An IPSO-RC-Based Study on Dynamic Coordination Excitation and Optimal Capacity Allocation for Marine Hybrid Energy Systems

Abstract

As a pivotal element in the maritime sector’s green transition, fuel-cell-powered ships have attracted increasing attention due to the energy efficiency and stability of their onboard powertrains. Yet, the dynamic coordination and capacity optimization of fuel cells and supercapacitors remain among the most formidable technological challenges. In this study, a hybrid marine power system pairing fuel cells with supercapacitors is devised by integrating robust control with a particle swarm optimization (PSO) algorithm. The results reveal that, under complex operating conditions, robust control effectively mitigates system uncertainties and secures reliable operation of the ship’s energy system. Optimally allocating component capacities via PSO markedly enhances the synergy between the fuel cell and the supercapacitor. Compared with conventional schemes, optimized architecture boosts energy efficiency by 12.5%, shortens response time by 8.4%, and demonstrates clear superiority in robustness and stability. This robust-control-based hybrid configuration therefore delivers outstanding performance and offers compelling guidance for the refined design of marine propulsion systems.

1. Introduction

In the maritime sector, traditional solar and wind energy systems cannot be applied to domains such as submarines. Against this backdrop, conventional hybrid propulsion systems have been widely adopted. This propulsion system primarily relies on high-efficiency diesel engines to drive generators, which in turn charge batteries and perform other functions. Fuel cells have rapidly attracted attention as a cutting-edge clean-energy technology. They offer markedly superior advantages, notably higher power density and exceptional operational reliability. By converting hydrogen or other renewable gases into electricity, fuel cells furnish ships with a continuous, dependable power supply while delivering zero emissions [1]. Yet, their sluggish dynamic response and limited output stability prevent them from single-handedly satisfying the highly variable power demands encountered at sea. Consequently, they are commonly integrated with energy-storage devices to form hybrid propulsion systems that guarantee stable navigation while minimizing fuel consumption and greenhouse-gas emissions [2,3]. Effective operation further depends on intelligent energy-management strategies that allocate power among the sources in accordance with real-time system requirements and component states. Although significant progress has been achieved in recent years, the field still confronts several formidable technical challenges.

Most current energy-management studies for hydrogen-fuel-cell hybrid marine systems are conducted under static, idealized sea conditions. Under dynamic disturbances, load uncertainties, or extreme weather, the prevailing schemes often suffer from inadequate robustness and diminished energy efficiency. In addition, the sizing of energy-storage components is typically based on empirical rules or static models, which struggle to meet the need for rapid, adaptive responses during actual voyages [4,5]. Compared to lithium batteries, supercapacitors offer higher power density and longer cycle life, making them more suitable for the focus of this study. To address these issues, this work combines robust control with a PSO to design a fuel-cell/supercapacitor hybrid propulsion system for marine applications. Robust control techniques are employed to counter system uncertainties and dynamic perturbations inherent in complex ocean environments, while PSO determines the optimal capacities of the fuel cell and supercapacitor, significantly enhancing overall energy efficiency and transient performance.

The remainder of this paper is organized as follows: Section 2 presents a comprehensive literature review; Section 3 details the construction of the marine fuel-cell/supercapacitor hybrid model based on robust control [6]; Section 4 describes the joint optimization of capacity sizing and operational strategy; Section 5 evaluates the proposed model experimentally and compares it with existing approaches; Section 6 concludes the paper.

2. Literature Review

2.1. Optimization Method for the Capacity Configuration of Energy-Storage Systems

Traditional solar and wind energy sources are not suitable for use as primary pro-pulsion systems for ships. Consequently, current mature research primarily focuses on conventional hybrid propulsion systems, typically combining high-efficiency diesel engines to drive generators that charge batteries and supply power to the propulsion system [7] proposes a numerical analysis method focused on optimizing the hybridization of a speed-coupled parallel hybrid powertrain for tracked vehicles. Dynamic programming algorithms and a custom driving cycle are employed to determine the optimal hybridization factor and evaluate parameter sensitivity.

Another research direction involves replacing diesel generators with fuel cells, integrating fuel cells with energy storage systems to form hybrid energy systems. Integrating energy-storage units into a fuel-cell-driven propulsion system secures power quality and overall stability. Given a fixed electrical topology, judicious selection of storage parameters not only offsets the fuel cell’s sluggish dynamics but also improves economic performance. Hence, determining the proper storage capacity is pivotal to the reliable operation of a ship’s power network. Because the optimal sizing problem is tightly intertwined with energy-management strategy, it is typically tackled through either coupled or decoupled optimization frameworks.

Within a decoupled framework, capacity sizing and energy-management are treated as two independent optimization tasks. In [8], Boveri established distinct models for each and solved them with the fmincon and CPLEX optimizers, respectively. To ease computational demands, Accetta and Pucci [9] first derived the storage capacity directly from the load profile, then introduced an optimization-based power-allocation strategy that minimizes both fuel consumption and generator output fluctuations. In [10], Focusing on a leisure vessel, Wang Jun determined the optimal battery capacity by analyzing load distributions across an entire voyage; their results confirmed that, under varying initial states of charge, the hybrid storage system reliably sustained ship operations. Seeking the best configuration, Pang Shui formulated a multi-objective problem that incorporated system cost and service life [11], treated the storage system’s rated capacity and power as decision variables, and resolved it through a differential-evolution algorithm.

Although the decoupled approach is relatively straightforward, it seldom furnishes a truly optimal solution [12]. Consequently, researchers have championed a coupled methodology to determine the optimal capacity of the energy-storage system. Within this framework, multi-objective functions are formulated so that the two problems can be addressed concurrently. In [13], Haseltalab proposed an optimization-based energy-management strategy that treats the ship’s available space and the storage system’s mass as constraints and seeks to maximize fuel-cell efficiency, thereby deriving the storage system’s optimal capacity. Letafat advanced a multi-objective optimization scheme that tackles both problems while factoring in the ship’s operating expenses and initial capital investment [14]. In [15], Bao integrated the capacity-sizing task with optimal power allocation in a unified mathematical model and examined how service life, investment cost, and current rate influence storage capacity; refs. [16,17] likewise adopted a joint multi-objective optimization approach, obtaining the optimal capacity by minimizing both the ship’s operating costs and battery degradation. To refine capacity sizing and power distribution still further, Valera-Garcia constructed a nonlinear multi-objective optimization model and solved it with a non-dominated sorting genetic algorithm [18].

In summary, the decoupled strategy treats the two problems in isolation and, although it can swiftly produce an optimal result, it rarely achieves the global optimum. Conversely, the coupled approach constructs a suite of multi-objective functions, enabling a far more faithful representation of real-world demands.

2.2. Energy Management Strategy for Fuel Cell Ships

Fuel cell ships typically integrate multiple energy systems to ensure stable operation while minimizing fuel consumption and greenhouse gas emissions [19]. Energy management strategies form the core of hybrid energy systems, enabling rational planning and control of power output from various sources based on system demands and the status and characteristics of power sources, thereby guaranteeing stable system operation [20]. Energy management strategies can be categorized into rule-based and optimization-based approaches [21].

Rule-based energy management strategies represent static control approaches. These strategies establish control rules based on prior experience to allocate load power among different power sources, offering robust performance and strong real-time capabilities. For ship hybrid energy systems [22,23] proposed a frequency-based energy management strategy. This approach employs a low-frequency filter to separate load power fluctuations into low-frequency and high-frequency components. Lithium batteries supply low-frequency loads, supercapacitors supply high-frequency loads, and the FCS handles the stable portion of load power. However, previous studies typically set the filter time constant as fixed. In [24], to ensure stable operation of the ship’s power system, researchers proposed an energy management strategy with variable filter time constants. This strategy enables real-time adjustment of the filter time constant based on the state of charge and operating conditions of the energy storage system. Wang et al. [25] proposed a droop-based energy management strategy to achieve rational load power distribution among generators. A small-signal model of the power system was established to analyze system stability.

Global optimization-based energy management strategies require knowledge of the entire load distribution across the operational cycle. They simultaneously consider optimization objectives and constraints, performing offline global optimization within a specified timeframe [26]. The core concept of dynamic programming involves decomposing the problem into distinct stages. By defining the state and interrelationships for each stage, subproblems are solved sequentially, enabling recursive resolution of the optimization problem [27]. In [28], Yuan studied a hybrid ship equipped with a diesel generator and batteries, proposing an energy management strategy combining dynamic programming and model predictive control. The study also examined the impact of noise disturbances on the control strategy. Dynamic programming algorithms can address energy management for loads but suffer from high computational complexity and poor handling of coupled constraints. Compared to dynamic programming, intelligent optimization algorithms demonstrate significant advantages in tackling complex nonlinear problems. In [29], Rafiei studied ferry equipped with fuel cells, lithium batteries, and shore power systems, considering both underway and docked states. To reduce operational costs, an improved cosine algorithm was employed to optimize the energy system scheduling.

When establishing energy management strategies for ship hybrid power systems, full consideration should also be given to the integration between fuel cells and the Energy Management System (EMS) to enable real-time monitoring and precise control of the fuel cell system. A fuel cell management system (FCMS) based on digital twins was proposed in Reference [30]. By integrating the fuel cell system with an energy management system (EMS), the system collects sensor data in real time and uploads it to the cloud, dynamically updating the battery twin model. Through the fusion of the digital twin model with the cloud platform, the system achieves intelligent management and optimized control of the fuel cell. Furthermore, the platform innovatively integrates high-precision digital twin models with cloud computing technology to establish a scalable, iterative smart management framework [31].

However, existing studies rarely investigate EMS performance under extreme marine conditions such as violent sea states, sudden load surges, or component failures. These scenarios often lead to rapid fluctuations in power demand and system instability, yet current strategies remain limited to steady or moderate operating states. Therefore, it is imperative to develop an adaptive, dynamically coordinated EMS capable of ensuring efficient energy distribution and system survivability under extreme conditions.

To address these limitations, this paper proposes a dynamic matching and capacity allocation method for marine hybrid energy systems based on IPSO-RC. To address the stability requirements of marine hybrid energy systems under nonlinear disturbances and parameter-varying conditions, a nonlinear function approximator combined with a multi-input regression model is employed to optimize matching coefficients online. Through fuzzy logic inference and Bayesian optimization, self-learning updates of matching coefficients are achieved. Simultaneously, time-varying filtering is introduced to suppress high-frequency oscillations in load power during marine operations. Within a robust feedback framework, the state feedback gain matrix K is made adjustable. By jointly optimizing MRAC and fuzzy inference, real-time compensation for parameter drift is achieved.

Table 1. Comparison of recent representative studies and the positioning of this study.

Research Objective/Method Summary	Control Type	Optimization Included	Validation Method	Reported Efficiency/Advantages	Limitations Compared with IPSO-RC
DC microgrid EMS for intelligent ships: aims to minimize fuel consumption and compensate for faults	Rule-/droop- based + optimization scheduling	Yes (convex/ numerical optimization)	Simulation	Reduced fuel consumption and fluctuation suppression	Focuses on fuel economy and fault compensation; lacks handling of strong nonlinearity, time-varying uncertainty, and coordinated capacity design
Cruise ship: capacity configuration of fuel cell and battery	Rule-/static strategy	Yes (capacity optimization)	Voyage simulation	Ensures power supply over full voyage	Mostly offline capacity optimization; lacks robustness to dynamic disturbances
Marine microgrid battery capacity multi-objective optimization	Rule-based	Yes (multi- objective evolutionary algorithm)	Simulation	Balances cost and lifetime	Not deeply coupled with online energy allocation or robust control
SOFC ship: integrated capacity sizing and EMS optimization	Optimization-oriented	Yes (coupled optimization)	Simulation	Improved fuel-cell efficiency	Targets steady-state efficiency; lacks adaptive and filtering mechanisms for harsh sea conditions
Zero-emission ferry: joint optimization of energy management and sizing	Optimization/scheduling	Yes (multi-objective)	Feasibility analysis + simulation	Balances OPEX and CAPEX	Limited adaptive learning and robustness; lacks compensation for parameter drift
Offshore ship hybrid system optimal design	Planning/multi-objective	Yes (NSGA-II)	Simulation	Balanced design trade-offs	Focuses on offline design; lacks online adaptation and time-varying filtering
Fuel cell ship: battery/supercapacitor hybrid with frequency-based load division	Rule-based (fixed or semi-variable filtering)	Partly (capacity/parameter)	Simulation	Effectively suppresses high-frequency load	Filtering parameters fixed or semi-fixed; poor adaptability to strong nonlinearity or time-varying operation
Fuel cell + battery + supercapacitor: SOC optimization and rule-based EMS	Rule + optimization	Yes (SOC-based objectives)	Simulation	Balances response and lifetime	Still rule-driven; lacks robust control and online identification coupling
Diesel–electric hybrid: combined DP and MPC EMS	MPC/optimization	Yes (offline DP + online MPC)	Simulation + noise analysis	High economy; considers noise	Based on predictive model; lacks online capacity coordination and adjustable robust feedback gain K
Modern zero-emission ship: stochastic MPC-based EMS	MPC (with degradation constraints)	Yes (stochastic optimization)	Simulation	Balances cost and battery degradation	Relies on accurate modeling and probabilistic assumptions; lacks online self-learning of matching coefficients

presents the control and optimization characteristics of representative hybrid energy research in recent years, highlighting the comprehensive innovative advantages of IPSO-RC in self-learning, robust feedback, and real-time capacity matching in this study. IPSO-RC is linked with constant capacity and energy matching, introducing self-learning and adjustable robust feedback to suppress high-frequency disturbances, achieving parameter drift compensation and high energy efficiency.

3. Modeling of Ship Hybrid Energy Systems Based on Robust Control

The energy system of the fuel cell vessel proposed in this paper consists of a supercapacitor and a fuel cell. The control method for this hybrid energy system is based on robust control. This section introduces the theoretical background and implementation of both components.

3.1. Ship Power System

This paper adopts a similar radially distributed marine power system architecture based on the IEEE Std. 1709–2010 standard for medium-voltage DC marine power systems (1 kV–35 kV) [32,33]. As shown in Figure 1, the hybrid energy system of the fuel cell vessel comprises two FCS and two ESS. FCS serves as the primary power source, while the ESS functions as an auxiliary power source to buffer fluctuations in load power.

In the FCS, the hydrogen fuel cell connects to the DC bus via a unidirectional DC/DC converter. This paper employs a proton exchange membrane fuel cell system (PEMFC) with an efficiency of 50–60%, primarily consisting of a fuel cell stack, hydrogen supply system, air supply system, and cooling system; the ESS, possessing both charging and discharging capabilities, connects to the DC bus via a bidirectional DC/DC converter, while the load connects to the DC bus via a DC/AC converter.

3.2. System State Feedback Controller Based on Robust Control Theory

Robust control theory focuses on ensuring system stability and performance in the presence of parameter perturbations, external disturbances, and modeling inaccuracies. In contrast to classical control theory—which assumes a precise mathematical model and fixed parameters—robust control explicitly accounts for uncertainties during the design phase by introducing structured uncertainty models and quantitative performance criteria. While classical control methods such as PID or optimal control perform effectively under nominal conditions, they often fail when faced with nonlinearities, time-varying parameters, or unpredictable disturbances typical of marine environments. Robust control, by incorporating worst-case analysis and guaranteeing stability margins, enables systems to maintain consistent performance even under substantial variations in model parameters or operational conditions. Therefore, it combines mathematical rigor with practical adaptability, making it particularly suitable for marine hybrid energy systems operating in uncertain and dynamically changing sea states.

To ensure stable system operation under dynamic disturbances and uncertainties, the state-space equations of the system are constructed using robust control theory. However, in actual ship navigation, several internal parameters of the fuel cell—such as cathode oxygen supply and stack temperature—exhibit slow time-varying characteristics. Relying solely on static robust control may therefore be insufficient to maintain optimal efficiency. To address this limitation, a self-tuning oxygen excess ratio control strategy inspired by “Self-tuning oxygen excess ratio control for PEMFCs under dynamic conditions” is introduced [34]. This adaptive approach employs online parameter identification to update air-path and fuel cell models in real time, enabling the control system to track variations in the oxygen supply and load demand. A second-order active disturbance rejection controller dynamically adjusts system parameters to compensate for unmodeled disturbances and time delays, maintaining the desired oxygen excess ratio under varying sea conditions. Integrating this adaptive mechanism within the robust control framework enhances the system’s resilience and ensures that the fuel cell consistently operates at high efficiency even under fluctuating and uncertain maritime environments.

The state space equation of the system constructed by robust control theory is as follows:

$$x(t)=A(\Delta )x(t)+B(\Delta )u(t)+E(\Delta )w(t)$$

(1)

$x\left(t\right)$ denotes the system state vector, which includes: $x_1$ denotes the output voltage of the fuel cell, reflecting the cell’s charge/discharge state; $x_2$ is the battery’s output current, which directly affects power output and battery health, which directly affecting power output and cell health; $x_3$ and $x_4$ represent the current and voltage of the supercapacitor, influencing its energy storage and instantaneous power output; load power $x_5$, which determines system power allocation; and system temperature $x_6$, indicating thermal stability during operation. To avoid over-simplifying the PEMFC by treating stack voltage/current as fixed “states,” we adopt a physics-informed, semi-empirical PEMFC voltage model. This model employs the MS-TSO method from [35] to estimate the parameters of the PEMFC voltage model. Experiments demonstrate that this parameter estimation method accurately identifies the kinetic (activation), ohmic, and concentration-loss parameters across the entire operating range. These parameters are handled slowly time-varying to reflect aging, humidity, and temperature drift, and the stack voltage is computed as an output of the internal electrochemical states rather than a static state variable. Accordingly, the state vector is augmented to include air-path and cathode oxygen partial-pressure proxies (and a double-layer/charge dynamics proxy), while online identification updates the parameter set in real time. This refinement provides a reliable mathematical basis for control and integrates consistently with the adaptive oxygen-excess-ratio loop described earlier.

These state variables collectively form the system’s state space, and robust control methods ensure stable operation under dynamic disturbances and uncertainties.

$u\left(t\right)$ represents the control input, $w\left(t\right)$ denotes external disturbance, and $\Delta \in D$ signifies the structural uncertainty of the system. $A(\Delta )$, $B(\Delta )$ and $E(\Delta )$ are uncertainty-related system matrices. To ensure bounded system states under arbitrary disturbances, an $H_{\infty }$ robust control method is employed for feedback gain design. Its objective is to minimize the maximum gain from disturbance $w\left(t\right)$ to output $z\left(t\right)$.

An $H_{\infty }$ control method is used for feedback gain design. In the generalized plant, tracking, control, and disturbance weighting functions are introduced, with the goal of minimizing the upper bound γ of the disturbance-to-performance output gain. System uncertainties are modeled as structured, norm-bounded variations. The baseline feedback gain $K_0$ is obtained through solving LMI conditions. On this basis, fuzzy logic and MRAC modules adjust $K_0$ adaptively—operating in parallel for rapid fine-tuning, or hierarchically where the upper layer modifies weights and γ while the lower layer refines the gain—thus balancing robustness and performance.

Satisfying the following conditions:

$${‖T_{zw}(s)‖}_{\infty }<\gamma$$

(2)

Introducing the Lyapunov function $V(x)=x^{T}Px$, and constructing a robust stability criterion that satisfies the following inequality:

$$\left[\begin{array}{l}{A}^T(\Delta )P+PA(\Delta )+C^{T}C PB(\Delta )\\ B^T(\Delta )P-\gamma \mathrm{I}\end{array}\right]<0$$

(3)

In this study, the stability criteria for (1) and (3) are constructed through the Lyapunov function. The system state is constrained by the positive definite matrix and the robust gain limit, and convergence and drift-free can be guaranteed without explicit boundary conditions. where $P>0$ and C is the system output matrix. By solving the linear matrix inequality, the feedback gain matrix $K$ satisfying robust stability requirements can be obtained, thereby constructing the state feedback controller $u\left(t\right)=-Kx\left(t\right)$, $u\left(s\right)$ is associated with $K_0$.

$K_0$ represents the feedback gain matrix in the time-domain formulation, while $K$ denotes the gain matrix employed in the frequency-domain expression, it represents the feedback gain matrix.

Considering the nonlinear and periodic disturbance characteristics of ship loading, a dynamic weighting function $W\left(s\right)$ is introduced into the robust framework to form a generalized plant structure. This ensures the system exhibits excellent attenuation capability against high-frequency disturbances. The final controller implementation structure is as follows:

$$u(s)=-K(x)x(s)=-(K_0\Delta K(x))x(s)$$

(4)

$∆K\left(s\right)$ represents the frequency-dependent adjustment term, which enhances the system’s environmental adaptability and stability margin through collaborative updates driven by fuzzy inference and adaptive laws. The structure of the ship energy management strategy is shown in Figure 2.

4. Cooperative Method for Capacity Configuration and Operational Strategy of Marine Hybrid Energy Systems

4.1. Optimization of Energy Storage Capacity Allocation Based on PSO

In hybrid energy systems, fuel cells and supercapacitors have significant differences in dynamic response characteristics and power density. By rationally allocating the capacity ratio of the two through the particle swarm optimization algorithm, the power response capability and energy density of the system can be effectively balanced. This method reduces operational energy consumption and component loss, thereby optimizing the capacity allocation model of the target system. However, the traditional particle swarm optimization algorithm has problems of premature convergence and limited search efficiency when dealing with complex nonlinear optimization problems with multiple conflicting objectives. To overcome these limitations, the research combines the adaptive inertia weighting operator and the local perturbation operator to dynamically adjust the velocity and position of particles according to the convergence state of the population, enhancing the balance between global exploration and local mining, and effectively preventing the stagnation of the local optimal state. In addition, the adaptive mechanism of this algorithm enables it to better handle the nonlinear coupling between energy capacity and power demand under different sea conditions.

Set the rated power of the fuel cell to $P_{fc}$ and the equivalent capacitance of the supercapacitor bank to $C_{sc}$. The ultimate objective is to minimize the energy usage redundancy rate η under a given load condition $P_l\left(t\right)$, while constraining the system voltage deviation δV and current peak $I_p$, forming the objective function as shown in (5):

$$\underset{{\mathrm{P}}_{\mathrm{fc}},{\mathrm{C}}_{\mathrm{sc}}}{\min }J=\alpha \eta +\beta \delta V_{rms}+\kappa I_p$$

(5)

α, β, and k represent weighting factors. System performance metrics under different capacity combinations are evaluated through simulation models to construct a fitness function.

α, β, and k represent weighting factors. The sum is 1, the base Settings are α = 0.4, β = 0.35, k = 0.25 and the sensitivity verification is conducted within the range The configuration parameters of the basic IPSO algorithm are as follows: population size N = 30 (ring topology, degree 2); The maximum number of iterations is 200. The inertia weight decreases linearly from 0.9 to 0.4. Individual and social learning factors adaptively adjust within the range of [1.5, 2.0] along with group diversity; The speed limit is 20% of the variable range. The probability of local disturbance in the global optimal neighborhood is 0.2. If the improvement is less than 1 × 10⁻⁴ after 30 consecutive iterations, it should be terminated early.

To enhance the optimization capability of traditional PSO in non-convex search spaces, adaptive inertia weights and local perturbation terms are introduced to modify the positions of standard particles. Through multi-round iterations and convergence criteria control, the optimization algorithm ultimately outputs the optimal capacity combination, enabling the system to achieve the synergistic goals of power stability and energy efficiency optimization under a typical navigation task spectrum.

The adaptive inertia weight $\overline{\omega }$ is dynamically adjusted according to the convergence progress of the population to balance exploration and exploitation. It decreases linearly from a higher initial value to a smaller final value as the iteration proceeds, ensuring global search capability at the early stage and local refinement near convergence. The adaptive weighting function is expressed as follows:

$$\overline{\omega }={\overline{\omega }}_{\max }-({\overline{\omega }}_{\max }-{\overline{\omega }}_{\min })\times {\displaystyle \frac{t}{t_{\max }}}$$

(6)

where ${\overline{\omega }}_{max}$ and ${\overline{\omega }}_{min}$ represent the maximum and minimum inertia weights, is the current iteration number, and $t_{max}$ denotes the maximum iteration count. A larger $\overline{\omega }$ encourages global exploration to avoid premature convergence, while a smaller $\overline{\omega }$ enhances local search precision, thereby improving the algorithm’s overall optimization efficiency and stability in non-convex capacity allocation problems.

4.2. Design and Optimization of Dynamic Energy Matching Strategies

Considering the critical impact of fuel cell health conditions in maritime applications on lifespan and safety, this paper introduces an online health estimation method based on polarization loss decomposition in the IPSO-RC framework [36]. Referring to the relevant research ideas, the decreasing of the reactor voltage is decomposed into two parts: activation loss and ohmic loss. A simplified voltage model is adopted for online parameter identification to obtain healthy parameters such as ohmic internal resistance and activation correlation coefficient that change slowly with the operating conditions. Based on this, the heap health status index (SOH_FC) and its rolling trend are constructed to characterize the changes in performance degradation and remaining lifetime.

At the optimization layer, the objective and constraint set of IPSO introduces health information in real time: in addition to energy efficiency and dynamic response indicators, health penalties (such as the increase rate of ohmic internal resistance and the decrease range of SOH_FC) are added, and soft and hard constraints are imposed on the climb rate of fuel cells, peak current density, temperature rise, etc. Meanwhile, SOH_FC is fed into the adaptive oxygen excess ratio (OER) loop and the power distribution strategy of the two-stage filter: When deterioration of health indicators or an increase in ohmic loss is detected, the output of the fuel cell should be appropriately derated, the proportion of supercapacitors in transient conditions should be increased, and the steady-state efficiency-durability trade-off of the OER should be relaxed, so that the control strategy can balance efficiency, robustness, and lifespan in complex sea conditions

The synergistic power supply of fuel cells and supercapacitors require consideration of their heterogeneous dynamic characteristics [37,38]. FCS offer high energy density but exhibit sluggish dynamic response, while supercapacitors deliver outstanding instantaneous power output yet have limited sustained energy supply capability. To fully leverage their respective strengths and dynamically schedule their power contributions in real-time across varying operating conditions, a dynamic matching strategy must be established to achieve energy allocation based on system state and load demand [39,40]. To this end, the instantaneous power allocation functions for fuel cells and supercapacitors are defined as follows.

The instantaneous power distribution model for fuel cells and supercapacitors is shown in (6):

$$P_{fc}(t)=\lambda (t)P_{load}(t),P_{sc}(t)=(1-\lambda (t))P_{load}(t)$$

(7)

Here, $\lambda \left(t\right)$ is the dynamic matching coefficient, which must be jointly determined based on the load change rate ${\displaystyle \frac{d{P}_{load}}{dt}}$, the state of charge of the supercapacitor $SO{C}_{SC}(t)$, and the stack current change rate ${\displaystyle \frac{d{I}_{fc}}{dt}}$. Its value ranges between 0 and 1.

To enhance the system’s adaptive capability under nonlinear disturbance conditions, a nonlinear function approximator is employed to optimize matching coefficients. Through fuzzy logic reasoning, the fuzzy rule set intelligently adjusts weight parameters based on system input variations and current state, enabling rapid response to load dynamics.

In the multi-input regression function, a regression model considers multiple input variables (load change rate, stack current, etc.) to precisely fit the system’s nonlinear behavior and improve matching accuracy. Through the above methods, the optimized matching coefficients undergo self-learning updates via a Bayesian optimization mechanism, dynamically adjusting during operation to ensure stable system performance under various load conditions.

In marine environments, load power typically exhibits periodic fluctuations and random disturbances. To address these challenges, a time-varying filtering strategy is introduced to smooth matching coefficients and suppress high-frequency oscillations. The specific weighted moving average filter can be expressed as follows:

$${\lambda }_{\mathrm{filt}}(t)={\displaystyle \sum _{k=1}^N{w}_k\lambda (t-k\Delta t),{\displaystyle \sum _{k=1}^N{w}_k}=1}$$

(8)

In Equation (7), $w_k$ represents the weighting factor. By appropriately selecting the weight and window width, a good balance can be achieved between response speed and matching smoothness. Optimized instantaneous power distribution models for fuel cells and supercapacitors will be updated based on the improved matching coefficient ${\lambda }_{filt}\left(t\right)$. The new power allocation formula is shown in Equation (8):

$$P_{FC}(t)={\lambda }_{\mathrm{filt}}(t)P_{total}(t), P_{sc}(t)=(1-{\lambda }_{\mathrm{filt}}(t))P_{total}(t)$$

(9)

Through these optimization steps, the system can reduce fuel cell power output and increase the proportion of energy supplied by supercapacitors during sudden load changes, enabling rapid response to load variations. During steady-state navigation phases, the fuel cell maintains primary power supply to ensure system energy efficiency and long-term operational reliability.

4.3. Adaptive Adjustment and Optimization of State Feedback Controller Parameters

The feedback system constructed based on robust control in the preceding section primarily addressed the impact of external disturbances. However, during the actual operation of marine hybrid energy systems, significant time-varying characteristics of system parameters render conventional controllers inadequate for fully accommodating dynamic system changes [41]. Therefore, an adaptive controller parameter adjustment mechanism must be designed to address the significant parameter time-varying nature of marine hybrid energy systems during operation [42].

To cope with the time-varying characteristics of system parameters, the IPSO-RC controller adopts a hierarchical structure: a fast robust state feedback loop and a slow supervision layer of the IPSO self-optimization module. The actual deployment parameters are as follows:

(1) Sampling frequency: RC/ADRC inner loop 100–200 Hz; Energy Management and scheduling layer: 5–20 Hz; The IPSO update frequency is 0.2–1 Hz (or event-triggered) to reduce the computing load. (2) Sensor: DC bus voltage, current; Fuel cell stack voltage and current; Air compressor airflow and pressure (for oxygen excess ratio OER control); Hydrogen temperature and pressure; Humidity; Supercapacitor voltage and current (for SOC estimation); The speed and torque of the propulsion motor, etc. (3) Computing load: It can be met by using industrial-grade PC or ARM SoC (>1 GFLOPS). The IPSO optimization cycle of 30 particles and 200 iterations is completed within a supervision window of 1 to 5 s. (4) Communication delay: Controlled within 10 to 20 ms via CAN/Ethernet bus, the RC closed-loop stability is ensured by using timestamps and the “hold the last sampling value” mechanism. Hydrogen safety measures include: leakage detection (installing hydrogen sensors in compartments and pipelines), forced ventilation, automatic isolation and purging, overpressure/over-temperature interlock, and emergency shutdown mechanism. In the system design, N + 1 redundancy (dual backup for key sensors and dual power paths), watchdog monitoring and backup EMS mode (rule-based droop control) are adopted to maintain “degraded operation” capability when IPSO or parameter identification fails.

In terms of certification, the system should comply with the IMO IGF Code (Safety Requirements for Low Flash Point Fuels), classification society standards, IEC 60092-504, IEEE 1709 and IEC 62443 standards.

The initial controller structure is built upon a robust feedback framework, with its performance highly dependent on the state feedback gain matrix $K$. To adapt to environmental changes and system parameter drift, a joint optimization strategy combining Model Reference Adaptive Control (MRAC) and fuzzy rule inference is employed [43,44].

The real-time controller gain adjustment structure is defined as follows:

$$K(t)=K_0+\Delta K(t)$$

(10)

Among these, $K_0$ represents the initial feedback gain matrix, while $∆K\left(t\right)$ denotes the error-dynamic adaptive adjustment term. The study considers that the system’s operating states exhibit distinct regional characteristics, including typical state intervals such as acceleration, cruising, deceleration, and docking. To adapt to these state changes, a fuzzy logic control method is employed to adjust the controller parameter $K\left(t\right)$ in a segmented manner.

Specifically, within each typical state interval, the system adjusts the increment $∆K\left(t\right)$ of the feedback gain matrix through fuzzy reasoning based on changes in state variables. The fuzzy logic control strategy executes segmented adjustments in accordance with the following steps:

Stage 1: The fuzzy input variables are delineated by the system’s state variables.

Stage 2: A fuzzy rule base is devised to accommodate fluctuations in speed and load. When both velocity and payload increase, the gain is amplified; when both decrease, the gain is attenuated; and when their variations are slight, the gain remains unchanged.

Stage 3: From the fuzzy sets of the input variables, fuzzy inference derives the control increment $K\left(t\right)$, which is subsequently defuzzified into a definitive gain adjustment value.

Therefore, using the navigation state variable $\theta \left(t\right)$ as the fuzzy input, the gain adjustment strategy is inferred from the rule set, as shown in Equation (10):

$$\Delta K(t)={\displaystyle \sum _{i=1}^M{\mathsf{\mu }}_{\mathrm{i}}(\theta (t))\cdot \Delta K_i}$$

(11)

${\mu }_i$ function is a subordinate function, and $∆K_i$ represents the adjustment value under the corresponding rule. Through the fuzzy regulator, the system achieves optimal controller parameter matching across different operating states, enhancing the synergistic effect of energy scheduling and power response.

Therefore, the algorithm flow designed in this study is illustrated in Figure 3:

As shown in Figure 3, the system addresses dynamic disturbances and uncertainties in complex environments by employing robust control to construct a state feedback controller with interference resistance. During the capacity allocation phase, an IPSO is introduced to perform multi-objective optimization of fuel cell power and supercapacitor energy storage capacity, yielding an optimal combination [45,46]. Subsequently, a dynamic matching strategy is designed based on load trend and state of charge to enable real-time intelligent energy allocation between the fuel cell and supercapacitor. An adaptive mechanism continuously adjusts controller parameters, ensuring the system maintains robustness and responsiveness under varying sea conditions and operational scenarios.

Each optimization process in Section 4.2 and Section 4.3 corresponds to a specific target function (TF), representing the performance criterion to be minimized or maximized during capacity allocation and control parameter tuning.

5. Modeling and Analysis

5.1. Experimental Design and Test Environment

The experimental platform of this study is a hardware-in-the-loop (HIL) test bench, which includes a real bidirectional DC/DC converter and an embedded controller, and is connected to the simulated ship load and sea condition disturbance models through MATLAB/Simulink (2022b). The simulation solver adopts the ode45 variable step solver with a step size of 1 ms and a total simulation duration of 6 h. The boundary condition is set to have the propulsion power demand vary between 1 and 5 kW, reflecting typical navigation conditions.

To validate the effectiveness of the designed robust control strategy and capacity configuration optimization method under actual sea conditions, an experimental platform was established. During the experiment, the following operations were performed sequentially:

Stage 1: Energy responsiveness testing.

Stage 2: Capacity configuration comparison experiment.

Stage 3: Dynamic load simulation matching test.

Stage 4: Controller Parameter Perturbation Stability Verification.

Stage 5: Safety and reliability assessment under thermal and electrical stress.

The safety and stability assessment stage is being carried out to ensure the robustness and engineering applicability of the proposed system. This stage focuses on analyzing the peak temperature rise, current ripple factor and voltage fluctuation amplitude under dynamic load conversion. These parameters are associated with the component life models of fuel cell stacks and supercapacitors to assess long-term reliability. Before analysis, all experimental data were preprocessed using outlier removal (3σ filtering) and moving average smoothing to eliminate sensor noise and transient peaks. The data sampling frequency is set to 10 Hz to ensure accurate capture of system dynamics without aliasing.

The entire experiment generated a total of 3762 sample records, with data covering dimensions such as voltage, current, power, state of charge (SOC), system temperature rise, and controller output. Table 1 presents the statistical information and variable fluctuation patterns of the collected samples [47,48].

Two-step IPSO parameter tuning was used: coarse grid search for range estimation and k-fold fine tuning for robustness verification. A ring topology was applied. Default settings: population size = 30, iterations = 200 (early stop after 30 stagnant runs), inertia weight 0.9 → 0.4, adaptive c₁/c₂, velocity limit 20% of range, local perturbation 0.2. Sensitivity: w affects early convergence, c₂ affects final accuracy.

Table 2. Model performance with different learning rates.

Parameter Name	Data Type	Unit	Min Value	Max Value	Mean	Standard Deviation
Fuel Cell Voltage	Continuous	V	700	1000	850	50
Fuel Cell Current	Continuous	A	0.5	95	45	15
Supercapacitor Voltage	Continuous	V	400	460	430	10
Supercapacitor SOC	Percentage	%	20	98	60	15
Load Power	Continuous	W	1000	5000	3000	1500
DC Bus Voltage	Continuous	V	690	1100	900	100
Controller Output Gain	Control Value	-	0	1	0.55	0.2
System Temperature	Continuous	°C	25	60	40	7

presents the statistical information and variable fluctuation patterns of the collected samples.

The detailed parameter data for each component of the experimental verification platform is shown in

Table 3. Key Components and Model Parameters of the Experimental Platform.

Component	Parameter	Symbol	Value/ Range	Unit	Data Source
Fuel Cell Stack (PEMFC)	Rated Power	P_FC	5 kW	W	Manufacturer spec (Ballard Nexa)
	Nominal Voltage	V_FC	850	V	Experimental measurement
	Max Current	I_FC, max	90	A	Experimental
	Efficiency (rated)	$\eta$_FC	52–58%	-	Manufacturer
	Stack Temp Range	T_FC	25–60	°C	Sensor record
Supercapacitor Bank	Equivalent Capacitance	C_SC	450 F	F	Maxwell data sheet
	Rated Voltage	V_SC	450	V	Experimental
	ESR	R_SC	25 m$\Omega$	$\Omega$	Data sheet
Bidirectional DC/DC Converter	Switching Frequency	f_sw	10	kHz	Controller spec
	Efficiency	$\eta$_conv	0.96	-	Bench measurement
Load Model	Dynamic Load Range	P_L	1–5	kW	Simulated propulsion curve
	Frequency Component	f_L	0.1	Hz	Derived from sea-state data
Controller	Sampling Frequency	f_s	100–200	Hz	Design value
	Communication Delay	$\tau$_com	10–20	ms	Bench measurement

.

5.2. Verification of a Marine Hybrid System Based on Robust Control

5.2.1. Comparative Analysis of Capacity Configuration Optimization Experiment Results

To validate the performance of capacity configuration optimization strategies under different algorithms, this study selected four capacity allocation algorithms for comparative analysis: the traditional manual heuristic method, the basic PSO, the hybrid genetic algorithm with particle swarm optimization (HGAPSO), the grey wolf optimization (GWO)-based strategy, and the IPSO proposed in this paper. All algorithms were implemented on a unified experimental platform with a continuous 6 h testing cycle simulating complex dynamic load variations.

To ensure the consistency and fairness of the comparison algorithm, all optimization algorithms adopt the same initialization conditions: population size N = 30, maximum number of iterations 200, and early termination if the improvement rate is less than 1 × 10⁻⁴ after 30 consecutive iterations. Each algorithm uses the same random seed, capacity boundary and fitness evaluation metrics (energy utilization efficiency, response time, overcharge rate). The inertia weight, crossover and mutation operators are set according to the algorithm standard, but the search space constraints are consistent. In the computational overhead evaluation, each algorithm was run 10 times on the same industrial-grade embedded processor (ARM Cortex-A72, 1.8 GHz, 4-core), and the average time consumption of a single optimization cycle and the iterative convergence rate were recorded. The results show that the average convergence times of HGAPSO and GWO are approximately 1.35 times and 1.28 times that of PSO, respectively, but both complete the verification of the real-time feasibility of the algorithm within a 5 s supervision period.

The definitions of each indicator are as follows:

$$EU=100\times {\displaystyle \frac{{\displaystyle {\int }_0^T{P}_{load}(t)dt}}{{\displaystyle {\int }_0^T(P_{FC,in}(t)+P_{SC,dis}(t))dt}}}$$

(12)

$$FCLR=100\times {\displaystyle \frac{{\max }_t{P}_{FC}(t)}{P_{FC,rated}}}$$

(13)

$$OCR=100\times {\displaystyle \frac{time\{SO{C}_{SC}(t)>SO{C}_{\max }\}}{T}}$$

(14)

$$PTR={\max }_t(T_{sys}(t)-T_{amb})$$

(15)

$$ERR=100-EU$$

(16)

$$CRF=\sqrt{{\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T{(I(t)-{\overline{I}}_{\tau }(t))}^{2}dt}}$$

(17)

${\overline{I}}_{\tau }$ is a 1–2 s moving average.

LRT—Load Response Time (ms): It refers to the average of the 10–90% rise time of the bus voltage (or load power error) in each standard step. Unit: ms Window: ±5 s per step, then take the average.

CAC—Controller Adjustment Times (times per hour): The number of times the control gain or parameters are significantly updated per hour. Unit: Times per hour Window: Full experimental cycle.

Experimental results are presented in

Table 4. Performance Comparison of Capacity Optimization Algorithms.

Algorithm	EU (%)	LRT (ms)	FCLR (%)	OCR (%)
Heuristic	82.31	134.5	96.2	4.7
PSO	88.67	105.2	91.4	3.1
HGAPSO	89.23	98.6	89.5	2.6
GWO	90.01	94.3	88.2	2.3
IPSO-RC	94.75	81.7	85.9	1.4

and Figure 4.

The evaluation metrics include: Average Energy Utilization Efficiency (EU), Load Response Time (LRT), Fuel Cell Load Ratio (FCLR), and Overcharge Rate (OCR).

Experimental results are shown in Table 4. It can be observed that the Heuristic method, configured based on manual empirical rules, exhibits the lowest overall performance. Its energy utilization efficiency is only 82.31%, with a response time as high as 134.5 milliseconds, resulting in sluggish response under dynamic loads and significant resource waste. The fuel cell is under considerable pressure, with a peak load ratio as high as 96.2%, posing a significant risk of aging. The supercapacitor overcharge rate is 4.7%, which also adversely affects its lifespan.

The basic PSO algorithm improved EU to 88.67% and reduced LRT to 105.2 ms, demonstrating a significant increase in response speed. However, FCLR remained at 91.4%, indicating that fuel cells still bear excessive load fluctuations.

HGAPSO significantly enhanced performance, achieving EU over 89%, response time shortened to 98.6 ms, and OCR reduced to 2.6%, exhibiting greater stability. Nevertheless, this algorithm remains constrained by population complexity, affecting convergence efficiency.

The GWO strategy outperforms the preceding methods, achieving an EU of 90.01%, reducing LRT to 94.3 ms, and further compressing OCR to 2.3%.

FCLR decreased to 88.2% yet still exhibited redundancy relative to the optimal value. Comprehensive evaluation reveals more balanced performance, though the algorithm may face local convergence issues under complex high-frequency disturbances.

The IPSO-RC algorithm developed in this paper demonstrates outstanding integrated optimization. EU surged to 94.75%, achieving optimal energy utilization, while LRT was controlled below 81.7 ms, leading in response speed. FCLR decreased to 85.9%, effectively alleviating stack load. OCR compression reached 1.4%, significantly protecting the supercapacitors. By integrating adaptive search, disturbance adjustment, and robust control feedback mechanisms, IPSO-RC demonstrates exceptional stability and scheduling precision in dynamic environments, offering substantial practical value. Overall, IPSO-RC strategies hold promises for replacing traditional configurations in maritime operations or complex navigation scenarios, maximizing system energy efficiency and ensuring long-term stable operation.

5.2.2. Comparative Analysis of System Security and Stability Test Results

After completing the basic performance evaluation of capacity configuration optimization, different algorithms were compared from the perspectives of system safety and operational load distribution [49].

The selected metrics are as follows. Peak Temperature Rise (PTR, unit: °C): Represents the maximum temperature increase within the system during continuous high-load operation, reflecting thermal stability. Energy Redundancy Ratio (ERR, unit: %): Indicates the proportion of energy within the system that remains unused per unit time. Current RMS Fluctuation (CRF, unit: A): Measures the smoothness of current response under dynamic loads; Controller Adjustment Count (CAC, unit: times/hour): Reflects the controller’s sensitivity to dynamic disturbances and adaptive frequency.

Experiments employed identical dynamic operating conditions and task profiles, collecting statistical metrics over a complete 6 h load variation cycle as shown in

Table 5. Comparison of Safety and Stability Indicators among Algorithms.

Algorithm	PTR (°C)	ERR (%)	CRF (A)	CAC (Times/Hour)
Heuristic	29.6	17.3	12.8	42
PSO	24.8	11.2	9.4	65
HGAPSO	22.3	9.1	8.1	71
GWO	21.7	8.4	7.8	76
IPSO-RC	18.2	5.6	5.3	84

and Figure 5.

After adopting the Heuristic method, the system’s PTR reaches 29.6 °C (maximum temperature rise), indicating poor thermal management capability. The system’s ERR is 17.3%, meaning nearly one-fifth of the energy is not effectively utilized, resulting in significant energy waste. The CRF reaches 12.8 A, indicating substantial current fluctuations and unstable load response. The CAC is 42 times per hour, indicating a slow response to disturbances. The Heuristic method performs the worst, exhibiting high temperature rise, low energy utilization efficiency, and sluggish system response to dynamic loads.

After applying PSO, the system’s PTR reached 24.8 °C, showing improved temperature rise and demonstrating certain thermal management capabilities compared to the Heuristic method. The ERR is 11.2%, indicating enhanced energy efficiency. The CRF is 9.4 A, reflecting significant current fluctuations though reduced compared to the Heuristic method. CAC reaches 65 cycles per hour, showing slightly improved response speed but still exhibiting lag. The PSO method partially improves energy utilization and thermal management yet remains inadequate in dynamic load response and current fluctuation control.

After applying HGAPSO, the PTR reached 22.3 °C, further reducing temperature rise and enhancing thermal management capabilities. The ERR decreased to 9.1%, lowering the energy redundancy rate while improving efficiency. The CRF achieved 8.1 A with minimal current fluctuation, demonstrating greater stability than the optimized PSO performance. CAC: Reached 71 cycles per hour, demonstrating improved response speed. HGAPSO exhibits enhanced performance in thermal management, energy efficiency, and dynamic response, offering advantages over PSO.

After adopting the GWO method, the PTR reached 21.7 °C, showing a reduced temperature rise but slightly higher than HGAPSO. The ERR was 8.4%, indicating high energy utilization efficiency. The CR was 7.8 A, demonstrating reduced current fluctuations. CAC achieved 76 cycles per hour, reflecting a rapid response. The GWO method performed well in thermal management, energy efficiency, and dynamic response, though regulation accuracy still requires improvement under extreme operating conditions.

The IPSO-RC method was ultimately adopted, achieving a PTR of 18.2 °C, the lowest temperature rise among all approaches, indicating optimal thermal management performance. With an ERR of 5.6%, it delivered the highest energy utilization efficiency, minimizing energy waste. The CRF reached a mere 5.3 A, exhibiting minimal current fluctuation and the most stable load response. The CAC had 84 cycles per hour, delivering the fastest response speed. The IPSO-RC method demonstrated the most outstanding performance across all metrics: lowest temperature rise, optimal energy efficiency, minimal current fluctuation, and fastest response speed. This makes it an effective solution.

Overall, the IPSO-RC method demonstrates optimal thermal management capabilities with the lowest temperature rise, ensuring system stability during prolonged operation. It exhibits the lowest energy redundancy rate, enabling efficient utilization of energy. The IPSO-RC method produces minimal current fluctuations, delivering more stable load response. Furthermore, it achieves the highest controller adjustment frequency, allowing rapid and precise response to load variations and disturbances.

5.2.3. Verification and Evaluation of System Robustness

To evaluate the system’s adaptability to external disturbances and model biases, the RPI was set to dynamically change throughout the iteration process. A lower RPI value indicates greater stability in maintaining system performance under complex disturbances. The results are shown in Figure 6.

The results indicate that the initial RPI value of the Heuristic approach was close to 35. Although it decreased slightly during the first 300 iterations, it subsequently exhibited significant fluctuations with multiple rebounds. The minimum value remained above 27, demonstrating insufficient stability.

The PSO strategy showed marked improvement in the initial phase but experienced a noticeable decline in convergence rate during the later stages. The RPI remained persistently above 25, indicating local stability but overall inadequate adjustment capability.

HGAPSO and GWO demonstrated enhanced robustness. After 400 iterations, HGAPSO’s fluctuations converged within a stable range around 22.5, while GWO achieved a brief plateau below 22.

However, both exhibited rebound phenomena following increased disturbance sensitivity. In contrast, IPSO-RC demonstrated high robustness throughout iterations. Its initial RPI remained below 22, with a notably faster decline rate than other algorithms. After 500 iterations, it entered a steady-state range, maintaining an average around 17.5 with significantly reduced fluctuation amplitude and no upward trend. IPSO-RC demonstrates remarkable dynamic self-tuning capability and stability margin under multi-source disturbances, parameter uncertainties, and task variations. It effectively enhances the system’s robust tracking capability against nonlinear disturbances, exhibiting superior iterative stability and dynamic disturbance rejection.

Overall, in the comparison of RPI, the IPSO-RC algorithm stands out most prominently. with an initial RPI controlled below 22. Its rate of decline during iteration was markedly faster than other algorithms, ultimately stabilizing around 17.5. This significantly improves system stability and robustness under multi-source disturbances and parameter uncertainties.

6. Conclusions

This paper presents an IPSO-RC framework for capacity allocation and energy scheduling of marine fuel cell/supercapacitor hybrid power systems under dynamic and uncertain sea conditions. This method combines the improved particle swarm optimization algorithm with robust control, fuzzy reasoning, and Bayesian update to achieve intelligent real-time coordination among sources. The experimental results show that, compared with the baseline, the performance has been significantly improved: the average energy utilization efficiency reaches 94.75%, the load response time is 81.7 ms, and the overcharge rate of the supercapacitor is 1.4%. During the iterative process, the average robustness index remained stable below 17.5, demonstrating strong anti-interference ability and self-tuning capability.

This strategy alleviates the peak load pressure on fuel cells, reduces overall energy consumption and temperature rise, and enhances the operational stability and reliability of typical navigation profiles. The limitation is that some PEMFC models are still simplified and cannot fully capture long-term degradation. Although the IPSO hyperparameters have been adjusted by the system, they may still be sensitive in some scenarios. The computing budget for embedded deployment requires further analysis. In the future, it will integrate online health estimation of PEMFC based on polarization loss to achieve health-aware scheduling and lifetime constraint optimization; explore deep reinforcement learning and safe MPC to improve generalization in high-dimensional controller spaces; and expand to multiple energy solutions with clear economic and safety constraints (such as photovoltaic/shore power). Research interpretable tuning strategies to reduce parameter sensitivity and simplify certification in maritime applications.