Research on Multi-Objective Parameter Matching and Stepwise Energy Management Strategies for Hybrid Energy Storage Systems

Abstract

Electric vehicle technologies present promising solutions for achieving energy conservation and emission reduction goals. However, efficiently distributing power across hybrid energy storage systems (HESSs) remains a major challenge in enhancing overall system performance. To address this, this paper proposes an energy management strategy (EMS) based on stepwise rules optimized by Particle Swarm Optimization (PSO). The approach begins by applying a multi-objective optimization method, utilizing the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to fine-tune the parameters of lithium-ion batteries and ultracapacitors for an optimal balance in system performance. Additionally, an innovative stepwise-based EMS has been designed using adaptive PSO. This strategy builds a real-time control mechanism by dynamically adjusting the power distribution gradient threshold, taking into account the compensation for the state of charge (SOC). Comparative analysis across three typical operating conditions—urban, suburban, and highway—demonstrates that the stepwise-rule optimized strategy reduces the energy consumption of the HESS by 3.19%, 7.9%, and 5.37%.

1. Introduction

With the increasingly severe environmental issues, electric vehicles (EVs) have emerged as a core solution for transforming the transportation energy system thanks to their zero emissions, high energy efficiency, and low noise characteristics [1,2,3]. The power supply system, being the core subsystem of EVs, plays a crucial role in determining overall vehicle performance, with the energy utilization efficiency of the lithium-ion battery–ultracapacitor hybrid architecture having a direct impact [4,5,6]. This study focuses on optimizing multi-objective parameter matching and energy management strategies (EMSs) for hybrid energy storage systems (HESSs), aiming to address the inherent limitations of traditional methods in terms of adaptability to dynamic conditions and global optimization capabilities.

In general, EMSs for HESSs can be broadly categorized into three types: rule-based, optimization-based, and artificial intelligence-based strategies [7,8,9]. A comparative table of existing studies is shown in Appendix A 

Table A2. Research Comparison.

Research Method	Results	Contribution
Energy management system (EMS) based on rule-inserted deep reinforcement learning.	The proposed EMS outperforms EMS without prior knowledge and other state-of-the-art deep reinforcement learning methods.	Utilizes embedded knowledge to address multi-objective optimization problems and simplify large control spaces.
Basic multi-objective Particle Swarm Optimization (PSO) algorithm to solve multi-objective optimization problems.	Achieves an effective balance among improved economy, reduced emissions, and extended battery lifespan.	Establishes a multi-objective optimization model considering energy consumption, emissions, and battery life.
Energy management strategy based on nonlinear model predictive control (NMPC) technology.	The NMPC strategy demonstrates better economy and durability than rule-based EMS, approaching the global optimal results obtained by dynamic programming.	Develops a multi-scale working condition prediction model. Innovatively applies “constant speed prediction” during transition phases, enhancing prediction accuracy and enabling online model implementation.
Multi-objective optimization energy management strategy for fuel cell hybrid electric vehicles based on rule learning.	The proposed strategy effectively reduces hydrogen consumption, extends fuel cell life, and shows potential for real-time applications.	Combines rule-based and optimization-based multi-objective EMS.
Model predictive control (MPC) based on EMS and deep Q-learning (DQL) to allocate power among multiple sources in plug-in hybrid electric vehicles (PHEV).	The proposed strategy achieves fuel economy close to that of offline stochastic dynamic programming strategies and adapts well to different state-of-charge (SOC) reference trajectories.	Integrates reinforcement learning-based and optimization-based EMS.

. Rule-based EMSs, due to their simple structure and strong real-time performance, play a crucial role in engineering applications. For example, Yuan H. B. et al. [10] introduced genetic algorithms to achieve adaptive adjustments of rule variables, improving power distribution efficiency. As application scenarios became more complex, the Schupbach R. M. team [11] innovatively developed a multi-mode switching mechanism based on demand power and ultracapacitor state of charge (SOC) mapping, which successfully reduced hybrid system acceleration time. To achieve more precise energy decoupling, Jaafar A. [12] and García P. [13] proposed a new energy distribution mechanism based on filtering principles. Their frequency-domain separation algorithm reduced the fluctuation of lithium-ion battery charge and discharge currents by 57%. However, these methods still have limitations in adapting to nonlinear time-varying systems.

Expanding upon this framework, optimization-based EMSs achieve global optimal control by constructing precise mathematical models, making significant theoretical breakthroughs. Currently, optimization-based EMSs [14] include methods such as dynamic programming (DP) [15], Pontryagin’s Minimum Principle (PMP), genetic algorithm (GA), and Particle Swarm Optimization (PSO) [16]. Compared to rule-based strategies, optimization-based approaches can more accurately account for various constraints, enabling the identification of the best balance between multiple objectives and adapting to more complex system requirements. For example, Zhang S. et al. [17] proposed a dynamic programming model based on driving behavior pattern recognition, achieving a 12.5% improvement in fuel efficiency under real-time conditions. Therefore, Peng H. et al. [18] innovatively embedded the dynamic programming results into the rule-based framework, controlling fuel economy error within 3%. Notably, the model predictive control strategy developed by Hredzak B. et al. [19] reduced the peak current of the lithium-ion battery by 61%, successfully compressing energy loss under high-load conditions to 0.8 kWh/100 km. This breakthrough provides a technological foundation for real-time optimization. Further advancing the research, Liu R. [20] developed a temperature-coupled GA model and introduced an environmental adaptability evaluation index, making a crucial step toward bridging theoretical research and engineering applications. However, when dealing with multi-objective optimization and complex constraints, the optimization process may fail to meet real-time requirements.

In contrast to the previous approaches, artificial intelligence-based energy management strategies utilize machine learning and deep learning techniques to automatically learn from data and optimize energy management, offering enhanced adaptability and flexibility [21,22,23]. For instance, Chen Z. [24] introduced a reinforcement learning-driven stochastic model predictive control method which dynamically adjusts battery power to closely match the fuel economy of the DP benchmark, overcoming the limitations of traditional MPC in prediction accuracy. Building on this, Liu Y. [25] combined simulated reinforcement learning with optimal guidance techniques, resulting in a 37% improvement in algorithm solving speed and an 8.6% reduction in overall vehicle energy loss under standard conditions. However, these methods require substantial training data and computational resources, and their lack of interpretability makes it challenging to provide transparent decision-making processes [26].

As mentioned above, the existing research has made significant progress in rule-based, optimization-based, and artificial intelligence-based energy management strategies (EMS), but these methods still face challenges in handling nonlinear time-varying systems, multi-objective optimization, and real-time performance. To achieve highly adaptable and precise energy management under varying operating conditions, there is an urgent need for a method that can automatically adjust rule parameters within a defined range, ensuring the strategy’s effectiveness and flexibility. Therefore, during the optimization process, it is essential to consider multiple constraints and objectives to achieve a globally optimal control strategy. This study proposes a hybrid energy management strategy that combines the PSO algorithm with stepwise rules. It not only retains the computational efficiency and real-time responsiveness of rule-based strategies but also enhances strategy flexibility and accuracy by optimizing gradient values through PSO. Additionally, it offers global optimization capabilities, effectively addressing the limitations of existing methods. The proposed approach demonstrates superior energy efficiency and system responsiveness, particularly under complex driving conditions.

To achieve the objectives outlined above, this paper first develops models for key elements of the HESS, including its topology, vehicle parameters, and the models of the lithium-ion battery and ultracapacitor. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is then applied to optimize the multi-objective parameter matching of the system, aiming to determine the optimal configuration of the lithium-ion battery and ultracapacitor parameters. Following this, the PSO algorithm is used to refine the gradient values in the stepwise rule, minimizing energy loss and enhancing energy utilization efficiency. Finally, the proposed energy management strategy is evaluated by comparing the performance of traditional rule-based strategies, stepwise-rule-based strategies, and the optimized stepwise-rule strategy across three typical driving conditions. This comparison demonstrates the advantages of the proposed method. The main contributions of this study are as follows: (1) The development of a multi-objective parameter matching optimization approach based on the NSGA-II algorithm for optimizing HESS parameters. (2) The introduction of a PSO-enhanced stepwise-rule energy management strategy that improves system energy efficiency and responsiveness.

The structure of this paper is organized as follows: Section 2 introduces the system modeling of the HESS for electric vehicles. Section 3 analyzes the multi-objective parameter matching method for the HESS. Section 4 provides a detailed explanation of the Particle Swarm Optimization-based stepwise EMS. Section 5 presents comparative analyses to validate the effectiveness and advantages of the proposed method. Finally, Section 6 concludes with a summary of the key findings and contributions of this study.

2. System Modeling

2.1. HESS Topology

The overall performance of the HESS largely depends on its topology, which directly influences system cost, control accuracy, and operational efficiency [27]. A well-designed topology not only enhances the power performance and energy economy of electric vehicles but also determines the synergy between energy storage units. Currently, the most common system topologies can be classified into four types: semi-active lithium-ion battery with DC/DC connection, semi-active ultracapacitor with DC/DC connection, fully active, and passive parallel topologies. This study comprehensively considers system energy loss, DC/DC converter efficiency, and structural design complexity, ultimately selecting the semi-active topology shown in Figure 1. This architecture leverages the real-time voltage regulation capability of the DC/DC converter to effectively control the voltage at the lithium-ion battery side and limit peak currents, thus extending the battery’s lifespan. Meanwhile, the ultracapacitor, with its high-power density and fast charge/discharge characteristics, effectively handles instantaneous high-power loads, significantly improving the overall energy management efficiency of the system.

2.2. Modeling of EV and Configuration

Once the power system topology is determined, the near-matching of component parameters becomes the key to optimizing the overall energy management of the vehicle [28]. The vehicle’s driving power is primarily influenced by rolling resistance, air resistance, gradient resistance, and acceleration resistance. To address this, the study establishes mathematical models based on dynamic principles to describe the vehicle’s driving state and power requirements, as shown in Equations (1) and (2), thereby enabling a quantitative representation of various resistances.

$$F_t=F_f+F_w+F_i+F_j=mgf\cos \alpha +{\displaystyle \frac{C_{d}A{v}^2}{21.15}}+mg\sin \alpha +\delta m{\displaystyle \frac{du}{dt}}$$

(1)

$$P_{req}={\displaystyle \frac{1}{{\eta }_t{\eta }_m{\eta }_{DC}}}\left({\displaystyle \frac{mgf}{3600}}v+{\displaystyle \frac{C_{d}A{v}^2}{76140}}v+{\displaystyle \frac{mg\sin \alpha }{3600}}v+{\displaystyle \frac{\delta m\frac{du}{dt}}{3600}}v\right)$$

(2)

where $F_t$ represents the vehicle’s driving force. $F_f$ denotes rolling resistance. $F_w$ represents air resistance. $F_i$ is gradient resistance. $F_j$ represents acceleration resistance. $P_{req}$ denotes the power required for vehicle operation in kilowatts, and $m$ is the total vehicle mass in kilograms. $g$ represents gravitational acceleration. $f$ is the rolling resistance coefficient, and $C_d$ is the air resistance coefficient. $\delta$ is the rotational mass conversion factor, and $A$ represents the frontal area. ${\eta }_t$ denotes the transmission efficiency, ${\eta }_m$ is the efficiency of the driving motor, and ${\eta }_{DC}$ represents the efficiency of the DC/AC converter. The vehicle’s key technical parameters and design performance specifications are summarized in

Table 1. Vehicle technical parameters.

Parameters	Values
Vehicle mass (m)	2170 kg
Gravitational acceleration (g)	9.80
Air resistance coefficient (Cd)	0.233
Rolling resistance coefficient (f)	0.233
Rotational inertia conversion factor (δ)	1.02
Frontal area (A)	2.27 m2
Transmission efficiency (ηt)	0.90
Motor efficiency (ηm)	0.90
DC/AC converter efficiency (ηCD)	0.95

and

Table 2. Design performance indicators.

Parameters	Values
Maximum vehicle speed	≥185 km/h
0–100 km/h Acceleration time	≤7.9 s
Maximum gradeability	≥20%
Driving range	≥506 km

, providing a solid data foundation for the subsequent system design and energy matching.

The model incorporates key parameters such as vehicle mass, drag coefficient, transmission, and motor efficiency, ensuring theoretical rigor, computational efficiency, and real-time responsiveness. Due to its simple structure and the generality of its parameters, the model can be flexibly adapted to different vehicle types and driving conditions. This makes it suitable for designing energy management strategies for electric vehicles, evaluating and predicting driving states, and providing a reliable theoretical foundation for the optimal design of vehicle energy management systems.

2.3. Lithium-Ion Battery and Ultracapacitor Model

To accurately assess the performance of the HESS, this study selects a 3.6 V-2 Ah 18650 nickel–cobalt–manganese ternary lithium-ion battery and a 2.7 V-1500 F ultracapacitor as the key energy storage components. The capacity, voltage, and charge/discharge characteristics of these components are measured through experiments, including capacity (CAP), Hybrid Pulse Power Characterization (HPPC), and Urban Dynamometer Driving Schedule (UDDS) tests, conducted at temperatures of 10 °C, 25 °C, and 40 °C [28]. The results show that the lithium-ion battery’s capacity slightly increases with temperature, while the ultracapacitor ’s capacity decreasing marginally with temperature. These data provide a reliable basis for the extraction of model parameters.

High-precision modeling is typically achieved through three main methods: electrochemical models, artificial neural network models, and equivalent circuit models. Electrochemical models offer the highest accuracy by describing the internal electrochemical reactions of lithium-ion batteries using a series of partial differential and algebraic equations. However, due to their complexity and the large number of parameters involved, they are computationally expensive. Artificial neural network models, which rely on extensive experimental data for training, enable efficient predictions. However, these models are highly dependent on data and face limitations when adapting to complex operating conditions. In contrast, equivalent circuit models use simple components such as voltage sources, capacitors, and resistors to simulate the external characteristics of lithium-ion batteries. These models are simple in structure and computationally efficient, offering distinct advantages in real-time energy management.

Considering the need for real-time performance and computational efficiency, this study selects the Thevenin model from the equivalent circuit models to represent the lithium-ion battery and ultracapacitor, based on experimental data [29,30]. Figure 2 shows the equivalent circuit models of the lithium-ion battery and ultracapacitor.

The discrete circuit equations for the Thevenin models of the lithium-ion battery and ultracapacitor are given by Equation (3):

$$\left\{\begin{array}{l}{U}_d(k)=U_d(k-1)e^{-{\displaystyle \frac{\Delta t}{\tau }}}+i_{L,k}{R}_d\left(1-e^{-{\displaystyle \frac{\Delta t}{\tau }}}\right)\hfill \\ U_t(k)=U_{ocvb}(k)-U_d(k)-i_{L,k}{R}_i\hfill \end{array}\right.$$

(3)

where $\Delta t$ represents the sampling time interval. $k$ is the sampling time. $U_d(k)$ denotes the polarization voltage. $U_t(k)$ is the terminal voltage, and $\tau$ is the time constant, which is the product of $R_d$ and $C_d$. Additionally, the ampere-hour integration method is used to estimate the SOC, as shown in Equation (4).

$$SOC(t)=SO{C}_0(t_0)-{\displaystyle \frac{{\displaystyle {\int }_{t_0}^t\eta }{i}_{L}dt}{C_a}}$$

(4)

where $SOC(t)$ represents the estimated SOC at time t, and $SO{C}_0(t_0)$ is the initial SOC value at $t_0$. $\eta$ and $C_a$ denote the charge/discharge efficiency and rated capacity of the lithium-ion battery or ultracapacitor. $i_L$ is the charge/discharge current of the lithium-ion battery or ultracapacitor at time $\tau$.

The validation results based on the UDDS driving cycle show that, regardless of the environmental temperature or SOC range, the model consistently maintains low error and fully meets the requirements for real-world vehicle applications. This provides solid theoretical support for the subsequent optimization of EMS.

2.4. DC/DC Converter Modeling

Given the significant differences in voltage characteristics between the lithium-ion battery and ultracapacitor, using a DC/DC converter to achieve effective electrical isolation and independent operation of the two is crucial. On the one hand, the converter not only enables precise control of charge and discharge currents but also dynamically adjusts the charging and discharging strategy, optimizing energy distribution and enhancing system efficiency and stability. On the other hand, it effectively prevents surge currents and voltage fluctuations, mitigating their adverse impact on battery lifespan and overall vehicle performance. Although complex models may increase the computational burden of EMSs, this study, through reasonable simplifications, still achieves precise voltage control and stable power output while meeting the 30 kW rated power requirement, as demonstrated by the calculations in

Table 3. The efficiency of the DC/DC converter.

${\mathit{\epsilon }}_{\mathbf{dc}}\left({\mathit{I}}_{\mathit{d}\mathit{c}},{\mathit{P}}_{\mathit{d}\mathit{c}}\right)$	0	5 kW	10 kW	20 kW	30 kW	40 kW	50 kW	≥120 kW
0 A	50%	50%	50%	50%	50%	50%	50%	50%
5 A	63%	67%	71%	73%	74%	73%	72%	72%
10 A	75%	84%	92%	95%	97%	95%	94%	94%
50 A	73%	82%	91%	93%	96%	93%	92%	92%
100 A	72%	80%	88%	91%	95%	92%	91%	91%
150 A	70%	76%	82%	89%	92%	91%	90%	90%
≥300 A	70%	76%	82%	89%	92%	91%	90%	90%

.

3. Multi-Objective Parameter Matching Method for HESS

3.1. Performance Requirement Analysis

The HESS of an electric vehicle directly affects the overall operational efficiency and driving experience, making it essential for the design to meet both energy and power demands. For example, during long-distance cruising or high-speed driving, the system must have a high energy density to ensure prolonged driving range. In contrast, in urban driving conditions with frequent starts, stops, accelerations, and decelerations, the ability to quickly release and recover energy becomes particularly important. Additionally, different driving scenarios have varying instantaneous power requirements. For instance, highway driving and climbing demand that the system can quickly provide sufficient output power, while steady driving focuses on maintaining a stable power output.

To address these requirements, this study calculates and analyzes six typical driving cycle conditions—REP05, NYCC, US06, UDDS, IM240, and SC03—using simulation software based on the vehicle’s technical parameters provided in Equation (2) and Table 1. The positive average power reflects the power demand in scenarios such as acceleration or climbing, where energy is output, while the negative average power corresponds to energy recovery scenarios, such as braking or descending. As illustrated in Figure 3 and

Table 4. The power and energy requirements of the vehicle for the HESS under six typical cycle conditions and the efficiency of the DC/DC converter.

Performance Parameters	UDDS	NYCC	US06	REP05	IM240	SC03
Positive energy demand/kJ	7680.1	1576.2	11128	23824	2393.6	4061.1
Negative energy demand/kJ	−3991.3	−1329.2	−4617	−7140.2	−1276.6	−2294.7
Positive power duration/s	761	222	396	1022	161	327
Negative power duration/s	350	167	160	332	68	157
Positive average power/kW	10.09	7.1	28.10	23.31	14.96	12.41
Negative average power/kW	−11.10	−7.95	−28.85	−21.50	−18.77	−14.61
Positive peak power/kW	49.91	41.68	120.41	116.53	49.91	65.35
Negative peak power/kW	−43.25	−39.01	−94.52	−95.75	−72.92	−69.02

, these results provide clear and valuable reference standards for the design and optimization of the HESS.

3.2. Energy and Power Configuration

Since electric vehicles have diverse energy and power requirements under different operating conditions, to fully exploit the advantages of both lithium-ion batteries and ultracapacitors, this section conducts an in-depth analysis of the energy and power configuration of the HESS from a system-wide perspective. Lithium-ion batteries, with their high energy density, support the vehicle’s long driving range, and their series configuration ensures that the system voltage meets load requirements. On the other hand, ultracapacitors, due to their high-power density and fast charge/discharge capabilities, provide high power output or enable efficient energy recovery during transient conditions such as starting, acceleration, and braking.

To enable the lithium-ion battery and ultracapacitor to work together synergistically and complement each other’s strengths, this study proposes four key constraint equations. The energy matching constraint equation for the lithium-ion battery pack determines the total energy that the lithium-ion battery pack needs to provide and the number of series-connected cells required to ensure that the range and voltage matching requirements are met. The first constraint equation is shown in Equation (5).

$$N_{bat,p}\ge {\displaystyle \frac{1000E_{506}}{C_{bat,cell}{U}_{bat,cell}{N}_{bat,s}{\eta }_{load}{\eta }_{dis}}}$$

(5)

where $E_{506}$ represents the energy required for the vehicle to travel a distance of 506 km. $C_{bat,cell}$ denotes the capacity of a single lithium-ion battery cell. $U_{bat,cell}$ is the rated voltage of a single lithium-ion battery cell. $N_{bat,s}$ is the number of series-connected lithium-ion battery cells. $N_{bat,p}$ is the number of parallel branches in the lithium-ion battery pack. ${\eta }_{load}$ represents the maximum depth of discharge of the lithium-ion battery, and ${\eta }_{dis}$ is the energy transfer efficiency of the lithium-ion battery.

Based on the demand for instantaneous high power during conditions such as starting, acceleration, and braking, the constraint equation for the number of ultracapacitor cells in parallel is established to ensure that the ultracapacitor pack has sufficient energy reserves at critical moments. This ensures that the system’s energy requirements align with the characteristics of the components. This is represented by Equation (6).

$$N_{uc,p}\ge {\displaystyle \frac{720000\max \left(E_{start\_max}-E_{diss\_max}-E_{reg\_max}\right)}{C_{uc,cell}{N}_{uc,s}\left(U_{uc,max}^2-U_{uc,min}^2\right)}}$$

(6)

where $N_{uc,s}$ and $N_{uc,p}$ represent the number of series-connected cells in each branch of the ultracapacitor pack and the number of parallel branches in the ultracapacitor pack. $C_{uc,cell}$ denotes the capacity of a single ultracapacitor cell. $U_{uc,max}$ is the maximum voltage of a single ultracapacitor cell, and $U_{uc,min}$ is the minimum cutoff voltage of a single ultracapacitor cell.

Considering the maximum positive peak power demand of the vehicle under various typical operating conditions, the power output constraint equation for the vehicle is established to quantify the power distribution capability of the HESS. This represents the third constraint equation, as shown in Equation (7).

$$\left\{\begin{array}{l}{P}_{bat}+P_{uc}\ge P_{veh,\max }\\ P_{bat}=P_{hess,avg}\\ P_{uc}=N_{uc,p},N_{uc,s},m_{uc,cell}{p}_{uc}\end{array}\right.$$

(7)

where $P_{bat}$ represents the specific power of the lithium-ion battery. $P_{uc}$ denotes the specific power of the ultracapacitor. $m_{uc,cell}$ is the mass of a single ultracapacitor cell. $P_{veh,\max }$ is the maximum positive peak power under the six typical operating conditions, and $P_{hess,avg}$ is the maximum positive average power under the six typical operating conditions.

To prevent damage to the lithium-ion battery from high-current discharges, a safety discharge constraint equation is introduced to impose a reasonable limit on the discharge rate of the battery, as shown in Equation (8).

$$k{C}_{bat}{U}_{bat}\ge P_{hess,avg}$$

(8)

where $k$ represents the discharge rate of the lithium-ion battery pack. $C_{bat}$ denotes the capacity of the lithium-ion battery pack, and $U_{bat}$ is the DC bus voltage.

3.3. Multi-Objective Parameter Natching Optimization

3.3.1. Multi-Objective Optimization Model

The parameter matching problem of the HESS for electric vehicles is considered a multi-objective optimization problem. This study takes into account three optimization objectives: total cost, total mass, and total volume. To address this problem, a multi-objective optimization function is constructed, with the number of parallel-connected lithium-ion battery cells and ultracapacitor cells as optimization variables. The goal of the optimization is to simultaneously minimize these metrics, ensuring a balance between the vehicle system’s economy and efficiency. The optimization model is represented by Equation (9).

$$\underset{x\in \Omega }{\min }=[J_c(x),M_z(x),V_z(x)]$$

(9)

where $J_c$ represents the total cost of the electric vehicle. $M_z$ and $V_z$ denote the total mass and total volume of the HESS.

The total cost of the electric vehicle, $J_c$, consists of three components: energy consumption cost $C_{ele}$, purchase cost $C_{buy}$, and maintenance cost $C_{wh}$. The mathematical model for the total cost is represented by Equation (10).

$$\left\{\begin{array}{c}{J}_c=C_{ele}+C_{buy}+C_{wh}\\ C_{ele}={\displaystyle \frac{S}{365}}\cdot {\displaystyle \frac{Q_{bat}{m}_{ele}U}{1000s_{car}}}\\ C_{buy}=\left(Q_{bat}\cdot C{a}_{bat}+Q_{uc}\cdot C{a}_{uc}\right)\cdot CRF\\ C_{wh}=\left(C_{batwh}+C_{ucwh}\right)/365\\ CRF={\displaystyle \frac{I{\left(1+I\right)}^{T_{sys}}}{{\left(1+I\right)}^{T_{sys}}-1}}\end{array}\right.$$

(10)

where $S$ represents the total annual driving distance of the electric vehicle. $Q_{bat}$ is the capacity of the lithium-ion battery pack; $m_{ele}$ is the electricity price; and $U$ is the voltage of the lithium-ion battery pack. $C{a}_{bat}$ and $C{a}_{uc}$ are the purchase costs of the lithium-ion battery and ultracapacitor, respectively. $CRF$ denotes the capital recovery factor [23]; $I$ is the interest rate. $T_{sys}$ represents the service life of the electric vehicle. $C_{batwh}$ and $C_{ucwh}$ represent the annual maintenance costs of the lithium-ion battery pack and ultracapacitor pack, respectively.

The total mass and total volume of the electric vehicle’s HESS are calculated using Equation (11).

$$\left\{\begin{array}{l}{M}_z=N_{bats}\cdot N_{batp}\cdot m_{bat}+N_{ucs}\cdot N_{ucp}\cdot m_{uc}\\ V_z=N_{bats}\cdot N_{batp}\cdot V_{bat}+N_{ucs}\cdot N_{ucp}\cdot V_{uc}\end{array}\right.$$

(11)

where $m_{bat}$ and $m_{uc}$ represent the mass of a single lithium-ion battery and ultracapacitor cell. $V_{bat}$ and $V_{uc}$ represent the volume of a single lithium-ion battery and ultracapacitor cell.

3.3.2. Parameter Matching Optimization Results

To optimize the parameter matching, this study experiments with multiple models of lithium-ion batteries and ultracapacitor cells. Different models of lithium-ion batteries and ultracapacitors have varying capacity, mass, volume, and cost characteristics, allowing the system design to flexibly balance multiple objectives. Considering the electric vehicle’s needs for driving range, weight, and space, using various battery and capacitor models enables a more comprehensive exploration of the optimization results under different configurations.

This study uses the NSGA-II algorithm to conduct a comprehensive optimization of three objectives [31,32]: total cost, total mass, and total volume of the system. The optimization parameters for the NSGA-II algorithm are set as shown in

Table 5. NSGA-II algorithm parameter settings.

Parameters	Values
Population Size	200
Maximum Iterations	100
Optimization Variables	2
$N_{batp}$	[1,15]
$N_{u\mathrm{c}p}$	[1,15]
Lithium-ion Battery Purchase Cost (CNY/kWh)	2000
Ultracapacitor Purchase Cost (CNY/kWh)	4500
Electricity Price (CNY/kWh)	0.725
Maximum Vehicle Lifetime (years)	30
Interest Rate (I)	0.025

.

Through simulation calculations, multiple Pareto optimal solution sets for the HESS were obtained, and the trade-offs between the objective functions under different configurations were analyzed. The optimization results are shown in Figure 4. It is observed that these three objectives cannot be simultaneously optimized in a single configuration and require reasonable trade-offs between them. By substituting the matching schemes corresponding to the Pareto solutions into the simulation model, the HESS parameter matching optimization results, as shown in

Table 6. HESS parameter matching optimization results.

Parameters	Results
configuration/parallel	(11,5)	(11,8)	(3,8)	(3,5)	(8,8)	(11,5)
specifications	25 Ah + 1200 F	25 Ah + 650 F	90 Ah + 650 F	90 Ah + 1200 F	35 Ah + 650 F	25 Ah + 1200 F
Total cost (CNY)	232,683	166,033	141,046	207,695	153,133	232,683
Total mass (kg)	856.83	865.53	920.4	911.7	942.72	856.83
Total volume (L)	445.02	463.99	587.68	568.71	652.78	445.02

, can be obtained.

Comparing five different HESS configurations, the following conclusions were drawn: The 25 Ah + 1200 F configuration shows a clear advantage in total mass and total volume, while the 90 Ah + 1200 F configuration is the most economical in terms of total cost. After considering all three objective functions, the 25 Ah + 650 F configuration was ultimately chosen as the optimal solution. The HESS configuration is as follows: 93 series-connected 25 Ah lithium-ion battery cells, with 11 parallel branches; 87 series-connected 650 F ultracapacitor cells, with 8 parallel branches. This configuration achieves a good balance between total cost, total mass, and total volume, effectively meeting the performance requirements of the electric vehicle while optimizing the overall system design.

4. PSO-Based Stepwise Energy Management Strategy for HESS

4.1. PSO Algorithm

The PSO algorithm is a swarm intelligence algorithm inspired by the information sharing and collaborative search behavior observed in bird flocks during foraging [33]. In PSO, each particle represents a potential solution, and it continuously adjusts its position and velocity in the solution space to search for the global optimal solution. The algorithm primarily guides the particle’s updates through the individual’s historical best position (pbest) and the global best position (gbest).

During each iteration, the particle dynamically updates its velocity and position based on the current fitness value and the distance to pbest and gbest, thereby enabling global search. The update formulas for the particle’s velocity and position in the PSO algorithm are shown in Equations (12) and (13).

$$V_{ij}(t+1)=V_{ij}(t)+c_1{r}_1(pbes{t}_{ij}(t)-X_{ij}(t))+c_2{r}_2(gbes{t}_j(t)-X_{ij}(t))$$

(12)

$$X_{ij}(t+1)=X_{ij}(t)+V_{ij}(t+1)$$

(13)

Although standard PSO exhibits strong global search capabilities for continuous optimization problems, it often struggles with discrete or complex problems, as it can easily become trapped in local optima and lacks sufficient local search ability. To address this limitation, an inertia weight is introduced. The inertia weight adds a momentum term to the velocity update formula, allowing the particle to maintain its previous search direction while adjusting its search pace based on environmental feedback during the exploration process. The update formula with inertia weight is shown in Equation (14).

$$V_{ij}(t+1)=\omega V_{ij}(t)+c_1{r}_1(pbes{t}_{ij}(t)-X_{ij}(t))+c_2{r}_2(gbes{t}_j(t)-X_{ij}(t))$$

(14)

where $\omega$ represents the inertia weight, which directly influences the algorithm’s search behavior: larger $\omega$ favors global search, while smaller $\omega$ enhances local exploration. To provide the algorithm with broad exploration capabilities in the early stages and focus on fine-tuning in later stages, this study adopts a dynamic inertia weight mechanism. A common strategy is to use a linear decreasing method, where the inertia weight gradually decreases from a larger value to a smaller value as the number of iterations increases, as shown in Equation (15).

$$\omega ={\omega }_{\max }-({\omega }_{\max }-{\omega }_{\min })\times {\displaystyle \frac{i}{T_{\max }}}$$

(15)

where ${\omega }_{\max }$ represents the maximum inertia weight. ${\omega }_{\min }$ represents the minimum inertia weight. $i$ is the current iteration number, and $T_{\max }$ is the maximum number of iterations. By dynamically adjusting the inertia weight, the PSO algorithm can thoroughly explore the solution space in the early stages to avoid premature convergence, while in the later stages, it enhances local search, further improving optimization accuracy.

4.2. Determination of Optimization Variables and Objective Functions

The EMS based on logical thresholds achieves energy switching between the lithium-ion battery and ultracapacitor by presetting fixed threshold values. This approach offers high computational efficiency and robustness. However, the threshold parameters in this strategy are entirely reliant on expert experience, and the static threshold setting makes it difficult to adapt to dynamic operating conditions. Such a static design may result in the ultracapacitor’s “peak shaving and valley filling” function being underutilized, which, in turn, affects the overall system performance and accelerates the aging process of the lithium-ion battery.

To address this issue, an EMS based on stepwise rules has emerged. This strategy smooths the lithium-ion battery’s output through stepwise power distribution, effectively mitigating instantaneous current shocks, thereby extending the battery’s lifespan. However, its stepwise gradient parameters still need to be manually set, and it lacks a dynamic adjustment mechanism, making it difficult to achieve optimal energy distribution efficiency under complex and changing operating conditions.

To overcome the inherent limitations of manually set parameters in both strategies, this study proposes a new method that aims to minimize energy loss in the HESS as the objective function. The PSO algorithm is introduced to dynamically optimize the stepwise gradient parameters in real-time, enabling adaptive energy distribution under dynamic conditions. The energy loss in the HESS primarily arises from three aspects: energy loss in the lithium-ion battery pack, energy loss in the ultracapacitor pack, and energy loss in the DC/DC converter. This can be expressed by Equation (16).

$$LOS{S}_{all}(\Delta Pi)=P_{bat}loss(\Delta Pi)+P_{uc}loss(\Delta Pi)+P_{dc}loss(\Delta Pi)$$

(16)

where $LOS{S}_{all}(\Delta Pi)$ represents the total energy loss, $s$ denotes the stepwise gradient value. $P_{bat}loss(\Delta Pi)$ represent the energy losses in the lithium-ion battery pack, $P_{uc}loss(\Delta Pi)$ is the energy losses in ultracapacitor pack, and $P_{dc}loss(\Delta Pi)$ denotes the energy losses in DC/DC converter. The energy loss functions for each part are expressed by Equation (17).

$$\left\{\begin{array}{l}{P}_{bat}loss=i_b^2\times R_i+U_d^2/R_d\\ P_{uc}loss=i_u^2\times R_c+U_R^2/R_u\\ P_{dc}loss=P_{bat}\times (1-{\eta }_{dc})\times {\eta }_{dc}^{-m}\\ LOS{S}_{all}=\min (P_{bat}loss+P_{uc}loss+P_{dc}loss)\end{array}\right.$$

(17)

4.3. Parameter Settings

Setting the stepwise gradient value appropriately is crucial for enhancing the overall performance of the HESS. A suitable gradient value not only allows the system to effectively respond to changes in power demand, enabling efficient energy distribution between the lithium-ion battery and ultracapacitor, thereby reducing unnecessary energy loss, but also maximizes the ultracapacitor’s rapid charge/discharge characteristics under high power demand conditions, significantly relieving the load on the lithium-ion battery. This helps reduce the high-frequency charge/discharge cycles of the lithium-ion battery, lowers the charge/discharge amplitude, and extends the battery’s lifespan. In this study, the stepwise gradient value Pi is considered an optimization variable, and through extensive numerical simulations, the reasonable value range is determined to be between 2000 and 4000. Values outside this range will result in reduced ultracapacitor utilization efficiency.

The optimization objective of this study is to minimize the total energy loss in the HESS. The energy loss primarily arises from the lithium-ion battery pack, ultracapacitor pack, and DC/DC converter. After determining the optimization variables and their value range, the key parameters for the PSO algorithm, including the initial population size, maximum number of iterations, speed factors $C_1$ and $C_2$, speed boundaries, variable dimensions, and inertia weight, are set. The specific parameter configurations are shown in

Table 7. Algorithm parameter settings.

Parameters	Values
Initial population size	50
Maximum number of iterations	25
$\mathrm{Velocity} \mathrm{factor} C_1$	2
$\mathrm{Velocity} \mathrm{factor} C_2$	2
Velocity boundary value	−50~+50
Variable dimension	1
Maximum inertia weight	0.9
Minimum inertia weight	0.4

.

5. Results and Discussion

5.1. Optimization Results and Operating Condition Selection

This study optimizes the stepwise gradient value using three different operating conditions: UDDS, INDIA_HWY_SAMPLE, and WVUSUB. By programming the corresponding algorithm based on PSO, the objective function values are calculated, and the solution variations are recorded to obtain the optimal solution. After running the simulation, the particle swarm algorithm iteration curve is obtained. After 20 iterations, the objective function value stabilizes around 1.03702 × 10⁶, and the improvement in subsequent iterations is minimal, indicating that the algorithm has converged.

Due to the differences in data characteristics across various operating conditions, the optimized stepwise gradient values and corresponding objective function values are different for each condition. The results are shown in

Table 8. Optimization results under different working conditions.

Parameters	$\mathit{P}\mathit{i}$ Values	Objective Function Value
UDDS	2501.9	1037.04 kJ
INDIA_HWY_SAMPLE	2714.3	893.84 kJ
WVUSUB	2000.6	1011.6 kJ

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5.2. Comparison of SOC

The SOC is a critical metric for evaluating the effectiveness of energy management between the lithium-ion battery and ultracapacitor. This study explores the impact of rule-based control strategies, stepwise-rule control strategies, and stepwise-rule optimization control strategies on the performance of the lithium-ion battery and ultracapacitor under different operating conditions. The evaluation criteria include the terminal SOC values of the lithium-ion battery and ultracapacitor, as well as the maximum discharge and charge currents. The specific results are shown in

Table 9. DC/DC convertor efficiency table.

Name	Rule Based			Stepwise-Rule Based			Stepwise-Rule Optimization Based
Operating condition	City	Highway	Suburban	City	Highway	Suburban	City	Highway	Suburban
Battery terminal SOC	0.7760	0.7749	0.7872	0.7766	0.7827	0.8482	0.7777	0.7850	0.7906
ultracapacitor terminal SOC	0.9008	0.8898	0.6753	0.9022	0.7652	0.8603	0.8917	0.7254	0.6276
Battery maximum discharge current	38.61	44.87	29.86	36.68	39.52	39.35	35.29	38.30	28.22
ultracapacitor maximum discharge current	134.91	75.41	80.53	140.27	90.93	81.29	145.19	96.04	84.48
Battery maximum charge current	−74.98	−46.58	−60.72	−30.63	−32.45	−24.05	−28.94	−11.51	−23.10
ultracapacitor maximum charge current	−149.63	−118.40	−167.61	−154.37	−133.17	−167.19	−157.75	−138.66	−169.68

.

The variations in the SOC of the lithium-ion battery and ultracapacitor under different control strategies for three operating conditions are illustrated in Figure 5. The results show that the ladder-rule-based control strategy significantly reduces the energy consumption of the lithium-ion battery compared to the traditional rule-based strategy. The final SOC values of the lithium-ion battery were 0.7766, 0.7827, and 0.8482, which represent improvements of about 0.5%, 0.44%, and 0.42%, respectively, over the rule-based control strategy. For the ultracapacitor, the stepwise-rule optimization control strategy maintained its SOC between 0.5 and 0.9, demonstrating more stable and reliable performance. In the UDDS, INDIA-HWY-SAMPLE, and WVUSUB conditions, the maximum differences between the two strategies were 1.64%, 7.2%, and 7.1%, respectively, indicating that the ladder-rule-based strategy makes better use of the ultracapacitor’s ability to respond to rapid energy demands during fast charging and discharging.

Under the stepwise-rule optimization control strategy, the terminal SOC of the lithium-ion battery was 0.7777, 0.7850, and 0.7906, showing improvements of 0.2%, 1.3%, and 0.4% compared to the traditional rule-based strategy. This suggests that the stepwise optimization strategy more effectively maintains the lithium-ion battery’s charge, thus improving overall energy efficiency. While the terminal SOC of the ultracapacitor showed slight decreases in UDDS and INDIA-HWY-SAMPLE conditions, dropping to 0.8917 and 0.7254, respectively, the SOC decreased by 7.1% in the WVUSUB condition, reaching 0.6276. This suggests that the stepwise-optimization strategy relies more heavily on the ultracapacitor’s fast charge/discharge characteristics to handle instantaneous energy needs. These results demonstrate that the stepwise-optimization control strategy provides clear advantages in optimizing energy management for the HESS, ensuring efficient and reliable energy distribution under varying conditions.

5.3. Comparison of Current

Comparing the current characteristics under different operating conditions, the SOC indicator intuitively reflects the energy distribution effectiveness. Meanwhile, the current indicator further reveals the actual performance of each control strategy in terms of reducing the load on the lithium-ion battery, protecting the battery, and utilizing the ultracapacitor ’s “peak shaving and valley filling” function.

Based on the data in Figure 6a,c,d, the maximum discharge currents of the lithium-ion battery under the ladder-optimization control strategy were 35.29 A, 38.30 A, and 28.22 A in the three operating conditions, with the fluctuation range being smaller compared to the ladder-rule control strategy. When compared to the traditional rule-based control strategy, the maximum discharge current of the lithium-ion battery decreased by 8.5%, 14.6%, and 5.5%, respectively. This suggests that after stepwise optimization, the lithium-ion battery experiences significantly less current shock under high-load conditions, which improves its operational stability and contributes to a longer lifespan.

As shown in Figure 6b,d,f, it can be seen that the ultracapacitor primarily handles the task of providing and absorbing high-frequency currents. Under the ladder-optimization control strategy, the peak currents of the ultracapacitor were 145.19 A, 96.04 A, and 84.48 A in the three operating conditions, which are 7%, 27%, and 4.8% higher, respectively, than those under the rule-based control strategy. This shows that with the optimized strategy, the ultracapacitor is able to take on a greater role in energy regulation, effectively shifting the instantaneous high-current loads away from the lithium-ion battery, thus offering better protection for it.

Furthermore, comparing the maximum discharge and charging current data for both the lithium-ion battery and ultracapacitor reveals the same result. Under the rule-based control strategy, the lithium-ion battery’s maximum discharge currents were 38.61 A, 44.87 A, and 29.86 A. In contrast, the ladder-optimization strategy reduced these values by 8.5%, 14.6%, and 3%, respectively. Meanwhile, the ultracapacitor’s maximum charging currents under the rule-based strategy were 5.4%, 17.1%, and 1.2% lower than those under the ladder-optimization strategy, further highlighting that the optimized strategy allowed the ultracapacitor to take on more of the braking energy recovery task, thus easing the load on the lithium-ion battery.

The experimental data show that the ladder-optimization control strategy is effective in reducing the peak currents and discharge shocks to the lithium-ion battery in all operating conditions. It also enables the ultracapacitor to more fully perform its role in “peak shaving and valley filling,” which not only improves the battery’s operational stability but also reduces its load under high-demand conditions, significantly contributing to extending its lifespan.

5.4. Comparison of Energy Loss

Energy loss is an important measure of system efficiency and component load, offering a direct insight into how effectively different control strategies allocate energy between the lithium-ion battery and ultracapacitor. The experimental results in Table 4 and Table 5 show that under various operating conditions, the energy loss in the HESS with the traditional rule-based control strategy is consistently higher than that with the ladder-rule control strategy and the optimized stepwise control strategy. As shown in

Table 10. Comparison of energy loss under different working conditions.

Operating Condition	Strategy	Composite Power	Battery	Ultracapacitor	DC/DC
UDDS	Rule (kJ)	1071.26	217.25	20.39	833.62
	Stepwise rule (kJ)	1041.64	164.55	33.38	843.72
	Stepwise-rule optimization (kJ)	1037.04	156.67	35.86	844.51
INDIA-HWY-SAMPLE	Rule (kJ)	970.60	212.23	11.18	747.19
	Stepwise rule (kJ)	910.50	181.15	15.70	713.66
	Stepwise-rule optimization (kJ)	893.84	172.51	17.16	704.18
WVUSUB	Rule (kJ)	80.7%	1069.01	128.22	23.29
	Stepwise rule (kJ)	14.3%	1028.40	124.32	23.85
	Stepwise-rule optimization (kJ)	65.8%	1011.6	117.74	25.73

, with the rule-based control strategy, the system energy loss values were 1071.26 kJ, 970.6 kJ, and 1068.01 kJ. However, when the stepwise-rule strategy was applied, these values dropped to 1041.64 kJ, 910.50 kJ, and 1028.40 kJ, respectively. Further optimization with the PSO-adjusted stepwise control strategy resulted in a reduction in energy loss by 3.19%, 7.9%, and 5.37% across the different operating conditions.

Additionally, the data reveal that the energy loss of the lithium-ion battery is much higher than that of the ultracapacitor, and the energy loss in the DC/DC converter accounts for around 70% of the total. This points to the need, in future optimizations of EMSs, to not only focus on the energy consumption of the lithium-ion battery but also to work on minimizing the energy loss in the converter.

6. Conclusions

This study presents stepwise EMSs optimized by the PSO algorithm, aiming to efficiently manage peak energy demands through the ultracapacitor, thereby extending the lithium-ion battery’s stable output time. The research begins by analyzing the key performance aspects of the lithium-ion battery, ultracapacitor, and bidirectional DC/DC converter within the HESS, based on a semi-active topology. A multi-objective optimization framework, which incorporates both economic and physical constraints, is then developed, and the NSGA-II algorithm is employed to find the Pareto optimal solutions. Building on this, the PSO algorithm is applied to adaptively optimize the stepwise control parameters in the stepwise EMS. The results show that, compared to the traditional rule-based control strategy, this method reduces energy consumption by 3.19%, 7.9%, and 5.37% under UDDS, HWFET, and US06 conditions, respectively, demonstrating its superiority.

The main theoretical contributions of this study include the following: First, it innovatively proposes a parameter matching optimization model that considers multi-physical-field constraints and uses a multi-objective optimization algorithm to effectively balance system performance with economic considerations, offering new insights into the design of HESS for electric vehicles. Second, by combining the dynamic inertia weight characteristics of the PSO algorithm with the discrete control advantages of the ladder-rule energy management strategy, a dynamic threshold adjustment mechanism is developed, which adapts to varying operating conditions.

However, there are some limitations in this study: On the one hand, the current optimization model focuses primarily on cost, weight, and volume, without considering factors like battery aging and lifespan. Future research should incorporate a battery aging model into the multi-objective optimization framework to more comprehensively balance performance and durability. On the other hand, although the strategy performs well under standard conditions, the ultracapacitor may still face challenges in energy absorption under extreme conditions. Future work could explore more flexible control rules or introduce dynamic adjustment mechanisms to further enhance system stability and energy efficiency in such scenarios.