Multi-Objective Optimization of Energy Storage Configuration and Dispatch in Diesel-Electric Propulsion Ships

Abstract

This study investigates the configuration of an energy storage system (ESS) and the optimization of energy management strategies for diesel-electric hybrid ships, with the goal of enhancing fuel economy and reducing emissions. An integrated mathematical model of the diesel generator set and the battery-based ESS is established. A rule-based energy management strategy (EMS) is proposed, in which the ship operating conditions are classified into berthing, maneuvering, and cruising modes. This classification enables coordinated power allocation between the diesel generator set and the ESS, while ensuring that the diesel engine operates within its high-efficiency region. The optimization framework considers the number of battery modules in series and the upper and lower bounds of the state of charge (SOC) as design variables. The dual objectives are set as lifecycle cost (LCC) and greenhouse gas (GHG) emissions, optimized using the Multi-Objective Coati Optimization Algorithm (MOCOA). The algorithm achieves a balance between global exploration and local exploitation. Numerical simulations indicate that, under the LCC-optimal solution, fuel consumption and GHG emissions are reduced by 16.12% and 13.18%, respectively, while under the GHG-minimization solution, reductions of 37.84% in fuel consumption and 35.02% in emissions are achieved. Compared with conventional algorithms, including Multi-Objective Particle Swarm Optimization (MOPSO), Non-dominated Sorting Dung Beetle Optimizer (NSDBO), and Multi-Objective Sparrow Search Algorithm (MOSSA), MOCOA exhibits superior convergence and solution diversity. The findings provide valuable engineering insights into the optimal configuration of ESS and EMS for hybrid ships, thereby contributing to the advancement of green shipping.

1. Introduction

Maritime transport is a pillar of the global economy, carrying approximately 90% of international trade shipments, while also serving as a major source of carbon dioxide (CO₂) and other greenhouse gas (GHG) emissions. At present, diesel engines remain the dominant power source for ships, leading to significant annual emissions. It is estimated that by 2050, CO₂ emissions from the shipping industry will increase by 150–250% [1]. The resulting energy consumption and environmental challenges have attracted increasing global attention [2,3,4,5,6]. Meanwhile, the International Maritime Organization (IMO) has introduced progressively stricter emission regulations and energy efficiency standards, placing clear demands on the industry to improve fuel economy and reduce emissions. Against this backdrop, hybrid power systems have emerged as a promising development trend in maritime propulsion, with ESS playing an increasingly critical role. The adoption of clean energy combined with advanced energy efficiency optimization strategies has become an effective means to comply with international conventions and achieve emission reduction targets [7].

Nevertheless, conventional diesel-powered systems face inherent problems such as high emissions and low efficiency. To address these issues, the integration of a lithium-ion energy storage system (ESS) with diesel engines, forming Diesel–ESS hybrid power systems, has become an important development direction for ship propulsion [8]. Such systems can alleviate efficiency losses and emissions caused by diesel engines under high load conditions, but they also introduce complex energy management challenges. In particular, under variable load navigation scenarios, how to optimally coordinate the power distribution between diesel generators and ESS to achieve reduced fuel consumption and lower emissions has become a research hotspot [9].

In recent years, researchers worldwide have conducted extensive studies to address the environmental and energy challenges in shipping, aiming to reduce GHG emissions and enhance energy utilization efficiency [10,11,12,13,14]. Moreover, the exploration of diverse renewable and alternative energy sources for shipboard applications has expanded, including batteries, supercapacitors, photovoltaic systems, fuel cells, and flywheel storage [15,16,17]. These efforts provide a solid technological foundation for the optimization and scheduling of hybrid ship power systems.

For the capacity matching of ship energy storage systems, Li et al. [18] employed a strategy combining operational condition segmentation with empirical mode decomposition to investigate multiple configuration schemes. Experimental validation demonstrated that the error between the matched lithium-ion batteries and theoretical power values was approximately 0.8%, confirming the stability of lithium-ion battery output power. In a related study, Zhang et al. [19] proposed a two-stage model predictive control MPC-based energy management strategy for diesel–electric hybrid ships integrated with battery storage and supercapacitors, achieving a 17.2% reduction in fuel consumption.

Energy management strategies for hybrid propulsion vessels encompass rule-based [20,21,22], optimization [13,23,24,25,26,27,28,29], and learning-based approaches [13,20,21,22,23,24,25,26,27,28,29,30,31,32]. Rule-based Energy Management Strategies (EMS). These methods feature low computational cost, high real-time performance, and ease of implementation in practical ship applications. Wu et al. [20] addressed issues related to energy density, load response, and ESS integration by normalizing optimization variables and applying a rule-based EMS, achieving 17.1% and 14.8% reductions in fuel consumption and emissions, respectively. Wang et al. [21] combined a rule-based EMS with multi-objective optimization to jointly minimize fuel consumption, life cycle cost, and maximize all-electric navigation time. Bonkile et al. [22] applied a hybrid strategy combining MPPT and rule-based EMS in a PV–battery system, improving both storage performance and system lifetime.

Optimization-based EMS. These strategies rely on mathematical modeling of hybrid systems and employ optimization algorithms to determine the optimal energy allocation. Yang et al. [24] proposed a two-stage EMS combining Mixed-Integer Nonlinear Programming (MINLP) and Model Predictive Control (MPC). Validated using a diesel-electric hybrid tugboat, the system demonstrated a deviation not exceeding 0.456% compared to global optimization. Zhao [25] developed a DP-based fuzzy logic controller integrated with wavelet analysis and PI control to optimize the power output of a system comprising fuel cells, photovoltaics, batteries, and supercapacitors. Compared to rule-based strategies, hydrogen consumption was reduced by 14.39%. Zhu et al. [26] employed the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) for the dual-objective optimization of component sizing in plug-in hybrid systems. Compared to single-objective optimization, this approach achieved a 37.9% reduction in fuel consumption and a 4.6% decrease in emissions. [23] Hu et al. [23] employed dynamic programming to optimize power distribution in dual-engine lithium-ion battery systems. Simulations utilized a conventional single-engine hybrid vehicle, demonstrating that DP achieves a 7.3% reduction in fuel consumption compared to the Charge Discharge Cycle Strategy.

Xu et al. [13] applied an improved Particle Swarm Optimization (PSO) algorithm for ship energy scheduling, achieving a 7.459% reduction in total costs using Comprehensively Improved Particle Swarm Optimization (CIPSO) and an 8.072% reduction in GHG emissions using Multi-populations Particle Swarm Optimization MPPSO. For the various energy storage forms employed in ship hybrid propulsion systems, Vu et al. [28] combined the ramp rate characteristics of different energy sources with a receding horizon optimization (RHO) algorithm coupled with Gradient Descent (GD). Compared to Fixed Horizon Optimization (FHO), this approach achieved execution speeds four times faster, thereby alleviating the burden of real-time optimization. Haseltalab et al. [27] integrated solid oxide fuel cells (SOFC) into a marine propulsion system. By establishing a simulation model of the ship’s electrical system through an optimization-based approach, they achieved a 53% reduction in carbon dioxide emissions and a 21% improvement in fuel efficiency compared to conventional diesel-electric vessels. Liu et al. [29] employed a hierarchical distributed electromagnetic control approach, conducting model-based in-the-loop and hardware-in-the-loop simulations. Their experiments demonstrated that, compared to DP-MPC-based strategies, the FDP-MPC-based approach achieved 99.89% energy savings and emission reductions.

Learning-based Energy Management Strategies. Reinforcement learning (RL) and deep learning have been increasingly adopted in recent studies. However, Deep Q-Network (DQN) methods often suffer from Q-value overestimation and poor convergence. To address this, Xiao et al. [30] proposed an improved DRL algorithm (DQN-CE), achieving a 4.11% cost reduction, a 24.4% increase in ESS participation, and a 31.3% reduction in training time. Song et al. [31] employed a Hybrid-Penalty Proximal Policy Optimization (HPO3) algorithm, formulating the scheduling problem as a constrained Markov Decision Process (MDP) to achieve near-optimal scheduling effectiveness. This approach achieved a nearly optimal scheduling performance, surpassing the global optimum by 1.22%, while PSO and GA fell short by 6.55% and 7.11% respectively. Shang et al. [32] reformulated the scheduling problem within a reinforcement learning framework, proposing an enhanced DRL approach termed the DQN-CE algorithm. This achieved a 4.11% reduction in economic costs, a 24.4% increase in energy storage system utilization, and a 31.3% decrease in training time.

Although previous studies have made progress in improving ship energy efficiency and reducing GHG emissions, several limitations and challenges remain:

Lack of systematic ESS sizing optimization: In most studies, ESS capacity configuration is determined empirically or based on a single objective, without fully considering the multi-objective trade-off between LCC and GHG emissions.

Insufficient application of multi-objective optimization algorithms: Although intelligent optimization methods such as Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) have been introduced, challenges remain in multi-objective coupled optimization (cost–emissions), including slow convergence, uneven solution set distribution, and susceptibility to local optima.

Limited adaptability to real operating conditions: Most existing studies focus on simplified scenarios or single-voyage conditions, lacking validation under variable load, long-distance navigation, and environmental disturbances, thereby limiting the reliability and generalizability of optimization results.

To address these gaps, this study proposes a joint optimization method for ESS sizing and energy scheduling, combining a rule-based EMS with the MOCOA. The main contributions are as follows:

Combining ESS capacity matching, rule-based EMS, and multi-objective optimization algorithms to achieve an integrated optimization framework.

A mathematical model for ship hybrid propulsion systems has been established, with simulation based on real-world vessel navigation data enhancing its engineering application value.

Employing the novel optimization algorithm MOCOA, a comparative analysis was conducted against MOPSO, NSDBO, and MOSSA. The results demonstrated improvements of at least 6.30% and 0.087% in the $IGD$ and $HV$ metrics, respectively.

By adopting dual-objective optimization of LCC and GHG emissions, fuel consumption and GHG emissions can be reduced by at least 16.12% and 13.18% respectively.

2. Hybrid Energy System Modeling and Dispatch Strategy

The Hybrid Electric Propulsion System (HEPS) consists of a diesel generator set, ESS, auxiliary units, and a propulsion system. The mathematical model provides the foundation for analyzing energy flow and optimization. In practical ship operations, maintaining power balance is essential for achieving energy optimization. In this study, mathematical modeling is established to describe the key components of the HEPS. As illustrated in Figure 1, the model simulates the generation and flow of energy during real operating conditions. The energy generation units, located on the upper side of the power bus, include two identical diesel engines, two identical generators, a battery system, and shore power. On the lower side of the power bus, the energy consumption units comprise the auxiliary load and propulsion load.

Figure 1 presents the hybrid propulsion system using the training ship Xin Hongzhuan of Dalian Maritime University as a case study. This medium-sized diesel-electric vessel (length ~69.8 m, displacement ~1450 tons, service speed 18 kn) is equipped with two 1500 kW podded propulsors and three 1520 kW diesel-electric generators. For the simulation experiments, two diesel generator sets were considered, and thus only two generator sets are shown in the figure. Such vessels are commonly used for training and research, where load fluctuations are significant, making them well-suited for hybrid diesel-ESS systems to improve efficiency and reduce emissions.

2.1. Diesel Engine Model

The accuracy of diesel engine modeling varies considerably across different studies. In this study, a mathematical model of the diesel engine was developed using brake-specific fuel consumption (BSFC) data from the Wärtsilä 8L20 engine. The data were obtained from the manufacturer’s Factory Acceptance Test (FAT) under various load levels (see Table A1). This model is employed to calculate the fuel mass flow rate ${\dot{m}}_{fuel}$ and to identify the optimal operating load range of the engine. BSFC is a key parameter for evaluating the fuel consumption of diesel generator sets, and it is commonly approximated by fitting the hourly fuel consumption of the engine with a quadratic polynomial function [33]. As the primary power source of fully electric-propulsion ships, the diesel generator plays a decisive role in overall energy efficiency. Its fuel consumption characteristics are described by the Specific Fuel Consumption (SFC) curve [20,34,35,36]. In the simulation process, the performance data curves were fitted and rationalized using multiple data points. The results indicate that within the main operating load range (60–95%), the use of manufacturer-provided performance data does not significantly affect the reliability of the research conclusions. The relationship can be expressed as:

(1)$SFC=0.01283P^2-2.1645P+277.67$

(2)$P={\displaystyle \frac{P_D(t)}{P_{\max }}}\times 100$

(3)${\dot{m}}_{fuel}={\displaystyle \frac{P_D(t)\cdot SFC}{1000}}$

where $P$ represents the engine load percentage; $P_D\left(t\right)$ denotes the diesel engine output power; $P_{\max }$ is the maximum output power of the diesel engine; and ${\dot{m}}_{fuel}$ is the mass flow rate of fuel consumption.

Figure 2 illustrates the fitted SFC curve. The results indicate that the diesel engine achieves relatively high efficiency when operating within a load range of 60–95%, with the maximum fuel efficiency occurring at an 85% load ratio. Operating the engine near this optimal load condition enables maximum fuel savings and significantly enhances fuel economy.

2.2. Generator Model

The permanent magnet synchronous generator is coupled with the diesel engine to form the diesel generator set (DG) for generating electrical power. The specifications of the generator are matched with those of the diesel engine to ensure reliable and efficient operation of the system. For each synchronous generator, the output power $P_{DG}(t)$ can be expressed as [37]:

(4)$P_{DG}(t)={\eta }_G(t)P_D(t)$

(5)${\eta }_G(t)=-0.08078{\left({\displaystyle \frac{P_D(t)}{1600}}\right)}^2+0.16142\left({\displaystyle \frac{P_D(t)}{1600}}\right)+0.89483$

where ${\eta }_G(t)$ is the generator efficiency, and the fitted data used is shown in Table A2. This data originates from experimental measurements conducted at the shipyard. $P_{DG}(t)$ is the output power of the DG set.

2.3. Energy Storage System

This study focuses on the configuration of the onboard energy storage system and the optimization of energy scheduling, without involving the electrochemical reactions within the battery. Accordingly, an equivalent circuit model is employed to characterize the behavior of the battery. In this research, a lithium-ion battery pack is selected due to its high energy density, long cycle life, low self-discharge rate, high reliability, and superior efficiency [34]. The ESS is composed of multiple battery modules connected in series, where each module consists of 50 cells connected in parallel. The key aspect of ESS matching is to determine the number of series-connected modules $N_{bat}$, which directly determines the system’s overall voltage level and capacity. The SOC of the ESS at time t is calculated as [21,37]:

(6)$SOC(t)=SO{C}_0(t)-{\displaystyle \int \frac{I_{bat}(t)}{Q_{bat}}}$

(7)$I_{bat}(t)={\displaystyle \frac{V_{OC}(t)}{2R_{bat}(t)}}-\sqrt{{\left({\displaystyle \frac{V_{OC}(t)}{2R_{bat}(t)}}\right)}^2-{\displaystyle \frac{P_{ESS}(t)}{R_{bat}(t)}}}$

(8)$Q_{bat}=50\cdot Q_{cell}$

(9)$V_{OC}(t)=V_{cell}(t)N_{bat}$

(10)$R_{bat}(t)=N_{bat}{R}_{bat}/50$

where $V_{cell}(t)$ and $R_{cell}$ represent the open-circuit voltage and internal resistance of the battery cell, respectively. $Q_{cell}$ denotes the capacity of a single cell, while $Q_{bat}$ is the total battery capacity. $N_{bat}$ is the number of battery modules connected in series. $V_{OC}(t)$ is the open-circuit voltage of the ESS. $R_{bat}$ is the internal resistance of the energy storage system. $SO{C}_0(t)$ is the initial state of charge, $I_{bat}(t)$ is the instantaneous battery current, and $P_{ESS}(t)$ represents the charging/discharging power of the ESS. ESS charge/discharge efficiency is set to 0.98.

For the study of shipboard ESS configuration, the parameters of the selected lithium-ion batteries are listed in Table A3, which [38].

2.4. Bidirectional Converter

In hybrid electric propulsion systems, the bidirectional DC converter (Bi-DC) with a charging controller serves as a critical interface that enables energy exchange between the battery storage system and the power bus. The Bi-DC operates in two distinct modes:

Inverter mode: converts DC power into AC power to supply the propulsion or auxiliary loads.

Rectifier mode: converts AC power into DC power to charge the battery energy storage system.

The integrated charging control function is essential to prevent overcharging or over-discharging of the battery pack, thereby prolonging battery life and ensuring system reliability [34]. The rated power of the bidirectional converter is determined based on the maximum charging/discharging requirement of the ESS, and can be expressed as:

(11)$P_{Bi-DC}=1.1P_{ESS,\max }$

where $P_{Bi-DC}$ is maximum power throughput of the converter, $P_{ESS,\max }$ is maximum charge/discharge power of the ESS, 1.1 is converter overload capacity factor (10% overload capability, with reasonable assumptions made for the purpose of simulation studies). The bidirectional converter conversion loss was set to 0.98.

2.5. Ship Load Model

To validate the effectiveness of the proposed methodology, actual propulsion load, auxiliary load, and navigation speed data of the target vessel (As shown in Figure 3) were obtained from its sea trial database. The data were sampled at a time interval of 300 s, covering an operational duration of approximately 83 h, and were employed as inputs for the simulation analysis. Accordingly, no additional mathematical modeling of ship loads was required.

In comparison to deriving load profiles through ship dynamics modeling, employing measured operational data more accurately reflects the real operating conditions of the vessel. For hybrid electric propulsion ships, such measured operational data not only enhances the accuracy and reliability of the ESS sizing and optimization but also provides valuable engineering references for future retrofitting projects. Therefore, the derivation of ship dynamics-based load models is not pursued in this study.

2.6. Rule-Based EMS

The core of energy dispatch optimization lies in achieving coordinated power allocation between the DG and the ESS. In this study, a rule-based EMS is adopted to determine power distribution according to the ship’s actual load demand and operational conditions. The strategy operates by classifying ship speeds and identifying operating modes [20], which are divided into three categories according to the measured navigation speed:

Berthing (0–3 knots): Auxiliary loads are mainly supplied by the ESS, while the DG is either shut down or maintained at minimal load.

Maneuvering (3–5 knots): Both the DG and ESS operate jointly to ensure power stability under frequent load fluctuations.

Cruising (>5 knots): The DG provides the main propulsion power, while the ESS performs peak shaving and maintains SOC within an appropriate range.

The measured ship speed profile of the target vessel, as shown in Figure 4, covers the above operating modes and provides a practical basis for implementing the proposed EMS.

The number of operating diesel generator sets is dictated by the ship’s total power demand. When berthed, the vessel primarily utilizes shore power for its main energy supply. The ESS is charged by shore power until the SOC reaches 80%, while the diesel generator sets remain shut down. In the maneuvering condition, one DG set is started as a standby unit, while the load demand is mainly supplied by the ESS. When the ship enters the cruising condition, the power demand is jointly supplied by the diesel generator sets and the ESS, with the ESS output regulated according to the SOC. When both the diesel generators and the ESS share the load, the diesel generators operate according to the SFC curve, ensuring that their load distribution remains in the high-efficiency region. As long as the SOC remains within the defined limits, the ESS provides the residual power demand. If the SOC falls below the lower threshold, a second DG set is started and operated near its rated power, while simultaneously charging the battery. Once the SOC reaches the upper threshold, the load demand is supplied jointly by a single DG and the ESS. The detailed operation of this rule-based EMS is illustrated in Figure 5.

3. Multi-Optimization Implementation

For the optimization of the ESS capacity, this study evaluates the trade-offs among different configuration cases using a bi-objective framework with the LCC and GHG emissions as the objective functions. The optimization problem is solved using a heuristic multi-objective optimization algorithm. The formulation includes decision variables, objective functions, and constraints, as defined in the following subsections.

3.1. Decision Variables

To account for the impact of integrating the ESS into the hybrid propulsion system, three decision variables are introduced:

Number of series-connected battery modules $N_{bat}$, which determines the nominal voltage and total capacity of the ESS;

Lower limit of state of charge $SO{C}_{\min }$, to prevent deep discharge and extend battery life;

Upper limit of state of charge $SO{C}_{\max }$, to avoid overcharging and ensure safe operation.

The decision variable vector is expressed as:

$X=\left\{N_{bat},SO{C}_{\min },SO{C}_{\max }\right\}$

3.2. Objective Functions

This study evaluates the performance of different energy storage system configurations using two metrics: GHG emissions and LCC. These are represented by the objective functions $f_1(X)$ and $f_2(X)$, defined as follows:

(12)$f_1(X)=GH{G}_{voy}(X)$

(13)$f_2(X)=LCC(X)$

where $f_1(X)$ quantifies the total GHG emissions over the operational life cycle of the hybrid propulsion system, which are influenced by the diesel generator efficiency and ESS dispatch strategy.$f_2(X)$ represents the LCC of the ESS, composed of the installation investment cost, the power system maintenance cost, and the energy consumption cost. The objective function can be expressed as:

$\min \{f_1(X),f_2(X)\}$

3.2.1. Greenhouse Gas Emission Function

GHG emissions consist of the emissions generated by fuel consumption during navigation, as well as the emissions associated with shore power usage during charging. The total amount of GHG emissions over the voyage can be expressed as [26,37]:

(14)$f_1(X)=GH{G}_{voy}=E_{fuel}{G}_{fuel}+E_{ele}{G}_{ele}$

(15)$E_{fuel}={\displaystyle \frac{m_{fuel}{L}_{CV}}{F_{kWh\_J}}}$

(16)$m_{fuel}={\displaystyle \int {\dot{m}}_{fuel}dt}$

where $E_{fuel}$ and $E_{ele}$ represent the energy consumed by diesel and shore power during navigation. $G_{fuel}$ and $G_{ele}$ represent the GHG emission factors for diesel consumption and shore power generation [20], respectively. $L_{CV}$ represents the Lower Heating Value of diesel, and $m_{fuel}$ denotes the fuel consumption mass.

3.2.2. Life-Cycle Cost Function

The LCC primarily consists of three components: the installation investment cost of the matched ESS, the maintenance cost of the power system, and the energy consumption cost. The mathematical formulation is expressed as follows [13,18,21,34]:

(17)$f_2(X)=C_{inv}+C_{mai}+C_{ene}$

The energy storage component of the hybrid system is formed by connecting individual battery cells in parallel into modules, which are then connected in series. An increase in the number of series-connected modules leads to a higher initial investment cost. Considering the finite life cycle of lithium-ion batteries, it is necessary to account for periodic replacements, and the associated replacement costs should be incorporated into the economic assessment. Therefore, taking into account the initial installation cost, subsequent replacement cost, and replacement frequency, the mathematical formulation of the installation investment cost can be expressed as follows:

(18)$C_{inv}=50C_{price}{N}_{bat}\left[1+{{\displaystyle {\sum }_{i=1}^{n_{re}}\left(\frac{1+g_{bat}}{1+I_a}\right)}}^{i\cdot T_{bat}}\right]$

(19)$n_{re}=\mathrm{int}\left[{\displaystyle \frac{Y_{ear}}{T_{bat}}}\right]$

where $C_{inv}$ is represents the installation investment cost of the energy storage system. $C_{price}$ is the price of the battery cell, $I_a$ is the annual interest rate, $g_{bat}$ is the annual inflation rate for battery prices, $Y_{ear}$ is the design life of the ship, and $T_{bat}$ is the battery life cycle. $n_{re}$ is the number of replacements, rounded to the nearest integer.

The mathematical formulation of the maintenance cost $C_{mai}$ is expressed as:

(20)$C_{mai}=50{\displaystyle {\sum }_{i=1}^{Y_{ear}}{c}_{mai}{N}_{bat}{C}_{price}}$

The annual maintenance cost coefficient of the power system, denoted as $c_{mai}$, is assumed to be 2% of each installation cost [34].

The energy consumption cost, denoted as $C_{ele}$, refers to the shore power replenishment when the ship is berthed. Regarding shore power charging, the following assumptions are made: The shore power charging cycle is fixed at 100 times per year, with each charge replenishing 50% of the total capacity of the energy storage system.

(21)$C_{ele}=100\cdot C_e\cdot Y_{ear}\cdot 0.5\cdot E_{ESS}$

(22)$E_{ESS}=Q_{bat}\cdot V_{cell}\cdot N_{bat}$

where $C_e$ represents the price of shore power, while $E_{ESS}$ denotes the total energy capacity of the energy storage system.

3.3. Constraints

3.3.1. Power Balance Constraint

The power supplied from the energy sources must remain balanced with the total load demand of the vessel. Specifically, the overall power demand of the ship, denoted as $P_{req}(t)$, should equal the power generated on the supply side. The power is provided jointly by the DG and the ESS. The total ship load demand consists of propulsion load $P_{pro}(t)$ and auxiliary load $P_{aux}(t)$. The mathematical formulation of the power balance between generation and demand is expressed as:

(23)$P_{pro}(t)+P_{aux}(t)=P_{req}(t)$

(24)$P_{req}(t)=P_{DG,1}(t)+P_{DG,2}(t)+P_{ESS}(t)$

where $P_{DG,1}(t)$ signifies the output power of diesel generator set No. 1, $P_{DG,2}(t)$ denotes the output power of diesel generator set No. 2, and $P_{ESS}(t)$ indicates the output power of the ESS.

3.3.2. Constraints of Diesel Generator Sets

In the actual operation of ships, it is necessary to impose restrictions on the output power of diesel generator sets in order to avoid sudden load increases or decreases, which may reduce the service life of the diesel generator sets [13].

(25)$P_{DG}^{\min }\le P_{DG,i}(t)\le P_{DG}^{rated}$

(26)$-R_{DG}^{\max }\le R_{DG,i}(t)\le R_{DG}^{\max }$

where $P_{DG,i}(t)$ and $R_{DG,i}(t)$ denote the output power and the power ramp rate of the i-th diesel generator at time $t$, respectively. $R_{DG,i}^{\max }$ represents the maximum permissible ramp rate of the generator. To ensure that the diesel engine operates within its high-efficiency region, $P_{DG}^{\min }$ and $P_{DG}^{rated}$ are defined as the minimum output power and the rated power of the diesel engine, respectively.

3.3.3. ESS Constraints

To ensure that the ESS maintains stable power output throughout the entire operational cycle of the vessel and prolongs its service life, constraints are imposed on the variation range of the SOC and the output power of the ESS. The mathematical formulations are expressed as follows:

(27)$SO{C}_{\min }(t)\le SOC(t)\le SO{C}_{\max }(t)$

(28)$-P_{ESS,\max }\le P_{ESS}(t)\le P_{ESS,\max }$

where $SO{C}_{\min }(t)$ and $SO{C}_{\max }(t)$ represent the lower and upper bounds of the SOC of the ESS, the scope definitions of both are based on assumptions, respectively, while $P_{ESS,\max }$ denotes the maximum output power of the ESS.

To ensure that the battery system can be effectively installed on the vessel, the number of series-connected battery modules is constrained based on the vessel’s physical space and weight-bearing capacity limitations. The mathematical expression is given as:

(29)$50\le N_{bat}\le 200$

3.4. Algorithm Principle

Heuristic multi-objective optimization algorithms have demonstrated excellent performance in solving complex engineering optimization problems. In this study, the MOCOA, which simulates the foraging and cleaning behaviors of coatis, is employed [39,40,41,42]. The algorithm iteratively alternates between two phases: exploration and exploitation, thereby balancing global search and local optimization. By incorporating pareto dominance relations, crowding distance, and grid partitioning mechanisms, MOCOA ensures both diversity and uniformity of the solution set. Through a non-dominated solution archive, the algorithm outputs an approximate Pareto front upon termination.

Exploration Phase: The foraging behavior of coatis is simulated. Each individual has two possible strategies: (i) randomly selecting a solution from the archive as a reference, (ii) searching within the population for a candidate solution superior to the current one and randomly selecting it as a reference. This phase emphasizes global search, enabling individuals to escape from local regions through randomness and reference solutions, thereby avoiding premature convergence [39].

(30)$X_i^{k+1}=X_{res}+\alpha (X_{res}-X_{best,i})+\gamma (X_{rand}-X_i^k)$

where $X_i^k$ is the position of the i-th individual at iteration $k$, $X_{res}$ is the reference solution selected from the archive or the population, $X_{best,i}$ is the Personal best solution of the i-th individual, $X_{rand}$ is the randomly generated solution in the decision space. The exploration factor $\alpha$ is set within the interval [0, 0.5] to prevent excessive dispersion while maintaining convergence. The random perturbation factor $\gamma$ is sampled from a uniform distribution within the interval [0, 1], thereby enhancing solution set diversity and reducing the risk of getting stuck in local optima. This constitutes a common configuration for meta-heuristic algorithms.

Exploitation Phase: A solution is randomly selected from the archive as a resource for the cleaning behavior. As the number of iterations increases, the algorithm gradually transitions from the exploration phase to the exploitation phase, focusing on local refinement.

(31)$X_i^{k+1}=X_{best,i}+\beta \left(X_{ref}-X_i^k\right)$

(32)$\beta =0.5+0.5\times \left(1-{\displaystyle \frac{k}{K}}\right)$

where $X_{ref}$ is the reference solution selected from the archive, $\beta$ is the cleaning factor that decreasing linearly with iteration count from 1 to 0.5. The purpose of linear decay is to maintain robust global search and perturbation capabilities during the early stages of the algorithm, while progressively enhancing local convergence in later phases, thereby striking a balance between exploration and exploitation. $K$ denotes the maximum number of iterations. $k$ denotes the number of iterations.

In MOCOA, each individual in the population represents a candidate design of the hybrid power system, with its decision vector defined as:

$X=\left\{N_{bat},SO{C}_{\min },SO{C}_{\max }\right\}$

For each candidate solution, the ship’s power system is simulated using the measured navigation load data to evaluate two objective functions: LCC and GHG. The feasibility of a solution is jointly determined by the constraints of power balance, diesel engine ramp rate, SOC dynamic range, and installation limits on volume and weight.

During the iterative process, the exploration phase of MOCOA perturbs $N_{bat}$ and the SOC boundary values to search for better battery configurations and operating ranges. The exploitation phase then progressively refines feasible solutions by guiding the diesel engine toward its high-efficiency operating range while optimizing the battery’s charging and discharging strategy. Ultimately, the algorithm achieves a balanced Pareto-optimal solution set between LCC and GHG, providing a robust basis for subsequent scheme selection.

As illustrated in Figure 6, the adopted optimization algorithm employs a dual-phase updating mechanism, which ensures strong global search capability during the early iterations and robust local convergence ability in the later iterations. By integrating the crowding distance and grid-based mechanisms, the algorithm achieves both uniformity and diversity of the Pareto-optimal solutions. Furthermore, the adaptive cleaning factor effectively balances the exploration and exploitation phases, thereby preventing premature convergence. The archive trimming strategy retains boundary and sparsely distributed solutions, which enhances the practical usability of the obtained Pareto front.

3.5. Optimization Framework

Based on the established mathematical models, constraints, and optimization algorithm, this study develops a closed-loop optimization framework for battery sizing in diesel-electric hybrid ships. Within the proposed optimization framework, three core components—battery capacity design, the EMS, and MOCOA—operate in a hierarchical and collaborative manner. The first tier comprises battery capacity design, which determines key parameters of the ESS, including capacity, output power, and the SOC operating range. The second tier comprises the EMS, which dynamically coordinates power allocation between the diesel engine and ESS across varying operating conditions. This ensures system compliance with operational requirements while adhering to constraints imposed by the battery design. The third tier, MOCOA, serves as the global optimization mechanism. It iteratively adjusts battery capacity design variables within EMS constraints, thereby achieving dual-objective optimization of LCC and GHG emissions. Through this hierarchical interaction, the framework establishes a closed-loop “design-operation-optimization” process: design parameters constrain operational strategies, operational feedback influences the optimization process, and the optimization algorithm further refines design variables. This collaborative mechanism ensures the framework balances short-term operational efficiency with long-term economic and environmental optimization, thereby enhancing its engineering applicability in shipboard energy management.

The rule-based EMS dynamically allocates energy based on load demand, battery SOC status, and the ship’s operating conditions. Specifically, it enables the use of the battery at low loads, triggers the diesel generator when SOC approaches its lower bound, and adjusts the diesel generator’s operating point to remain within its high-efficiency region.

The MOCOA optimization algorithm is then applied within this framework to balance life-cycle cost and GHG emissions, generating a set of optimal solutions. The overall implementation process is illustrated in Figure 7.

The three components together form an integrated framework of parameter design, operational control, and global optimization. Through this approach, the hybrid power system of the ship can achieve stable and safe operation, while simultaneously ensuring fuel economy and environmental sustainability over its life cycle.

4. Algorithm Validation and Case Study

In this section, the performance of the proposed MOCOA multi-objective optimization algorithm is evaluated in comparison with other optimization methods, such as MOPSO. Various benchmark functions and performance metrics are employed in the simulation tests for comparative analysis. Furthermore, the optimization problem defined in the previous section is solved using these algorithms, where the measured propulsion and auxiliary load data of the target vessel operating on the Dalian–Shenzhen route are utilized. Based on these data, the study focuses on the optimal sizing of the energy storage system and the development of a rule-based energy management strategy.

4.1. Multi-Objective Algorithm Comparative Analysis

The performance of the MOCOA algorithm is further evaluated against NSDBO, MOSSA, and MOPSO. Two widely used performance metrics, Inverted Generational Distance ($IGD$) and Hypervolume ($HV$), were employed to assess the convergence, diversity, and coverage of the obtained solution sets. The benchmark functions $ZDT1$, $ZDT3$ and $ZDT4$ [43] were selected as the test problems. These three functions exhibit complementary characteristics: $ZDT1$ examines convergence through convex Pareto frontier analysis, $ZDT3$ employs discontinuous frontier evaluation to measure diversity, while $ZDT4$ assesses robustness in scenarios featuring multiple local optima. Their combined application ensures balanced and representative validation of algorithms.

(33)$\begin{array}{l}ZDT1=\left\{\begin{array}{l}{f}_1(x)=x_1\\ f_2(x)=g(x)(1-\sqrt{{\displaystyle \frac{f_1(x)}{g(x)}}})\\ g(x)=1+{\displaystyle \frac{9}{n-1}}{\displaystyle {\sum }_{i=2}^n{x}_i}\end{array}\right.\\ s.t.0\le x_i\le 1,i=2,3,\dots ,30\end{array}$

(34)$\begin{array}{l}ZDT3=\left\{\begin{array}{l}{f}_1(x)=x_1\\ f_2(x)=g(x)\left(1-\sqrt{{\displaystyle \frac{f_1(x)}{g(x)}}}-{\displaystyle \frac{f_1(x)}{g(x)}}\sin (10\pi f_1(x))\right)\\ g(x)=1+{\displaystyle \frac{9}{n-1}}{\displaystyle {\sum }_{i=2}^n{x}_i}\end{array}\right.\\ s.t.0\le x_i\le 1,i=2,3,\dots ,30\end{array}$

(35)$\begin{array}{l}ZDT4=\left\{\begin{array}{l}{f}_1(x)=x_1\\ f_2(x)=g(x)(1-\sqrt{{\displaystyle \frac{f_1(x)}{g(x)}}})\\ g(x)=1+10(n-1)+{\displaystyle {\sum }_{i=2}^n(x_i^2-10\cos (4\pi x_i))}\end{array}\right.\\ s.t.0\le x_1\le 1,-5\le x_i\le 5,i=2,3,\dots ,10\end{array}$

$IGD$: The $IGD$ metric evaluates the convergence and distribution of the obtained Pareto front approximation A relative to the true Pareto front $A^{\ast }$. It is defined as [43]:

(36)$IGD(A,P^{*})={\displaystyle \frac{1}{\left|P^{*}\right|}}{\displaystyle \sum _{v\in P^{*}}d(v,A)}$

where $d(v,A)$ represents the minimum Euclidean distance between a reference point $v\in P^{\ast }$ and the obtained solution set $A$. A smaller $IGD$ value indicates better convergence and diversity.

$HV$: The $HV$ indicator measures the volume of the objective space dominated by the solution set $A$ and bounded by a reference point $r$. It is formulated as [43]:

(37)$HV(A)=volume\left({\displaystyle \underset{x\in A}{\cup }\left[f_1(x),r_1\right]\times \left[f_2(x),r_2\right]\times \cdots \times \left[f_m(x),r_m\right]}\right)$

where $f_i(x)$ is the i-th objective value of solution $x$, and $r$ is a predefined reference point. A larger $HV$ value represents a better spread and coverage of the Pareto front.

In the simulation experiments, all algorithms were configured with the same iteration number, initial population size, and archive capacity. Each test was repeated 20 times, and the average values were adopted as the final results. The obtained solution sets and their corresponding true Pareto fronts for the three test functions are shown in Figure 8a–c.

Figure 9a–f illustrates the comparative results of the four algorithms in terms of $IGD$ and $HV$. For the $IGD$ metric, MOCOA achieves substantial improvements over the other three algorithms, reducing the $IGD$ value by at least 8.50%, 12.83%, and 6.30%, respectively. This demonstrates that MOCOA yields solutions closer to the true Pareto front. Regarding the $HV$ metric, MOCOA also shows superior performance, with improvements of at least 0.097%, 0.087%, and 0.18%, respectively. The higher $HV$ values confirm that MOCOA provides better solution coverage and distribution in the objective space.

In summary, MOCOA demonstrates significant superiority over NSDBO, MOSSA, and MOPSO in terms of convergence and diversity maintenance. The comparative results validate the effectiveness and robustness of MOCOA in addressing multi-objective optimization problems.

The problem of energy storage capacity sizing and scheduling optimization in diesel-electric ships is essentially a constrained multi-objective optimization problem, requiring the simultaneous minimization of life-cycle cost (LCC) and GHG emissions (GHG) under a nonlinear system. The superior convergence and solution diversity demonstrated by MOCOA in benchmark test functions indicate its ability to maintain global search capability while avoiding premature convergence, which is critical for identifying high-quality trade-off solutions in engineering applications. Moreover, the more uniform Pareto solution set provides a wider range of design options, enabling decision-makers to flexibly choose according to different priorities between cost and emission reduction. Therefore, the benchmark results not only validate the performance advantage of the algorithm but also provide strong support for its effectiveness in the ship application studied in this work.

4.2. Case Study: Application to Hybrid Diesel-Electric Ships

This section applies MOCOA to the optimal design of hybrid diesel-electric ship systems. The focus is placed on battery sizing, rule-based energy management strategies, and overall system performance evaluation. By integrating MOCOA into the design and optimization framework, the study aims to balance LCC and GHG emissions. This ensures both economic and environmental benefits throughout the ship’s operation.

4.2.1. Ship Load Data

To investigate the impact of different energy storage system configurations on LCC and GHG emissions, actual ship load data during navigation is used for energy management simulation of the ship’s power system. The ship load is divided into propulsion load and auxiliary load, with the propulsion load being the most critical component of the total energy demand. When applying the energy management strategy, the objective is to dynamically coordinate the power output of diesel generator sets and batteries to meet the energy requirements of both the propeller and onboard service loads. At the same time, the strategy seeks to minimize fuel consumption and emission levels. Analysis of the target vessel’s actual operational load data indicates that the selected voyage segment covers typical operating conditions, including departure, port approach, and cruising at sea.

Figure 10 illustrates the distribution of propulsion and auxiliary loads under different operating conditions, including departure, port approach, and cruising at sea.

4.2.2. Configuration Case Analysis

In this study, the heuristic multi-objective optimization algorithm MOCOA is employed to address the capacity configuration of the ESS and the optimization of power allocation among different energy sources. The parameters of the optimization algorithm are set as follows: population size $N_p=100$, repository size $N_r=80$, maximum number of generations $N_{\max }=150$, and number of grids per dimension $N_r=30$.

Based on these settings, the MOCOA algorithm is applied to optimize the system under varying load demands during the voyage. The resulting optimization outcomes are represented as Pareto front solutions, which clearly illustrate the trade-off relationship between LCC and GHG emissions. These Pareto fronts provide a valuable reference for evaluating the rationality of different storage capacity configuration cases. The obtained Pareto fronts are shown in the following figure.

As shown in Figure 11, the optimization algorithm reveals the impact of different configurations on the objective functions. In the process of determining the battery capacity for ship energy storage system matching, six representative configuration cases are selected from the Pareto front for further analysis and comparison. Representativeness refers to the fact that the selected cases cover different regions of the Pareto front, reflecting the trade-off characteristics between LCC and GHG emissions.

Specifically, the six chosen cases include the LCC-optimal solution, the GHG-optimal solution, a compromise solution, and three intermediate solutions with varying ESS capacities and SOC constraints. This selection ensures that both extreme optima and more practically feasible intermediate schemes are considered. Given the large number of Pareto solutions, analyzing each one individually would be redundant and reduce readability; therefore, six representative cases are deemed sufficient to capture the overall trend in this study. To address the influence of different weightings of LCC and GHG emissions on energy storage system capacity configuration and EMS [38], the weights are equally assigned as $\alpha =0.5$ to achieve a balance between economic viability and environmental sustainability to prevent any single objective from dominating the optimization process. The detailed parameters and performance indicators of the six selected cases are presented in Table 1.

This table summarizes the main parameters of six representative energy storage system configurations obtained from the optimization algorithm, along with their corresponding LCCs and GHG emissions. The comparison provides useful insights for capacity matching and energy management in practical ship applications. LCC and GHG exhibit sensitivity to fuel prices, battery lifespan, and operational conditions. Rising fuel costs substantially enhance the economic value of fuel savings; reduced battery lifespan significantly increases replacement expenses, undermining lifecycle economics; operational conditions (such as load fluctuations and docking periods) directly impact ESS peak shaving and emission reduction efficacy. Within the equation system, fuel consumption parameters (SFC curve), battery capacity parameters ($Q_{bat}$, $N_{bat}$), and SOC constraints ($SO{C}_{\min }$, $SO{C}_{\max }$) are paramount to the results, constituting the core factors determining the balance between cost and emissions reduction.

4.2.3. Analysis of Costs and Greenhouse Gas Emissions

LCC of the energy storage system is primarily composed of three components: the initial investment cost, maintenance cost, and energy consumption cost. GHG emissions consist of those generated from diesel fuel consumption during navigation, as well as indirect emissions from shore power charging when the energy storage system is integrated.

Figure 12 illustrates the variation in LCC across different configuration schemes, whilst Figure 13 compares the total GHG emissions for each scheme over an identical voyage distance. Collectively, these results provide a clear assessment of the trade-offs between economic and environmental benefits for various energy storage configuration options.

The variation in energy storage system capacity exerts a significant influence on the distribution of LCCs. A comparative analysis of the six configuration cases is summarized as follows:

Case 1: Features the smallest storage capacity and the lowest LCC, only about one-third of Case 2. However, its GHG emissions are 33.61% higher than those of Case 2. Case 2: Has the largest storage capacity among the six cases, resulting in the highest LCC. Although emissions are minimized, its economic performance is poor. Cases 3 and 4: Show similar capacities and LCC. Compared with the system without lithium battery integration, their GHG emissions are reduced by 15.3% and 19.3%, respectively. Notably, Case 4 achieves an additional reduction of 5208 kg in emissions compared with Case 3, but incurs an extra 200,000 CNY in costs. Cases 5 and 6: Exhibit comparable GHG reduction, achieving 28.66% and 29.63% lower emissions than the system without storage, respectively. However, Case 5 increases the LCC by approximately 4.18 million CNY, making it significantly less economical than Case 6.

The divergence between LCC and GHG optimization outcomes arises from the dual influence of battery capacity on both diesel engine operating range and investment cost. Medium-capacity configurations tend to be more economical, whereas larger capacities yield superior environmental performance. For LCC, the key cost drivers include the initial investment, replacement, and maintenance expenses of the ESS. A large-capacity ESS substantially increases upfront installation costs as well as replacement costs over the vessel’s lifetime. Although greater storage capacity reduces fuel consumption, the cost of shore power replenishment partially offsets the economic benefits. Consequently, the LCC-optimal solution corresponds to a medium-capacity ESS, which balances fuel savings against battery investment. For example, Case 1 achieves a 16.12% reduction in fuel consumption and a 13.18% reduction in GHG emissions. In contrast, GHG emissions are more sensitive to operational efficiency than to cost. By mitigating load fluctuations, the ESS reduces the time diesel generators operate in low-efficiency zones, enabling them to remain longer within their high-efficiency range. Additionally, the use of shore power during port stays further decreases emissions. In this regard, larger ESS capacities are advantageous, as they extend the duration of high-efficiency operation. However, once the load profile has been sufficiently smoothed, the marginal emission reduction effect diminishes. Therefore, the GHG-optimal solution favors a large-capacity ESS configuration. For instance, Case 2 achieves a 37.84% reduction in fuel consumption and a 35.02% reduction in emissions, albeit at a substantially higher capital cost.

4.2.4. Fuel Consumption Analysis

In the optimization of energy storage system configuration and energy management strategies for diesel-electric ships, analyzing fuel consumption during voyages is equally essential. By comparing the six selected configuration cases, the influence of different storage configurations on fuel consumption can be further evaluated, providing a reference for improving fuel economy. The specific fuel consumption profiles for each case during the voyage are shown in Figure 14.

As illustrated in the figure, integrating an energy storage system leads to a noticeable reduction in fuel consumption during the voyage compared with the system without storage. Specifically, Case 2 achieves a reduction of approximately 12.3 tons compared to Case 1, and 15.2 tons compared to the configuration without lithium battery storage. However, relative to Cases 5 and 6, the reduction is only 0.37 tons and 2.5 tons, respectively, indicating a certain similarity in fuel-saving performance among these three cases. In addition, Cases 1, 3, and 4, which feature similar total storage capacities, achieve fuel savings of 6.5 tons, 7.3 tons, and 8.9 tons, respectively. This highlights that variations in storage system capacity have a significant influence on the effectiveness of fuel consumption reduction.

In Cases 2 and 5, although the ESS capacities differ significantly, both configurations provide sufficient capacity to accommodate most load fluctuations, and the improvement in diesel generator efficiency has essentially reached saturation. Further increasing the ESS capacity (as in Case 5) yields only marginal benefits in reducing low-load operation. As a result, despite the difference in battery size, the diesel generators in both cases operate at a similarly efficient level, leading to comparable fuel consumption outcomes. This finding indicates that, in practical ship design, a larger ESS capacity is not always better; rather, a balance must be struck among investment cost, emission reduction, and efficiency improvement.

4.2.5. Characteristics of Different Configuration Cases

As discussed earlier, GHG emissions during ship operation primarily originate from two sources: (1) fuel consumption of diesel generators during voyages and (2) indirect emissions from shore power charging. The differences in emission reduction effectiveness across various ESS configurations are largely determined by the variation in these two emission sources.

Meanwhile, fuel consumption is primarily attributed to the diesel generators operating under different load conditions. A properly sized ESS can supply power during low-load periods, reducing the operating time or output levels of diesel generators. This ultimately decreases both fuel consumption and emissions. Conversely, when the ESS capacity is insufficient, the initial investment and LCC are lower, but the reduction in fuel consumption and emissions is limited.

The impacts of different configuration cases on GHG emissions and fuel consumption are illustrated in Figure 15. It can be observed that as the ESS capacity increases, fuel consumption is gradually reduced, leading to a significant decrease in GHG emissions. However, this effect does not increase linearly; once a certain capacity is reached, the marginal benefits of fuel savings and emission reduction diminish. The differences highlight the trade-off between economic performance and environmental benefits in ESS configuration.

As shown in Figure 16, different configuration cases generally demonstrate that fuel savings become more significant with increasing LCC. However, Cases 5 and 6 exhibit an opposite trend. Analysis of the Pareto front indicates that although higher lifecycle investments tend to reduce both GHG emissions and fuel consumption, under certain ESS capacity configurations, applying the same energy management strategy may lead to higher fuel consumption and emissions despite increased investment.

According to simulation results, we recommend controlling the ESS capacity within the range of 2–5 MWh during the matching process for energy storage systems in medium-sized diesel-electric vessels. Within this range, a balance between LCC and GHG emissions can be achieved. Beyond this interval, fuel savings tend towards saturation while investment costs rise significantly. Recommended State of Charge (SOC) limits are: lower bound 0.10–0.30, upper bound 0.80–0.95. This approach effectively extends battery life while maintaining operational flexibility. Diesel generator sets should operate within an 80% to 95% load range. When SOC falls below the lower limit, the second unit should be activated, and battery charging should commence. During berthing operations, shore power should be prioritized to charge the energy storage system to 80% capacity, thereby reducing subsequent diesel engine start-stop cycles and emissions during maneuvering phases.

5. Conclusions

This study addresses the energy optimization problem of diesel-electric ships by establishing a hybrid propulsion system model that integrates diesel generator sets, an ESS, and onboard loads. A rule-based energy EMS was proposed and coupled with the novel MOCOA to achieve joint optimization of ESS sizing and energy scheduling.

Different configuration options present a clear trade-off between cost-effectiveness and emission reduction. For scenarios where economic efficiency is the primary objective, the life-cycle cost–optimal configuration (Case 1) is recommended. If the focus is on meeting stricter emission reduction requirements, the emission-optimized configuration (Case 2) is preferable. Case 5, which provides a more balanced compromise, is recommended for practical engineering applications. Despite these promising results, certain limitations remain. The following research will combine the following directions: Incorporating Reinforcement Learning (RL), Deep Reinforcement Learning (DRL), or Model Predictive Control (MPC) to enhance the intelligence and adaptability of EMS. MPC, RL, and DRL demonstrate significant potential for ship energy management in addressing non-linearity, uncertainty, and multi-objective optimization. MPC possesses robustness and constraint-handling capabilities, making it suitable for real-time scheduling. RL and DRL exhibit strong adaptive capabilities but require substantial data and training; Integrating battery degradation mechanisms and lifetime prediction models into the optimization framework for more realistic lifecycle assessments.

In summary, the proposed ESS sizing and energy scheduling optimization method not only enriches the theoretical research framework for hybrid ship energy optimization but also provides practical insights for real-world applications.

Author Contributions

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

Data Availability Statement

The Dataset is available on request from the authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ESS	Energy storage system
GHG	Greenhouse gas
EMS	Energy Management Strategy
SOC	State of charge
MPC	Model Predictive Control
LCC	Lifecycle cost
IMO	International Maritime Organization
HEPS	Hybrid Electric Propulsion System
BSFC	Brake-specific fuel consumption
SFC	Specific Fuel Consumption
DG	Diesel generator
IGD	Inverted Generational Distance
HV	Hypervolume
MOCOA	Multi-Objective Coati Optimization Algorithm
MOPSO	Multi-Objective Particle Swarm Optimization
NSDBO	Non-dominated Sorting Dung Beetle Optimizer
MOSSA	Multi-Objective Sparrow Search Algorithm
RL	Reinforcement Learning
DRL	Deep Reinforcement Learning

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 Table

Figure 1 The architecture of the HEPS.

Figure 2 Diesel Engine SFC Curve.

Figure 3 The training ship Xin Hongzhuan of Dalian Maritime University.

Figure 4 Speed profile throughout the trip.

Figure 5 Rule-based Energy Management Strategy.

Figure 6 Algorithm flowchart.

Figure 7 Optimization framework.

Figure 8 Comparison of solution sets and true Pareto fronts of test functions under different optimization algorithms.

Figure 9 Comparison of Test Functions Based on IGD and HV Metrics.

Figure 10 Load profile throughout the trip.

Figure 11 Comparison of optimization results.

Figure 12 Lifecycle cost of different cases.

Figure 13 Greenhouse gas emissions during the voyage.

Figure 14 Fuel consumption during the voyage.

Figure 15 Comparison of greenhouse gas emission reductions under different scenarios.

Figure 16 Comparison of fuel reduction between different cases.

Table 1

Parameters and performance indicators of six representative configuration cases.

Parameters	${\mathit{N}}_{\mathit{b}\mathit{a}\mathit{t}}$	$\mathit{S}\mathit{O}{\mathit{C}}_{\mathbf{min}}$	$\mathit{S}\mathit{O}{\mathit{C}}_{\mathbf{max}}$	${\mathit{E}}_{\mathit{E}\mathit{S}\mathit{S}}$
Case 1	50	0.177	0.886	2240
Case 2	148	0.171	0.901	6630
Case 3	55	0.124	0.915	2464
Case 4	56	0.130	0.906	2509
Case 5	108	0.177	0.899	4838
Case 6	93	0.122	0.923	4166

Appendix A

Table A1

BSFC test data for diesel engines.

Load (%)	BSFC FAT (g/kWh)
50	201.5
75	187.3
85	186.5
100	189.4

Table A2

Generator efficiency test data.

Shaft Load (%)	η
25	0.9250
50	0.9490
75	0.9600
85	0.9620
100	0.9635

Table A3

Main parameters of the energy storage device.

Parameter	Li-Ion Battery (Signal)
Manufacturer	EVE
Type	LF 280K (1P52S)
Capacity	280 (Ah)
Rate Voltage	3.2 (V)
Internal Resistance	2.5 × 10−3 (Ω)
Continuous Charge/Discharge Current	1C (A)
Size (L× W × H)	0.173 × 0.072 × 0.204 (m × m × m)
Price of Unit	725 (¥)

Table A4

The key parameter values used in the simulation [21,26].

Parameter	Value	Parameter	Value
$L_{CV}$	42.7 × 106 (J/kg)	$F_{kWh\_J}$	3.6 × 106
$G_{fuel}$	0.266 (kg/kWh)	$C_{price}$	725 (¥/cell)
$G_{ele}$	0.32 (kg/kWh)	$Y_{ear}$	30 (year)
$C_e$	0.60 (¥/kWh)	$P_{DG}^{rated}$	1520 (kW)
$T_{bat}$	8 (year)	$R_{DG}^{\max }$	500 (kW)
$g_{bat}$	5 (%)	$SO{C}_{\min }$	[0.10, 0.50]
$I_a$	2 (%)	$SO{C}_{\max }$	[0.60, 0.95]
$P_{DG}^{\min }$	960 (kW)	$P_{ESS,\max }$	1500 (kW)
$P_{\max }$	1600 (kW)	$SO{C}_0$	0.80

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