2). Let be the value of the optimal solution of DMOP for edge , when the and , which can be obtained by dynamic programming as explained in Sec. V-A. For every node and every discrete level in the range , we create a node . If then we have an edge with weight , for every555Note that, since battery charging at each node is free, it suffices to construct only one edge corresponding to ; however, defining the graph in this general form allows to extend the dynamic program for the case when there is a cost for charging each unit of battery at node . discrete levels in such that


In addition, we create a source node with edges , for each in the range , having weight and cost ; and a destination node with edges having weight , for all in the range .

Fig. 2: An illustration of and , where is a set of discrete levels .

Then the optimal solution to uPPDM can be obtained by finding an -shortest path in the graph , with the (non-negative) weights interpreted as distances.

Vii-A2 General Case

Next, we consider the more general case when the fuel cost per unit may not be equal at all nodes , and with the additional restriction that the PHEV can make at most stops between and to do a fuel refill. We assume that battery charging (at the stop) is allowed only when the vehicle stops for fuel refill666This assumption can be removed, if at all nodes..also, we assume that battery charging at node costs per unit, and the objective is to minimize the combined fuel cost and battery charging cost. The basic idea is to adopt the dynamic program for the so-called Gas Station Problem in [14], and apply it to the graph constructed above. We define the graph as the subgraph of such that for all . (That is, in (46), takes only one value, namely , which corresponds to the case when no battery charging is allowed at node .)

As we have discretized the battery level in , we may also discretize the gas level in . However, following [14], we can already define a discrete set , as the set of fuel levels that are sufficient to consider at node :


See Lemma 1 in the Appendix for a justification. Here, is the shortest distance (with respect to the weights ) between (not necessarily adjacent nodes) and in the graph (note that we use as it is assumed that the PHEV does not make any stop between and ).