Minimum Hydrogen Consumption Energy Management for Hybrid Fuel Cell Ships Using Improved Weighted Antlion Optimization

Abstract

Energy management in hybrid fuel cell ship systems faces the dual challenges of optimizing hydrogen consumption and ensuring power quality. This study proposes an Improved Weighted Antlion Optimization (IW-ALO) algorithm for multi-objective problems. The method incorporates a dynamic weight adjustment mechanism and an elite-guided strategy, which significantly enhance global search capability and convergence performance. By integrating IW-ALO with the Equivalent Consumption Minimization Strategy (ECMS), an improved weighted ECMS (IW-ECMS) is developed, enabling real-time optimization of the equivalence factor and ensuring efficient energy sharing between the fuel cell and the lithium-ion battery. To validate the proposed strategy, a system simulation model is established in Matlab/Simulink 2017b. Compared with the rule-based state machine control and optimization-based ECMS methods over a representative 300 s ferry operating cycle, the IW-ECMS achieves a hydrogen consumption reduction of 43.4% and 42.6%, respectively, corresponding to a minimum total usage of 166.6 g under the specified load profile, while maintaining real-time system responsiveness. These reductions reflect the scenario tested, characterized by frequent load variations. Nonetheless, the results highlight the potential of IW-ECMS to enhance the economic performance of ship power systems and offer a novel approach for multi-objective cooperative optimization in complex energy systems.

1. Introduction

In recent years, rising awareness of energy challenges and environmental degradation has emerged. The burning of fossil fuels generates emissions that impair ecological stability and amplify global warming, posing substantial risks to both economic systems and natural environments [1]. Renewable energy solutions, particularly hydrogen fuel cells, wind-assisted propulsion, and solar photovoltaic systems, are being increasingly implemented in contemporary marine applications. For hybrid-powered vessels employing multiple energy sources, optimal power allocation between different generation units across varying operational modes represents a critical determinant of overall system efficiency. Consequently, intelligent energy management strategies for marine hybrid propulsion systems have emerged as a focal point in maritime research. The design and implementation of an advanced shipboard Energy Management System (EMS), aiming to minimize hydrogen consumption, extend vessel lifespan, and enhance operational safety and stability, has become the core solution for coordinated power distribution in multi-source propulsion systems [2,3,4].

In current research practice, energy management systems are typically classified into three principal paradigms: rule-based heuristic methods, deterministic optimization algorithms, and intelligent optimization frameworks incorporating machine learning. Rule-based strategies determine the operating mode of hybrid power systems based on driving conditions, battery state-of-charge (SOC) ranges, and system parameters, ensuring operation within an optimal range. Ma et al. [5] proposed an energy management strategy utilizing an adaptive time-constant filter for energy storage systems, which significantly improved both the response speed and balancing accuracy of state-of-charge through an optimized dynamic compensation mechanism. These studies demonstrate that rule-based strategies are simple in logic and easy to implement, but they heavily rely on prior experience and exhibit slow responsiveness to dynamic system changes, making it difficult to ensure optimal control. Intelligent optimization-based strategies, such as reinforcement learning, deep learning, and optimal control, have been employed to construct energy management systems. Yang et al. [6] designed a reinforcement learning method to enable real-time power distribution. They proposed a multi-objective reward function incorporating system safety constraints, economic efficiency metrics, and fuel cell degradation characteristics. A double Q-learning strategy was adopted to update the Q-value function, using the battery charge state as an index to design a real-time reference path for power distribution, thereby improving battery lifespan. Li et al. [7] formulated a comprehensive total cost of ownership (TCO) model that integrates energy consumption, equivalent fuel consumption, and power source degradation metrics. To optimize the energy management system (EMS), they implemented a novel constrained double Q-learning reinforcement learning algorithm featuring an adaptive action space mechanism, optimized energy flow distribution. Although intelligent optimization-based strategies offer superior control performance, they involve high design complexity, require accurate ship modeling, and suffer from a lack of training datasets for ship operating conditions, indicating considerable room for further research.

Compared to the aforementioned control strategies, optimization-based energy management approaches offer not only superior control performance but also maintain real-time responsiveness of the power system. The concept of ECMS was originally proposed by Paganelli [8]. The core methodology employs an energy-equivalence transformation, where real-time electrical consumption is converted to equivalent fuel consumption through dynamic conversion factors, enabling the reduction of a multi-objective global optimization problem to a tractable instantaneous optimization framework. In contrast to conventional global optimization algorithms, ECMS offers advantages such as low computational complexity and no requirement for future driving condition prediction, while still achieving near-optimal control performance. Therefore, it holds significant potential for real-time control in hybrid marine power systems [9,10,11]. The equivalence factor is a key parameter in ECMS, directly affecting the power distribution strategy between the engine and the battery. Variations in the equivalence factor lead to different energy allocation patterns, thereby influencing overall fuel economy and battery charging/discharging behavior in ships [12]. Xie et al. [13] developed an optimization-based power management system (PMS) for shipboard power systems (SPS), where model predictive control (MPC) is applied to dynamically adjust the equivalence factor. The proposed method enables real-time optimal power allocation among hybrid energy sources while minimizing fuel consumption. Kim et al. [14] proposed a neural network-based predictive ECMS method, in which the neural network issues generator commands considering the ship’s nonlinear load and battery SOC. By adjusting the neural network weights, the difference between predicted and actual load is minimized. Experimental results demonstrate that the optimal energy control strategy ensures stable performance of the propulsion motor by adapting to real-time variations in load and battery SOC.

Previous optimization efforts regarding the equivalence factor have primarily focused on reducing fuel consumption. However, in practical applications, frequent charging and discharging of the battery to satisfy load demands can result in excessive cycling, significantly accelerating battery degradation. Therefore, in the design of ship energy management strategies, the original single-objective optimization problem must be extended to a multi-objective one that considers both fuel economy and battery degradation. Among various multi-objective optimization methods, the ALO algorithm has been proven effective due to its simplicity, ease of implementation, robust global exploration ability, and robust performance, making it suitable for handling diverse objective functions and constraints [15,16]. The ALO algorithm, introduced by Mirjalili et al. [17], inspired by the natural predation behavior of antlions. Its core principle lies in achieving global optimization through simulated hunting mechanisms, enhancing search ability via the random walk of ants, maintaining population diversity through roulette wheel selection, and improving convergence through elite strategies. This study proposes an Equivalent Consumption Minimization Strategy (ECMS)-based energy management method that incorporates an Improved Weighted Antlion Optimization (IW-ALO) algorithm. To demonstrate its effectiveness, the hybrid fuel cell ship Alsterwasser is selected as a representative case study. The methodology follows a layered structure: first, the traditional ECMS is applied to allocate ship load power, where the equivalence factor converts fuel cell output into equivalent hydrogen consumption. Building upon this foundation, fitness functions are then established under different load conditions, capturing operational scenarios such as start-stop operations, berthing, and unberthing. Next, the IW-ALO algorithm is introduced to adaptively optimize the power allocation strategy. Through the integration of elite strategies and roulette wheel selection, IW-ALO identifies the optimal equivalence factor for the current operating state. This stepwise process ensures that each methodological component contributes to the next, ultimately achieving reduced hydrogen consumption, improved overall energy management efficiency, and extended battery lifespan.

The key contributions of this work are summarized as follows:

(1)

An energy management strategy based on IW-ALO is proposed to minimize equivalent hydrogen consumption, enhance ship operation economy, and reduce battery degradation.

(2)

The IW-ALO algorithm dynamically determines the optimal equivalence factor according to real-time operating conditions, enabling IW-ECMS to achieve globally optimal power distribution.

(3)

Compared with rule-based SMC and conventional ECMS strategies, IW-ECMS effectively smooths battery power fluctuations and reduces hydrogen consumption by 42.6% and 43.4%, respectively.

The structure of the rest of this paper is as follows: Section 2 presents the hybrid modeling of the fuel cell-powered ship. Section 3 presents the proposed IW-ECMS, which is based on the IW-ALO. Section 4 provides simulation experiments to validate the effectiveness of the proposed method. In this section, the proposed strategy is evaluated against rule-based and conventional optimization-based energy management strategies under typical operating conditions. Finally, Section 5 provides the conclusions of this study.

2. Hybrid Energy Storage System Modeling

Fuel cells offer high energy density but exhibit relatively slow dynamic response, which can adversely affect the ship’s propulsion performance during frequent load fluctuations. In contrast, lithium-ion batteries provide fast response and long cycle life. When integrated with fuel cells, they significantly enhance the overall performance of the ship’s energy system. Currently, hybrid energy storage systems (HESS) are primarily implemented in passive, semi-active, or fully active configurations. Although passive topologies are straightforward and economical, they inherently lack the capability to regulate power flow. Fully active topologies offer precise power allocation control but are associated with higher costs and increased control complexity. The propulsion system in this study adopts a semi-active topology, which combines the advantages of passive and fully active configurations while mitigating their respective drawbacks. This structure achieves a desirable balance between performance and cost, as illustrated in Figure 1.

2.1. Fuel Cell Model

This study adopts the fuel cell stack model proposed by Motapon et al. [18], which integrates both electrical and electrochemical characteristics while neglecting the effects of internal fluid dynamics. This simplification facilitates the construction of the simulation model and has been widely applied in fuel cell system simulations.

A fuel cell is an electrochemical device that directly converts chemical energy into electrical energy through a sequence of coupled physicochemical processes. As illustrated in Figure 2a, a Proton Exchange Membrane Fuel Cell (PEMFC) comprises an anode, a polymer electrolyte membrane, and a cathode. The solid polymer membrane selectively conducts protons (H⁺ ions) while electrically isolating the electrodes. Hydrogen is continuously supplied to the anode, whereas oxygen is delivered to the cathode [19]. The ideal open-circuit voltage of the fuel cell can be determined by the Nernst equation, which depends on the partial pressures, temperature, and concentrations of the reactants and products. The actual cell voltage accounts for the Nernst voltage minus losses due to activation, ohmic resistance, and mass transport limitations. Therefore, the fuel cell stack can be effectively modeled as a controllable voltage source in series with a fixed internal resistance.

The output voltage of the fuel cell can be represented as:

$$V_{\mathrm{fc}}=E_{\mathrm{OC}}-NA\mathrm{l}\mathrm{n}\left({\displaystyle \frac{i_{\mathrm{fc}}}{i_0}}\right)-i_{\mathrm{fc}}{r}_{\mathrm{fcin}}$$

(1)

where $V_{\mathrm{fc}}$ denotes the fuel cell stack voltage (V), $E_{\mathrm{OC}}$ is the open-circuit voltage (V), N is the number of cells in the stack, A is the Tafel slope (V), $i_{\mathrm{fc}}$ is the stack output current (A), $i_0$ is the exchange current (A), and $r_{\mathrm{fcin}}$ represents the internal resistance (Ω). The logarithmic term accounts for activation polarization losses, while the ohmic loss is modeled by the internal resistance term. The open-circuit voltage is given by the following expression:

$$E_{\mathrm{OC}}=K_{\mathrm{C}}{E}_n$$

(2)

where $E_n$ denotes the theoretical cell voltage calculated from the Nernst equation, and $K_{\mathrm{C}}$ is a dimensionless correction factor (≤1) that accounts for deviations between the theoretical and actual open-circuit voltage under rated operating conditions. This factor reflects combined physical and empirical effects, including minor activation losses and cell-to-cell variations, and is determined based on the relationship between internal current characteristics and the Tafel slope. Its value and applicability range are obtained through calibration against experimental data collected under steady-state conditions, ensuring that the corrected voltage model accurately represents the practical performance of the fuel cell stack.

$$E_n=1.229+\left(T-298\right){\displaystyle \frac{-44.43}{2F}}{\displaystyle \frac{\mathrm{R}T}{2\mathrm{F}}}\ln \left(P_{{\mathrm{H}}_2}{P}_{{\mathrm{O}}_2}^{1/2}\right) T\le {100}^{°}\mathrm{C}$$

(3)

where R is the universal gas constant (8.3145 J/mol·K), T is the operating temperature (K), F is the Faraday constant (96,485 A·s/mol), and $P_{{\mathrm{H}}_2}$ and $P_{{\mathrm{O}}_2}$ represent the partial pressures of hydrogen and oxygen, respectively. These parameters are directly related to the thermodynamic potential of the hydrogen–oxygen electrochemical reaction in the fuel cell. R, T and F are used to correct the standard potential, where $P_{{\mathrm{H}}_2}$ and $P_{{\mathrm{O}}_2}$ reflect the influence of reactant gas concentrations on the reversible voltage output.

Figure 2b illustrates the polarization curves obtained from the fuel cell model and the considered experimental data. The similarity between the curves is clearly evident from the figure.

2.2. Battery Model

In the proposed hybrid propulsion system, the fuel cell serves as the primary power source, while the lithium-ion battery functions as an auxiliary energy unit. In this work, the lithium-ion battery model provided by MATLAB/Simulink Simscape Power Systems is employed. The model adopts a modified Shepherd curve-fitting method [20]. In this framework, a voltage polarization component is incorporated into the discharge voltage expression to more accurately reflect the impact of SOC on battery behavior. To enhance numerical stability, the measured battery current is replaced by a filtered current, thereby considering the influence of polarization resistance. The corresponding battery voltage is expressed as follows:

$$V_{\mathrm{b}\mathrm{a}\mathrm{t}}=E_0-K{\displaystyle \frac{Q}{Q-it}}·it-R_b·I+A_b\mathrm{e}\mathrm{x}\mathrm{p}\left(-B·it\right)-K{\displaystyle \frac{Q}{Q-it}}·i*$$

(4)

where $E_0$ denotes the constant voltage of the battery (V), K is the polarization constant (V/Ah), Q corresponds to the rated capacity (Ah), $i*$ represents the filtered current (A), reflecting the effective charge transfer (Ah), $A_b$ indicates the exponential voltage term (V), B refers to the reciprocal of the exponential time constant (Ah⁻¹), and $R_b$ characterizes the internal resistance (Ω).

2.3. DC/DC Converter Model

The DC/DC converter is a key power conversion component in hybrid energy systems, enabling bidirectional energy flow and interconnecting components operating at different voltage levels. Isolated DC/DC converters utilize transformer-based structures to achieve electrical isolation, offering enhanced safety and fault protection, which makes them suitable for applications with stringent safety requirements. However, their relatively large size, lower efficiency, and structural complexity limit their applicability in systems where space and weight are constrained, such as ships and unmanned aerial vehicles (UAVs). In contrast, non-isolated DC/DC converters feature a compact structure, simple design, and high conversion efficiency, making them especially well-suited for lightweight and space-constrained hybrid power systems [21].

Based on system integration and efficiency requirements, a non-isolated DC/DC converter is selected in this study as the energy interface unit. Common modeling approaches for DC/DC converters include the switching model and the average model. The switching model accurately captures the dynamic behavior of power electronic devices and is suitable for high-fidelity simulations, but it entails a high computational burden and long simulation time. In contrast, the average model replaces switching elements with controlled sources, preserving the system’s dynamic characteristics while significantly improving simulation efficiency. Therefore, the average model is adopted in this work. Figure 3 illustrates the average model of the non-isolated DC/DC converters used in this study. On the fuel cell (FC) side, a unidirectional boost converter is implemented to boost the low output voltage to match the DC bus voltage. On the lithium-ion battery side, a bidirectional DC/DC converter is deployed, combining both boost and buck functions to enable bidirectional power flow between the battery and the DC bus.

As shown in Figure 3, V_H and V_L represent the voltages on the DC bus side and the power source side, respectively, D is the duty cycle, and η denotes the converter efficiency.

3. Energy Management Strategy

As the central control mechanism of the fuel cell–lithium battery hybrid energy storage system, the energy management strategy ensures coordinated power sharing between the fuel cell and the lithium-ion battery through real-time optimization of power allocation. This approach minimizes hydrogen consumption while keeping the battery SOC within the designated optimal range. The subsequent sections provide a detailed description of the proposed strategy.

3.1. Equivalent Consumption Minimization Strategy

The Equivalent Consumption Minimization Strategy is a classical instantaneous optimization control method widely applied in the transportation sector and has been demonstrated to effectively reduce energy consumption [22,23]. The objective of ECMS is to minimize fuel consumption by reducing the fuel used by the fuel cell and the equivalent fuel required to maintain the battery’s SOC. The equivalent hydrogen consumption of the battery is governed by an equivalent factor θ, which is determined by the SOC and is expressed as follows:

$$\theta =1-2\mu {\displaystyle \frac{\left(\mathrm{SOC}-0.5\left({\mathrm{SOC}}_{\mathrm{H}}+{\mathrm{SOC}}_{\mathrm{L}}\right)\right)}{{\mathrm{SOC}}_{\mathrm{H}}-{\mathrm{SOC}}_{\mathrm{L}}}}$$

(5)

where $\mu$ represents the SOC penalty coefficient, typically set to 0.6 to regulate SOC balance during operation, as this value offers an effective trade-off between minimizing hydrogen consumption and maintaining SOC stability [8]; SOC_H and SOC_L denote the upper and lower limits of the SOC, respectively.

In the ECMS framework, the load demand and SOC serve as inputs, while the outputs are the reference powers of the fuel cell and the battery. The associated optimization problem is expressed as minimizing the objective function f(x), given by:

$${f\left(x\right)=(P}_{\mathrm{fc}}+{\theta ·P}_{bat})+△T$$

(6)

The constraints of the optimization problem are defined as follows:

$$\left\{\begin{array}{c}{P}_{\mathrm{f}\mathrm{c}\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{\mathrm{f}\mathrm{c}}\le P_{\mathrm{f}\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}\\ P_{\mathrm{bat} \min }\le P_{\mathrm{bat}}\le P_{\mathrm{bat} \mathrm{m}\mathrm{a}\mathrm{x}}\\ P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}=P_{\mathrm{f}\mathrm{c}}+P_{\mathrm{bat}}\\ 0\le \alpha \le 2\end{array}\right.$$

(7)

where $P_{\mathrm{bat}}$ denotes the power of the lithium-ion battery, and $P_{\mathrm{f}\mathrm{c}}$ represents the power of the fuel cell, $△T$ is the time interval, and $\alpha$ is an equivalent factor in ECMS that balances and optimizes the power distribution between the fuel cell and the lithium-ion battery [8].

Based on Equation (6), the optimal fuel cell power is computed and limited within the minimum and maximum bounds to prevent operation in low-efficiency regions. The battery power is then obtained by subtracting it from the total load demand, after which the current for each system is calculated by dividing the respective power by its voltage. This allows the battery’s state of charge to be determined.

3.2. Improved Weighted Antlion Optimization Algorithm

In 2015, Mirjalili et al. [17] proposed the ALO algorithm, inspired by the natural predatory behavior of antlions hunting ants. The core concept of ALO is to achieve global optimization through the antlion’s hunting mechanism, enhance search capability via the ants’ random walks, maintain population diversity using roulette wheel selection, and improve optimization performance with an elitism strategy. To address the traditional ALO algorithm’s limitations in adapting to dynamic system states and its inclination to settle in local optima, this study proposes an IW-ALO algorithm. IW-ALO further enhances optimization performance by incorporating a dynamic equivalent factor and multi-objective constraints. The workflow of the IW-ALO algorithm is as follows:

(1)

Population Initialization

The positions of ants and antlions are initialized within the parameter space, with the initialization computed as follow

$${\theta }_{i,j}^t={\theta }_L^i+r\times ({\theta }_U^i-{\theta }_L^i)i=1,2,\dots ,D;j=1,2,\dots ,N$$

(8)

where ${\theta }_{i,j}^t$ represents the i dimension parameter of the j individual at iteration t, and ${\theta }_L^i$, ${\theta }_U^i$ represent the lower and upper limits of the i parameter dimension, respectively; r is a random variable uniformly distributed over the range [0, 1] D is the dimension of the parameter space, and N denotes the size of the population. This formulation follows the boundary-constrained initialization principle of the original ALO [17], while adopting a generalized representation to enhance clarity and ensure compatibility with the multi-objective optimization framework employed in this study.

After initialization, the fitness of each individual is evaluated based on system performance, using a multi-objective weighted function that accounts for hydrogen consumption and battery degradation:

$$fitness\left(\theta \right)={\omega }_1\cdot f_{\mathrm{H}2}\left(\theta \right)+{\omega }_2\cdot f_{\mathrm{S}\mathrm{O}\mathrm{H}}\left(\theta \right)$$

(9)

where $f_{\mathrm{H}2}\left(\theta \right)$ denotes the hydrogen consumption, $f_{\mathrm{S}\mathrm{O}\mathrm{H}}\left(\theta \right)$ represents the degree of battery health degradation. The weighting coefficients ${\omega }_1$, ${\omega }_2$ are adjusted iteratively based on the relative significance of hydrogen consumption and battery health, while always satisfying the constraint ${\omega }_1$ + ${\omega }_2$ = 1 to ensure consistent scaling. This adaptive adjustment enables the optimization process to balance hydrogen economy with battery degradation, thereby enhancing the overall performance of the energy management strategy.

The population individuals are ranked in descending order of fitness, and the current best fitness value $f_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ along with its corresponding parameters ${\theta }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ are recorded.

Figure 4 illustrates the Pareto front obtained by the weighted-sum method under different weight distributions. The color bar represents the fuel economy weight ${\omega }_1$, while the annotated points on the Pareto curve explicitly indicate the corresponding values of ${\omega }_1$. It can be observed that when ${\omega }_1$ approaches 0, the optimization tends to minimize the SOC deviation, resulting in negligible fluctuation of battery SOC but relatively higher hydrogen consumption. Conversely, as ${\omega }_1$ increases, the solutions shift toward reduced hydrogen consumption at the expense of larger SOC deviation. This clearly demonstrates the trade-off relationship between hydrogen consumption and SOC regulation, and confirms the capability of the proposed strategy to generate a continuous set of optimal solutions according to different operational preferences.

(2)

Population Iterative Update

In each iteration, the position of each ant is updated based on the roulette wheel selection mechanism and the guidance of the elite antlion. The position update is computed as follows:

$${\theta }_j^{t+1}=w(t)\cdot R_E^t(l)+(1-w(t))\cdot R_A^t(l)+\eta \cdot \mathcal{N}(\mathrm{0,1})$$

(10)

where ${\theta }_j^{t+1}$ denotes the position of the j ant at iteration t + 1, $R_E^t(l)$ and $R_A^t\left(l\right)$ represent the positions obtained through the elite strategy and roulette-wheel-based random walk, respectively, and $w\left(t\right)$ is a dynamic weighting factor, defined as:

$$w(t)=w_{max}-(w_{max}-w_{min})\times {\displaystyle \frac{t}{T_{max}}}$$

(11)

where $\eta \cdot \mathcal{N}\left(\mathrm{0,1}\right)$ are Gaussian perturbation terms introduced to prevent premature convergence, t is the current iteration index, and $T_{max}$ denotes the maximum number of iterations. In addition, to enhance the algorithm’s ability to escape local optima and improve its adaptability to dynamic system states, a dynamic equivalent factor $\theta \left(t\right)$ is introduced. The fitness function is dynamically adjusted based on load variations and battery SOC fluctuations. After each update, boundary constraints are applied to the population individuals to ensure all parameters remain within allowable limits. The fitness of the updated population is then evaluated. If the elite antlion’s fitness is superior to that of the corresponding ant, it is considered a capture, and the elite antlion’s position replaces that of the ant. Subsequently, the population is re-ranked, and the optimal solution is updated accordingly.

(3)

Conclude the iterations and return the optimal solution

The process iterates repeatedly until the conditions for termination are met, either reaching the maximum iteration count or satisfying the convergence criterion based on optimal fitness variation, yielding the optimal parameters ${\theta }_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$ and the corresponding optimal fitness value $f_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$.

Figure 5 illustrates the optimization performance of IW-ALO compared with standard ALO, PSO, and GA in terms of Best Score evolution over time. As shown, IW-ALO converges significantly faster and achieves a substantially lower Best Score (8.08 × 10⁻¹³) than standard ALO (3.12 × 10⁻⁷), GA (5.36 × 10²), and PSO (3.47 × 10⁴). This demonstrates that IW-ALO not only provides superior solution quality but also exhibits high computational efficiency, highlighting its practical applicability for real-time energy management in hybrid fuel cell ship systems.

The Pareto front characterizes the set of optimal trade-off solutions in multi-objective optimization problems [24]. In this study, battery degradation is modeled using a cumulative energy throughput-based approach, which assumes that degradation is proportional to the total energy cycled through the battery. Although this simplified model does not explicitly consider temperature effects, depth-of-discharge (DoD), or detailed cycle aging, previous studies have demonstrated its reliability for comparative evaluation of energy management strategies [25,26]. Figure 6 illustrates a comparison of Pareto fronts obtained from mainstream multi-objective optimization algorithms based on normalized battery degradation and hydrogen consumption as dual objectives, where both battery degradation and hydrogen consumption are normalized and thus dimensionless to ensure comparability as dual objectives. As shown in Figure 6, the proposed IW-ALO algorithm significantly outperforms mainstream multi-objective optimization algorithms in terms of convergence speed and solution diversity. Although SMPSO performs better in battery degradation, its solution set exhibits considerable dispersion in the hydrogen consumption metric. Furthermore, the Pareto fronts of NSGA-III, MOEA/D, and SPEA2 algorithms deviate overall from the reference front. Particularly, in the dual optimal region of low battery degradation and low hydrogen consumption, the solution sets are sparse, making it difficult to achieve ideal global trade-off solutions. In contrast, the Pareto front generated by IW-ALO exhibits a more compact distribution and closely approximates the reference front, demonstrating a favorable balance between battery degradation and hydrogen consumption. This highlights the algorithm’s effectiveness in handling multi-objective trade-offs. Notably, in the low battery degradation region, IW-ALO effectively suppresses battery degradation while maintaining low hydrogen consumption, which presents a distinct advantage in practical applications that require balancing battery lifespan and energy efficiency.

Figure 7 presents a comparison of five representative performance metrics for multi-objective optimization among NSGA-III, MOEA/D, SPEA2, SMPSO, and the proposed IW-ALO algorithm. The metrics include Generational Distance (GD), Inverted Generational Distance (IGD), Mean Spread (MS), Spacing (SP), and Hypervolume (HV), which are commonly used to evaluate solution set convergence, distribution breadth, uniformity, and overall quality [27]. As shown in Figure 7, The GD and IGD results indicate that IW-ALO achieves the closest approximation to the reference Pareto front, demonstrating superior convergence performance. This is attributed to the dynamic weight adjustment and elite-guided search mechanisms, which enable the algorithm to explore the solution space comprehensively while avoiding local optima. The MS and SP metrics show that IW-ALO maintains high diversity and uniform distribution of the solution set, effectively preventing clustering or uneven dispersion, which ensures a balanced trade-off between hydrogen consumption and battery degradation. The HV metric, representing overall solution quality and coverage, is highest for IW-ALO, indicating that the obtained solution set achieves both high convergence and diversity across all objectives. Overall, these results confirm that IW-ALO not only outperforms other mainstream algorithms in convergence, distribution uniformity, and multi-objective trade-off capability but also provides a practical and reliable approach for energy management in hybrid fuel cell ship systems, enabling optimized power allocation that can enhance system stability and operational economy.

3.3. The Proposed Energy Management Strategy Based on IW-ECMS

Under conventional ECMS control, the equivalence factor is typically fixed. The selection of different equivalence factors θ can significantly affect the control performance of ECMS. Therefore, selecting an appropriate θ under varying operating conditions and driving modes is critical to achieving optimal ECMS control. An effective θ enables the derivation of optimal solutions under different constraint conditions.

To address the challenge of determining an effective θ, this study introduces an improved weighted antlion-based multi-objective global optimization method and designs an Equivalent Consumption Minimization Strategy based on the Improved Weighted Antlion Optimization. IW-ALO is employed to globally search for optimal solutions and determine appropriate equivalence factors based on specific ship operating conditions, thereby achieving optimal ECMS control performance.

In the proposed strategy, the overall fuel consumption C of the hybrid power system is defined by summing the actual hydrogen usage of the fuel cell C_fc and the lithium battery’s equivalent hydrogen usage C_bat. To minimize the total equivalent hydrogen consumption, both the fuel cell and battery hydrogen consumptions must be calculated for the system under investigation. According to [12], the lithium battery’s equivalent hydrogen consumption is determined as:

$$C_{\mathrm{bat}}=\left\{\begin{array}{ll}{\displaystyle \frac{P_{\mathrm{bat}}}{{\eta }_{\mathrm{dis}}{\stackrel{-}{\eta }}_{\mathrm{chg}}}}{\displaystyle \frac{C_{\mathrm{fc},\mathrm{avg}}}{P_{\mathrm{fc},\mathrm{avg}}}}({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{P}_{\mathrm{bat}}),\hfill & P_{\mathrm{bat}}\ge 0\hfill \\ \hfill & \hfill \\ P_{\mathrm{bat}}{\eta }_{\mathrm{chg}}{\stackrel{-}{\eta }}_{\mathrm{dis}}{\displaystyle \frac{C_{\mathrm{fc},\mathrm{avg}}}{P_{\mathrm{fc},\mathrm{avg}}}}({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{P}_{\mathrm{bat}}),\hfill & P_{\mathrm{bat}}<0\hfill \end{array}\right.$$

(12)

where $P_{\mathrm{bat}}$ indicates the output power of the lithium-ion battery pack; ${\eta }_{\mathrm{dis}}$ and ${\eta }_{\mathrm{chg}}$ correspond to the discharge and charge efficiencies of the battery, respectively, which in this study are taken as 0.95 and 0.90, consistent with values reported in [12]; ${\stackrel{-}{\eta }}_{\mathrm{chg}}$ and ${\stackrel{-}{\eta }}_{\mathrm{dis}}$ refer to the average charging and discharging efficiencies of the battery. The conditions $(P_{\mathrm{bat}}\ge 0)$ and ${(P}_{\mathrm{bat}}<0)$ correspond to the battery charging and discharging modes, respectively. The term $({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{P}_{\mathrm{bat}})$ reflects the proportional contribution of battery power to the system. Average efficiencies are calculated based on instantaneous power flows. $C_{\mathrm{fc},\mathrm{avg}}$ is the average hydrogen consumption rate of the fuel cell; and $P_{\mathrm{fc},\mathrm{avg}}$ is the average output power of the fuel cell.

Considering the time-varying and nonlinear characteristics of the fuel cell, the hydrogen consumption rate can be determined from the instantaneous output power and voltage of the fuel cell, as well as the overall system efficiency. The relationship is expressed as:

$$C_{\mathrm{f}\mathrm{c}}={\displaystyle \frac{5.25P_{\mathrm{f}\mathrm{c}}}{5V_{\mathrm{f}\mathrm{c}}F{E}_{\mathrm{f}\mathrm{c}}}}={\displaystyle \frac{{1.05P}_{\mathrm{f}\mathrm{c}}}{V_{\mathrm{f}\mathrm{c}}{E}_{\mathrm{f}\mathrm{c}}}}$$

(13)

where $C_{\mathrm{f}\mathrm{c}}$ represents the hydrogen consumption rate of the fuel cell (g/s); $P_{\mathrm{f}\mathrm{c}}$ is the instantaneous output power (W); $V_{\mathrm{f}\mathrm{c}}$ denotes the instantaneous voltage of a single fuel cell under current operating conditions. F is the Faraday constant; and $E_{\mathrm{f}\mathrm{c}}$ represents the overall electrical efficiency of the fuel cell system.

The proposed IW-ECMS improves the energy management objective function by introducing an equivalence factor θ, which enables the alignment of fuel cell consumption efficiency with battery consumption efficiency [28]. This formulation aims to minimize the total cost function $J_{\mathrm{m}\mathrm{i}\mathrm{n}}$, and the corresponding objective function is defined as follows:

$$\left\{\begin{array}{c}{J}_{\mathrm{m}\mathrm{i}\mathrm{n}}=min\left(C_{\mathrm{f}\mathrm{c}}+C_{\mathrm{b}\mathrm{a}\mathrm{t}}\right)=min\left(C_{fc}+\theta {\displaystyle \frac{P_{\mathrm{b}\mathrm{a}\mathrm{t}}}{Q_{\mathrm{L}\mathrm{H}\mathrm{V}}}}\right)\\ (P_{\mathrm{f}\mathrm{c}},P_{\mathrm{b}\mathrm{a}\mathrm{t}})=\underset{\left(\right(P_{\mathrm{f}\mathrm{c}},P_{\mathrm{b}\mathrm{a}\mathrm{t}}\left)\right)}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\{C_{\mathrm{f}\mathrm{c}}+C_{\mathrm{b}\mathrm{a}\mathrm{t}}\}\end{array}\right.$$

(14)

where J denotes the total hydrogen consumption of the ship; $C_{\mathrm{f}\mathrm{c}}$ is the hydrogen consumption of the fuel cell; $C_{\mathrm{b}\mathrm{a}\mathrm{t}}$ is the equivalent hydrogen consumption of the lithium battery system; $Q_{\mathrm{L}\mathrm{H}\mathrm{V}}$ is the lower heating value of hydrogen (j/g); θ is the equivalence factor obtained through IW-ALO optimization according to the ship’s operating conditions; and $P_{\mathrm{f}\mathrm{c}}$ and $P_{\mathrm{b}\mathrm{a}\mathrm{t}}$ represent the optimal output powers of the fuel cell and lithium battery, respectively.

In summary, the overall framework of the proposed IW-ECMS is illustrated in Figure 8. The powertrain model receives the optimal battery power $P_{\mathrm{b}\mathrm{a}\mathrm{t}\_\mathrm{o}\mathrm{u}\mathrm{t}}$ and fuel cell power $P_{{\mathrm{f}\mathrm{c}}_{\_}\mathrm{o}\mathrm{u}\mathrm{t}}$ generated by the objective function to perform energy management. Real-time feedback on the ship’s load power $P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$ and battery SOC is used to update system operation status. Under system constraints, the initial power outputs of the battery $P_{\mathrm{b}\mathrm{a}\mathrm{t}}$ and fuel cell $P_{\mathrm{f}\mathrm{c}}$ are converted into the corresponding hydrogen consumption rates $C_{\mathrm{b}\mathrm{a}\mathrm{t}}$ and $C_{\mathrm{f}\mathrm{c}}$ via the equivalent hydrogen consumption model. These values are then input into the IW-ECMS objective function. The equivalence factors are optimized using the IW-ALO to determine the optimal equivalence factors corresponding to the current load $P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$, with the optimal solution indicated by an asterisk (*) in Figure 8. Finally, the objective function outputs the optimal power allocation between the fuel cell and the battery.

4. Simulation and Results Analysis

4.1. Simulation Setup

To evaluate the performance and efficacy of the proposed energy management strategy, a simulation model of a fuel cell-powered ship propulsion system was established in MATLAB/Simulink 2017b. As illustrated in Figure 9, the system primarily comprises four core components: the fuel cell, the lithium-ion battery, the energy management system (EMS), and the load. The load module is based on operational data from the hybrid fuel cell vessel Alsterwasser, which is regarded as representative of typical marine applications, as shown in Figure 10. The vessel operates under two conditions: during 90–200 s it undergoes maneuvering navigation, where the load demand varies significantly between 0 and 115 kW, while in the remaining period it performs constant-speed cruising with a nearly steady load of approximately 42 kW. This load profile reflects complex operating scenarios relevant to small-scale hybrid vessels. The corresponding power distribution obtained with the proposed IW-ECMS is presented in Figure 11. In the maneuvering stage, the sharp load fluctuations cause a dynamic interaction between the fuel cell and the battery, with the fuel cell output adaptively adjusted and the battery compensating for transient demands. In contrast, during constant-speed cruising, the fuel cell predominantly supplies the steady load, while the battery provides minor support to ensure system stability. These results demonstrate that the proposed strategy enables effective power allocation under both dynamic and steady operating conditions.

The proposed IW-ECMS energy management strategy was implemented in the MATLAB/Simulink SimPowerSystems environment (version 2023b). The hybrid system is modeled as a modular configuration consisting of proton exchange membrane fuel cell (PEMFC) stacks, each rated at 10 kW, together with a 48 V, 40 Ah lithium-ion battery pack. In practice, multiple fuel cell stacks of this type are combined in parallel to achieve the required power level for ship propulsion, while the battery provides transient support and load balancing. The PEMFC is connected through a unidirectional boost converter, while the battery pack is interfaced with two bidirectional DC/DC converters. Key parameters of both components are listed in

Table 1. Parameters of the Fuel cell and Battery.

	Parameter	Value
Fuel cell parameter	Output Voltage Range	52.5–52.46 V
	Rated Operating Point	(41.15 V, 250 A)
	Maximum Operating Point	(39.2 V, 320 A)
	Rated Fuel Cell Power	50%
	Operating Temperature	45 °C
Battery parameter	Output Voltage	48 V
	Rated Capacity	40 Ah
	Full Charge Voltage	55.88 V
	Rated Discharge Current Internal Resistance Battery Voltage Response Time	17.4 A 0.012 Ω 30 s

, and their characteristics are shown in Figure 10. The PEMFC polarization curve (Figure 12a) illustrates voltage variations under different current densities, indicating the activation, ohmic, and concentration polarization regions. This verifies the accuracy of the PEMFC model and defines its operational limits. The battery discharge curves (Figure 12b) present voltage responses under different rates, reflecting capacity retention and dynamic behavior, which determine its state-of-health (SOH) and power distribution constraints. These results support the model validation and the subsequent optimization of the IW-ECMS.

4.2. Comparative Simulation Analysis

To highlight the superiority of the proposed IW-ECMS, this study conducts a comparative analysis with two benchmark strategies: SMC and ECMS. These methods were selected as representative rule-based and optimization-based control strategies, respectively, and are widely used in existing literature for experimental validation. As shown in Figure 13, the SMC strategy determines the power split between the fuel cell and the battery through predefined operating modes and switching rules, which ensures system stability and protects the battery under dynamic load conditions. In contrast, the ECMS, illustrated in Figure 14, formulates the power distribution as an instantaneous optimization problem by introducing an equivalent hydrogen consumption factor. This factor balances fuel cell and battery usage in each time step, aiming to minimize hydrogen consumption while constraining battery degradation. In this study, the ECMS baseline was implemented strictly following standard formulations reported in the literature to ensure a fair and consistent comparison with the proposed IW-ECMS.

Figure 15a,b illustrate the time-domain dynamic responses of fuel cell output power under three energy management strategies. Frequent fluctuations in fuel cell output power are a critical factor leading to performance degradation and shortened lifespan [29]. Suppressing power fluctuations is crucial for enhancing the stability and reliability of the fuel cell system. Comparative results indicate that the system based on the IW-ECMS demonstrates superior dynamic performance in regulating fuel cell output power. Compared with traditional SMC and ECMS strategies, the fuel cell output power curve under the IW-ECMS exhibits a more stable trend with significantly reduced peak power fluctuations. During the period of frequent power fluctuations from 80 s to 115 s, the maximum fluctuation amplitude of the fuel cell output power under the IW-ECMS decreases by approximately 33.11% compared to the ECMS and by about 20.26% relative to the SMC strategy. Moreover, the local magnified plots more intuitively demonstrate that IW-ECMS effectively mitigates short-term large amplitude charging and discharging fluctuations, smoothing the power output. This result fully validates the advantage of IW-ECMS in suppressing fuel cell power fluctuations and reducing dynamic load shocks, thereby contributing to delaying fuel cell performance degradation and enhancing the system’s long-term stability and economic efficiency.

Figure 16 illustrates the dynamic response characteristics of battery output current under three energy management strategies. The dynamic response of battery current directly reflects the system’s adaptability and control performance in meeting varying load power demands [30]. An optimal current response not only satisfies the instantaneous power requirements of the load but also effectively suppresses dynamic shocks within the system, thereby mitigating battery degradation. As shown in Figure 16, the proposed IW-ECMS achieves a more agile dynamic response, enabling rapid tracking of load fluctuations and reducing excessive response delays. Meanwhile, the current fluctuation range under the IW-ECMS is significantly narrowed, resulting in a smoother output process. This ensures prompt response to sudden load changes while effectively mitigating the performance impact caused by frequent and large current fluctuations. In contrast, the traditional ECMS and SMC strategies exhibit certain degrees of response lag or excessive oscillations in battery current, which compromises the balance between response speed and output stability. These findings further confirm the superior comprehensive capability of the IW-ECMS in improving the dynamic behavior performance and stability during propulsion system operation.

Figure 17a compares the equivalent hydrogen consumption and final SOC under three control strategies. The total hydrogen consumption during system operation was 290.2 g, 294.2 g, and 166.6 g, respectively. Notably, the IW-ECMS achieved the lowest equivalent hydrogen consumption, reducing it by 42.6% and 43.4% compared with SMC and ECMS strategies, respectively. It should be emphasized that these reductions are specific to the tested scenario, which features frequent and sharp load variations that amplify the limitations of conventional ECMS with static equivalence factors. Managing the battery SOC helps maintain sufficient charge to meet propulsion demands [31]. Figure 17b illustrates the battery SOC evolution under the different control strategies. It is evident that after applying the three strategies to typical operating conditions, the terminal SOC levels of the system battery were 43.8%, 52.9%, and 55.8%, respectively. The proposed IW-ECMS effectively maximizes battery energy utilization within SOC boundaries, restoring it close to the initial state, which is conducive to system cyclic control. Moreover, the IW-ECMS exhibited a narrower SOC fluctuation range, helping to prevent over-discharge or over-charge of the battery, thereby reducing performance degradation and prolonging its lifespan.

5. Conclusions

This paper proposes an ECMS based on the IW-ALO algorithm, which achieves a balance between real-time performance and global optimization in energy management. Firstly, the enhanced IW-ALO algorithm effectively reduces hydrogen consumption while suppressing battery degradation. Secondly, the proposed IW-ECMS significantly mitigates fuel cell power fluctuations and lowers the equivalent hydrogen consumption. Finally, comparative simulations on the MATLAB/Simulink platform among IW-ECMS, SMC, and conventional ECMS strategies demonstrate that the proposed IW-ECMS outperforms both SMC and traditional ECMS in maintaining power stability, improving dynamic response, and reducing energy consumption. Moreover, IW-ECMS demonstrates strong anti-interference capability and robustness. Nevertheless, the present study remains confined to simulation-based validation. Although the use of real-world load profiles from the hydrogen-powered vessel Alsterwasser ensures practical relevance, the reported benefits are scenario-dependent and may be influenced by parameter uncertainties arising from component aging and environmental fluctuations, as well as by computational demands in large-scale or highly dynamic conditions. Future work will extend validation to concatenated multi-cycle tests and random load variations to confirm system robustness over longer horizons, complemented by experimental verification on a scaled hybrid powertrain test bench.