Adaptive Equivalent Consumption Minimization Strategies for Plug-In Hybrid Electric Vehicles: A Review

Abstract

Adaptive Equivalent Consumption Minimization Strategies (A-ECMSs) are one of the best methodologies to optimize fuel consumption of plug-in hybrid vehicles (PHEVs) coupled with low computational requirements. In this paper, a review of A-ECMSs is proposed. Starting from an economic-environmental contextualization, hybrid vehicles are presented and classified, together with their modeling methodologies and the physical-mathematical representation of their components. Next, the control theory for hybrid vehicles is introduced and classified, deriving the A-ECMS approach. Several works accounting for different A-ECMS implementations, based on technology integration, time horizon, adaptivity mechanism, and technique, are addressed. The literature analysis shows a broad coverage of possibilities: the simple proportional-integral (PI) rule for equivalence factor adaptivity is often used, imposing a given battery state-of-charge (SoC); it is possible to optimally plan the battery SoC trajectory through offline optimization with optimal algorithms or by predicting ahead conditions with model predictive control (MPC) or neural networks (NNs); the integration with emerging technologies such as Vehicle-To-Everything (V2X) can be helpful, accounting also for car-following data and GPS information. Moreover, speed prediction is another common technique to optimally plan the battery SoC trajectory. Depending on available on-board computational power and data, it is possible to choose the best A-ECMS according to its application.

1. Introduction

Recent regulations and policy directives concerning the reduction in pollutant emissions in the transport sector have driven increasing interest in advanced technologies for alternative and sustainable mobility; while the electric vehicle market demand is struggling to grow, a promising mid-term solution lies in hybrid electric vehicle (HEV) mobility.

The main evidence of climate change is the increasing global average temperature anomaly as depicted in Figure 1: while the average temperature anomaly fluctuation before 1930 can be associated with environmental changes being 0.3 °C lower on average, starting from the early 1940s, the mean temperature anomaly has steadily increased up to over 1 °C [1].

Scientific boards agree on a causal relationship between average temperature and CO₂ emissions due to the greenhouse effect: an increase in CO₂ content in the Earth’s atmosphere increases thermal energy storage and, hence, temperature rise. The annual CO₂ emissions are reported in Figure 2: while older world regions such as North America and Europe are reducing annual greenhouse gas (GHG) emissions with green regulations, increasing economic growth in Asia pushes further away the produced emissions peak [2].

GHG emissions are produced by several sectors in different proportions, as shown in Figure 3. Starting from 1995, the transportation sector is responsible for most of the GHG emissions, after electricity and heat production, with almost 8 billion tons produced (around 16% of the total value) only in 2021 [3].

Looking at the transportation sector in 2018, of the 8 billion tons, 3.6 billion tons of CO₂ were emitted by passenger vehicles, including cars, motorcycles, buses, and taxis [4].

Plug-in HEVs (PHEVs) are a promising mid-term solution to rapidly decrease total vehicle emissions; however, sales numbers still struggle to rise due to high cost, heterogeneous purchasing power, and limiting charging time [5]. In addition, not all countries have installed proper charging infrastructures. In 2019, Norway had the European sales relative record for plug-in HEVs, with almost 14% share of new vehicle registrations, and this number continues to rise today. In 2023, 4.3 million cars sold all over the world were plug-in, with Europe’s share being about 20% [6]. It is notable to remember that the depletion of fossil fuels, which PHEVs still rely on, will have direct consequences on HEVs due to the lower availability of stable energy sources and higher fuel prices, pushing regulations and markets towards pure Electric Vehicles (EVs) [7].

The research community has shown great interest in HEV technologies: starting from 2000, more than 67,000 papers with the keyword “hybrid vehicles” have been published, while almost 8700 papers with “plug-in hybrid vehicles” are available. The number of publications per year is reported in Figure 4, with the orange bars for “plug-in hybrid vehicles” as search keyword. An increasing trend in publications is clearly visible, with around 14% more publications each year compared to the previous year.

The purpose of this work is to analyze the current literature concerning control strategies required for the suitable energy management of the plug-in HEV powertrain, with a particular focus on adaptive Equivalent Consumption Minimization Strategies (A-ECMS). Firstly, a proper state-of-the-art assessment is proposed in Section 2, with a focus on hybrid vehicle classification, modeling approaches, and applied control strategies. Afterwards, Section 3 presents the analysis of existent possibilities for A-ECMS, with insights into relevant solutions.

2. State-of-the-Art Assessment

In this section, an overview on HEVs definition, classification, and control strategy statement is given, also briefly discussing the models widely used to represent powertrain components.

2.1. Hybrid Electric Vehicles Overview

A hybrid vehicle is a vehicle powered by more than one power source: fixing the power demand by the driver or the road load, the delivered power is divided between power sources, leading to additional degrees of freedom of the powertrain [8]. Most HEVs consider an Internal Combustion Engine (ICE) coupled with a battery capable of delivering a certain amount of power comparable to that of the ICE, unlike traditional vehicles, where the battery serves only for start-up phases. In the past decade, the role of ICEs has been replaced by Proton Exchange Membrane Fuel Cell (PEMFC) systems, using hydrogen as a primary chemical energy source [9]. The main advantage of using more than one power source is the possibility to run the ICE in better efficient operating conditions, e.g., by decoupling the power request from the ICE or reducing the ICE power output. In both cases, a reduction in fuel consumption and pollutant emissions is achieved, while the power request by the driver can still be satisfied [10]. Furthermore, in some powertrain architectures, the ICE can be downsized, leading to a reduction in vehicle weight, manufacturing costs, and fuel consumption [11]. On the other hand, an increase in vehicle complexity and, often, component number occurs. It should be noted that, to achieve a reduction in fuel consumption, a proper Energy Management Strategy (EMS) that manages the power outputs of each powertrain component must be correctly designed and implemented [12].

2.1.1. Powertrain Configuration

The layout of a hybrid vehicle powertrain is characterized by multiple energy flows exchanged among the power systems [13]. With reference to ICE-based HEVs, three main layouts can be identified: series, parallel, and mixed.

Series Layout

Within a series layout, the conventional engine (ICE) directly recharges the battery (B) through an electric generator (EG). The battery is connected to an electric motor-generator (EMG) that can work in both charge and discharge modes. The EMG delivers the mechanical power to the drivetrain, taking the required power from the battery in driving mode. Regenerative braking can also be accomplished: the battery can be charged through the EMG during vehicle speed reduction (i.e., braking), motivating the choice of a more complex electric machine [14]. For plug-in HEVs, battery charging can also be fulfilled through a plug port connected to an external power source (e.g., the electric grid) (P). In Figure 5, a schematic configuration of a series plug-in HEV is reported: mechanical and electrical energy flows are sketched with solid and dashed arrows, respectively.

In this case, the ICE is decoupled from the road load: this allows the engine to operate in its optimal operating conditions, maximizing efficiency and reducing fuel consumption and pollutant emissions; furthermore, transient operations can be avoided or optimally controlled [15]. The plug port can be directly linked to the battery or to an electrical node together with the electrical generator for the ICE. The main disadvantage of this configuration is the high powertrain complexity: this can cause an increase in vehicle weight, reducing the beneficial effect of hybridization on fuel consumption; more specialized and frequent maintenance is required, and a high number of components leads to an increase in vehicle costs compared to other architectures.

Parallel Layout

In the case of a parallel layout, the ICE and the battery can independently satisfy the road load demand since they are both mechanically connected to the drivetrain (W) [16]. The battery (B) feeds the electric motor-generator (EMG) linked to the traction axle, while the engine (ICE) needs no additional components since it is mechanically coupled directly to the traction axle [17]. For plug-in HEVs, battery charging operations can again be fulfilled through a plug port connected to an external power source (P). In Figure 6, a schematic configuration of a parallel plug-in HEV is reported: mechanical and electrical energy flows are sketched with solid and dashed arrows, respectively.

Differently from the series layout, the parallel configuration presents less components, with lower vehicle weight and complexity. The ICE and the battery can independently satisfy the road load demand, allowing the possibility to explore several control strategies to optimize the entire powertrain [18]. However, the ICE is affected by road load fluctuations, causing a worsening in fuel consumption. In addition, ICE optimization is difficult to achieve since its performances are strongly connected to the road load profile over time.

Mixed Series-Parallel Layout

In the mixed configuration, the ICE can both charge the battery (B) with an electric generator (EG) and drive the traction axle: the mechanical power split is managed with a planetary gear that divides the mechanical power output of the ICE towards the battery and the road load (W) [19]. The battery charging for plug-in HEVs is still represented through a plug port (P). In Figure 7, a schematic configuration of the mixed series-parallel HEV is reported: mechanical and electrical energy flows are sketched with solid and dashed arrows, respectively.

This layout is the most complex of the three: the ICE can work in series mode through the EG, achieving best efficiency, and in parallel mode, satisfying road load in case of high-power demand. The main advantages of series and parallel layouts are combined in this configuration, at the expense of the highest powertrain complexity, vehicle weight, and component number.

2.1.2. Degree of Hybridization

A crucial design parameter to further characterize HEVs is the Degree of Hybridization (DoH) [20], defined as the ratio between maximum power deliverable by the battery over the total maximum power of the powertrain as shown in Equation (1):

$$DoH={\displaystyle \frac{P_{Batt}^{max}}{P_{ICE}^{max}+P_{Batt}^{max}}}$$

(1)

where $P_{Batt}^{max}$ and $P_{ICE}^{max}$ are the maximum power deliverable by the battery and the ICE, respectively. Based on the DoH [21], a functional classification of HEVs is proposed in

Table 1. Functional classification of hybrid vehicles based on degree of hybridization.

Classification	Degree of Hybridization	Features
Zero hybrid	0%	ICE start-up
Micro hybrid	<10%	Start/Stop
		Regenerative breaking
Mild hybrid	10–30%	Start/Stop
		Regenerative breaking
		Power assistance
Full hybrid	>40%	Start/Stop
		Regenerative breaking
		Power assistance
		Short electric drive
Plug-in hybrid	>50%	Start/Stop
		Regenerative breaking
		Power assistance
		Extended electric drive
		Plug-in capability
Full electric	100%	Start/Stop
		Regenerative breaking
		Power assistance
		Full electric drive
		Plug-in capability
		No ICE

: from micro-hybrids to plug-in hybrids, many features are added to the powertrain with the reduction in engine size and an increase in battery capacity.

Together with DoH, the hybrid vehicle design is also characterized by the battery capacity: lower battery capacity is used only for ICE start-up and, in some cases, for start-stop features for micro-hybrids; to take advantage of hybridization, the battery pack must be able to store enough energy. Typically, the battery capacity increases with DoH. When it comes to pure EVs, the battery alone must accomplish every power requirement together with guaranteeing enough mileage to the driver; hence, the recharging technology for this class of batteries is more sophisticated with respect to HEVs and PHEVs, allowing fast recharging times. On the other hand, fast recharging technologies are critical for battery life and durability, while slow charging preserves battery life.

2.1.3. Mathematical Modeling of HEVs

In order to coherently represent the performances of HEVs, several mathematical models have been developed and proposed in the literature, ranging in complexity and computational burden. For every component, a suitable mathematical representation is needed: once the overall mathematical model is set up, control strategies can be tested on the HEV model for software-in-the-loop and hardware-in-the-loop validation procedures [22]. Hereafter, the main models widely adopted in the literature [23] are presented for backward approaches, in which it is assumed that the desired speed is perfectly matched by the vehicle, representing the external input to the powertrain. In the case of forward dynamics, the calculation procedure considers as input the power delivered by the powertrain sources, i.e., battery and ICE, and computes the vehicle speed as output [24]. This latter formulation is more adherent to real-life applications, but it is not suited for global optimization algorithms; the backward formulation, instead, leaves space for optimal control strategies and real-time optimization.

Longitudinal Vehicle Dynamics

To characterize road load and, hence, power demand, typically the longitudinal vehicle dynamics equation is written [25]. The force balance for longitudinal dynamics is shown in Figure 8, where the main forces acting on a vehicle are considered (i) the powertrain traction/braking force $F_t$; (ii) the aerodynamic friction $F_d$; (iii) the rolling resistance $F_r$; (iv) the gravitational force $mg$; and (v) the inertia force $F_i$.

The longitudinal vehicle dynamics force balance is shown in Equation (2).

$$F_i=F_t-F_d-F_r-mg\sin \alpha$$

(2)

where α represents the road slope, and m is the vehicle mass.

Every term in Equation (2) can be expressed as a function of the known parameters of the vehicle. The inertia force is associated with the vehicle mass and that of its rotating parts, the latter being expressed as an equivalent additional mass m_eq:

$$F_i=\left(m+m_{eq}\left(\gamma \right)\right){\displaystyle \frac{dv}{dt}}$$

(3)

It is important to note that the equivalent additional mass is a function of the gear ratio $\gamma$, as the transmission ratio between moving parts and wheels influences the equivalent inertia; for low gear ratios (equivalent to high gear numbers), the influence can be neglected, while for high gear ratios (equivalent to low gear numbers), the effect of the rotating parts is evident.

The aerodynamic drag force is typically expressed as being proportional to the square of the vehicle speed:

$$F_d={\displaystyle \frac{1}{2}}{\rho }_{amb}{c}_x{A}_f{v}^2$$

(4)

where ${\rho }_{amb}$ is the ambient air density, $c_x$ is the drag coefficient, and $A_f$ is the frontal area. The drag coefficient can also be expressed as a function of the vehicle speed, together with other parameters involving actual flow conditions [26], but for typical test cycles, it is generally assumed to be constant.

Rolling tire resistance is often modeled as the gravity force longitudinal projection:

$$F_r=c_{r}mg\cos \alpha$$

(5)

where $c_r$ is the rolling resistance coefficient, often assumed to be as a quadratic polynomial function of the vehicle speed [27].

Electric Motor-Generator Unit

Different approaches can be adopted to model the EGMs. The first widely used approach is to model this component through its efficiency [28], which is defined differently in traction and generator modes:

$$\left\{\begin{array}{c}{\eta }_m\left(T,\omega \right)={\displaystyle \frac{P_{mech}}{P_{el}}}\\ {\eta }_g\left(T,\omega \right)={\displaystyle \frac{P_{el}}{P_{mech}}}\end{array}\right.$$

(6)

where $P_{mech}$ is the mechanical power generated/absorbed by the EMG, $P_{el}$ is the electrical power absorbed/produced by the EMG, while ${\eta }_m$ and ${\eta }_g$ are the efficiency in traction and generation modes, respectively.

Another approach is through the Willans line method [23], in which the generated power is calculated considering the energy conversion efficiency and friction power losses:

$$P_{gen}=e_n\left(P_{abs},\omega \right)P_{abs}-P_{friction}\left(\omega \right)$$

(7)

where $e_n$ is the energy conversion efficiency function of the electrical power and rotational speed, $P_{gen}$ is the generated power that can be both mechanical and electrical, $P_{abs}$ is the absorbed power that can be both electrical and mechanical, and $P_{friction}$ is the friction power losses function of the rotational speed. The energy conversion coefficient and friction power losses can be calculated with polynomial interpolations, typically up to second-order polynomials. Finally, the maximum torque for traction mode, the maximum mechanical power, and the maximum rotational speed must be known to ensure the feasibility of the results: the maximum torque defines an upper limit at constant torque, the maximum power defines a hyperbolic upper limit, and the maximum speed defines a threshold speed limit that cannot be accomplished by the EMG.

DC/DC Converter

Every high-power electrical architecture needs proper power electronics to ensure power transmission at different voltage values. The DC/DC converter is typically an electronic circuit in which several switches are controlled together with inductors [29]: acting on the switches’ duty cycle, it is possible to modulate the output voltage from the source voltage with low power dissipation. In the latest applications, DC/DC converters are reduced to complete integrated circuits [30]. The duty cycle timing is orders of magnitude lower than other characteristic times involving vehicle components dynamics: concerning powertrain modeling, to avoid model stiffness, a behavioral model is preferred for DC/DC converters, representing the converter efficiency as a constant or mapped value, or with a circuital representation. In the case of efficiency approach, the relationships between source power and load power in both directions are shown:

$$\left\{\begin{array}{c}{P}_l=P_s{ \eta }_{DCDC}^{s\to l}, \mathrm{source} \mathrm{to} \mathrm{load}\\ P_l={\displaystyle \frac{P_s}{{\eta }_{DCDC}^{l\to s}}}, \mathrm{load} \mathrm{to} \mathrm{source}\end{array}\right.$$

(8)

where $P_s$ is the source power, $P_l$ is the load power, and ${\eta }_{DCDC}^{s\to l}$ and ${\eta }_{DCDC}^{l\to s}$ are the source-to-load and load-to-source power conversion efficiencies, respectively. Alternatively, the circuital scheme in Figure 9 can be considered.

In this case, instead of modeling converter efficiency, a fixed power absorption $P_{fix}$ and the internal resistance $R_{int}$ can be identified [31]. Kirchoff’s laws can be written for the two circuits, coupling them with power transmission between the two ideal generators to obtain the following power balance:

$$P_s=P_{fix}+R_{int}{\displaystyle \frac{P_l^2}{V_l^2}}+P_l$$

(9)

This approach allows us to model power losses in both directions without switching through different formulations, since the power losses are mirrored in both energy flow directions.

Internal Combustion Engine

The ICE is the only component that consumes fuel to operate; hence, it needs to be properly modeled. Considering its efficiency, the simple power balance of Equation (10) can be written:

$$P_{mech}=P_{chem}{\eta }_g={\dot{m}}_{f}LH{V}_f{\eta }_g\left(T,\omega \right)$$

(10)

where ${\dot{m}}_f$ is the instantaneous fuel consumption, $LH{V}_f$ is the fuel lower heating value, and ${\eta }_g$ is the engine Brake Thermal Efficiency (BTE). The BTE is typically mapped as a function of brake torque $T$ and engine angular speed $\omega$; alternatively, the Brake Specific Fuel Consumption (BSFC), defined with Equation (11), is available from the technical data.

$$BSFC={\displaystyle \frac{{\dot{m}}_f}{P_{mech}}}={\displaystyle \frac{1}{{\eta }_{g}LH{V}_f}}$$

(11)

If technical data are not available, the Willans line approach can be preferrable to model ICE’s performance [23]: the mechanical power is assumed to be equal to the algebraic sum between a fraction of chemical power with an energy conversion coefficient and friction power losses:

$$P_{mech}=e_f\left(T,\omega \right)P_{chem}-P_{loss}\left(\omega \right)$$

(12)

where $e_f$ is the energy conversion efficiency as a function of brake torque and engine speed, and $P_{loss}$ is the friction power losses depending on engine speed. This equation can also be written in normalized terms with mean effective pressures:

$$p_{me}=e_f\left(p_{ma},c_m\right)p_{ma}-p_{mf}\left(c_m\right),$$

(13)

where $p_{me}$ is the mean effective pressure, $p_{ma}$ is the mean available pressure, $p_{mf}$ is the mean friction pressure, and $c_m$ is the mean piston speed.

The energy conversion efficiency is assumed equal to the engine indicated efficiency, since it does not account for mechanical losses; typically, it is modeled with second-order polynomial interpolation of the available mean pressure and mean piston speed. The advantage of identifying normalized engine variables is the possibility to scale available data of similar engines: if the technical data of the actual engine are not available, scaling up or down with the Willans line approach allows us to solve this problem of data lacking. The Whole Open Throttle (WOT) torque curve must also be considered to ensure engine feasibility: this is typically interpolated as an engine speed second order polynomial envelope.

Battery

In the literature, a circuital model scheme is widely adopted to effectively represent battery performance [32]. Considering quasi-static modeling, the battery can be modeled at the single-cell level as an ideal voltage source coupled with an internal resistance, both of which are functions of the battery State-of-Charge (SoC), as depicted in Figure 10.

From Kirchoff’s law, the voltage equation of the battery cell is written as:

$$U_b=U_{OC}\left(\xi \right)-R_{int}\left(\xi \right)I_b$$

(14)

where $U_{OC}$ is the open-circuit voltage, $R_{int}$ is the internal resistance, $I_b$ is the cell current, $U_b$ is the cell terminal voltage, and $\xi$ is the cell SoC. The functional dependence of the open-circuit voltage and internal resistance can be available from manufacturer data or with interpolations; the most common interpolation is the linear interpolation through the Willans line approach [23]:

$$\left\{\begin{array}{c}{U}_{OC}\left(\xi \right)=a_1\xi +a_2\\ R_{int}\left(\xi \right)=b_1\xi +b_2\end{array}\right.,$$

(15)

where $a_1$, $a_2$, $b_1$, and $b_2$ are coefficients to be determined experimentally or from the technical datasheet; the model parameters can also be identified with respect to cell temperature, if the thermal management is included. The single cell SoC can be computed by integrating the battery current in the following equation:

$$\dot{\xi }=-{\displaystyle \frac{I_b}{C_b}},$$

(16)

where $C_b$ is the battery single cell capacity. Equation (21) can also be manipulated to be written as power balance multiplying both terms by battery current in order to directly calculate the current from the battery discharge/charge power:

$$I_b={\displaystyle \frac{U_{OC}\left(\xi \right)-\sqrt{U_{OC}^2\left(\xi \right)-4R_{int}\left(\xi \right)P_b}}{2R_{int}\left(\xi \right)}},$$

(17)

where $P_b$ is the battery single cell power. The battery performance should also be limited by maximum discharge and charge currents, together with maximum discharge and charge voltages: these constraints translate into limits on battery delivered or absorbed power. More sophisticated models include dynamic components such capacitors to represent the dynamic effects of batteries [33]. The presented model has no information about the battery state of health, although it is known that the real battery capacity tends to reduce over time due to aging and cycling fatigue: to improve SoC estimation, several sub-models to correct the available battery capacity $C_b,$ based on historical data, Kalman filters, or AI-based models, can be found in the literature [34].

2.2. Control Strategies Overview

The additional degree of freedom given by the coexistence of more power components requires a proper control strategy implemented within the ECU. Since the battery acts as an energy buffer for the powertrain, the control strategy typically assumes battery power as control variable to manage fuel consumption while satisfying vehicle power demand [35]. The ECU should accomplish different objectives while optimizing components at different complexity levels: this is generally referred to as multilevel control architecture, in which two or more levels are defined as objectives and actuation strategies [36].

In Figure 11, the schematization of a general multilevel control architecture is depicted. The architecture is divided into three main levels described hereafter:

• High control level: starting from the power demand requested at the wheels ($P_{load}^{req}$), the power split controller establishes the amount of power that must be delivered by the battery ($P_{battery}^{req}$) and the ICE ($P_{ICE}^{req}$); the power split logic is the main core of the overall control strategy, and it has direct effects on fuel consumption.
• Medium control level: this level defines the combination of component parameters to ensure the single power delivery as established by the high-level controller; for the ICE, the control typically involves the combination of the brake torque ($T_{ICE}^{req}$) and speed (${\omega }_{ICE}^{req}$) to minimize fuel consumption (${\dot{m}}_f$), while for the battery, the main control variable is the current ($I_b^{req}$), since the voltage ($U_b^{req}$) is generally constrained by the open-circuit voltage and battery SoC variation ($\dot{\xi }$).
• Low control level: it involves the actuators for the ICE and the battery, actuating the required component parameters.

The high-level control logic can be based on heuristic approaches, or it can be derived from an optimal algorithm, for which a description is provided below.

2.2.1. Heuristic Approach

With the heuristic approach, the developed control strategy neither follows an optimality criterion nor uses past or future information; instead, it is based on reasonable assumptions coming from real-life practice or simple considerations about powertrain behavior and limits. Mainly, a deterministic rule-based strategy is applied: simple activation rules and state machines are defined to satisfy a given condition based on the vehicle state, in particular the battery SoC [37]. Normally, the rule-based control strategies ensure that the SoC does not fall outside a given range with a thermostat control activation on the ICE to compensate for every discharge (i.e., charge sustaining); alternatively, it is possible to define a given battery discharge rate, using the battery as an electrical assist to the powertrain (i.e., charge depleting). Rule-based strategies are suitable for micro and mild hybrid, mainly series-layout configurations, but they are neither capable of minimizing fuel consumption nor of correctly managing the hybrid powertrain, especially in the case of plug-in configurations [38]. Another possibility is represented by fuzzy logic control strategies, in which the rule-based concept is extended to overcome said limitations. Fuzzy controllers still need predefined rules, but the transition between rules is not step-like in the case of simple rule-based controllers: this time, the control output is a mix of different rules, known as fuzzification, allowing a more coherent control of powertrain dynamics [39]. To manage the fuzzification process, different membership functions in the literature can be found, such as trapezoidal rules, Singleton fuzzifiers, Gaussian-type fuzzifiers, etc.

2.2.2. Optimal Control

In the case of optimal control strategies, a mathematical constrained minimization problem is defined for each specific application. In general, a cost function $J$ that involves fuel consumption must be minimized for a given driving cycle:

$$J={\int }_0^{t_{end}}{\dot{m}}_f\left(t,x\left(t\right),u\left(t\right)\right)\mathrm{d}t$$

(18)

where $x\left(t\right)$ is a given state variable, and $u\left(t\right)$ is a given control variable. The cost function can also be improved by implementing penalty factors for pollutant emissions, moving from pure fuel consumption to minimization to a multi-objective problem [40]. For HEVs, battery SoC is usually assumed as state variable, with a given initial condition $\xi \left(0\right)$, while battery power is used as control variable; furthermore, a penalty function depending on the final SoC value is usually considered to ensure a certain discharge rate over the entire driving cycle, leading to the following formulation:

$$J={\int }_0^{t_{end}}{\dot{m}}_f\left(t,\xi \left(t\right),P_b\left(t\right)\right)\mathrm{d}t+\varphi \left({\xi }_{end}^{req}\right)$$

(19)

The minimization problem should also account for time-dependent powertrain constraints, especially on the battery SoC, battery power, ICE (or FC) power, and power rate. These constraints can be formulated as follows:

$$\left\{\begin{array}{c}{\xi }_{min}\le \xi \left(t\right)\le {\xi }_{max}\\ P_{b_{chg}}\le P_b\left(t\right)\le P_{b_{dis}}\\ P_{IC{E}_{min}}\le P_{ICE}\left(t\right)\le P_{IC{E}_{max }}\\ {\dot{P}}_{IC{E}_{min}}\le {\dot{P}}_{ICE}\left(t\right)\le {\dot{P}}_{IC{E}_{max }} \end{array}\right.,$$

(20)

Dynamic Programming

Dynamic programming (DP) is one of the main techniques to approach the fuel consumption minimization problem for hybrid vehicles [41]: its implementation guarantees a globally optimal solution over the drive cycle within the constraints; on the other hand, it requires a heavy computational burden both for memory usage and CPU operations, with computational time exponentially growing with the problem complexity. In addition, DP allows the calculation of an optimal solution for fuel consumption minimization, but it is not implementable on-board, leaving only an offline minimization [42].

To implement DP, a discretization of time, state, and control variables must be introduced, preferably with constant time step: thanks to the discretization, the cost function of Equation (26) can be calculated as a simple summation over the cycle:

$$J={\sum }_{k=0}^{N-1}{\dot{m}}_f\left(t^k,P_b^k\right)+\varphi \left({\xi }^N\right),$$

(21)

with $k$ as time step index, and $N$ is the total number of time-steps. The penalty function on the final SoC value is a simple step function that is equal to infinity in the case of the constraint violation; otherwise, it is equal to zero. To compute the optimal solution, the algorithm proceeds backward over time: for a given $k$ time instant, the span of possible SoC values for the next time step is calculated with an explicit numerical difference scheme as in Equation (22):

$$f_i^{k+1}={\xi }_i+\dot{\xi }\left({\xi }_i,P_b^k\right)\Delta t,$$

(22)

where ${\xi }_i$ is the given value of the SoC grid. The optimal battery power is the value corresponding to the minimum cost function according to Equation (23):

$$\begin{array}{c}{P_b^k}_{opt}\left({\xi }_i\right)=\underset{P_b^k}{\mathrm{argmin}}\left(J^{k+1}\left(f_i^{k+1}\right)+{\dot{m}}_f\left(t^k,P_b^k\right)\right)\end{array},$$

(23)

Once the optimal battery power is known over the drive cycle, the optimal battery SoC trajectory can be calculated explicitly in a similar way as in Equation (22):

$${\xi }_{opt}^{k+1}={\xi }_{opt}^k+\dot{\xi }\left({\xi }_{opt}^k,{P_b^k}_{opt}\left({\xi }_{opt}^k\right)\right)\Delta t,$$

(24)

It is worth noting that the optimal trajectory values may not correspond to the grid values ${\xi }_i$ imposed before, so an interpolation for ${P_b^k}_{opt}$ would be required.

A forward approach for DP was proposed in a previous work of the authors [43]: since the DP calculates the control strategy profile backward, it is not guaranteed that the final solution computed forward would still be feasible. To overcome this issue, a time-independent formulation of minimization, which can run both forward (hence the name Forward Approach to Dynamic Programming, or FADP) and backward, is introduced, with a bijective correspondence between state variable (i.e., battery SoC) and control variable (i.e., battery power). In addition, another reason for using FADP is to compute the states of a dynamical system. Indeed, the computation of the system dynamics is only possible by performing a forward time marching to integrate the differential equations (e.g., the thermal dynamics of a battery). Since this algorithm is still computationally difficult to implement, an iterative variation was also proposed (namely Iterative Approach to Dynamic Programming, or IFADP) to narrow the SoC boundary grid while keeping the number of points intact, hence increasing the grid fitting with no expense on computational burden [43].

Pontryagin’s Minimum Principle

When the computational requirements of DP cannot be satisfied, Pontryagin’s Minimum Principle (PMP) comes in hand: this algorithm allows us to find a sub-optimal solution, often close to the global optimum of DP, with drastically low computation requirements and time [44]. The PMP algorithm starts with the definition of the Hamiltonian function derivable from the cost function of Equation (19):

$$H\left(\xi \left(t\right),P_b\left(t\right),\lambda \left(t\right)\right)={\dot{m}}_f\left(t,P_b\left(t\right)\right)+\lambda \left(t\right)\dot{\xi }\left(\xi \left(t\right),P_b\left(t\right)\right),$$

(25)

where $\lambda \left(t\right)$ is the co-state variable introduced for the hard constraint on the final SoC value, whose dynamics are calculated as:

$$\dot{\lambda }\left(t\right)=-\lambda \left(t\right){\displaystyle \frac{\partial \dot{\xi }}{\partial \xi }},$$

(26)

The PMP states that the optimality of the solution is guaranteed only when the Hamiltonian is minimized at each time instant as:

$$H\left({\xi }_{opt}\left(t\right),{P_b}_{opt}\left(t\right),{\lambda }_{opt}\left(t\right)\right)\le H\left(\xi \left(t\right),P_b\left(t\right),\lambda \left(t\right)\right) \forall t\in \left[0,t_{end}\right],$$

(27)

The optimality condition is a necessary but not sufficient mathematical condition for the global problem optimality; hence, the PMP algorithm can only reach a sub-optimal solution, differently from DP. The initial co-state value must be calculated by imposing the respect of the final SoC value constraint: this leads to a non-linear problem that can be solved with classical methods known in the literature, such as the shooting or bi-section methods.

Equivalent Consumption Minimization Strategy

In many applications in which the PMP algorithm is implemented, the co-state shows slight variations over time, especially when it is required that the SoC initial value must be maintained at the end of the cycle; hence, it can be assumed constant to simplify the mathematical description. Hence, the Hamiltonian can be rewritten as:

$$H\left(\xi ,P_b,s_0\right)={\dot{m}}_f+s_0{\dot{m}}_{EQ},$$

(28)

where ${\dot{m}}_f$ is the engine fuel consumption, $s_0$ is the constant co-state treated as an equivalence factor, and ${\dot{m}}_{EQ}$ is the equivalent fuel consumption due to the state dynamics:

$${\dot{m}}_{EQ}=\dot{\xi }\left(P_b\right),$$

(29)

The influence of the equivalence factor $s_0$ on the Hamiltonian is simply demonstrated by Equation (28): a high value of $s_0$ increases the weight on battery charging, while a low value of $s_0$ would lead to battery depleting. In general, the ECMS can be seen as a direct derivation of the PMP algorithm [45].

3. Adaptive Equivalent Consumption Minimization Strategies

In general, when thinking about hybrid vehicles, it is common to consider only ICE fuel consumption as the minimization objective: this is true from a pure chemical energy point of view, but to obtain better results, the entire powertrain efficiency must be considered; hence, the electrical energy consumed from the battery is assumed as an additional fuel consumption [46]. The battery can be considered as short-term energy storage with different efficiencies for charging and discharging operations: in some operative conditions, it can be better to allow worse efficiency for the ICE to take advantage of battery performance in better conditions [47]. This is the main idea of ECMS: there is an optimal energy balance between battery and ICE; hence, the minimization problem must also include the electrical energy delivered from the battery. To introduce electrical energy consumption, a key role is covered by the equivalence factor that establishes the weight of battery electrical power in the minimization process [48]: a low equivalence factor would promote battery discharging since giving less importance to battery operations, while a high equivalence factor would ensure a high battery SoC, guaranteeing continuous recharging operations.

The main limitation of the ECMS approach is that a constant equivalence factor is not capable of managing powertrain balance optimality in dynamic vehicle operative conditions, such as, for example, (i) road profiles with frequent slope variations; (ii) driving conditions, e.g., urban or highway; (iii) vehicle weight variation, such as in the case of the public transportation sector or delivery vans; and (iv) variable traffic conditions. Changing the weight factor on battery usage would increase the powertrain efficiency in situations in which the electric motor would be preferrable with respect to the ICE or vice versa, still respecting component constraints.

Adaptive-ECMS (A-ECMS) is used to address the aforementioned issue of static ECMS: the equivalence factor is not constant, but it can vary over time based on different criteria. Considering the battery power as control variable [43], a typical A-ECMS can be represented with its Hamiltonian function as:

$$H\left(\xi ,P_b,t\right)={\dot{m}}_f\left(P_b\left(t\right)\right)+s\left(t\right){\dot{m}}_{EQ}\left(\xi \left(t\right),P_b\left(t\right)\right),$$

(30)

where the equivalence factor $s\left(t\right)$ is a function of time.

3.1. A-ECMS Classification

The classification of A-ECMS methodologies is based on different features, from algorithm adaptivity to needed information. In

Table 2. Functional classification of A-ECMS for plug-in HEVs.

Criterion	Focus
Adaptivity mechanism	SoC dependent
	Drive cycle
	Operative conditions
Time horizon	Short-term
	Long-term
Adaptation technique	Parametric optimization
	Real-time optimization
	Data-driven approach
Emerging technology integration	V2X communication system
	GPS integration
	AI-supported

, a functional classification of A-ECMS is reported, with main features highlighted.

The choice of one A-ECMS approach over another mainly depends on several factors such as data availability, computational resources, and scalability. The development of an A-ECMS strategy based on battery SoC is widely accepted, since it has direct consequences on fuel consumption and recharging strategies, as well as being easy to implement on-board; in addition, battery SoC is easy to estimate both with model- and data-based estimators. SoC-based only A-ECMS strategies, on the other hand, can achieve only a sub-optimal minimization of fuel consumption, with low consistency. Implementing information about the drive cycle, with prediction or recognition techniques, is an added value in the SoC trajectory planning, but it requires higher computational resources, together with external technology integration (e.g., GPS). Another possibility is to calculate the optimal long-term SoC trajectory offline with parametric or, if possible, real-time optimization, exploiting the planned SoC trajectory within the A-ECMS: this is one of the preferred solutions, but it requires an external computation process with IoT systems integration on-board. The main strength of every A-ECMS solution is the high implementation flexibility to achieve better powertrain optimizations; however, sophisticated A-ECMS integrations come along with high computational burden and costs.

3.1.1. Adaptivity Mechanism

The adaptivity of the equivalence factor is typically based on battery SoC. Low values of $\mathrm{s}\left(\mathrm{t}\right)$ promote battery discharging operations, while high values improve battery SoC rate on cost function, favoring battery charging by the ICE. A simple adaptation mechanism can be a rule-based or fuzzy controller, in which an inverse proportionality between equivalence factor and battery SoC is formulated: for low battery SoC, the equivalence factor must increase to sustain battery charging at the expense of fuel consumption by ICE; for high battery SoC, the equivalence factor must be low to promote electric operations, saving fuel and taking advantage of the higher electric motor efficiency with respect to ICE. This strategy is easy to implement and requires low computational burden, but it is ineffective on global optimization and with complex driving cycles.

Information on driving cycle in adaptivity mechanisms can improve overall powertrain efficiency. Starting from a given vehicle speed profile, an optimal instantaneous equivalence factor can be determined to maximize powertrain efficiency over the entire drive cycle using Model Predictive Control (i.e., MPC) or Machine Learning (i.e., ML) approaches; the vehicle speed profile is typically predicted from previous information based on driving style or with route-planning techniques using GPS data, if available. This approach reduces fuel consumption through more precise control on the powertrain, but it can require high computational burden, often with offline optimization.

Vehicle sensor information can also be incorporated into the equivalence factor adaptivity: recognizing driving patterns, traffic situations, and driving style guarantees high flexibility on the equivalence factor variation, promoting battery operations, e.g., in case of high traffic situations at low speed or ICE operations during high torque demands. Furthermore, Vehicle-To-Everything (i.e., V2X) communication infrastructure allows vehicle interconnectivity advantages, from electrical grid information to charging stations availability for battery charging operations. High hardware-software integration is required to handle increasing sensor information, with high data transfer speed and communication interfaces.

3.1.2. Time Horizon

The time span considered for energy consumption minimization is a crucial strategy design parameter since it drastically affects the A-ECMS performance. In general, short time horizons are considered to minimize fuel consumption instantaneously or over brief time spans: this approach is coherent for strategies with low complexity and suitable for real-time control. However, this approach does not allow us to optimize the powertrain operations for long distances, and it often provides a sub-optimal solution.

Long-term time horizons are the best solution for fuel consumption minimization: choosing an entire drive cycle time span, the overall fuel consumption can be minimized at its best, with optimal or quasi-optimal solutions; to do so, vehicle speed planning is required, with GPS information (if available) or with speed profile prediction. The final solution is better than that of the short time horizon in terms of fuel consumption, at the expense of higher computational burden.

3.1.3. Adaptation Technique

One of the crucial factors on A-ECMS performance is the functional dependence of the equivalence factor. A simple adaptivity mechanism can depend on battery SoC variation: for a given SoC target profile in time, the equivalence factor can be expressed as summation of a constant reference term corrected with a SoC-dependent term as:

$$s\left(t\right)=s_0+\Delta s\left(\xi \left(t\right),t\right),$$

(31)

The static term $s_0$ can be estimated with offline optimization, for example, by applying PMP algorithm; in a previous work [43], the authors proposed a time-variant $s_0$ as function of the Urban Degree that expresses the variation rate of a given portion of drive cycle. The correction term $\Delta s\left(\xi \right)$ can be expressed in different ways: in the literature [49], it is common to determine the correction term with a PI controller logic:

$$\Delta s\left(\xi \left(t\right),t\right)=k_p\left(\xi \left(t\right)-{\xi }_{ref}\left(t\right)\right)+k_i{\int }_0^t\left(\xi \left(\tau \right)-{\xi }_{ref}\left(\tau \right)\right)d\tau ,$$

(32)

where $k_p$ and $k_i$ are the proportional and integral gains of the PI controller, respectively, while ${\xi }_{ref}\left(t\right)$ is the requested battery SoC over time. The reference SoC value can also be defined with a given charging/discharging rate:

$${\xi }_{ref}\left(t\right)={\xi }_0+{\displaystyle \frac{{\xi }_{end}-{\xi }_0}{L_{ref}}}{\int }_0^{t}v\left(\tau \right)d\tau ,$$

(33)

where ${\displaystyle \frac{{\xi }_{end}-{\xi }_0}{L_{ref}}}$ is the charge/discharge rate for a given distance $L_{ref}$, while $v\left(t\right)$ is the vehicle speed. According to closed-loop control theory, PI gains tuning determines the adaptivity sensibility: high values of proportional gain increase adaptivity sensibility to the SoC target, while high values of the integral gain ensure SoC target reaching. Simpler formulations account only for P controller, not including the integral part. Alternatively, offline-developed maps of the equivalence factor as a function of battery SoC and other parameters with pre-defined rules can be simply implemented on-board.

The target SoC profile can be either defined with simple rules or with more sophisticated approaches that leverage information from driving cycle prediction. The actual driving conditions can be obtained with vehicle sensors, such as road slope, vehicle speed, vehicle weight, etc. V2X infrastructure is the next step for better integration with IoT technologies, combining road information (e.g., charging infrastructure distance, electrical grid information, traffic jams, etc.) and also suggesting target speed to minimize fuel consumption.

3.1.4. Emerging Technologies Integration

To improve the benefits of A-ECMS implementation on-board, several emerging technologies are being implemented. One of the main solutions is the adoption of V2X communication interface: this technology allows us to gain useful information about vehicles, infrastructures, and pedestrians, leading to more powertrain efficiency and sustainability [50]. The V2X technology can be specialized in (i) Vehicle-To-Vehicle (V2V); (ii) Vehicle-To-Infrastructure (V2I); (iii) Vehicle-To-Pedestrian (V2P); (iv) Vehicle-To-Network (V2N); and (v) Vehicle-To-Grid (V2G). For plug-in HEVs, the V2G interface leaves room for optimization strategies based on electrical grid energy availability and costs, with even more sophisticated rules to discharge/charge the battery when it pays off. Applying V2X communication technologies in A-ECMS, it is possible to have more flexible control on powertrain, improving efficiency and reducing fuel consumption; on the other hand, high computational burden is required, together with proper IoT infrastructure on vehicles and with more expensive instrumentation. V2G integration within A-ECMS design has several implications on the control strategy itself, both contemplating efficiency losses and preserving grid sustainability: the battery charging phases at recharging infrastructures must be accomplished without affecting grid electrical balance, meaning that A-ECMS should plan recharging phases, maximizing vehicles, grid, and system synergy.

Implementing GPS allows more predictive energy management by providing road conditions, slopes, speed limits, and traffic jams. For example, in case of incoming downhill, the A-ECMS can improve battery recharging with regenerative braking and no fuel consumption by the ICE; GPS information can be coupled with MPC algorithms to suggest an optimal speed profile based on traffic conditions and speed limits; furthermore, with GPS, it is possible to plan the next stop at charging stations. The GPS effectiveness on powertrain efficiency is strongly linked to maps accuracy and signal reception, together with continuous communication with cloud interfaces: this can lead to wrong or poor solutions on charging strategies.

Artificial Intelligence (i.e., AI) has played a major role with great developments in recent years: it is possible to implement an AI-based approach to further optimize energy management of plug-in HEVs with high flexibility and precision in dynamic situations. With AI, it is possible to reduce the modeling effort and, hence, the mathematical complexity to represent powertrain performance with data-driven techniques. A simple implementation of AI methodology is the Supervised Machine Learning with Deep Neural Networks (NNs), in which the model is fed with historic datasets of vehicles of interest in different conditions of road, traffic, operations, and battery SoC. A limitation of Supervised ML is the non-capability to adapt in new situations not included in training datasets: the Reinforcement Learning (RL) comes in hands, with its high adaptivity to new dynamic situations. A hybrid MPC-AI approach can also be followed, partially reducing the modeling effort with Physical-Informed NN (i.e., PI-NN). The main drawback of AI implementation is the strong need for data, requiring offline training sessions, which can be time-consuming and with high computational burden; furthermore, the obtained models are black boxes with low information content.

3.2. A-ECMS Implementation

In Figure 12, a flow chart example for an A-ECMS based on drive cycle prediction is depicted.

Starting from actual driving conditions or, if available, GPS route planning data, the predicted vehicle speed can be estimated: with speed profile as input, the intermediate optimization layer estimates the target battery SoC profile, which minimizes the equivalent energy consumption, typically with PMP algorithms (other optimal algorithms, such as DP and Gradient-Based, can still be used, but they would be time- and computationally consuming). Vehicle simulation can be performed with mathematical models (i.e., MPC) or with data-driven models (i.e., ML). From the SoC target profile, the adaptive equivalence factor can be updated in time for its on-board implementation within the A-ECMS.

A first application of an A-ECMS based on road load feedback is reported in the work of Musardo et al. [51]: the authors, starting from a standard ECMS, introduce an equivalence factor adaptor based on predicted torque from GPS road load information. The application of the proposed A-ECMS results in improvements in fuel economy ranging from 6% to 18% for standard regulatory drive cycles. Other approaches based solely on SoC can be found in [52,53]. The equivalence factor can also be calculated developing map datasets based on battery SoC and vehicle distance, as performed in [54,55]. A combination of optimal algorithm and A-ECMS is proposed by Shen et al. [56], who apply an equivalence factor tuning process with the implementation of Adaptive Dynamic Programming (ADP) for every instant, coupling it with a simple PI controller for battery SoC trajectory following.

Regarding the V2X communication infrastructure, in the work by Ha and Lee [57], a V2X-based A-ECMS for plug-in HEVs simulated in a hardware-in-the-loop environment is proposed: together with speed assistance guidance for eco-driving, the authors develop a supervisor control strategy implementing route prediction, hence, defining the target battery SoC over the route. Thanks to the V2X communication interface, the authors are able to reduce fuel consumption by 60%, with 36% reduction of CO₂ emissions compared to standard PHEV, with no V2X technology on-board. The paper leaves room for further improvements, from infrastructure enhancement to autonomous driving, aiming at better results. A simpler interface can be obtained by integrating V2N technology on-board only to reconstruct the road ahead, as proposed by [58]. Travel prediction is possible also with other stochastic techniques based on previous GPS data, as performed by [59], or implementing NNs for intelligent driving recognition, as performed by [60,61].

Works on A-ECMS based on speed forecasting can be found in [62,63]: the authors propose a long short-term memory NN trained on real-world driving data for a heavy-duty fuel cell parallel hybrid vehicle; the NN is implemented to predict the following speed profile, in order to optimally plan the battery SoC profile, hence reducing the fuel consumption. To predict the following road load, it is also possible to consider it as a Markovian process as performed by Lin et al. [64]: the authors develop a correction coefficient for the equivalence factor based on battery SoC, vehicle distance, and road typology with an offline minimization process; the correction coefficient is then implemented on-board to adapt the A-ECMS calculated on predicted speed profile generated with a Montecarlo approach. Route preview-based A-ECMS is one of the preferred solutions to develop PHEV control strategies: if the following road profile is predicted, it is possible to optimally plan the battery SoC trajectory with optimization algorithm, such as PMP, as performed by [65]. Different implementations can also be found in [66,67,68].

NN-based A-ECMS are widely diffused in the literature [69]: the increasing computational power of modern computers allows training NNs with big datasets covering different operative conditions, drivers, road conditions, etc. In the work of Wu et al. [70], a mode-recognition NN trained on DP and PMP optimization results is proposed: the recurrent NN is used to estimate online the co-state for speed prediction-based model predictive control framework. A different implementation of ML is performed by Xia et al. [71], who propose a short-term speed forecast implementing a long short-term memory NN trained on multiple driving cycles; the speed forecast is used to optimally plan the battery SoC profile with a given discharge rate, adapting the equivalence factor with a PI controller. The proposed A-ECMS is compared with results coming from DP optimization, showing a slight increase in fuel consumption of roughly 5%, while the computational burden is drastically reduced as expected. It is possible to implement recurrent NNs to account for battery aging in equivalence factor adaptivity as performed by Han et al. [72]. NNs can also be implemented for more detailed fuel consumption modeling, such as in [73], in which the authors trained an artificial NN to represent transient behavior of ICE.

In a previous work of the authors [74], an NN-based A-ECMS for a hybrid Range Extender (REx) hydrogen ICE/battery vehicle is proposed, whose flow chart is sketched in Figure 13. The power commanded to the REx is computed as a function of the battery SoC and the power required for the wheels; to account for a large variety of road load conditions, the NN is trained on the WMTC-L6e standard drive cycle, and it is tested on a random driving cycle generated from a stochastic method based on Markovian chains [37]. To perform an optimal energy management, the PMP solution in terms of power commanded to the REx is used as target dataset to train the NN.

To make the energy management strategy adaptive, a PI control rule is used to manage the operation of the range extender [74]. The proposed A-ECMS adopts a “sliding window” mechanism that can be observed in Figure 14: the requested power provided by the REx is dynamically computed based on the residual of target SoC with respect to the actual SoC as a sliding window moving with requested road load.

In terms of fuel consumption, the authors reported that the NN A-ECMS can be compared to the traditional EMS obtained with PMP; furthermore, a significant reduction in computational burden can be achieved with the NN A-ECMS compared to the traditional EMS, making it suitable for real-time applications.

Real-time traffic conditions can be useful to plan proper speed profile, hence, optimizing the battery SoC trajectory. In the work of Yu et al. [75], the authors propose a short-term speed prediction and driving cycle recognition to adapt the equivalence factor based on driver patterns and GPS information, continuously adapting the battery SoC reference trajectory using a non-linear Auto-Regressive NN. The proposed algorithm is able to reduce fuel consumption by 8%. Other authors implement traffic information to correct the battery SoC trajectory, such as [76,77]. It is also possible to reproduce driver behavior with a classification method implementing offline questionnaire and online NNs, as proposed by [78]: knowing the driver behavior and traffic conditions, the authors can reduce fuel consumption of roughly 7% under aggressive driver behavior with respect to driver non-informed A-ECMS.

A different approach for buses is proposed by Song et al. [79], who identify passengers’ load using a recursive least square algorithm, while a genetic algorithm optimization is used to determine the optimal equivalence factor accounting for passengers’ load and bus stations’ distribution in terms of relative distance and load. The algorithm can reduce fuel consumption by roughly 2.5% with respect to traditional ECMS. Another work focused on bus optimization with A-ECMS is reported in [80].

The car-following process can be another aspect of interest when designing A-ECMS. In the work of Xue et al. [81], the authors propose an A-ECMS in which the cost function also accounts for the car-following distance, ensuring safety other than fuel consumption minimization, optimizing the equivalence factor with particle swarm optimization; the algorithm proposed maintains a good fuel economy together with safe driving and contingency adaptivity. A similar work is proposed by Liu et al. [82]: a gray NN predicts the following vehicle speed, adjusting the target vehicle speed with a fuzzy adaptive control algorithm; with a subsequent fuzzy logic, the equivalence factor is adapted to respect the planned battery SoC trajectory, reaching finally roughly 95% optimality of the solution.

Alternative approaches to A-ECMS implementation and usage can be found in the literature. In the work by Yang et al. [83], an equivalent factor adaptive regression is introduced integrating the bisection method and polynomial penalties derived from a battery SoC target following ECMS. The authors propose a Markovian process speed forecast, and the predicted speed is used to optimally plan the battery SoC trajectory; the equivalence factor is adjusted with a simple proportional rule with respect to SoC residual, with an adaptive gain penalized if the actual battery SoC is far from the reference. The results show a slight worsening of fuel consumption of roughly 1% with respect to PMP optimization, but with enhanced SoC robustness against disturbances differently from standard A-ECMS. A different implementation is proposed by Li et al. [84], in which a Dragonfly optimization algorithm is introduced to find the best equivalence factor for each time instant for a fuel cell PHEV. The baseline equivalence factor adaptivity rule can also be influenced by power mode transition frequency as performed by Wang et al. [85]: the authors propose an optimized EMS with a balance between fuel consumption and power mode transition frequency, leaving room for a slight increase in fuel consumption to avoid unnecessary mode transitions; however, the authors do not propose a procedure implementation for A-ECMS.

4. Conclusions

Plug-in hybrid vehicles are one of the best technological solutions to reduce pollutant emissions of the transport sector according to new strict regulations. To ensure that fuel consumption is effectively optimized, there is a need for a proper on-board control strategy, hence, with a significant interest in researching, designing, and developing new control strategies using data coming from sensors and mathematical models. A-ECMS comes in handy: its mathematical principle is versatile, suitable for real-time on-board control, and integrable with other technologies.

This paper presented an overview of A-ECMS methodologies for plug-in HEVs. First, a wide definition of HEVs was given, with classification and model equations to fully represent vehicle performance, hence, determining the fuel consumption. Then, an introduction and a classification of multi-level control architecture is proposed, with the definition of optimal control algorithms from which the A-ECMS is derived. Finally, an insight into A-ECMS is proposed, with classification criteria and implementation examples.

A-ECMS applications are many and varied: in the literature, it is possible to find methodologies ranging from simple approaches, e.g., following a given battery SoC trajectory, to complex integrations interacting with V2X technology or supported by NNs. The cited research works investigate various aspects of vehicle optimization, often including additional cost functions for safety or avoiding unnecessary power transients; NNs are widely included in A-ECMS implementation, from the optimization of the equivalence factor to the representation of driver behavior or powertrain performances; several works integrate an offline optimization process to find the best equivalence factor. The choice of what to include within the cost function to be optimized is strictly related to data availability, data typology, and computational power available on-board.

Several improvements can be achieved in the future by looking at emerging technologies. IoT devices are becoming faster and cheaper, promoting online integration of A-ECMS with route planning, including recharging infrastructures: for example, it would be possible to suggest the driver where to stop to recharge the battery based on distance, electricity cost, and renewable electrical energy fraction, favoring also environmental-friendly solutions. In addition, machine learning techniques are playing a major role over time thanks to higher data availability, leading to semi- or full-autonomous driving solutions that can benefit from properly designed A-ECMS integration.