Enhancing Fuel Economy and Emission Performance of Dual-Motor PHEVs via a Robust Adaptive ECMS Control Framework

Abstract

The energy efficiency of plug-in hybrid electric vehicles (PHEVs) is critically influenced by their energy management strategies, particularly in dual-motor architectures. This study proposes a novel dual static Adaptive Equivalent Consumption Minimization Strategy (A-ECMS) controller to enhance fuel economy, emission performance, and the robustness of this controller designed to optimize fuel economy and emission performance while maintaining robustness against external disturbances in dual-motor PHEVs. The proposed method dynamically adjusts equivalent consumption factors based on driving conditions, addressing limitations of traditional fixed-factor ECMS approaches. Comparative experiments were conducted under standard and disturbed driving cycles, using rule-based (RB) control and conventional ECMS as benchmarks. The results demonstrate that A-ECMS improves fuel economy by 7.8% and reduces CO₂ emissions by 15.14 g/km under nominal conditions. Even with 5% random noise introduced to test robustness, A-ECMS maintains a 6.9% fuel economy improvement and achieves a 14.39% CO₂ reduction relative to ECMS. Furthermore, in two representative scenarios, the A-ECMS strategy yields cost savings of 30.17% and 28.56%, respectively. These findings confirm the robustness, adaptability, and economic advantage of A-ECMS, offering valuable insights for the real-time control of hybrid powertrains in future low-carbon mobility solutions.

1. Introduction

1.1. Literature Review

As is well-known, the consumption of fossil fuels has become a primary driver of both pollutant emissions and global energy scarcity. , This dual crisis of energy scarcity and ecological imbalance has catalyzed a global transition toward cleaner and more sustainable energy solutions. To achieve this transition, recent research has branched into two main directions: optimizing fuel properties and advancing powertrain electrification. In terms of fuel optimization, significant progress has been made in utilizing low-carbon alternatives (e.g., methanol and alcohol fuels) to mitigate emissions at the source. For instance, Jin et al. provided a comprehensive bibliometric analysis of low-carbon alcohol fuels for decarbonizing road transportation. Furthermore, recent studies have delved into the combustion characteristics of methanol/diesel dual-fuel engines and investigated the energy analysis of hybrid power systems under cold-start conditions. Within this context, the automotive industry, as a major consumer of fossil fuels and a substantial contributor to CO₂ emissions, faces mounting pressure to adopt greener and more energy-efficient technologies.

Over the past decade, vehicle electrification has been widely recognized as an effective solution to address these challenges, and numerous manufacturers have developed various types of electrified vehicles, including hybrid electric vehicles (HEVs), battery electric vehicles (BEVs), and plug-in hybrid electric vehicles (PHEVs). In particular, PHEVs, with their dual power systems, enable flexible selection of power sources according to actual driving demands, making them one of the most promising technologies for the future automotive market.

Depending on the design of the powertrain system, energy storage systems (ESSs) can be configured in various forms, such as battery-engine combinations, or the combination of fuel cells and batteries. Based on the type of power source, HEVs can be classified into petrol-electric, hydraulic, and electro-hydraulic hybrid vehicles. These systems offer high cycle efficiency, rapid energy release, and excellent power density, meeting the short-term high-power energy requirements of modern vehicles. Electro-hydraulic hybrid vehicles employ batteries and hydraulic accumulators as energy storage devices. The combination of batteries’ high energy density and hydraulic accumulators’ high power density enables superior adaptability to dynamic road conditions with frequent starts and shifts.

One of the primary challenges with hybrid electric vehicles is to fully utilize their advantages and energy sources efficiently, considering the complexity of this system. This entails allocating energy to different ESSs while satisfying physical constraints and predefined objectives. , Constraints refer to variables of onboard components, such as the state of charge (SOC) and health status of the battery, speed, and torque limits of the internal combustion engine (ICE) and generator unit, output current and power of the battery pack, and the dynamic characteristics of the powertrain system. , Furthermore, one of the most challenging issues in the field of hybrid electric vehicle energy management is that the vehicle control unit (VCU) typically lacks knowledge of upcoming driving conditions.

At the vehicle’s control level, certain control algorithms are adopted to determine the power required by the vehicle under different operating conditions and to coordinate the rational allocation of this power among various power sources. Although these strategies comprehensively consider both emissions and economic performance, they often involve substantial computational complexity, limiting their applicability in real-time vehicular systems and leading to higher implementation costs.

The core of the global optimal control strategy lies in optimization methods and optimal control theory. Hence, it is mainly used for evaluating other control strategies and providing a theoretical basis for the formulation of other control strategies. Neural networks can achieve close-to-optimal control effects with relatively low computational complexity but require a large amount of training data. The performance of HEVs is closely tied to their EMSs, with fuel economy and emissions reduction representing critical objectives that conventional vehicles cannot achieve.

Energy management strategies for HEVs are generally classified into rule-based, instantaneous optimization-based, and global optimization-based approaches. These strategies aim to minimize energy losses, reduce equivalent fuel consumption, and achieve optimal trade-offs between fuel economy and emissions. However, instantaneous optimization methods are prone to local optima and cannot guarantee global optimality. Conversely, global optimization techniques, although theoretically superior, are computationally intensive and unsuitable for real-time vehicle control.

In recent years, many studies have contributed to obtaining optimal control strategies for a series of hybrid electric vehicles. For instance, researchers developed an online corrective predictive EMS for the optimal control strategy of a series of hybrid electric tracked vehicles (HETV). Deep Reinforcement Learning (DRL) is currently one of the most popular research methods, combining Reinforcement Learning (RL) and Neural Networks (NN). In ref. , the authors utilized the Deep Q-Learning (DQL) algorithm to address the power management problem of series HETVs. Simulation results demonstrate that the proposed method can reduce fuel consumption by 5.96% compared to traditional RL algorithms. In ref. , the research group found that the Dyna-H algorithm can achieve convergence faster than Dyna, but incomplete training of state-action pairs in Dyna-H may lead to poor adaptability.

Recent advancements in machine learning methodsincluding supervised learning, unsupervised learning, reinforcement learning, deep learning, and DRLhave expanded EMS capabilities. For example, researchers validated the optimality and adaptability of RL-based EMSs against dynamic programming (DP) benchmarks. In ref. , Markov chains were utilized to derive transition probabilities for RL-based EMSs, while ref. employed fuzzy encoding predictors for power demand forecasting. To accommodate diverse driving scenarios, researchers proposed data-driven DRL and actor-critic architectures for PHEV EMS design. Likewise, they integrated a stochastic driver model with DRL for power allocation, demonstrating significant potential in enhancing EMS intelligence and adaptability.

In practical manufacturing and application, differences in the State of Charge (SOC) of batteries are inevitable and constitute one of the crucial factors affecting battery lifespan. In ref. , the authors applied the DRL method to find the optimal strategy for balancing the SOC of all batteries, thereby extending battery life and reducing maintenance. Ref. proposed a combined RL and NN approach and applied it to power allocation control in electric vehicles. The research results demonstrate that this method can improve the efficiency of hybrid electric vehicles. In ref. DQL was employed to address the power allocation problem for hybrid electric buses. Simulation results indicate that the adopted method exhibits good adaptability and optimality, with faster convergence speed compared to Q-learning.

In recent years, Hybrid Energy Storage Systems (HESS), which include supercapacitors and batteries, have garnered widespread attention due to their role in extending battery lifespan. Supercapacitors, characterized by long lifespan and high instantaneous power, serve as important complements to energy storage systems. The power density of supercapacitors is two to three times that of batteries. Supercapacitors operate within a temperature range of −40 to 70 °C, exhibiting better adaptability than batteries. In recent years, researchers have systematically integrated batteries and supercapacitors into Hybrid Energy Storage Systems (HESS) − to simultaneously meet the high energy and power requirements of energy storage systems. It is evident that as auxiliary energy sources, supercapacitors contribute to improving the efficiency and dynamic response of energy storage systems. In ref. , a novel method was proposed for real-time allocation of load power in HESS and predicting battery lifespan.

In ref. , the torque control strategy of a planetary gear dual-motor transmission system was studied, and a stiffness-dynamics model of the transmission system was established to avoid drastic changes in motor speed and reduce impact during shifting. Ref. employed a multispeed transmission to enhance the operational efficiency of electric vehicles and designed a shifting control algorithm to ensure seamless and rapid shifting. In ref. , a clutchless automatic transmission and control method for electric vehicles was proposed and experimentally validated on a test bench. Ref. focused on the study and development of dual-control strategies to enhance the performance of a 48 V P3 hybrid AMT vehicle. They formulated electric mode switching decisions and shifting process strategies to improve the driving experience. Ref. investigated a single-motor dual-clutch hybrid powertrain system. To enhance shifting performance, they introduced the nonlinear characteristics of the first clutch and stroke control system. Refs. , investigated the operation of dual clutches during the shifting process in dual-clutch transmissions. Ref. established a 3-degree-of-freedom transmission dynamics model and, taking upshift as an example, analyzed the transient dynamics of the upshift process.

Although predecessors have carried out extensive research and applications on the energy management and transmission control of hybrid powertrains, most existing studies predominantly focus on single-motor configurations or rely on rule-based strategies that lack adaptability. Specifically, there are two main deficiencies in previous works: First, limited attention has been given to the complex coupling control required for dual-motor architectures integrated with AMT, which offers higher flexibility but demands more sophisticated energy allocation. Second, traditional ECMS strategies typically employ fixed equivalence factors, rendering them insufficient to handle the stochastic nature of real-world driving conditions and external disturbances. Consequently, there is an urgent need for a robust control framework that can dynamically adjust to uncertain driving environments while optimizing multiple objectives. To address these gaps, this paper proposes a novel dual-static Adaptive ECMS (A-ECMS) framework. This approach not only fills the gap in multimotor energy management but also introduces a robustness enhancement mechanism to ensure optimal performance under perturbed driving cycles.

1.2. Main Contributions

• 1)Current research on plug-in hybrid electric vehicles (PHEVs) predominantly focuses on single-motor configurations, with limited attention given to the implications of multimotor architectures on energy management strategies. This study enhances the conventional P2 architecture by integrating an additional motor between the transmission and differential. This secondary motor serves dual functions: it can deliver power to the differential and, under specific operational modes (as detailed in Section of this paper), act as a generator for regenerative braking. This work addresses a significant research gap concerning energy management strategies for multimotor PHEV systems.
• 2)The primary challenge in the current Equivalent Consumption Minimization Strategy (ECMS) lies in balancing multiple optimization objectives while ensuring real-time computation and adaptive strategy adjustment. To overcome these limitations, we propose a dual static adaptive methodology integrated into the ECMS framework, resulting in the development of an Adaptive Equivalent Consumption Minimization Strategy (A-ECMS) controller. Experimental results demonstrate that the proposed A-ECMS significantly improves fuel efficiency compared to traditional ECMS approaches.
• 3)Furthermore, this study evaluates the environmental impacts of PHEV energy management strategies. The experimental findings indicate that the A-ECMS effectively reduces CO₂ emissions, contributing to improved environmental sustainability.

2. Exploring Dual Static Adaptive Optimization Strategy Using PMP Framework

2.1. Construct Experimental Cycle Conditions

Cycle conditions provide the essential baseline data required for conducting this research. All simulations in this study are performed using a MATLAB Simulink environment. The simulation model comprises subsystems for the powertrain, battery, and energy management strategy, and it is evaluated under two scenarios: a standard UDDS driving cycle and a UDDS cycle with 5% random noise introduced. To investigate system performance under different operating environments, two distinct experimental scenarios are constructed as follows:

2.1.1. Scenario 1: UDDS Driving Cycle

In this scenario, the standard UDDS (Urban Dynamometer Driving Schedule) cycle is utilized to evaluate and compare the performance of the three energy management strategies under consistent conditions. Figure illustrates the speed distribution of the UDDS driving cycle used in this study. Additionally, Figure presents the engine input-output power and the corresponding battery power observed under the UDDS driving cycle.

1.

UDDS driving cycle.

2.

Vehicle power demand under UDDS.

2.1.2. Scenario 2: UDDS Driving Cycle with 5% Random Noise

To simulate stochastic variations encountered during real-world driving, this scenario introduces 5% random noise into the original UDDS cycle. Subsequently, smoothing functions are applied to the noise-perturbed data to approximate realistic driver behavior. To maintain the authenticity of the modified driving conditions, constraints are imposed by setting maximum and minimum acceleration boundaries. This ensures that both the total driving distance and the average speed under noisy conditions remain consistent with the original UDDS cycle. The detailed modeling process for generating the noise-augmented cycle is depicted in Figure .

3.

Mathematical modeling process of random driving conditions.

At this stage, the driving conditions are represented as

$$\begin{array}{l}{v}_n=v_{\mathrm{spl}}+wgn(\mathrm{card}(v_{\mathrm{spl}}),1,p_n)\\ n=1,2,3,4\end{array}$$ 1

In the equation, v_n denotes the velocity of the driving conditions after the addition of random noise, v _spl represents the velocity under the original driving sample conditions, card(·) is a function of the number of elements in the computation vector, and p_n signifies the noise intensity, indicating the complexity of traffic conditions, measured in [dBW] units. A larger value ofp_n indicates a greater influence of random factors on vehicle velocity, reflecting more complex traffic scenarios.

Due to the significant oscillations in velocity introduced by the added random noise, which are inconsistent with real-world driving behavior, a locally weighted scatterplot smoothing (LOWESS) method is applied to smooth the velocity data. The smoothing process is defined as follows:

$$v_n^{\prime }=\frac{1}{m}\sum _{s=-m}^m\psi \left(\frac{s}{m}\right)·v_n(t+s)$$ 2

In this equation, m denotes the window width of the smoothing function, which determines the smoothness of the velocity curve and reflects the driver’s driving style. A larger value of m leads to a smoother driving process. ψ(x) represents the weighting function applied to the velocity data at time t, which can be expressed as

$$\psi (x)=\{\begin{array}{ll}\frac{m^2-1}{m^2}(1-x^2)\hfill & x^2<1\hfill \\ 0\hfill & x^2\ge 1\hfill \end{array}$$ 3

The maximum acceleration capability of the powertrain system under the current torque depends not only on vehicle parameters such as mass, slope resistance, and wheel rotational inertia but also on the current driving speed and road gradient. Accordingly, the maximum acceleration of the vehicle at a given moment is a function of velocity and road gradient variations, expressed as

$${a_{n,\max }}^{\prime }=\frac{\frac{T_{v\max }}{r}-\frac{1}{2}A{C}_d{\rho }_{\mathrm{air}}{({v_n}^{\prime })}^2-m_{v}gf\cos \theta -m_{v}g\sin \theta }{m_v\delta }$$ 4

In the equation, T _em,max represents the maximum torque that the powertrain system can provide at the current velocity and can be further expressed as

$$T_{v\max }=(T_{\mathrm{eng},\max }+T_{\mathrm{em},\max })i_T{i}_0$$ 5

In the equation, T _em,max and T _em,max represent the maximum torque outputs of the internal combustion engine (ICE) and the electric motor (EM), respectively, at the current vehicle speed.

To maintain the authenticity of driving conditions, this study sets the maximum acceleration boundary under random conditions 5m/s². Based on the modeling process illustrated in Figure , a 5% random noise was introduced to Scenario 1. Subsequently, a smoothing process and maximum acceleration boundary were applied to ensure the feasibility and consistency of the generated driving profiles. The resulting random conditions are illustrated in Figure .

4.

Utilizes a cycle’s operation condition.

Figure illustrates the vehicle power demand under Scenario 2. In Figure b, the comparison is presented between the motor power demand with 5% added random noise and the motor power demand in Scenario 1.

5.

Vehicle power demand with added random noise.

2.2. Modeling the Structure of Plug-In Hybrid Electric Vehicles

In modern society, plug-in hybrid electric vehicles (PHEVs) play a critical role in promoting sustainable transportation. Despite the powerful driving force for socioeconomic development provided by automobiles, the significant energy consumption and severe air pollution they bring about have undeniable negative impacts on the environment. PHEVs are recognized for their superior fuel economy and their potential to significantly reduce air pollutant emissions. The object of this study is a plug-in dual-motor hybrid electric vehicle based on electronically controlled mechanical automatic transmission (AMT). This novel hybrid vehicle consists of an engine, a battery pack, two drive motors, and an electronically controlled mechanical automatic transmission, as illustrated in Figure .

6.

Structure diagram of PHEV vehicle.

The selection of AMT for this hybrid architecture is motivated by its high transmission efficiency, cost-effectiveness, and seamless integration with hybrid powertrains. Derived from manual transmissions, AMT employs a simplified structure, which reducing manufacturing and maintenance costs while minimizing energy losses to enhance fuel economy. Its lightweight design contributes to vehicle weight reduction and enhanced driving range. Electric motor assistance optimizes gear transitions, reducing shift shock and improving ride comfort. AMT’s electronic control enables efficient engine-motor coordination, making it well-suited for hybrid architectures. Compared to automatic transmissions (AT) and dual-clutch transmissions (DCT), AMT offers higher reliability, lower failure rates, and reduced maintenance costs, providing an optimal balance between performance and affordability in hybrid vehicle applications.

In this powertrain system, two primary pathways are designed for power transmission. The first pathway involves the internal combustion engine (ICE) transmitting power through the clutch to Motor 1, then to the AMT, followed by Motor 2, and finally to the differential, which delivers torque to the wheels to propel the vehicle forward. Motor 1 and Motor 2 are identical in terms of design parameters. The second pathway originates from the battery, which supplies electrical energy to Motor 1. The transmitted power then follows the same route as the first pathway, passing sequentially through the AMT, Motor 2, and the differential before reaching the wheels. Additionally, the Energy Management System (EMS) is integrated with the Engine Control Unit (ECU), motor control unit (MCU), and battery control unit (BCU). The EMS continuously monitors system states and dynamically adjusts power flow and component coordination based on optimization strategies, ensuring that the vehicle operates in an energy-efficient and performance-optimized state.

2.3. Operating Mode Division and Torque Distribution

The operating modes of a hybrid electric vehicle (HEV) are critical for evaluating its performance. This study delineates the operating modes and torque distribution as follows:

2.3.1. Operating Mode

2.3.1.1. Set-Up Mode

Characterized by low vehicle speed and low engine efficiency, the vehicle operates in pure electric mode.

2.3.1.2. Gentle Acceleration

To improve overall energy efficiency, the engine operates in its optimal range, regulated by the ISG (Integrated Starter Generator) motor. If the engine’s optimal output torque T _best exceeds the required torque T _need and the battery’s state of charge (SOC) is below the target SOC_target, the vehicle enters charge-sustaining mode during driving. If the SOC is above the target SOC_target, it switches to pure electric mode.

2.3.1.3. Normal Acceleration

If the SOC is above SOC_target, the vehicle operates in pure electric mode. If the SOC is below SOC_target and T_need is less than the T _best, the vehicle enters charge-sustaining mode during driving. If T _need reaches the optimal range, the vehicle switches to engine-only mode. If the SOC is low and T _need exceeds T _best, the vehicle enters hybrid drive mode.

2.3.1.4. Emergency Acceleration

Under high torque demands, the vehicle operates in hybrid drive mode. If the SOC drops below the minimum threshold (SOC_d), the vehicle can discharge excessively to the lowest SOC (SOC _low). If further discharge is needed, the vehicle switches to engine-only mode to meet the driver’s torque demand by increasing engine load.

2.3.1.5. Low-Speed Cruising

If the SOC is above SOC_target, the vehicle is driven by the motor alone. If the SOC_need is below SOC_target, the vehicle enters charge-sustaining mode during driving.

2.3.1.6. Medium-Speed Cruising

The engine operates in its optimal range and drives the vehicle solely.

2.3.1.7. High-Speed Cruising

With a higher torque demand, if the SOC is above the lower limit SOC_d for the Charge Sustaining (CS) mode, the vehicle operates in hybrid drive mode. If the SOC is below SOC_d, the vehicle enters engine-only mode, increasing engine load to meet the driver’s torque demand.

2.3.1.8. Regenerative Braking and Coasting Regeneration

The motor operates in generator mode to recover energy. If the clutch is engaged, it remains engaged, and the engine undergoes fuel cutoff control, providing negative torque to avoid frequent engine starts and stops, which could increase fuel consumption.

2.3.1.9. Mechanical Braking

The motor operates in zero-torque mode, with the rotor acting as an inertial flywheel. If the clutch is engaged, it remains engaged, and the engine undergoes fuel cutoff control, providing negative torque. If the vehicle speed decreases to the point where the engine speed drops below the minimum idle value, the clutch disengages, and the engine shuts down.

2.3.2. Torque Distribution

2.3.2.1. Pure Electric Mode

When the PHEV operates in pure electric mode, the motor torque T _m is equal to the vehicle torque T _v, while the engine torque T _e is zero.

2.3.2.2. Engine-Only Mode

In this mode, the motor torque T _m is zero, and the engine torque T _e equals the vehicle torque T _v.

2.3.2.3. Hybrid Drive or Charge-Sustaining Mode

To enhance energy efficiency and appropriately distribute torque between the engine and the motor, the PHEV utilizes the dual-state adaptive ECMS (Equivalent Consumption Minimization Strategy) designed in this study for torque allocation.

2.3.2.4. Regenerative Braking Mode

During regenerative braking, the engine torque T _e is zero, and the motor torque T _m is T _b1(v). In coasting regenerative braking mode, T _e remains zero, and T _m is T _b2(v). Here, T _b1(v) and T _b2(v) represent the ISG motor’s generation torque as functions of vehicle speed, with regenerative braking torque being higher than coasting regenerative braking torque to avoid affecting the coasting distance.

2.4. Exploring Single Static Adaptive Mechanisms Using PMP Framework

The fundamental principle of the ECMS is to ensure that the vehicle meets prescribed driving tasks (such as predefined speed profiles) while intelligently allocating power output between the engine and the electric motor. The energy cost can be assigned to electricity in this way. Essentially, this equates to using (or saving) a certain amount of fuel by using electrical energy. When employing the ECMS method, the power battery can be viewed as a reversible auxiliary fuel tank. The power consumed during discharge can be replenished in the future through engine fuel consumption, regenerative braking energy, and external grid charging.

Figure illustrates the ECMS control principle. Under battery discharge, the electric motor provides mechanical energy (Figure a), while the dashed lines represent processes related to the recovery of electrical energy for future use. Since the operating point at which the motor charges the battery is unpredictable, an approximate average efficiency is set. Under battery charging (Figure b), the motor converts received mechanical energy into electrical energy stored in the battery, and the dashed lines represent processes related to generating mechanical energy from this stored electrical energy at some future moment. The mechanical energy received by the electric motor does not necessarily come from the engine; hence, it can be considered as fuel saving. In this scenario, the equivalent fuel consumption of the motor is negative. Overall, treating the battery as a reversible auxiliary fuel tank enables the ECMS method to more efficiently manage the energy flow of hybrid vehicles and minimize total energy consumption.

7.

Control principles of ECMS.

To systematically analyze the optimization capability of the dual-static adaptive ECMS control strategy, identify its underlying causes of error, and provide a more accurate description of the energy state of the battery (state of charge, SOC), we introduce a concept called the equivalent factor, s­(t). The equivalent factor plays a crucial role in the energy allocation exchange between the engine and the electric motor in the control strategy. By introducing this concept, the consumed electrical energy can be equated to an equivalent amount of fuel. This approach not only simplifies calculations but also allows us to compute the equivalent fuel consumption rate, which is the sum of the actual fuel consumption and the fuel equivalent of the consumed electricity.

By introducing the equivalent factor, the energy exchange between the internal combustion engine and the electric motor is standardized, achieving consistency in system energy management. This methodology not only streamlines the energy management process but also enhances the overall efficiency and performance of the hybrid powertrain system.

The equivalent factor s(t) establishes a correlation between the cost of electrical power and fuel consumption cost. The transient optimization energy cost function in ECMS can be described as

$$J_{\mathrm{ECMS}}=c_fṁ(t)+s(t)\frac{c_e{P}_{\mathrm{bat}}(t)}{3600}$$ 6

Using the PMP principle, the optimal solution s(t)­can be derived. For the optimization problem that only considers energy consumption, the state equation of the system can be rewritten according to the definition of the battery circuit model:

$$\stackrel{.}{\mathrm{SOC}}(t)=-\frac{I_{\mathrm{bat}}}{Q_{\mathrm{bat}}}=-\frac{I_{\mathrm{bat}}{U}_{\mathrm{oc}}}{Q_{\mathrm{bat}}{U}_{\mathrm{oc}}}=-\frac{P_{\mathrm{bat}}}{3.6E_{\mathrm{bat}}}$$ 7

In this equation, E _bat represents the amount of energy that can be stored in the battery, and its units Wh.

The Hamiltonian function can be reformulated as

$$\begin{array}{l}H=c_f{ṁ}_f(t)+c_e\frac{P_{\mathrm{bat}}(t)}{3600}+\lambda (t)\stackrel{.}{\mathrm{SOC}}(t)\\ =c_f{ṁ}_f(t)+(1-\frac{1000\lambda (t)}{c_e{E}_{\mathrm{bat}}})c_e\frac{P_{\mathrm{bat}}(t)}{3600}\end{array}$$ 8

Combining eqs and , we can obtain

$$s(t)=1-\frac{1000\lambda (t)}{c_e{E}_{\mathrm{bat}}}$$ 9

Equation represents the connection between ECMS and PMP. Although ECMS is an engineering-based heuristic algorithm, its optimization theory can be explained using the PMP theory. In the PMP strategy, the Hamiltonian function in eq is solved by hitting the target method with covariance t, adjusting the covariance value at each moment as the operating conditions change to obtain the optimal solution. Similarly, the optimal equivalent factor in ECMS can be obtained through iterative search, but it also requires prior knowledge of driving condition information. How to find an approximation of the optimal equivalent factor under unknown operating conditions is a focus of ECMS research. Adaptive ECMS based on SOC feedback is a simple yet effective method that can be implemented using a PI (Proportional and Integral) controller:

$$s(t)=s_0+k_p(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)-\mathrm{SOC}(t))+k_i{\int }_0^t(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(\mathit{t})-\mathrm{SOC}(\mathit{t}))dt$$ 10

In the equation, s ₀ represents the initial equivalent factor, k _p and k _i are the proportional and integral coefficients of the controller, respectively. Based on the driving distance of the driving task, the reference SOC of the PHEV after one charge can be planned as follows:

$$\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)=\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{initial}}-\frac{d_n(t)}{D_{\mathrm{spl}}}(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{initial}}-\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{target}})$$ 11

In the equation, D _spl represents the total distance traveled between two charging tasks of the PHEV. d_n (t) denotes the distance of a driving task, which is the distance traveled by the vehicle after charging.

In the equivalent consumption minimization strategy (ECMS), the electrical energy consumed by the battery is typically converted into an equivalent fuel consumption by multiplying it with an appropriate conversion factor, known as the equivalence factor. As shown in eq , a larger equivalence factor implies that the same battery discharge power represents a higher equivalent fuel consumption. In such cases, the Energy Management System (EMS) tends to reduce the battery discharge power or even switch to charging the battery. Conversely, if the equivalence factor is too low, the battery discharge power will increase, and the motor will provide more power to drive the vehicle.

As indicated in eq , the equivalence factor is a variable related to the battery’s internal resistance and open-circuit voltage, making it challenging to obtain an analytical solution. To determine an appropriate equivalence factor, it is commonly assumed that the battery’s internal resistance and open-circuit voltage do not vary with the state of charge (SOC). Therefore, the covariant is considered a constant, leading to a fixed equivalence factor. Through extensive simulation, the equivalence factor is iteratively adjusted to ensure that the terminal SOC meets predefined constraints, providing an approximate range of feasible values. The optimized EMS then enables real-time energy management in hybrid electric vehicles, achieving an effective balance between energy efficiency and system performance.

2.5. Developing a Two-State Adaptive Optimization Strategy Using PMP (Pontryagin’s Minimum Principle) Framework

After studying the adaptability of the PI controller, to explore a simple and effective multiobjective adaptive solution, we first directly incorporate the battery aging term into the ECMS optimization strategy. We embed the aging model into the optimization framework to obtain a reasonable weighting factor based on PMP. The performance of the single-state adaptive strategy in the multiobjective adaptive optimization process was tested in simulation scenarios composed of random conditions 1 and random conditions 4. The instantaneous cost function can be represented as

$$G=c_f{ṁ}_f+s(t)c_e\frac{P_{\mathrm{bat}}}{3600}={\omega }_r{c}_{bat}\frac{\sigma |I_{\mathrm{bat}}|}{3600A{h}_{\mathrm{nom}}}$$ 12

To track the battery capacity degradation trajectory of the ideal compromise solution, the state variable Ah _eff is introduced. According to the PMP principle, the Hamiltonian function can be transformed into:

$$H=c_f{ṁ}_{\mathrm{eff}}+c_e\frac{P_{\mathrm{bat}}}{3600}+{\omega }_r{c}_{bat}\frac{\sigma |I_{\mathrm{bat}}|}{3600A{h}_{\mathrm{nom}}}+{\lambda }_0(t)\mathrm{SOC}+{\lambda }_1(t)A{ḣ}_{\mathrm{eff}}$$ 13

In the equation, P _bat = I _bat U _oc, $A{ḣ}_{\mathrm{eff}}=\frac{\sigma |I_{\mathrm{bat}}|}{3600}$ , and λ₁(t) is the covariance vector corresponding to the state variable Aḣ_eff. Then, the Hamiltonian function can be transformed as follows:Aḣ_eff

$$H=c_f{ṁ}_f+(1-\frac{1000{\lambda }_0(t)}{E_{\mathrm{bat}}{c}_e})\frac{c_e{P}_{\mathrm{bat}}}{3600}+(1+\frac{{\lambda }_1(t)A{h}_{\mathrm{nom}}}{{\omega }_r{c}_{\mathrm{bat}}}){\omega }_r\frac{c_{\mathrm{bat}}\sigma |I_{\mathrm{bat}}|}{3600A{h}_{\mathrm{nom}}}$$ 14

The instantaneous optimization function with dual adaptive factors can be represented as

$$G=c_f{ṁ}_f+s_1(t)c_e\frac{P_{\mathrm{bat}}}{3600}+s_2(t){\omega }_r{c}_{\mathrm{bat}}\frac{\sigma |I_{\mathrm{bat}}|}{3600A{h}_{\mathrm{nom}}}$$ 15

Combining eqs and , we obtain

$$\{\begin{array}{l}{s}_1(t)=(1-\frac{1000{\lambda }_0(t)}{E_{\mathrm{bat}}{c}_e})\hfill \\ s_2(t)=(1+\frac{{\lambda }_1(t)A{h}_{\mathrm{nom}}}{{\omega }_r{c}_{\mathrm{bat}}})\hfill \end{array}$$ 16

For a multi-input multioutput (MIMO) system, decentralized PID control is one of the most commonly used control schemes. In this paper, dual PI controllers are employed for DA-ECMS with dual-state adaptive control, and their adaptive rules are designed as follows:

$$\{\begin{array}{l}{s}_1(t)=s_{10}+k_{p1}(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)-\mathrm{SOC}(t))+k_{i1}{\int }_0^t(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)-\mathrm{SOC}(t))dt\hfill \\ s_2(t)=s_{20}+k_{p2}(A{h}_{\mathrm{ref}}(t)-A{h}_{\mathrm{eff}}(t))+k_{i2}{\int }_0^t(A{h}_{\mathrm{ref}}(t)-A{h}_{\mathrm{eff}}(t))dt\hfill \end{array}$$ 17

In the equation, k _p1 and k _p2 represent the scaling factors of the two PI controllers, k _i1 and k _i2 represent the integral factors, SOC _ref and Ah _ref represent the reference values of the effective throughput. s ₁₀ and s ₂₀ are the initial equivalent factors and can be expressed as

$$\{\begin{array}{l}{s}_{10}=1-\frac{1000{\lambda }_0^{\mathrm{opt}}(0)}{E_{\mathrm{bat}}{c}_e}\hfill \\ s_{20}=1+\frac{{\lambda }_1^{\mathrm{opt}}(0)A{h}_{\mathrm{nom}}}{{\omega }_r{c}_{\mathrm{bat}}}\hfill \end{array}$$ 18

In the equation, ${\lambda }_1^{\mathrm{opt}}(0)$ and λ₁(t) are the initial values of the optimal covariance, and it is noteworthy that only one state variable, battery SOC, is involved in the PMP offline optimization process. Therefore, the ideal compromise solution obtained according to PMP. ${\lambda }_1^{\mathrm{opt}}(0)=0$ is the desired SOC terminal value at the end of the driving task, and the reference SOC trajectory can be planned as

$$\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}=\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{initial}}-(\frac{d_n(t)}{D_{\mathrm{spl}}}-R_d\left(\frac{d_n(t)}{D_{\mathrm{spl}}}\right))(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{initial}}-\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{target}})$$ 19

In the equation, R_d (·)­represents the floor function, which returns the largest integer less than or equal to the specified value. d_n represents the actual distance traveled by the vehicle, and D _spl represents the total distance traveled by the vehicle between charging tasks.

Ah _eff accumulates with the increase of effective throughput ah driving distance. If it is assumed that there is an approximate linear relationship between effective throughput ah and driving distance, the reference trajectory can be expressed as

$$A{h}_{\mathrm{ref}}(t)=(\frac{d_n(t)}{D_{\mathrm{spl}}}-R_d\left(\frac{d_n(t)}{D_{\mathrm{spl}}}\right))A{h}_{\mathrm{eff}}^{\mathrm{opt}}+R_d\left(\frac{d_n(t)}{D_{\mathrm{spl}}}\right)(A{h}_{\mathrm{eff}}^{\mathrm{opt}}+A{h}_{\mathrm{eff}}^c)$$ 20

In the equation, $A{h}_{\mathrm{eff}}^{\mathrm{opt}}$ is the optimal effective throughput corresponding to the ideal compromise solution at the end of the driving task, and $A{h}_{\mathrm{eff}}^c$ is the effective throughput Ah generated as the SOC rises from SOC_target to SOC_initial once it is charged.

2.6. Enhancing Two-State Adaptive Optimization Strategy

This paper proposes a novel approach based on kernel density estimation and entropy-power Bayesian theory for the classification and identification of severe working conditions. Additionally, the DA-ECMS strategy is further enhanced through the integration of PI control.

The variation at a given moment is highly random, and the specific functional form of the overall probability density is often unavailable. Therefore, it is impossible to match the real data distribution with a particular statistical model. Kernel density estimation is a widely used method for computing nonparametric probability densities. The idea behind kernel density estimation is to use a kernel function to measure the similarity between data points. In the data space ${\mathbb{R}}^d$ , we assume that the given set of multidimensional training sample data is $\mathrm{\Omega }={\{C_m\}}_{m=1}^{\mathrm{M}}$ , where $C_m={\{c_n^m\}}_{n=1}^N,c_n^m=\{x_1^{m,n},...,x_{\mathrm{I}}^{m,n}\}$ , and $m,n\in {\mathbb{N}}^{+}$ . If we find the probability density of the i-th element in an unknown set of observed data c = {x ₁, x ₂, ... , x_i }, i = 1,2, ... , I, it can be expressed as

$$\mathrm{f̂}(x_i|C_m)=\frac{1}{h{N}_n}\sum _{n=1}^{\mathrm{N}}K(x_i-x_i^{m,n})$$ 21

In the equation, h is the bandwidth parameter, and K is the kernel function. The Gaussian kernel function is commonly used.

$$K(x)=\frac{1}{\sqrt{2\pi }}\exp (-\frac{1}{2}{(x)}^2)$$ 22

The resulting kernel density estimation function is

$$\mathrm{f̂}(x_i|C_m)=\frac{1}{hN\sqrt{2\pi }}\sum _{n=1}^{\mathrm{N}}\exp (-\frac{1}{2}{(\frac{x_i-x_i^{m,n}}{h})}^2)$$ 23

The optimal bandwidth can be determined by minimizing the mean integrated squared error function. The specific form of the mean integrated squared error function is as follows:

$$\mathrm{MISE}(h)=E\left\{\int {[\mathrm{f̂}(x_i|C_m)-\mathrm{f̅}(x_i|C_m)]}^2\mathrm{d}{x}_i\right\}$$ 24

In the equation, f̅(x_i |C _m ) is the true probability density function value, E(·) is the mathematical expectation.

When the probability of occurrence of an event C _m is assumed to be p(C _m), the probability of occurrence of an event C _m is expressed as p(C _m|x _i) under the premise that the event x_j occurs. In statistics, p(C _m) is referred to as the prior probability and p(C _m |x _i ) is referred to as the posterior probability or conditional probability. In statistics, p(C _m) is referred to as the prior probability and p(C _m |x _i ) is referred to as the posterior probability or conditional probability. According to Bayes’ theorem, the posterior probability of the event C _m given the occurrence of x_i can be directly expressed as

$$p(C_m|x_i)=\frac{p(x_i|C_m)·p(C_m)}{\sum _{m=1}^{\mathrm{M}}p(x_i|C_m)p(C_m)}$$ 25

In the equation, the value of p(x_i |C_m ) can be calculated by kernel density estimation, namely, p(x_i |C_m ) = f̂(x_i |C_m ).

The concept of “severity of working conditions” is proposed to characterize the random fluctuations of speed. If this variable x_i is used to represent the severity of speed variation, the posterior probability p(C_m |x_i ) will also exhibit uncertainty, and information entropy is an appropriate method to describe this uncertainty. The concept of information entropy originates from Shannon’s information theory, which is used to quantify information measurement and its effects. To quantitatively describe the influence of the randomness of variables on the posterior probability, this study employs the entropy Bayesian method to compute the integral posterior probability c of the observed data set c:

$$p(C_m|c)=\sum _{i=1}^{\mathrm{I}}p(C_m|x_i){\omega }_i^m$$ 26

In the equation, ${\omega }_i^m$ represents the entropy weight coefficient, essentially indicating the relative competitiveness of each indicator. The magnitude of the entropy coefficient is closely related to the evaluation of the object. Specifically, the above entropy value represents the effectiveness of the indicator, with the maximum value of the entropy coefficient being 1 and the minimum value being 0. A smaller entropy value indicates that the indicator does not provide sufficient information for decision-makers and is relatively weak in competitiveness in evaluation. However, a larger entropy weight value indicates that the indicator provides a significant amount of useful information and deserves attention. The entropy weight coefficient of each indicator can be expressed as

$${\omega }_i^m=\frac{1}{I-1}|1-\frac{1-H_i^m}{\sum _{i=1}^I(1-H_i^m)}|$$ 27

$$H_i^m=-\frac{1}{\ln (N)}\sum _{n=1}^{\mathrm{N}}{p}_i^{m,n}\ln (p_i^{m,n})$$ 28

$$p_i^{m,n}=\frac{e_i^{m,n}}{\sum _{n=1}^{\mathrm{N}}{e}_i^{m,n}}$$ 29

In the equation, $H_i^m$ is the error entropy of the posterior probability of the i-th variable in C _m , $p_i^{m,n}$ is the error proportion of the posterior probability of the variable $x_i^{m,n}$ , N is the number of samples in the training sample cm, and ${\mathrm{e}}_i^{m,n}$ is the posterior probability error of the variable $x_i^{m,n}$ . The posterior probability error of each variable can be described as

$$e_i^{m,n}=\min \{1,\left|\frac{p(C_m)-p(C_m|x_i^{m,n})}{p(C_m)}\right|\}$$ 30

The severity of operation conditions depends mainly on the driver’s actions of pressing the brake or accelerator pedals, which can be described by the magnitude and velocity of the pedal. The amplitude of the accelerator or brake pedal determines the vehicle’s acceleration, while the pedal’s velocity determines the rate of acceleration change or velocity (i.e., the degree of impact). The definition of severity is as follows:

$$\mathrm{Jerk}(t)=\frac{{\mathrm{d}}^{2}v(t)}{\mathrm{d}{t}^2}$$ 31

In the equation, υ(t) is the vehicle’s travel speed.

The present study selects acceleration and impact as the identification features for assessing the severity of workplace conditions. These two indicators are effective in quantitatively describing driving behavior. The intensity defined in this study is strongly correlated with driving behavior.

The severity of the situation between smooth and severe conditions can be represented by

$$L(c)=\{L|\mathrm{\Delta }P=P_s(C_s|c)-P_g(C_g|c)\in ({\chi }_{\mathrm{u}\mathrm{p}},{\chi }_{\mathrm{l}\mathrm{w}})\}$$ 32

In the equation, P _s (C _s |c) and P_g (C_g |c) represent the posterior probabilities of the vehicle’s current condition being in severe condition and smooth condition, respectively. Formula L denotes the severity level of the working condition, and (χ_up,χ_lw) is a threshold value for determining the severity level.

$$L(t+T)=L\left(\frac{1}{\mathrm{N}}\sum _{n=1}^{\mathrm{N}}c(t+n{T}_0)\right)$$ 33

In the equation, T is the recognition period, T ₀ is the sampling period, which is consistent with the sampling period of the adaptive policy and is set to 1s. C is the sampled data in the recognition period, and N is the number of sampled data in a recognition period.

Based on the results of identifying the severity of the working conditions, the PI adaptive mechanism (eqs. 38 and 39) developed in Chapter 4 is now improved as follows:

$$\{\begin{array}{l}{s}_1(t)=s_{10}(t)+k_{p1}(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)-\mathrm{SOC}(t))+k_{i1}{\int }_0^t(\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{ref}}(t)-\mathrm{SOC}(t))dt\hfill \\ s_2(t)=s_{20}(t)+k_{p2}(A{h}_{\mathrm{ref}}(t)-A{h}_{\mathrm{eff}}(t))+k_{i2}{\int }_0^t(A{h}_{\mathrm{ref}}(t)-A{h}_{\mathrm{eff}}(t))dt\hfill \end{array}$$ 34

$$\{\begin{array}{l}{s}_{10}(t)=[1-\frac{1000{\lambda }_0^{opt}(0)}{E_{\mathrm{bat}}{c}_e}]+{\tau }_{1}L(t+T)\hfill \\ s_{20}(t)=[1+\frac{{\lambda }_0^{opt}(0)A{h}_{\mathrm{nom}}}{{\omega }_r{c}_{\mathrm{bat}}}]+{\tau }_{2}L(t+T)\hfill \end{array}$$ 35

In the equation, τ ₁ and τ ₂ are the correction coefficients for the severity level of the working condition. The proportional coefficients k _p1, k _p2 and integral coefficients k _i1, k _i2 are consistent with the DA-ECMS strategy, and the planning methods for SOC_ref and Ah _ref are the same as those of the DA-ECMS strategy.

2.7. Vehicle Parameters

After the above modeling process, we have determined the general situation of the vehicle, and, on this basis, the vehicle parameters are determined. The vehicle parameters are determined based on common SUVs popular in the market, and the detailed specifications are listed in Table .

1. Vehicle Parameters.

Item	Parameter	Value
Vehicle	Curb weight (kg)	1920
	Length (mm)	4708
	Final gear ratio	6.733
Engine	Displacement (L)	2.78
	Max power (kW)	184
	Max speed (rpm)	5400–5700
Electrical motor1/2	Max power (kW)	300
	Max speed (rpm)	8000
	Max torque (Nm)	200
Batter cell	Capacity (Ah)	5.3
	Voltage (V)	37.992
	Number of cells	72

3. Results and Discussion

3.1. Study on the Influence of Parameters in A-ECMS Strategy

3.1.1. Discussion on the Influence of Initial Equivalent Factor on Experiments

All simulations in this study were conducted using MATLAB/Simulink. To investigate the impact of the initial equivalence factor s ₀on battery SOC, the optimization performance of ECMS strategies with varying initial equivalence factors was analyzed without incorporating the PI controller. Six initial equivalence factors were selected for comparison: 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6. The results (Figure ) indicate that with the increase in the equivalence factor, the vehicle’s power performance slightly improves. Specifically, as shown in Table , when the equivalence factors are 2.1 and 2.6, the vehicle’s power performance is 4.46 and 4.94, respectively, representing an increase of 9.717%. However, the MPGe decreases with the increase in the initial equivalence factor. For instance, when the equivalence factors are 2.1 and 2.6, the MPGe is 52.70 and 47.62, respectively, demonstrating a decrease of 9.639%.

8.

Effects of different initial equivalence factors on SOC, performance, MPGe, and fuel price.

2. SOC, Performance, MPGe, and Fuel Price for Different Initial Equivalence Factors.

S 0	SOC	Performance	MPGe	Fuel price(¥/km)
2.1	15.7%	4.46	52.70	0.3836
2.2	23.0%	4.59	51.27	0.3779
2.3	28.4%	4.68	50.27	0.3706
2.4	33.5%	4.77	49.35	0.3664
2.5	38.9%	4.86	48.38	0.3624
2.6	43.1%	4.94	47.62	0.3555

The variation in the initial equivalence factor results in deviations in the dual-static adaptive ECMS control strategy. Therefore, selecting the appropriate initial equivalence factor is crucial for studying the optimized vehicle performance and fuel economy of dual-motor plug-in hybrid electric vehicles. In this study, we use the initial value of the optimal covariance trajectory as the basis for calculating the optimal initial equivalence factor:

$$s_0^{\mathrm{opt}}=1-\frac{1000{\lambda }_0^{\mathrm{opt}}(0)}{c_e{E}_{\mathrm{bat}}}$$ 36

3.1.2. Discussion on the Impact of Control Factors of PI Controller on Experiments

To ensure convergence to the reference value, it is necessary to adjust the adaptive parameters, proportional factor k_P and integral factor k _i , in eq appropriately. Therefore, it is necessary to study the influence mechanism of each adaptive parameter on the optimization results. To make the comparison more evident, the initial equivalence factor is set to s ₀ = 2.5. In the driving task, the influence of the proportional factor on SOC and the adaptive equivalence factor was studied with the integral factor of the PI controller set to zero. As shown in Figure , with the increase of the proportional factor, the rate of SOC decrease gradually decreases. The principle of SOC feedback-based adaptive ECMS execution is illustrated in Figure .

10.

Influence of different PI scale factors on SOC.

9.

Implementation principle of adaptive ECMS based on SOC feedback.

Figure and Figure respectively illustrate the influence of different PI proportional factors on SOC and the equivalence factor.

11.

Influence of different PI control factors on ΔSOC.

When studying the impact of the PI controller on the performance and fuel economy of plug-in hybrid electric vehicles in the dual-static adaptive ECMS control strategy, we separately treat the proportional factor and the integral factor as single variables to better observe their influence on the performance of the dual-static adaptive ECMS control strategy. In this study, two sets of experiments will be conducted: the first set involves keeping the integral gain at zero and varying the proportional gain K _P; the second set involves keeping the proportional gain at zero and adjusting the integral gain K _i.

From Table , it can be observed that keeping the integral factor K _i constant, as the proportional factor K _P increases, the value of ΔSOC gradually decreases. This implies that the vehicle can accomplish the driving task using less electrical energy. Furthermore, with an increase in the integral factor K _i, the difference in SOC caused by the proportional factor K _P is 10.15%, while their combined effect on battery performance reaches 19.87%. When the integral factor K _i is 0.75, the variation in battery SOC is only 1.1% with K _P values of 0 and 0.75, while the impact on battery performance is 4.73%.

3. Different ΔSOC Variations Under Various PI Control Factors.

PI control factors	KP = 0	K P = 0.25	K P = 0.5	K P = 0.75
K i = 0	51.08%	48.79%	44.46%	40.93%
K i = 0.25	32.72%	30.69%	29.42%	27.45%
K i = 0.5	25.54%	30.69%	24.75%	24.52%
K i = 0.75	23.27%	22.94%	22.50%	22.17%

Figure depicts a three-dimensional schematic diagram illustrating the impact of the control factors of the PI controller on SOC. The four pictures in Figure all show the same trend. Overall, from the graph, it can be observed that SOC is more sensitive to the integral factor K _i. As the integral factor K _i increases, the control capability of the PI controller gradually strengthens, resulting in smaller variations in SOC. Along the Y-axis (K _i), it can be noted that as the integral factor K _i decreases, the influence of the proportional factor K _P on the PI controller gradually increases.

To better investigate the impact of PI control factors on the dual-static adaptive ECMS strategy, it is important to select the sensitive region of the PI controller in the experiment. From previous experiments, we learned that the integral factor K _i has a greater influence on the PI controller, and when the integral factor K _i = 0, the battery SOC is most sensitive to changes in the proportional factor K _P, with the fastest response. Therefore, our experiment is conducted with a fixed integral factor K _i = 0.

We set four proportional factorsK _P, namely 0, 0.25, 0.5, and 0.75. The results, as shown in Figure , indicate that by increasing the proportional factor K _P, the growth rate of the equivalent factor gradually increases over the same period. This implies that the cost of using electrical energy will increase, and the powertrain system is more inclined to use the engine to drive the vehicle, which is not conducive to unleashing the potential of plug-in hybrid electric vehicles.

12.

Effect of different PI scale factors on equivalent factors.

3.2. Comparison of Experimental Results for A-ECMS Strategy, ECMS Strategy, and RB Strategy

In this study, to evaluate the effectiveness of the designed dual-static adaptive ECMS controller, it will be compared with two widely used energy strategies to investigate potential fuel economy and carbon emission reduction. The two strategies chosen for comparison are rule-based strategy and ECMS strategy.

3.2.1. The RB Strategy

Parameters of the rule-based controller are typically obtained through optimization techniques. The battery’s State of Charge (SOC) and load power determine the operating points of the engine and generator, as shown in Table .

4. Rule-Based Control Strategy.

Judgment	State of the Battery	Power supply
$$\begin{array}{l}{P}^{*}<P_{e\_\max }\hfill \\ \mathrm{SOC}<\mathrm{S}\mathrm{O}{\mathrm{C}}_{\max }\hfill \end{array}$$	Charging	$$\begin{array}{l}{P}_E={\eta }_e{P}_e\\ P_b=P_a-P_E\end{array}$$
$$\begin{array}{l}{P}^{*}<P_{e\_\max }\hfill \\ \mathrm{SOC}\ge \mathrm{S}\mathrm{O}{\mathrm{C}}_{\max }\hfill \end{array}$$	Not working	$$\begin{array}{l}{P}_E={\eta }_e{P}_e\\ P_b=0\end{array}$$
$$\begin{array}{l}{P}^{*}>P_{e\_\max }\hfill \\ \mathrm{SOC}<\mathrm{S}\mathrm{O}{\mathrm{C}}_{\min }\hfill \end{array}$$	Discharging	$$\begin{array}{l}{P}_E={\eta }_e{P}_{e\_\max }\\ P_b=P_a-P_E\end{array}$$
$$\begin{array}{l}{P}^{*}>P_{e\_\max }\hfill \\ \mathrm{SOC}\le \mathrm{S}\mathrm{O}{\mathrm{C}}_{\min }\hfill \end{array}$$	Not working	$$\begin{array}{l}{P}_E={\eta }_e{P}_e\\ P_b=0\end{array}$$

Although the Rule-Based (RB) strategy boasts advantages such as clear logic and ease of implementation, its formulation relies on predefined rules and thresholds. Consequently, static rules cannot dynamically adjust to varying driving conditions and vehicle states, making it challenging to ensure globally optimal energy management. The RB strategy typically only considers the current vehicle state and driving demands, without forecasting future driving conditions, such as upcoming inclines or traffic situations, which limits its ability to optimize energy management.

In the table, P* represents the target power demand; P_e denotes the rated power of the engine; P_{e _max} indicates the maximum power that the engine can provide; η_e represents the efficiency of the engine when operating; P_E denotes the effective output power of the engine; P_b represents the rated power of the battery; P_a represents the total power that the engine and the battery can provide (ignoring the impact of vehicle electronic devices on the total power in the experiment); SOC_min and SOC_max represent the minimum and maximum charging states of the battery.

3.2.2. The ECMS Strategy

The equivalent consumption minimization strategy (ECMS) is derived from Pontryagin’s Minimum Principle (PMP) and is a chronological method. Due to its fast computation speed and the absence of a requirement for the global power request of the drive configuration, ECMS is practically implementable. Generally, ECMS is able to identify a series of candidate solutions to the PHEV energy management problem by establishing a set of necessary conditions from PMP. These candidate solutions must not only satisfy these necessary conditions but also be subject to restrictions imposed by defined state constraints. Through the interaction of these two principles, the optimal control strategy can be found using a trial-and-error method.

However, in ECMS, the optimal control solution can only be achieved through perfect coordination of equivalent factors. Equivalent factors are used to scale between electrical cost and fuel cost in the Hamiltonian function defined by the minimum principle and cannot be predetermined a priori. Moreover, ECMS is highly stringent, and even slight deviations from the optimal equivalent factor values may result in unacceptable operation of the vehicle outside the SOC boundaries.

3.2.3. Experimental Results

In Chapter 2, we established two test scenarios to simulate different usage environments. In this chapter, we will compare the experimental results of the A-ECMS strategy, the ECMS strategy, and the RB strategy based on these two scenarios.

3.2.3.1. Scenario 1: UDDS Driving Cycle

Under the typical UDDS driving cycle, the traditional ECMS strategy, RB strategy, and the dual-static adaptive A-ECMS strategy designed in this paper were tested separately. For the RB strategy, the target SOC was set to 45%. Therefore, when the SOC of the vehicle battery reached around 45%, the RB strategy would initiate the battery sustainment strategy. According to the experiment, it was observed that at the end of the test, the battery SOC value under the RB strategy was 42.5%.

Under the same conditions as depicted in Figure , the fuel economy, battery SOC, CO₂ emissions, and vehicle performance achieved by the dual-static adaptive A-ECMS algorithm were compared with the ECMS strategy and rule-based (RB) algorithm.

As shown in Table , the improved dual-static adaptive A-ECMS strategy can reduce fuel consumption by 7.8% compared to the traditional ECMS strategy. Since the RB strategy is applicable only to known road conditions and is difficult to use in reality, it is generally used as a reference optimization target for other research strategies. Analyzing its behavior can provide possible improvements for controller design.

5. Fuel Consumption, Electricity Consumption, CO₂ Emissions, and Vehicle Performance under Scenario 1.

Control Strategy	Fuel Consumption (L/100 km)	Fuel Economy (%)	Electric Consumption (ΔSOC)	Electric Economy (%)	CO2 Emission (g/km)	MPGe
ECMS	4.98	100%	61.9%	100%	95.6	45.4
A-ECMS	4.59	92.2%	30.3%	48.9%	80.4	49.1
RB	5.51	120%	47.5%	76.7%	57.9	62.1

From the four images depicted in Figure , it can be observed that the dual-static adaptive A-ECMS strategy designed in this paper achieved excellent results in terms of battery SOC reduction, with a decrease of only 30.3%. The energy consumption of the ECMS strategy was 204.3% of that of the A-ECMS strategy, while the RB strategy consumed 156.8% of the energy consumed by the A-ECMS strategy. MPGe, or Miles Per Gallon Equivalent, represents the distance a vehicle can travel per equivalent gallon of gasoline. A higher MPGe value indicates greater fuel efficiency, meaning the vehicle can travel further under the same conditions. This metric is used as a performance indicator for the vehicle.

13.

Results of ECMS, A-ECMS, and RB under Scenario 1.

In the experiment, we also tested the CO₂ emissions under the three strategies. The emission under the ECMS-based strategy was 95.6 g/km, while under the improved dual-static adaptive A-ECMS strategy, it was reduced to 80.4 g/km. For the RB strategy, the emission was 57.9 g/km. Compared to the ECMS strategy, the A-ECMS strategy reduced CO₂ emissions by 15.9%.

Through experimental testing, the improved dual-static adaptive A-ECMS strategy not only improved vehicle performance but also optimized the allocation of engine power and battery power during PHEV operation. This resulted in a 7.8% increase in fuel economy and a reduction of approximately 31.6% in energy consumption. Furthermore, it also lowered CO₂ emissions, reducing them by 15.2 g per kilometer compared to the ECMS strategy.

3.2.3.2. Scenario 2: Adding 5% Random Noise

From Table , it can be observed that under Scenario 2, the fuel consumption of the improved dual-static adaptive A-ECMS strategy proposed in this paper is reduced by 6.9% compared to the ECMS strategy. From the four images shown in Figure , we can see that the final SOC values for the three strategies are as follows: for the ECMS strategy, SOC is 24%; for the A-ECMS strategy, it is 58.95%; and for the RB strategy, it is 39.68%. Comparing the SOC consumption based on the A-ECMS strategy as the baseline, the SOC consumption for the ECMS strategy is 100%, and for the RB strategy, it is 76.24%. The PHEV based on the A-ECMS strategy, while maintaining good vehicle performance during driving, also reduces CO₂ emissions by 14.39 g/km compared to the ECMS strategy.

6. Results of the Three Strategies under Scenario 2.

Control Strategy	Fuel Consumption (L/100 km)	Fuel Economy (%)	Electric Consumption (ΔSOC)	Electric Economy (%)	CO2 Emission (g/km)	MPGe
ECMS	6.39	100%	66%	100%	116.84	35.66
A-ECMS	5.95	93.1%	31.05%	47%	102.45	38.18
RB	6.40	100%	50.32%	76.24%	90.89	44.91

14.

Experimental results of ECMS, A-ECMS, and RB under Scenario 2.

The initial conditions for the three strategies are consistent with those of Scenario 1. From Figure b, it can be observed that the differences in battery SOC among the three strategies are not significant until around 250 s. Under the RB strategy, the battery SOC reaches 45%, the set threshold, at 500 s, after which it operates in charge-sustaining (CS) mode. Under the ECMS strategy, the battery SOC decreases rapidly, and by the end of the experiment, it has already dropped to around 24%. Compared to the former two strategies, the improved A-ECMS strategy proposed in this paper optimizes the power distribution of the battery, resulting in a slower decrease in SOC.

Figure a, c, and d depicts the performance of the three strategies in terms of fuel consumption, CO₂ emissions, and vehicle performance. Within the three test ranges, the performance of the A-ECMS strategy is slightly better than that of the ECMS strategy. However, it is evident from the three plots that the stability of the RB strategy is not as good as that of the A-ECMS strategy during long-term testing. Starting from around 500s, the PHEV based on the RB strategy shows a significant increase in fuel consumption and CO₂ emissions, as well as a decline in vehicle performance. This is because after 500 s, the vehicle based on the RB strategy transitions from an energy consumption mode to a charge-sustaining mode. In the charge-sustaining mode, the vehicle primarily relies on the engine as the main power source, while the battery acts as a power auxiliary, resulting in increased fuel consumption, higher CO₂ emissions, and decreased vehicle performance.

3.2.4. Economic Benefits

To validate the accuracy of the proposed improved dual-static adaptive ECMS strategy in the paper, we simulated vehicle dynamics and economics using an optimization framework simulation model and compared it with the ECMS strategy and RB strategy. Ultimately, we obtained the results shown in Figures and .

Figures and illustrate the schematic diagrams of battery SOC, fuel consumption per 100 km, and CO₂ emissions for the improved dual-static adaptive ECMS strategy proposed in this paper, as well as for the ECMS strategy and RB strategy under different application scenarios. Through analysis, we found that the improved dual-static adaptive ECMS strategy can achieve adaptive adjustment of equivalent factors. By optimizing the distribution of vehicle kilometers more reasonably and optimizing the working states of the engine and battery, the hybrid powertrain system of the engine-battery can drive the vehicle more efficiently. This not only improves fuel economy but also effectively reduces CO₂ emissions.

In order to intuitively observe the potential economic benefits that the proposed A-ECMS strategy may bring in the future, we formulated energy cost unified eqs (41) and (42) to obtain the electricity consumption cost per kilometer A C _elec (RMB) and the fuel consumption cost C _fuel (RMB). Specifically, the total consumption cost is calculated by summing the fuel cost and electricity cost based on their respective unit prices and consumption rates observed in the experiments.

Where E_m is the voltage per battery terminal, E_B is the capacity per battery, N is the total number of batteries, and m _fuel is the fuel consumption per 100 km. The battery pack used in this experiment consists of 72 small batteries connected in series, with each small battery having a rated capacity of 5.3 Ah.

To obtain the energy consumption and fuel consumption, we consulted the National Bureau of Statistics and determined the electricity price to be 0.61 RMB per kWh and the fuel price to be 7.98 RMB per liter. We calculated the cost by adding the fuel consumption per 100 km to the experiment’s electricity consumption. The total consumption calculation results are shown in Table .

7. ECMS, A-ECMS, and RB Consumption Costs in Scenarios 1 and 2 .

	Scenario	Fuel Comsumption (L/km)	Electric Consumption (ΔSOC)	Total Consumption (¥)
ECMS	Scenario 1	4.98	61.9%	82.24
	Scenario 2	6.39	66%	96.31
A-ECMS	Scenario 1	4.59	30.3%	57.43(30.17%↓)
	Scenario 2	5.59	31.05%	68.80(28.56%↓)
RB	Scenario 1	5.51	47.5%	76.59(6.87%↓)
	Scenario 2	6.40	50.32%	85.63(11.09%↓)

^aNote: A decrease of 30.17%↓ indicates a reduction in cost by 30.17%.

In the same scenario, through comparison, we found that the improved dual-static adaptive ECMS strategy proposed in this paper, compared to the traditional ECMS strategy, reduces fuel consumption costs by 7.8% and 6.9%, and increases battery capacity by 31.6% and 34.95%, respectively. Additionally, the total consumption is reduced by 30.17% and 28.56% in the two scenarios, respectively. This indicates the effectiveness and feasibility of the energy recovery method based on the electrically controlled mechanical automatic transmission for the dual-motor plug-in hybrid electric vehicle proposed in this paper.

To better demonstrate the research findings, Chapter Four of this paper proposes a comparative experimental study between the A-ECMS strategy and the ECMS strategy, as well as the RB strategy, in two different usage scenarios. First, the structure and modeling of a plug-in hybrid electric vehicle (PHEV) with an electrically controlled mechanical automatic transmission were designed. Based on this model, three energy management approaches were proposed. Simulation results indicate that under the UDDS driving cycle, the fuel economy of the A-ECMS algorithm designed in this paper is 7.8% higher than that of the ECMS algorithm, with a reduction of 15.14 g/km in CO₂ emissions.

Furthermore, to validate the robustness of the A-ECMS strategy under perturbations, this paper introduced 5% random noise into the typical operating conditions. Simulation results demonstrate that under perturbations, the fuel economy of the A-ECMS strategy improved by 6.9% compared to the ECMS strategy. Additionally, CO₂ emissions were reduced by 14.39g/km. This indicates that the developed energy management controller exhibits good robustness.

In the experiments conducted in Scenario 2, it is evident that the overall stability of the A-ECMS strategy surpasses that of the RB strategy. This is because the RB strategy enters a state of battery charge maintenance mode when the SOC of the battery reaches the set threshold. During this mode, the engine becomes the primary power source while the battery serves as an auxiliary, leading to a significant increase in fuel consumption and CO₂ emissions. Consequently, the stability of the power system is compromised.

In terms of economic benefits, the A-ECMS strategy proposed in this paper achieves cost savings of 30.17% and 28.56% compared to the ECMS strategy in the two scenarios, respectively. This demonstrates that the bistatic adaptive ECMS strategy proposed in this paper outperforms the traditional ECMS strategy, highlighting its significant implications for related research.

4. Conclusions

This study presents a novel bistatic adaptive equivalent consumption minimization strategy (A-ECMS) for plug-in hybrid electric vehicles (PHEVs) with a dual-motor architecture and an electronically controlled automated manual transmission. By introducing dual-state adaptive control and real-time equivalence factor adjustment, the strategy aims to balance power distribution between the engine and electric motors while enhancing fuel economy, emission reduction, and control robustness. Simulation results demonstrate that A-ECMS outperforms conventional ECMS and rule-based (RB) strategies under both standard and disturbed driving conditions. In the UDDS cycle, A-ECMS reduced fuel consumption by 7.8%, CO₂ emissions by 15.14 g/km, and total energy cost by 30.17% relative to ECMS. Even with 5% random noise, A-ECMS achieved 6.9% fuel savings and a 14.39 g/km CO₂ reduction, validating its robustness against driving uncertainties. Furthermore, the strategy maintained more balanced battery SOC trajectories and superior vehicle performance indicators such as MPGe.

These experimental results validate that the A-ECMS strategy not only enhances energy efficiency and reduces emissions under ideal conditions but also performs reliably in stochastic environments, which is critical for practical vehicle applications. The integration of dual adaptive mechanisms and equivalence factor control expands the methodology of hybrid vehicle energy management and offers a scalable approach for real-time control implementation. Future research will focus on integrating predictive driving condition recognition and V2X-based information sharing to enhance adaptability and system-level optimization.

Glossary

Nomenclature

PHEV

plug-in hybrid electric vehicle

A-ECMS

adaptive equivalent consumption minimization strategy

ECMS

equivalent consumption minimization strategy

HEV

hybrid electric vehicle

BEV

battery electric vehicles

SOC

state of charge

ICE

internal combustion engine

HETV

hybrid electric tracked vehicles

DRL

deep reinforcement learning

RL

reinforcement learning

NN

neural networks

DQL

deep Q-learning

HESS

hybrid energy storage systems

AMT

automated manual transmissions

PMP

pontryagin minimum principle

UDDS

urban dynamometer driving schedule

EM

electric motor

EMS

energy management system

ECU

engine control unit

MCU

motor control unit

BCU

battery control unit

MIMO

multi-input multioutput

MPGe

miles per gallon equivalent

RB

rule-based

The data presented in this study are available within the article. (Specifically, the vehicle parameters are listed in Table , and the simulation results are presented in Figure – and Tables – ).

H.C.L. conceived and designed the study. H.L.W. and Y.L collected and analyzed the data and were major contributors in writing the manuscript. M.Y and Y.Z are mainly responsible for presenting the data in the form of pictures and tables. All authors read and approved the final manuscript.

This work was supported by the Technology Project of Henan Province (232102240058).

The pictures are created by myself and the team, and everyone has agreed. The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

The authors declare no competing financial interest.