The Application of an Improved LESS Dung Beetle Optimization in the Intelligent Topological Reconfiguration of ShipPower Systems

Abstract

To address the shortcomings of the Dung Beetle Optimization (DBO) algorithm in ship power-system fault reconfiguration, such as low population diversity and an imbalance between global exploration and local exploitation, the authors of this paper propose an improved Dung Beetle Optimization (LESSDBO) algorithm. The improvements include optimizing the initial population using Latin hypercube sampling and an elite population strategy, optimizing parameters with an improved sigmoid activation function, introducing the sine–cosine algorithm (SCA) for position update optimization, and performing multi-population mutation operations based on individual quality. The LESSDBO algorithm was applied to simulate the fault reconfiguration of a ship power system, and it was compared with the traditional DBO, Genetic Algorithm (GA), and Modified Particle Swarm Optimization (MSCPSO) methods. The simulation results showed that LESSDBO outperformed the other algorithms in terms of convergence accuracy, convergence speed, and global search capability. Specifically, in the reconfiguration under Fault 1, LESSDBO achieved optimal convergence in seven iterations, reducing convergence iterations by more than 30% compared with the other algorithms. In the reconfiguration under Fault 2, LESSDBO achieved optimal convergence in eight iterations, reducing convergence iterations by more than 23% compared with the other algorithms. Additionally, in the reconfiguration under Fault Condition 1, LESSDBO achieved a minimum of four switch actions, which is 33% fewer than the other algorithms, on average. In the reconfiguration under Fault Condition 2, LESSDBO achieved a minimum of eight switch actions, which is a 5.9% reduction compared with the other algorithms. Furthermore, LESSDBO obtained the optimal reconfiguration solution in all 50 trials for both Faults 1 and 2, demonstrating a 100% optimal convergence probability and significantly enhancing the reliability and stability of the algorithm. The proposed method effectively overcomes the limitations of the traditional DBO in fault reconfiguration, providing an efficient and stable solution for the intelligent topology reconfiguration of ship power systems.

1. Introduction

The rapid development of the modern marine industry has placed higher demands on the stability and intelligence of ship power systems. On 26 March 2024, the Singaporean-registered container ship MV Dali experienced a series of power-system failures while departing from the Port of Baltimore, which led to a collision with the Francis Scott Key Bridge, causing the partial collapse of the bridge and significant casualties, as shown in Figure 1. Prior to the incident, the MV Dali’s generators tripped multiple times, resulting in a complete blackout of the vessel [1]. Not only do ship power systems provide power for propulsion and various critical equipment, but their reliability is also directly linked to the safety and operational efficiency of ships. In the complex marine environment, where power systems face extreme conditions and long-distance operations, intelligence has become an inevitable trend [2]. Intelligent topology reconfiguration is a key part of power-system intelligence, enabling quick reconfiguration of power topology during faults to restore the power supply and minimize interruptions to critical equipment [3]. Traditional methods struggle to effectively respond to complex fault environments, while intelligent reconfiguration technology, through advanced algorithms, enhances emergency response capabilities and power reliability, thereby ensuring the safe operation of ships and promoting the efficient development of the marine industry.

The key task for researchers is to restore the power supply to critical loads as quickly and efficiently as possible while minimizing power loss [4]. To this end, various strategies have been proposed. For example, Zhang Xinyue et al. described an improved Grey Wolf Optimization strategy that prioritizes restoring critical loads while considering practical constraints such as reconfiguration time [5]. Lan H et al. combined expert system theory and GIS fault detection to verify whether cable current and load voltage limits were violated [6]. Jia Lirei introduced an improved dual-particle swarm algorithm to enhance the optimization accuracy of local search [7]. Wu Qihuan et al. proposed a dynamic reconfiguration optimization strategy for ship power systems, with weighted load shedding and minimal voltage deviation as the optimization objectives, and they used an improved inertia particle swarm optimization algorithm to solve the optimization model [8]. Su Li et al. proposed a constrained multi-objective optimization method based on the two-stage differential evolution (TSDE) algorithm, using different control strategies at different stages for ship microgrid reconfiguration based on a multi-objective optimization algorithm [9]. Xu Zehua designed a fault reconfiguration method based on regional multi-agents, which showed significant effects in distributed structure reconfiguration [10].

Although an increasing number of researchers tend to use precise and efficient artificial intelligence algorithms to solve the topology reconfiguration problem of ship power systems, these algorithms often suffer from issues such as excessive parameter tuning, poor convergence results, slow convergence speed, and low optimization accuracy.

The authors of this study propose an improved LESS Dung Beetle Optimizer (introducing Latin hypercube sampling, the elite population strategy, a sigmoid function for parameter optimization, and the sine–cosine algorithm for position update optimization, abbreviated as LESSDBO, the abbreviations and symbols of such terms in this paper are listed in

Table A1. Nomenclature and abbreviations.

Serial No.	Symbol	Definition
1	LESSDBO	Improved Dung Beetle Optimization Algorithm
2	DBO	Dung Beetle Optimization Algorithm
3	GA	Genetic Algorithm
4	MSCPSO	Modified Particle Swarm Optimization
5	sigmoid	S-shaped activation function
6	LHS	Latin hypercube sampling
7	EPS	elite population strategy
8	Popsize	population size
9	SCA	sine–cosine algorithm
10	ABT	automatic transfer switch
11	MBT	manual transfer switch
12	MSB	main switchboard
13	DB	distribution board
14	ACB	main switch of the generator
15	TL	link cable
16	☆	load
17		normally closed switch
18		normally open switch
19		cable connection endpoint

at the end of the document.) to address the problem of fault reconfiguration in shipboard power systems. The Dung Beetle Optimizer (DBO), proposed in 2022, is a metaheuristic optimization algorithm with advantages such as fast convergence speed, a combination of global exploration and local exploitation, and strong optimization capability [11]. However, when handling complex fault reconfiguration problems, the DBO tends to fall into local optima, making it difficult to effectively search for the global optimum. This limitation is particularly critical in the intelligent topology reconfiguration of power systems, as the system must quickly and effectively reconfigure during a fault to ensure power supply to critical loads and minimize power loss. First, Latin hypercube sampling (LHS) and the elite population strategy (EPS) are used to optimize the initial population, ensuring a more uniform distribution of the population to improve global search efficiency and increase population diversity. Next, an improved sigmoid function is introduced to optimize control factors, enhancing the adaptability of the algorithm during the search process—ensuring high global exploration ability in the early iterations and improving local exploitation accuracy in the later stages. Subsequently, the sine–cosine algorithm (SCA) is used to enhance the position update strategy, leveraging the oscillatory properties of the sine and cosine functions to maintain population diversity and effectively balance global exploration and local exploitation in the latter stages of the algorithm. Finally, a multi-population mutation strategy is introduced, classifying the population into different types based on individual fitness and designing different differential mutation operators for each type to help the algorithm escape local optima and improve the overall search capability.

The goal of improving the DBO algorithm is to enhance its global search and local exploitation capabilities in shipboard power-system fault reconfiguration, particularly when dealing with complex power-system topologies and multimodal problems, ensuring fast and effective optimization and ultimately achieving a reliable reconfiguration of the system. This method aims to minimize load loss and the number of switching actions while ensuring system stability and efficiency. Through these improvements, the LESSDBO algorithm effectively overcomes the problems of low population diversity and susceptibility to local optima in the traditional DBO, significantly improving the efficiency and effectiveness of fault reconfiguration.

This article is structured as follows:

• Introduction: This section introduces the background and significance of the research, the current status of domestic and international studies, the motivation and measures for algorithm improvement, and the advantages of solving the proposed problems.
• Improved Dung Beetle Optimization: First, the traditional Dung Beetle Optimization is introduced, followed by a detailed explanation of the improvement concepts and the improvement process of the Dung Beetle Optimization.
• Modeling of Shipboard Power-System Fault Reconfiguration: This section includes the reconstruction objective function, constraints, the establishment of a load–branch adjacency matrix model, and the discretization of the algorithm for reconfiguration.
• Simulation Verification of Fault Reconfiguration: The improved algorithm is used for the simulation verification of fault reconfiguration in a shipboard power system. The reconfiguration results are compared with those of different algorithms and analyzed.
• Conclusion: This section summarizes the content of the article and provides prospects for future research directions.

2. Improved LESS Dung Beetle Optimization (LESSDBO)

2.1. Dung Beetle Optimization

The Dung Beetle Optimizer (DBO) is a swarm intelligence optimization algorithm proposed by Jiankai Xue et al. in 2022. It draws inspiration from the biological behavior of dung beetles, characterized by strong optimization capabilities and fast convergence. Its main characteristics are as follows:

Dung beetles feed on animal dung. They roll dung into balls and move them to safe locations to prevent other beetles from stealing them. The primary goal is to move the dung ball quickly and effectively.

Dung beetles use celestial clues (especially the sun, moon, and polarized light) for navigation. This helps them roll the dung ball in a straight line. In the absence of light (in dark environments), they do not move in a straight line but instead follow curved paths, sometimes even circular. Several factors, such as wind or uneven ground, may cause the dung beetle to deviate from its original direction.

When dung beetles encounter obstacles that prevent them from moving forward, they often climb onto the dung ball and perform a “dance” (which involves a series of rotations and pauses) to decide their next move.

Dung beetles bury their dung balls, and female beetles lay eggs in them. The dung ball becomes both the developmental environment and the essential food source for the larvae. Thus, dung balls are vital for the survival of dung beetles.

• Population Initialization

The population initialization formula is as follows [11]:

$$X^{t+1}=rand\times (Ub-Lb)+Lb$$

(1)

Here, $rand$ is a random number between 0 and 1, $Ub$ is the upper boundary, and $Lb$ is the lower boundary.

2.

Rolling Dung Beetles

Rolling dung beetles are those that mold dung into balls and roll them to safe locations for storage, as shown in Figure 2. As stated above, dung beetles use celestial clues such as the sun, moon, and polarized light for navigation, which allows them to roll the dung ball in a straight line. When environmental conditions change, the beetle’s position also changes, and the rolling behavior can be represented as follows [11]:

$$\begin{array}{l}{x}_i(t+1)=x_i(t)+\alpha ·k·x_i(t-1)+b·\Delta x\\ \Delta x=\mid x_i(t)-X^w\mid \end{array}$$

(2)

Here, $t$ represents the current iteration, $x_i(t)$ is the position of the $i$-th dung beetle at iteration $n$, $k\in (0,0.2]$ is a constant deflection factor, $b\in (0,1)$ is a constant, $\alpha$ is a natural factor assigned the value −1 or 1, $X^w$ represents the global worst position, and $\Delta x$ represents environmental change. A value of α = 1 indicates no deflection, and α = −1 indicates a deflection from the original direction. The higher the value of $\Delta x$, the weaker the environmental influence. If $r<0.9$, the dung beetle updates its position using an obstacle-free strategy. If $r\ge 0.9$, the beetle updates its position using an obstacle strategy.

3.

Dancing Dung Beetles

When $r\ge 0.9$, the dung beetle encounters an obstacle and cannot proceed. In this case, it performs a “dance” to reposition itself and determine a new path, as shown in Figure 3. Using the tangent function, it identifies a new rolling direction. Once the dung beetle finds a new direction, it continues to roll the dung ball forward. The position update formula is as follows [11]:

$$x_i(t+1)=x_i(t)+\tan (\theta )\mid x_i(t)-x_i(t-1)\mid$$

(3)

If $\theta \in [0,{\displaystyle \frac{\pi }{2}},\pi ]$, the tangent value is undefined, and the position of the dung beetle does not update.

4.

Egg-laying Dung Beetles

Female dung beetles bury dung balls to provide a safe environment for their offspring, simulating a boundary selection strategy for the egg-laying region, as shown in Figure 4. The strategy is as follows [11]:

$$\begin{array}{l}L{b}^{\ast }=\max \{X^{\ast }·(1-R),Lb\}\\ U{b}^{\ast }=\min \{X^{\ast }·(1+R),Ub\}\end{array}$$

(4)

$$R=1-{\displaystyle \frac{t}{T_{\max }}}$$

(5)

Here, $X^{\ast }$ represents the local optimal solution, $T_{\max }$ is the maximum number of iterations, $t$ represents the current iteration, $Ub$ is the upper boundary, and $Lb$ is the lower boundary.

Once the egg-laying area is determined, the female dung beetle will choose the egg ball in this area to lay eggs, as shown in Figure 5. Each female dung beetle only lays one egg in each iteration. The boundary range of the egg-laying area changes dynamically, which is mainly determined by the R value. It can be seen that the position of the egg ball is also dynamic during the iteration process and is strictly limited to a certain range.

Its position update formula is

$$x_i(t+1)=X^{\ast }+b_1·(x_i(t)-L{b}^{\ast })+b_2·(x_i(t)-U{b}^{\ast })$$

(6)

Here, $X^{\ast }$ is the local optimal solution, $x_i(t)$ is the location information of the second spawning, $b_1$ and $b_2$ are 1 × D dimensional random vectors, and D represents the dimension of the optimization problem. The spawning location is strictly limited to the boundary of the spawning area.

5.

Foraging Dung Beetles

After hatching, the dung beetle larvae forage collectively. Therefore, an optimal foraging zone is established to guide the young beetles, as shown in Figure 6. The foraging area is dynamically simulated using a boundary-selection strategy similar to the egg-laying zone.

$$\begin{array}{l}L{b}^b=\max \{X^b·(1-R),Lb\}\\ U{b}^b=\min \{X^b·(1+R),Ub\}\end{array}$$

(7)

$$R=1-{\displaystyle \frac{t}{T_{\max }}}$$

(8)

After determining the foraging boundary, the foraging dung beetle will forage in this area, and its position update formula is

$$x_i(t+1)=x_i(t)+C_1·(x_i(t)-L{b}^b)+C_2·(x_i(t)-U{b}^b)$$

(9)

Here, $x_i(t)$ is the foraging beetle’s position at iteration $t$, $C_1$ is a normally distributed random number, and $C_2$ is a random vector belonging to $(0,1)$.

6.

Thieving Dung Beetles

Some dung beetles steal dung balls from others in globally optimal locations. The updated formula for thieving beetles is as follows:

$$x_i(t+1)=X^b+S·g·(\left|x_i(t)-X^b\right|+\left|x_i(t)-X^{\ast }\right|$$

(10)

Here, $S$ is a constant, $g$ is a random vector of a normal distribution, and $X^b$ is the best food source in the vicinity.

2.2. Improved Dung Beetle Optimization

To better analyze and solve the problem of fault reconfiguration in ship power systems and to build an intelligent topological reconfiguration model for these systems with the optimization goals of minimizing recovery time, maximizing load recovery, achieving a uniform load distribution, and balancing generator efficiency, an improved LESS Dung Beetle Optimization is proposed. This improved algorithm addresses the shortcomings of the traditional Dung Beetle Optimization, such as low population diversity, imbalanced global exploration and local exploitation capabilities, and weak global exploration abilities [12].

• Latin Hypercube Sampling and Elite Population Strategy

In the original dung beetle population, a randomized approach is generally used for initialization. Although this method is simple, it may result in an uneven distribution of the initial population in the search space, as shown in Figure 7. This uneven distribution reduces population diversity, which increases the risk of the algorithm falling into a local optimum and introduces significant uncertainty in its early stages. If the initial population distribution can be improved to make it more uniform, the global search efficiency and coverage of the solution space can be effectively enhanced.

To address this issue, the Latin hypercube sampling (LHS) method is introduced. Latin hypercube sampling was proposed by Mckay et al. in 1979 as an approximate random sampling method from multivariate parameter distributions [13]. Unlike traditional random sampling, LHS divides the sampling range into equal subintervals for each dimension, and then it randomly selects one sample point within each subinterval to ensure that each interval has a representative sample. Therefore, LHS has the characteristic of uniform stratification, which allows it to spread sample values well across the entire range, even when the number of samples is small, thus covering the entire parameter space. This characteristic is particularly suitable for multivariable and high-dimensional optimization problems, as it can effectively reduce sample bias and make each sample of the initial population more representative, as shown in Figure 8.

To further improve the quality of the initial population, an elite population strategy was also adopted. First, a portion of dung beetle individuals was generated using Latin hypercube sampling to ensure that these individuals uniformly covered the entire search space. Then, additional dung beetle individuals were randomly generated to increase the randomness and diversity of the population. All initial individuals were evaluated for fitness, and then they were ranked according to their fitness values. The top “Popsize” individuals with the highest fitness were selected as the elite population, as shown in Figure 9. This approach ensures a more uniform distribution of the initial population while maintaining high diversity. This not only increases the diversity of the initial population and prevents the algorithm from getting stuck in local regions during early searches but also improves the global optimization performance and enhances the exploration capability for different solutions, thereby increasing the convergence speed and accuracy of the algorithm [14].

2.

Nonlinear Control Factor R Based on Improved Sigmoid Function

Extensive research has found that swarm intelligence algorithms often suffer from imbalanced global and local search capabilities, leading to reduced optimization performance and convergence precision. Therefore, a generalized and widely applicable swarm intelligence algorithm should include a robust coordination mechanism to balance global and local search capabilities, thereby improving convergence precision and speed.

In the DBO algorithm, R is the update parameter of the spawning area/foraging area used in the brood ball/small dung beetle update strategy. These two dung beetles are used to characterize the transition stage of the algorithm from exploration to development.

In the DBO algorithm, when facing complex multimodal problems, the linear decrease in the control factor R cannot accurately reflect the complex search process when multimodal situations occur. The control factor based on the change in the sigmoid function can obtain better search capabilities [15]. Therefore, on this basis, a control factor based on the improved sigmoid is introduced as follows:

$$R=R_{init}\cdot {\displaystyle \frac{1}{1+e^{\frac{10\cdot iter-5\cdot MaxIt}{MaxIt}}}}$$

(11)

Here, $R_{init}$ is the initial value of the control factor, $iter$ is the current iteration count, and $MaxIt$ is the maximum number of iterations. The function graph is shown below.

As can be seen in Figure 10, the variation curve based on the corrected sigmoid function gradually decreases at the beginning of the iteration, keeping R at a high level for a longer period of time to enhance the algorithm’s early global exploration capability. R then rapidly declines in the middle phase, allowing it to remain low during the later iterations, which improves the accuracy of the algorithm’s local exploitation and balances its ability to perform global and local searches.

3.

Thieving Dung Beetle Improved by Sine–Cosine Algorithm

The thieving dung beetle primarily moves around the optimal position (i.e., the best individual) during the stealing stage. This search strategy is too singular and does not adequately explore the neighborhood of the current best individual, which can reduce population diversity and cause stagnation at local optima.

The sine–cosine algorithm (SCA), proposed by Mirjalili in 2016, is a metaheuristic algorithm that utilizes the oscillatory nature of sine and cosine functions to continuously approach the global optimum [16]. The position update formula for the SCA is

$$X_{ij}^{t+1}=\left\{\begin{array}{cc}{X}_{ij}^t+r_1\times cos(r_2)\times \left|r_3{P}_{ij}^t-X_{ij}^t\right|& r_4\ge 0.5\\ & \\ X_{ij}^t+r_1\times sin(r_2)\times \left|r_3{P}_{ij}^t-X_{ij}^t\right|& r_4<0.5\end{array}\right.$$

(12)

$$r_1=a-t{\displaystyle \frac{a}{T}}$$

(13)

The core purpose of the SCA is to use the oscillation adjustment of the sine–cosine model to achieve global and local optimization in order to obtain the global optimal value. Therefore, the authors of this paper introduce the SCA into DBO, using the oscillation of the sine–cosine function to maintain the diversity of the algorithm’s later population and enhance the algorithm’s local development capabilities. At the same time, on the basis of the SCA, r1 is adjusted to have a nonlinear decreasing characteristic, thereby balancing the exploration and development of the SCA. A high weight is started with in the initial stage, and then it slowly decreases to achieve better globality. As the iteration proceeds, the low weight slowly decreases in the later stage, which enhances the algorithm’s local convergence ability. In summary, the updated formula of the thief dung beetle is as follows:

$$x_i(t+1)=\left\{\begin{array}{ll}w\cdot X^b+r_1\cdot \cos (r_2)\cdot \left|r_3{X}^b-x_i(t)\right|\hfill & {\mathrm{r}}_4\ge 0.5\hfill \\ w\cdot X^b+r_1\cdot \sin (r_2)\cdot \left|r_3{X}^b-x_i(t)\right|\hfill & {\mathrm{r}}_4<0.5\hfill \end{array}\right.$$

(14)

$$r_1={(1-(Iter/MaxIt))}^{2\times Iter/MaxIt}$$

(15)

4.

Multi-Population Mutation Strategy

To help the algorithm escape local optima, a multi-population mutation strategy combining various forms of differential evolution is designed. A single mutation method may not satisfy the evolutionary needs of all individuals [17]. For example, highly fit individuals tend to cluster near the current best individual, emphasizing local exploitation, while less fit individuals may be farther from the optimum, requiring global exploration. Based on this observation, the population is divided into three groups, each of which applies a different differential evolution operator, enhancing the search capabilities of the improved DBO algorithm and helping it escape local optima. The specific strategies are as follows:

(1)

Elite Group—DE/current-to-best/1:

This group consists of individuals in the top one-third of the population in terms of fitness. Due to their good fitness, they are more likely to be close to the global optimal solution. Therefore, this group mainly conducts development searches and small disturbance exploration in the local area where the dominant population is located. Although DE/current-to-best/1 lacks sufficient global exploration capabilities, its local development performance is excellent and can meet the needs of the dominant population. Therefore, the DE mutation strategy is introduced, and its calculation formula is

$$X{D}_i(t)=X_i(t)+F_n(t)\times [X_{best}(t)-X_i(t)]+F_n(t)\times [X_{r1}(t)-X_{r2}(t)]$$

(16)

Here, $X{D}_i(t)$ is the mutant individual, $X_i(t)$ is the individual before mutation, $F_n(t)$ is the scaling factor, $X_{best}(t)$ is the best individual, and $r_1$ and $r_2$ are random integers between $\left[1,Popsize\right]$.

After mutation, the position is updated according to the greedy strategy; that is,

$$X_i(t+1)=\left\{\begin{array}{ll}X{D}_i(t),\hfill & f(X{D}_i(t))<f(X_i(t))\hfill \\ X_i(t),\hfill & else\hfill \end{array}\right.$$

(17)

(2)

Middle Group—DE/mean-current/2:

This group consists of individuals in the top one-third to two-thirds of the population in terms of fitness. The fitness of this group is between good and bad. It can not only learn locally from the dominant population but can also explore the global space from a distance to help the algorithm balance exploration and development. DE/mean-current/2’s strategy is different from the previous strategy [18]. It retains the ability to explore and develop, and it can achieve a good compromise between the two. Therefore, the DE mutation strategy is introduced, and the calculation formula is

$$\begin{array}{l}X{D}_i(t)=X_{c1}+F_n(t)\times [X_{c1}-X_i(t)]+F_n(t)\times [X_{c2}-X_i(t)]\\ X_{c1}={\displaystyle \frac{X_{r1}(t)+X_{r2}(t)}{2}},X_{c2}={\displaystyle \frac{X_{r1}(t)+X_{best}(t)}{2}}\end{array}$$

(18)

After mutation, the position is updated according to the crossover strategy in differential evolution; that is,

$$X_{i,j}(t+1)=\left\{\begin{array}{cc}X{D}_{i,j}(t),\hfill & \hspace{1em}if\hspace{1em}randi = j \mathrm{or} \mathrm{rand}<=\mathrm{CR}\hfill \\ X_{i,j}(t),\hfill & \hspace{1em}\mathrm{else}\hfill \end{array}\right.$$

(19)

Here, $randi\in [1,\dim ]$, and CR is the crossover probability.

(3)

Weak Group—DE/rand/1:

This group consists of individuals in the last one-third of the population in terms of fitness. Due to their poor fitness, they can perform global exploration in a wide range of areas to find potential global optimal solutions without relying on the position of any other individual in the group, helping the algorithm to escape from the local optimum. DE/rand/1 combines the mutation strategy of two random difference vectors to widely search the domain range of the current individual, which not only improves the population diversity but also helps the algorithm escape from the local optimum and shows good global search performance. Therefore, the DE mutation strategy is introduced, and its calculation formula is

$$X{D}_i(t)=X_{r1}(t)+F_n(t)\times [X_{r2}(t)-X_{r3}(t)]$$

(20)

Here, $r_1$, $r_2$, and $r_3$ are random integers between $\left[1,Popsize\right]$.

After mutation, the position is updated according to the greedy strategy; that is,

$$X_i(t+1)=\left\{\begin{array}{ll}X{D}_i(t),\hfill & f(X{D}_i(t))<f(X_i(t))\hfill \\ X_i(t),\hfill & else\hfill \end{array}\right.$$

(21)

3. Fault Reconfiguration Modeling of Ship Power Systems

In the network reconstruction problem of ship power systems, the main mathematical problem is to find the optimal solution of a function with multiple objectives and constraints. This involves building mathematical models according to different objective functions. This process involves seeking the best balance between multiple objectives while complying with specific system constraints to achieve the best effect of network reconstruction [19].

The subject of this study is a typical ring-type power system of a large vessel, as shown in Figure 11. The system includes four main generators supporting 20 different loads. To ensure a continuous power supply to critical loads, a backup power supply path is designed, allowing switching between the normal and backup power supply paths via automatic transfer switches (ABTs) and manual transfer switches (MBTs). After a fault occurs in the power grid and is isolated, these switches can be operated to restore power to the non-faulty area as much as possible without violating the constraints. In the figure, MSB represents the main switchboard, DB represents the distribution board, ACB is the main switch of the ship generator, TL is the connecting cable, and the dashed line represents the backup power supply line.

To further study the power grid structure in depth, the system is symbolized. Figure 12 shows a schematic diagram of the symbolized structure of the ship power system. Here, G1–G4 represent the four main generators with a rated capacity of 710A. The red symbols “☆” represent the 20 loads, with the solid lines connecting the loads representing normal power supply lines and the dashed lines representing backup power lines. The green symbols “” represent normally closed switches, the yellow symbols “” represent normally open switches, and the black symbols “” represent cable connection endpoints. The branches are numbered 1–102, with some branches (such as 11, 20, 38, 45, 64, 70, 87, and 94) connecting multiple loads, and the branch capacity is 420A. The load working currents and grades are shown in

Table 1. System load working current and load level.

Load	Current (A)	Category	Switch Type	Load	Current (A)	Category	Switch Type
L1	325	1	1	L11	72	3	0
L2	100	1	1	L12	200	2	0
L3	225	1	1	L13	120	3	0
L4	80	3	0	L14	30	2	0
L5	185	3	0	L15	87	2	0
L6	44	2	0	L16	205	2	1
L7	150	3	1	L17	165	3	0
L8	160	2	1	L18	200	3	0
L9	205	3	0	L19	70	1	1
L10	110	2	1	L20	100	1	1

, where a switch grade of “0” indicates a manual switch, and “1” indicates an automatic switch. Class 1 loads have a higher priority than Class 2 loads, and Class 2 loads have a higher priority than Class 3 loads.

3.1. Establishment of Objective Functions for Ship Power-System Network Reconfiguration

1.

Minimizing Power Loss

In the fault reconfiguration process of a ship power system, Class 1 loads, which are essential for ship and personnel safety, are given the highest priority for restoration. At the same time, minimizing the number of power-lost loads is essential. Based on the classification of ship load levels, the objective function for minimizing power loss $f_1$ is as follows [20]:

$$\min f_1=k_1{\displaystyle \sum _{a=1}^l(1-x_a)}{L}_{ga}+k_2{\displaystyle \sum _{b=1}^m(1-x_b)}{L}_{gb}+k_3{\displaystyle \sum _{c=1}^n(1-x_c)}{L}_{gc}$$

(22)

Here, $L_{ga}$, $L_{gb}$, and $L_{gc}$ represent the power values of the Class 1, 2, and 3 loads, respectively. Additionally, $l$, $m$, and $n$ represent the number of Class 1, 2, and 3 loads. $x_a$, $x_b$, and $x_c$ represent the power supply status of the respective loads, where “1” indicates power, and “0” indicates power loss. $k_1$, $k_2$, and $k_3$ are the priority coefficients for the three load levels, with $k_1=1$, $k_2=0.6$, and $k_3=0.1$. After normalization, the minimized objective $f_1$ becomes

$$\min f_1=k_{\lambda }{\displaystyle \sum _{s=1}^t{L}_g,\lambda =1,2,3}$$

(23)

$L_g$ represents the loss, $t$ represents the minimum number of loss loads, and, in extreme cases, when all loads cannot work, the value is the largest; that is,

$$\max f_1=k_1{\displaystyle \sum _{a=1}^l{L}_{ga}}+k_2{\displaystyle \sum _{b=1}^m{L}_{gb}}+k_3{\displaystyle \sum _{c=1}^n{L}_{gc}}$$

(24)

The normalization process is as follows:

$$U\left(f_1\right)={\displaystyle \frac{f_1-\min f_1}{\max f_1-\min f_1}}$$

(25)

2.

Minimizing Recovery Time

In ship power-system fault recovery, faster recovery is better when other conditions are equivalent. The speed of recovery depends on both the optimization search speed and the time needed to operate different types of switches. The objective function is [21]

$$\min f_2=k_i{\displaystyle \sum _{i=1}^m{S}_{Mi}}+k_j{\displaystyle \sum _{j=1}^n{S}_{Aj}}$$

(26)

Here, $S_{Mi}$ is the status of the manual switches (0 or 1, where 0 indicates no action, and 1 indicates switching to the backup power line), and $S_{Aj}$ is the status of automatic switches (0 or 1, where 0 indicates no action, and 1 indicates switching to the backup power line). $m$ is the total number of operated manual switches, and $n$ is the total number of automatic switches. Based on experience, the time ratio between manual and automatic switches is approximately 10:1; thus, $k_1=10$, and $k_2=1$. In extreme cases, all switches are operated. Then,

$$\max f_2=k_i{S}_{Msum}+k_j{S}_{Asum}$$

(27)

$S_{Msum}$ and $S_{Asum}$ represent the total number of manual switches and automatic switches, respectively. When $\min f_2=0$, it means that no switch is operated. The normalized model is

$$U\left(f_2\right)={\displaystyle \frac{f_2-0}{\max f_2-0}}={\displaystyle \frac{f_2}{\max f_2}}$$

(28)

3.

Maximizing Load Distribution Uniformity

After reconfiguration, the topology of the transmission lines changes, which leads to inevitable fluctuations and network losses. To improve the adaptability of the ship’s power grid after reconfiguration, the load distribution should be as uniform as possible. The objective function for load distribution uniformity is [22]

$$\min f_3={\displaystyle \sum _{i=1}^{Bs}{\left(I_{Bi}/I_{Bt}\right)}^2}$$

(29)

In the formula, $B_s$ represents the total number of lines, $I_{Bi}$ represents the current of line $i$, and $I_{Bt}$ is the total current of the ship’s power grid. In the most ideal case, the load is completely evenly distributed; in the least ideal case, the load distribution is extremely uneven: only one line bears all the current, and the current of the other lines is 0. At this time, $\max f_3=1$. The objective function itself is between [${\displaystyle \frac{1}{B_s}},1$]. Here, the reference value normalization method is used to normalize the formula, thereby obtaining

$$U(f_3)={\displaystyle \frac{1-f_3}{1-1/B_s}}$$

(30)

4.

Maximizing Generator Efficiency Balance

During network reconfiguration, the efficiency of the generators should also be considered. Each generator must be used efficiently to avoid sudden drops in efficiency for any particular generator [23]. The objective function for generator efficiency balance is

$$\min f_4=\min {\displaystyle \frac{P_{Gi}}{P_{Gin}}}$$

(31)

Here, $P_{Gin}$ is the actual active power output of generator $i$, and $P_{Gin}$ is the rated active power output of generator $i$. $\min f_4=0$ indicates that all generators have stopped working, and $\max f_4=1$ means that all generators are running at their rated power. The normalization process is

$$U(f_4)={\displaystyle \frac{f_4-0}{1-0}}=f_4$$

(32)

5.

Comprehensive Objective Function

The comprehensive objective function for ship power-system reconfiguration can be expressed as

$$\min f=k_{1}U(f_1)+k_{2}U(f_2)+k_{3}U(f_3)+k_{4}U(f_4)$$

(33)

Here, $k_1$, $k_2$, $k_3$, and $k_4$ are the weighting coefficients determined using the analytic hierarchy process. The scaled matrix is

$$A=\left[\begin{array}{cccc}1& 3& 5& 7\\ 1/3& 1& 3& 5\\ 1/5& 1/3& 1& 3\\ 1/7& 1/5& 1/3& 1\end{array}\right]$$

(34)

The eigenvalues of matrix A are ${\lambda }_1=4.1320$, ${\lambda }_2=0.2814$, ${\lambda }_3=0.0472$, and ${\lambda }_4=0.0027$. Then, ${\lambda }_{\max }=4.1320$. The eigenvector is

$$X=\left(\begin{array}{l}0.8264\\ 0.3872\\ 0.1886\\ 0.0905\end{array}\right)$$

(35)

The normalization process is

$$k=\left(\begin{array}{l}0.5537\\ 0.2595\\ 0.1263\\ 0.0606\end{array}\right)$$

(36)

It is found that $CR=0.0489<0.1$, which meets the consistency check.

Thus, the comprehensive objective function becomes

$$\min f=0.5537U(f_1)+0.2595U(f_2)+0.1263U(f_3)+0.0606U(f_4)$$

(37)

3.2. Constraints for Ship Power-System Network Reconfiguration

1.

Network Structure Constraints

For loads that have both normal and backup power supply paths, only one path can be operational at a time to avoid the occurrence of loops or islands in the reconfigured grid. The constraint formula is as follows:

$$Z_{\alpha i}+Z_{\beta i}=1\hspace{1em}(i\in \mathsf{\Omega })$$

(38)

Here, $\mathsf{\Omega }$ is the set of loads with backup power lines, $Z_{\alpha i}$ represents the normal power supply state for load $i$ (0 for disconnected), and $Z_{\beta i}$ represents the backup power supply state for load $i$ (1 for connected).

2.

System Capacity Constraints

During the reconfiguration process, care must be taken not to overload the branch or generator capacity. If an overload occurs, unnecessary loads should be shed. The constraint formula is as follows:

$$x_i{C}_i\le C_{Ni}\hspace{1em}(i\in N_f)$$

(39)

3.

Branch Current Constraints

System branch currents must be restricted to ensure that cable currents remain within the operational range. The constraint formula is as follows:

$$I_i\le I_{Ni}$$

(40)

Here, $I_i$ is the current on branch $i$, and $I_{Ni}$ is the maximum allowable current for that branch.

3.3. Establishment of an Adjacency Matrix for the Loads and Branches in the Ship Power System

Considering the constraints mentioned and the uncertainty of load recovery paths before reconfiguration, an adjacency matrix for the loads and branches in the power system is constructed to assess the effectiveness of the reconfiguration scheme. The load–branch adjacency matrix method is an improved technique in power-system analyses based on matrix extension that evaluates the impact of loads on the system. This method comprehensively considers backup power paths and describes the interconnections of components in the system, and it can be applied to fault location, state assessment, and steady-state analysis. The matrix is established based on the system’s topology and electrical parameters.

1.

Principles for Generating the Load–Branch Matrix

(1)

Identifying Nodes and Branches: Nodes are the endpoints formed by the interconnection of two or more circuit components, while branches are the cable lines linking those endpoints.

(2)

Establishing the Topology: The topology of the power system is constructed based on the connections between the nodes and branches, which outlines the connections and paths in the system.

(3)

Constructing the Matrix: Based on the topology, a load–branch adjacency matrix is constructed. The connections between the nodes and branches are represented using 0 (disconnected) and 1 (connected), with the rows and columns of the matrix representing the nodes and branches, respectively.

Using Figure 13, which is a simplified diagram of the grid, the principle of generating the node matrix can be explained. In the diagram, there are 10 branches. Nodes 1 and 2 are feeder branches, and, among the four loads, L1 and L2 are critical loads with backup power supply lines.

In the diagram, it can be observed that feeder branch 1’s downstream branches are 3, 4, 5, 6, and 7, while feeder branch 2’s downstream branches are 3, 7, 8, 9, and 10. By listing the feeder branches as the first row of the matrix and searching downstream for connected branches, matrix A is obtained as follows:

$$A=\left[\begin{array}{cccccccccc}1& 0& 0& 0& 0& 2& 0& 0& 0& 0\\ 3& 4& 5& 6& 7& 3& 7& 8& 9& 10\end{array}\right]$$

(41)

By replacing the branch numbers with their corresponding capacities, the system capacity matrix CCC is obtained:

$$C=\left[\begin{array}{cccccccccc}{c}_1& 0& 0& 0& 0& c_2& 0& 0& 0& 0\\ c_3& c_4& c_5& c_6& c_7& c_3& c_7& c_8& c_9& c_{10}\end{array}\right]$$

(42)

The load–branch-related matrix analysis is as follows: If L1 is powered by backup branch 10, then $c_4=0$. If L2 is powered by normal branch 8, then $c_6=0$. If L3 is powered normally and L4 is powered off, then $c_9=0$, and matrix C is obtained:

$$C=\left[\begin{array}{cccccccccc}{c}_1& 0& 0& 0& 0& c_2& 0& 0& 0& 0\\ c_3& 0& c_5& 0& c_7& c_3& c_7& c_8& 0& c_{10}\end{array}\right]$$

(43)

In the reconstruction scheme, the total power consumption of the branch should be less than the capacity of the root branch, that is, ${\displaystyle \sum _{i\in \mathsf{\Omega }}{c}_i}\le C_n$, where $c_n$ is the maximum capacity of the branch, and $\mathsf{\Omega }$ is the set of all conducting branches under branch $c_n$. That is, $c_3+c_5+c_7\le c_1$, $c_3+c_7+c_8+c_{10}\le c_2$ in the matrix.

According to the simplified diagram of the ship power grid structure studied in this paper, the ship power grid matrix with each generator as the root can be obtained by using the load–branch adjacency matrix generation principle as follows:

$$G1=\left[\begin{array}{cccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 5& 6& 11& 0& 0& 20& 0& 0& 29& 32\\ 0& 7& 12& 17& 18& 21& 23& 24& 30& 33\\ 0& 8& 13& 0& 19& 22& 0& 25& 31& 0\\ 0& 0& 14& 0& 98& 44& 0& 26& 0& 0\\ 0& L1& L2& L4& L8& L10& L5& L6& L3& 0\end{array}\right]$$

(44)

$$G2=\left[\begin{array}{cccccccccc}2& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 34& 35& 38& 0& 0& 45& 0& 0& 55& 57\\ 0& 36& 39& 40& 41& 46& 47& 52& 56& 58\\ 0& 37& 28& 0& 42& 0& 48& 53& 63& 0\\ 0& 0& 27& 0& 43& 0& 49& 54& 0& 0\\ 0& L3& L6& L9& L10& L11& L15& L16& L20& 0\end{array}\right]$$

(45)

$$G3=\left[\begin{array}{cccccccccc}3& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 59& 60& 64& 0& 0& 70& 0& 0& 79& 81\\ 0& 61& 65& 66& 69& 71& 72& 77& 80& 82\\ 0& 62& 0& 67& 51& 0& 73& 78& 87& 0\\ 0& 0& 0& 68& 50& 0& 74& 93& 0& 0\\ 0& L20& L18& L16& L15& L17& L14& L12& L19& 0\end{array}\right]$$

(46)

$$G4=\left[\begin{array}{cccccccccc}4& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 83& 84& 88& 0& 0& 95& 0& 0& 102& 103\\ 0& 85& 89& 90& 94& 100& 96& 101& 10& 104\\ 0& 86& 0& 91& 76& 0& 97& 16& 9& 0\\ 0& 0& 0& 92& 75& 0& 98& 15& 0& 0\\ 0& L19& L13& L12& L14& L7& L8& L2& L1& 0\end{array}\right]$$

(47)

Its state matrix S is as follows:

$$S=\left[\begin{array}{cccccccccccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15& 16& 17& 18& 19& 20\\ 1& 1& 1& 0& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 1& 1& 0& 0& 1& 1\\ x_1& x_2& x_3& x_4& x_5& x_6& x_7& x_8& x_9& x_{10}& x_{11}& x_{12}& x_{13}& x_{14}& x_{15}& x_{16}& x_{17}& x_{18}& x_{19}& x_{20}\\ y_1& y_2& y_3& 0& 0& y_6& 0& y_8& 0& y_{10}& 0& y_{12}& 0& y_{14}& y_{15}& y_{16}& 0& 0& y_{19}& y_{20}\end{array}\right]$$

(48)

The first row of the matrix represents the load number. The second row uses 0 or 1 to indicate whether the load has a backup power supply path, where 1 represents the presence of a backup path, and 0 represents its absence. The third row represents the status of the normal power supply path, with 1 indicating that the line is closed (in use) and 0 indicating that the line is open (not in use). The fourth row represents the status of the backup power supply path, where 1 indicates that the backup power path is being used, and 0 indicates that it is either not in use or that the load does not have a backup power supply path. As the usage status of the power supply paths (represented by the third and fourth rows) is uncertain, the variables $x$ and $y$ are used to represent this uncertainty.

2.

Applications of the Load–Branch Adjacency Matrix

(1)

Fault Localization: The load–branch matrix can clearly indicate the relationship between loads and branches, allowing researchers to quickly determine fault points in the grid. After a fault occurs in the ship’s power grid, the matrix can be used to identify the fault node and its corresponding number, enabling quick determination of the reconfiguration path and reducing calculation time and complexity [24].

For example, if branches 88 and 20 have a circuit-breaker fault, the power supply status matrix of 20 loads is as follows:

$$\left[\begin{array}{cccccccccccccccccccc}L1& L2& L3& L4& L5& L6& L7& L8& L9& L10& L11& L12& L13& L14& L15& L16& L17& L18& L19& L20\\ x_1& x_2& x_3& 1& 0& 2& 1& x_8& 1& 1& 1& 2& 0& 1& x_{15}& x_{16}& 1& 1& x_{19}& x_{20}\end{array}\right]$$

$x$ indicates that the load power supply line is optional, 0 indicates that the load loses power, 1 indicates that the load is powered by the normal line, and 2 indicates that the load is powered by the backup power supply line. Loads 4, 5, 7, 9, 11, 13, 17, and 18 are the third type of loads, which only have normal power supply lines, so the corresponding matrix elements are all 1. Among them, loads 5 and 13 are out of power due to branch failures, and the element is 0. Loads 6 and 12 are powered by the backup branch, and the element is 2. Loads 10 and 14 can only be powered by the normal line due to backup branch failures, and the element is 1.

(2)

Capacity Constraints Expression: According to the matrix, the method of limiting the capacity in the load–branch matrix is as follows: search the right neighbor of the matrix element from the second row of the matrix to the right. If the right neighbor is 0, record this node until a non-zero element is found. For example, the root branches are 1, 2, 3, and 4. The main branches under the root branch are 11, 21, 38, 45, 64, 70, 88, and 95. Take branch 88 as an example. L13 is powered by an independent line, and L12 and L14 have backup power supply lines. Then, the power situation of branch 88 is

$$P_{88}=k_{13}\times p_{13}+(2-k_{12})\times k_{12}\times p_{12}+(k_{14}-1)\times k_{14}/2\times p_{14}$$

(49)

$k_{12}$, $k_{13}$, and $k_{14}$ represent the power supply of the corresponding loads and $p_{12}$, $p_{13}$, and $p_{14}$ represent the power of the corresponding loads. The remaining branches can be introduced in sequence.

(3)

Generator Efficiency Balance: The output power of the generator can be adjusted by controlling the state of the switch to achieve a balanced load distribution and maximize the efficiency of the generator. Take generator No. 1 as an example:

$$P{G}_1=P_{11}+P_{20}+(2-k_1)\times k_1\times p_1+(k_3-1)\times k_3/2\times p_3$$

(50)

The same can be said for the remaining generators.

The generator balance rate formula is

$${\eta }_i={\displaystyle \frac{P{G}_i}{P{G}_N}}$$

(51)

(4)

Uniqueness of load power supply line

In the S matrix,

$$x_i+y_i=1$$

(52)

3.4. Discretization of the Improved Dung Beetle Optimization for Fault Reconfiguration of Ship Power Systems

As the algorithm is ultimately applied to the multi-objective optimization problem of reconfiguring ship power systems, the reconfiguration scheme provided involves combinations of switch actions. Therefore, it is necessary to discretize the improved Dung Beetle Optimization, which deals with continuous variables [25], to adapt it to the 20-dimensional variable composed of switches.

For critical loads (Classes 1 and 2),

$$X_{i,j}=\left\{\begin{array}{ll}0,\hfill & \mathrm{if}\hspace{1em}{x}_{i,j}\in [0,0.5)\hfill \\ 1,\hfill & \mathrm{else}\hspace{1em}\mathrm{if}\hspace{1em}{x}_{i,j}\in [0.5,1.5]\hfill \\ 2,\hfill & \mathrm{else}\hspace{1em}\mathrm{if}\hspace{1em}{x}_{i,j}\in (1.5,2]\hfill \end{array}\right.$$

(53)

For general loads (Class 3),

$$X_{i,j}=\left\{\begin{array}{ll}0,\hfill & \mathrm{if}\hspace{1em}{x}_{i,j}\in [0,0.25)\hfill \\ 1,\hfill & \mathrm{else}\hspace{1em}\mathrm{if}\hspace{1em}{x}_{i,j}\in [0.25,1]\hfill \end{array}\right.$$

(54)

Here, $X_{i,j}$ represents the discretized dung beetle individual. A value of 0 indicates a power loss for the load, 1 indicates supply through the normal path, and 2 indicates supply through the backup path.

4. Simulation Verification of Fault Reconfiguration in Ship Power Systems

In this simulation, the population size is set to N = 40, with four types of dung beetle roles in the population distributed in a 1:1:1:1 ratio. The upper and lower boundaries are [0, 1], and the maximum number of iterations is 50. Each algorithm is tested 50 times to ensure the reliability of the results. The optimal individual is described by a combination of 20 numbers composed of 0, 1, and 2, where 0 represents a power loss, 1 represents the normal power supply, and 2 represents the backup power supply. This combination represents the power supply status of 20 loads under the constraints of the power grid when the comprehensive objective function is minimized. Additionally, curves showing the relationship between the number of iterations and the number of switching actions, as well as the relationship between the number of iterations and the best fitness value, are output to analyze the effectiveness of the algorithm in the reconfiguration of the ship’s power system. This serves as a basis for analyzing and comparing the algorithm’s performance in the reconfiguration of the ship power system.

4.1. Specific Steps of Algorithm Implementation

Step 1: Load Encoding

The final result of fault reconfiguration in ship power systems is a combination of switch actions. Loads are encoded as follows: 0 represents a power loss, 1 represents power supplied through the normal path, and 2 represents power supplied through the backup path.

Step 2: Connectivity Check

The fault nodes are input, and a connectivity check is performed.

Step 3: Population Initialization

The following are set: the population size N, the maximum number of iterations T, the proportions for the four types of dung beetles, the upper boundary Ub, the lower boundary Lb, and the dimensionality D of the objective function. Using the elite population strategy, the initial population is created by combining Latin hypercube sampling with random initialization. The fitness of each initial dung beetle is calculated and ranked, and the top Popsize elite individuals are selected. The best position is recorded [26].

Step 4: Position Update and Fitness Calculation

Rolling dung beetles update their positions according to Equation (2). If an obstacle is encountered, their positions are updated according to Equation (3). Egg-laying dung beetles update their positions based on Equations (4), (6) and (11). Foraging dung beetles update their positions using Equations (7), (9) and (11). Thieving dung beetles update their positions according to Equations (14) and (15). The fitness of the updated population is recalculated, and the global optimum is found. Boundaries are checked, and any individuals that exceed the boundaries are removed.

Step 5: Multi-strategy Differential Mutation

The population is divided into three categories based on fitness, and each category applies a different differential evolution operator. The high-fitness population updates their positions using Equations (16) and (17), the medium-fitness population updates their positions using Equations (18) and (19), and the low-fitness population updates their positions using Equations (20) and (21).

Step 6: Iteration Check

If the maximum number of iterations TTT is reached, proceed to Step 7; otherwise, return to Step 4.

Step 7: Optimization Complete

The algorithm ends, outputting the best position (switch combination) and fitness value, as well as the number of switch actions [27].

A flowchart is shown in Figure 14.

4.2. Optimization Results and Analysis for Fault Reconfiguration of Ship Power Systems

4.2.1. Fault Condition 1

Based on the model diagram, the conditions for Fault 1 are as follows: lines 38 and 70 have a fault and are disconnected. Fault nodes are identified for connectivity. Due to the disconnection at Node 38, L9 has no backup circuit and cannot receive power. L10 is supplied through a backup line, while L6, with its backup circuit faulted, can only be supplied through the normal line. Due to the damage to line 70, L17 has no backup circuit and cannot receive power, while L14 can be supplied through a backup line, and L12, with its backup circuit faulted, can only be supplied through the normal line. The partial restoration paths for some switches are as follows:

$$\left[\begin{array}{cccccccccccccccccccc}L1& L2& L3& L4& L5& L6& L7& L8& L9& L10& L11& L12& L13& L14& L15& L16& L17& L18& L19& L20\\ x_1& x_2& x_3& x_4& x_5& 1& x_7& x_8& 0& 2& x_{11}& 1& x_{13}& 2& x_{15}& x_{16}& 0& x_{18}& x_{19}& x_{20}\end{array}\right]$$

Reconfiguration simulations were performed for Fault Condition 1 using LESSDBO, DBO, MSCPSO, and the GA, with 50 repeated trials. The results are as follows: in Figure 15, Figure 16, Figure 17 and Figure 18, the vertical axis values 0, 1, and 2 represent the power loss of the load, the normal circuit supply, and the backup circuit supply, respectively. Figure 19 shows a comparison of the number of iterations and optimal switch actions for the four algorithms under this fault condition, while Figure 20 shows a comparison of the number of iterations and the best fitness values for the four algorithms. Each of the four algorithms is represented by a different colored curve. Table 2 summarizes the optimization comparison of the four algorithms under this fault condition.

1.

Comparison of Optimal Load Supply Schemes for Four Optimization Algorithms under Fault Condition 1

Figure 15. DBO optimal individual.

Figure 15. DBO optimal individual.

Figure 16. MSCPSO optimal individual.

Figure 16. MSCPSO optimal individual.

Figure 17. GA optimal individual.

Figure 17. GA optimal individual.

Figure 18. LESSDBO optimal individual.

Figure 18. LESSDBO optimal individual.

2.

Comparison of iterations and switch actions for different algorithms

Figure 19. Comparison of iterations and switch actions for different algorithms.

Figure 19. Comparison of iterations and switch actions for different algorithms.

3.

Comparison of fitness values over iterations for different algorithms

Figure 20. Comparison of fitness values over iterations for different algorithms.

Figure 20. Comparison of fitness values over iterations for different algorithms.

4.

Comparison of optimization results of several algorithms under fault conditions

Table 2. Comparison of optimization results of several algorithms under fault conditions.

Algorithm	DBO	MSCPSO	GA	LESSDBO
Optimal individual	11111111021112110121	12111111021112110121	11111111021102110121	11111111021112110111
Optimal reconstruction scheme	L10, L14, and L19 backup power supply; L9 and L17 power failure	L2, L10, L14, and L19 backup power supply; L9 and L17 power failure	L10, L14, and L19 backup power supply; L9 and L17 power failure	L10 and L14 backup power supply; L9 and L17 power failure
Minimum number of switches	6	8	6	4
Optimal convergence algebra	13	10	15	7
Power loss	370	370	370	370
Optimal convergence probability	90%	85%	100%	100%

Table 2. Comparison of optimization results of several algorithms under fault conditions.

Algorithm	DBO	MSCPSO	GA	LESSDBO
Optimal individual	11111111021112110121	12111111021112110121	11111111021102110121	11111111021112110111
Optimal reconstruction scheme	L10, L14, and L19 backup power supply; L9 and L17 power failure	L2, L10, L14, and L19 backup power supply; L9 and L17 power failure	L10, L14, and L19 backup power supply; L9 and L17 power failure	L10 and L14 backup power supply; L9 and L17 power failure
Minimum number of switches	6	8	6	4
Optimal convergence algebra	13	10	15	7
Power loss	370	370	370	370
Optimal convergence probability	90%	85%	100%	100%

From the simulation results, it can be seen that the reconfiguration schemes of the power system obtained using all four algorithms under Fault Condition 1 are “11111111021112110121”, which means that L10, L14, and L19 are supplied by backup power, while L9 and L17 are without power, with a total power outage of 370A. Additionally, the simulation results show that DBO converged in 13 iterations, achieving the optimal reconfiguration scheme with a minimum of six switch actions and a convergence probability of 90%; MSCPSO converged in 10 iterations, achieving the optimal reconfiguration scheme with a minimum of eight switch actions and a convergence probability of 85%; the GA converged in 15 iterations, achieving the optimal reconfiguration scheme with a minimum of six switch actions and a convergence probability of 100%; and LESSDBO converged in 7 iterations, achieving the optimal reconfiguration scheme with a minimum of four switch actions and a convergence probability of 100%.

4.2.2. Fault Condition 2

In Fault Condition 2, generator unit G4 fails to start. Fault nodes are identified for connectivity, resulting in L7 and L13 having no backup circuits and being unable to receive power. The normal circuits of L8, L12, and L19 are damaged, while the backup supply lines for L1, L2, and L14 are damaged. The partial restoration paths for some switches are as follows:

$$\left[\begin{array}{cccccccccccccccccccc}L1& L2& L3& L4& L5& L6& L7& L8& L9& L10& L11& L12& L13& L14& L15& L16& L17& L18& L19& L20\\ 1& 2& x_3& x_4& x_5& x_6& 0& 2& x_9& x_{10}& x_{11}& 2& 0& 1& x_{15}& x_{16}& x_{17}& x_{18}& 2& x_{20}\end{array}\right]$$

Using the same parameters, reconfiguration simulations for the fault were conducted with LESSDBO, DBO, MSCPSO, and the GA, with 50 repeated trials. The results are as follows: In Figure 21, Figure 22, Figure 23 and Figure 24, the vertical axis values 0, 1, and 2 represent the power loss of the load, the normal circuit supply, and the backup circuit supply, respectively. Figure 25 shows a comparison of the number of iterations and optimal switch actions for the four algorithms under this fault condition, while Figure 26 shows a comparison of the number of iterations and the best fitness values for the four algorithms. Each of the four algorithms is represented by a different colored curve. Table 3 summarizes the optimization comparison of the four algorithms under this fault condition.

(1)

Optimal Load Supply Schemes for Four Algorithms under Fault Condition 2

Figure 21. Optimal load supply scheme for DBO under Fault Condition 2.

Figure 21. Optimal load supply scheme for DBO under Fault Condition 2.

Figure 22. Optimal load supply scheme for MSCPSO under Fault Condition 2.

Figure 22. Optimal load supply scheme for MSCPSO under Fault Condition 2.

Figure 23. Optimal load supply scheme for GA under Fault Condition 2.

Figure 23. Optimal load supply scheme for GA under Fault Condition 2.

Figure 24. Optimal load supply scheme for LESSDBO under Fault Condition 2.

Figure 24. Optimal load supply scheme for LESSDBO under Fault Condition 2.

(2)

Comparison Chart of the Relationship Between Iteration Counts and Optimal Switch Actions Under Fault Condition 2

Figure 25. Comparison chart of the relationship between iteration counts and optimal switch actions for the four algorithms under Fault Condition 2.

Figure 25. Comparison chart of the relationship between iteration counts and optimal switch actions for the four algorithms under Fault Condition 2.

(3)

Comparison Chart of Iteration Counts and Optimal Fitness Values Under Fault Condition 2

Figure 26. Comparison chart of iteration counts and optimal fitness values for the four algorithms under Fault Condition 2.

Figure 26. Comparison chart of iteration counts and optimal fitness values for the four algorithms under Fault Condition 2.

(4)

Optimization Comparison of the Four Algorithms Under Fault Condition 2

Table 3. Optimization comparison of the four algorithms under Fault Condition 2.

Algorithm	DBO	MSCPSO	GA	LESSDBO
Optimal individual	1111120011 1201110122	1111010211 1201110021	1111010211 1201110021	11111200111201110122
Optimal reconstruction scheme	L6, L12, L19, and L20 backup power supply; L7, L8, L13, and L17 power failure	L8, L12, and L19 backup power supply; L5, L7, L13, L17, and L18 power failure	L8, L12, and L19 backup power supply; L5, L7, L13, L17, and L18 power failure	L6, L12, L19, and L20 backup power supply; L7, L8, L13, and L17 power failure
Minimum number of switches	9	8	9	8
Optimal convergence algebra	13	11	14	9
Power loss	595	820	820	595
Optimal convergence probability	96%	92%	94%	100%

Table 3. Optimization comparison of the four algorithms under Fault Condition 2.

Algorithm	DBO	MSCPSO	GA	LESSDBO
Optimal individual	1111120011 1201110122	1111010211 1201110021	1111010211 1201110021	11111200111201110122
Optimal reconstruction scheme	L6, L12, L19, and L20 backup power supply; L7, L8, L13, and L17 power failure	L8, L12, and L19 backup power supply; L5, L7, L13, L17, and L18 power failure	L8, L12, and L19 backup power supply; L5, L7, L13, L17, and L18 power failure	L6, L12, L19, and L20 backup power supply; L7, L8, L13, and L17 power failure
Minimum number of switches	9	8	9	8
Optimal convergence algebra	13	11	14	9
Power loss	595	820	820	595
Optimal convergence probability	96%	92%	94%	100%

From the simulation results under Fault Condition 2, it can be seen that the power-system reconfiguration schemes obtained by the four algorithms are as follows: DBO and LESSDBO obtained the reconfiguration scheme “11111200111201110122”, which means that L6, L12, L19, and L20 are supplied by backup power, while L7, L8, L13, and L17 are without power, with a total power outage of 595A. MSCPSO and the GA obtained the reconfiguration scheme “11110102111201110021”, which means that L6, L12, L19, and L20 are supplied by backup power, while L8, L12, and L19 are also supplied by backup power, and L5, L7, L13, L17, and L18 are without power, with a total power outage of 820A. Additionally, the simulation results show that DBO converged in 13 iterations, achieving the optimal reconfiguration scheme with a minimum of 13 switch actions and a convergence probability of 96%; MSCPSO converged in 11 iterations, achieving the optimal reconfiguration scheme with a minimum of 8 switch actions and a convergence probability of 92%; the GA converged in 14 iterations, achieving the optimal reconfiguration scheme with a minimum of 9 switch actions and a convergence probability of 99%; and LESSDBO converged in 9 iterations, achieving the optimal reconfiguration scheme with a minimum of 8 switch actions and a convergence probability of 100%.

5. Conclusions and Future Work

In summary, in the application of fault reconfiguration for ship power systems, LESSDBO outperformed the other algorithms in terms of convergence accuracy, convergence speed, and global search capability. Specifically, in the reconfiguration under Fault 1, LESSDBO achieved optimal convergence in seven iterations, reducing the convergence iterations by more than 30% compared with the other algorithms. In the reconfiguration under Fault 2, LESSDBO achieved optimal convergence in eight iterations, reducing the convergence iterations by more than 23% on average compared with the other algorithms. Additionally, in the reconfiguration under Fault Condition 1, LESSDBO achieved a minimum of four switch actions, which is 33% fewer on average than the other algorithms. In the reconfiguration under Fault Condition 2, LESSDBO achieved a minimum of eight switch actions, which is a 5.9% reduction compared with the other algorithms. Furthermore, LESSDBO obtained the optimal reconfiguration solution in all 50 trials for both Faults 1 and 2, demonstrating a 100% optimal convergence probability, which significantly enhanced the robustness and superiority of the algorithm. This study is suitable for power-system researchers, marine engineers, and intelligent algorithm developers. It provides valuable insights for ship power-system fault reconfiguration, intelligent optimization, and system reliability improvement, and it is expected to be applied and promoted in practice [28].

However, there is still room for improvement in existing research. In practical applications, it is also necessary to consider continuous dynamic mutations of the system. Faced with the complex and variable marine environment, load and fault modes are characterized by randomness and dynamics, making it crucial to consider continuous dynamic mutations to improve the robustness and adaptability of the algorithm [29]. Ensuring stability and efficiency under different working conditions will enhance the applicability and practical value of the algorithm in real engineering scenarios.

The next step could involve establishing an engineering verification platform. A hardware environment simulating a ship power system could be built in a laboratory, integrating a software system for fault detection, optimization control, and result analysis. Various typical fault scenarios could be designed for experimental verification to evaluate the response capability and recovery effectiveness of the LESSDBO algorithm under different faults. Based on engineering verification, representative small- and medium-sized vessels could be selected for real-ship verification to validate the applicability and superiority of the algorithm in real environments. Testing could include fault simulation and load recovery under different conditions, recording key performance metrics such as fault response time and recovery efficiency, to ensure sufficient reliability of the algorithm in complex real environments. In addition, emergency plans should be developed to ensure testing safety, and the advantages of the LESSDBO algorithm could be verified through comparative data analysis.

The future direction of the development of intelligent topology reconfiguration technology for ship power systems includes real-time monitoring and data-driven approaches, autonomous distributed reconfiguration systems, the integration of energy efficiency and green technologies, and system optimization using simulation and digital twin technologies. Intelligent research on ship power systems will also become an important part of enhancing the level of ship intelligence [30].