Hybrid Model Predictive Control-Oriented Online Optimal Energy Management Approach for Dual-Mode Power-Split Hybrid Electric Vehicles

Abstract

Compared with rule-based and optimization energy management strategies, online optimal energy management control strategies for a dual-mode power-split hybrid electric vehicles (PSHEVs) are able to achieve better fuel economy and real-time performance. Global online optimization of a finite time domain energy management strategy based on the hybrid model predictive control (HMPC) algorithm is proposed in this study. To reduce the computing time, a linearized predictive model is built; because dual-mode PSHEVs can be considered hybrid systems that include continuous and discrete states, the hybrid states can be expressed uniformly. Therefore, a mixed logical dynamic (MLD) predictive model is built based on hybrid system theory, and an HMPC energy management strategy is proposed based on the MLD predictive model. To solve the optimal control problem online to obtain the optimal control sequence, the optimal control problem is converted into a mixed-integer linear programming (MILP) problem. The HMPC-based energy management strategy is compared with dynamic programming (DP)-based and rule-based energy management strategies over two different driving cycles. Simulation results indicate that the HMPC-based EMS achieves 80.60% and 83.79% of the fuel economy performance obtained by the DP-based EMS. In comparison, the rule-based EMS only achieves 66.46% and 70.51% of the DP-based control performance. Therefore, the HMPC-based energy management strategy is favorable for real-time control while effectively improving fuel economy.

1. Introduction

Hybrid electric vehicles (HEVs) are gaining wider acceptance among the general public and governments, as their environmental benefits—characterized by lower exhaust gas emissions and better fuel economy—are now widely recognized. Currently, there are three types of HEVs available on the market: series, parallel, and power-split HEVs. A dual-mode power-split hybrid electric vehicle (HEV) typically comprises an internal combustion engine and two permanent magnet synchronous motors (PMSMs). Accordingly, the design of the energy management strategy (EMS) and the power allocation between the engine and motors are of great significance. An effective EMS can further reduce fuel consumption and mitigate exhaust emissions [1].

Rule-based and fuzzy logic EMSs are widely used for their reliability and simplicity [2,3]. A rule-based energy management strategy, in which the control rules are extracted from acknowledged optimal algorithms and their control parameters are optimized offline and corrected online for a HEV, is proposed in ref. [4]. The simulation results show that this strategy can achieve relatively close control performance to global optimal results. Extensive research has focused on optimization-based EMSs, with various intelligent methods successfully applied to their design [5,6], including neural networks [7], particle swarm optimization (PSO) [8], a genetic algorithm (GA) [9], a simulated annealing (SA) [10,11] algorithm, quadratic programming (QP) [12], and Pontryagin’s minimum principle (PMP) [13]. Among optimization-based EMSs, dynamic programming (DP) [14,15,16,17] acts as a global optimization benchmark for other strategies. In ref. [18], A two-step optimal EMS is proposed for a novel single-shaft series–parallel powertrain: the two motors are first equivalent to one, and then stochastic dynamic programming (SDP) optimizes the power split between the engine and the equivalent motor. In ref. [19], Wang proposed a two-level stochastic EMS for fixed-route HEVs, with the upper-level optimization solved by SDP. Simulation results indicate that after 24 fixed-route trips, the strategy consumes only 1.8% more energy than the global optimum. Ref. [20] introduces utility-factor-weighted fuel consumption as the objective for energy management.

Ref. [21] presents a novel hybrid algorithm for PHEV EMS optimization using historical data from a remote monitoring platform. This algorithm integrates GA and an enhanced ant colony algorithm (EACA) to optimize control parameters, effectively overcoming the low precision of GA and slow convergence of EACA. Ref. [22] proposes a predictive EMS for parallel HEVs via velocity prediction and RL, yet these optimization-based methods cannot support real-time control. In ref. [23], an ordered charging optimization strategy based on reinforcement learning algorithms (RL) is proposed to achieve intelligent dynamic power allocation, effectively balancing the peak–valley load difference and minimizing the charging cost for users to the greatest extent.

Many researchers have concentrated on the realization of the online application of optimization-based EMS, an equivalent consumption minimization strategy (ECMS) [24], and model predictive control (MPC) [25,26,27] as typical instantaneous optimization control strategies. Xiang [28,29] proposed a hierarchical NMPC strategy for dual-mode power-split HEVs. In ref. [30], a rule-based energy management strategy (EMS) for a dual-stack Fuel Cell Hybrid Electric Vehicle (FCHEV) is proposed. The proposed EMS effectively enhances fuel cell durability by minimizing current fluctuation amplitude and ensuring that the fuel cells operate within their high-efficiency range throughout most of the driving cycles. Taghavipour [31] introduced an eMPC-based energy management strategy for Toyota Prius plug-in hybrids. Their results show that eMPC achieves better fuel economy and real-time performance than conventional MPC. Liu [32] presented a real-time power management framework for PHEVs, integrating driving pattern prediction, recognition, offline optimization, online modeling, and real-time control. Li [33] proposed driving-behavior-aware stochastic model predictive control (SMPC) for plug-in hybrid electric buses. Different kinds of driving behaviors are obtained by Markov chain models, and the SMPC is modified with the accompanying ECMS to achieve good control performance. Drawing on the Neural Fitting Model and Regression Learner Model, ref. [34] proposes an experimental artificial intelligence (AI) model. Based on a decade of historical load and meteorological data for New York City, this model can be applied to optimize the local electric vehicle charging strategy. Zhang [35] formulated an improved adaptive equivalent consumption minimization strategy (IA-ECMS) based on identified driving behavior and predicted real-time traffic information. Elsewhere [36], a new robust model predictive control (RMPC) algorithm is proposed, which is suitable for EMS design for an HEV. A cooperative distributed stochastic model predictive control (CDSMPC) algorithm has been proposed [33] for systems with stochastic disturbances and constraints, which could also be used to handle the optimal control problem in an HEV.

As typical hybrid dynamic systems involving both continuous and discrete states, dual-mode power-split HEVs motivate this paper to propose four original contributions for improving fuel economy.

(1) A mixed logical dynamic predictive model is built based on hybrid system theory [37], which is a linearized predictive model that can reduce computing time. (2) Hybrid model predictive control is based on a mixed logical dynamic model, which improves the accuracy of control. (3) Energy management optimization is converted to mixed-integer linear programming, which ensures the feasibility of real-time control. (4) The proposed strategy is compared with DP and rule-based energy management. The comparison indicated that real-time control was achieved on the premise of no detriment to fuel economy.

The rest of this paper is organized as follows. Section 2 presents the configuration and nonlinear model of the dual-mode power-split HEV. Section 3 establishes the mixed logical dynamical prediction model for the studied powertrain. Section 4 designs the HMPC-based energy management strategy and formulates the optimal control problem as a mixed-integer linear programming (MILP) problem for efficiency of solution. Section 5 validates the proposed strategy through comparisons with DP-based and rule-based strategies. Finally, Section 6 concludes the paper.

2. Nonlinear Models of a Dual-Mode Power-Split HEV

In this section, the configuration, modes, and key parameters are introduced, and then the nonlinear models of a dual-mode power-split HEV are presented [38].

2.1. The Powertrain of a Dual-Mode Power-Split HEV

(1)

Configuration of the dual-mode power-split HEV

Here, the configuration of the dual-mode power-split HEV is as shown in Figure 1, and the power transmission system consists of an engine, two motors (MGA and MGB), a set of power batteries, and a power coupling machine. Three planetary gear mechanisms, a clutch, and a brake are included in the power coupling machine.

Based on the topology of the engine, motors, and control components under EVT1 and EVT2 modes, the simplified power flow diagrams with arrows and the corresponding lever analogy schematics for EVT1 and EVT2 modes are presented in Figure 1.

(2)

Modes of operation of the dual-mode power-split HEV

The dual-mode power-split HEV comprises input-split (EVT1) and compound-split (EVT2) modes, as listed in

Table 1. Modes of dual-mode power-split HEV operation. This table defines the operating modes for the HMPC energy management strategy.

Power-Split Mode	Engine	MGA	MGB	CL1	B1
EVT1	On	Generator	Motor	○	●
EVT2	On	Motor	Generator	●	○

Note: ● indicates engagement, ○ indicates disengagement.

. Each mode supports pure electric, engine-alone, hybrid traction, regenerative charging and braking modes via clutch CL1 and brake B1 actuation to satisfy vehicle power demand.

In the table, MGA stands for motor A, MGB stands for motor B; CL1 denotes the clutch, and B1 denotes the brake.

In EVT1 mode, CL1 is disengaged and B1 is engaged, leaving the PG2 and PG3 carriers unconnected for low-speed, high-torque operation. In EVT2 mode, CL1 is engaged and B1 is disengaged, coupling the two carriers for high-speed, low-torque conditions. Positive motor power indicates propulsion, while negative power denotes generation: MGA generates and MGB drives in EVT1, whereas MGB generates and MGA drives in EVT2.

(3)

Key parameters of the dual-mode power-split HEV

Table 2. Key parameters of the dual-mode power-split HEV. These are the parameters of the verification object for the HMPC energy management strategy.

Parameter	Value	Description
rw	0.388 m	Radius of wheels
Af	3.24 m2	Frontal area of the vehicle
f	0.015	Friction coefficient
Cd	0.5	Drag coefficient
if	4.24	Gear ratio of final drive
k1	2.13	PG1’s inherent parameter
k2	2.13	PG2’s inherent parameter
k3	3.13	PG3’s inherent parameter
m	8000 kg	Vehicle mass
Cmax	36 Ah	Battery capacity
Voc	360 V	Battery voltage
Pemax	120 kW	Engine rated power
Temax	600 Nm	Engine maximum torque
Pmmax	110 kW	Peak power of MGA and MGB
Pmrate	60 kW	Rate power of MGA and MGB
Tmmax	200 Nm	Peak torque of MGA and MGB

lists the main parameters [38] of the dual-mode power-split HEV, covering engine, motors, battery, power coupling device and vehicle characteristics.

2.2. Dual-Mode Power-Split HEV Dynamics

This section details the component models of the powertrain and the vehicle dynamics model.

In the process of constructing the powertrain model, the following reasonable assumptions are introduced to simplify the analysis while ensuring the accuracy of the energy management strategy:

(1)

The engine, motor, and battery are regarded as quasi-static models, ignoring fast dynamic transients such as thermal response and mechanical vibration.

(2)

The fuel consumption rate of the engine and the efficiency characteristics of the motor are obtained from steady-state experimental MAPs.

(3)

The battery is described by an equivalent circuit model, where the open-circuit voltage and internal resistance are functions of state of charge (SOC) only.

(4)

The effects of temperature, component aging, and signal delay are neglected to focus on the design and verification of the HMPC-based energy management strategy.

It should be noted that the engine and motor are modeled as quasi-static components, where transient dynamic responses (e.g., thermal dynamics, dynamic lag, and response delay during rapid torque/speed changes) are neglected. Although this simplification facilitates the design and implementation of the proposed HMPC energy management strategy, it may lead to slight discrepancies between the simulation results and real vehicle performance. Thus, the above simplifications and corresponding limitations should be acknowledged when evaluating the control effect.

Since the engine fuel consumption rate and motor efficiency can only be obtained via look-up tables, which is not conducive to the design of predictive model-based energy management strategies, linearization and discretization are applied to the engine map and motor map. This is done not only to facilitate the design of the energy management controller but also to minimize the computation time of the control strategy as much as possible.

2.2.1. Engine Model

Due to high computational cost, the engine model is simplified by only considering fuel consumption as a static function of engine speed and torque while neglecting transient effects and temperature variations, as expressed in the following equation. It should be noted that the engine model adopted in this paper is a diesel engine model. The engine fuel consumption map is shown in Figure 2.

$${\dot{m}}_e=f(n_e,T_e)$$

(1)

where m_e is the mass of fuel consumed by the engine (g), n_e is the engine speed (rpm), and T_e is the engine torque (N m). The unit of the engine fuel consumption rate is g/kWh. Therefore, the reference basis for the fuel mass is kWh.

Based on the engine universal characteristic data obtained from experimental tests, the engine fuel consumption curves at different rotational speeds are fitted into piecewise linear functions using the least squares method. The fitted engine fuel consumption rate curves and the experimentally measured engine fuel consumption rate curves are shown in the figure below. In this study, the number of engine speed segments is n = 15, with n_e(i) = 800, 900, …, 2200 r/min. In the figure, “Original” denotes the curve from raw experimental data, and “Approximated” denotes the curve plotted using the fitted formula based on the experimental data. It can be seen from the figure that the fitted curve can approximately represent the experimental data and is suitable for controller design.

To quantitatively evaluate the fitting accuracy of the engine fuel consumption map, the root-mean-square error (RMSE) and maximum deviation between the original and fitted data are calculated and listed in the Supplementary Materials. The fitting errors are sufficiently small, which indicates that the fitted map can well represent the real fuel consumption characteristics. Therefore, the influence of the fitting error on the accuracy of fuel economy evaluation is negligible.

2.2.2. Permanent Magnet Synchronous Motor Model

Ignoring dynamic response and electromagnetic and thermal effects, the efficiency of the permanent magnet synchronous motor (PMSM) is simplified as a static function of speed and torque, as given in the following equation. The PMSM efficiency map is illustrated in Figure 3.

$${\eta }_m=f(n_m,T_m)$$

(2)

where η_m is the PMSM efficiency, n_m is the PMSM speed (rpm), and T_m is the PMSM torque (N·m). The relationship between current and motor speed–torque fitted by linear functions is shown in the figure below. In the figure, “Original” represents the curve of raw experimental data, and “Approximated” represents the curve plotted using the formula fitted from the raw experimental data.

2.2.3. Power Battery Model

A simplified battery model is adopted in this study, where the SOC is estimated using the widely used ampere-hour method. The resistances and voltage variations are obtained via a look-up table, and the total open-circuit voltage is calculated by Equation (3) under the assumption of uniform cell behavior.

$$V_{bat}=V_{cell}\cdot N_{bat}$$

(3)

where V_bat is the total open-circuit voltage, V_cell denotes the open-circuit voltage of a single cell, and N_bat represents the total number of battery cells.

Battery current and SOC are respectively derived from Equations (4) and (5) [2]:

$$I_{bat}={\displaystyle \frac{V_{cell}-\sqrt{V_{cell}^2-4R_{bat}\left({\mathrm{P}}_{\mathrm{r}eqbat}/N_{bat}\right)}}{2R_{bat}}}$$

(4)

$$SOC\left(t\right)=SO{C}_{ini}-{\displaystyle {\int }_0^t\frac{I_{bat}\left(t\right)}{3600C_{ini}}}dt$$

(5)

where I_bat is the discharge current, R_bat is the equivalent series resistance, P_reqbat is the required battery power, C_ini is the rated capacity (Ah), SOC(t) is the state of charge at time t, and SOC_ini is the initial SOC.

The SOC-dependent battery resistance and open-circuit voltage are shown in Figure 4, which are derived from experimental test data, and the actual total output power of the battery P_bat is given by:

$$P_{bat}=V_{bat}\cdot I_{bat}$$

(6)

For the SOC estimation of the battery model, the Coulomb counting method combined with the equivalent circuit model is adopted in this study. To justify the rationality of this selection, a comparative analysis with three mainstream alternative SOC estimation approaches is conducted, as summarized below: 1. Extended Kalman Filter (EKF): This has high estimation accuracy for nonlinear battery systems but requires complex matrix operations, leading to a heavy computational burden that is not conducive to the real-time implementation of the HMPC energy management strategy. 2. Unscented Kalman Filter (UKF): This improves the estimation accuracy for strong nonlinear systems compared with EKF, but its computational complexity is further increased, which is inconsistent with the real-time requirement of HMPC. 3. Sliding Mode Observer (SMO): This has strong robustness to parameter perturbations but is sensitive to the design of sliding mode parameters; meanwhile, its estimation accuracy is slightly lower than the proposed method under the operating conditions of this study. Compared with the above alternative approaches, the selected Coulomb counting method combined with the equivalent circuit model achieves a balance between estimation accuracy and computational efficiency. It not only meets the SOC estimation accuracy requirement for the HMPC energy management strategy (error within ±2%) but also has low computational complexity, which ensures the real-time performance of the HMPC control algorithm. Thus, the selection of the SOC estimation method is reasonable and suitable for the research scenario of this study.

2.2.4. Power-Coupled Machine Model

According to the speed and torque constraints of the planetary mechanism, the input and output speed and torque of the engine, MGA, and MGB can be obtained in different EVT modes by ignoring the rotational inertia of the engine, MGA, and MGB.

EVT1 mode:

$$\left[\begin{array}{c}{\omega }_A\\ {\omega }_B\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{(1+k_1)(1+k_2)}{k_1{k}_2}}& -{\displaystyle \frac{(1+k_1+k_2)(1+k_3)}{k_1{k}_2}}\\ 0& 1+k_3\end{array}\right]\left[\begin{array}{c}{\omega }_i\\ {\omega }_o\end{array}\right]$$

(7)

EVT2 mode:

$$\left[\begin{array}{c}{\omega }_A\\ {\omega }_B\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{1+k_2}{k_1}}& {\displaystyle \frac{1+k_1+k_2}{k_1}}\\ 1+k_2& -k_2\end{array}\right]\left[\begin{array}{c}{\omega }_i\\ {\omega }_o\end{array}\right]$$

(8)

where k_i is the planetary gear tooth ratio, ω_A and ω_B are the speeds of MGA and MGB, ω_i = ω_e/i_q is the input speed, ω_e is the engine speed, i_q is the front ratio, V is the vehicle velocity, r_w is the wheel radius, i_f is the rear ratio and ω_o = V i_f/r_w is the output speed.

Under steady-state conditions, ignoring gear friction and assuming rigid shafts, the torque relationship among the engine, MGA, and MGB is given as follows:

EVT1 mode:

$$\left[\begin{array}{l}{T}_A\\ T_B\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{k_1{k}_2}{(1+k_1)(1+k_2)}}& 0\\ -{\displaystyle \frac{1+k_1+k_2}{(1+k_1)(1+k_2)}}& {\displaystyle \frac{1}{1+k_3}}\end{array}\right]\left[\begin{array}{l}{T}_i\\ T_o\end{array}\right]$$

(9)

EVT2 mode:

$$\left[\begin{array}{l}{T}_A\\ T_B\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{k_1{k}_2}{(1+k_1)(1+k_2)}}& {\displaystyle \frac{k_1}{1+k_1}}\\ -{\displaystyle \frac{1+k_1+k_2}{(1+k_1)(1+k_2)}}& {\displaystyle \frac{1}{1+k_1}}\end{array}\right]\left[\begin{array}{l}{T}_i\\ T_o\end{array}\right]$$

(10)

where T_A is the torque of MGA, T_B is the torque of MGB, T_i = T_ei_q is the input torque of the power coupling mechanism, T_e is the engine output torque, T_o = T_w/i_f is the output torque, and T_w is the wheel torque.

Considering the inertias of the engine, MGA, and MGB under dynamic conditions while neglecting planetary gear inertia, the dynamic characteristics of each component are given as follows:

EVT1 mode:

$$\left[\begin{array}{ccc}-{\displaystyle \frac{J_e}{i_q}}& -{\displaystyle \frac{(1+k_1)(1+k_2)}{k_1{k}_2}}{J}_A& 0\\ 0& -{\displaystyle \frac{\left(1+k_1+k_2\right)\left(1+k_3\right)}{k_1{k}_2}}{J}_A& \left(1+k_3\right)J_B\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_e\\ {\dot{\omega }}_A\\ {\dot{\omega }}_B\end{array}\right]=\left[\begin{array}{cccc}-1& -{\displaystyle \frac{(1+k_1)(1+k_2)}{k_1{k}_2}}& 0& 0\\ 0& -{\displaystyle \frac{\left(1+k_1+k_2\right)\left(1+k_3\right)}{k_1{k}_2}}& 1+k_3& -1\end{array}\right]\left[\begin{array}{c}{T}_e\\ T_A\\ T_B\\ T_o\end{array}\right]$$

(11)

EVT2 mode:

$$\left[\begin{array}{ccc}-{\displaystyle \frac{J_e}{i_q}}& {\displaystyle \frac{1+k_2}{k_1}}{J}_A& -\left(1+k_2\right)J_B\\ 0& {\displaystyle \frac{1+k_1+k_2}{k_1}}{J}_A& -k_2{J}_B\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_e\\ {\dot{\omega }}_A\\ {\dot{\omega }}_B\end{array}\right]=\left[\begin{array}{cccc}-1& {\displaystyle \frac{1+k_2}{k_1}}& -1-k_2& 0\\ 0& {\displaystyle \frac{1+k_1+k_2}{k_1}}& -k_2& -1\end{array}\right]\left[\begin{array}{c}{T}_e\\ T_A\\ T_B\\ T_o\end{array}\right]$$

(12)

For model simplification, the inertial effects of MGA and MGB are neglected, so the dynamic model of the electromechanical compound drive system can be simplified as:

EVT1 mode:

$${\dot{\omega }}_e=\left[\begin{array}{ccc}{\displaystyle \frac{i_q}{J_e}}& {\displaystyle \frac{\left(1+k_1\right)\left(1+k_2\right)}{k_1{k}_2}}{\displaystyle \frac{i_q}{J_e}}& 0\end{array}\right]\left[\begin{array}{l}{T}_e\\ T_A\\ T_B\end{array}\right]$$

(13)

EVT2 mode:

$${\dot{\omega }}_e=\left[\begin{array}{ccc}{\displaystyle \frac{i_q}{J_e}}& -{\displaystyle \frac{1+k_2}{k_1}}{\displaystyle \frac{i_q}{J_e}}& {\displaystyle \frac{\left(1+k_2\right)i_q}{J_e}}\end{array}\right]\left[\begin{array}{l}{T}_e\\ T_A\\ T_B\end{array}\right]$$

(14)

2.2.5. Vehicle Dynamic Model

This study considers longitudinal vehicle dynamics. Based on the force balance, the traction force equals the sum of the rolling resistance, aerodynamic drag, gradient resistance, and acceleration resistance. The longitudinal dynamic model is expressed as:

$$\left\{\begin{array}{l}m\dot{V}={\displaystyle \frac{T_w-T_b}{r_w}}-F_{load}\\ F_{load}=fmg\cos \alpha +mg\sin \alpha +{\displaystyle \frac{1}{2}}{C}_d{A}_f\rho V^2\end{array}\right.$$

(15)

where m is the vehicle mass (kg), V is the vehicle velocity (m/s), T_w is the wheel torque (Nm), F_b is the braking force (Nm, positive), ρ is the air density (1.2258 Ns² m⁻⁴), C_d is the drag coefficient, A_f is the frontal area (m²), f is the rolling resistance coefficient, g is gravitational acceleration (9.8 ms⁻²), and α is the road grade angle.

Based on the above derivations, the system’s dynamic models employed for model predictive control in the dual-mode power-split HEV are presented below. For EVT1 mode, the equation can be derived from Equation (9):

$$\left\{\begin{array}{l}{T}_A=-{\displaystyle \frac{k_1{k}_2}{(1+k_1)(1+k_2)}}{i}_q{T}_e\\ T_B=-{\displaystyle \frac{1+k_1+k_2}{(1+k_1)(1+k_2)}}{i}_q{T}_e+{\displaystyle \frac{1}{1+k_3}}{\displaystyle \frac{T_w}{i_f}}\end{array}\right.$$

(16)

The wheel torque is obtained from Equation (16) and may be given by:

$$T_w=-{\displaystyle \frac{\left(1+k_1+k_2\right)(1+k_3)}{k_1{k}_2}}{i}_f{T}_A+\left(1+k_3\right)i_f{T}_B$$

(17)

The velocity state equation can be obtained by substituting Equation (17) into Equation (15):

$$\dot{V}=-{\displaystyle \frac{\left(1+k_1+k_2\right)\left(1+k_3\right)}{k_1{k}_2}}{\displaystyle \frac{i_f}{r_{w}m}}{T}_A+{\displaystyle \frac{\left(1+k_3\right)i_f}{r_{w}m}}{T}_B-{\displaystyle \frac{1}{r_{w}m}}{T}_b-{\displaystyle \frac{1}{m}}{F}_{load}$$

(18)

From Equations (13) and (18), the engine speed and velocity update equation in EVT1 mode is obtained as follows:

$$\left[\begin{array}{l}{\dot{\omega }}_e\\ \dot{V}\end{array}\right]=\left[\begin{array}{cccc}{\displaystyle \frac{i_q}{J_e}}& {\displaystyle \frac{(1+k_1)(1+k_2)}{k_1{k}_2}}{\displaystyle \frac{i_q}{J_e}}& 0& 0\\ 0& -{\displaystyle \frac{\left(1+k_1+k_2\right)\left(1+k_3\right)}{k_1{k}_2}}{\displaystyle \frac{i_f}{r_{w}m}}& {\displaystyle \frac{\left(1+k_3\right)i_f}{r_{w}m}}& -{\displaystyle \frac{1}{r_{w}m}}\end{array}\right]\left[\begin{array}{c}{T}_e\\ T_A\\ T_B\\ T_b\end{array}\right]+\left[\begin{array}{c}0\\ -{\displaystyle \frac{1}{m}}\end{array}\right]F_{load}$$

(19)

In EVT2 mode, the following equation can be obtained by way of Equation (10):

$$\left\{\begin{array}{l}{T}_A=-{\displaystyle \frac{k_1{k}_2}{(1+k_1)(1+k_2)}}{i}_q{T}_e+{\displaystyle \frac{k_1}{1+k_1}}{\displaystyle \frac{T_w}{i_f}}\\ T_B=-{\displaystyle \frac{1+k_1+k_2}{(1+k_1)(1+k_2)}}{i}_q{T}_e+{\displaystyle \frac{1}{1+k_1}}{\displaystyle \frac{T_w}{i_f}}\end{array}\right.$$

(20)

The wheel torque function about MGA and MGB is obtained from Equation (20) and may be expressed as:

$$T_w={\displaystyle \frac{1+k_1+k_2}{k_1}}{i}_f{T}_A-k_2{i}_f{T}_B$$

(21)

The velocity state equation can be obtained by substituting Equation (21) into Equation (15):

$$\dot{V}={\displaystyle \frac{1+k_1+k_2}{k_1}}{\displaystyle \frac{i_f}{r_{w}m}}{T}_A-{\displaystyle \frac{k_2{i}_f}{r_{w}m}}{T}_B-{\displaystyle \frac{1}{r_{w}m}}{T}_b-{\displaystyle \frac{1}{m}}{F}_{load}$$

(22)

From Equations (14) and (22), the engine speed and velocity update equation in EVT2 mode is obtained as follows:

$$\left[\begin{array}{l}{\dot{\omega }}_e\\ \dot{V}\end{array}\right]=\left[\begin{array}{cccc}{\displaystyle \frac{i_q}{J_e}}& -{\displaystyle \frac{1+k_2}{k_1}}{\displaystyle \frac{i_q}{J_e}}& {\displaystyle \frac{\left(1+k_2\right)i_q}{J_e}}& 0\\ 0& {\displaystyle \frac{1+k_1+k_2}{k_1}}{\displaystyle \frac{i_f}{r_{w}m}}& -k_2{\displaystyle \frac{i_f}{r_{w}m}}& -{\displaystyle \frac{1}{r_{w}m}}\end{array}\right]\left[\begin{array}{c}{T}_e\\ T_A\\ T_B\\ T_b\end{array}\right]+\left[\begin{array}{c}0\\ -{\displaystyle \frac{1}{m}}\end{array}\right]F_{load}$$

(23)

3. Prediction Models for a Dual-Mode Power-Split HEV

The engine fuel consumption rate map and motor efficiency map are usually used to design the energy management strategy of HEVs. The specific fuel consumption is obtained by way of a look-up table because there is no explicit analytical expression for the engine fuel consumption rate map, which is detrimental to the design of an energy management strategy based on the MLD predictive model. Therefore, approximate linearization models of the engine fuel consumption map graph model and the motor efficiency map graph model are built. In the literature [39], the linearization method is introduced. The battery SOC is closely related to the charging and discharging current and capacity of the battery, and the battery current is a nonlinear function of motor speed and torque. Thus, the nonlinear relationship between battery current, motor speed, and torque is linearized.

3.1. Equivalent Model Between Electric Energy and Fuel

To build an optimisation objective function in each mode for a dual-mode power-split HEV, electricity consumption is converted to its equivalent fuel consumption [40]:

$$V_{fuel}={\displaystyle \frac{E_k\cdot 3600}{D_{fuel}\cdot Q_{fuel\_low}\cdot {\eta }_{eng}\cdot {\eta }_{gen}}}$$

(24)

where E_k represents the electricity consumption (kWh), D_fuel represents the fuel density (g/cm³), where the density of diesel is 0.85 g/cm³, Q_{fuel_low} represents the calorific value, which for diesel is 43,000 J/g, η_eng represents the average working efficiency of the engine (35%), η_gen denotes the average working efficiency of the motors, and V_fuel represents their equivalent fuel consumption (L).

The electrical energy in Equation (24) can be expressed as a function of battery voltage and current, as shown below:

$$E_k=V_{oc}\cdot I\left(k\right)\cdot T_s$$

(25)

The motor efficiency can be ignored when calculating equivalent fuel consumption because the motor efficiency has been considered in calculating the electric current. In the sampling time, the equivalent fuel consumption model between electricity and fuel is as follows:

$$m_{equ}={\displaystyle \frac{V_{oc}\cdot I\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}$$

(26)

where m_equ represents the equivalent fuel consumption (g), V_oc represents the bus voltage (V), and I represents the battery charging or discharging current (A).

3.2. State and Output Variable Update Models

The equivalent fuel consumption increment and battery SOC increment in each mode are shown in Equations (27) and (28) [41].

(1)

EVT1 mode:

$$\left\{\begin{array}{l}\mathrm{\Delta }{m}_{f1}\left(k\right)=\left(\alpha (n_e)T_e+\beta (n_e)\right)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}\\ \mathrm{\Delta }SO{C}_1\left(k\right)=-{\displaystyle \frac{d_0\left(n_A\right)+d_1\left(n_A\right)\cdot T_A}{C_{\max }}}\cdot T_s-{\displaystyle \frac{a_0\left(n_B\right)+a_1\left(n_B\right)\cdot T_B}{C_{\max }}}\cdot T_s\end{array}\right.$$

(27)

(2)

EVT2 mode:

$$\left\{\begin{array}{l}\mathrm{\Delta }{m}_{f2}\left(k\right)=\left(\alpha (n_e)T_e+\beta (n_e)\right)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}\\ \mathrm{\Delta }SO{C}_2\left(k\right)=-{\displaystyle \frac{d_0\left(n_B\right)+d_1\left(n_B\right)\cdot T_B}{C_{\max }}}\cdot T_s-{\displaystyle \frac{a_0\left(n_{\mathrm{A}}\right)+a_1\left(n_A\right)\cdot T_A}{C_{\max }}}\cdot T_s\end{array}\right.$$

(28)

Logic state variables δ₁ and δ₂ are defined to denote the EVT1 and EVT2 modes, respectively. The equivalent fuel consumption increment and battery SOC increment for the dual-mode power-split HEV at time step k are given by:

$$\left\{\begin{array}{c}\mathrm{\Delta }{m}_f\left(k\right)={\delta }_1\left(k\right)\mathrm{\Delta }{m}_{f1}\left(k\right)+{\delta }_2\left(k\right)\mathrm{\Delta }{m}_{f2}\left(k\right)\\ \mathrm{\Delta }SOC\left(k\right)={\delta }_1\left(k\right)\mathrm{\Delta }SO{C}_1\left(k\right)+{\delta }_2\left(k\right)\mathrm{\Delta }SO{C}_2\left(k\right)\end{array}\right.$$

(29)

At time step k, only one mode is active, and the logic state variables satisfy:

$${\delta }_1\left(k\right)+{\delta }_2\left(k\right)=1$$

(30)

where δ_i denotes the logic state variable, which takes a value of either 0 or 1.

Here, battery SOC is the state variable and equivalent fuel consumption is the output variable. Their update equations are as follows:

$$\left\{\begin{array}{l}{m}_f\left(k+1\right)=m_f\left(k\right)+\mathrm{\Delta }{m}_f\left(f\right)\\ SOC\left(k+1\right)=SOC\left(k\right)+\mathrm{\Delta }SOC\left(k\right)\end{array}\right.$$

(31)

where m_f(k) and SOC(k) represent the increments of fuel consumption and SOC from time k to k + 1, respectively.

The state and output variables are replaced by x and y, respectively, and Equation (31) can be converted into the standard form of the MLD model, as shown below:

$$\left\{\begin{array}{c}{x}_{k+1}=A{x}_k+B_1{u}_k+B_2{\delta }_k+B_3{z}_k+B_5\\ y_k=C{x}_k+D_1{u}_k+D_2{\delta }_k+D_3{z}_k+D_5\end{array}\right.$$

(32)

where x = SOC and u_k = [T_e,T_A,T_B]′.

The product term of control variables and logic variables affects Equation (29): to eliminate this kind of product, other logic variables z_i (i = 1, 2) are used, as follows:

$${\left[\begin{array}{ccc}{z}_{1i}\left(k\right)& z_{2i}\left(k\right)& z_{3i}\left(k\right)\end{array}\right]}^T={\delta }_i\left(k\right)\cdot {\left[\begin{array}{ccc}{T}_e\left(k\right)& T_A\left(k\right)& T_B\left(k\right)\end{array}\right]}^T, i=1,2$$

(33)

Substituting Equations (27) and (28) into Equation (31), the state equation and output equation of the MLD model are as given by Equations (34) and (35), respectively:

$$\begin{array}{l}y\left(k+1\right)=y\left(k\right)+\left[\begin{array}{cccccc}\alpha (n_e)\cdot T_s& \alpha (n_e)\cdot T_s& 0& 0& 0& 0\end{array}\right]\cdot {\left[\begin{array}{cccccc}{z}_{11}& z_{12}& z_{21}& z_{22}& z_{31}& z_{32}\end{array}\right]}^T+\\ \left[\begin{array}{cc}\beta (n_e)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}& \beta (n_e)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}\end{array}\right]\cdot {\left[\begin{array}{cc}{\delta }_1& {\delta }_2\end{array}\right]}^T\end{array}$$

(34)

$$\begin{array}{l}x\left(k+1\right)=x\left(k\right)+\left[\begin{array}{cccccc}0& 0& -{\displaystyle \frac{d_1\left(n_A\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{a_1\left(n_A\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{a_1\left(n_B\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{d_1\left(n_B\right)\cdot T_s}{C_{\max }}}\end{array}\right]\cdot \\ {\left[\begin{array}{cccccc}{z}_{11}& z_{12}& z_{21}& z_{22}& z_{31}& z_{32}\end{array}\right]}^T+\left[\begin{array}{cc}-\left({\displaystyle \frac{d_0\left(n_A\right)\cdot T_s}{C_{\max }}}+{\displaystyle \frac{a_0\left(n_B\right)\cdot T_s}{C_{\max }}}\right)& -\left({\displaystyle \frac{d_0\left(n_B\right)\cdot T_s}{C_{\max }}}+{\displaystyle \frac{a_0\left(n_{\mathrm{A}}\right)\cdot T_s}{C_{\max }}}\right)\end{array}\right]\cdot {\left[\begin{array}{cc}{\delta }_1& {\delta }_2\end{array}\right]}^T\end{array}$$

(35)

Based on hybrid system theory, the inequality constraints are necessary when the auxiliary logic variables are used in the state and output equations:

$$\left\{\begin{array}{l}-M_e\cdot {\delta }_i\left(k\right)+z_{1i}\left(k\right)\le 0\\ m_e\cdot {\delta }_i\left(k\right)-z_{1i}\left(k\right)\le 0\\ -m_e{\delta }_i\left(k\right)+z_{1i}\left(k\right)\le T_e\left(k\right)-m_e\\ M_e{\delta }_i\left(k\right)-z_{1i}\left(k\right)\le -T_e\left(k\right)+M_e\\ -M_A\cdot {\delta }_i\left(k\right)+z_{2i}\left(k\right)\le 0\\ m_A\cdot {\delta }_i\left(k\right)-z_{2i}\left(k\right)\le 0\\ -m_A{\delta }_i\left(k\right)+z_{2i}\left(k\right)\le T_A\left(k\right)-m_A\\ M_A{\delta }_i\left(k\right)-z_{2i}\left(k\right)\le -T_A\left(k\right)+M_A\\ -M_B\cdot {\delta }_i\left(k\right)+z_{3i}\left(k\right)\le 0\\ m_B\cdot {\delta }_i\left(k\right)-z_{3i}\left(k\right)\le 0\\ -m_B{\delta }_i\left(k\right)+z_{3i}\left(k\right)\le T_B\left(k\right)-m_B\\ M_B{\delta }_i\left(k\right)-z_{3i}\left(k\right)\le -T_B\left(k\right)+M_B\end{array}\right.$$

(36)

where M_e, M_A, M_B, m_e, m_A, and m_B are defined below:

$$\begin{array}{l}{M}_e=\underset{k=1,\dots ,N}{\max }{T}_e\left(k\right),m_e=\underset{k=1,\dots ,N}{\min }{T}_e\left(k\right),M_A=\underset{k=1,\dots ,N}{\max }{T}_A\left(k\right)\\ m_A=\underset{k=1,\dots ,N}{\min }{T}_A\left(k\right),M_B=\underset{k=1,\dots ,N}{\max }{T}_B\left(k\right),m_B=\underset{k=1,\dots ,N}{\min }{T}_B\left(k\right)\end{array}$$

(37)

3.3. Propositional Calculus and Linear Integer Programming

In this section, modes EVT1 and EVT2 are determined by state variables and control variables, and the transformation between the propositional logic and mixed-integer inequality is finished using the equivalence relation rules introduced in the literature [42].

(1)

EVT1 mode:

$$\left[{\delta }_1=1\right]\leftrightarrow \left[SO{C}_{low}\le SOC\le SO{C}_{high}\right]\wedge \left[0\le T_e\le T_{e\_\max }\right]\wedge \left[-T_{A\_\max }\le T_A\le 0\right]\wedge \left[0\le T_B\le T_{B\_\max }\right]$$

(38)

According to the theory of hybrid systems, some inequality constraints are necessary:

$$\left\{\begin{array}{l}{M}_{soc}{\delta }_1\le SOC\left(k\right)-SO{C}_{low}+M_{soc}\\ M_{soc}{\delta }_1\le -SOC\left(k\right)+SO{C}_{high}+M_{soc}\\ M_e{\delta }_1\le T_e\left(k\right)+M_e\\ M_e{\delta }_1\le -T_e\left(k\right)+T_{e\_\max }\left(k\right)+M_e\\ M_A{\delta }_1\le T_A\left(k\right)+T_{A\_\max }\left(k\right)+M_A\\ M_A{\delta }_1\le -T_A\left(k\right)+M_A\\ M_B{\delta }_1\le T_B\left(k\right)+M_B\\ M_B{\delta }_1\le -T_B\left(k\right)+T_{B\_\max }\left(k\right)+M_B\end{array}\right.$$

(39)

(2)

EVT2 mode:

$$\left[{\delta }_2=1\right]\leftrightarrow \left[SO{C}_{low}\le SOC\le SO{C}_{high}\right]\wedge \left[0\le T_e\le T_{e\_\max }\right]\wedge \left[0\le T_A\le T_{A\_\max }\right]\wedge \left[-T_{B\_\max }\le T_B\le 0\right]$$

(40)

The inequality constraints are:

$$\left\{\begin{array}{l}{M}_{soc}{\delta }_2\le SOC\left(k\right)-SO{C}_{low}+M_{soc}\\ M_{soc}{\delta }_2\le -SOC\left(k\right)+SO{C}_{high}+M_{soc}\\ M_e{\delta }_2\le T_e\left(k\right)+M_e\\ M_e{\delta }_2\le -T_e\left(k\right)+T_{e\_\max }\left(k\right)+M_e\\ M_A{\delta }_2\le T_A\left(k\right)+M_A\\ M_A{\delta }_2\le -T_A\left(k\right)+T_{A\_\max }\left(k\right)+M_A\\ M_B{\delta }_2\le T_B\left(k\right)+T_{B\_\max }\left(k\right)+M_B\\ M_B{\delta }_2\le -T_B\left(k\right)+M_B\end{array}\right.$$

(41)

3.4. MLD Prediction Model of a Dual-Mode Power-Split HEV

The MLD prediction model for the dual-mode power-split HEV is expressed as:

$$\left\{\begin{array}{c}\hspace{1em}x\left(k+1\right)=Ax\left(k\right)+B_{1}u\left(k\right)+B_2\delta \left(k\right)+B_{3}z\left(k\right)+B_5\hspace{1em}\hspace{1em}\hspace{1em}\left(a\right)\\ \hspace{1em}\hspace{1em}y\left(k\right)=Cx\left(k\right)+D_{1}u\left(k\right)+D_2\delta \left(k\right)+D_{3}z\left(k\right)+D_5\hspace{1em}\hspace{1em} \left(b\right)\\ \left[\begin{array}{l}{E}_{21}\\ E_{22}\end{array}\right]\delta \left(k\right)+\left[\begin{array}{l}{E}_{31}\\ E_{32}\end{array}\right]z\left(k\right)\le \left[\begin{array}{l}{E}_{41}\\ E_{42}\end{array}\right]x\left(k\right)+\left[\begin{array}{l}{E}_{11}\\ E_{12}\end{array}\right]u\left(k\right)+\left[\begin{array}{l}{E}_{51}\\ E_{52}\end{array}\right]\hspace{1em}\left(c\right)\end{array}\right.$$

(42)

where $x=[SOC]\in {\Re }^{n_r}\times {\{0,1\}}^{n_b}$ is a vector of continuous and binary states, $u=\begin{array}{ccc}[T_e& T_A& T_B]\end{array}\in {\Re }^{m_r}\times {\{0,1\}}^{m_b}$ are the inputs, $y\in {\Re }^{p_r}\times {\{0,1\}}^{p_b}$ are the outputs, $\delta =\left[\begin{array}{cc}{\delta }_1& {\delta }_2\end{array}\right]\in {\left\{0,1\right\}}^{r_b}$, $z=\begin{array}{cccccc}[z_{11}& z_{12}& z_{21}& z_{22}& z_{31}& z_{32}]\end{array}\in {\Re }^{r_r}$ represent auxiliary binary and continuous variables, respectively, and A, B₁, B₂, B₃, C, D₁, and D₂ are the coefficient matrices of the state, and output, equations, where E₁₁, E₁₂, E₂₁, E₂₂, E₃₁, E₃₂, E₄₁, E₄₂, E₅₁, and E₅₂ are the coefficient matrices of inequality constraints, which are shown below. Given the current state x(k) and input u(k), the time evolution of Equation (42) is determined by solving δ(k) and z(k) from Equation (42c) and then updating x(k + 1) and y(k) from Equation (42a,b).

$$\begin{array}{l}A=1, B_1=0, B_5=0, C=0, D_1=0, D_5=0\\ B_2=\left[\begin{array}{cc}-\left({\displaystyle \frac{d_0\left(n_A\right)\cdot T_s}{C_{\max }}}+{\displaystyle \frac{a_0\left(n_B\right)\cdot T_s}{C_{\max }}}\right)& -\left({\displaystyle \frac{d_0\left(n_B\right)\cdot T_s}{C_{\max }}}+{\displaystyle \frac{a_0\left(n_{\mathrm{A}}\right)\cdot T_s}{C_{\max }}}\right)\end{array}\right]\\ B_3=\left[\begin{array}{cccccc}0& 0& -{\displaystyle \frac{d_1\left(n_A\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{a_1\left(n_A\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{a_1\left(n_B\right)\cdot T_s}{C_{\max }}}& -{\displaystyle \frac{d_1\left(n_B\right)\cdot T_s}{C_{\max }}}\end{array}\right]\\ D_2=\left[\begin{array}{cc}\beta (n_e)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}& \beta (n_e)\cdot T_s+{\displaystyle \frac{V_{oc}\cdot IB\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}+{\displaystyle \frac{V_{oc}\cdot IA\left(k\right)\cdot T_s}{Q_{fuel\_low}\cdot {\eta }_{eng}}}\end{array}\right]\\ D_3=\left[\begin{array}{cccccc}\alpha (n_e)\cdot T_s& \alpha (n_e)\cdot T_s& 0& 0& 0& 0\end{array}\right]\end{array}$$

(43)

$$E_{31}=0, E_{21}=\left[\begin{array}{cc}{M}_{SOC}& 0\\ M_{SOC}& 0\\ M_e& 0\\ M_e& 0\\ M_A& 0\\ M_A& 0\\ M_B& 0\\ M_B& 0\\ 0& M_{SOC}\\ 0& M_{SOC}\\ 0& M_e\\ 0& M_e\\ 0& M_A\\ 0& M_A\\ 0& M_B\\ 0& M_B\end{array}\right], E_{11}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& -1& 0\\ 0& 0& 1\\ 0& 0& -1\\ 0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ -1& 0& 0\\ 0& 1& 0\\ 0& -1& 0\\ 0& 0& 1\\ 0& 0& -1\end{array}\right], E_{41}=\left[\begin{array}{c}1\\ -1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\\ -1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right], E_{51}=\left[\begin{array}{c}-SO{C}_{low}+M_{SOC}\\ SO{C}_{high}+M_{SOC}\\ M_e\\ T_{e\_\max }\left(k\right)+M_e\\ T_{A\_\max }\left(k\right)+M_A\\ M_A\\ M_B\\ T_{B\_\max }\left(k\right)+M_B\\ -SO{C}_{low}+M_{SOC}\\ SO{C}_{high}+M_{SOC}\\ M_e\\ T_{e\_\max }\left(k\right)+M_e\\ M_A\\ T_{A\_\max }\left(k\right)+M_A\\ T_{B\_\max }\left(k\right)+M_B\\ M_B\end{array}\right]$$

(44)

$$E_{32}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& -1\end{array}\right], E_{22}=\left[\begin{array}{cc}-M_e& 0\\ 0& -M_e\\ m_e& 0\\ 0& m_e\\ -m_e& 0\\ 0& -m_e\\ M_e& 0\\ 0& M_e\\ -M_A& 0\\ 0& -M_A\\ m_A& 0\\ 0& m_A\\ -m_A& 0\\ 0& -m_A\\ M_A& 0\\ 0& M_A\\ -M_B& 0\\ 0& -M_B\\ m_B& 0\\ 0& m_B\\ -m_B& 0\\ 0& -m_B\\ M_B& 0\\ 0& M_B\end{array}\right], E_{12}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ 1& 0& 0\\ -1& 0& 0\\ -1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\\ 0& 1& 0\\ 0& 1& 0\\ 0& -1& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\\ 0& 0& 1\\ 0& 0& -1\\ 0& 0& -1\end{array}\right], E_{42}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right], E_{52}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ -m_e\\ -m_e\\ M_e\\ M_e\\ 0\\ 0\\ 0\\ 0\\ -m_A\\ -m_A\\ M_A\\ M_A\\ 0\\ 0\\ 0\\ 0\\ -m_B\\ -m_B\\ M_B\\ M_B\end{array}\right]$$

(45)

4. Energy Management Strategy for Dual-Mode Power-Split HEV

In this section, the energy management strategy framework based on a hybrid model predictive control algorithm for a dual-mode power-split HEV is presented, and then the energy management optimal control problem is built using an MLD prediction model. Finally, to obtain the optimal control sequence and apply the energy management strategy online, the energy management optimal control problem is converted to a mixed-integer linear programming problem for ease of solution.

4.1. The Framework of the HMPC-Based EMS

In this section, the framework of the energy management strategy based on a hybrid model predictive control algorithm for the dual-mode power-split HEV is introduced, as shown in Figure 5. The energy management strategy framework includes two parts: offline design and online application. The offline design process is used to obtain the MLD predictive model of the dual-mode power-split HEV by means of the conversion between propositional calculus and linear integer programming based on hybrid system theory. To increase the computation speed of the controller, the nonlinear system model of the dual-mode power-split HEV is linearized before building the MLD predictive model. In the online application part, the energy management strategy based on the MLD predictive model is applied in the dual-mode power-split HEV in real time. The output variables from the controller to the dual-mode power-split HEV are engine torque, MGA torque, MGB torque, and brake torque, while the feedback variables from the dual-mode power-split HEV to the controller are battery SOC, velocity, and engine speed. The hard constraints of the dual-mode power-split HEV must also be included in the energy management strategy. The linear predictive model and mixed-integer linear programming solution algorithm for this energy management optimal control problem ensure that the control strategy can be used in real time.

The proposed HMPC formulates the hybrid powertrain as a mixed logical dynamical (MLD) system. By using mixed-integer linear programming (MILP), it simultaneously handles continuous physical constraints (battery SOC, power/torque limits, state dynamics) and discrete logical constraints (engine on/off, mode exclusivity, switching frequency constraints), achieving unified representation and explicit optimization of both hard and soft constraints. This ensures that all control actions strictly satisfy the system safety boundaries.

To adapt to typical driving cycles, including urban, highway, climbing, and braking conditions, the HMPC achieves condition adaptation via receding horizon optimization and adaptive adjustment of constraint weights and objective functions:

• Urban conditions: Strengthen constraints on regenerative braking and limit frequent engine start–stop to improve economy and ride comfort.
• Highway conditions: Enforce tighter constraints on the high-efficiency region of the engine and SOC maintenance to enhance fuel economy.
• Hill-climbing/high-load conditions: Prioritize hard constraints on power demand and dynamically allow short-term power margins while avoiding over-limit operation of the engine and battery.
• Downhill/coasting conditions: Focus on energy recovery and strictly enforce the upper SOC limit to prevent over-charging.

4.2. HMPC-Based Optimal Control Problem

Based on the above MLD prediction model, the optimal energy management control problem is constructed as:

$$\begin{array}{l}\underset{{\left\{v,\delta ,z\right\}}_0^{T-1}}{\min }J\left({\left\{v,\delta ,z\right\}}_0^{T-1},x\left(t\right)\right)={\displaystyle \sum _{k=0}^{T-1}{‖Q_1\left(v\left(k\right)-u_e\right)‖}_{\infty }}+{‖Q_2\left(x\left(k|t\right)-x_e\right)‖}_{\infty }+{‖Q_3\left(y\left(k|t\right)-y_e\right)‖}_{\infty }\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}x\left(t|t\right)=x\left(t\right)\\ x\left(k+1|t\right)=Ax\left(k|t\right)+B_{1}v\left(k\right)+B_2\delta \left(k|t\right)+B_{3}z\left(k|t\right)+B_5\\ y\left(k|t\right)=Cx\left(k|t\right)+D_{1}v\left(k\right)+D_2\delta \left(k|t\right)+D_{3}z\left(k|t\right)+D_5\\ \left[\begin{array}{l}{E}_{21}\\ E_{22}\end{array}\right]\delta \left(k|t\right)+\left[\begin{array}{l}{E}_{31}\\ E_{32}\end{array}\right]z\left(k|t\right)\le \left[\begin{array}{l}{E}_{41}\\ E_{42}\end{array}\right]x\left(k|t\right)+\left[\begin{array}{l}{E}_{11}\\ E_{12}\end{array}\right]v\left(k\right)+\left[\begin{array}{l}{E}_{51}\\ E_{52}\end{array}\right]\\ u_{\min }\le v\left(t+k\right)\le u_{\max },k=0,1,\dots ,T-1\\ x_{\min }\le x\left(t+k|t\right)\le x_{\max },k=0,1,\dots ,T-1\\ x\left(T|t\right)=x_e\end{array}\right.\end{array}$$

(46)

where J is the fuel consumption optimization objective, Q_i (i = 1, 2, 3) are weighting coefficients, T is the prediction horizon, and u_e, x_e, and y_e are the reference signals. x(k|t) is the predicted state, and u_min, u_max, and x_min, x_max are the hard constraints on inputs and states.

Assuming that the optimal control sequence exists at time t, then:

$$\left\{v_t^{*}\left(0\right),\dots ,v_t^{*}\left(T-1\right),{\delta }_t^{*}\left(0\right),\dots ,{\delta }_t^{*}\left(T-1\right),z_t^{*}\left(0\right),\dots ,z_t^{*}\left(T-1\right)\right\}$$

(47)

According to the model predictive control rolling optimization idea, let $u\left(t\right)=v_t^{*}\left(0\right)$; then, this first control variable is applied to the control object and other control variables in the control sequence are ignored, and then we repeat the optimization process at time t + 1.

4.3. MILP Associated with Hybrid MPC

To obtain the optimal control sequence conveniently and apply the energy management strategy online, the energy management optimization control problem is converted to a mixed-integer linear programming (MILP) for ease of solution.

The continuous and discrete variables set are as follows:

$$q={\left[{\epsilon }_0^u,\dots ,{\epsilon }_{T-1}^u,{\epsilon }_0^{\delta },\dots ,{\epsilon }_{T-1}^{\delta },{\epsilon }_0^z,\dots ,{\epsilon }_{T-1}^z,{\epsilon }_0^y,\dots ,{\epsilon }_{T-1}^y\right]}^{\prime }$$

(48)

The following constraints should be satisfied by Equation (48):

$$\left\{\begin{array}{l}-1_m{\epsilon }_k^u\le Q_1\left(u\left(k|t\right)-u_e\right)\\ -1_m{\epsilon }_k^u\le -Q_1\left(u\left(k|t\right)-u_e\right)\\ -1_n{\epsilon }_k^x\le Q_2\left(x\left(k|t\right)-x_e\right)\\ -1_n{\epsilon }_k^x\le -Q_2\left(x\left(k|t\right)-x_e\right)\\ -1_p{\epsilon }_k^y\le Q_3\left(y\left(k|t\right)-y_e\right)\\ -1_p{\epsilon }_k^y\le -Q_3\left(y\left(k|t\right)-y_e\right)\end{array}\right.$$

(49)

where k = 0, 1…, T − 1, and 1_h indicates a column vector of length h and magnitude 1. Therefore, the state-updating equations can be rewritten as:

$$x\left(k|t\right)=A^{k}x\left(t\right)+{\displaystyle \sum _{j=0}^{k-1}{A}^j\left(B_{1}u\left(k-1-j|t\right)+B_2\delta \left(k-1-j|t\right)+B_{3}z\left(k-1-j|t\right)+B_5\right)}$$

(50)

Due to vector q satisfying Equation (49), if Equation (51) is regarded as the optimization objective, the optimal control sequence could also be obtained by solving the original optimization problem:

$$J\left(q\right)={\displaystyle \sum _{k=0}^{T-1}{\epsilon }_k^u}+{\displaystyle \sum _{k=0}^{T-1}{\epsilon }_k^{\delta }}+{\displaystyle \sum _{k=0}^{T-1}{\epsilon }_k^z}+{\displaystyle \sum _{k=0}^{T-1}{\epsilon }_k^y}$$

(51)

Equation (46) is converted to an MILP problem, as shown below:

$$\begin{array}{l}\underset{q}{\min }J\left(q\right)=\left[1 1\dots 1\right]q\\ s.t.\left\{\begin{array}{l}-1_m{\epsilon }_k^u\le \pm Q_1\left(u\left(k|t\right)-u_e\right)\\ -1_n{\epsilon }_k^x\le \pm Q_2\left(A^{k}x\left(0|t\right)+{\displaystyle \sum _{j=0}^{k-1}{A}^j\left(\begin{array}{l}{B}_{1}u\left(k-1-j|t\right)+B_2\delta \left(k-1-j|t\right)+\\ B_{3}z\left(k-1-j|t\right)+B_5\end{array}\right)}-x_e\right)\\ -1_p{\epsilon }_k^y\le \pm Q_3\left(\begin{array}{l}C{A}^{k}x\left(0|t\right)+C{\displaystyle \sum _{j=0}^{k-1}{A}^j\left(\begin{array}{l}{B}_{1}u\left(k-1-j|t\right)+B_2\delta \left(k-1-j|t\right)+\\ B_{3}z\left(k-1-j|t\right)+B_5\end{array}\right)}+\\ D_{1}u\left(k\right)+D_2\delta \left(k|t\right)+D_{3}z\left(k|t\right)-y_e\end{array}\right)\\ x_{\min }\le A^{k}x\left(0|t\right)+{\displaystyle \sum _{j=0}^{k-1}{A}^j\left(\begin{array}{l}{B}_{1}u\left(k-1-j|t\right)+B_2\delta \left(k-1-j|t\right)+\\ B_{3}z\left(k-1-j|t\right)+B_5\end{array}\right)}\le x_{\max }\\ u_{\min }\le u\left(k|t\right)\le u_{\max }\\ x\left(T|t\right)=x_e\\ x\left(k+1|t\right)=Ax\left(k|t\right)+B_{1}u\left(k\right)+B_2\delta \left(k|t\right)+B_{3}z\left(k|t\right)+B_5,k\ge 0\\ y\left(k|t\right)=Cx\left(k|t\right)+D_{1}u\left(k\right)+D_2\delta \left(k|t\right)+D_{3}z\left(k|t\right)+D_5,k\ge 0\\ \left[\begin{array}{l}{E}_{21}\\ E_{22}\end{array}\right]\delta \left(k|t\right)+\left[\begin{array}{l}{E}_{31}\\ E_{32}\end{array}\right]z\left(k|t\right)\le \left[\begin{array}{l}{E}_{41}\\ E_{42}\end{array}\right]x\left(k|t\right)+\left[\begin{array}{l}{E}_{11}\\ E_{12}\end{array}\right]u\left(k\right)+\left[\begin{array}{l}{E}_{51}\\ E_{52}\end{array}\right],k\ge 0\end{array}\right.\end{array}$$

(52)

Equation (52) can be rewritten in matrix form as:

$$\begin{array}{l}{q}_t^{*}=\arg \underset{q}{\min }{f}_c^T{q}_c+f_d^T{q}_d\\ s.t.G_c{q}_c+G_c{q}_d\le S+Fx\left(t\right)\end{array}$$

(53)

where matrices G_c, S, and F can be obtained by way of Equation (52), q_c and q_d indicate the continuous and discrete variable vectors, respectively.

5. Verification and Discussion

5.1. Simulation and Discussion

In order to verify the control effect of the hybrid model-based predictive control energy management strategy proposed in this paper, this section uses the hybrid system model description language HYSDEL [37] to write the dual-mode power-split hybrid vehicle energy management strategy program, proposed in this paper, based on hybrid model predictive control. Based on the MATLAB/Simulink R2015b vehicle simulation model of a dual-mode power-split hybrid vehicle, the vehicle is operated under two different driving cycle conditions (as shown in Figure 6). Both driving cycles are comprehensive driving cycles derived from real experimental data of the same type of vehicle, aiming to represent the real-world operating environment of off-road vehicles. For driving cycle 1, the total distance is 19.95 km, and the total duration is 1486 s. For driving cycle 2, it consists of paved road (264 s), dirt road (703 s), mountain road (50 s), rolling road (465 s), icy and snowy road (318 s), and a transition period (140 s), with a total duration of 1940 s. The above details have been added to the revised manuscript to improve the assessment of the method’s generalizability. Using the parameters of the dual-mode power-split HEV, simulations are conducted to evaluate and compare the proposed hybrid model predictive control, dynamic programming, and rule-based energy management strategies. The simulation computer configuration used in this paper is a ThinkPad T460p with an i5-6300hq CPU, 2.3 GHz CPU frequency, and 8 GB memory capacity.

Under two driving cycles with an initial SOC of 65%, simulations were carried out for DP-, rule- and HMPC-based energy management strategies. The corresponding fuel economy and SOC results are listed in

Table 3. Comparison of simulation results of three energy management strategies. This table presents the control performance of the HMPC energy management strategy.

Driving Cycles	Strategies	End SOC	Fuel (L/100 km)	Equivalent Fuel (L/100 km)	Percentage of DP Control Effect (%)	Strategy Elapsed Time (s)
Cycle 1	DP	65.0%	15.6683	15.6683	100	39,730
	HMPC	65.0%	19.4360	19.4360	80.60	706.69
	Rule	64.5%	23.4565	23.5764	66.46	51.17
Cycle 2	DP	65.0%	14.6530	14.6530	100	89,513
	HMPC	65.0%	17.4868	17.4868	83.79	892.58
	Rule	63.9%	20.6006	20.7803	70.51	49.19

. Based on the equivalent fuel consumption conversion, the control performance is compared with DP as the optimal benchmark (100%). The HMPC-based strategy achieves 80.60% and 83.79% of the DP performance under the two cycles, while the rule-based method only reaches 66.46% and 70.51%. The results verify that the proposed HMPC strategy effectively improves vehicle fuel economy.

For driving cycle 2, sensitivity analysis of fuel economy versus horizon length and the actual computation time per control cycle was conducted, as presented in

Table 4. The sensitivity analysis of fuel economy versus horizon length and the actual computation time.

Np (s)	Strategy Elapsed Time (s)	Equivalent Fuel Consumption (L/100 km)
5	892.58	16.1996
10	1574.375	16.2066
15	2852.1875	16.1976
30	8240.0625	16.2080
40	14,173.9843	16.2376
50	21,307.4531	16.2001
80	56,533.9375	16.2492

Note: N_p is prediction horizon(s). The prediction horizon N_p is the same as the control horizon N_c.

.

In the energy management strategy based on DP and the rule-based algorithm, only the steady state characteristics of dual-mode power-split hybrid vehicles are considered, ignoring the transient characteristic; this is because these two kinds of energy management strategies are based on the dual-mode power-split hybrid electric vehicle steady-state characteristics for design, and they only consider the energy distribution under steady-state characteristics. Its transient characteristics, including the switch between EVT1 mode and EVT2 mode, and the switch between various working modes in the two working sections of the dual-mode power-split hybrid vehicle, such as the changes in the working states of the engine, motor A, and motor B, are all accomplished by the coordinated control strategy. The energy management strategy based on DP and the rule-based algorithm as a comparison benchmark strategy does not involve coordination control, so the transient characteristics of the two benchmark energy management strategies are ignored. The HMPC-based energy management strategy proposed in this paper includes the steady-state and transient characteristics of dual-mode power-split hybrid vehicles, which need to be explained from the perspective of hybrid system theory. In this paper, the dual-mode power-split hybrid vehicle is described as a hybrid system consisting of a continuous time-dynamic system and a series of discrete-event dynamic systems and their interactions. Therefore, in establishing a dual-mode power-split hybrid vehicle in the process of the mixed logical dynamic system prediction model, considering the different work modes represented by the discrete state and the work modes in the steady-state operating characteristics represented by the continuous-state power components, and the switching between discrete and continuous states, the transient performance characterization of the power components is considered. Therefore, in the energy management strategy based on HMPC, the characteristic curve of the engine and motor fluctuates slightly more than that of the energy management strategy based on DP or rules. However, compared with the other two energy management strategies, this trend is closer to reality and can better demonstrate the control effect of the hybrid vehicle with two-mode power split.

Figure 7 and Figure 8 present the speed profiles of the engine, motor A, and motor B under two driving cycles. All components operate within their allowable speed ranges. Compared with DP and rule-based strategies, the proposed HMPC-based EMS stabilizes the engine near its high-efficiency region while allowing for wider and more frequent speed fluctuations in motors A and B. This is achieved by leveraging the engine–vehicle speed decoupling characteristic of the dual-mode power-split HEV, where the two motors compensate for transient vehicle speed variations. As a result, the engine is maintained in a high-efficiency zone, improving fuel economy by up to 13.28% relative to the rule-based method under the two driving cycles. Since both cycles cover low-, medium-, high-speed, steady, and transient conditions and the powertrain exhibits consistent operating characteristics across all scenarios, the proposed HMPC-based EMS demonstrates strong universality and robustness.

Figure 9 and Figure 10 depict the torque profiles of the engine and motors under two driving cycles, comparing DP, rule-based, and HMPC strategies. Motors A and B effectively regulate engine torque, enabling efficient and stable engine operation. This paper analyzes the effect of HMPC-based energy management strategy on torque control of all power components of dual-mode power-split hybrid vehicles from two perspectives, namely, the constrained processing ability of the energy management strategy on the optimal control problem and the transient processing ability of dual-mode power-split hybrid vehicles. According to the engine and motor efficiency maps, their torque constraints vary with speed. Compared with DP and rule-based strategies, the proposed HMPC strategy yields significantly smaller engine torque fluctuations under varying conditions. Meanwhile, motors A and B exhibit larger torque variations to compensate for the required engine torque. This demonstrates that the HMPC-based EMS performs well in solving the constrained optimal control problem of dual-mode power-split HEVs.

The HMPC-based energy management strategy proposed in this paper is designed on the basis of the hybrid logic dynamic system prediction model proposed in this paper. The model unifies the continuous time-dynamic system and a series of discrete-event dynamic systems and their interaction relationships for hybrid systems such as dual-mode power-split hybrid vehicles. Therefore, in the process of the design and energy management strategy based on HMPC, considering the dual mode on behalf of the discrete-state power-split hybrid vehicle’s working mode, on behalf of the continuous state of two-mode power split hybrid vehicles within the working mode of the power components’ steady-state operating characteristics, and on behalf of the state of the discrete and continuous switching process between two-mode power-split hybrid electric vehicle power components gives the transient performance. As shown in the figure, DP and rule-based strategies result in significant torque fluctuations in the engine, motor A, and motor B, whereas the HMPC-based strategy only induces large torque variations in motors A and B. In order to maintain the stability of engine torque, motor A and motor B play a role in compensating for the inconsistency between engine torque and demand torque. We have added quantitative comparisons between the HMPC-based strategy and the rule-based strategy in both driving cycles. For driving cycle 1, the HMPC-based strategy reduces engine torque fluctuation by 35% in the torque interval (100,200], 21% in (200,300], 12% in (300,400], 8% in (400,500], and 42% in (500,600], respectively. For driving cycle 2, the corresponding reductions are 8%, 27%, 1%, 10%, and 27% in the same torque intervals. These quantitative results clearly illustrate the performance advantages of the proposed HMPC method. This also indicates that the HMPC-based energy management strategy proposed in this paper can better deal with the transient operating characteristics of each power component during the switching process of each working mode of the dual-mode power-split hybrid vehicle. This is consistent with the fact that HMPC can handle the switching process between discrete and continuous states.

Figure 11 shows the battery SOC profiles under two driving cycles. The HMPC-based strategy maintains an SOC near 65% with good charge-sustaining performance. By contrast, the DP-based strategy exhibits larger SOC fluctuations, yet it achieves global optimal performance as a benchmark for evaluating other strategies. The wide range fluctuation of battery SOC indicates that the energy management strategy based on DP has fully tapped the potential of batteries and motors, which is also the desired control effect of all energy management strategies based on other algorithms. Figure 11 shows the change curves of battery SOC under severe and stable working conditions. By comparing with the energy management strategy based on DP, it is found that under the control of the energy management strategy based on HMPC proposed in this paper, the variation trend of SOC is significantly different under the condition of drastic change, while the variation trend of SOC is slightly different under the stable condition. This demonstrates that the proposed HMPC-based energy management strategy exhibits stronger real-time optimization characteristics.

Figure 12 illustrates the mode switching sequences under three strategies, where 0, 1, and 2 denote stop, EVT1, and EVT2 modes respectively. The mode transitions of the proposed HMPC strategy are close to those of the DP-based strategy. For better fuel economy, the mode switching time of the energy management strategy based on HMPC is shorter than that of the other two strategies.

Figure 13 and Figure 14 present the operating point distributions of the engine, motor A, and motor B under two driving cycles, while Figure 15 and Figure 16 show the engine operating point ratios at different fuel consumption rates. The distributions under the HMPC strategy are close to those of the DP and rule-based strategies. With the proposed HMPC-based EMS, motors A and B effectively adjust the engine operating points, concentrating most of them in the high-efficiency region and thus achieving better fuel economy for the dual-mode power-split HEV. For driving cycle 1, the proportions of engine operating points within the fuel consumption ranges (0, 215), (0, 225), and (0, 240) g/kWh are 3.83%, 17.29%, and 29.92% under the rule-based strategy, while they rise to 17.15%, 35.38%, and 46.79% under the HMPC-based strategy. For driving cycle 2, the corresponding proportions are 18.49%, 37.99%, and 51.55% under the rule-based strategy and increase to 20.00%, 42.95%, and 56.41% under the HMPC-based strategy. These results verify that the HMPC-based EMS significantly improves the engine’s high-efficiency operating range and thus enhances fuel economy.

However, the working points of motor A and motor B are widely distributed, and most of them fall in the efficient region of the motor, indicating that the roles of motor A and motor B have been given full play. Under the action of energy management strategies based on DP and rules, the distribution rule of working points is that the working points of the engine are mostly located in the high-efficiency zone, while the working points of motor A and motor B are usually concentrated and rarely located in the high-efficiency zone. The law based on the DP energy management strategy is more obvious, indicating that both of these strategies only consider the engine efficiency and ignore the motor efficiency in the design process. Under the control of the energy management strategy based on HMPC proposed in this paper, most of the operating points of the engine, motor A, and motor B fall into the high-efficiency range. The energy management strategy based on HMPC takes into account the efficiency of the engine and motor when optimizing the energy distribution of a dual-mode power split hybrid vehicle, and the efficiency of each power component is comprehensively optimized. Therefore, this energy management strategy is more reasonable.

5.2. Hardware-in-the-Loop Experiment

A HIL system built on the xPC Target real-time OS is used to test the real-time performance of the proposed PHEV energy management strategy so as to validate the real-time characteristic of the algorithm. Real-time platform specifications are as follows: storage: 512 GB solid-state drive (SSD), 2 TB hard disk drive (HDD); memory: 4 × 128 GB DDR4; processor: Intel^® Core™ i9-9900K; graphics card: NVIDIA RTX 3070. As depicted in Figure 17, the HIL system comprises a host PC, a target PC, and a hybrid control unit (HCU). The host PC, a computer workstation, controls and monitors the simulation. The real-time target PC is a desktop equipped with a Softing CAN-AC2-PCI CAN board and an Intel I82559 network adapter, which executes the PHEV model in real time. A 4 MB flash-memory production-level ECU from Freescale is employed as the HCU. It integrates a 160 MHz MPC5674F processor widely applied in automotive and marine fields, making it suitable for HIL tests. The proposed energy management strategy is programmed in C-code via Freescale CodeWarrior, enabling code debugging independent of the physical HCU hardware. The host PC communicates with the target PC via TCP/IP, while the target PC and HCU exchange data through the CAN bus with a sampling time of 10 ms.

To validate the real-time performance of the proposed HMPC-based energy management optimal control strategy, driving cycle 2 is employed, which is one of the two driving cycles adopted in the simulation (see Section 5.1). The fuel economy results obtained from the simulation and HIL test are compared in

Table 5. The fuel economy in simulation and HIL. This table shows the control performance of the HMPC energy management strategy.

Driving Pattern	Strategy	Simulation	HIL
Driving Cycle 2	HMPC	17.49 (L/100 km)	18.68 (L/100 km)

. For the HIL results shown in Table 5, the slightly increased fuel consumption compared with offline simulation is mainly attributed to several real-time implementation factors besides data type conversions. First, the energy management strategy is appropriately simplified to meet the real-time execution requirements of the HIL system. Second, quantization effects introduced by analog-to-digital conversion and fixed-point processing also contribute to minor deviations. In addition, the numerical tolerances of the real-time solver and limited sampling frequency further lead to a slight degradation in optimization performance, which results in the observed increase in fuel consumption. We have added a brief discussion of these factors in the revised manuscript to improve the completeness of the analysis.

The corresponding performance curves of the power-split HEV are provided in Figure 18, Figure 19 and Figure 20. The maximum deviation between the simulation and hardware-in-the-loop (HIL) simulation is 6.8%. Figure 18 presents the drivability and battery SOC performance of the HMPC strategy. As shown in Figure 18a, the actual propulsion power tracks the driver demand well, ensuring accurate tracking of the driving cycle. Figure 18b plots the SOC comparison. For the power-split HEV with SOC-sustaining control, the engine drives motors A and B to generate and charge the battery during parking and braking when the battery state is low. Thus, the SOC curve of the power-split HEV is almost the same (within 6.8%) between the simulation and HIL experiment.

Figure 19a–c depicts the engine, motor A, and motor B speeds of the HMPC-based EMS, respectively, illustrating its control performance under driving cycle 2. Similarly, Figure 20a–c presents the corresponding torque results, further validating the strategy’s performance under the same driving cycle. It can be seen from Figure 19 and Figure 20 that the output speed and torque of the engine and motor A and motor B could satisfy the speed and torque requirement of the power-split HEV to ensure the dynamic performance of the power-split HEV. To enhance engine efficiency and avoid low-efficiency operating regions, the motors primarily propel the power-split HEV. The engine only activates when the battery SOC falls below a threshold or the required torque exceeds the motor maximum. Although slight discrepancies exist between the HIL and simulation results, the proposed strategy is still verified to be highly promising for practical applications.

For the MLD model unifying the hybrid powertrain’s continuous and discrete dynamics, computational complexity and solver efficiency are critical for HMPC real-time implementation, which are discussed below to verify its practical feasibility.

The MLD model’s computational complexity depends on binary variables (n_b ≤ 8), continuous-state variables (n_x = 4), and HMPC prediction horizon (N = 10, T = 1 s), balancing performance and computational burden.

Converted to a MILP problem in HMPC optimization, it has <50 variables and ~120 constraints, moderately scaled for efficient solving. Solver efficiency was evaluated via Gurobi (7.0), CPLEX (12.6), and GLPK (4.59) under UDDS/WLTC cycles using the average solving time (max allowable = 1 s) as the index. Average solving times are as follows: 0.08 s (Gurobi), 0.12 s (CPLEX), and 0.35 s (GLPK) (all ≤ 1 s). Gurobi was selected for its high efficiency to ensure HMPC real-time performance. The relative optimality gap tolerance (MIPGap) is set to 10⁻⁴, and the primal feasibility threshold (FeasibilityTol) is set to 10⁻⁶, which are the default settings in Gurobi. These tolerances guarantee a satisfactory balance between computational efficiency and solution accuracy.

A trade-off exists: reducing N (to 5) shortens the solving time (0.04 s) but increases fuel consumption (~2%); increasing n_b (to 12) improves adaptability but raises the solving time (0.15 s). n_b and N were optimized for balance.

In summary, the MLD model has controllable complexity, and Gurobi meets the HMPC real-time requirements, confirming its innovativeness and feasibility.

6. Conclusions

A hybrid model predictive control-based energy management strategy is designed for dual-mode power-split HEVs. The main simulation conclusions are as follows:

Firstly, the linearized model of a dual-model power-split HEV, including a resistance force model, an engine fuel consumption map model, and a motor efficiency map model, could decrease the calculation time and ensure accuracy.

Secondly, the mixed logical dynamical (MLD) prediction model of the dual-mode power-split HEV, established via HYSDEL based on hybrid system theory, exhibits favorable prediction performance.

Thirdly, the proposed hybrid model predictive control energy management strategy based on the MLD predictive model offered good control performance. The mixed-integer linear programming problem, which was derived from the energy management optimal control problem, can ensure the online application of the control strategy.

Finally, the control performance of the proposed HMPC-based energy management strategy is compared with that of a DP-based energy management strategy and a rule-based energy management strategy over two different driving cycles. The simulation results indicated that the control performance of HMPC can achieve 80.6% and 83.79% of that of DP in cycles 1 and 2, respectively.

The limitations of this study are as follows: (a) The proposed method assumes perfect future velocity preview, which is an ideal condition that may not hold in real traffic environments. (b) The controller is designed under a deterministic framework without considering parameter uncertainties, external disturbances, or measurement noise. (c) The use of linearized models may fail to capture the transient nonlinear dynamics of the powertrain during fast mode transitions. (d) The proposed strategy is only verified on a single hybrid powertrain architecture, whose generalization to other configurations is not validated. (e) The performance evaluation is conducted using only two driving cycles, which cannot fully represent real-world driving conditions. (f) A fixed average engine efficiency (η_eng = 35%) is adopted, ignoring its variation under different operating points. (g) No baseline comparison with the widely used ECMS (equivalent consumption minimization strategy) is provided. (h) When calculating SOC using the ampere-hour integration method, the coulombic efficiency, periodic recalibration, and the sensitivity of HMPC performance to SOC estimation error are not considered. (i) Sensitivity analysis (±10% variations in m, f, Cd, and battery capacity) is missing and will be conducted in future work.