Trajectory optimization for auxiliary power unit operation of an extender range electric vehicle.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 11;16:16893. doi: 10.1038/s41598-026-48203-0 Trajectory optimization for auxiliary power unit operation of an extender range electric vehicle Ye Yang 1, ✉ , Jingyi Tian 2 , Boru Jia 3, 4 Author information Article notes Copyright and License information 1 School of Automotive and Transportation, Tianjin University of Technology and Education, Tianjin, 300222 China 2 School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin, 300222 China 3 School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081 China 4 National Key Laboratory of Multi-perch Vehicle Propulsion Systems, Beijing, 100081 China ✉ Corresponding author. Received 2026 Jan 15; Accepted 2026 Apr 7; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13230910  PMID: 41963503 Abstract This paper focuses on the trajectory optimization based Auxiliary Power Unit (APU) for an Extended Range Electric Vehicle (EREV). Firstly, a new three-zone control strategy has been proposed to optimize the APU steady-state operation, and the simulation result show significant enhancements in fuel efficiency compared to the traditional thermostat and power-following control strategy. Furthermore, low-frequency noise and vibrations generated by the APU during transient operations are the primary factors affecting passenger comfort, which are closely associated with transient operational trajectory of the APU. In order to address the fuel consumption and NVH (Noise, Vibration, and Harshness) challenges in EREV during dynamic operation, a multi-objective optimization model is developed. The multi-objective particle swarm algorithm and weighted scaling method are employed to solve the model, obtaining the transient operating trajectory of the APU. Finally, the method proposed is validated on a test bench and compared with single-objective optimization method based on the best BSFC (Brake Specific Fuel Consumption) and traditional optimization method. The results indicate that the multi-objective operational trajectory proposed can effectively improve the NVH performance of the APU at the cost of a slight increase(3.9%) in fuel consumption. Although a slight increase in fuel consumption occurs during the transient switching process, the overall energy efficiency of the APU system remains improved due to the high-efficiency steady-state operation enabled by the three-zone control strategy. Keywords: extended range electric vehicle, trajectory optimization, particle swarm optimization, NVH, fuel consumption Subject terms: Energy science and technology, Engineering, Mathematics and computing Introduction Faced with an increasingly severe energy crisis and growing concerns about air pollution, electric vehicles are gaining global recognition for their significant contributions to energy conservation and environmental protection. However, the widespread adoption of electric vehicles is hindered by current limitations in power battery technology. The low energy density, long charging times, and high costs associated with power batteries make it challenging to achieve significant performance improvements in the short term. Consequently, extended-range electric vehicles (EREVs), as series hybrid electric vehicles, have attracted widespread attention for their high efficiency, extended driving range, cost-effectiveness, simple structure, and enhanced portability 1 – 3 . EREVs comprise both an APU and a power battery, forming a multidisciplinary nonlinear system encompassing mechanical, electrical, and thermodynamic aspects 4 , 5 . The existence of these two power sources increases the complexity of the powertrain system. Therefore, ensuring the dynamic performance and high efficiency of EREVs crucially depends on rational and effective energy management strategies and precise APU operational control. Literature review Energy management strategies(EMSs)for hybrid electric vehicles have been a hot topic of concern for scholars worldwide. Existing EMSs are primarily classified into rule-based and optimization-based approaches. Rule-based strategies mainly limit the working area of power system components and determine the working mode through preset control parameters. Depending on the type of selected control parameters, rule-based EMSs can be further classified into deterministic rules and fuzzy rules. Refs 6 ] and [ 7 respectively employ thermostatic and power-following EMSs for energy distribution in hybrid electric vehicles, reducing fuel-to-electric conversion losses and extending battery life. Won et al. 8 and Wassif et al. 9 designed an EMS based on basic fuzzy control, achieving good fuel economy. EMSs based on optimization algorithms can be classified into real-time optimization and global optimization strategies. Global optimization strategies are based on the global optimality theory and employ relevant optimization methods to determine the optimal energy allocation. Notably, dynamic programming (DP) 10 , 11 and Pontryagin’s Minimum Principle (PMP) 12 , 13 represent the two most widely applied global optimization methods. Regarding real-time optimization control, Equivalent Consumption Minimization Strategies (ECMS) 14 , 15 and Model Predictive Control (MPC) 16 , 17 are the two most representative methods. Additionally, a range of intelligent optimization algorithms, including the Quadratic Programming Algorithm 18 , Simulated Annealing 19 , and Mixed Integer Convex Programming Algorithm 20 , have also achieved satisfactory results. With the rapid development of machine learning technology, some reinforcement learning approaches in hybrid powertrain control have been increasingly adopted in researches 21 – 24 . Although these strategies differ in formulation and implementation, most of them are developed under a steady-state efficiency-oriented paradigm, in which the APU is expected to operate near its optimal efficiency region defined by static characteristic maps. Existing studies primarily focus on the optimal allocation of energy between the power battery and the APU, or on maintaining the APU operation at high-efficiency operating points. However, such a paradigm exhibits inherent limitations. First, most existing methods emphasize steady-state operating point optimization, while neglecting the dynamic characteristics of the APU during transient processes. In real driving conditions, the APU operates under highly dynamic environments, where frequent variations in vehicle power demand lead to continuous switching between different operating points. Second, the optimization is typically based on static efficiency maps, implicitly assuming quasi-steady operation of the engine-generator system. This assumption ignores the nonlinear and inertial characteristics of the engine, resulting in discrepancies between theoretical optimality and actual performance. Third, frequent operating point switching may introduce additional fuel consumption and dynamic losses, which cannot be captured by steady-state optimization alone. To tackle the operation optimization problem of the APU, several studies have attempted to improve its efficiency by adjusting engine speed and torque trajectories. For example, Perez et al. 25 treated the engine’s efficiency as a nonlinear function of power and utilized nonlinear programming techniques for trajectory optimization. Massimo et al. 26 and Carlos et al. 27 proposed an optimal control strategy of the engine–generator subsystem to generate a desired amount of energy within a given period of time. Ref 28 introduced an optimization method for identifying the maximum efficiency operating point of the APU based on dynamic MAP, resulting in a 7.15% efficiency enhancement. In pursuit of heightened APU efficiency, Hong et al. 29 utilized PID and fuzzy control theory to govern the engine, attaining an optimal APU operating trajectory. Likewise, Lan 30 , Xu 31 , and other researchers explored similar methodologies 32 , 33 . Nino et al. 34 presented a calculation method for optimizing the trajectory of an APU, which aimed to obtain the optimal trajectory based on the maximum efficiency of the engine and generator subsystems according to the energy demand within a given time period. In response to the fuel consumption and emission challenges during the mode transition of the EREV, Ref 35 employed an improved genetic algorithm to achieve an optimal trajectory during the dynamic switching process. Refs 36 – 39 have explored trajectory optimization methods considering fuel consumption and emissions during dynamic processes. Despite these efforts, the transient operation of the APU remains insufficiently addressed. The operating point switching process plays a crucial role in determining overall system performance. Due to the inherent nonlinearity and inertia of the engine, key variables such as air–fuel ratio, torque output, and combustion state cannot respond instantaneously to rapid changes in operating conditions. During transitions from low power to high power, the engine may temporarily operate under fuel-rich conditions, leading to deteriorated combustion, increased fuel consumption, and significant torque fluctuations. Moreover, for a given transition between two operating points, multiple feasible trajectories exist in the speed–torque domain, and different trajectories may result in substantially different performance outcomes. This indicates that the problem should be reformulated from conventional operating point optimization to transient trajectory optimization. In addition to fuel economy and emissions, Noise, Vibration, and Harshness (NVH) is another critical performance index that has been largely overlooked in existing studies. This issue is particularly important for EREVs. Due to frequent APU start–stop events and operating point switching, vibration and noise become more perceptible to passengers 40 , 41 , thereby significantly affecting ride comfort. During transient processes, rapid changes in engine speed and torque can induce strong mechanical shocks, structural vibrations, and radiated noise. These NVH characteristics are highly sensitive to the transient trajectory and are strongly coupled with the dynamic behavior of the system, yet they cannot be effectively evaluated using steady-state performance metrics alone. Limitations of existing research Based on the above analysis, it can be concluded that existing studies on APU still exhibit several limitations. First, most research focuses on steady-state operating point optimization, with insufficient attention paid to the transient switching process between operating points. Second, although some studies have explored trajectory optimization methods, their objectives are primarily limited to fuel consumption and emissions, without considering ride comfort-related performance. Third, NVH characteristics, which are highly sensitive to transient dynamics, are rarely incorporated into the optimization framework. As a result, the coupling relationship between transient trajectories and NVH performance has not been adequately addressed. Motivation and innovation The transient operating point switching process of the APU is a crucial factor that impacts fuel consumption and NVH. During the transition from low power to high power, the engine must rapidly adjust throttle opening, intake air, and fuel injection to meet the increased load demand. However, due to the inherent delay in air–fuel ratio dynamics and the nonlinear characteristics of the engine, the combustion process may deteriorate, leading to increased vibration and noise, as well as reduced fuel efficiency. To address these challenges, this paper proposes a trajectory optimization method for the APU transient operation process. The fuel consumption rate, body vibration intensity, and average sound pressure level (ASPL) of envelope surface radiated noise (ESRN) are considered simultaneously as optimization objectives. A multi-objective particle swarm optimization (MOPSO) algorithm combined with a weighted scaling method is employed to solve the optimization problem, thereby obtaining the optimal transient operating trajectory of the APU. Organization of this paper This paper is organized as follows. Section  2 discusses the powertrain and parameters of the EREV. Section  3 proposes a three-zone control strategy for the APU. Section  4 describes the APU operation points transient switching problem and introduces the multi-objective particle swarm algorithm and weighted scaling method to solve the optimization problem. Section  5 discusses and analyzes the experiments results. Conclusions are made in Sect.  6. Powertrain configuration The powertrain system structure of the EREV is shown in Fig. 1 , mainly consisting of a power battery, a driving motor, an APU, and various controllers. The APU is composed of an engine, generator, and APU controller, with the engine and generator co-axially connected. When the State of Charge (SOC) of the power battery falls below a predetermined threshold, the vehicle controller (VCU) initiates the APU based on the required power and other auxiliary information, sending the vehicle power demand command P cmd to the APU controller. The APU controller takes P cmd as an input parameter, executes an optimization algorithm, and adjusts the engine speed and the generator torque to n cmd and T cmd , respectively, based on the optimization results. Fig. 1. Open in a new tab Powertrain architecture of the studied EREV. Since the APU is not mechanically connected to the vehicle, the engine can operate at any point on its speed-torque-power characteristic curve. Therefore, optimization of vehicle fuel consumption and NVH can be achieved by adjusting the working points of the engine and generator. The specific parameters of EREV are list in Table 1 . Table 1. The EREV parameters. Component Parameter Name Value Unit EREV Curb weight 1798 kg Frontal area 2.4 m 2 Air drag coefficient 0.3 — Air density 1.2 Kg/m 3 Wheel radius 0.274 m Rolling resistance coefficient 0.015 — Engine Maximum power 72 kW Maximum speed 5500 rpm Generator Rated power 30 kW Maximum speed 3500 rpm Battery Capacity 107 Ah Voltage range 355 V Drive Motor Rated power 42 kW Peak torque 150 kW Open in a new tab Three-zoo control strategy Presently, thermostat control, power following control, and multi-operation point control are commonly used in engineering 42 , 43 . In the thermostat control strategy, the engine operates at a fixed point and outputs constant power, resulting in high work efficiency. However, constant power output may lead to repeated charging and discharging of the battery, affecting its health and reducing its lifespan. For the power following control strategy, the output power of the APU changes with the target power, enabling it to closely follow the power demand of the vehicle and avoid repeated charging and discharging of the power battery. However, this strategy often requires the APU to operate dynamically, leading to increased dynamic fuel consumption. The multi-operation point control strategy combines the advantages of power following and thermostat control. It allows the APU to operate at several working points, reducing the APU’s dynamic operation while extending the lifespan of the power battery. However, improper selection of working points can still lead to frequent engine operating point transitions. Based on the above analysis 6 , 44 , this paper proposes a three-zone control strategy, which selects a reasonable torque variation range at three fixed speed points to form three operating zones, thereby achieving the goal of APU following the target power. The division of operating zones is shown in Table 2 , and the schematic diagram of the operating zones is depicted in Fig. 2 . Table 2. Segmentation method of APU. Load zone Power range(kW) Engine speed( n /min) A 0–10 1500 B 8–18 2000 C 16–30 2750 Open in a new tab Fig. 2. Open in a new tab APU working area diagram. The division of the three operating regions and the selection of the fixed engine speeds were determined based on statistics of the vehicle power demand under the three major cycles of FTP/HWFET/NEDC, determining that 0-10 kW, 8-18 kW and 16-30 kW are the three core power ranges, accounting for 92.3% of the total power demand. The three speed points of 1500 r/min, 2000 r/min, and 2750 r/min were chosen because they are located in relatively high-efficiency regions of the engine and allow the APU to cover the required power range (0–30 kW) with reasonable torque variation while satisfying the speed and torque constraints of both the engine and generator. Based on the vehicle power demand and battery SOC status, the APU operating point switching strategy has been formulated, as depicted in Fig. 3 . Fig. 3. Open in a new tab APU operating point switching scheme. (1) When , the power battery provides the energy, so the EREV Works in pure electric mode and the APU is off. (2) When , the power battery stops discharging, APU outputs maximum power and charges the battery. (3) When & , APU works in zone A. (4) When & , APU works in zone B. (5) When & , APU works in zone C. (6) When & , APU works at the maximum power point. The Federal Test Procedure (FTP), Highway Fuel Economy Test (HWFET), and New European Driving Cycle (NEDC), were selected to simulate and analyze three-zone control strategy in MATLAB/Simulink, respectively. To validate the advantages of the three-zone control strategy, simulations are compared with traditional thermostat, power-following and DP control strategy. Due to variations in the terminal SOC among the four control strategies, a direct assessment of the vehicle’s fuel consumption becomes unfeasible. To ensure a fair assessment of fuel economy, it becomes imperative to adjust the influence of the terminal SOC. Subsequently, the relative fuel consumption efficiency is computed. Considering the article length limitation, the details of the simulation have been simplified, which can be found in Ref 14 . 1 Where, FC represents the fuel consumption of each strategy, and FC opt is the optimal value computed using DP among the four strategies. The smaller the , the better the fuel economic of the control strategy. Table 3 shows the 100 km fuel consumption of four control strategies after terminal SOC correction under different cycle conditions. As seen from the table, the three-zone control strategy exhibits superior fuel economy compared to thermostat control strategies and power-following control strategies. Compared with the thermostat control strategy, the three-zone can lower 8.1% in FPT, 3.5% in HWFET, and 2.8% in NEDC. Compared with the power following control strategy, the three-zone can lower 2.7% in FTP, 8.8% in HWFET, and 1.4% in NEDC. This superiority is attributed to the fact that thermostat control strategies, while operating stably at the engine’s highest efficiency point, result in increased fuel consumption due to frequent start-stop events and secondary energy losses from battery charging and discharging. Power-following control strategies, although achieving power following, lead to increased fuel consumption due to the frequent operating points switching. The three-zone control strategy adjusts the APU to operate at three specified speed points, which not only significantly reduces the number of battery charging and discharging, but also reduces the transient fuel consumption of the engine due to dynamic operation. Table 3. Comparison of fuel economy under four control strategies. Drive Cycle Strategy Fuel(L/100km) SOC FTP Thermostat 8.2 0.35 10.8% Power-following 7.8 0.32 5.4% Three-zone 7.6 0.31 2.7% DP 7.4 0.30 0 HWFET Thermostat 6.1 0.36 7.0% Power-following 6.7 0.31 12.3% Three-zone 5.9 0.32 3.5% DP 5.7 0.30 0 NEDC Thermostat 7.6 0.37 5.6% Power-following 7.5 0.31 4.2% Three-zone 7.4 0.29 2.8% DP 7.1 0.30 0 Open in a new tab Trajectory optimization for operation point transient switching Modeling of the APU The engine is a nonlinear and variable system, which is difficult to describe using an accurate physical model. However, the main purpose of the engine model in this paper is to predict the output torque of the engine. Therefore, the engine was modeled by the average output torque of the engine based on the experimental data. According to energy conservation, the average torque of the engine can be expressed by the following formula: 2 where T e (t) is the engine output torque, Q is the low calorific value of gasoline, m f (t) is the fuel injection quantity of the engine at time t, η e is the engine efficiency. For the injection quantity response delay, an inertial element was added to describe the dynamic change process of the engine. 3 where τ is the nominal delay time of the engine torque output, which mainly comes from intake delay and stroke delay. Due to the fast torque response speed, short delay time, and accurate feedback torque calculation of the generator, its inertia and delay links can be ignored compared to the engine. The torque response model of the generator can be expressed as follows: 4 where T g is the torque of the generator; p 0 is the number of pole pairs; Ψ f is the magnetic flux of the permanent magnet; L d , L q , i d and i q represent the inductance and current of the d-axis and q-axis, respectively. Due to the coaxial connection between the engine and generator, their rotational speeds are the same. The physical relationship between engine torque and generator torque can be described by the APU speed model. The APU speed model can be represented as follows: 5 where J m is the system inertia, B is the rotation damping which includes the system friction, ω is the speed of the APU. Presentation of the optimization problem When there are substantial changes in vehicle power demand, the APU must adjust its speed operating points, transitioning between the A, B, and C regions. The transient trajectory during engine speed switching significantly influences engine fuel consumption, vibration and noise. Currently, some literatures only focus on the fuel economy during APU dynamic transitions. The problem is that the vibration and noise of the APU become particularly strong when switching over a large speed range, which seriously affects the ride comfort. Therefore, it is necessary to develop appropriate transient trajectory to enhance the fuel economy and reduce vibration and noise during APU speed transitions. In order to address this issue, this paper proposes a multi-objective optimization method for the APU’s transient trajectory, with the APU’s fuel consumption rate, body vibration intensity, and ASPL of the ESRN as the objective functions. To obtain the optimal dynamic trajectory during the APU speed transition, point M (2000r/min, 72Nm) in region B and point N (2800r/min, 85Nm) in region C were selected as the study objects. Optimization research was conducted on the trajectory of the APU transitioning from point M to point N. The specific switching operation points of the APU are shown in Fig. 4. In real APU operation, multiple trajectories exist from point M to point N. Figure 4 depicts five transient operation trajectories. The challenge is how to choose the most optimal trajectory to minimize fuel consumption and vibration noise during the APU speed transition. Fig. 4. Open in a new tab APU switching point. To optimize the vibration noise performance of the APU, it is necessary to conduct vibration noise level testing experiments. This involves establishing a vibration intensity MAP and an envelope surface radiated noise sound pressure level MAP of the APU, similar to the fuel consumption rate MAP. Subsequently, fitting functions that describe the relationships between body vibration intensity, ASPL of the ESRN, and fuel consumption rate with APU speed and torque are essential. The fitting functions and parameters of the fitting results are detailed in Appendix. These fitted functions act as objective functions in multi-objective optimization, ultimately identifying the optimal transient trajectory. The specific details of the APU vibration noise and fuel consumption testing experiments are illustrated in Fig. 14 . Five measurement points are set at the center positions on the measurement surfaces in front, rear, left, right, and above the APU for noise testing, as denoted by yellow boxes labeled “A, B, C, D, E” in the figure. Six points on the APU are selected for measuring vibration signals in three directions at each measurement point, as indicated by red boxes labeled “1, 2, 3, 4, 5, 6” in the figure. During the testing process, the APU’s speed varies from 2000 to 2800r/min, and the torque ranges from 50 to 102Nm. Within this range, testing conditions are set at intervals of [100 r/min, 10 Nm], resulting in a total of 52 testing points. Fig. 14. Open in a new tab Sensitivity analysis of key variables. Body vibration acceleration, ASPL of the ESRN, and fuel consumption rate of the APU are measured at each operating point. Subsequently, the sampled data is subjected to bicubic interpolation to obtain the APU’s speed-torque-equivalent vibration intensity characteristics (as shown in Fig. 5 ), speed-torque-radiated noise average sound pressure level characteristics (as shown in Fig. 6 ), and speed-torque-fuel consumption rate characteristics (as shown in Fig. 7 ). It can be observed that the optimal fuel consumption rate region of the APU does not coincide with the optimal vibration intensity region or the optimal radiated noise sound pressure level region. The best fuel consumption rate point does not necessarily correspond to the best vibration noise, and vice versa. Therefore, it is necessary to perform multi-objective optimization of the APU’s transient operating points to balance fuel consumption and vibration noise. Fig. 5. Open in a new tab APU operation trajectory. Fig. 6. Open in a new tab APU vibration intensity MAP. Fig. 7. Open in a new tab APU radiated noise average sound pressure level MAP. The multi-objective optimization problem of APU fuel consumption and vibration noise can be described as a minimization multi-objective optimization problem with three objective functions, including 、 and , as shown in Eq. ( 6 ), using APU speed and torque as optimization variables. 6 In practical APU systems, the speed and torque of the APU are typically subject to two types of constraints. The first constraint arises from the mechanical characteristics of the engine and generator, where the minimum torque of the APU ( T min ) is the greater value between the minimum torque of the engine ( Te min ) and the minimum torque of the generator ( Tg min ). The maximum torque of the APU ( T max ) is jointly determined by the engine’s maximum torque curve ( Te max (n) ) and the generator’s maximum torque characteristic curve ( Tg max (n) ), 7 In addition, the minimum speed ( n min ) and maximum speed ( n max ) of the APU should meet the following conditions: 8 Where ne min and ne max represent the respective minimum and maximum speeds of the engine, while ng min and ng max denote the corresponding minimum and maximum speeds of the generator. The second constraint originates from the power demand commands issued by the vehicle’s control strategy, 9 Where dP apu_max and dP apu_min represent the maximum and minimum power change rates of APU, respectively. Since this paper investigates the optimal trajectory of the APU transitioning from speed point M to speed point N, the permissible range for the APU operating point is [2000-2800r/min, 15-25 kW]. The optimization variables need to meet the following constraints: 10 Multi-objective Optimization algorithm In recent years, owing to the various advantages of swarm intelligence-based evolutionary algorithms, the use of multi-objective swarm intelligence-based evolutionary algorithms has become mainstream in scientific research and application domains for solving multi-objective optimization problems. Commonly used algorithms include the Non-dominated Sorting Genetic Algorithm (NSGA), Differential Evolution (DE), Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO). Among these, the PSO algorithm, due to its excellent convergence, unique search mechanism, and ease of computer program implementation, has been widely applied in many engineering fields. Therefore, this method is used to optimize the transient trajectory during the APU speed switching process. PSO is a method that achieves optimization through collaboration and information sharing among individuals within a population. Assuming the search space of the objectives is an S-dimensional space and there exist m particles, the i -th particle is represented as a S-dimensional vector . The optimal solution may be the position of each particle, evaluated using function to calculate the fitness value and measure the quality of the solutions. The velocity of the i -th particle is denoted as , and the best position this particle has explored so far is recorded as . Meanwhile, the current best position of the entire particle swarm is denoted as . 11 PSO begins with an initialization phase where a group of random particles is set. Subsequently, it goes through multiple iterations to search for the optimal solution. During each iteration update, each particle needs to track two extremes ( ). Once these two optimal values are obtained, the particle’s velocity and position are updated using Eqs. ( 12 ) and ( 13 ). 12 13 Where s =[1, S ], i =[1, m ]; and are mutually independent pseudo-random numbers༛ and are learning factors. The multi-objective Particle Swarm Optimization (MOPSO) algorithm extends the basic Particle Swarm Algorithm by introducing an external elite archive, dominance comparison, and an adaptive grid mechanism to solve multi-objective optimization problems. The ultimate goal of the MOPSO algorithm is to obtain a non-dominated solution set that covers the entire search space and approaches the true Pareto frontier with as few computational resources as possible. While finding a set of Pareto solutions with good convergence and distribution cannot completely solve multi-objective optimization problems. In engineering practice, only one final solution needs to be determined. Weighted scaling is a straightforward and effective approach that clearly reflects the weights of optimization objectives, allowing the transformation of a multi-objective problem into a single-objective one for optimization. Due to the different units and scales used for APU fuel consumption rate, body vibration intensity, and ASPL of the ESRN, it is necessary to normalize each objective function before applying the weighted scaling method. First, the sound pressure level ( dB , logarithmic scale) is converted into a linear sound pressure value P using the formula: 14 where, P 0 = 2 × 10 − 5 Pa is the reference sound pressure. Then, 15 where, . Finally, the weighted scaling values of the normalized Pareto optimal solutions are calculated using Eq. ( 11 ). 16 where, , , ω m represents the weighting factor. Optimization of multi-objective models Based on the characteristics of APU fuel consumption rate, vibration intensity, and ASPL of the ESRN shown in Figs. 5 , 6 and 7 , the MOPSO algorithm is applied in MATLAB to solve the multi-objective model. The parameter settings of the MOPSO algorithm are as follows: the inertia weight ω was linearly decreased from 0.9 to 0.4 during the iteration process to enhance global exploration in the early stage and improve local convergence in the later stage. The learning factors were set to c1 = 2 and c2 = 2, which are widely used in PSO-based multi-objective optimization studies. The population size was set to N = 200, and the size of the external elite archive was set to N_elite = 100, corresponding to a ratio of 0.5. This ratio was determined through preliminary tests to ensure sufficient diversity of the Pareto solution set while maintaining acceptable computational cost. To evaluate the convergence performance and distribution quality of the Pareto front obtained by the MOPSO algorithm, the Generational Distance (GD) and Inverted Generational Distance (IGD) were used as convergence indicators. The calculated GD = 0.082 and IGD = 0.095 of the MOPSO algorithm in this paper are both much less than 0.2, which indicates that the algorithm’s Pareto front has a high approximation degree with the true front and excellent convergence performance. The specific process of multi-objective optimization is shown in Fig. 8 . Fig. 8. Open in a new tab APU fuel consumption rate MAP. In order to obtain a continuous trajectory curve, the target models of APU fuel consumption rate, equivalent vibration intensity, and ASPL of the ESRN were optimized at intervals of 1 kW. Figure 9 lists the lowest fuel consumption point, lowest vibration intensity point, lowest sound pressure level point, and the final solution when in the Pareto solution at specific power points of 17 kW, 19 kW, 21 kW, and 23 kW, respectively. Fig. 9. Open in a new tab Multi-objective optimization flow char. The optimal transient trajectory is obtained by connecting the final solution at each kilowatt with a smooth curve, as shown in Fig. 10 . Additionally, Fig. 11 presents the optimal trajectory curve (BSFC) obtained with APU fuel consumption rate as the optimization objective. Comparing the two curve, significant differences are observed at certain power points between 2200 r/min and 2400 r/min. This discrepancy arises because the optimal BSFC curve is the result of single-objective optimization, considering only the best fuel consumption rate, whereas the optimal multi-objective curve takes into account fuel consumption rate, vibration intensity, and radiation noise sound pressure level. Fig. 10. Open in a new tab Distribution of Pareto optimal solution: ( a )17 kW; ( b )19 kW; ( c )21 kW; ( d )23 kW. Fig. 11. Open in a new tab Multi-objective optimization curve. Figure 12 displays the transient trajectory of the APU switching from point M to point N under traditional control methods. As shown in Fig. 12 , when the transition begins, the APU torque remains constant, allowing the speed to accelerate from 2000 r/min to 2800 r/min, and then maintaining a constant speed while the torque increases from 72 Nm to 85 Nm. Fig. 12. Open in a new tab BSFC curve. Robustness and sensitivity analysis To ensure its practical applicability, robustness analysis under varying scenarios and sensitivity analysis of critical variables are essential. The robustness of the optimization model was evaluated across diverse driving cycles and environmental conditions. Three representative scenarios were considered: Urban Driving (FTP Cycle): Frequent stops and low-speed operation increase transient APU activations. Highway Driving (HWFET Cycle): Sustained high speeds demand stable APU operation at higher power levels. Mixed Driving (NEDC Cycle): Combines urban and highway characteristics with moderate acceleration/deceleration. Table 4 compares the optimization results for fuel consumption, vibration intensity, and ASPL across these cycles. The results show that the proposed method achieves stable NVH improvement and fuel consumption reduction under different working conditions, with an average fuel consumption reduction of 9.75%, vibration intensity reduction of 10.59% and noise reduction of 12.22%, which fully verifies the significant robustness improvement of the proposed method. Table 4. Robustness analysis of optimization result. Driving cycle Method Fuel Consumption (g) Vibration Intensity (mm/s) ASPL (dB) FTP Traditional 14.21 296 107.6 103.6 94.5 Single-objective 12.35 312 Proposed 12.83 265 HWFET Traditional 14.58 289 106.8 102.9 93.7 Single-objective 12.62 305 Proposed 13.12 258 NEDC Traditional 14.35 293 107.2 103.2 94.1 Single-objective 12.28 308 Proposed 12.98 262 Open in a new tab A sensitivity analysis was conducted to identify how changes in key constraints affect optimization outcomes. The variables analyzed included: APU Speed Range (Δn): ±100 r/min deviation from [2000–2800 r/min]. Torque Limit (ΔT): ±5 Nm deviation from [50–100 Nm]. Power Change Rate (ΔP/dt): ±0.5 kW/s deviation from 3 kW/s. Figure 13 illustrates the sensitivity of the objective functions to these changes. The results show that the model is most sensitive to variations in torque limits (ΔT = ± 5 Nm), leading to a 4.2% increase in fuel consumption and 3.8% increase in vibration intensity. Conversely, adjusting the power change rate within ± 0.5 kW/s had minimal impact (≤ 1.5% change in all objectives). This indicates that torque constraints are critical for balancing fuel economy and NVH, while power change rate constraints primarily affect system stability. Fig. 13. Open in a new tab Transient trajectory curve under traditional control. Experimental verification To validate the effectiveness of the multi-objective optimized trajectory for APU, experiments were conducted on the APU test bench and compared with the best BSFC trajectory under single-objective optimization and the dynamic trajectory under traditional control methods. The schematic diagram and physical representation of the experimental bench are shown in Figs. 14 and 15 , respectively. The experimental procedure is as follows: Fig. 15. Open in a new tab Experimental platform block diagram. Firstly, based on the three dynamic trajectories shown above, the engine speed curve and generator torque curve were obtained separately. Subsequently, the acquired speed and torque curves were input into the APU controller. Finally, the fuel consumption meter and vibration noise testing system were used to record the fuel consumption, vibration intensity, and radiation noise sound pressure level. To ensure the stability of the APU system during dynamic transitions, the maximum power increase rate of the APU was limited to 3 kW/s. Additionally, to ensure the fairness of the experimental results, the dynamic transition time was kept consistent for all three methods. Based on this, the speed and torque curves for the three dynamic trajectories are shown in Figs. 16 , 17 and 18 . Fig. 16. Open in a new tab Platform test of the APU. Fig. 17. Open in a new tab APU speed and torque curves of traditional control. Fig. 18. Open in a new tab APU speed and torque curves of single objective optimization. Experimental results analysis The vibration and noise test results of the APU under the three dynamic trajectories are shown in Fig. 19 . Figure 19 (a)-(c) represent the vibration acceleration results at three vibration measurement points, while Fig. 19 (d)-(f) show the radiation sound pressure level results at three noise measurement points. The experimental testing duration exceeded 20 s, and the most valuable 10 s of data were selected for analysis. This 10-second data was divided into three stages: Stage 1, spanning from 10 s to 13s, represents the test results at a constant speed of 2000 r/min. Stage 2, from 13 s to 17s, showcases the experimental results during the acceleration process under different switching strategies. Stage 3, from 17 s to 20s, portrays the test results after reaching 2800 r/min during acceleration. Fig. 19. Open in a new tab Speed and torque curves of multi-objective optimization. Fig. 20. Open in a new tab Vibration and noise test results of APU under different strategies. Figure 19 (a) presents the vibration acceleration results along the Z-axis at measurement point 1, (b) shows the results along the Y-axis at measurement point 2, and (c) displays the results along the X-axis at measurement point 3. From the results of the three vibration tests, it can be observed that during the acceleration process in Stage 2, different switching trajectories have a significant impact on the vibration in all three directions. The traditional switching trajectory exhibits a higher vibration acceleration between 15s and 18s, while the switching trajectory from single-objective optimization results in a noticeable increase in vibration acceleration after the 13s switch, surpassing the vibration levels observed in the other two trajectories. In comparison, the switching trajectory proposed in this paper results in relatively lower vibration acceleration. Figure 19 (d) - (f) show the ASPL test results at three different measurement points of the APU. The results indicate that there are significant differences in radiation noise at different structural components of the APU. Moreover, during the acceleration phase, it is evident that different switching trajectories have a notable impact on noise levels, with the multi-objective optimization approach resulting in relatively lower noise. To mitigate the influence of measurement points variations on the results, the vibration intensity and ASPL obtained from the six vibration measurement points and five noise measurement points on the APU are used as evaluation criteria. The calculation formula for equivalent vibration intensity is: 17 Where is the vibration intensity, in mm/s; 、 、 are the root mean square values of vibration velocities in three directions, in mm/s; 、 、 are the measurement points in three directions. The calculation formula for the average sound pressure level is: 18 Where is the average sound pressure level, is the weighted average sound pressure level of the measured sound source A, K 1A is the background noise correction value, K 2A is the environmental correction value. 19 20 21 Where is the A-weighted time average sound pressure level of the i -th microphone background noise, is the A-weighted average sound pressure level of the microphone array on the measurement surface during the operation of the measured sound source, is the A-weighted time average sound pressure level of the background noise captured by the loudspeaker array on the measurement surface. By sampling and processing the experimental data, Table 5 lists the test results of APU fuel consumption, body vibration intensity, and ASPL under three different switching trajectories. And the experimental results were compared according to formula ( 22 ), 22 Table 5. Experimental results of three optimization methods. Methods Fuel consumption(g) Comparison Vibration intensity(mm/s) Comparison Sound pressure(dB) Comparison Traditional method 14.21 ± 0.18 - 296 ± 9 - 107.6 ± 0.8 - Single-objective optimization 12.35 ± 0.15 -13.09% 312 ± 11 5.40% 103.6 ± 0.7 5.67% Multi-objective optimization 12.83 ± 0.16 -9.71% 265 ± 8 -10.47% 94.5 ± 0.6 -12.17% Open in a new tab Where E i represents the experimental results of the multi-objective optimization model proposed in this paper and the single objective optimization mode based on optimal fuel consumption, E 0 represents the experimental results under traditional control models. Table 5 reveals that the traditional optimization method has the highest fuel consumption, while the single objective optimization method has the lowest fuel consumption, which is 13.09% lower than the traditional optimization. The multi-objective optimization method shows a slight increase in fuel consumption compared to the single-objective optimization model but remains 9.71% lower than traditional optimization. The vibration intensity and ASPL under multi-objective optimization are the lowest among the three methods, with a 10.47% reduction in vibration intensity and a 12.17% reduction in radiation noise sound pressure level compared to traditional methods. The reason for such a large gap in results is that the engine in the APU does not switch continuously and slightly like traditional engine. Instead, in switches abruptly between two operating points with large differences in speed, torque, and power values. During the switching process, the engine operating point continuously changes along the dynamic characteristic curve, deviating from the conventional steady-state characteristic curve. In the traditional control method, transitioning from low-power to high-power operation points requires the engine to quickly increase the throttle opening to ensure acceleration to a higher target speed value in a short period of time. In this acceleration process, the engine is in a fuel rich injection state, and there is a significant overshoot in the engine speed, which is inevitably accompanied by significant torque impact and mechanical noise, causing damage to the smoothness and fuel economy of the entire vehicle. For the single-objective optimization method, the engine runs along the optimal fuel economy trajectory, ignoring the vibration and noise during the switching process. Each trajectory test was repeated five times under identical conditions. The values listed in Table 5 represent the mean results, and the standard deviations are also provided to evaluate the repeatability of the transient experiments. The variations are within the normal range of engine bench transient tests and do not affect the comparison among different control strategies. In summary, the MOPSO algorithm proposed can effectively addresses the vibration and noise problems during the transient operation points switching process of the APU at the cost of a slight increase in fuel consumption compared to the single objective optimization method. When compared to the traditional optimization method, it exhibits significant advantages in both fuel economy and the reduction of vibration and noise. Conclusion In order to improve fuel economy and reduce NVH, a trajectory optimization-based three-zone control strategy is proposed in this paper. The strategy comprehensively considers the APU operating points transient switching process, which significantly affects the fuel economy and NVH. Initially, a novel three-zone control strategy is introduced for the EREV to optimize the APU steady-state operation. Simulation results demonstrate that this control strategy show significant enhancements in fuel efficiency compared to the traditional thermostat and power-following control strategy. Subsequently, in order to address the fuel consumption and NVH challenges, this study creatively optimizes not only the fuel economy but also the vibration and noise of the APU’s transient trajectory. The fuel consumption rate, body vibration intensity and ASPL of the ESRN are involved as optimization goals. The multi-objective particle swarm algorithm and weighted scaling method are employed to determine the operating trajectory. Finally, the proposed method is validated on a test bench and compared with two other methods. The results indicate that the transient trajectory obtained in this paper exhibited the lowest vibration intensity and ASPL, which effectively enhanced the NVH performance of the APU during the transient switching process, although fuel consumption increased slightly. Although this study achieved significant results in optimizing the APU operating trajectory of EREVs, several areas remain for further exploration. For example, future research could integrate machine learning techniques with multi-objective optimization algorithms to further improve optimization accuracy and computational efficiency. Additionally, with the development of autonomous driving technology, incorporating dynamic demands from autonomous driving scenarios into the optimization model is another important direction for future research. Acknowledgements The authors gratefully acknowledge the financial support from the Open Foundation of National Key Laboratory of Multi-perch Vehicle Propulsion Systems (Project number: QDXT-WY-202407-19). Author contributions Ye Yang: Writing – original draft, Methodology, Conceptualization. Jingyi Tian: Software, Data curation. Boru Jia: Visualization, Investigation. 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