Multi-Temporal Energy Management Strategy for Fuel Cell Ships Considering Power Source Lifespan Decay Synergy

Abstract

With increasingly stringent maritime environmental regulations, hybrid fuel cell ships have garnered significant attention due to their advantages in low emissions and high efficiency. However, challenges related to the coordinated control of multi-energy systems and fuel cell degradation remain significant barriers to their practical implementation. This paper proposes an innovative multi-timescale energy management strategy that focuses on optimizing the lifespan decay synergy of fuel cells and lithium batteries. The study designs an attention-based CNN-LSTM hybrid model for power prediction and constructs a two-stage optimization framework: The first stage employs Model Predictive Control (MPC) for long-term power planning to optimize equivalent hydrogen consumption, while the second stage focuses on real-time power allocation considering both power source degradation and system operational efficiency. The simulation results demonstrate that compared to single-layer MPC and the Equivalent Consumption Minimization Strategy (ECMS), the proposed method exhibits significant advantages in reducing single-voyage costs, minimizing differences in power source degradation rates, and alleviating power source stress. The overall performance of this strategy approaches the global optimal solution obtained through Dynamic Programming, comprehensively validating its superiority in simultaneously optimizing system economics and durability.

1. Introduction

As a primary mode of maritime transportation, ships serve as a foundational support for human exploration of ocean resources. However, traditional propulsion systems relying on fossil fuels pose significant environmental pollution challenges [1]. The International Maritime Organization (IMO) has emphasized a future focus on the efficient utilization of clean energy in shipping. In response, shipping enterprises worldwide are actively advancing the integration of renewable energy technologies such as solar power, wind energy, and fuel cells to reduce carbon emissions and enhance energy efficiency [2,3]. Despite their advantages, such as high power density and efficiency [4], fuel cells face challenges including slow startup times and limited operational lifespan. The development of energy storage technologies has provided solutions for power dispatch in shipboard electrical systems [5]. Hybrid energy storage systems, combining energy-dense and power-dense storage components, significantly improve the performance and reliability of renewable energy-based ship power systems [6].

Advanced energy management strategies are essential for multi-energy hybrid power systems to achieve optimal power and torque distribution among various power sources while coordinating electric propulsion systems and ensuring the efficient synergistic operation of multiple power sources. According to the source and implementation method of actual energy allocation control, control strategies can be classified into three categories: rule-based control strategies oriented to engineering applications, optimal control strategies focusing on optimization modeling and solving, and learning-based control strategies driven by data [7]. Rule-based control strategies primarily rely on predetermined rules based on engineering experience, characterized by their simple structure and ease of implementation. The rule-based control strategy in reference [8], which utilizes batteries and shuts down engines at the minimum allowable power based on specific fuel consumption, demonstrated superior cost-effectiveness among all tested control strategies. In comparison, energy management strategies based on global optimization represent a more sophisticated approach, leveraging optimal control theory and artificial intelligence optimization techniques to design optimal energy allocation schemes for specific navigation conditions, thereby achieving a balance among multiple objectives. An optimization strategy was developed using dynamic programming (DP) in reference [9] that considered both battery degradation and electricity costs, significantly outperforming rule-based methods. While global optimization control strategies possess theoretical optimality, with DP strategies frequently serving as benchmarks for evaluating the global optimality of other algorithms, their practical application is somewhat limited due to the stochastic and unpredictable nature of vessel operating conditions. The ECMS, initially proposed by Paganelli [10], employs an equivalence factor to convert electrical energy consumption into fuel consumption, transforming global optimization problems into instantaneous optimization problems [11]. An efficient Energy Management System based on ECMS was proposed in reference [12], maintaining fuel cell system efficiency above 60% under most operating conditions while effectively suppressing fluctuations in fuel cell power output. Compared to ECMS and DP, MPC demonstrates unique advantages in hybrid power systems by predicting system behavior over a future time horizon while balancing computational efficiency and real-time capability. In marine hybrid power systems, predictive control achieves an energy balance among multiple power sources through online rolling optimization of multivariable problems. Reference [13] introduced an MPC-based coordinated control strategy that exhibited significant advantages in reducing fluctuations while maintaining autonomous operation. Furthermore, the energy management strategy designed based on nonlinear MPC models in [14] reduced fuel consumption and carbon emissions within the optimization period while maintaining robust disturbance rejection capabilities.

Furthermore, Multi-stack Fuel Cell Systems (MFCS) have attracted increasing attention due to their advantages in reduced hydrogen consumption, minimal degradation, and enhanced durability [15]. Power distribution strategies for MFCS primarily focus on system efficiency and hydrogen consumption optimization [16,17]. A hierarchical EMS was proposed for hybrid multi-stack fuel cell (FC) systems in the reference [18]. The first layer employs Sequential Quadratic Programming to determine power distribution ratios among multiple FC stacks while maintaining the battery State of Charge within acceptable bounds. The second layer optimizes an objective function based on overall system efficiency using genetic algorithms, achieving optimal power distribution among different FC stacks within the multi-stack system. Research findings demonstrate that this hierarchical EMS significantly reduces system hydrogen consumption. However, fuel cells are complex multi-physical systems characterized by relatively slow dynamic response characteristics and operational parameters that evolve dynamically with environmental variations and cell aging processes. This complexity can lead to performance inconsistencies among individual stacks within MFCS [19]. Consequently, strategies that solely prioritize system efficiency enhancement or fuel consumption optimization may not effectively sustain long-term system performance.

Ship power demands exhibit significant variations across different operational modes, including departure, cruising, port entry, and berthing, resulting in substantial load fluctuations and uncertainties [20]. These characteristics make long-term stable power demand prediction challenging, limiting forecasting capabilities to short-term load predictions. In MPC-based energy management strategies, load power prediction accuracy substantially influences power distribution effectiveness. ECMS requires dynamic calculations based on real-time data, introducing considerable computational complexity that increases controller burden and demands high hardware specifications. This computational load becomes particularly pronounced under high-frequency operational variations. Moreover, ECMS emphasizes instantaneous optimal distribution while inadequately addressing long-term energy system states (such as battery and supercapacitor State of Charge levels), potentially leading to the excessive utilization of energy storage devices during prolonged operation, thereby compromising system longevity and energy allocation flexibility. Furthermore, the EMS primarily focuses on optimizing either fuel economy or individual power source lifetime degradation [21,22]. However, such strategies fail to adequately account for the disparate degradation characteristics between fuel cells and power batteries. When one power source prematurely reaches its end-of-life due to excessive utilization, it disrupts the hybrid power system’s equilibrium, forcing the remaining power source to bear additional loads, thereby accelerating its performance deterioration. This non-synergistic operation not only compromises the vessel’s economic benefits but also leads to reduced durability of the entire power system, making it challenging to achieve an optimal balance between fuel economy and system durability.

Based on the aforementioned analysis, this paper proposes a multi-temporal energy management strategy for fuel cell ships that considers power source lifespan decay synergy. The primary innovations and contributions of this research include:

• A novel multi-temporal two-layer energy management strategy that achieves synergistic optimization across multiple time scales through an innovative hierarchical architecture. The upper-level controller manages long-term power distribution planning, while the lower-level controller executes precise real-time high-frequency power regulation, effectively balancing system economics and durability.

• An attention-enhanced CNN-LSTM power prediction model that leverages the complementary advantages of CNN in feature extraction and LSTM networks in temporal sequence modeling. The incorporation of attention mechanisms further enhances prediction accuracy, providing reliable decision support for energy management strategy implementation.

• Integration of differential fuel cell degradation characteristics and coordinated power source lifetime management into the optimization framework: The strategy considers varying degradation patterns among fuel cells and incorporates a lifetime-synchronized power distribution approach, achieving synchronized degradation control between fuel cells and power batteries.

The paper is structured as follows: Section 2 elaborates on the topological design of the hybrid ship power system. Section 3 presents a comprehensive examination of the theoretical framework and implementation methodology for the proposed multi-temporal energy management system. Section 4 validates the effectiveness of the proposed strategy through simulation experiments and provides comprehensive comparisons with existing methods. Section 5 summarizes the main research findings and innovative contributions.

2. Modeling of Fuel Cell Ship Propulsion System

2.1. Hybrid Power System Topology for Marine Applications

This study focuses on the “Hydrogen Vessel No. 1” fuel cell ship, which integrates fuel cells, lithium batteries, and supercapacitors as power sources. To enhance the economic viability and reliability of fuel cell vessels, a multi-stack fuel cell hybrid power system was designed based on the original power system architecture, with the primary power source parameters detailed in Table 1. The system employs a composite energy storage system (ESS) to replace the original single lithium battery configuration. This composite ESS comprises supercapacitors and lithium batteries, where lithium batteries offer high energy density but experience significant performance degradation over time and cycling. The degradation of lithium batteries primarily stems from complex electrochemical reactions during operation, encompassing irreversible processes such as electrode material structural changes, interfacial film growth, and active material loss [23,24]. These degradation mechanisms necessitate the careful consideration of performance deterioration characteristics in lithium battery applications. In contrast, supercapacitors, utilizing physical adsorption principles for energy storage, demonstrate exceptional cycling stability with negligible capacity degradation [25]. Their operational mechanism, avoiding chemical bond breaking and reformation, enables higher charge–discharge currents and extended operational lifetimes. The hybrid power system implements a fully active topology, as illustrated in Figure 1. The fuel cell subsystem consists of cell stacks and auxiliary equipment, including air compressors, hydrogen circulation pumps, and cooling water pumps. The dual fuel cell system employs a parallel topology, enabling independent control of each fuel cell stack and enhancing system stability. Each fuel cell connects to the DC bus through a unidirectional DC/DC converter, while lithium batteries and supercapacitors interface through bidirectional DC/DC converters. Within this architecture, the Energy Management System (EMS) coordinates the operation of fuel cells, lithium batteries, and supercapacitors to optimize electrical energy utilization efficiency and effectively reduce vessel operational costs.

2.2. Fuel Cell Model

This study employs the HYT-21-1346 Proton Exchange Membrane Fuel Cell (PEMFC). The fuel cell model is developed based on experimentally derived $P_{fc}$ − $C_{fc}$ and $P_{fc}$ − ${\eta }_{fc}$ characteristic curves. Let $P_{fc}$ represent the fuel cell output power after DC/DC conversion, ${\eta }_{fc}$ represent the system efficiency, and $C_{fc}$ represent the actual hydrogen consumption rate. The relationship between these three parameters can be expressed as:

(1)$C_{fc}={\displaystyle \frac{P_{fc}}{{\eta }_{fc}LH{V}_{H_2}}}$

where $LH{V}_{H_2}$ is the lower heating value of hydrogen, which equals 120 MJ/kg. With the prolonged operation of the system, fuel cell performance gradually deteriorates, leading to a decline in system durability. The fuel cell lifetime significantly dominates the total operational costs of the system. Generally, the factors contributing to fuel cell performance degradation can be categorized into four main aspects: start–stop cycles, high-load operation, low-load operation, and load fluctuations [26,27]. Therefore, fuel cell performance degradation patterns can be quantified by analyzing operational data during system runtime. The degradation rate of the fuel cell is defined as follows:

(2)$D_{fc}={\displaystyle \frac{d_1{T}_{low}+d_2{T}_{high}+d_3{N}_{tran}+d_4{N}_{ss}}{V_{init}\times 10\%}}$

where $T_{low}$, $T_{high}$, $N_{tran}$, and $N_{ss}$ denote the operational durations at low and high power states, cumulative power variations, and start–stop cycles of the fuel cell, respectively; $d_1\sim d_4$ represents the degradation rates under these four operating conditions (as shown in Table 2); and $V_{init}$ is the initial single-cell voltage of the fuel cell, where the end-of-life criterion is defined as a 10% voltage degradation from the initial value.

Due to the dynamic power output characteristics of fuel cells under varying operating conditions, the performance degradation among individual stacks often exhibits non-uniform patterns [28]. To investigate the impact of fuel cell performance degradation, two operational scenarios were established for comparison: Fuel Cell 1 represents a new stack without degradation (healthy state $S_{OH}=1$), while Fuel Cell 2 operates in a state of health $S_{OH}=0.7$. Here, the state of health is characterized by the voltage degradation level at the rated current, where the maximum allowable voltage drop is defined as 10% relative to the initial output voltage. The system efficiency and hydrogen consumption rate curves under different SOH values were calculated using the degradation formula proposed in [29]. The fuel cell system efficiency comprises three components, defined as follows:

(3)${\eta }_{fc}={\eta }_{fuel}\cdot {\eta }_{conv}\cdot {\eta }_{elec}$

(4)${\epsilon }_{conv}=0.9+0.1\cdot S_{OH}$

where ${\eta }_{fuel}$ is the fuel utilization rate, which can be approximated as 100% due to the hydrogen circulation mode implemented through hydrogen injectors and hydrogen circulation pumps, ${\eta }_{fuel}=1$; ${\eta }_{conv}$ represents the conversion efficiency, defined as the ratio between the electrical energy generated by the fuel cell and the chemical energy of the consumed hydrogen. This efficiency decreases with fuel cell degradation, and its degradation level can be quantified by the coefficient of variation given in Equation (4); ${\eta }_{elec}$ denotes the electrical efficiency, which is determined by the DC/DC converter efficiency and parasitic power consumption of auxiliary equipment. Its degradation level can be expressed by the coefficient of variation as defined in Equation (5).

(5)${\epsilon }_{elec}={\displaystyle \frac{{\eta }_{DC/DC}-\frac{E_{auz}}{{\epsilon }_{conv}\cdot E_{stack}}}{{\eta }_{DC/DC}-\frac{E_{auz}}{E_{stack}}}}$

where ${\eta }_{DC/DC}$ represents the conversion efficiency of the DC/DC converter, $E_{auz}$ denotes the parasitic power consumption of auxiliary equipment, and $E_{stack}$ is the stack output power. The degradation level of the fuel cell system efficiency can be characterized by the combined degradation of ${\eta }_{conv}$ and ${\eta }_{elec}$:

(6)${\eta }_{fc\_degraded}={\epsilon }_{elec}\cdot {\epsilon }_{conv}\cdot {\eta }_{fc}$

where ${\eta }_{fc\_degraded}$ represents the fuel cell system efficiency after performance degradation. The degraded fuel cell system efficiency is calculated based on Equations (3)–(6). The hydrogen consumption rate at various net power outputs of the fuel cells is derived from Equation (1). The system efficiency curves and hydrogen consumption rate curves of the fuel cells are illustrated in Figure 2.

2.3. Lithium Battery Model

The lithium battery possesses dual functionality in energy storage and charge–discharge capabilities. As an energy storage component, the battery capacity is determined by the total charge it can transfer. The battery capacity Q can be calculated using the following expression:

(7)$Q=I\cdot t$

Equation (7) indicates that a battery with capacity Q can sustain a constant discharge current $I$ for a duration $t$. The State of Charge (SOC) of the battery is defined as the ratio between the remaining capacity and the total capacity, expressed as:

(8)$SOC=1-{\displaystyle \frac{{\displaystyle \int I(t)dt}}{Q}}$

In this study, the Rint equivalent circuit model is adopted for the traction battery, which consists of an ideal voltage source in series with internal resistance, as illustrated in Figure 3:

According to the Rint equivalent circuit model, the mathematical expression for the battery output power is given as:

(9)$P_{batt}=U_{oc}{I}_b-R_b{I}_b^2$

where $U_{oc}$ denotes the open-circuit voltage, while $I_b$ and $R_b$ represent the battery current and internal resistance, respectively. Both the open-circuit voltage and internal resistance are functions of the battery SOC. According to Equation (9), the current $I_b$ is derived as:

(10)$I_b={\displaystyle \frac{U_{oc}-\sqrt{U_{oc}^2-4R_b{P}_{batt}}}{2R_b}}$

The dynamic behavior of the lithium battery SOC can be described by the following expression, derived from the combination of Equations (8) and (10):

(11)$SOC(k+1)=SOC(k)-{\displaystyle \frac{U_{oc}-\sqrt{U_{oc}^2-4R_b{P}_{batt}}}{2Q{R}_b}}$

The degradation characteristics of lithium batteries are governed by their SOC and transient power profiles. High current operations, especially during charging processes, lead to accelerated lifetime reduction. The rate of performance degradation exhibits a positive correlation with the magnitude of SOC variations. The corresponding semi-empirical degradation model [30] is formulated as:

(12)$D_{batt}={\displaystyle \frac{1}{Q_{batt}}}{\displaystyle {\int }_0^t\left|F(SO{C}_{batt})G(i_{batt})I_{batt}(t)\right|dt}$

where $Q_{batt}$ represents the battery capacity, $F(SO{C}_{batt})$ is a weighting function that penalizes battery degradation due to excessive SOC utilization, and $G(i_{batt})$ is a weighting function that penalizes battery degradation caused by excessive transient power. These functions are expressed as follows:

(13)$F(SO{C}_{batt})=1+3.25{(1-SO{C}_{batt})}^2$

(14)$\left\{\begin{array}{l}G(I_{batt})=1+0.45{\displaystyle \frac{I_{batt}}{I_{batt\_nom}}},i_{batt}\ge 0\\ G(I_{batt})=1+0.55{\displaystyle \frac{\left|I_{batt}\right|}{I_{batt\_nom}}},i_{batt}<0\end{array}\right.$

where $I_{batt\_nom}$ represents the rated current of the battery, and the degradation function $D_{batt}$ ranges between 0 (beginning of life) and 1 (end of life).

2.4. Supercapacitor Model

Among the diverse equivalent circuit models available for supercapacitors, this investigation utilizes a widely adopted simplified model for analytical modeling, as depicted in Figure 4:

Based on the equivalent circuit, the mathematical expressions for the output power and State of Charge (SOC) of the supercapacitor can be formulated as:

(15)$P_{sc}=U_{sc}{I}_{sc}-I_{sc}^2{R}_{sc}$

(16)$SOC={\displaystyle \frac{U_{sc}-U_{scmin}}{U_{scmax}-U_{scmin}}}$

where $P_{sc}$ represents the supercapacitor output power, $U_{sc}$ denotes the terminal voltage, $I_{sc}$ is the supercapacitor current, $R_{sc}$ represents the supercapacitor resistance, and $U_{scmax}$ and $U_{scmin}$ denote the maximum and minimum operating voltages of the supercapacitor, respectively.

3. Proposed Energy Management Strategy

This section presents a two-stage energy management strategy based on Model Predictive Control (MPC) with multiple time scales. The strategy aims to optimize power distribution among fuel cells, lithium batteries, and supercapacitors based on power demand data during navigation (as illustrated in Figure 5), ensuring the economically efficient operation of the ship’s hybrid power system while satisfying operational constraints. The hierarchy of the proposed algorithm is shown in Figure 6. The comprehensive methodology of the proposed energy management strategy proceeds as follows: Initially, future power load profiles are predicted based on historical power data, followed by power allocation for the predicted load data. The resulting reference power values are then dynamically computed in conjunction with real-time load data. The strategy primarily comprises a prediction phase and two optimization phases: In the first optimization phase, an MPC controller optimizes the fuel cell power and lithium battery charge–discharge power based on low-frequency power components over extended time scales, minimizing equivalent hydrogen consumption. The optimized battery and supercapacitor charge–discharge powers, along with the fuel cell power, serve as reference inputs for the lower-level real-time power optimization controller. In the second phase, the lower-level real-time power optimization controller performs secondary optimization of high-frequency power components based on the reference values from the upper-level MPC. The optimized results are then implemented as actual control outputs. Figure 7 illustrates the framework of the proposed energy management strategy.

3.1. CNN-LSTM Prediction Model with Attention Mechanism

This section presents an attention-enhanced CNN-LSTM model for power prediction. The model incorporates an attention-based CNN architecture that extends the standard CNN framework with parallel attention pathways for salient feature extraction. The attention pathway employs expanded input dimensions to broaden the receptive field, thereby comprehensively capturing temporal contextual information and effectively learning the significance of local sequence features. By amplifying the weights of crucial temporal features while suppressing the influence of non-essential features, the attention module effectively addresses the limitations of traditional models in discriminating temporal feature importance. The multi-scale input approach, where both the standard CNN module and attention mechanism module process input sequences of varying lengths, enables the more robust extraction of short-sequence features. In terms of feature extraction, the LSTM architecture extracts coarse-grained features from the fine-grained features obtained from the front end, providing refined processing of multi-dimensional features. This architectural design mitigates the memory loss and gradient vanishing problems typically associated with extended step lengths. The attention-enhanced CNN-LSTM model achieves comprehensive temporal data characterization [31] through the fusion of both coarse and fine-grained features.

The architecture of the attention-enhanced CNN-LSTM model comprises five primary layers: the input layer, CNN layer, LSTM layer, attention layer, and output layer, as illustrated in Figure 8. Historical load data serve as the input to the CNN layer, where convolution operations are performed to increase the depth and compress the parameter quantity while pooling operations reduce feature dimensionality. The fully connected layer transforms features into a one-dimensional structure, completing feature vector extraction. The LSTM and attention layers learn the internal variation patterns of the load from the extracted features, thereby enabling predictive functionality. The output layer generates the prediction results. The model has been trained using historical navigation data from the vessel and subsequently employed to predict the power profile shown in Figure 5, with a sampling interval of 2 s and a prediction horizon of 20 s. The prediction results are presented in Figure 9.

As demonstrated in Figure 8, the predicted power demand curve closely aligns with the actual power demand profile. The Root Mean Square Error (RMSE) is employed as the evaluation metric for assessing the accuracy of the ship’s load power prediction. RMSE represents the square root of the ratio between the squared deviations of predicted values from actual values and the number of observations, as expressed in Equation (17):

(17)$\begin{array}{l}RMSE={\displaystyle \frac{{\displaystyle \sum _{k=1}^{n}RMSE(k)}}{n}}\\ RMSE(k)=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^{t_p}{(p(k+i)-p_0(k+i))}^2}}{t_p}}}\end{array}$

where $n$ represents the number of sampling points in the operating condition, $RMSE(k)$ denotes the value at time $k$ within prediction horizon $t_p$,$p(k+i)$ represents the predicted value at $i$ seconds after time $k$, and $p_0(k+i)$ indicates the actual value at $i$ seconds after time $k$. The prediction results for horizons of 10 s, 20 s, and 40 s were analyzed, with the outcomes presented in Table 3. From the perspective of RMSE, prediction horizons of 10 s, 20 s, and 40 s correspond to error values of 1.01 kW, 1.87 kW, and 3.98 kW, respectively. In terms of average prediction time, the computational requirements are approximately 20.12 ms, 57.34 ms, and 150.56 ms, respectively. While the 10 s prediction horizon exhibits the lowest RMSE and computational overhead, it proves insufficient to fully capture power variation trends. Conversely, although the 40 s horizon offers an extended prediction range, its significant error accumulation and 150 ms computational latency make it unsuitable for real-time control requirements. The 20 s prediction horizon, however, demonstrates an optimal balance across all metrics: its RMSE is only marginally higher than that of the 10 s horizon, its 57.34 ms computational latency remains acceptable for real-time systems, and it provides an adequate prediction range to support energy management system decisions. Therefore, this study adopts a 20 s prediction horizon, a selection that ensures both prediction accuracy and compliance with the real-time requirements of marine power systems.

3.2. Model Predictive Control Principles

Model Predictive Control (MPC), emerging as an innovative computational control algorithm in the late 1970s, is also known as receding horizon control [32]. The operational mechanism of MPC can be succinctly characterized as follows: At each sampling instant, the current control action is determined by solving a finite-horizon open-loop optimal control problem, utilizing the current process state as the initial condition for the optimization. Only the first element of the optimal control sequence is implemented. At the subsequent sampling instant, this process is repeated, with new measurements serving as the initial conditions for predicting future system dynamics, thereby refreshing and resolving the optimization problem. The fundamental architecture of MPC comprises four essential components: the prediction model, feedback correction, receding horizon optimization, and reference trajectory [33].

The operational principle of MPC is schematically illustrated in Figure 10. Let $H_p$ denote the prediction horizon, $H_q$ represent the control horizon, $x_r$ designate the reference trajectory for state variable $x$, and $x_p$ represent the predicted trajectory for state variable $x$. Generally, constraints must be imposed on the predicted trajectory, which are formulated as follows:

(18)$min{J}_k={\displaystyle {\sum }_{t=k}^{k+p}L(x(t),u(t))}$

(19)$\left\{\begin{array}{l}{x}_{min}(t)\le x(t)\le x_{max}(t)\\ u_{min}(t)\le u(t)\le u_{max}(t)\\ k\le t\le k+p\end{array}\right.$

In Equations (18) and (19), $x(t)$ represents the state variable at time $t$,$u(t)$ denotes the control variable at time $t$, $L$ represents the performance index at time $k$, and $J_k$ denotes the performance index at time instant k over the prediction horizon $k-k+p$.

From Equation (18), it can be observed that the performance index is calculated solely within the prediction horizon. Similarly, the performance index at time instant $k+1$ can be expressed as:

(20)$min{J}_{k+1}={\displaystyle {\sum }_{t=k+1}^{k+p+1}L(x(t),u(t)})$

3.3. System Constraints

At any given time instant, the total grid output power can be expressed as follows:

(21)$P_{total}=P_{fc1}+P_{fc2}+P_{Batt}+P_{sc}$

In the design of system operational constraints, trade-offs must be considered among performance, lifetime, and safety. The power constraints of the fuel cell are intended to meet load demands while preventing overload operation, thereby extending its service life. The power constraints and SOC range of the lithium battery are employed to prevent excessive charging and discharging cycles, thus enhancing cycle life and ensuring thermal management stability. The power constraints and SOC range of the supercapacitor are established to provide rapid transient power response while minimizing efficiency losses and thermal issues. These constraints comprehensively incorporate the performance characteristics of the fuel cell, battery, and supercapacitor, aiming to meet system load demands while achieving an optimal balance between energy efficiency and component longevity. The operational constraints for the fuel cell, lithium battery, and supercapacitor are defined as follows:

(22)$\left\{\begin{array}{l}0\mathrm{kW}\le P_{fc}\le 120\mathrm{kW}\\ -100\mathrm{kW}\le P_{batt}\le 100\mathrm{kW}\\ -120\mathrm{kW}\le P_{sc}\le 120\mathrm{kW}\\ 20\%\le SO{C}_{batt}\le 80\%\\ 10\%\le SO{C}_{sc}\le 90\%\end{array}\right.$

3.4. First Optimization Stage

Due to the distinct characteristics of supercapacitors and batteries, the ship’s load power is decomposed according to frequency components. The high-frequency components of the load power should be absorbed by the supercapacitor. In this study, low-pass filtering is employed to decompose the high-frequency power signals. The low-pass filtering control strategy is illustrated in Figure 11.

In the figure, $T_1$ represents the filtering time constant, $P_{low}$ and $P_{high}$ denote the low-frequency and high-frequency components of the load power, respectively, while $P_{batt}$, $P_{fc}$, and $P_{sc}$ represent the output powers of the lithium battery, fuel cell, and supercapacitor, respectively. A larger $T_1$ value results in smoother low-frequency power components, consequently directing more high-frequency power components to the supercapacitor for absorption. The $T_1$ value should be synchronized with the control period of the MPC in the primary optimization phase, where larger MPC control periods, corresponding to extended time scales, necessitate larger τ values to achieve smoother low-frequency power profiles.

In this phase, the MPC controller focuses on optimizing low-frequency power distribution over the prediction horizon. It determines reference values for battery charge–discharge power, supercapacitor charge–discharge power, and fuel cell power output. The optimization objective function for this phase aims to minimize overall hydrogen consumption while accounting for the performance degradation of both fuel cells and lithium batteries. In this primary optimization phase, the total fuel cell power is equally distributed between two fuel cells using a balanced allocation method, with the controller sampling time set to 2 s and the prediction horizon equal to the control horizon. The MPC controller objective function is formulated as:

(23)$\begin{array}{c}\min J_1={\displaystyle \sum _{i=1}^N{C}_{fc1}}+{\displaystyle \sum _{i=1}^N{C}_{fc2}}+{\displaystyle \sum _{i=1}^N{C}_{batt}}+{\displaystyle \sum _{i=1}^N{k}_{fc\_deg}{C}_{fc\_deg}}+{\displaystyle \sum _{i=1}^N{k}_{batt\_deg}{C}_{batt\_deg}}\\ \mathrm{s}.\mathrm{t}.\left(21\right),\left(22\right)\end{array}$

where $C_{fc1}$ and $C_{fc2}$ represent the hydrogen consumption rates of fuel cell 1 and fuel cell 2, respectively; $C_{batt}$ denotes the equivalent hydrogen consumption rate of the lithium battery; $k_{fc\_deg}$ and $k_{Batt\_deg}$ are degradation weights that indicate the optimization priorities for fuel cell and lithium battery degradation within the overall system performance. Based on the data presented in Figure 2, the hydrogen consumption rate function is obtained through quadratic polynomial fitting:

(24)$C_{fc}=a\cdot P_{fc}^2+b\cdot P_{fc}+c$

where a, b, and c are the fitting coefficients of the fuel cell. To minimize hydrogen consumption in the hybrid power system, the electrical energy consumed by the lithium battery can be equivalently expressed in terms of the chemical energy consumed by the fuel cell [34], calculated as:

(25)$C_{batt}=\left\{\begin{array}{l}{\displaystyle \frac{k{P}_{batt}}{{\eta }_{dis}{\eta }_{fc\_avg}LH{V}_{H_2}}},P_{batt}\ge 0\\ {\displaystyle \frac{k{P}_{batt}{\eta }_{chg}}{{\eta }_{fc\_avg}LH{V}_{H_2}}},P_{batt}\le 0\end{array}\right.$

where ${\eta }_{dis}$ and ${\eta }_{chg}$ represent the charging and discharging efficiencies of the lithium battery, respectively; c denotes the average efficiency of the fuel cell. To ensure the lithium battery operates within an appropriate SOC range and prevent damage from excessive charging or discharging, a correction function $k$ is introduced. The magnitude of $k$ determines the rate of convergence, where larger values indicate faster convergence and smaller values indicate slower convergence. This can be calculated using the following expression:

(26)$k=1-2\mu {\displaystyle \frac{(SOC-0.5(SO{C}_{max}+SO{C}_{min}))}{SO{C}_{max}-SO{C}_{min}}}$

where $SO{C}_{max}$ and $SO{C}_{min}$ denote the upper and lower SOC thresholds for the lithium battery, established at 0.8 and 0.2, respectively, in this investigation, and $\mu$ represents the balance coefficient. Recognizing lifetime degradation as a critical determinant of fuel cell performance and system economics, this study implements an equivalence methodology derived from the ECMS (Equivalent Consumption Minimization Strategy) to quantify fuel cell degradation in terms of equivalent hydrogen consumption. This approach achieves dual objectives: minimizing dynamic load fluctuations in fuel cell operation while simultaneously decelerating the degradation process through the explicit consideration of performance deterioration mechanisms, ultimately enhancing overall system durability. The equivalent hydrogen consumption $C_{\mathrm{deg}}$ attributable to fuel cell degradation is formulated as:

(27)$C_{\mathrm{deg}}={\displaystyle \frac{D_{fc}{c}_{fc}}{c_{H_2}}}$

where $c_{fc}$ and $c_{H_2}$ represent the unit costs of the fuel cell and hydrogen, respectively, while $D_{fc}$ denotes the fuel cell degradation rate as derived from Equation (2). Since the exclusive consideration of fuel cell lifetime degradation could result in accelerated deterioration and premature end-of-life of the lithium battery, the objective function incorporates battery degradation metrics. The associated degradation cost of the lithium battery is formulated as:

(28)$C_{batt\_deg}=D_{batt}{Q}_{batt}$

The objective function (Equation (23)) of the MPC controller is solved using the Sequential Dynamic Programming (SDP) algorithm.

3.5. Second Optimization Stage

The initial solution set obtained from the first optimization phase serves as the reference power for the second phase:

(29)$P^{ref}=\left[\begin{array}{l}{P}_{fc1}^{ref}\\ P_{fc2}^{ref}\\ P_{batt}^{ref}\\ P_{sc}^{ref}\end{array}\right]$

The real-time power optimization controller in the secondary phase performs power allocation based on instantaneous load demands. This phase’s optimization objective is to achieve optimal power distribution by comprehensively considering the characteristics of energy storage devices and high-frequency components of power demands while ensuring fuel economy. Given the differential degradation rates among units in the dual fuel cell system, the uniform distribution method employed in the primary phase cannot achieve optimal efficiency. Therefore, this phase necessitates further consideration of maximizing the overall efficiency of the fuel cell system. The objective function for the secondary optimization phase is expressed as:

(30)$\begin{array}{c}\min J_2=-{\alpha }_1\left({\displaystyle \frac{P_{fc1}+P_{fc2}}{\frac{P_{fc1}}{{\eta }_1}+\frac{P_{fc2}}{{\eta }_2}}}\right)+{\alpha }_2(C_{fc\_\mathrm{deg}}+C_{batt\_\mathrm{deg}})+{\alpha }_3{\left(P_{batt}-P_{batt}^{ref}\right)}^2\\ +{\alpha }_4{\left(\left(P_{fc1}+P_{fc2}\right)-\left(P_{fc1}^{ref}+P_{fc2}^{ref}\right)\right)}^2+{\alpha }_5{\left(P_{uc}-P_{uc}^{ref}\right)}^2\\ s.t.(21),(22)\end{array}$

where coefficients ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_4$, and ${\alpha }_5$ represent penalty factors. The first term aims to maximize the efficiency of the dual fuel cell system; the second term reflects the degradation costs of both fuel cells and the lithium battery; and the third, fourth, and fifth terms represent the tracking requirements for reference values generated by the upper-level controller, including fuel cell power references and charge–discharge power references for the battery pack and supercapacitor. These tracking terms ensure the maintenance of fuel economy optimization performance established at the macro time scale in the first phase.

The nonlinear programming problem described by Equation (30) is solved using the GUROBI commercial optimization solver.

4. Simulation Results

To comprehensively evaluate the optimization effectiveness of the proposed strategy, this study conducts a multi-dimensional comparative analysis. The proposed strategy is initially benchmarked against both the conventional single-level Model Predictive Control (MPC) strategy and the Equivalent Consumption Minimization Strategy (ECMS). Furthermore, to validate the global optimality of the proposed strategy over the entire operational cycle, comparisons are made with the Dynamic Programming (DP) strategy. This multi-tiered comparative validation approach effectively demonstrates the optimization performance of the proposed strategy.

4.1. Evaluation Criterion

To objectively evaluate the economic performance of the power system, this study defines the total equivalent hydrogen consumption $C_{H_2\_equ}$, which not only encompasses the direct hydrogen consumption of fuel cells but also incorporates the equivalent hydrogen consumption derived from SOC deviations of both the lithium battery and the supercapacitor from their initial states, as expressed in Equation (23):

(31)$C_{H_2\_equ}=C_{fc}+{\displaystyle \frac{(SO{C}_{batt\_init}-SO{C}_{batt\_end})Q_{batt}+(SO{C}_{sc\_init}-SO{C}_{sc\_end})Q_{sc}}{{\eta }_{fc\_avg}LH{V}_{H_2}}}$

where $SO{C}_{batt\_end}$ and $SO{C}_{sc\_end}$ represent the final SOC of the lithium battery and supercapacitor, respectively; ${\eta }_{fc\_avg}$ denotes the average efficiency of the fuel cell system. To objectively evaluate the optimization performance of various control strategies, this study establishes a single-voyage cost assessment index for the vessel’s power system. This index comprehensively incorporates three critical dimensions: hydrogen fuel consumption cost, fuel cell system degradation cost, and lithium battery degradation cost, enabling a holistic characterization of the power system’s economic performance, as shown in Equation (24).

(32)$C_{total}=C_{H_2\_equ}\cdot c_{H_2}+D_{fc}\cdot c_{fc}\cdot P_{fc\_rated}+D_{batt}\cdot c_{batt}\cdot Q_{batt}$

where $c_{batt}$ represents the unit cost coefficient, which is determined through market research to be 35 CNY/kg for hydrogen, 3500 CNY/kW for fuel cells, and 2000 CNY/kWh for lithium batteries; $P_{fc\_rated}$ and $Q_{batt}$ denote the rated power of the fuel cell system and the nominal capacity of the lithium battery system, respectively.

4.2. Power Distribution

Figure 12a–c illustrates the operational characteristics of the vessel’s hybrid power system under four different strategies, depicting the power profiles of the fuel cell system, lithium battery, and supercapacitor. The simulation results demonstrate that the proposed strategy achieves optimal real-time power control based on power demand forecasting, enabling coordinated optimization across both long and short time scales. The fuel cell system predominantly operates in a power-following mode with enhanced operational efficiency. This approach not only ensures sufficient power redundancy in the hybrid system to accommodate load variations but also effectively minimizes the dynamic load fluctuations of the fuel cell, maximizing the peak-shaving and valley-filling capabilities of the energy storage system. The results indicate that the lithium battery power profile exhibits relatively smooth characteristics, while the high-frequency power components of the load demand are primarily absorbed by the supercapacitor. This effectively prevents high-frequency power fluctuations in battery output, achieving the intended control objectives. In contrast, the conventional single-layer MPC strategy, which must solve for both real-time and predicted power simultaneously, demonstrates suboptimal power distribution due to prediction accuracy limitations and system constraints. This results in underutilization of the energy storage system’s auxiliary function and increases fuel cell power output fluctuations. The traditional ECMS strategy, which only addresses real-time power distribution without future power information, tends to converge to local optima during optimization. To maintain the SOC of the energy storage system, this approach leads to higher fuel cell power output, resulting in increased power variation rates and hydrogen consumption. Consequently, it fails to ensure fuel economy and yields a suboptimal overall performance.

The proposed strategy accounts for the differential degradation states of fuel cells, with the power distribution between two fuel cells illustrated in Figure 12d. The fuel cell with inferior performance (SOH = 0.7) exhibits a significantly lower power fluctuation amplitude compared to the one with superior performance (SOH = 1). This demonstrates that the proposed strategy effectively mitigates the aging rate of the Multi Fuel Cell System (MFCS), thereby reducing the frequency and costs associated with fuel cell replacement. Furthermore, according to calculations using Equation (33), the overall efficiency of the fuel cell system reaches 58.5% when using the strategy proposed in this paper, while it is 57.1% when using the equal distribution strategy. The comparison results show that the method in this paper improved the overall efficiency of the fuel cell system by 1.4%, effectively enhancing the system’s energy utilization rate.

(33)${\eta }_{total}={\displaystyle \frac{{\displaystyle \sum (P_{FC1}+P_{FC2})}}{{\displaystyle \sum (\frac{P_{FC1}}{{\eta }_{FC1}}+\frac{P_{FC2}}{{\eta }_{FC2}})}}}$

4.3. Energy Storage Unit State of Charge

Figure 13 illustrate the State of Charge (SOC) profiles of the lithium battery and supercapacitor under four different control strategies. The simulation results demonstrate that all strategies maintain the SOC of both energy storage units within their constraint boundaries. Specifically, the lithium battery SOC fluctuation ranges are 46.6–60% for the single-layer MPC strategy, 26.9–60% for the ECMS strategy, and 27.1–71.2% for the DP strategy, which exhibits the widest variation. The proposed strategy maintains the battery SOC within 34.2–60%. As a locally optimizing approach, the proposed strategy deliberately restricts SOC deviations from median values to reserve sufficient capacity for subsequent power demands. The single-layer MPC strategy demonstrates a narrower battery SOC fluctuation range compared to the proposed strategy, primarily due to increased battery power output in the first optimization stage of our strategy, which aims to limit fuel cell power variations. From the perspective of energy storage system performance evaluation, the proposed strategy demonstrates favorable balanced characteristics: Regarding lithium battery degradation, the capacity loss is only marginally higher than the single-layer MPC strategy, indicating effective capacity loss control while maximizing battery performance utilization. For the supercapacitor, this strategy exhibits the relatively largest SOC fluctuation range, an inherent consequence of the dual-layer optimization structure. This effectively accomplishes power smoothing for both the lithium battery and fuel cell system. It should be noted that due to the strategy’s emphasis on mitigating fuel cell and lithium battery degradation, there exists a significant deviation between the final and initial SOC values of the energy storage elements. This represents a deliberate trade-off within the system’s global optimization process.

4.4. Comparative Analysis of Coordinated Degradation of Fuel Cell and Lithium Battery Systems

Based on Equations (2) and (12), the degradation rates of the fuel cell system and lithium battery system were calculated. Figure 14 shows the degradation rate curves of the fuel cell and lithium battery under four different energy management strategies. For the DP, single-layer MPC, and ECMS strategies, fuel cell total power distribution follows an equal allocation approach. At the end of the operating cycle, the differential degradation rates between the dual power sources under the proposed strategy, DP, single-layer MPC, and ECMS are 0.0019%, 0.101%, 0.0218%, and 0.0545%, respectively.

Compared to the DP, single-layer MPC, and ECMS strategies, the proposed strategy achieves remarkable reductions in the differential degradation rates between dual power sources of 98.12%, 91.36%, and 96.48%, respectively. This superior optimization performance can be attributed to several key factors: First, although the DP strategy achieves the lowest total voyage cost, it excessively relies on the lithium battery to minimize fuel cell degradation. This leads to premature battery life depletion, failing to achieve balanced optimization between voyage economics and coordinated power source degradation. Second, the single-layer MPC strategy’s insufficient utilization of the lithium battery’s regulatory capabilities results in a fuel cell degradation rate exceeding that of the battery, thereby exacerbating the degradation inconsistency between the two systems. Third, the ECMS strategy, in its pursuit of minimizing equivalent hydrogen consumption, is constrained by its tendency to converge to local optima. This limitation increases power fluctuations in both fuel cell and lithium battery systems, consequently amplifying the disparity in degradation rates between the two power sources.

This strategy achieves the synchronized degradation control of fuel cell and lithium battery systems by organically integrating fuel economy with coordinated power degradation management. From the perspectives of long-term system economics and maintenance, this coordinated control strategy has significant practical implications: First, by achieving synchronized degradation of lithium batteries and fuel cells, this strategy can significantly extend the overall service life of the ship’s power system. When fuel cells and lithium battery systems degrade at similar rates, it prevents the need for complete system replacement due to premature failure of a single component, thereby reducing equipment renewal costs. Second, coordinated degradation control significantly reduces maintenance requirements. By avoiding uneven aging between components, the frequency of maintenance inspections can be reduced. This not only directly saves maintenance costs but also reduces vessel downtime for maintenance, improving operational efficiency. These research findings demonstrate that while achieving coordinated power source degradation, this strategy brings significant economic benefits to vessel operators and has practical guiding significance for advancing ship power systems toward more sustainable and economical development.

To further validate the effectiveness of the proposed strategy, the simulation results were compared with existing experimental studies. Lu et al. [35] built a fuel cell stack durability test platform to experimentally study the coordinated life degradation of dual power sources. Their experimental results showed that under the CWC1 combined working conditions, after adopting the PEMS strategy, the difference in degradation rates between the fuel cell and battery decreased from 0.00538% to 0.00021% compared to the ECMS strategy. The strategy proposed in this paper reduced the degradation rate difference from 0.0545% to 0.0019%, with a relative improvement (96.48%) basically consistent with the experimental results (96.1%). Through comparisons with experimental studies, it can be found that the performance improvement demonstrated by the proposed strategy in simulation and HIL testing shows good consistency with actual test results.

4.5. Comparative Analysis of Power Source Stress

Power source stress analysis primarily examines the high-frequency power output characteristics of both fuel cell and lithium battery systems, serving as a critical factor influencing their respective service life [36]. Lower power source operational stress, characterized by the reduced frequency and amplitude of power output fluctuations, is conducive to maintaining optimal source performance. The power source stress can be effectively quantified through the standard deviation $\sigma$ of high-frequency components obtained via Haar wavelet transformation. Figure 15 illustrates the comparative analysis of power source stress under different control strategies.

Under the proposed strategy, the average stress per fuel cell unit is 0.1537, while the power source stress indices for DP, single-layer MPC, and ECMS strategies are 0.0782, 0.242, and 0.2462, respectively. Compared to real-time optimization strategies (single-layer MPC and ECMS), the proposed strategy achieves significant reductions in individual fuel cell stress of 36.5% and 37.5%, respectively. While the DP strategy exhibits the lowest fuel cell stress, this is achieved at the expense of lithium battery longevity. Regarding lithium battery stress performance, the proposed strategy demonstrates a characteristic pattern of “superior macroscopic optimization with elevated microscopic stress levels”. This distinctive feature primarily stems from the strategy’s optimization mechanism: during the first optimization phase, the lithium battery assumes the critical role of compensating for fuel cell power fluctuations. Although this design reduces the system’s overall energy demand, it results in a relatively aggressive battery power allocation strategy, manifested as more frequent charge–discharge transitions, leading to higher power source stress compared to single-layer MPC and ECMS strategies.

However, considering the online real-time optimization nature of the proposed strategy, this stress distribution characteristic actually facilitates synchronized degradation control between fuel cells and lithium batteries. Through the rational allocation of power source stress, the strategy achieves the comprehensive optimization of voyage economics and system durability while maintaining real-time response capabilities. These results further validate the superiority of the proposed strategy in coordinated power source management.

4.6. Comprehensive Performance Comparison

Table 4 presents a comprehensive comparison of key performance metrics across different strategies. The proposed strategy achieves comparable equivalent hydrogen consumption to single-layer MPC and ECMS strategies. Although the DP strategy demonstrates the lowest hydrogen consumption, its high computational complexity and poor real-time performance make it impractical for real-world applications. In this trade-off, the proposed strategy opts to restrict fuel cell operation within high-efficiency variable load ranges to suppress degradation, sacrificing some instantaneous efficiency while laying the foundation for overall performance improvement. The proposed strategy significantly reduces fuel cell degradation costs by 30.4% and 8.9% compared to single-layer MPC and ECMS strategies, respectively. However, due to the reduced fuel cell degradation leading to a relatively increased battery load, the battery degradation cost of the proposed strategy is higher than that of single-layer MPC but notably lower than DP and ECMS strategies. From an economic perspective, the proposed strategy demonstrates significant advantages, saving 12.6% and 11.7% in costs compared to single-layer MPC and ECMS strategies, respectively. Although the total cost is 1.6% higher than the DP strategy, the proposed strategy overcomes the limitations of high computational complexity and poor real-time performance associated with DP, providing a more practical solution for real-world applications. The proposed strategy shows far lower power source degradation rate differences compared to other strategies, confirming its more balanced and rational energy distribution. Additionally, the fuel cell high-frequency stress is significantly lower than single-layer MPC and ECMS strategies, contributing to the extended fuel cell life; although the lithium battery high-frequency stress is relatively higher, this trade-off is a necessary compromise to achieve overall low degradation rates, as battery systems possess superior dynamic response capabilities and better tolerance compared to fuel cells.

Comprehensive analysis indicates that the proposed strategy achieves multi-objective coordinated optimization through its control algorithm: while maintaining energy efficiency, it significantly improves the system’s overall economic performance through the reasonable distribution of dynamic loads and balanced component degradation rates while maintaining near-globally optimal control effects.

5. Conclusions

This paper presents a multi-timescale energy management strategy that incorporates coordinated power source degradation. The strategy features an innovative dual-layer optimization architecture that synergistically integrates Model Predictive Control (MPC)’s predictive optimization capabilities with real-time control responsiveness. An attention-enhanced CNN-LSTM prediction model was developed, achieving high-precision forecasting with a power prediction RMSE of merely 3.69 kW, thus providing robust decision support for strategy implementation. The simulation results demonstrate that compared to single-layer MPC strategy and the ECMS strategy, the proposed strategy reduced total voyage costs by 12.6% and 11.7%, respectively, while mitigating fuel cell high-frequency stress by 36.5% and 37.5%, with a performance approximating the globally optimal DP strategy solution. By incorporating differentiated fuel cell degradation and lithium battery aging characteristics, the strategy significantly enhanced dual power source lifetime synchronization, reducing degradation rate disparities by 98.12%, 91.36%, and 96.48% compared to DP, single-layer MPC, and ECMS strategies, respectively. Furthermore, effective power smoothing was achieved through coordinated energy storage system control. This research contributes a solution for fuel cell hybrid vessels that combines optimization performance with real-time operability, demonstrating substantial potential for engineering implementation.

To address the current limitations of this study, future research directions may focus on the following aspects:

• Development of adaptive multi-mode energy management strategies tailored to different voyage phases (departure, cruising, port entry, and berthing), enhancing system operational efficiency throughout the entire voyage.

• Development of adaptive power distribution strategies for multi-stack fuel cell systems that consider differential degradation rates among individual stacks, incorporating real-time performance monitoring and fault diagnosis capabilities to optimize both system efficiency and longevity.

• Development of load power prediction models integrating multidimensional information (vessel speed, heading, and sea conditions), improving prediction accuracy and robustness.

Author Contributions

Conceptualization, X.Z., X.Y., and M.Z.; methodology, X.Z.; software, Y.W. and L.L.; validation, X.Z. and X.Y.; formal analysis, S.N.; investigation, X.Z.; resources, X.Y.; data curation, C.Z.; writing—original draft preparation, X.Z. and X.Y.; writing—review and editing, X.Y. and M.Z.; visualization, L.L. and S.N.; supervision, X.Y. and M.Z.; project administration, X.Y. and M.Z.; funding acquisition, X.Y. and M.Z. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data will be made available on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Power system topology of the hybrid ship.

Figure 2. (a) Efficiency; (b) consumption rate.

Figure 3. Rint equivalent circuit diagram of battery.

Figure 4. Equivalent circuit of supercapacitor.

Figure 5. Fuel cell ship load power data.

Figure 6. The algorithm hierarchy of the proposed algorithm.

Figure 7. Energy management strategy framework.

Figure 8. CNN-LSTM based on attention mechanism model framework.

Figure 9. Load power prediction results.

Figure 10. Model Predictive Control (MPC) principle.

Figure 11. Low-pass filtering control strategy.

Figure 12. (a) Total fuel cell power; (b) lithium battery power; (c) supercapacitor power; (d) fuel cell power distribution.

Figure 13. (a) Lithium battery SOC; (b) supercapacitor SOC.

Figure 14. Comparison of life decay rates of dual power sources: (a) Proposed strategy; (b) DP strategy; (c) Single-layer MPC strategy; (d) ECMS strategy.

Figure 15. Stress analysis and comparison of power source under different strategies: (a) Proposed strategy; (b) DP strategy; (c) Single-layer MPC strategy; (d) ECMS strategy.

Table 1

Power device parameters.

Power Supply Device	Parameter	Value
Fuel cell	Rated power (kW)	120
Lithium battery	Capacity (Ah)	120
	Rated voltage (V)	500
	Initial SOC (%)	60
Supercapacitor	Capacity (kWh)	5
	Rated voltage (V)	500
	Initial SOC (%)	60

Table 2

Degradation rate of fuel cell.

Operating Conditions	Degradation Rate
Low power operation	${\mathrm{d}}_1=10.17\mathsf{\mu }\mathrm{V}/\mathrm{h}$
High power operation	${\mathrm{d}}_2=11.74\mathsf{\mu }\mathrm{V}/\mathrm{h}$
Variable load	${\mathrm{d}}_3=0.0441\mathsf{\mu }\mathrm{V}/\mathrm{k}\mathrm{W}$
On-Off	${\mathrm{d}}_4=23.91\mathsf{\mu }\mathrm{V}/\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}$

Table 3

Different time domain prediction results.

Prediction Time Domain	10 s	20 s	40 s
REMS	1.01	1.87	3.98
Average time per prediction(ms)	20.12	57.34	150.56

Table 4

Comparison of the results of each strategy.

Result	ProposedStrategy	DP Strategy	Single-Layer MPC	ECMS Strategy
Equivalent hydrogen consumption (kg)	10.9	10.5	10.9	10.9
Cost of FC degradation (CNY)	251.5	149.9	361.5	275.6
Battery degradation cost (CNY)	38.2	143.5	25.5	104.7
Total cost (CNY)	672.5	661.8	769.4	761.2
Difference in attenuation rate between FC and BAT (%)	0.0019%	0.101%	0.022%	0.054%
FC High frequency stress ($\sigma$)	0.1537	0.0782	0.242	0.2462
BAT High frequency stress ($\sigma$)	0.3593	0.4315	0.2466	0.2035

References

1. Wang, K.; Yan, X.; Yuan, Y.; Jiang, X.; Lin, X.; Negenborn, R.R. Dynamic Optimization of Ship Energy Efficiency Considering Time-Varying Environmental Factors. Transp. Res. Part D Transp. Environ.; 2018; 62, pp. 685-698. [DOI: https://dx.doi.org/10.1016/j.trd.2018.04.005]

2. Pan, P.; Sun, Y.; Yuan, C.; Yan, X.; Tang, X. Research Progress on Ship Power Systems Integrated with New Energy Sources: A Review. Renew. Sustain. Energy Rev.; 2021; 144, 111048. [DOI: https://dx.doi.org/10.1016/j.rser.2021.111048]

3. Qu, J.; Wang, H.; Zhou, J.; Zhang, B. Research on Hybrid Energy Management Strategy for Zero-Carbon Ships. Proceedings of the 2023 IEEE/IAS Industrial and Commercial Power System Asia (I&CPS Asia); Chongqing, China, 7 July 2023; pp. 837-843.

4. Becherif, M.; Claude, F.; Hervier, T.; Boulon, L. Multi-Stack Fuel Cells Powering a Vehicle. Energy Procedia; 2015; 74, pp. 308-319. [DOI: https://dx.doi.org/10.1016/j.egypro.2015.07.613]

5. Belmonte, N.; Luetto, C.; Staulo, S.; Rizzi, P.; Baricco, M. Case Studies of Energy Storage with Fuel Cells and Batteries for Stationary and Mobile Applications. Challenges; 2017; 8, 9. [DOI: https://dx.doi.org/10.3390/challe8010009]

6. Roh, G.; Kim, H.; Jeon, H.; Yoon, K. Fuel Consumption and CO2 Emission Reductions of Ships Powered by a Fuel-Cell-Based Hybrid Power Source. JMSE; 2019; 7, 230. [DOI: https://dx.doi.org/10.3390/jmse7070230]

7. Fan, A.; Li, Y.; Liu, H.; Yang, L.; Tian, Z.; Li, Y.; Vladimir, N. Development Trend and Hotspot Analysis of Ship Energy Management. J. Clean. Prod.; 2023; 389, 135899. [DOI: https://dx.doi.org/10.1016/j.jclepro.2023.135899]

8. Roslan, S.B.; Tay, Z.Y.; Konovessis, D.; Ang, J.H.; Menon, N.V. Rule-Based Control Studies of LNG–Battery Hybrid Tugboat. JMSE; 2023; 11, 1307. [DOI: https://dx.doi.org/10.3390/jmse11071307]

9. Liu, C.; Wang, Y.; Wang, L.; Chen, Z. Load-Adaptive Real-Time Energy Management Strategy for Battery/Ultracapacitor Hybrid Energy Storage System Using Dynamic Programming Optimization. J. Power Sources; 2019; 438, 227024. [DOI: https://dx.doi.org/10.1016/j.jpowsour.2019.227024]

10. Sun, C.; Sun, F.; He, H. Investigating Adaptive-ECMS with Velocity Forecast Ability for Hybrid Electric Vehicles. Appl. Energy; 2017; 185, pp. 1644-1653. [DOI: https://dx.doi.org/10.1016/j.apenergy.2016.02.026]

11. Zeng, Y.; Cai, Y.; Kou, G.; Gao, W.; Qin, D. Energy Management for Plug-In Hybrid Electric Vehicle Based on Adaptive Simplified-ECMS. Sustainability; 2018; 10, 2060. [DOI: https://dx.doi.org/10.3390/su10062060]

12. Ge, Y.; Zhang, J.; Zhou, K.; Zhu, J.; Wang, Y. Research on Energy Management for Ship Hybrid Power System Based on Adaptive Equivalent Consumption Minimization Strategy. JMSE; 2023; 11, 1271. [DOI: https://dx.doi.org/10.3390/jmse11071271]

13. Hou, J.; Sun, J.; Hofmann, H. Mitigating Power Fluctuations in Electrical Ship Propulsion Using Model Predictive Control with Hybrid Energy Storage System. Proceedings of the 2014 American Control Conference; Portland, OR, USA, 4–6 June 2014; pp. 4366-4371.

14. Chen, L.; Gao, D.; Xue, Q. Energy Management Strategy of Hybrid Ships Using Nonlinear Model Predictive Control via a Chaotic Grey Wolf Optimization Algorithm. JMSE; 2023; 11, 1834. [DOI: https://dx.doi.org/10.3390/jmse11091834]

15. Moghadari, M.; Kandidayeni, M.; Boulon, L.; Chaoui, H. Operating Cost Comparison of a Single-Stack and a Multi-Stack Hybrid Fuel Cell Vehicle Through an Online Hierarchical Strategy. IEEE Trans. Veh. Technol.; 2023; 72, pp. 267-279. [DOI: https://dx.doi.org/10.1109/TVT.2022.3205879]

16. Macias Fernandez, A.; Kandidayeni, M.; Boulon, L.; Chaoui, H. An Adaptive State Machine Based Energy Management Strategy for a Multi-Stack Fuel Cell Hybrid Electric Vehicle. IEEE Trans. Veh. Technol.; 2020; 69, pp. 220-234. [DOI: https://dx.doi.org/10.1109/TVT.2019.2950558]

17. Xie, P.; Asgharian, H.; Guerrero, J.M.; Vasquez, J.C.; Araya, S.S.; Liso, V. A Two-Layer Energy Management System for a Hybrid Electrical Passenger Ship with Multi-PEM Fuel Cell Stack. Int. J. Hydrogen Energy; 2023; 50, pp. 1005-1019. [DOI: https://dx.doi.org/10.1016/j.ijhydene.2023.09.297]

18. Kandidayeni, M.; Kelouwani, S.; Boulon, L.; Trovão, J.P. Designing a Hierarchical Energy Management Strategy for a Hybrid Multi-Stack Fuel Cell System. Proceedings of the 2023 IEEE Vehicle Power and Propulsion Conference (VPPC); Milan, Italy, 24 October 2023; pp. 1-5.

19. Ahmadi, P.; Torabi, S.H.; Afsaneh, H.; Sadegheih, Y.; Ganjehsarabi, H.; Ashjaee, M. The Effects of Driving Patterns and PEM Fuel Cell Degradation on the Lifecycle Assessment of Hydrogen Fuel Cell Vehicles. Int. J. Hydrogen Energy; 2020; 45, pp. 3595-3608. [DOI: https://dx.doi.org/10.1016/j.ijhydene.2019.01.165]

20. Gao, J.; Lan, H.; Cheng, P.; Hong, Y.-Y.; Yin, H. Optimal Scheduling of an Electric Propulsion Tugboat Considering Various Operating Conditions and Navigation Uncertainties. JMSE; 2022; 10, 1973. [DOI: https://dx.doi.org/10.3390/jmse10121973]

21. Fletcher, T.; Thring, R.; Watkinson, M. An Energy Management Strategy to Concurrently Optimise Fuel Consumption & PEM Fuel Cell Lifetime in a Hybrid Vehicle. Int. J. Hydrogen Energy; 2016; 41, pp. 21503-21515. [DOI: https://dx.doi.org/10.1016/j.ijhydene.2016.08.157]

22. Tang, L.; Rizzoni, G.; Onori, S. Energy Management Strategy for HEVs Including Battery Life Optimization. IEEE Trans. Transp. Electrific.; 2015; 1, pp. 211-222. [DOI: https://dx.doi.org/10.1109/TTE.2015.2471180]

23. Xu, B.; Oudalov, A.; Ulbig, A.; Andersson, G.; Kirschen, D.S. Modeling of Lithium-Ion Battery Degradation for Cell Life Assessment. IEEE Trans. Smart Grid; 2018; 9, pp. 1131-1140. [DOI: https://dx.doi.org/10.1109/TSG.2016.2578950]

24. Han, X.; Lu, L.; Zheng, Y.; Feng, X.; Li, Z.; Li, J.; Ouyang, M. A Review on the Key Issues of the Lithium Ion Battery Degradation among the Whole Life Cycle. eTransportation; 2019; 1, 100005. [DOI: https://dx.doi.org/10.1016/j.etran.2019.100005]

25. Şahin, M.; Blaabjerg, F.; Sangwongwanich, A. A Comprehensive Review on Supercapacitor Applications and Developments. Energies; 2022; 15, 674. [DOI: https://dx.doi.org/10.3390/en15030674]

26. Ren, P.; Pei, P.; Li, Y.; Wu, Z.; Chen, D.; Huang, S. Degradation Mechanisms of Proton Exchange Membrane Fuel Cell under Typical Automotive Operating Conditions. Prog. Energy Combust. Sci.; 2020; 80, 100859. [DOI: https://dx.doi.org/10.1016/j.pecs.2020.100859]

27. Schmittinger, W.; Vahidi, A. A Review of the Main Parameters Influencing Long-Term Performance and Durability of PEM Fuel Cells. J. Power Sources; 2008; 180, pp. 1-14. [DOI: https://dx.doi.org/10.1016/j.jpowsour.2008.01.070]

28. Wang, X.; Li, Q.; Wang, T.; Han, Y.; Chen, W. Optimized Energy Management Strategy Based on SQP Algorithm for PEMFC Hybrid Locomotive. Proceedings of the 2019 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific); Seogwipo-si, Republic of Korea, 8–10 May 2019; pp. 1-5.

29. Song, K.; Ding, Y.; Hu, X.; Xu, H.; Wang, Y.; Cao, J. Degradation Adaptive Energy Management Strategy Using Fuel Cell State-of-Health for Fuel Economy Improvement of Hybrid Electric Vehicle. Appl. Energy; 2021; 285, 116413. [DOI: https://dx.doi.org/10.1016/j.apenergy.2020.116413]

30. Ghaderi, R.; Kandidayeni, M.; Soleymani, M.; Boulon, L.; Trovao, J.P.F. Online Health-Conscious Energy Management Strategy for a Hybrid Multi-Stack Fuel Cell Vehicle Based on Game Theory. IEEE Trans. Veh. Technol.; 2022; 71, pp. 5704-5714. [DOI: https://dx.doi.org/10.1109/TVT.2022.3167319]

31. Wan, A.; Chang, Q.; AL-Bukhaiti, K.; He, J. Short-Term Power Load Forecasting for Combined Heat and Power Using CNN-LSTM Enhanced by Attention Mechanism. Energy; 2023; 282, 128274. [DOI: https://dx.doi.org/10.1016/j.energy.2023.128274]

32. Darby, M.L.; Nikolaou, M. MPC: Current Practice and Challenges. Control Eng. Pract.; 2012; 20, pp. 328-342. [DOI: https://dx.doi.org/10.1016/j.conengprac.2011.12.004]

33. Schwenzer, M.; Ay, M.; Bergs, T.; Abel, D. Review on Model Predictive Control: An Engineering Perspective. Int. J. Adv. Manuf. Technol.; 2021; 117, pp. 1327-1349. [DOI: https://dx.doi.org/10.1007/s00170-021-07682-3]

34. Truong, H.V.A.; Do, T.C.; Dang, T.D. Enhancing Efficiency in Hybrid Marine Vessels through a Multi-Layer Optimization Energy Management System. JMSE; 2024; 12, 1295. [DOI: https://dx.doi.org/10.3390/jmse12081295]

35. Lu, D. Online Optimization of Energy Management Strategy for FCV Control Parameters Considering Dual Power Source Lifespan Decay Synergy. Appl. Energy; 2023; 348, 121516. [DOI: https://dx.doi.org/10.1016/j.apenergy.2023.121516]

36. Wang, T.; Li, Q.; Wang, X.; Chen, W.; Breaz, E.; Gao, F. A Power Allocation Method for Multistack PEMFC System Considering Fuel Cell Performance Consistency. IEEE Trans. Ind. Appl.; 2020; 56, pp. 5340-5351. [DOI: https://dx.doi.org/10.1109/TIA.2020.3001254]