Optimization of Sailing Speed for Inland Electric Ships Based on an Improved Multi-Objective Particle Swarm Optimization (MOPSO) Algorithm

Abstract

Sailing speed is a critical factor affecting the ship’s energy consumption and operating costs for a voyage. Inland waterways present a complex navigation environment due to their narrow channels, numerous curved segments, and significant variations in water depth and flow speed. This paper constructs a model of a ship’s energy consumption based on an analysis of ship resistance and the energy transfer relationship of ships. The K-means clustering algorithm is introduced to divide the Yangtze River waterway into multiple segments based on the similarity of navigation environments. Considering the constraints of the ship’s main engine and the desired arrival time, a multi-objective particle swarm optimization (MOPSO) algorithm, improved with cosine decreasing inertial weight and Gaussian random mutation, is employed to optimize segmented navigation speeds to achieve different goals. Finally, four cases are studied with a fully electric ship navigating the reaches of the Yangtze River. The results indicate that the optimized speed can reduce ship energy consumption by up to 6.18% and significantly reduce ship energy consumption and operational costs under different conditions.

1. Introduction

1.1. Background

Shipping is one of the most important modes of trade transport, accounting for more than 80% of international trade. However, with the development of maritime trade and the increase in ship tonnage, energy consumption and air pollution problems have become increasingly severe. The Fourth International Maritime Organization (IMO) GHG Study 2020 estimated that GHG emissions from shipping in 2018 accounted for some 2.89% of global anthropogenic GHG emissions and that such emissions could represent between 90% and 130% of 2008 emissions by 2050 [1]. The 80th session of the IMO Marine Environment Protection Committee (MEPC 80) proposed that by 2030, the annual greenhouse gas emissions from international maritime trade should be reduced by at least 20%, with an aim to achieve a 30% reduction compared to 2008 levels [2].

Strategies for energy saving and emission reduction in ships mainly include hull design optimization, operational optimization, and the application of new green energy. Hull design optimization involves optimizing the shape of the hull through hydrodynamic design before ship building to minimize resistance and improve seaworthiness [3]. In addition to the optimization of the hull design, operational optimization is also a cost-effective and quick method, including speed optimization, route optimization, operational scheduling optimization, and trim optimization [4].

Among various optimization strategies, including ship hull optimization, loading condition optimization, and sailing speed optimization,  sailing speed optimization is the most direct and effective method. By reasonably adjusting the sailing speed, fuel consumption can be significantly reduced, and greenhouse gas emissions can be accordingly decreased. And speed optimization can further enhance the energy-saving and emission-reduction effects based on hull optimization. Greenhouse gas emissions are approximately proportional to fuel consumption, and fuel consumption per unit time increases with higher speeds. Studies have shown that the fuel consumption per unit time is approximately a cubic function of speed [5]. Fuel costs account for more than 60% of total operating costs, greatly affecting the economic benefits and market competitiveness of shipping companies [6]. Higher speeds enable faster arrival at ports, reducing inventory costs of goods in transit and increasing trade volume per unit of time. However, this also leads to increased fuel consumption and gas emissions [7]. Therefore, considering the current competitive shipping market and rising fuel prices, shipowners or charterers often adopt slow steaming strategies. Nevertheless, extremely low speeds can lead to engine stalling, and ships must arrive at ports within stipulated times. Hence, the optimization of the  sailing speed is a complex nonlinear optimization problem with multiple objectives and constraints.

The optimization of sailing speed during navigation depends on factors such as shipping mode, ship payload, fuel costs, inventory costs of goods, freight rates, and weather conditions [8]. The Maersk Line conducted slow steaming trials with over 100 container ships, confirming that ships can operate normally with engine loads reduced to 10%, corresponding to about half of the design speed [9]. Wen et al. developed a constrained programming model to solve multi-ship routing and speed optimization under time, cost, and environmental constraints but did not consider weather factors [7]. Considering weather factors, meteorological ship route optimization mainly focuses on the impact of wind and waves on navigation, aiming to find the optimal route and speed to minimize operating costs, fuel consumption, and navigation risks [10]. Moreover, speed optimization also needs to consider maritime regulations. IMO has established four emission control areas (ECAs), including the Baltic Sea, the North Sea, the English Channel and all European ports, the North American coast, and the US Caribbean coast. Within the areas of ECAs, ships must use more expensive fuels with a sulfur content of less than 1%, a standard reduced to 0.5% from 2020 onwards. The Chinese government has established the Yangtze River Delta ECA, stipulating that ships entering this control area must use fuel with a sulfur content of less than 0.1% after 1 January 2022. Many scholars have conducted speed optimization and route planning studies to mitigate the adverse effects of ECAs [11,12,13]. Although inland ships and ocean-going ships both aim to reduce fuel consumption and operational costs, the factors influencing optimization are significantly different. The speed optimization of inland ships considers complex and variable environmental conditions such as fluctuating water depths, water flow, narrow waterways, and navigation rules specific to inland waterways. In contrast, the speed optimization of ocean-going ships focuses more on weather conditions, the impact of ECAs, and long-distance route planning. Overall, most current research on speed optimization concentrates on ocean-going ships, while research in the field of speed optimization for inland ships is relatively scarce.

In the context of the significant development of the shipping industry and the emphasis on energy saving and emission reduction, all-electric ships (AESs) represent the future trend. Currently, an increasing number of AES are being put into use. However, research on how to save electrical energy for inland AES is still insufficient. Studying the speed optimization for AESs in the complex inland waterway environment to reduce energy consumption is of great significance for lowering inland shipping costs and reducing environmental pollution.

1.2. Related Works

Scholars have conducted many studies to optimize sailing speed, reduce ships’ operating costs, and lower emissions. Wang and Meng develop a mixed-integer nonlinear programming model to study speed optimization in the container shipping network, taking routes into account [14]. Du et al. established an artificial neural network model based on ships’ navigation report data, which minimizes fuel consumption by adjusting the ship’s speed and trim [15]. Norstad et al. consider speed as an input to study the scheduling and speed optimization of tramp shipping, significantly improving the profitability of shipping companies [16]. Lashgari et al. developed a stochastic linear integer programming model to jointly optimize route decisions, speed optimization, and refueling strategies in liner shipping, aiming to reduce total costs [17]. These studies demonstrate the effectiveness of various optimization models and techniques in improving fuel efficiency and operational costs for maritime transportation.

Other scholars have considered the impact of ECA regulations on costs and emissions in their speed optimization studies. By analyzing actual ship route data, Fagerholt et al. find that ship operators often choose to detour longer routes to avoid ECA. Additionally, ships tend to sail at reduced speeds within ECA and at higher speeds outside ECA [18]. Ma et al. optimize both ship routes and speeds while considering the impact of ECA, finding that reducing emissions within ECA might increase total voyage emissions [19]. Zhen et al. develop a dual-objective model to minimize total fuel costs and SO₂ emissions, optimizing routes and speeds both inside and outside ECAs [20]. These studies indicate that while it is possible to reduce emissions within ECAs through various methods, they may lead to higher overall emissions and operational costs. Other studies have focused on speed optimization based on weather conditions and Estimated Time of Arrival (ETA). ETA is determined according to the navigation plan, and within the allowable range of ETA, the ship’s route planning needs to meet the requirements of low energy consumption and short navigation time as much as possible [21]. To address the issue of decreased accuracy in long-term weather forecasting, Tzortzis and Sakalis propose a method that divides the total time of a fixed route into several smaller time intervals to dynamically determine the speed for each segment [22]. Du et al. thoroughly consider meteorological conditions, ship main engine power, and navigation safety constraints, analyzing the differences in speed, ETA, energy consumption, and emissions on multiple pre-set routes, and providing different optimal routes based on navigation objectives [23]. These studies have contributed to speed optimization through model construction, combining relevant maritime regulations and ship operations. However, their focus and scenarios are mainly on ocean-going ships, and there is currently little research on speed optimization for inland ships.

Unlike ocean navigation, inland waterways are narrow, with significant variations in water depth and flow speed, presenting a more complex navigation environment. Some scholars have conducted research on ship energy efficiency and speed optimization by considering the characteristics of inland waterways. Wang et al. consider the time-varying nature of inland environmental factors (wind, water flow, water depth, etc.), use a Model Predictive Control (MPC) method to design an optimized controller, testing the optimal speed under time-varying environmental factors with different time steps. This approach significantly improves ship energy efficiency and reduces fuel consumption and CO₂ emissions compared to traditional static optimization methods [6]. Yan et al. consider the effects of wind speed, wind direction, water depth, and flow speed, employing a distributed parallel K-means algorithm to achieve finer segmentation of inland navigation segments and determine the optimal engine speed for each segment, ultimately improving ship energy efficiency [24]. Fan et al. use speed as a decision variable and construct a multi-stage speed optimization model considering environmental factors, showing that flow speed and direction are key factors affecting the navigation speed of ships on the Yangtze River [25]. Yuan et al. use a density-based spatial clustering method to cluster environmental information such as wind speed, wind direction, water level, and water flow on the Yangtze River trunk line, segmenting the Yangtze River waterway to reduce the impact of the highly variable inland navigation environment on speed optimization [26].

With the increasing density and performance of large-capacity lithium-ion batteries, the application of electric propulsion ships has become possible. Although fully electric ships cannot currently meet the energy demands of ocean navigation, they can be used for port-to-port transportation and berthing tasks in shorter inland voyages with the advent of high-capacity marine batteries and the increase in the number of charging stations along inland waterways [27]. Fan et al. analyze the green decarbonization needs of China’s inland ships and the characteristics of new power systems, proposing a solution to replace traditional power systems with electric propulsion [28]. Zhang et al. consider environmental factors such as cargo load, flow speed, and water depth, constructing an energy-consumption model based on the energy transfer process of electric ships and optimizing the speed and energy under battery swapping mode, reducing battery replacement costs [29]. Sun et al. consider the dynamic changes in electricity prices and the complex navigation environment of inland waterways to establish a speed and energy optimization model, minimizing ship operating costs. They conducted a case study on AES in inland navigation, providing a reference for the operation and management of inland electric propulsion ships [30]. These scholars’ studies have contributed to the development of inland AES, but overall, the application of and research on inland AES ships are still relatively limited. More comprehensive and in-depth research on AES for inland navigation needs to be conducted on green, low-carbon, and efficient development of inland shipping.

1.3. Major Contribution

To address the current deficiencies in inland waterway shipping research, particularly in the context of energy saving, emission reduction, and the rise of new energy sources, this paper studies an electric propulsion ship on the Yangtze River and constructs an energy-consumption model. This model considers the complex environment of inland waterways, navigation constraints, and the energy transfer relationship of electric ships. The Yangtze River waterway is segmented based on environmental similarity using the K-means clustering algorithm to improve optimization accuracy. Subsequently, the optimal speed for each segment is solved by using an improved multi-objective particle swarm optimization (MOPSO) algorithm, and the effectiveness of this method is validated through comparative experiments. The main contributions of this paper are as follows:

• A dynamic energy-consumption model for AES is built, which comprehensively considers the inland waterway environment, navigation constraints, navigation effect (e.g., shallow water), characteristics of the electric ship’s motors, and the energy transfer relationship.
• Automatic waterway segmenting for sailing speed is realized by introducing the K-means clustering algorithm.
• An improved MOPSO algorithm is proposed, which solves the multi-objective speed optimization problem through Pareto optimal solutions, significantly reducing ship energy consumption and operational costs.

1.4. Structure

In Section 2, we determine the ship-resistance calculation methods under different environmental factors by analyzing the inland waterway environment and construct a ship energy-consumption model based on the ship energy transfer relationship. Section 3 introduces a K-means clustering segmentation method for inland waterways and provides a detailed explanation of the improvement strategies for the MOPSO algorithm. In Section 4, we conduct four simulation case studies to test the optimization capabilities of the MOPSO algorithm in multi-objective optimization, with different ship loads, and under different ETA conditions, thereby verifying the algorithm’s effectiveness. Section 5 summarizes the research work presented in this paper and outlines future research directions.

2. Ship Energy Consumption Modeling

2.1. Analysis of Inland Navigation Environment

As important waterways for transportation, inland waterways possess unique characteristics and complexities in their navigation environment. In inland waterway transportation, there may be various risk factors such as narrow channels, insufficient water depth, and changing water flow [31]. Ships constantly navigate through ever-changing environments, making it difficult to predict their operational status under persistent disturbances [32]. To optimize the navigation of inland ships, it is necessary to thoroughly analyze the inland navigation environment. The following provides a detailed analysis of the inland navigation environment in terms of aspects such as meteorological and hydrological conditions and waterway characteristics.

The water depth and flow speed of inland waterways varies significantly, which is influenced greatly by seasonal changes and rainfall. During the dry season, the water depth is relatively shallow, while in the flood season, the water depth is much deeper. Flow speed is a key factor affecting speed and maneuverability; it is generally higher in the upper reaches of a river and gradually slows down in the middle and lower reaches. Meteorological conditions are an important factor affecting inland navigation, with wind speed and direction directly impacting the ship’s speed and stability. Figure 1 summarizes the key factors that need to be considered for optimizing the navigation of inland ships.

Inland waterways are usually narrow with numerous bends, and the meteorological and hydrological conditions vary significantly across different segments. Therefore, it is necessary to conduct a detailed analysis of the inland navigation environment and divide the waterway into several segments based on environmental characteristics. Furthermore, the navigation rules are adjusted according to the different navigation scenarios. For instance, the sailing speed of ships in the bridge area is limited to guarantee navigation safety. The lower limit of sailing speed is also regulated to avoid traffic congestion. Under different navigation conditions, ships may need to adjust their sailing strategies to achieve various objectives. The MOPSO algorithm can handle complex multi-objective optimization problems, providing better navigation plans for inland ships. Resistance is a crucial factor affecting ships’ energy consumption. Therefore, before conducting speed optimization, it is essential to first analyze the factors influencing the resistance of inland ships.

2.2. Ship Resistance Analysis

During the navigation of a ship, the resistance generated by air and water against the hull must be overcome by the main engine power to propel the ship forward. The primary components of ship resistance include calm water resistance, wind resistance, and wave resistance. To estimate ship resistance, scholars have proposed various methods, mainly including Computational Fluid Dynamics (CFD) numerical simulation methods, empirical formula methods, model test chart methods, and parent ship methods. For inland ships, calm water resistance is typically calculated using the Zvankov formula [33], as shown in Equation (1).

$$R_0=\left(0.17S{V}^{1.83}+\xi C_B{A}_M{V}^{1.7+4F_r}\right)g$$

(1)

where $R_0$ represents ship calm resistance, S is the wetted surface area of the ship (m²), V is the speed through the water (m/s), $C_B$ is the block coefficient, $A_M$ is the midship section area (m²), g is the gravitational acceleration (m/s²), $\xi$ is the residual resistance coefficient, and $F_r$ is the Froude number, as shown in Equations (2) and (3).

$$F_r={\displaystyle \frac{V}{\sqrt{gL}}}$$

(2)

$$\xi ={\displaystyle \frac{1.77C_B^{2.5}}{{\left(\frac{L}{6B}\right)}^3+2}}$$

(3)

where L (m) represents the length of the ship an B (m) represents the ship’s breadth.

The wind resistance on the superstructure of the ship during navigation can be described by the average wind pressure on the hull, as shown in Equation (4).

$$R_{\mathrm{wind}}={\displaystyle \frac{1}{2}}{C}_{\mathrm{wind}}{\rho }_{\mathrm{air}}{A}_T{V}_{\mathrm{wind}}^2$$

(4)

where $C_{\mathrm{wind}}$ represents the air resistance coefficient, ${\rho }_{\mathrm{air}}$ represents the air density, $A_T$ (m/s²) represents the windward area, and $V_{\mathrm{wind}}$ (m/s) represents the relative wind speed.

The wave resistance experienced by the ship is shown in Equation (5)

$$R_{\mathrm{wave}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{0.065}{{\left(F_r\right)}^2}}{\left({\displaystyle \frac{H}{L_{\mathrm{wl}}}}\right)}^2{\rho }_{\mathrm{water}}S{V}^2$$

(5)

where H (m) represents the wave height, $L_{\mathrm{wl}}$ (m) denotes the waterline length of the ship, and ${\rho }_{\mathrm{water}}$ (10³ kg/m³) represents the water density.

Therefore, the total resistance R experienced by the ship is shown in Equation (6).

$$R=R_0+R_{\mathrm{wind}}+R_{\mathrm{wave}}$$

(6)

2.3. Ship Engine–Propeller Energy Transfer Relationship

The principle of ship propulsion can be simply described as the main engine generating power, which is transmitted through the transmission equipment and shaft system to the propeller. The propeller then produces thrust to overcome the resistance encountered by the ship, allowing the ship to maintain a certain speed during navigation.

The methods for calculating propeller thrust T and torque Q are shown in Equation (7).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}T=(1-t)K_T\times \rho n^2{D}^4\hfill \\ Q=K_Q\times \rho n^2{D}^5\hfill \end{array}\right.\end{array}$$

(7)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{K}_T=f_T\left(J\right)\hfill \\ K_Q=f_Q\left(J\right)\hfill \end{array}\right.\end{array}$$

(8)

where $K_T$ is the propeller thrust coefficient, $K_Q$ is the torque coefficient, n is the propeller speed, D is the diameter of the propeller, and J is the advance coefficient, as shown in Equation (9).

$$J={\displaystyle \frac{(1-w)\times V}{n\times D}}$$

(9)

where w represents the wake fraction coefficient. For twin-propeller, twin-rudder ships, w needs to account for the interaction between the propellers, as shown in Equation (10).

$$w={\displaystyle \frac{w_{p0}\times exp(-4{\beta }_p^2)}{r^{\prime }}}$$

(10)

where $w_{p0}$ is the initial coefficient, ${\beta }_p$ is the initial phase angle of the propeller, and $r^{\prime }$ is the adjustment coefficient, taking a positive value for the right propeller and a negative value for the left propeller.

The effective power of the ship is the power required for the ship to overcome resistance and advance at a certain speed. Effective power can be expressed as the product of the ship’s resistance R and speed V. Due to losses during power transmission from the main engine, the effective power is generally less than the main engine power. The main engine power $P_{\mathrm{B}}$ is shown in Equation (11).

$$P_{\mathrm{B}}={\displaystyle \frac{R\times V}{{\eta }_S{\eta }_O{\eta }_R{\eta }_H}}$$

(11)

where ${\eta }_S$ represents the shaft transmission efficiency, ${\eta }_R$ represents the relative rotating efficiency; ${\eta }_O$ represents the open water propeller efficiency, and ${\eta }_H$ represents the hull efficiency, as shown in Equations (12) and (13).

$${\eta }_H={\displaystyle \frac{1-t}{1-w}}$$

(12)

$${\eta }_O={\displaystyle \frac{K_T}{K_Q}}\times {\displaystyle \frac{J}{2\pi }}$$

(13)

Therefore, $P_{\mathrm{B}}$ can be expressed as Equation (14).

$$P_{\mathrm{B}}={\displaystyle \frac{2\pi nQ}{{\eta }_S{\eta }_R}}$$

(14)

The ship’s energy consumption E is shown in Equation (15).

$$E=\sum _{i=1}^n{P}_{\mathrm{B}i}\times \Delta T_i$$

(15)

where $\Delta T_i$ represents the sailing time of each segment.

3. Segmented Speed Optimization Based on an Improved Multi-Objective Particle Swarm Optimization Algorithm

3.1. Inland Waterway Segmentation Method Using K-Means Clustering Method

The K-means algorithm is a commonly used unsupervised machine learning algorithm primarily used for data clustering analysis. Its core idea is to partition a dataset into K clusters through iterative optimization, minimizing the sum of distances from each data point to its respective cluster center. The basic principle is illustrated in Algorithm 1.

In the long-distance speed optimization of inland ships, it is crucial to segment the waterway reasonably to better manage and optimize the navigation process. The K-means clustering method can effectively segment inland waterways [34]. First, environmental information such as the geographic location, water depth, flow speed, wind speed, and direction is obtained, and  these environmental factors are selected as clustering features. To eliminate the differences in dimensions of different features, the Z-score formula is used for standardization, as shown in Equation (16).

$$Z={\displaystyle \frac{X-\mu }{\sigma }}$$

(16)

where X is the original data, i.e., the clustering features listed above; $\mu$ is the mean of each feature; and $\sigma$ is the standard deviation of each feature.

Algorithm 1 K-means Algorithm

Initialize: Randomly select K points from X as initial centroids $\mu$ Set $iter=0$ while convergence is not reached and $iter<max\_iters$ do    Assign clusters:    for each data point $x_i$ in X do      Compute distance $d(x_i,{\mu }_j)$ for each centroid ${\mu }_j$      Assign $x_i$ to the cluster $C_j$ with the nearest centroid ${\mu }_j$    end for    Update centroids:    for each cluster $C_j$ do      Compute the new centroid ${\mu }_j$ as the mean of all points in $C_j$    end for    $iter=iter+1$ end while Return: Clusters C and centroids $\mu$

Here, we take the waterway from Wuhan to Nanjing in the Yangtze River as an example; the number of clusters is set to 10, mainly based on variations in water flow speed. The results of the K-means clustering segmentation of the middle and lower reaches of the Yangtze River waterway are shown in Figure 2.

3.2. Improved Multi-Objective PSO Algorithm

The particle swarm optimization (PSO) algorithm was proposed by J. Kennedy and R. C. Eberhart in 1995 as a population-based optimization algorithm [35]. PSO simulates the behavior of bird flocking or fish schooling, leveraging the collaborative effects of individuals and the swarm to find the global optimum. Each particle represents a candidate solution, moving through the search space, with its position and speed updated according to Equations (17) and (18):

$$v_i(t+1)=\omega v_i\left(t\right)+c_1{r}_1(pBes{t}_i\left(t\right)-x_i\left(t\right))+c_2{r}_2(gBes{t}_i\left(t\right)-x_i\left(t\right))$$

(17)

$$x_i(t+1)=x_i\left(t\right)+v_i(t+1)$$

(18)

The PSO algorithm is primarily used for solving single-objective optimization problems. To handle multi-objective optimization problems, the MOPSO algorithm is developed. MOPSO algorithm is an extension of the PSO algorithm aimed at solving multi-objective optimization problems. The particle position update strategy in the MOPSO algorithm is the same as in the PSO algorithm, but non-dominated solutions are stored in an external archive during the iteration process, ultimately yielding the Pareto optimal solution set. The crowding distance is a metric for measuring the density of solutions in the archive. By calculating the distance of each solution in each objective dimension, the solution with the largest crowding distance is selected as the global best solution ($gBest$) to ensure uniform coverage of the search space. A larger crowding distance indicates fewer solutions around it, and selecting these solutions helps maintain diversity.

To improve the convergence speed and global search capability of the algorithm, a cosine decreasing strategy is used to modify the inertial weight. The cosine decreasing strategy is a method for dynamically adjusting the inertial weight, gradually reducing the weight according to a cosine function during each iteration, as shown in Equation (19). At the early stage of iterations, a larger inertial weight enhances the global search capability, allowing particles to explore a larger search space and increasing the probability of finding the global optimal solution. As the number of iterations increases, the inertial weight gradually decreases, enhancing local search capability, which helps particles conduct a fine search in the neighborhood of the solution, improving the solution’s precision and convergence speed. Compared to the linear decreasing inertial weight, the cosine decreasing strategy better achieves these goals, as shown in Figure 3.

$$\omega ={\omega }_{\min }+{\displaystyle \frac{1}{2}}({\omega }_{\max }-{\omega }_{\min })\left(1+cos\left({\displaystyle \frac{t\times \pi }{t_{\max }}}\right)\right)$$

(19)

where ${\omega }_{\max }$ and ${\omega }_{\min }$ represent the maximum and minimum weight coefficients, t represents the current iteration number, and $t_{\max }$ represents the maximum number of iterations.

To enhance the global search capability of the algorithm and increase the diversity of solutions, preventing the algorithm from falling into local optima, Gaussian distribution random perturbations are added to mutate the speed and position of the particles. The mutation operation introduces new search directions and positions, increasing the diversity of the search space and better adapting to more complex multi-objective optimization problems. This approach helps in finding a more balanced and comprehensive Pareto optimal solution set. The specific mutation method is shown in Equations (20)–(22).

$$\begin{array}{c}\hfill {x^{\prime }}_i^j=\left\{\begin{array}{cc}{x}_i^j+G(0,{\sigma }^2),\hfill & \hfill \mathrm{if}P\le Q\left(t\right)\\ x_i^j,\hfill & \hfill \mathrm{if}P>Q\left(t\right)\end{array}\right.\end{array}$$

(20)

$$\begin{array}{c}\hfill {v^{\prime }}_i^j=\left\{\begin{array}{cc}{v}_i^j+G(0,{\sigma }^2),\hfill & \hfill \mathrm{if}P\le Q\left(t\right)\\ v_i^j,\hfill & \hfill \mathrm{if}P>Q\left(t\right)\end{array}\right.\end{array}$$

(21)

$$Q\left(t\right)={\displaystyle \frac{-t^2}{e^{\pi t_{\max }}}}$$

(22)

where $x_i^j$ represents the position of the i-th particle in the j-th dimension and ${x^{\prime }}_i^j$ represents the new position of the mutated particle. $G(0,{\sigma }^2)$ is a Gaussian random variable with a mean of 0 and a variance of ${\sigma }^2$, where ${\sigma }^2$ represents the variance of all particles in the current population. P follows a uniform distribution within the interval $[0,1]$, and $Q\left(t\right)$ is the criterion for determining whether a particle undergoes mutation. The mutation method for the speed of each particle is the same.

By using cosine decreasing inertial weight and Gaussian mutation, the MOPSO algorithm is improved. Figure 4 shows the changes in fitness values at different iteration numbers before and after the improvement of the algorithm. It can be observed that before the improvement, the fitness values converge slowly, whereas after the improvement, the convergence speed of the fitness values is significantly faster, and the final fitness value achieved is lower. The improved MOPSO algorithm demonstrates a significant enhancement in both search efficiency and optimization effectiveness. The steps of the improved MOPSO algorithm are shown in Algorithm 2.

Algorithm 2 Improved MOPSO Algorithm

Initialize particle population P with random positions and velocities Initialize external archive A with non-dominated solutions Evaluate and update archive A with initial particles Set $iter=0$ while$iter<max\_iters$do    Evaluate and Update Archive:    for each particle p in P do      Evaluate fitness of p      if p dominates any solution in A then         Remove dominated solutions from A         Add p to A      end if    end for    Select Global Best:    Select $gBest$ from A based on crowding distance    Update Particles:    for each particle p in P do      Update speed $v_p$ using $gBest$ and personal best $pBest$      Apply cosine decreasing inertial weight      Apply Gaussian random mutation to $v_p$ and position $x_p$      Update position $x_p$ with new speed $v_p$    end for    $iter=iter+1$ end while Return: Pareto optimal solution set A

3.3. Constraints and Objective Function

When optimizing the long-distance speed of inland ships, various constraints and optimization objectives need to be considered, as detailed below:

The ship must meet the expected arrival time requirements. Specifically, the sailing time for each segment should satisfy the relationship shown in Equation (23).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\sum _{i=1}^n\Delta T_i\le T_{\max }\hfill \\ \Delta T_i={\displaystyle \frac{s_i}{v_i}}\hfill \end{array}\right.\end{array}$$

(23)

where $s_i$ is the distance of each segment, and $v_i$ is the ship’s speed over the ground for each segment.

To ensure navigation safety, the main engine speed must be restricted. The engine speed should not exceed the rated speed, nor should it be too low, as shown in Equation (24).

$$n_{\min }\le n_{\mathrm{e}}\le n_{\max }$$

(24)

where $n_e$ (rpm) is the main engine speed, $n_{\max }$ is the rated speed of the main engine, and $n_{\min }$ is the minimum safe speed.

The speed limits and transit times stipulated by the maritime authorities along the Yangtze River mainline vary from region to region. Therefore, the ship’s speed needs to be restricted within the prescribed range. In bridge areas and special segments, the corresponding speed requirements must also be observed, as shown in Equation (25).

$$v_{\min }\le v_i\le v_{\max }$$

(25)

where $v_{\min }$ (kn) and $v_{\max }$ are the minimum and maximum speed limits for the i-th segment, respectively.

To ensure navigation safety, the main engine’s power must also be restricted. The engine power should not exceed the safe range, as shown in Equation (26).

$$P\le P_{\max }$$

(26)

where P (kW) is the main engine power and $P_{\max }$ is the maximum allowable power of the main engine.

Multi-objective optimization problems usually involve multiple conflicting objectives, requiring a balance to be found among them. When setting up the objective function, it is necessary to consider these objectives simultaneously. In this study, the objective function of the MOPSO algorithm particularly focuses on two key outcomes in speed optimization: ship energy consumption and sailing time, as shown in Equation (27). This approach ensures that the optimization results can effectively reduce energy consumption while completing the navigation tasks within a reasonable time frame.

$$\underset{v}{min}F\left(v\right)=[f_1\left(v\right),f_2\left(v\right)]$$

(27)

where v represents a particle in an n-dimensional search space, $f_1(v$) and $f_2(v$) are the objective functions for minimum energy consumption and minimum time, respectively. MOPSO aims to find the Pareto optimal solution in the search space under the constraints of these two objectives, thereby achieving the best balance between energy consumption and sailing time.

4. Case Study

The subject of this study is a twin-propeller, twin-rudder electric propulsion ship navigating the Yangtze River. This ship primarily undertakes special transportation tasks in the middle and lower reaches of the Yangtze River. The electric propulsion system of the ship allows for the precise control of rotational speed and power output, significantly impacting the optimization method. Unlike diesel-powered ships, fully electric ships can reduce energy loss associated with changes in the main engine’s speed, further enhancing the accuracy of speed optimization. The ship’s main parameters are shown in

Table 1. The main parameters of the experimental ship.

Parameters	Value
Ship length	20.00 m
Design waterline length	19.80 m
Breadth	4.80 m
Designed draft	0.80 m
Designed prismatic coefficient	0.68
Designed block coefficient	0.60
Main engine power	100 kW×2
Designed speed	10.8 kn

.

The sailing characteristic curves of the electric ship at different speeds are shown in Figure 5. For instance, 600PP represents the propulsive power at a propeller speed of 600 RPM, 600EP represents the main engine power at a propeller speed of 600 RPM, and 90% TP represents the effective power of the ship at 90% load.

The parameter settings for the MOPSO algorithm are as follows: the initial number of particles is 30, the maximum number of iterations is 100, and the dimensions of each particle are 10, corresponding to the speed for 10 segments. Inertial weight coefficient: ${\omega }_{\max }=0.9$, ${\omega }_{\min }=0.4$. Learning factors: $c_1=2.0$, $c_2=2.0$.

4.1. Case 1: Comparison of PSO, MOPSO, and Improved MOPSO Algorithms

To further validate the advantages of the proposed improved MOPSO algorithm in terms of solving speed, we conduct 50 comparative simulation experiments with the PSO algorithm, the MOPSO algorithm, and the improved MOPSO algorithm. The experimental environment consisted of a Windows 11 operating system running on an AMD R7-5800H processor (AMD, sourced from Wuhan, China) with a 3.2 GHz main frequency and 16 GB of memory. The simulation software used is MATLAB R2022a. The results are shown in Figure 6.

The PSO algorithm, on average, found the optimal solution after approximately 27 iterations. The MOPSO algorithm is slightly slower, finding the optimal solution after an average of 29 iterations. In contrast, the improved MOPSO algorithm achieved the optimal solution the fastest, averaging 20 iterations to reach the optimal solution. This indicates that the MOPSO algorithm using the cosine decreasing and Gaussian random number mutation improvement strategies can significantly increase solving speed, finding the global optimal solution more quickly.

4.2. Case 2: Minimization of Energy Consumption and Sailing Time

In the long-distance speed optimization problem for inland navigation, sailing speed and sailing time are two conflicting objectives. Reducing sailing speed decreases energy consumption but simultaneously increases sailing time, making this a typical multi-objective optimization problem. The goal of this case study is to find a balance between energy consumption and sailing time using the MOPSO algorithm, thereby achieving the best compromise solution on the Pareto front.

To verify the effectiveness of the MOPSO algorithm designed in this paper for multi-objective optimization problems, we conducted simulation experiments under the conditions of a total length of 720 km for the middle and lower reaches of the Yangtze River, an ETA ≤ 48 h, and the experimental ship navigating downstream. The goals are set to minimize the ship’s energy consumption and sailing time. The experimental results show the Pareto front of energy consumption and sailing time corresponding to different speed combinations, as illustrated in Figure 7. Additionally, we selected the coordinates of the ideal solution A with the minimum energy consumption and the shortest time, and the coordinates of the negative ideal solution B with the maximum energy consumption and the longest time. By drawing a straight line connecting points A and B, we calculated the point on the Pareto front that is closest to this line, representing the best compromise solution between energy consumption and time.

Based on AIS data analysis, the average downstream speed of ships on the Yangtze River is between 7 kn and 8.6 kn [36]. To verify the speed optimization effect of the algorithm, we assume that the ship travels at a speed of 7.9 kn relative to the water along the entire voyage of 720 km. This makes the total voyage time approximately 48 h. Meanwhile, the ship engine maintains a constant speed to eliminate additional energy consumption caused by changes in engine conditions. Subsequently, we compare this with the minimum energy consumption obtained by the MOPSO algorithm proposed in this paper. The results are shown in

Table 2. Energy consumption comparison before and after optimization.

Segment	Speed before Optimization (kn)	Speed after Optimization (kn)	Energy before Optimization (kWh)	Energy after Optimization (kWh)
Channel I	7.9	7.9	151.36	148.56
Channel II	7.9	7.3	152.64	109.96
Channel III	7.9	7.8	191.62	180.58
Channel IV	7.9	7.9	230.92	228.44
Channel V	7.9	8.5	207.02	273.10
Channel VI	7.9	7.7	181.94	164.52
Channel VII	7.9	7.3	171.16	121.50
Channel VIII	7.9	8.5	185.12	244.20
Channel IX	7.9	7.2	239.08	157.86
Channel X	7.9	7.7	161.68	142.38
Sum			1872.54	1771.10

.

The optimal speed corresponding to the minimum energy consumption solved by the algorithm compared to the assumed constant speed during navigation is shown in Figure 8. From the data in Table 2, we can see that the energy consumption of the ship in segments V and VIII after optimization is higher than before optimization, while the energy consumption in the remaining segments is lower than before optimization. However, the overall energy consumption is lower than before optimization. The sailing duration before optimization is 48 h, and the sailing time after optimization is 47.58 h. Through speed optimization, the total energy consumption was reduced by 5.41% compared to before optimization. This result indicates that the MOPSO algorithm proposed in this paper significantly reduces ship energy consumption and improves operational efficiency while maintaining comparable sailing time.

4.3. Case 3: Minimization of Energy Consumption under Different Load Conditions

In inland navigation speed optimization, ship load has a significant impact on energy consumption. Different loads change the ship’s draft and resistance, thus affecting energy consumption. To more comprehensively evaluate the optimization effect of the MOPSO algorithm, it is necessary to conduct optimization analysis under different ship load conditions.

Based on the effective power curves of the hull under different ship loads shown in Figure 5, this section sets up speed optimization and energy consumption analysis under different load conditions (90% load, 100% load, and 110% load) to further evaluate the effectiveness and stability of the algorithm. In the experiments, other conditions (waterway length, water flow speed, ETA ≤ 48 h) were set the same.

Figure 9 shows the optimal sailing speed and energy consumption distribution under different load conditions. The ship’s energy consumption was calculated based on the speed optimization results, and a comparative analysis before and after optimization was conducted according to the assumptions in Case 2. By analyzing the energy consumption results of the ship before and after optimization in

Table 3. Energy consumption comparison under different loads.

	Energy at 90% load	Energy at 100% load	Energy at 110% load
Total Energy before optimization	1623.32 kWh	1832.44 kWh	2013.28 kWh
Total Energy after optimization	1536.64 kWh	1775.24 kWh	1955.06 kWh
Total Energy reduction	5.34%	3.16%	2.89%
Energy per hour before optimization	33.82 kW/h	38.18 kW/h	41.34 kW/h
Energy per hour after optimization	32.30 kW/h	37.32 kW/h	40.74 kW/h
Energy per hour reduction	4.49%	2.25%	1.45%

, it can be seen that under 90% load, 100% load, and 110% load conditions, the optimized energy consumption is reduced by 5.34%, 3.16%, and 2.89%, respectively. Additionally, the optimized energy consumption per hour decreased by 4.49%, 2.25%, and 1.45%, respectively. This indicates that the MOPSO algorithm can effectively reduce ship energy consumption under different load conditions.

4.4. Case 4: Minimization of Energy Consumption under Different ETA Conditions

In inland navigation, ETA is a critical factor in navigation planning and scheduling. Different ETA requirements directly affect the ship’s speed and energy consumption. A shorter ETA typically requires higher speeds, leading to increased energy consumption, whereas a longer ETA allows for lower speeds, helping to save fuel. To minimize energy consumption, it is necessary to optimize the ship’s speed while meeting different ETA requirements.

This section sets up different ETA scenarios and uses the improved MOPSO algorithm for speed optimization to compare the ship’s energy consumption under various ETA conditions, providing a reference for formulating navigation plans and scheduling strategies. Under consistent experimental conditions, we tested the ship’s energy consumption for ETA ≤ 46 h, ETA ≤ 48 h, and ETA ≤ 50 h.

Figure 10 shows the optimal speed distribution under different ETA conditions. The ship’s energy consumption is calculated based on the speed optimization results, as shown in

Table 4. Energy consumption comparison under different ETAs.

	Energy for ETA ≤ 46 h	Energy for ETA ≤ 48 h	Energy for ETA ≤ 50 h
Total Energy before optimization	2201.74 kWh	1832.44 kWh	1576.62 kWh
Total Energy after optimization	2065.60 kWh	1780.04 kWh	1525.50 kWh
Total Energy reduction	6.18%	2.86%	3.24%
Energy per hour before optimization	47.86 kW/h	38.18 kW/h	31.54 kW/h
Energy per hour after optimization	45.12 kW/h	37.32 kW/h	30.76 kW/h
Energy per hour reduction	5.35%	2.25%	2.47%

. It can be seen that under ETA ≤ 46 h, ETA ≤48 h, and ETA ≤ 50 h conditions, the optimized energy consumption is reduced by 6.18%, 2.86%, and 3.24%, respectively. Additionally, the optimized energy consumption per hour decreased by 5.35%, 2.25%, and 2.47%, respectively. This indicates that the MOPSO algorithm can effectively reduce ship energy consumption under different ETA conditions.

5. Conclusions

This paper considers the inland navigation environment and ship characteristics, constructing an energy-consumption model and using the K-means waterway clustering method to reasonably segment the middle and lower reaches of the Yangtze River, thereby achieving more precise speed optimization. Additionally, the MOPSO algorithm is improved by employing a cosine decreasing inertial weight and Gaussian random mutation strategy. Case studies verify the algorithm’s multi-objective optimization capability. Under different load and ETA conditions, the algorithm effectively reduces ship energy consumption.

However, there are some limitations in the research methods of this paper. Firstly, the issue of battery replacement due to insufficient battery capacity in the long-distance navigation of inland electric ships is neglected. Additionally, due to data deficiencies, the characteristic curves of the electric motor are not considered, limiting the depth of the research. In future work, more in-depth research should be conducted by combining more actual operational data of ships, adopting more precise ship energy-consumption modeling methods and more efficient optimization algorithms. Furthermore, research should be carried out in conjunction with the development of inland electric ships to provide more practical insights.