Hybrid multi-objective optimization for energy efficiency and battery health in distributed in-wheel motor electric vehicles

Abstract

Achieving both high energy efficiency and extended battery lifespan remains a major challenge in distributed in-wheel motor electric vehicles. This study proposes a hybrid multi-objective optimization framework combining the Asynchronous Advantage Actor–Critic (A3C) algorithm with an Improved Sparrow Search Algorithm (ISSA). The A3C algorithm performs real-time torque allocation to maximize instantaneous energy efficiency, while the ISSA optimizes long-term battery management strategies by explicitly embedding capacity fading and thermal safety constraints. To coordinate these conflicting objectives across different time scales, a dynamic weight adjustment mechanism and a cross-scale shared memory pool are introduced. Simulation results under NEDC, WLTC, and CLTC driving cycles demonstrate that the framework significantly outperforms conventional algorithms in terms of convergence speed, energy consumption reduction, and battery durability. Furthermore, Hardware-in-the-Loop (HIL) experiments confirm the framework’s excellent real-time performance and control accuracy on a physical controller, validating its engineering feasibility.

Introduction

Distributed hub motor drive systems, as a pivotal technology for next-generation electric vehicles, offer unparalleled advantages in transmission efficiency and dynamic control flexibility. By eliminating mechanical differentials and transmission shafts, these systems enable independent and precise torque regulation for each wheel, significantly enhancing vehicle handling stability and energy utilization. ¹ However, with the rapid commercialization of this technology, the dual challenge of optimizing energy efficiency while preserving battery health has emerged as a critical bottleneck. Recent studies indicate that high-frequency power fluctuations typical of distributed drives can accelerate battery degradation, necessitating strategies that optimize battery aging alongside energy consumption. ² Consequently, developing a comprehensive management system that harmonizes these conflicting objectives has become an urgent research imperative. ³

Existing energy management strategies generally fall into rule-based and optimization-based categories. While traditional approaches like fuzzy Q-learning controllers have shown promise in enhancing robustness, ⁴ they often rely on simplified models that struggle to adapt to complex stochastic environments. To address this, intelligent management strategies incorporating multi-source information have been proposed for range-extended vehicles. ⁵ Furthermore, integrated power and thermal management frameworks have been developed to mitigate battery aging in connected automated vehicles. ⁶ Despite these improvements, applying such methods to the specific architecture of distributed drive vehicles remains challenging. Louback et al. ⁷ reviewed energy systems for dual-motor configurations, highlighting the need for specialized optimization, while Yu et al. ⁸ proposed a real-time hierarchical control framework. However, these methods primarily focus on powertrain efficiency, often treating battery health as a secondary constraint. Broader energy management research, such as in microgrids^9,10 or electric bus fleets, ¹¹ offers insights into coordinating power and cooling subsystems, but these large-scale stationary or predetermined-route solutions are difficult to transpose directly to the dynamic constraints of passenger EVs. Similarly, strategies for heavy-duty applications like mining trucks ¹² lack general applicability to distributed hub-motor platforms.

With the advent of artificial intelligence, Deep reinforcement learning and meta-heuristic algorithms have shown great potential in handling non-linear optimization problems. DRL has been successfully applied to intelligent infrastructure ¹³ and integrated energy systems, ¹⁴ enhancing policy quality through knowledge embedding. However, comprehensive reviews suggest that pure DRL approaches still face hurdles regarding sample efficiency and safety generalization. ¹⁵ To overcome these limitations, hybrid methodologies combining evolutionary algorithms with RL have emerged. For instance, Ko ¹⁶ integrated genetic algorithms with RL for wireless charging trams, and Zhou ¹⁷ applied similar principles to intelligent warehouse management. These studies demonstrate that hybridizing global search algorithms with real-time learning can improve convergence and robustness, providing a theoretical basis for applying such architectures to vehicle control.

In the specific domain of distributed in-wheel motor vehicles, current research focuses heavily on torque distribution algorithms. Wu et al. ¹⁸ and Sun et al. ¹⁹ established foundational methods for energy-optimal torque allocation, yet their adaptability to dynamic degradation mechanisms remains to be validated. Other approaches prioritize handling stability; for example, Zhai et al. ²⁰ proposed a two-level torque distribution strategy for various adhesion conditions. While multi-objective optimization approaches ²¹ and centroid-adaptive methods ²² have improved overall performance, they often suffer from high computational complexity. Coordinated control strategies ²³ have further enhanced stability, but balancing real-time response with computational efficiency remains a persistent hurdle.

Crucially, a significant research gap remains in addressing the “time-scale mismatch” between objectives: motor torque control requires millisecond-level response (fast dynamics), whereas battery aging is a cumulative process spanning months or years (slow dynamics). Existing loosely coupled strategies fail to facilitate deep interaction between these conflicting objectives, often resulting in solutions that sacrifice long-term durability for short-term efficiency gains or vice versa.

To bridge this gap, this study proposes a hybrid multi-objective optimization framework that organically integrates the A3C algorithm with an ISSA. The main contributions are:

• (1) A Real-time Layer: The A3C algorithm handles high-frequency torque allocation, utilizing asynchronous parallel learning to adapt to dynamic driving conditions in real-time.

• (2) A Global Layer: An ISSA with mechanism-based constraints optimizes long-term battery management, explicitly embedding capacity fading and thermal safety into the search process.

• (3) Coupling Mechanism: A dynamic weight adjustment strategy and a cross-scale shared memory pool are introduced to coordinate fast and slow dynamics, ensuring short-term decisions align with long-term health objectives.

The remainder of this paper is organized as follows: Section 2 establishes the theoretical models for vehicle dynamics and battery health. Section 3 elaborates on the A3C-ISSA hybrid methodology. Section 4 presents a comparative analysis of simulation results under multiple driving cycles. Section 5 validates the framework via HIL experiments. Finally, Section 6 summarizes the conclusions.

Theoretical model

This section analyzes the structure and operational characteristics of distributed electric drive hub motors. Integrating vehicle dynamics and energy consumption mechanisms, it constructs an energy allocation and efficiency model for multi-motor drive systems. Furthermore, it incorporates battery health modeling covering key factors such as SOC, SOH, capacity degradation, internal resistance growth, and temperature effects, laying the mathematical and physical foundation for the subsequent hybrid optimization method.

Characteristics of distributed drive systems

The primary characteristics of distributed hub motor drive systems manifest in several aspects. First, regarding dynamic properties, since each wheel can independently regulate output torque, the vehicle achieves millisecond-level rapid power response and high-precision torque control. This feature not only enhances energy utilization efficiency but also ensures maneuvering stability and safety under complex operating conditions. Second, at the energy allocation level, the distributed drive system dynamically adjusts power output across motors based on vehicle states (e.g. acceleration, braking, steering) and road conditions (e.g. friction coefficient, gradient), achieving real-time energy efficiency optimization. Compared to centralized drives, the distributed architecture more effectively matches drive force demands with battery supply characteristics, thereby reducing overall vehicle energy consumption.

Due to coupling relationships between motors, energy distribution transforms into a multi-objective, multi-constraint optimization problem requiring simultaneous consideration of overall energy efficiency, battery health, and vehicle performance. Furthermore, as the sole energy source, the battery’s SOC, SOH, and temperature characteristics directly impact the effectiveness and feasibility of energy distribution strategies. Balancing short-term energy efficiency with long-term lifespan while ensuring real-time responsiveness remains a critical challenge for distributed drive systems. An overview of this research is presented in Figure 1.

Figure 1.

Overview of research content.

System dynamics modeling

To achieve efficient energy management and battery health optimization for distributed hub motor electric vehicles, this paper first establishes a theoretical model of the distributed electric drive system. This model comprehensively considers multiple critical factors, including dynamic characteristics, motor energy allocation, SOC, and battery aging characteristics, to provide an accurate mathematical foundation for subsequent optimization algorithm design.

Mathematical model of driving forces and energy consumption

To optimize the energy allocation strategy, the following mathematical model for energy distribution was established. It incorporates multiple factors such as vehicle dynamics, motor characteristics, and battery performance to maximize overall system energy efficiency.

First, the total driving force demand of the vehicle $F_t$ is defined as:

$$F_t=F_r+F_f+F_w+F_i$$

(1)

where $F_r$ , $F_f$ , $F_w$ , and $F_i$ represent rolling resistance, gradient resistance, aerodynamic resistance, and acceleration resistance, respectively.

For distributed hub motor systems, the total drive force $F_t$ is composed of the sum of the drive forces provided by each hub motor:

$$F_t=\sum _{i=1}^n{F}_i$$

(2)

where $n$ denotes the number of hub motors, and $F_i$ represents the driving force provided by the $i$ -th hub motor.

Considering the relationship between motor efficiency and its operating point, the motor efficiency function ${\eta }_i(T_i,{\omega }_i)$ is introduced, where $T_i$ and ${\omega }_i$ represent the torque and angular velocity of the $i$ -th motor, respectively. The motor input power $P_{\mathrm{in},i}$ can be expressed as:

$$P_{\mathrm{in},i}=\frac{T_i·{\omega }_i}{{\eta }_i(T_i,{\omega }_i)}$$

(3)

The total input power of the system $P_{\mathrm{in}}$ is:

$$P_{\mathrm{in}}=\sum _{i=1}^n{P}_{\mathrm{in},i}$$

(4)

The objective of energy allocation optimization is to minimize the total input power $P_{\mathrm{in}}$ while satisfying the total driving force requirement. The corresponding optimization objective and constraints can be expressed as:

$$\underset{\{T_i,{\omega }_i\}}{\min }{P}_{\mathrm{in}}=\underset{\{T_i,{\omega }_i\}}{\min }\sum _{i=1}^n\frac{T_i·{\omega }_i}{{\eta }_i(T_i,{\omega }_i)}$$

(5)

$$\{\begin{array}{c}\sum _{i=1}^n{F}_i=F_t\hfill \\ \multicolumn{1}{c}{T_{\min ,i}\le T_i\le T_{\max ,i}}\\ \multicolumn{1}{c}{{\omega }_{\min ,i}\le {\omega }_i\le {\omega }_{\max ,i}}\end{array}$$

(6)

where $T_{\min ,i}$ , $T_{\max ,i}$ , ${\omega }_{\min ,i}$ , and ${\omega }_{\max ,i}$ represent the minimum and maximum torque and angular velocity constraints for the $i$ -th motor, respectively.

Energy allocation and efficiency model

Since each motor possesses independent torque output capability and efficiency characteristics, the energy distribution problem is fundamentally a multi-constraint optimization problem. Its objective is to maximize overall system efficiency while satisfying vehicle dynamics requirements.

Considering the wheel radius $R$ , the relationship between motor output torque and driving force is expressed as:

$$T_i=\frac{F_i·R}{{\eta }_i}$$

(7)

where ${\eta }_i$ is the efficiency function for the i-th motor, whose value strongly correlates with the motor’s rotational speed ${\omega }_i$ and output torque $T_i$ . The motor’s input power is jointly determined by torque and angular velocity:

$$P_i=\frac{T_i·{\omega }_i}{{\eta }_i}$$

(8)

To measure overall operational efficiency, the system’s total efficiency is defined as the product of the battery output power and the efficiencies of each subsystem:

$${\eta }_{\mathrm{sys}}={\eta }_{\mathrm{bat}}·{\eta }_{\mathrm{mot}}·{\eta }_{\mathrm{trans}}$$

(9)

where ${\eta }_{\mathrm{bat}}$ , ${\eta }_{\mathrm{mot}}$ , and ${\eta }_{\mathrm{trans}}$ represent battery efficiency, motor efficiency, and transmission efficiency, respectively.

The objective of energy allocation optimization can be formalized as:

$$\{\begin{array}{c}\underset{\left\{{\lambda }_i\right\}}{\max }{\eta }_{\mathrm{sys}}\hfill \\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.\sum _{i=1}^N{\lambda }_i=1}\\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.T_i^{\min }\le T_i\le T_i^{\max }}\\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.{\omega }_i^{\min }\le {\omega }_i\le {\omega }_i^{\max }}\end{array}$$

(10)

Through this composite weighting function, the energy allocation strategy dynamically adjusts based on driving conditions and environmental factors, thereby maintaining high system energy efficiency and operational stability across diverse driving scenarios.

Battery health model

The operational state of the battery directly determines the effectiveness of the energy allocation strategy, while its long-term degradation process significantly impacts vehicle economy and reliability. This section establishes theoretical models for SOC and SOH, further incorporating descriptions of capacity decay, internal resistance changes, and temperature safety constraints to construct a systematic model of battery health.

SOC and SOH modeling

Battery state modeling focuses on two core metrics: SOC and SOH. SOC reflects the remaining usable capacity and serves as the foundational parameter for energy management; SOH measures battery degradation and lifespan, being critical for long-term performance optimization.

First, SOC is typically defined as the ratio of current charge to rated capacity. Its dynamic variation can be expressed using the Coulomb integration method:

$$\mathrm{SOC}\left(t\right)=\mathrm{SOC}\left(0\right)-\frac{1}{C_{\mathrm{n}}}{\int }_0^{t}I\left(\tau \right)d\tau$$

(11)

where $\mathrm{SOC}\left(0\right)$ is the initial state of charge, $C_{\mathrm{n}}$ is the battery’s rated capacity, and $I\left(\tau \right)$ is the current at time $\tau$ (positive for discharge, negative for charge).

In practical applications, SOC is not an isolated variable; its rate of change is influenced by factors such as charge/discharge rate (C-rate), temperature, and efficiency. Therefore, to enhance modeling accuracy, an efficiency correction factor ${\eta }_{\mathrm{coul}}$ can be introduced, resulting in the following modified SOC model:

$$\mathrm{SOC}\left(t\right)=\mathrm{SOC}\left(0\right)-\frac{1}{C_{\mathrm{n}}}{\int }_0^t\frac{I\left(\tau \right)}{{\eta }_{\mathrm{coul}}}d\tau$$

(12)

where ${\eta }_{\mathrm{coul}}$ is the coulombic efficiency coefficient, reflecting energy losses from non-ideal charging/discharging processes.

Unlike SOC, battery SOH primarily characterizes battery life degradation and is typically defined as the ratio of current maximum available capacity to initial rated capacity:

$$\mathrm{SOH}\left(t\right)=\frac{C_{\max }\left(t\right)}{C_{\mathrm{n}}}$$

(13)

where $C_{\max }\left(t\right)$ is the maximum available capacity at time $t$ , and $C_{\mathrm{n}}$ is the rated initial capacity. As charge-discharge cycles increase and operating conditions vary, battery capacity gradually degrades, causing SOH to decrease over time.

To more accurately describe the evolution of battery health status, SOH can be modeled as a function of multiple factors:

$$\mathrm{SOH}\left(t\right)=f(N_{\mathrm{cyc}},\mathrm{DoD},T,R_{\mathrm{int}})$$

(14)

where $N_{\mathrm{cyc}}$ represents cycle count, $\mathrm{DoD}$ denotes DOD, $T$ indicates battery operating temperature, and $R_{\mathrm{int}}$ signifies battery internal resistance. The rate of SOH degradation accelerates with increased DOD, higher operating temperatures, and faster internal resistance growth.

Capacity degradation and internal resistance variation model

During long-term operation, batteries undergo complex chemical and physical degradation processes, primarily manifested as capacity decay and internal resistance increase. These factors directly impact energy utilization efficiency and battery health status, forming the critical foundation for SOH modeling.

First, battery capacity degradation is typically closely related to cycle count, depth of discharge, and temperature. The capacity degradation model can be expressed as:

$$C_{\max }(t)=C_0-k_1{N}_{\mathrm{cyc}}^{\alpha }-k_2{\int }_0^{t}f(T(\tau ))d\tau$$

(15)

where $C_0$ is the initial battery capacity, $C_{\max }(t)$ is the maximum available capacity at time $t$ , $N_{\mathrm{cyc}}$ is the number of charge-discharge cycles, $k_1$ and $\alpha$ are cycle-related capacity decay coefficients, $k_2$ is the temperature decay factor, and $f(T(\tau ))$ is the function describing the impact of temperature on capacity degradation.

Simultaneously, the battery’s internal resistance gradually increases during use, leading to greater energy loss and thermal effects during high-rate charging and discharging. The internal resistance change model can be expressed as:

$$R_{\mathrm{int}}(t)=R_0+{\beta }_1{N}_{\mathrm{cyc}}+{\beta }_2{\int }_0^t|I(\tau )|d\tau$$

(16)

where $R_0$ is the initial internal resistance, $R_{\mathrm{int}}(t)$ is the internal resistance at time $t$ , ${\beta }_1$ is the cycle-related degradation coefficient, ${\beta }_2$ is the cumulative current effect factor, and $|I(\tau )|$ is the absolute value of the instantaneous current.

To comprehensively optimize battery health during the algorithmic search process, capacity degradation and internal resistance increase are combined into a composite health objective function, denoted as J _health . This objective serves to guide the optimization search and is expressed as:

$$J_{\mathrm{health}}={\gamma }_1\frac{C_{\max }(t)}{C_0}+{\gamma }_2\frac{R_0}{R_{\mathrm{int}}(t)}$$

(17)

where ${\gamma }_1$ and ${\gamma }_2$ are weighting coefficients.

Battery temperature effects

During actual operation of electric vehicles, the thermal characteristics of batteries critically impact their performance and lifespan. Excessively high operating temperatures not only accelerate material aging and capacity degradation but may also pose safety hazards such as thermal runaway and electrolyte decomposition.

During charging and discharging, batteries generate Joule heat and heat from side reactions. Their temperature changes can be modeled using the thermal equilibrium equation:

$$m_{\mathrm{bat}}{c}_{\mathrm{p}}\frac{\mathrm{dT}\left(t\right)}{\mathrm{dt}}=Q_{\mathrm{gen}}\left(t\right)-Q_{\mathrm{loss}}\left(t\right)$$

(18)

where $m_{\mathrm{bat}}$ is the battery mass, $c_{\mathrm{p}}$ is the specific heat capacity of the battery, $T\left(t\right)$ is the battery temperature at time $t$ , $Q_{\mathrm{gen}}\left(t\right)$ is the heat generated during charging/discharging, and $Q_{\mathrm{loss}}\left(t\right)$ is the heat dissipated by the battery through convection, conduction, and radiation.

Battery heat generation is primarily caused by internal resistance, expressed mathematically as:

$$Q_{\mathrm{gen}}\left(t\right)=I^2\left(t\right)·R_{\mathrm{int}}\left(t\right)+I\left(t\right)·(U_{\mathrm{oc}}-U_{\mathrm{term}})$$

(19)

where $I\left(t\right)$ is the battery current, $R_{\mathrm{int}}\left(t\right)$ is the battery internal resistance, $U_{\mathrm{oc}}$ is the open-circuit voltage, and $U_{\mathrm{term}}$ is the terminal voltage. The first term represents Ohmic heating, while the second term accounts for electrochemical polarization heat.

The heat dissipation of the battery can be approximated as:

$$Q_{\mathrm{loss}}\left(t\right)=\mathrm{hA}(T\left(t\right)-T_{\mathrm{env}})$$

(20)

where $h$ is the overall heat transfer coefficient, $A$ is the heat dissipation area, and $T_{\mathrm{env}}$ is the ambient temperature. This equation indicates that the heat dissipation rate is proportional to the difference between the battery temperature and the ambient temperature.

Methodology

Building upon the theoretical modeling of vehicle dynamics and battery electrochemistry established in Section 2, this chapter proposes a hybrid multi-objective optimization framework designed to bridge the gap between instantaneous energy control and long-term health management. Before detailing the specific algorithmic implementations, it is essential to clarify the methodological boundary between the proposed framework and existing state-of-the-art approaches.

Current energy management strategies for distributed drive vehicles predominantly employ decoupled or serial architectures, where the upper-level Energy management system allocates torque based on static look-up tables or instantaneous efficiency optimization, treating battery protection merely as a passive lower-level threshold constraint. This architecture essentially represents an “open-loop” aging management approach, which ignores the reverse impact of cumulative battery degradation on real-time control strategies.

In contrast, the framework proposed herein establishes a tightly coupled bi-level closed-loop architecture. By integrating the real-time decision-making capability of the A3C algorithm with the global optimization advantage of the ISSA, the system achieves a bidirectional information flow. Specifically, the A3C layer executes high-frequency torque allocation, while the ISSA layer—enhanced by chaotic initialization and a golden sine–cosine strategy—continuously updates the optimization weights based on mechanism-driven degradation constraints. This design enables the microsecond-level torque control to “perceive” month-level capacity fading trends via a cross-scale shared memory pool.

Compared to existing rule-based control or single-objective optimization algorithms, this hybrid framework offers four significant advantages:

• (1) Through the A3C deep reinforcement learning agent, the framework enables adaptive energy allocation that effectively handles complex and stochastic driving conditions in real-time.

• (2) The modified ISSA enhances global search capabilities by explicitly embedding battery degradation mechanisms (e.g. SEI film growth and active material loss) and thermal safety boundaries into the constraints, ensuring solutions are not only mathematically optimal but also physically plausible and engineering-feasible.

• (3) By introducing dynamic weight adjustment and a cross-scale shared memory pool, the framework coordinates short-term energy efficiency with long-term battery longevity, overcoming the “time-scale mismatch” inherent in traditional methods.

• (4) The architecture incorporates strict computational constraints to balance optimization performance with execution speed, guaranteeing real-time feasibility for deployment in onboard vehicle control systems.

Based on these methodological innovations, the comprehensive control flowchart for the proposed hybrid optimization method is illustrated in Figure 2.

Figure 2.

Control framework of the proposed method.

To more intuitively demonstrate the boundary between this study and existing literature, we have summarized the feature comparison between the proposed framework and typical SOTA methods in Table 1.

Table 1.

Feature comparison between existing SOTA methods and the proposed A3C + ISSA framework.

Method category	Typical representative	Control architecture	Battery health model	Multi-timescale coupling	Real-time capability
Fuzzy/logic-based	Fuzzy Q-learning 4	Open-loop/look-up table	None (threshold only)	None	High
Single-objective optimization	Instantaneous energy minimization 18	Static optimization	Simplified (resistance only)	Weak (ignores aging feedback)	Medium/Low
Traditional hierarchical strategy	Hierarchical control framework 8	Serial/decoupled	Linear aging model	Loose (fixed parameter update)	Low (Offline/Update)
Proposed method (A3C + ISSA)	DRL + Improved SSA	Bi-level closed-loop	High-fidelity (electrochemical + Thermal)	Strong (dynamic weights + shared memory)	High (ms-level response)

Real-time energy efficiency optimization strategy based on A3C

Distributed hub motor vehicles require rapid and precise energy allocation under complex operating conditions to balance overall powertrain performance and energy efficiency. Leveraging its parallel learning and real-time policy update capabilities, the A3C algorithm enables adaptive energy distribution among multiple motors in dynamic environments, making it suitable for short-term energy efficiency optimization in this study.

Basic framework and update mechanism of A3C

In energy management for distributed hub-motor electric vehicles, the highly dynamic and uncertain environment demands control strategies that balance real-time responsiveness with optimality. As a representative deep reinforcement learning method, the A3C algorithm leverages its multi-agent parallel learning and asynchronous update mechanism to rapidly converge and generate robust energy allocation strategies under complex driving conditions. It is therefore employed for real-time energy efficiency optimization in this study.

The A3C framework comprises two core networks: the policy network and the value network. The policy network outputs a probability distribution over actions $a_t$ based on the current environment state $s_t$ , guiding motor power allocation. The value network evaluates the value function of states, correcting the policy update direction to ensure learning stability. Specifically, the policy gradient update formula is:

$${\nabla }_{\theta }J(\theta )={\mathbb{E}}_{s_t,a_t~{\pi }_{\theta }}[{\nabla }_{\theta }\log {\pi }_{\theta }(a_t|s_t)A(s_t,a_t)]$$

(21)

where θ represents the parameters of the policy network, ${\pi }_{\theta }(a_t|s_t)$ denotes the probability of selecting action $a_t$ in state $s_t$ , and $A(s_t,a_t)$ is the advantage function measuring the superiority of the current action relative to the average policy.

The advantage function is defined as:

$$A(s_t,a_t)=Q(s_t,a_t)-V(s_t)$$

(22)

Among these, $Q(s_t,a_t)$ represents the state-action value function, and $V(s_t)$ denotes the state value function.

The objective of the value network is to minimize the difference between the predicted state value and the target reward, with its loss function defined as:

$$L_V(\varphi )=\frac{1}{2}{(R_t-V_{\varphi }(s_t))}^2$$

(23)

where $\varphi$ represents the value network parameters, and $R_t$ denotes the return based on temporal differences, defined as:

$$R_t=r_t+\gamma V_{\varphi }(s_{t+1})$$

(24)

where $r_t$ is the immediate reward, and $\gamma \in (0,1)$ is the discount factor, used to balance short-term energy efficiency with long-term battery health objectives.

To simultaneously optimize the policy network and value network, A3C’s total loss function is typically defined as:

$$L(\theta ,\varphi )=L_{\pi }(\theta )+c·L_V(\varphi )-\beta H({\pi }_{\theta })$$

(25)

where $L_{\pi }(\theta )$ is the policy loss, $L_V(\varphi )$ is the value loss, $H({\pi }_{\theta })$ is the entropy regularization term encouraging diversity during exploration; $c$ and β are weight coefficients.

Regarding the update mechanism, A3C employs a multi-threaded parallel approach. Each agent explores and computes gradients in an independent environment, then asynchronously transmits these gradients back to the global network for parameter updates. This asynchronous update mechanism not only significantly improves sampling efficiency but also enhances strategy robustness through the diversity of trajectories across different environments. For distributed hub motor vehicles, this mechanism enables the energy management system to rapidly learn effective power allocation schemes across diverse driving states (e.g. acceleration, deceleration, hill climbing, and high-speed cruising), thereby maximizing energy efficiency while meeting power demands.

Algorithm improvements

Although the A3C algorithm demonstrates good convergence performance in complex environments through its multi-agent parallel and asynchronous update mechanisms, it may still face issues such as insufficient sample utilization, excessive value function volatility, and sensitivity to learning rate selection when handling high-dimensional continuous state spaces and dynamic driving conditions. To further enhance the algorithm’s stability and generalization capability, this paper introduces three key improvements to the A3C framework: experience replay, target networks, and adaptive learning rate adjustment.

First, the experience replay mechanism effectively breaks the correlation between continuous samples by storing historical samples generated from agent-environment interactions and performing random sampling training. This enhances the independent and identically distributed nature of training data. Let the experience pool be denoted as $\mathcal{D}=\{(s_t,a_t,r_t,s_{t+1})\}$ . At each training iteration, a mini-batch of samples $(s_j,a_j,r_j,s_{j+1})$ is randomly sampled from this pool to update the policy and value networks. The temporal difference update formula for the value function can be expressed as:

$${\delta }_j=r_j+\gamma V(s_{j+1};{\varphi }^{-})-V(s_j;\varphi )$$

(26)

where ${\delta }_j$ is the temporal difference error, $\gamma$ is the discount factor, $V(s;\varphi )$ is the current value network, and $V(s;{\varphi }^{-})$ is the target value network (see below).

Second, to mitigate oscillations caused by frequent updates to the value function, this paper introduces a target network within the A3C framework. Specifically, alongside the current value network $V(s;\varphi )$ , a delayed-update target network $V(s;{\varphi }^{-})$ is maintained, whose parameters are periodically updated as follows:

$${\varphi }^{-}\leftarrow \tau \varphi +(1-\tau ){\varphi }^{-}$$

(27)

where $\tau \in (0,1]$ is the soft update coefficient. When $\tau =1$ , a hard update occurs; when $\tau <1$ , parameters undergo a gradual smoothing transition. This addresses the policy oscillation issue caused by sudden fluctuations.

Finally, addressing the high sensitivity of convergence speed and stability to learning rate, this paper designs an adaptive learning rate adjustment mechanism. Let the base learning rate be ${\alpha }_0$ . At iteration $k$ , the dynamic learning rate can be expressed as:

$${\alpha }_k=\frac{{\alpha }_0}{1+\lambda k}$$

(28)

where λ is the decay factor.

Battery health optimization based on ISSA

Maintaining battery health during long-term operation is critical for the reliability and lifespan of distributed electric drive systems. SSA tends to get stuck in local optima when handling high-dimensional complex problems. Therefore, this paper proposes ISSA, which enhances its global optimization capability through chaotic initialization, golden sine-cosine local search, and mechanism-based constraints.

Chaotic initialization and population diversity enhancement

Chaotic systems exhibit ergodicity, randomness, and non-repeatability, generating sequences with near-uniform distribution within finite intervals. Thus, chaotic mappings can generate initial population positions, overcoming limitations of traditional random initialization. Let the population size be $N$ and the optimization problem dimension be $d$ . The initial position of the $i$ -th individual in the $j$ -th dimension is represented as $X_{\mathrm{ij}}$ . The chaotic sequence generation formula based on the Logistic map is:

$$z_{k+1}=\mu z_k(1-z_k),z_k\in (0,1)$$

(29)

where μ is the chaotic control parameter, and $z_k$ represents the chaotic variable generated at iteration $k$ .

Mapping this sequence to the upper and lower bounds of the search space $[X_j^{min},X_j^{max}]$ yields the initialization position of the individual:

$$X_{\mathrm{ij}}=X_j^{min}+z_k·(X_j^{max}-X_j^{min})$$

(30)

Initial solutions generated through this method distribute more uniformly across the search space, effectively expanding the coverage of candidate solutions and providing a richer exploratory foundation for subsequent evolutionary iterations. Compared to traditional SSA, the improved algorithm incorporating chaotic initialization enables faster escape from local optima in battery health optimization tasks, enhancing adaptability to complex non-convex objective functions. For instance, in the multi-objective optimization problem of battery capacity degradation and internal resistance increase, the chaotic initialization mechanism enables the algorithm to cover a broader search area during early iterations, thereby improving the exploration efficiency for strategies balancing long-term battery lifespan and energy efficiency.

Golden sine-cosine local search strategy

While SSA demonstrates strong global search capabilities in the early stages, it tends to exhibit limitations in later iterations, such as narrow search directions and insufficient ability to escape local optima. This paper introduces a local search strategy based on SSA that combines the golden ratio with sine-cosine adjustment mechanisms. By dynamically adjusting individual search paths, this approach achieves multidirectional expansion of the solution space and localized refinement, thereby enhancing the algorithm’s convergence performance in complex multi-objective optimization tasks.

Specifically, the individual position update formula is expressed as:

$$\begin{array}{c}{X}_i^{t+1}=X_i^t\hfill \\ +\phi ·\left(\begin{array}{c}\alpha ·\sin ({\theta }_t)+\hfill \\ \multicolumn{1}{c}{(1-\alpha )·\cos ({\theta }_t)}\end{array}\right)·(X_{\mathrm{best}}^t-X_i^t)\hfill \end{array}$$

(31)

where $X_i^t$ denotes the position of the $i$ -th individual at iteration t, $X_i^{t+1}$ represents its updated position, $X_{\mathrm{best}}^t$ is the current global optimum, and φ is the scaling factor based on the golden ratio, which maintains convergence speed and solution distribution balance during the update process. The parameter $\alpha \in [0,1]$ controls the weight ratio between the sine and cosine terms. ${\theta }_t$ is the dynamically adjusted angle variable during iteration, typically exhibiting nonlinear variation with iteration count to ensure search diversity and gradual convergence.

In the above formula, the periodic oscillations of $\sin ({\theta }_t)$ and $\cos ({\theta }_t)$ provide individuals with opportunities to jump in different directions. This effectively avoids the monotony of search paths, enabling individuals to expand in multiple directions within the search space. When α approaches 1, the algorithm favors utilizing the oscillatory characteristics of the sine function for local exploration; when α is small, it leverages the smooth variations of the cosine function to enhance global search capabilities. Through this mechanism, the algorithm achieves an adaptive balance between global exploration and local exploitation during iteration.

Safety boundary function

To ensure the battery operates within a safe range, temperature constraints must be introduced. According to the thermal model, the battery temperature must satisfy:

$$T_{min}\le T(t)\le T_{max}$$

(32)

where $T_{min}$ and $T_{max}$ represent the lower and upper limits of the battery’s safe operating temperature range. Exceeding these thresholds significantly accelerates material degradation and may pose safety risks. Therefore, a temperature penalty function is constructed:

$$\begin{array}{c}{P}_T(t)=max(0,T(t)-T_{max})\hfill \\ +max(0,T_{min}-T(t))\hfill \end{array}$$

(33)

This function yields zero when temperature is within the safe range and generates positive values when exceeding boundaries, imposing penalties on non-compliant solutions during optimization.

Figure 3 illustrates the detailed optimization process of the proposed ISSA. The algorithm begins with chaotic initialization to enhance population diversity and evaluates the fitness function based on mechanism constraints. It then updates positions using the golden sine-cosine strategy, iteratively optimizing global and local optima, and terminates upon meeting convergence criteria to output the optimal battery charging/discharging strategy. To ensure the battery operates within a safe and physically consistent range, mechanism-based constraints must be introduced. In addition to the thermal model (equation (18)), we explicitly integrate the capacity degradation model (equation (15)) and internal resistance growth model (equation (16)) established in Section 2.3.2 into the optimization process. Specifically, the algorithm monitors the evolution trajectory of the SOH alongside temperature boundaries; strategies that meet power demands but result in capacity fade rates or internal resistance growth rates exceeding the bounds predicted by the physical models are penalized. This multi-dimensional constraint mechanism ensures that the strategies generated by ISSA are not only thermally safe but also physically consistent regarding electrochemical aging characteristics, thereby realizing the complete constraint architecture depicted in Figure 3.

Figure 3.

Optimization flow of the ISSA.

Hybrid optimization framework of A3C and ISSA

This paper constructs a hybrid optimization framework combining A3C and ISSA, organically integrating the real-time decision-making capability of reinforcement learning with the global search advantage of intelligent optimization. Through multi-objective function design, dynamic weight adjustment, cross-scale shared memory pool, and real-time constraints, this framework achieves synergistic optimization of energy efficiency and battery life.

Multi-objective function construction and optimization goals

During operation of distributed hub-motor electric vehicles, energy efficiency and battery health often exhibit conflicting demands: prioritizing instantaneous energy efficiency maximization may lead to frequent deep discharges, elevated temperatures, and increased internal resistance, thereby shortening battery lifespan. Conversely, excessive focus on battery protection may compromise the vehicle’s dynamic performance and energy efficiency. Therefore, this study incorporates both energy efficiency optimization and battery health management into the optimization framework, achieving a dynamic equilibrium between the two through the construction of a multi-objective function.

First, the energy efficiency optimization objective function is defined. Based on the energy allocation and efficiency model established in Section 2, the system’s energy efficiency target can be expressed as:

$$f_{\mathrm{eff}}=\frac{F_{\mathrm{total}}·v}{P_{\mathrm{total}}}$$

(34)

where $F_{\mathrm{total}}$ represents total driving force, v denotes vehicle speed, and $P_{\mathrm{total}}$ signifies system input power.

Next, define the battery health objective function. Integrating the modeling results from Chapter 2 on capacity degradation, internal resistance growth, and temperature constraints, the battery health evaluation function can be expressed as:

$$f_{\mathrm{bat}}={\lambda }_1\frac{C_{\max }(t)}{C_0}+{\lambda }_2\frac{R_0}{R_{\mathrm{int}}(t)}-{\lambda }_3{P}_T(t)$$

(35)

where $C_{\max }(t)$ is the maximum available capacity at time $t$ , $C_0$ is the initial battery capacity, $R_{\mathrm{int}}(t)$ is the battery internal resistance, $R_0$ is the initial internal resistance, $P_T(t)$ is the temperature penalty function, and λ₁, λ₂, λ₃ are weighting coefficients used to balance the effects of capacity retention, internal resistance growth, and temperature safety on battery health.

Under the joint consideration of energy efficiency and battery health, both are formulated as a combined multi-objective optimization problem:

$$maxF={\omega }_1{f}_{\mathrm{eff}}+{\omega }_2{f}_{\mathrm{bat}}$$

(36)

where ${\omega }_1$ and ${\omega }_2$ are dynamic weighting coefficients satisfying ${\omega }_1+{\omega }_2=1$ .

Additionally, this optimization problem must satisfy the following constraints:

$$\begin{array}{c}\sum _{i=1}^N{F}_i\ge F_{\mathrm{total}}\hfill \\ T_i^{min}\le T_i\le T_i^{max},{\omega }_i^{min}\le {\omega }_i\le {\omega }_i^{max}\hfill \\ \mathrm{SO}{C}_{min}\le \mathrm{SOC}(t)\le \mathrm{SO}{C}_{max}\hfill \\ T_{min}\le T(t)\le T_{max}\hfill \end{array}$$

(37)

where $T_i$ and ${\omega }_i$ represent the torque and angular velocity of the $i$ -th motor, respectively; $\mathrm{SOC}(t)$ and $T(t)$ denote the state of charge and temperature of the battery, respectively.

Dynamic weight adjustment mechanism

Within a multi-objective optimization framework, energy efficiency optimization and battery health management often exhibit mutually constraining relationships. Using fixed weights for solving these objectives fails to adapt to the changing importance of different goals under dynamic driving conditions. This paper designs a dynamic weight adjustment mechanism based on the construction of a multi-objective function to achieve adaptive balancing between energy efficiency and battery health objectives across different time scales.

The comprehensive optimization objective function is defined as:

$$F(t)={\omega }_1(t)f_{\mathrm{eff}}(t)+{\omega }_2(t)f_{\mathrm{bat}}(t)$$

(38)

where $f_{\mathrm{eff}}(t)$ and $f_{\mathrm{bat}}(t)$ represent the energy efficiency objective function and battery health objective function, respectively; ${\omega }_1(t)$ and ${\omega }_2(t)$ denote dynamic weights satisfying:

$${\omega }_1(t)+{\omega }_2(t)=1,{\omega }_1(t),{\omega }_2(t)\in [0,1]$$

(39)

The dynamic adjustment of weights is achieved through a time-dependent function to dynamically balance energy efficiency and battery health across different iteration stages. This paper employs a Sigmoid-type function to define the weight variation pattern:

$${\omega }_1(t)=\frac{1}{1+e^{-\kappa (t-t_0)}},{\omega }_2(t)=1-{\omega }_1(t)$$

(40)

where κ is a sensitivity parameter controlling the weight change rate, and t₀ denotes the iteration time at the equilibrium point.

Specifically, the variation of ${\omega }_1(t)$ undergoes three stages: during the initial iteration phase ( $t<<t_0$ ), ${\omega }_1(t)$ assumes a small value, prioritizing battery health protection. When $t=t_0$ , ${\omega }_1(t)$ equals 0.5, representing the equilibrium point where both objectives are balanced. Subsequently, as t increases beyond $t_0$ ( $t>>t_0$ ), ${\omega }_1(t)$ gradually approaches 1, shifting the optimization focus toward improving energy efficiency.

This dynamic mechanism enables prioritizing long-term battery longevity during initial training phases while progressively enhancing energy efficiency in later operational stages, achieving cross-timescale optimization balance.

Furthermore, to enhance responsiveness to real-time vehicle conditions, this paper introduces a SOC adjustment factor into the weight function. Let $\varphi (\mathrm{SOC})$ denote the SOC-related normalization function, then the weight can be modified as:

$$\begin{array}{c}{\omega }_1(t)=\frac{1}{1+e^{-\kappa (t-t_0)}}·\varphi (\mathrm{SOC})\hfill \\ {\omega }_2(t)=1-{\omega }_1(t)\hfill \end{array}$$

(41)

where $\varphi (\mathrm{SOC})$ assumes larger values at higher SOC levels to ensure energy efficiency gains when battery energy is abundant, while reducing weights as SOC approaches lower limits to prioritize battery health protection.

Cross-scale shared memory pool design

Within the multi-objective optimization framework for distributed hub motor vehicles, the A3C algorithm focuses on real-time optimization of short-term energy efficiency, while the improved ISSA emphasizes global regulation of long-term battery health. Without effective information exchange, the temporal scale and objective differences between these two approaches can lead to discrepancies between local and global optima, hindering the synergistic optimization of energy efficiency and lifespan. To address this, this paper proposes a cross-scale shared memory pool as an information bridge between A3C and ISSA. This enables positive feedback and complementarity between short-term strategy learning and long-term global search, thereby enhancing overall optimization performance.

The fundamental concept of the shared memory pool is to uniformly store the state-action-reward-next state quadruplet $(s_t,a_t,r_t,s_{t+1})$ generated by A3C during real-time decision-making alongside the candidate solution vector $X_i^t$ formed by ISSA during global search, then fuse them through a cross-scale indexing mechanism. Let the content of the memory pool at time t be represented as:

$$\mathcal{M}(t)=\{(s_{\tau },a_{\tau },r_{\tau },s_{\tau +1}),X_j^{\tau }|\tau \le t\}$$

(42)

where $\mathcal{M}(t)$ represents the shared memory pool at time $t$ , and N denotes the ISSA population size.

At the application level, A3C’s empirical data can guide ISSA’s local search. For instance, ISSA individuals’ position updates rely not only on the global optimal solution $X_{\mathrm{best}}^t$ but also reference optimal action sequences from A3C. Specifically, the individual update formula is modified as:

$$X_i^{t+1}=X_i^t+\rho ·(X_{\mathrm{best}}^t-X_i^t)+\eta ·g(a_t)$$

(43)

where $\rho$ denotes the global convergence factor, $\eta$ represents the interaction adjustment coefficient, and $g(a_t)$ is the energy distribution vector mapping function derived from A3C actions $a_t$ . This design enables ISSA to leverage A3C’s local experience during global search, thereby enhancing convergence efficiency in high-dimensional optimization problems.

Conversely, ISSA’s global optimization results can also inform A3C’s policy update process. Specifically, the high-quality solution vector $X_{\mathrm{elite}}^t$ identified by ISSA during iteration is mapped to a reward correction term, introduced into A3C’s value update formula:

$$R_t^{\prime }=r_t+\gamma V(s_{t+1})+\zeta ·h(X_{\mathrm{elite}}^t)$$

(44)

where $R_t^{\prime }$ denotes the modified temporal difference reward, ζ is the adjustment factor, and $h(X_{\mathrm{elite}}^t)$ represents the reward enhancement function derived from ISSA’s optimal solution mapping.

Real-time and computational complexity constraints

In the collaborative optimization of energy efficiency and battery health for distributed hub-motor vehicles, the optimization algorithm must not only possess global optimization capabilities but also meet the real-time requirements of the onboard control system. Within the hybrid optimization framework design, this paper further introduces real-time and computational complexity constraints to ensure the proposed method can operate effectively in embedded vehicle environments.

First, from the perspective of algorithm runtime, let τ denote the average runtime per iteration. The real-time constraint can be expressed as:

$$\tau \le {\tau }_{max}$$

(45)

where ${\tau }_{max}$ represents the maximum decision delay threshold permitted by the in-vehicle system (typically ranging from milliseconds to hundreds of milliseconds). Second, from the perspective of computational complexity, the overall complexity of the hybrid optimization framework combining A3C and ISSA can be expressed as:

$${\mathcal{O}}_{\mathrm{hybrid}}={\mathcal{O}}_{\mathrm{A}3\mathrm{C}}+{\mathcal{O}}_{\mathrm{ISSA}}$$

(46)

where ${\mathcal{O}}_{\mathrm{A}3\mathrm{C}}$ primarily depends on the number of parallel agents M, the state dimension $d_s$ , and the action dimension $d_a$ , with its complexity approximated as:

$${\mathcal{O}}_{\mathrm{A}3\mathrm{C}}=\mathcal{O}(M·d_s·d_a)$$

(47)

where ${\mathcal{O}}_{\mathrm{ISSA}}$ depends on the population size N, problem dimension d, and iteration count T, with its complexity expressed as:

$${\mathcal{O}}_{\mathrm{ISSA}}=\mathcal{O}(N·d·T)$$

(48)

After comprehensive consideration, the computational complexity constraint for the hybrid framework is:

$${\mathcal{O}}_{\mathrm{hybrid}}\le {\mathcal{O}}_{max}$$

(49)

where ${\mathcal{O}}_{max}$ represents the maximum tolerable computational complexity for in-vehicle systems. By rationally controlling the population size, number of parallel agents, and iteration steps in algorithm design, computational burden can be reduced while ensuring optimization performance, thereby achieving real-time deployability.

Finally, to further quantify the algorithm’s real-time capability and efficiency, this paper introduces a comprehensive performance metric:

$$J=\alpha ·{\eta }_{\mathrm{sys}}-\beta ·\frac{\tau }{{\tau }_{max}}$$

(50)

where ${\eta }_{\mathrm{sys}}$ represents the system energy efficiency metric, $\tau$ denotes the average runtime, ${\tau }_{max}$ indicates the latency upper bound, and α and β are weighting coefficients.

Simulation test and result analysis

To validate the effectiveness of the proposed method in distributed hub motor vehicles, testing and verification were conducted. The test environment encompassed hardware configurations and software tools to ensure comprehensive and accurate evaluation.

Simulation environment setup

The feasibility and effectiveness of the designed algorithm were simulated using the MATLAB/Simulink platform to validate the control strategy and optimization algorithm. The simulation model and algorithm parameter settings are shown in Table 2.

Table 2.

Simulation model and algorithm parameter settings.

Module name	Parameter name	Numerical
Vehicle module	Vehicle mass	1800 kg
	Wheelbase	2.8 m
	Track width	1.6 m
Vehicle dynamics module hub motor	Coefficient of aerodynamic drag	0.28
	Rated power	50 kW
	Peak torque	700 Nm
	Maximum speed	1500 rpm
Battery	Battery capacity	60kWh
	Nominal voltage	400 V
	Maximum charging power	100 kW
	Maximum discharge power	250 kW
A3C algorithm module	Learning rate	0.0005
	Discount factor	0.99
	Entropy regularization coefficient	0.01
	Number of parallel environments	16
	State space dimension	12
	Action space dimension	4
ISSA	Maximum iterations	600
	Forager ratio	0.2
	Watcher ratio	0.1
	Chaos mapping coefficient	4
	Local search probability	0.8

Result analysis

Figure 4 presents the convergence performance of different optimization methods under three standard driving cycles (NEDC, WLTC, and CLTC). All simulations were conducted over 500 iterations using consistent initial conditions and parameter settings, with the normalized objective value reflecting a weighted combination of energy efficiency, battery health, and system stability. Under the NEDC condition (Figure 4(a)), characterized by frequent acceleration–deceleration events and low-speed operation requiring fast response, the A3C + ISSA algorithm achieved convergence to 0.12 within approximately 100 iterations—a 50% improvement over standalone A3C (0.24). In contrast, GA, PSO, and PID converged only to 0.28, 0.30, and 0.45, respectively, highlighting their slower and less effective optimization performance. This improvement is attributed to the complementary strengths of A3C’s asynchronous parallel learning and ISSA’s chaotic initialization combined with the golden sine–cosine local search strategy.

Figure 4.

Convergence performance comparison of hybrid optimization algorithms under different driving cycles.

For the WLTC condition (Figure 4(b)), which covers low-, medium-, high-, and ultra-high-speed stages representative of real driving environments, A3C + ISSA demonstrated stable convergence and reached an objective value of approximately 0.15. Meanwhile, PSO exhibited sustained oscillations between 0.30 and 0.35, indicating limited robustness in high-dimensional and dynamic optimization spaces. Under the CLTC cycle (Figure 4(c)), featuring pronounced acceleration fluctuations that intensify dynamic adaptability requirements, A3C + ISSA converged from 1.0 to 0.35 within the first 50 iterations—3.2 times faster than GA and 4.5 times faster than PID—and remained stably near the optimal solution thereafter.

Figure 5 compares the energy consumption performance of different control strategies under three standard driving cycles (NEDC, WLTC, and CLTC), with energy consumption normalized to kWh/100 km. Under the NEDC condition (Figure 5(a)), A3C + ISSA achieved the lowest energy consumption of 12.7 kWh/100 km, representing reductions of 9.3%, 16.4%, 19.1%, and 23.0% compared with A3C, GA, PSO, and PID, respectively, thereby demonstrating clear superiority over all benchmark methods. For the WLTC cycle (Figure 5(b)), although overall energy consumption increased due to extended high-speed segments, A3C + ISSA maintained 14.0 kWh/100 km—10.7% lower than A3C (15.5 kWh/100 km). Moreover, its standard deviation (±0.3 kWh/100 km) was markedly smaller than those of PSO (±0.8) and PID (±1.2), indicating enhanced stability and robustness. Under the CLTC condition (Figure 5(c)), A3C + ISSA achieved 15.2 kWh/100 km, delivering a 23.2% energy reduction compared with PID’s 19.8 kWh/100 km, further validating its effectiveness across diverse driving conditions.

Figure 5.

Energy consumption comparison of different optimization strategies under three driving cycles.

Figure 6 presents the evolution of battery health metrics—including SOH, capacity retention, and internal resistance growth—over 5000 charge–discharge cycles under different driving conditions, comparing the long-term performance of A3C + ISSA with conventional control strategies. Under the NEDC condition (Figure 6(a)), A3C + ISSA maintained a battery SOH of 96.5% after 5000 cycles, while the traditional method declined to 94.2%. Considering the 80% SOH end-of-life threshold, the proposed strategy extended cycle life from approximately 3500 to over 5000 cycles, corresponding to a 42.8% improvement. Under the WLTC cycle (Figure 6(e)), capacity retention decreased more gradually with A3C + ISSA, maintaining 96.8% after 3000 cycles compared to 94.5% for the baseline method. This improvement is primarily attributed to A3C + ISSA’s adaptive SOC window management, which mitigates degradation from deep charge–discharge cycling. Under the CLTC condition (Figure 6(i)), internal resistance for the traditional control increased to 118 mΩ (18% growth), whereas A3C + ISSA limited it to 110 mΩ (10% growth), effectively reducing the resistance growth rate by 44.4%. This demonstrates superior mitigation of degradation caused by electrochemical side reactions. The shaded prediction region represents the 95% confidence interval of degradation trends, incorporating uncertainties from cell variability, temperature fluctuation, and driving behavior, thus providing statistical confidence for lifetime estimation. Overall, despite higher stress levels under CLTC conditions, A3C + ISSA consistently exhibited stable battery protection across all driving cycles. These improvements arise from the ISSA layer’s explicit modeling of degradation constraints and its global optimization capability, enabling a sustained balance between energy efficiency and battery longevity. The results confirm the framework’s effectiveness in extending battery life, reducing lifecycle costs, and supporting the sustainable operation of electric vehicles.

Figure 6.

Battery health state evolution characteristics and degradation mechanism comparison under different driving cycles.

Figure 7 illustrates the real-time control behavior and dynamic battery response of the A3C + ISSA strategy under NEDC, WLTC, and CLTC driving conditions. Under the NEDC condition (Figure 7(a)), power demand rises rapidly to 18 kW during acceleration. The A3C controller allocates approximately 60% of the power to the front axle to enhance acceleration while maintaining left–right symmetry for directional stability. During cruising, power decreases to 5–8 kW with nearly equal distribution among all four wheels, whereas the braking phase exhibits negative power due to coordinated front–rear axle operation that maximizes regenerative energy recovery. The SOC decreases from 90% to about 88%, and the battery temperature rises from 25°C to 35°C—both remaining within safe operating limits. Under the WLTC condition (Figure 7(b)), the power demand exceeds 40 kW during high-speed segments, with the rear axle carrying roughly 55% of the load to improve transmission efficiency and stability. SOC decreases by 3.5%, and the maximum temperature reaches 40°C, remaining below the 45°C safety threshold. In the CLTC scenario (Figure 7(c)), frequent acceleration and deceleration lead to high-frequency oscillations in the power curve, with peaks up to 30 kW. Nevertheless, the control update period remains within 10 ms, satisfying the real-time constraints of vehicle dynamics. Across all three cycles, the power distribution curves are smooth and continuous, with no abrupt fluctuations, ensuring both driving comfort and reduced mechanical stress on the drivetrain.

Figure 7.

Real-time power allocation response and battery thermal-electrical coupling characteristics under three driving cycles.

Figure 8 compares the Pareto frontiers of A3C + ISSA, NSGA-II, and SPEA2 under NEDC, WLTC, and CLTC conditions to evaluate their performance in the dual-objective optimization of energy consumption and battery life. The objectives were defined as minimizing energy consumption (kWh/100 km) and maximizing cycle life. Each algorithm was independently executed 30 times under identical computational settings to ensure statistically robust results.

Figure 8.

Pareto frontier comparison of multi-objective optimization algorithms in energy consumption-battery life trade-off space.

Under the NEDC condition (Figure 8(a)), A3C + ISSA achieves a clearly superior Pareto frontier, delivering longer battery life at comparable energy consumption levels. Its solution set is continuous, evenly distributed, and spans a broader range of trade-offs. In the WLTC scenario (Figure 8(b)), where overall energy consumption is higher, A3C + ISSA retains its dominance, providing the best compromise between energy use and lifespan. The framework achieves a hypervolume of 8.32 × 10⁴, surpassing NSGA-II and SPEA2 by 16.4% and 26.4%, respectively, demonstrating superior convergence and solution diversity. Under the CLTC condition (Figure 8(c)), A3C + ISSA again produces a more complete Pareto frontier and successfully explores extreme trade-off boundaries between energy saving and battery wear, indicating robust global search capability.

Overall, the A3C layer effectively learns fine-grained power allocation policies within the continuous state–action space to balance short-term energy efficiency and long-term battery life, while the ISSA layer refines global parameter optimization through enhanced search diversity and constraint handling. Their synergistic interaction yields consistently superior Pareto fronts across all driving conditions. In contrast, NSGA-II and SPEA2 exhibit slower convergence and uneven solution distributions under high-dimensional constraints. These results confirm that A3C + ISSA not only provides theoretically superior Pareto solutions but also delivers more reliable and diverse decision options for multi-objective vehicle control optimization.

Figure 9 compares the real-time response time and normalized computational complexity of different optimization strategies under the NEDC, WLTC, and CLTC driving cycles. Under the NEDC condition (Figure 9(a)), the A3C algorithm achieves the fastest response time of approximately 38 ms but lacks long-term optimization capability. Although ISSA provides strong global search ability, its 95 ms response time fails to meet real-time control requirements. Both GA and PSO exceed 150 ms with complexities approaching 1.0, rendering them unsuitable for embedded vehicle controllers. In contrast, A3C + ISSA maintains a 48 ms response time—satisfying the 50 ms real-time threshold—with a normalized complexity of 0.72 within an acceptable range. This balance results from a dual-layer temporal decoupling structure: the A3C layer executes high-frequency, real-time power allocation, while the ISSA layer performs low-frequency optimization of long-term parameters in the background. The shared memory interaction between layers ensures efficient coordination between performance and computational load. Under the WLTC condition (Figure 9(b)), where dynamic calculations are more complex, all algorithms exhibit longer response times; however, A3C + ISSA remains stable at 45 ms, while GA and PSO increase to 210 and 168 ms, further widening the performance gap. In the CLTC test cycle (Figure 9(c)), A3C + ISSA maintains a stable complexity of 0.72 ± 0.02, demonstrating scalability and insensitivity to varying test conditions, whereas GA and PSO exhibit sharp complexity growth as constraints increase.

Figure 9.

Real-time response performance and computational complexity trade-off analysis of different optimization algorithms.

Figure 10 depicts the three-phase stator current waveforms and key electrical parameters of the four in-wheel motors operating under the A3C + ISSA control strategy during steady-state cruising in the NEDC cycle. All four motors exhibit standard sinusoidal current waveforms with 120° phase separation, confirming accurate vector control. The RMS current ranges from 19 to 20 A, corresponding to approximately 6.2 kW per wheel, aligning well with the vehicle’s total power demand. The power factor remains 0.88, and the total harmonic distortion (THD) is maintained around 10%, both within industry standards, validating the precision of the current-loop control. The front axle motors show slightly higher current amplitudes than the rear, reflecting a front-wheel-priority energy distribution strategy, while minor left–right current asymmetries enable differential yaw-stability correction. The standard deviation of current amplitude across all motors is less than 2 A, and the waveforms remain smooth and continuous, indicating strong disturbance suppression and stable traction performance. The superimposed high-frequency ripple coincides with the PWM switching frequency but remains small in amplitude, exerting negligible impact on system efficiency and motor longevity.

Figure 10.

Three-phase current waveforms and electrical performance parameter validation of four independent in-wheel hub motors.

Figure 11 illustrates the hardware implementation of the A3C + ISSA control strategy at both the power electronics and upper-level power scheduling layers under NEDC conditions. Figure 11(a) and (b) show the PWM drive waveforms of the front-axle motors, exhibiting standard rectangular pulses with a 15 kHz switching frequency, an average duty cycle of approximately 50%, a dead time of 2 μs, and rise/fall times below 100 ns—all compliant with automotive-grade IGBT specifications. The duty cycle dynamically varies with power demand, highlighting the real-time scheduling capability of the A3C controller. The left and right motor waveforms remain nearly symmetrical, with minor duty cycle deviations enabling differential control for yaw stability.

Figure 11.

PWM control signal characteristics and power tracking response performance validation for four independent drive motors.

Figure 11(c) and (d) compare the target power commands and actual power outputs of the rear-axle motors. The results indicate a system response delay of approximately 25 ms, a steady-state error below 2%, and an update frequency of 1 kHz. The control system maintains ± 2% tracking accuracy across multiple step and ramp power transitions, ensuring both dynamic responsiveness and steady-state precision. Overall, the A3C + ISSA control framework demonstrates fast, stable, and precise performance in both PWM signal generation and closed-loop power tracking. Key indicators—25 ms response delay, 2% steady-state error, and 1 kHz update rate—meet or exceed automotive motor control standards, confirming the framework’s robustness and feasibility for real-time hardware deployment.

Figure 12 comprehensively illustrates the dynamic management performance of the A3C + ISSA strategy in regulating battery current, SOC, and temperature under NEDC conditions. The simulation was conducted using an electrochemical–thermal coupled model of a 32 kWh battery pack. Figure 12(a) shows the current waveform, where the peak discharge and regenerative currents are 154.5 A and –44.9 A, respectively—both below the 200 A design limit—demonstrating the strategy’s ability to smooth and suppress current spikes. The average ripple is only 5 A RMS, indicating minimal cyclic stress. Figure 12(b) presents the SOC evolution, which decreases from 85% to approximately 73%, remaining consistently within the 30%–90% health range and thus preventing accelerated degradation from deep charge–discharge cycling. The SOC trend also aligns with the previously reported energy consumption data, confirming model consistency. Figure 12(c) shows the temperature profiles from six sensors distributed across the battery pack. The temperature remains uniformly controlled at 25 ± 2°C, with a maximum spatial difference below 3°C, indicating that the thermal management system maintains effective temperature regulation with low energy consumption while preventing local hotspots and gradients. Overall, the results demonstrate that the proposed optimization framework achieves closed-loop coupled management of current, charge, and temperature, maintaining current peaks within limits, SOC within the optimal window, and temperature within safe operating ranges. This integrated control significantly mitigates degradation, extending battery lifespan and reducing overall lifecycle costs.

Figure 12.

Lithium-ion battery pack charge-discharge current characteristics, SOC dynamic estimation and multi-point temperature field distribution monitoring.

Figure 13 depicts the operating trajectories and temporal evolution of four in-wheel motors on the torque–speed plane under NEDC conditions, obtained through a coupled finite-element and vector-control simulation of PMSMs. The color gradient represents time progression, covering all major operating states including stationary, acceleration, cruising, and deceleration. Figure 13(a) and (b) show nearly identical trajectories for the front-axle motors, primarily distributed within the constant-torque region and the medium- to high-speed, low-torque zone. The dense clusters coincide with the motor’s high-efficiency area, confirming the efficiency-oriented behavior of the A3C + ISSA control strategy. The trajectories also exhibit proper dispersion in the braking zone, demonstrating effective regenerative energy recovery. Figure 13(c) and (d) illustrate that the rear-axle motors generate slightly lower peak torque than the front, reflecting a differentiated control strategy in which the front axle provides greater propulsion while the rear prioritizes stability and thermal balance. All four motors operate well within the safe torque–speed envelope, maintaining adequate safety margins. The clear front–rear differentiation and left–right symmetry indicate that the control framework effectively balances traction demand and vehicle stability. Collectively, these results confirm that the A3C + ISSA strategy realizes multi-level optimization in four-wheel independent drive—achieving differentiated front–rear torque allocation, symmetric left–right coordination, efficiency-oriented operation, and safe operating margins—thereby validating its engineering applicability for distributed in-wheel motor systems.

Figure 13.

Torque-speed full operating characteristic maps and dynamic response performance evaluation of four independent in-wheel hub motors.

Discussion

To objectively position the proposed A3C + ISSA framework within the current research landscape, this section provides a comparative analysis against mainstream approaches in the literature and discusses the boundaries of its application.

• (1) Compared to traditional evolutionary algorithms such as NSGA-II and SPEA2, A3C + ISSA demonstrates stronger convergence capabilities and broader solution coverage. As quantified in Section 4.2, under the WLTC cycle, the Hypervolume indicator of the proposed method reaches 8.32 × 10⁴, surpassing NSGA-II and SPEA2 by 16.4% and 26.4%, respectively. This indicates a more effective exploration of the non-convex trade-off boundary between energy consumption and battery life, providing superior candidate solutions for decision-making.

• (2) Existing methods often employ decoupled hierarchical strategies ⁸ or single-time-scale reinforcement learning, ¹³ failing to reconcile millisecond-level torque control with month-level battery aging. The proposed framework bridges these fast and slow dynamics via the dynamic weight adjustment mechanism and the cross-scale shared memory pool. Simulation results confirm that this coupling mechanism extends battery cycle life by 42.8% without compromising vehicle dynamics, significantly outperforming single-objective strategies that ignore aging feedback.

• (3) Although the global search of ISSA entails high computational complexity, this study adopts a “offline training-online inference” hierarchical deployment architecture. High-frequency torque control is executed by the trained A3C network, with a single-step inference time of less than 10 ms, fully meeting the real-time requirements of onboard controllers. This offers superior engineering practicality compared to model predictive control approaches ²¹ that rely heavily on intensive online iterative optimization.

Notwithstanding its superior performance, the method faces certain challenges in practical application:

• (1) The global optimization of the ISSA layer relies heavily on the accuracy of the electrochemical battery model. If actual battery parameters deviate significantly from the mechanistic model due to extreme environments or aging (i.e. model mismatch), the optimality of the global weights may degrade. Future research could incorporate adaptive state observers or digital twin technologies to enhance online model correction capabilities.

• (2) In contrast to rule-based methods ⁴ that require no training, the A3C agent demands substantial offline data interaction and computational resources to converge. While transfer learning may reduce training time for new driving cycles, improving sample efficiency remains a common challenge for reinforcement learning in vehicle control domains.

Hardware-in-the-loop experimental validation

To verify the real-time performance, control accuracy, and engineering feasibility of the proposed A3C + ISSA hybrid optimization framework on a physical controller, a Hardware-in-the-Loop (HIL) experimental platform was established. Compared to pure software simulations, HIL testing can introduce the computational delays, communication latencies, and electrical characteristics of a real controller and its interfaces, thereby providing a more accurate assessment of the algorithm’s actual performance when deployed in a vehicular environment. This experiment aims to validate the end-to-end effectiveness of the algorithm, from high-level strategy to low-level hardware execution.

The architecture of the HIL experimental platform constructed for this study is shown in Figure 14. The platform forms a complete closed-loop testing system composed of three parts: a Host PC, a Real-time Target Machine, and a physical test bench. The Host PC serves as the monitoring layer, responsible for setting driver profiles (e.g. NEDC/WLTC), issuing start/stop commands, and monitoring and storing data in real-time. The Real-time Target Machine (e.g. Speedgoat/dSPACE) is the core of the simulation, running a high-fidelity vehicle model with microsecond-level precision. This model integrates a driver model, four independent in-wheel motor and wheel models, a vehicle dynamics model, and a battery and energy storage system model.

Figure 14.

Hardware-in-the-loop simulation system architecture for the distributed drive electric vehicle.

On the physical test bench, the A3C + ISSA optimization algorithm proposed in this paper was deployed onto an automotive-grade embedded Controller Under Test (CUT) through automatic code generation. This controller communicates with the Real-time Target Machine in a closed loop via a 500 kbps CAN bus: it receives simulated sensor signals (such as vehicle speed, SOC, SOH, etc.) from the target machine and, based on the algorithm’s calculations, sends back four independent torque commands. Simultaneously, these commands are processed by a power amplifier to drive a physical motor connected to a load dynamometer. This architecture allows the CUT to operate as if it were in a real vehicle, enabling it to be tested under near-realistic electrical and mechanical loads in a safe and controlled laboratory environment, thus providing robust validation for the algorithm’s engineering application.

The ultimate effectiveness of the top-level optimization strategy depends on the execution fidelity of the underlying hardware. We first verified the dynamic tracking performance of the motor in response to the high-level power commands. As shown in Figure 15, the HIL experimental results indicate that the actual output power of the rear-axle motor (red solid line) can accurately follow the target power (blue solid line) and remains stable within the ±2% target error band. Specific performance metrics show that the system’s response delay is only 23.0 ms, the steady-state error is less than 1.9%, and the controller update frequency reaches 1 kHz. These key dynamic indicators meet or exceed in-vehicle motor control standards, strongly demonstrating that the commands from the proposed optimization strategy can be executed by the hardware quickly, stably, and precisely.

Figure 15.

Verification of motor power dynamic tracking performance in HIL experiment.

After verifying the precision of power tracking, we further evaluated the macroscopic electrical performance of the motor on the physical hardware. As shown in Figure 16, the three-phase current waveforms of the rear-axle motor exhibit a standard sinusoidal form, indicating that the current loop control is stable and effective. Key electrical parameters reveal that the motor’s Root Mean Square (RMS) current is 19.9 A, the power factor is as high as 0.92, and the Total Harmonic Distortion (THD) is controlled at 10.0%, all of which meet industry design standards. This result confirms that the optimization algorithm, when working in conjunction with the hardware, can ensure excellent power quality and efficient energy conversion.

Figure 16.

Verification of motor three-phase current electrical characteristics in HIL experiment.

Conclusion

This study addresses the dual challenge of improving energy efficiency and safeguarding battery health in distributed in-wheel motor electric vehicles by proposing a hybrid multi-objective optimization framework integrating the A3C algorithm with ISSA. At the theoretical level, comprehensive models of vehicle dynamics and battery health were developed, incorporating drive-force distribution, capacity degradation, internal resistance variation, and thermal constraints, providing the physical foundation for algorithmic design. Methodologically, A3C enables real-time energy distribution optimization among multiple motors with strong adaptability under dynamic conditions, while ISSA enhances global search performance through chaotic initialization, golden sine–cosine local search, and mechanism-driven constraints, thereby improving the optimality and feasibility of battery health management. The hybrid framework further integrates a dynamic weight adjustment mechanism and a cross-scale shared memory pool, enabling synergistic optimization between short-term energy efficiency and long-term battery longevity while explicitly satisfying real-time and computational efficiency requirements.

Simulation results under NEDC, WLTC, and CLTC driving conditions demonstrate that the proposed framework substantially outperforms traditional control methods and baseline algorithms in terms of convergence speed, energy efficiency, and battery life extension. Additionally, the hardware-in-the-loop validation demonstrated the framework’s outstanding real-time performance and control precision, confirming its strong potential for deployment in actual vehicle controllers and its overall engineering applicability.

Future research will focus on three directions: (1) validating the robustness of the framework under complex and non-ideal road conditions, including boundary scenarios such as extreme weather and motor faults; (2) developing lightweight, embedded implementations of the framework on in-vehicle hardware to meet stringent real-time and resource constraints; and (3) extending the methodology to fleet-level and vehicle-to-grid interaction scenarios to enhance the overall synergy between energy efficiency and battery health management at the transportation system level.