A Method for Reconstructing State Information of Ship Integrated Power System Distribution Networks Under Incomplete Data Conditions

Abstract

To address non-complete data issues such as missing, distorted, redundant and chaotic data in distribution network condition monitoring data easily caused by ship faults (e.g., sensor failure, cable short circuit, etc.), an improved optimization algorithm (IPS-SH5N1) considering the demand for incomplete information reconstruction in the distribution network of ship integrated power systems is proposed based on the SH5N1 meta-heuristic optimization algorithm. This algorithm is used to optimize the weights and thresholds of a BPNN, achieving accurate completion of non-complete information in the distribution network. The IPS-SH5N1-BPNN is applied to conduct information reconstruction verification on 2000 groups of incomplete data samples of ship integrated power systems. The results show that the information reconstruction accuracy (MSE) of this method is improved by more than 90% compared with the BPNN; the model training convergence time and global optimization time consumption are significantly shortened, and the average online single information reconstruction time is reduced to 4.45 ms. This method has core advantages of high precision, fast convergence, strong robustness and excellent real-time performance, which can provide reliable technical support for the intelligent operation and maintenance of ship integrated power systems.

1. Introduction

The ship integrated power system (SIPS) is the core development direction of marine power systems and has now gradually entered the engineering application stage of the second-generation medium-voltage direct current (MVDC) integrated power system. With advantages such as high power density, strong operational flexibility, and remarkable energy-saving effects, the ship integrated power system has been widely applied in various high-performance vessels including large ocean-going cargo ships, playing an irreplaceable role in promoting the upgrading of ship equipment.

However, the ship integrated power system operates for a long time in harsh environments characterized by high temperature, high humidity, high vibration and strong electromagnetic interference. It is affected by factors such as sensor hardware failures, complex electromagnetic interference, communication link defects and severe navigation conditions. This leads to problems including missing sensor data, noise and outlier contamination, and structural system damage, creating an incomplete information scenario of “unknown measurement + unknown topology”, which severely undermines the integrity and accuracy of data. The existence of incomplete information not only affects the authenticity of system condition monitoring and the reliability of stability analysis but may also lead to the failure of system control strategies in extreme cases, triggering serious safety accidents such as damage to power electronic equipment and power interruption to key loads, directly threatening the navigation safety of the ship [1,2].

Moreover, traditional state estimation methods rely on accurate system models and complete measurement data, making it difficult to handle extreme scenarios of “unknown measurements + unknown topology”. Data-driven methods represented by backpropagation neural networks (BPNNs) possess strong nonlinear fitting capabilities, yet they still suffer from issues such as high sensitivity of model performance to parameters like initial weights and thresholds and a strong tendency to fall into local optimal solutions during training. Therefore, conducting research on information reconstruction methods under incomplete information conditions carries important theoretical significance and urgent engineering application value. Efficient and accurate information reconstruction methods can achieve the precise completion of incomplete data and rapid reconstruction of system state information, helping to improve the safe and reliable operation capability of ship integrated power systems and promote the intelligent development of marine equipment.

To address this problem, this paper proposes an information reconstruction method based on the SH5N1 optimization algorithm (IPS-SH5N1) that considers the incomplete information reconstruction requirements of the ship integrated power system distribution network and the BPNN, providing a technical solution for the processing of incomplete state information of the ship integrated power system distribution network.

Foreign-related research started earlier, forming a research system from traditional state estimation to the integration of intelligent algorithms. The core focus is on algorithm optimization and engineering practicality, and the research focus has extended to scenarios with incomplete data [3]. Its methods are all derived from improvements in general power system state estimation technology.

In the traditional stage, foreign scholars mainly used interpolation methods, Kalman filtering and its improved algorithms to deal with data problems [4,5]. Although early interpolation methods were simple and efficient, they struggled to handle complex data missing in scenarios with incomplete data [6,7]; although Extended Kalman filter (EKF) and unscented Kalman filter (UKF) improved accuracy, they still could not meet engineering needs in cases of large-scale incomplete data [8,9,10].

In recent years, research has shifted towards the integration of intelligent algorithms. By optimizing model parameters such as neural networks through Particle Swarm Optimization (PSO) and Genetic Algorithms (GA), the reconstruction performance has been improved [3,11,12,13,14]. For example, PSO-optimized BPNNs can reduce reconstruction errors, and GA-optimized models can achieve rapid completion [15,16], both of which draw on research paradigms related to power systems [11,17].

In terms of the coordination between fault propagation and information reconstruction, foreign scholars have drawn on graph theory modeling methods for power systems to capture fault propagation paths and combined Bayesian networks to improve the robustness of reconstruction [18,19,20,21,22,23]. At the same time, attention is paid to engineering verification, and some achievements have been applied to high-performance ships [3,24]. Currently, its shortcomings are that specific optimization algorithms have not yet been applied to ship medium-voltage DC systems, and there is still a lack of research on collaborative reconstruction in extreme scenarios with incomplete data [25,26,27,28].

Domestic research started relatively late. In the early stage, it drew on onshore power systems and foreign achievements [3,25] and relied on the development of domestic state estimation technology [1,29] to gradually form its own characteristics. The core breakthroughs lie in the optimization of intelligent algorithms and the adaptation to scenarios with incomplete data. However, there are problems such as unbalanced research and insufficient engineering application [3,25].

Early domestic research focused on improving traditional methods, using interpolation, Kalman filtering, and other techniques to reconstruct data, which optimized the adaptability of the methods. However, it did not fully consider the particularity of incomplete ship data, making it difficult to handle complex incomplete data [1,4,5,6,7,8,30,31].

With the popularization of intelligent algorithms, domestic research has focused on the application of models such as BPNNs and LSTM. By improving the network structure, the risk of overfitting is reduced, and the completion accuracy is improved [11,18,21,32,33,34,35]. In terms of algorithm fusion, PSO and GA are used to optimize neural network parameters, and the anti-interference ability of the model is enhanced by combining the characteristics of fault propagation, which fills the gap in related research on medium-voltage DC systems [3,13,14,15,18,20,21,36].

There are four major problems in current domestic research: first, there is little research on extreme data incomplete scenarios [26,27,34,35]; second, the collaborative optimization of fault propagation and reconstruction is insufficient [25,26,27,37]; third, there is a lack of multi-algorithm integration models, and engineering costs are not fully considered [3,19,25,33]; fourth, the degree of engineering application is low, and most remain in the simulation stage [3,18,25,34].

Both domestic and foreign research have gone through the development process of “traditional methods—intelligent algorithms”, with the core goal of improving reconstruction accuracy and anti-interference ability [3,26]. Foreign countries have profound technical accumulation and focus on engineering verification, but there is insufficient research on extreme data incomplete scenarios [9,22,24,25]. Domestic development is rapid, with strong scenario adaptability, but the degree of engineering and algorithm integration is not as good as that of foreign countries [12,24,25,34].

The common research trends both domestically and internationally are: constructing multi-algorithm integrated models, focusing on extreme data incomplete scenarios, and promoting engineering applications [3,18,19,21,26,27,28,33,34]. Domestically, it is necessary to focus on breaking through key technologies of collaborative optimization and engineering in the future and learn from technologies such as PMU applications and bad data detection to improve data reliability [8,25,27,32].

2. Integrated Power System of Ships and Incomplete Data Types

2.1. Structure of Ship Integrated Power System

The topological structures of ship integrated power systems mainly include two typical types: radial and ring. Among them, the ring topology has the highest power supply reliability and is suitable for ships such as Ocean-going cargo ship that have high requirements for power supply reliability. The fault propagation paths of different topological structures vary significantly: the fault propagation path of the radial topology is single, while that of the ring topology is complex and has a wide range of influence. Therefore, this paper selects the integrated power system with a ring topology as the research object, and its topological structure is shown in Figure 1.

The integrated power system of a ship has inherent operating characteristics such as no external power grid support, large short-circuit current, frequent load fluctuations, and strong electromagnetic interference. When equipment failures or abnormal disturbances occur in the system, these characteristics are likely to cause incomplete phenomena such as data loss and distortion, which put forward higher requirements for the adaptability of information reconstruction methods, such as having strong anti-interference ability and fast convergence speed.

2.2. Incomplete Data Type

In an integrated shipboard power system, incomplete information refers to datasets that fail to fully and accurately reflect the actual operating state of the system. According to fault modes and damage characteristics, it can be classified into three categories: data missing, noise interference, and structural damage.

Data missing includes random scattered missing and large-scale continuous missing, caused by factors such as temporary sensor failures, instantaneous interference, and communication packet loss, characterized by no data uploaded from nodes with missing data.

Noise interference stems from complex electromagnetic interference, sudden load changes, and sensor parameter drift, which causes data values to deviate from the real operating state and easily leads to misjudgment of the system state. It is characterized by the addition of high-intensity Gaussian noise or multiple-type data distortion to the data of interfered nodes.

Structural damage is the most severe fault mode under extreme operating conditions, including distribution network line breakage, generator unit shutdown, failure of load modules and power electronic components, etc. It is characterized by topological changes, with the altered area unable to be sensed by sensors, forming an information black box of “unknown topology + unknown measurement”, posing the greatest difficulty for information reconstruction.

The three types of information incompleteness may occur individually or simultaneously. In actual damage scenarios of shipboard integrated power systems, the mixed incomplete scenario—where all three conditions occur at the same time—is often more likely to arise.

3. Information Reconstruction Method

3.1. BPNN Basic Model

The BPNN has strong nonlinear fitting ability and can achieve accurate mapping between input features of damage monitoring data and output targets of estimated values of system state parameters, making it the basic framework for information reconstruction of marine integrated power systems. The topological structure of the BPNN is shown in Figure 2. To adapt to the information reconstruction requirements of marine integrated power systems, a three-layer BPNN structure of input layer–hidden layer–output layer is adopted.

The training process of the BP neural network is as follows.

Construct the basic topological structure of a BP neural network and input the signals of each node of the system:

$$X_n=(x_1,x_2,\dots ,x_n)$$

(1)

the expected output is Y, representing different types of faults.

Initialize the parameters of the BP neural network. Set the initial weights between the input layer and the hidden layer as $w_{ij}$, the initial weights between the hidden layer and the output layer are $w_{jk}$, the number of neuron nodes in the input layer is $n$, the number of neuron nodes in the hidden layer is $l$, the number of neuron nodes in the output layer is $m$, the initial threshold of the hidden layer is $a$, and the initial threshold of the output layer is $b$.

The formula for the range of possible numbers of neuron nodes in the hidden layer is

$$l=\sqrt{m+n}+c$$

(2)

where $c$ is a constant with a value range between 1 and 10.

The calculation formula of the output function of hidden layer neurons is

$$z_j=f_1\left({\displaystyle \sum _{i=1}^l{w}_{ij}{x}_i}-a_k\right),\hspace{1em}k=1,2,\dots ,l$$

(3)

$f_1(x)$ is the activation function of the hidden layer.

The output layer represents the output type of residual current fault, and the output formula for each node in the output layer is

$$p_k=f_2\left({\displaystyle \sum _{j=1}^q{w}_{jk}}{z}_j+b_k\right),\hspace{1em}k=1,2,\dots n$$

(4)

$f_2(x)$ is the activation function of the output layer.

The error e is defined as the output value minus the expected value:

$$e_k=y_k-x_k,\hspace{1em}k=1,2,\dots n$$

(5)

The update formulas for weights and thresholds between layers are as follows:

$$\left\{\begin{array}{l}{w}_{jk}=w_{jk}+\eta z_j{e}_j\hfill \\ a_j=a_j+\eta z_j(1-z_j){\displaystyle \sum _{j=1}^m{w}_{jk}}{e}_k\hfill \\ b_j=b_k+e_k\hfill \end{array}\right.$$

(6)

where $\eta$ represents the learning rate.

During the training process, if the error reaches the expected value or the number of iterations reaches the upper limit, it indicates that the BPNN has completed the training. If neither of the above two conditions is met, the training continues.

The BPNN has strong nonlinear fitting ability and is suitable for the information reconstruction of ship integrated power systems. However, it has inherent limitations such as slow convergence speed and being prone to falling into local optimal solutions, which need to be optimized and improved.

3.2. IPS-SH5N1 Optimization Algorithm

The IPS-SH5N1 optimization algorithm is adaptively optimized based on the SH5N1 meta-heuristic optimization algorithm and is specifically designed to address the problem of incomplete information in the damage scenario of the integrated power system (IPS) shipboard distribution network. It aims to achieve the globally optimal configuration of the weights and thresholds of the BPNN for node parameter mapping of the entire system by simulating the completion rules of incomplete data and the reconstruction characteristics of state information in the ship power system. Meanwhile, it accurately quantifies the interference intensity of incomplete features under three core damage modes, namely data loss, noise interference, and structural damage, on system state monitoring and solves the defects of the BPNN such as being prone to local optima, slow convergence speed, and poor adaptability to information-incomplete scenarios.

Compared with the SH5N1 optimization algorithm, the IPS-SH5N1 optimization algorithm deeply integrates the inherent operating characteristics of the ship’s integrated power system with the incomplete data characteristics under three core damage modes. Through an adaptive search strategy and parameter adjustment mechanism, it achieves accurate completion of the ship’s incomplete data and efficient reconstruction of system state information, providing a targeted optimized solution for information reconstruction tasks in the emergency damage disposal and intelligent operation and maintenance of the ship’s integrated power system.

3.2.1. SH5N1 Optimization Algorithm [38]

Experimental results in reference [38] show that the SH5N1 optimization algorithm achieves significantly superior overall performance compared with 10 mainstream optimization algorithms. In tests on 33 standard benchmark functions, it ranks first in solution accuracy, optimization success rate and computational efficiency. It can effectively solve highly difficult optimization problems that other algorithms fail to handle and also demonstrates strong adaptability in high-dimensional scenarios and engineering design problems. Therefore, the SH5N1 optimization algorithm is adopted to optimize the initial weights and thresholds of the BPNN.

The SH5N1 optimization algorithm maps the search space of BPNN weights and thresholds to the host environment for virus transmission, treats the weight and threshold solutions to be optimized as individual viruses, and achieves a dynamic balance between global exploration and local exploitation of the optimal weights and thresholds for the BPNN by simulating viral infection and transmission, adaptive mutation, and environmental adaptation behaviors. The algorithm stores the positions of all virus individuals in a two-dimensional matrix $X=x_j^i$, where $i$ denotes the serial number of a virus individual and $j$ represents the dimension of the search space. The core mechanism consists of four parts: permutation mutation, position update, dynamic parameter adjustment, population position calibration, and global optimal update. The core parameters of the algorithm are shown in

Table 1. Algorithm control parameters.

Parameter Symbol	Physical Meaning of Parameters	Value Range
$P_1$	The infection probability threshold of viruses in poultry, controlling the switching between the development/exploration stages	0.8~0.95
$P_2$	Host survival probability (virus mutation threshold), controlling the triggering of mutation behavior	0~1
$P_{attack}$	The random probability of a virus attacking a host	[[0,1]
$P_{adapt}$	Random parameters of viral environmental adaptability	[0,1]
$F_g$	Global optimal solution (the most suitable environment for virus reproduction)	/
$F_p$	Population local optimal solution matrix	/
$w$	Weight coefficient	/
$c$	Balance coefficient	/
$p$	Adaptive coefficient	/
$T$	Maximum number of iterations	Custom
$t$	Current iteration count	1 ≤ t ≤ T

.

The core mathematical models and mechanisms are as follows:

• Permutation mutation mechanism

Permutation mutation is the core of algorithms to avoid local optima. Through three index permutations and shifts of the optimal weight and threshold positions of the BPNN, three independent optimal position matrices $pmp1$, $pmp2$, and $pmp3$ are generated, preventing excessive reliance on a single global optimal solution.

$$\left\{\begin{array}{l}{\alpha }_1=\left(j_1,j_2,\dots ,j_n\right)\\ r{t}_1=\left(ro+{\sigma }_1\right)\%n\\ {\alpha }_2={\alpha }_1*\left(r{t}_1+1\right)\\ r{t}_2=\left(ro+{\sigma }_2\right)\%n\\ {\alpha }_3={\alpha }_2*\left(r{t}_2+1\right)\end{array}\right.$$

(7)

Among them, $j_1,j_2,\dots ,j_n$ is a random integer within interval [1, n], and $n$ represents the dimension of the search space; $ro$ is the rotation offset reference value, a reference parameter used for generating the rotation index, and is employed to rearrange the indices of the search dimensions, $ro\in [1,n-1]$. In addition, ${\sigma }_1=(i_1,i_2,i_3)$, $i_1$, $i_2$ and $i_3$ are random integers ranging from 1 to 3, realizing the cyclic shift of the index vector. The symbol % denotes the modulo operation. $rt$ is the rotated index array. The rotated index is obtained by adding a random offset to $ro$ and then taking the modulus with respect to the number of dimensions, which is used to rearrange the original search dimension indices and generate index offsets for different search paths. $\alpha$ is the permuted position index array, used to extract different search path matrices from the local optimal population. Based on the index vectors ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, the corresponding rows of the population local optimal solution matrix ${\mathit{F}}_p$ are retrieved to generate three optimal position matrices for BPNN weights and thresholds $pmp1$, $pmp2$, and $pmp3$.

2.

Position Update Mechanism

The algorithm judges the suitability of the current environment by comparing the virus attack probability $P_{attack}$ with the infection rate threshold $P_1$. It updates the positions of virus individuals in two modes, “favorable environment (local exploitation)” and “unfavorable environment (global exploration)”, so as to achieve a preliminary balance between exploration and exploitation. The core position update formula is shown in Equation (8):

$$x_j^i=\left\{\begin{array}{cc}pm{p}_{3,j}^i+R_j^i*\left(F_{g,j}-F_{p,j}^i\right)& P_{attack}<P_1\\ x_j^i+{\displaystyle \frac{R_j^i*\left(pm{p}_{1,j}^i-pm{p}_{2,j}^i\right)+R_j^i*\left(F_{g,j}-x_j^i\right)}{2}}& P_{attack}\ge P_1\end{array}\right.$$

(8)

where $P_{attack}$ is a random number within the range [0,1], simulating the random attacking behavior of viruses against hosts; $F_{g,j}$ is the $j$-dimensional value of the global optimal position; $F_{p,j}^i$ is the $i$-dimensional value of the local optimal position of the $j$-th virus individual; and $R_j^i$ is a random number matrix within the range [0,1], which introduces random disturbances for position updates to avoid solidification of the search process.

When viruses encounter extremely unfavorable environments such as strong host immune responses, the algorithm triggers an adaptive mutation mechanism. Through the mutation and recombination of viral genes, large-scale jumps in the search space are realized, further enhancing the algorithm’s ability to escape local optima. The trigger condition for the mutation mechanism is the viral adaptive probability $P_{adapt}$ ≤ $P_2$, where $P_{adapt}$ is a random number within the range [0,1], simulating the environmental adaptability of viruses.

$$x_{new,j}^i=\left\{\begin{array}{l}{x}_j^i+R_j^i*\left(x_{g,j}-x_j^i\right),P_{attack}<P_1;P_{adapt}\le P_2\hfill \\ x_j^i+R_j^i*\left(x_{g,j}-x_{p,j}^i\right),P_{attack}\ge P_1;P_{adapt}\le P_2\hfill \end{array}\right.$$

(9)

$$x_{mutate,j}^i=c*{\displaystyle \frac{\left(x_{new,j}^i+x_{old,j}^i\right)*r_3*w}{2}}$$

(10)

where $x_{old,j}^i$ represents the original position of the virus individual before mutation, $c$ is the exploration–exploitation balance coefficient, $r_3$ is a random number within the range [0,1], $w$ is the weight coefficient, and the values of $c$ and $w$ are updated in real time through a dynamic parameter adjustment mechanism.

3.

Dynamic Parameter Adjustment Mechanism

The SH5N1 algorithm realizes the smooth transition between global exploration and local exploitation during the iteration process through three core dynamic parameters, the weight coefficient $w$, the balance coefficient $c$, and the adaptive coefficient $p$, avoiding the problems of excessive exploration in the early stage and excessive exploitation in the later stage of the algorithm.

The weight coefficient $w$ is used to control the exploitation intensity of the algorithm and decays exponentially with the increase in the number of iterations.

$$w=w\ast [w_{min}+\left(w_{max}-w_{min}\right)*e^{-(t/T)}]$$

(11)

where $w_{max}$ and $w_{min}$ are the upper and lower limits of the weight coefficient, respectively, taking $w_{max}$ = 0.9 and $w_{min}$ = 0.1.

The balance coefficient $c$ is used to offset the excessive convergence tendency of the weight coefficient.

$$c=r_4*e^{-4*(t/T)}$$

(12)

where $r_4$ is a random number within the range [0,1] and $c$ gradually decays as the number of iterations increases; it takes a larger value in the early stage to encourage global exploration and a smaller value in the later stage to ensure the stability of local exploitation.

The adaptive coefficient $p$ is used to control the selection probability of mutation results.

$$p={\displaystyle \frac{1}{1+e^{-10*\left(\frac{t}{T}-0.5\right)}}}+r_5*(1-p)$$

(13)

where $r_5$ is a random number in the range [0,1]. In the early stage of iteration, the value of $p$ is relatively small, making the algorithm more prone to mutation and improving population diversity; in the late stage of iteration, $p$ approaches 1, and only mutation results with better fitness are accepted, ensuring the convergence stability of the algorithm.

4.

Population Position Calibration and Global Optimum Update Mechanism

After each position update and mutation operation, the algorithm performs final calibration on the positions of virus individuals to avoid invalid search, as shown in Equation (14):

$$x_j^i=x_{p,j}^i*m_{xp}+x_j^i*m_x$$

(14)

where $m_{xp}$ represents the mutation coefficient vector of the entire population and $m_x$ represents the mutation coefficient vector of the current virus individual.

Through a random selection mechanism, it is determined whether to update the mutated individual to the global optimal solution, as shown in Equation (15).

$$\left[F_g,X_g\right]=\left\{\begin{array}{c}\left[F\left(x_{mutate}\right),X_{mutate}\right],P_{adapt}\le P_2;p_s\le p\\ min(F(x)),others\end{array}\right.$$

(15)

where $F\left(x\right)$ represents the fitness function, $x_{mutate}$ represents the position of the mutated virus individual, and $p_s$ is a random number within the range [0,1]. The mutated individual is updated as the global optimal solution only when the trigger condition is satisfied and its fitness is superior; otherwise, the individual with the minimum fitness in the population is selected as the global optimal solution.

In summary, the iterative process of the SH5N1 algorithm is shown in Figure 3.

3.2.2. Adaptive Optimization Considering the Reconstruction Requirement of Incomplete Information for Ship IPS Distribution Network

Although the SH5N1 optimization algorithm has excellent global optimization capability, it has three core adaptation defects when directly applied to the reconstruction of incomplete data for ship IPS and the optimization of BPNN weights and thresholds:

• The core control parameters $P_1$ and $P_2$ take fixed values, which cannot adapt to the optimization requirements under different damage degrees, easily leading to slow convergence in mild damage scenarios and premature convergence in severe damage scenarios;
• It adopts full-dimensional undifferentiated permutation mutation without considering the importance difference of BPNN weights and thresholds, which destroys the effective information of core dimensions, increases a large number of invalid calculations, and results in the low efficiency of high-dimensional optimization;
• It adopts completely random population initialization without integrating the prior knowledge of BPNN structure and IPS system operation, leading to low early-stage optimization efficiency and many convergence iterations.

To address the three core adaptation defects of the original algorithm, this paper completes adaptive optimization by combining the inherent operating characteristics of ship IPS and the damage mode of incomplete data, forming a dedicated IPS-SH5N1 optimization algorithm, with the specific design as follows.

• Adaptive parameter dynamic adjustment mechanism based on system damage degree

Define the system damage degree index $\delta$ to quantify the severity of system incomplete data, as shown in Equation (16):

$$\delta ={\displaystyle \frac{N_{loss}+N_{distort}}{N_{total}}}$$

(16)

where $N_{total}$ represents the total number of sensor nodes in the system, $N_{loss}$ represents the number of data-missing nodes, $N_{distort}$ represents the number of data-distorted nodes, and the larger the value of $\delta$, the more severe the system damage.

Based on the damage degree $\delta$, the core control parameters $P_1$ and $P_2$ are dynamically and adaptively updated, as shown in Equations (17) and (18):

$$P_1(t)=P_{1,min}+(P_{1,max}-P_{1,min})\cdot (1-\delta )\cdot e^{-t/T}$$

(17)

$$P_2(t)=P_{2,min}+(P_{2,max}-P_{2,min})\cdot \delta \cdot (1-t/T)$$

(18)

where $P_{1,max}$ = 0.95, $P_{1,min}$ = 0.6, $P_{2,max}$ = 0.8, and $P_{2,min}$ = 0.2.

The core logic of this mechanism is as follows: the higher the damage degree $\delta$, the smaller the value of $P_1$, and the easier the algorithm is to trigger the global exploration mode, which is suitable for high-difficulty optimization tasks; the larger the value of $P_2$, the higher the mutation trigger probability, improving the algorithm’s ability to escape from local optima. In the later stage of iteration, $P_1$ gradually increases and $P_2$ gradually decreases, strengthening the local exploitation capability to ensure convergence accuracy and stability.

2.

Directional Mutation and Search Mechanism for Core Dimensions of BPNN

The weights between the input layer and the hidden layer that are strongly correlated with incomplete damage features are divided into core optimization dimensions, while the remaining bias thresholds and non-core weights are classified as non-core dimensions. An importance mask matrix $M=[m_1,m_2,\dots ,m_D]$ for BPNN weights and thresholds (where D represents the total dimension of BPNN weights and thresholds) is constructed to realize directional mutation optimization with decoupled core and non-core dimensions, as shown in Equation (19).

$$m_d=\left\{\begin{array}{ll}1,\hfill & \mathrm{Core} \mathrm{dimension}\hfill \\ 0,\hfill & \mathrm{Non-core} \mathrm{dimension}\hfill \end{array}\right.$$

(19)

The following is based on the mask matrix, conduct targeted optimization on the permutation mutation, position update, and variation mechanisms of the original algorithm.

When generating the optimal position matrix for $pmp1$, $pmp2$, and $pmp3$, keep the values of non-core dimensions unchanged and only perform permutation operations on core dimensions.

During position update and variation, only implement step-size scaling and random perturbation on core dimensions. The optimized variation formula is shown in Equation (20):

$$x_{new,j}^i=\left\{\begin{array}{ll}{x}_j^i+(1+\delta )\cdot R_j^i\cdot (x_{g,j}-x_{p,j}^i),\hfill & m_j=1 \mathrm{and} \mathrm{meet} \mathrm{the} \mathrm{original} \mathrm{conditions}\hfill \\ x_j^i,\hfill & m_j=0\hfill \end{array}\right.$$

(20)

The more severe the damage, the larger the mutation step size and the stronger the global exploration capability. This mechanism avoids invalid mutations in non-core dimensions, preserves effective weight information, and reduces the computational load of a single iteration.

3.

Population Initialization Mechanism with Prior Fusion

By integrating the prior structural information of the BPNN and the prior operation information of the IPS system, a hierarchical population initialization strategy is constructed, as shown in Equation (21):

$$X_i=\left\{\begin{array}{ll}{X}_{pre}+\delta \cdot randn(1,D),\hfill & 70\% \mathrm{probability}\hfill \\ X_{base}+0.1\cdot randn(1,D),\hfill & 20\% \mathrm{probability}\hfill \\ rand(1,D)\cdot (X_{max}-X_{min})+X_{min},\hfill & 10\% \mathrm{probability}\hfill \end{array}\right.$$

(21)

where $X_{base}$ is the reference value of Xavier initialization based on the BPNN structure; $X_{pre}$ is the BPNN weight threshold obtained by pre-training the steady-state data of the system before damage, which possesses prior information of system operation; D is the total dimension of the weight threshold; and $X_{max}$ and $X_{min}$ are the upper and lower limits of parameter search. This strategy enables the algorithm to approach the optimal solution region at the early stage of iteration, reducing the number of convergence iterations, while retaining 10% random individuals to ensure population diversity.

3.2.3. Information Reconstruction Method Based on IPS-SH5N1-BPNN

The process of the information reconstruction method based on the IPS-SH5N1-BPNN is shown in Figure 4, with the specific procedures as follows:

• Collect the IPS full-system node power flow monitoring data of ships, mark three types of incomplete damage data, and calculate the system damage degree;
• Perform Min-Max normalization on the original data and construct an importance mask matrix for BPNN weights and thresholds;
• Divide the dataset into a training set and a test set at a ratio of 4:1, initialize the BPNN network, and determine the total dimension of weights and thresholds as well as the search boundaries;
• Generate the initial population of the IPS-SH5N1 algorithm based on a hierarchical initialization strategy, set the core parameters of the algorithm, calculate the fitness value of each individual, and update the global optimal solution;
• Adaptively update $P_1$ and $P_2$ based on the damage degree and implement the directional permutation mutation and position update;
• After the iteration terminates, output the optimal combination of weights and thresholds and assign them to the BPNN to complete model training;
• Input the test set data into the trained model, output the full-system information reconstruction results, and complete the performance evaluation.

Figure 4. Information reconstruction process of IPS-SH5N1-BPNN.

Figure 4. Information reconstruction process of IPS-SH5N1-BPNN.

4. Analysis of Simulation and Experimental Results

4.1. Simulation Settings

Based on the MATLAB platform, a power flow model of the shipboard medium-voltage DC integrated power system is constructed, with the core parameters of the model referring to the actual technical indicators of China’s first-and-a-half generation and second-generation shipboard integrated power systems.

Aiming at the typical fault modes of data acquisition during the operation of the shipboard power system, four types of incomplete data scenarios are designed to verify the algorithm performance. The total sample size of the dataset is 2000 groups, which are divided into a training set and a test set at a ratio of 4:1. The sample distribution and division rules of each scenario are shown in

Table 2. Number of samples in each scenario.

	Complete Data	Random Missing	Noise Interference	Structural Damage	Mixed Incomplete
Training set	400	400	400	400	50
Testing set	0	100	100	100	50

.

Comparative experiments are conducted on three methods: the BPNN method, the SH5N1-BPNN method, and the IPS-SH5N1-BPNN method. The number of iterations for the SH5N1 optimization algorithm is set to 20. The training parameters of the BPNN for the three methods remain consistent, with a maximum number of training iterations of 1000, a learning rate of 0.1, and a training target error of 0.00001.

Four types of core evaluation indicators are selected to quantify algorithm performance: ① accuracy: Mean Squared Error (MSE); ② convergence: number of stable convergence iterations; ③ real-time performance: average time consumption for single information reconstruction; ④ robustness: stable convergence rate, error standard deviation and coefficient of determination of the test samples. The computer configuration is as follows: CPU, AMD Ryzen 9 7945 HX with Radeon Graphics 2.50 GHz; GPU, NVIDIA GeForce RTX 4060 Laptop GPU 8 GB; Memory, 32 GB 5200 MT/s. The program runs under MATLAB R2022b.

4.2. Result Analysis

The information reconstruction results of the training set and test set were analyzed. The four core evaluation indicators are shown in

Table 3. Comparison of convergence indexes of three methods.

	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
Number of iterations	17	17	15
Convergence time	23.16 s	14.87 s	12.58 s
Time consumption in algorithm optimization	/	2344.60 s	1857.32 s

,

Table 4. Comparison of accuracy indicators of three methods.

	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
MSE (×10−7)	107.52	13.26	5.76

,

Table 5. Comparison of real-time performance indicators of three methods.

	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
Average time consumed per single information reconstruction	25.90 ms	5.28 ms	4.45 ms

,

Table 6. Comparison of robustness indicators of three methods (stable convergence rate).

Damage Scenario	Stable Convergence Rate
	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
Random missing	100.0%	100.0%	100.0%
Noise interference	98.8%	100.0%	100.0%
Structural damage	82.5%	99.5%	100.0%
Mixed incompleteness	60.3%	94.5%	99.8%

,

Table 7. Comparison of robustness indicators of three methods (standard deviation of errors).

	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
Standard deviation of errors (×10−7)	12.57	0.74	0.16

and

Table 8. Comparison of robustness indicators of three methods (coefficient of determination).

	BPNN	SH5N1-BPNN	IPS-SH5N1-BPNN
Coefficient of determination	0.938	0.975	0.992

.

The comparative analysis results of the training set convergence indicators of the three methods are shown in Table 3. The three methods show little difference in the number of iterations. In terms of convergence time, the IPS-SH5N1-BPNN takes only 12.58 s to converge, which is 45.7% shorter than the 23.16 s of the BPNN and 15.4% shorter than the 14.87 s of the SH5N1-BPNN, demonstrating a significant advantage in training efficiency in high-dimensional complex scenarios. In terms of algorithm optimization time consumption, the full-system optimization time of the SH5N1-BPNN reaches 2344.60 s, while that of the IPS-SH5N1-BPNN is reduced to 1857.32 s, a decrease of 20.8%, which greatly lowers the one-time global optimization cost during the model training phase.

A comparative analysis of the accuracy-type indicators of the three methods is shown in Table 4. The average MSE of all convergent samples in the test set is calculated, the data involved in the accuracy calculation is in per-unit value, and the power base value is 1 MW.

The test results show that the reconstruction accuracy of the IPS-SH5N1-BPNN in the test set is comprehensively superior, with three error indicators significantly lower than those of the other two methods: MSE is reduced by 94.6% compared with the BPNN and 56.6% compared with the SH5N1-BPNN.

Limited by the gradient descent mechanism, the BPNN is prone to falling into local optimal solutions and has the largest reconstruction error; the SH5N1-BPNN optimizes network weights and thresholds through meta-heuristic global optimization, achieving an order-of-magnitude improvement in accuracy compared with the BPNN, while the IPS-SH5N1-BPNN proposed in this paper further enhances the precision of global optimization through an optimization strategy, effectively reducing the reconstruction error and verifying the effectiveness of the optimization strategy.

The comparison of the real-time performance indicators of the three methods on the test set is shown in Table 5. The average single reconstruction time of the BPNN reaches 25.90 ms, while the two optimized methods, the SH5N1-BPNN and IPS-SH5N1-BPNN, optimize the network weight thresholds through global optimization. The single reconstruction time is reduced to 5.28 ms and 4.45 ms, respectively representing a reduction of 79.6% and 82.8% compared with the BPNN and achieving a leapfrog improvement in real-time performance. The single reconstruction time of the IPS-SH5N1-BPNN is 15.7% shorter than that of the SH5N1-BPNN. Overall, both intelligent optimization methods can effectively solve the problem of insufficient real-time performance of the BPNN, and the IPS-SH5N1-BPNN achieves a slight optimization in computational efficiency compared with the SH5N1-BPNN.

Based on the comparison results of the test sets of the three methods in Table 6, the following conclusions can be drawn: in damage scenarios with random missing data and noise interference, both the IPS-SH5N1-BPNN and SH5N1-BPNN still maintain a 100% convergence rate, while the BPNN fails to converge, showing a clear performance differentiation among the three methods; in structural damage scenarios, the SH5N1-BPNN also fails to converge, whereas the IPS-SH5N1-BPNN still maintains a 100% convergence rate; and in mixed incomplete damage scenarios, the convergence rates of all three methods decrease to varying degrees, with the BPNN exhibiting the most significant decline, resulting in a remarkable performance gap among them. Overall, the convergence stability of the three methods is ranked as IPS-SH5N1-BPNN > SH5N1-BPNN > BPNN. The IPS-SH5N1-BPNN demonstrates prominent advantages in damage resistance and convergence stability and can better meet the engineering application requirements of corresponding scenarios.

The error standard deviation index is calculated using all convergent test set samples selected by three methods. The comparison of the error standard deviation indicators among the three methods is shown in Table 7. From the perspective of this robustness index, the BPNN model has the largest error standard deviation, with a high degree of dispersion in prediction results and poor stability, showing weak robustness when facing incomplete data; the error standard deviation of the BPNN model optimized by SH5N1 is significantly reduced, the model output fluctuation is decreased, and the robustness is obviously improved; and the error standard deviation of the BPNN model optimized by IPS-SH5N1 is further reduced to the lowest level, with the most stable prediction results, the strongest anti-interference ability against data disturbance and incomplete information, and the optimal overall robustness.

The comparison of the error standard deviation indicators among the three methods is shown in Table 8. A comparative analysis of the coefficient of determination of the three information reconstruction methods for the same mixed incomplete scenario among the convergent samples in the test set shows that the coefficient of determination of the basic BPNN method is 0.938, with moderate fitting performance and information restoration accuracy; the coefficient of determination of the BPNN method optimized by SH5N1 is improved to 0.975, with a significantly enhanced model fitting effect and reliability of information reconstruction. In contrast, the IPS-SH5N1-BPNN method proposed in this paper achieves a coefficient of determination of 0.992, which is closer to 1, indicating that this method has the highest fitting degree to real data in the scenario of mixed incomplete data, the best consistency between information reconstruction results and actual values, and optimal robustness and reconstruction accuracy.

5. Conclusions

Aiming at the technical bottleneck of information reconstruction under the scenario of incomplete data in ship integrated power systems, this paper proposes an SH5N1 optimization algorithm (IPS-SH5N1) that takes into account the demand for incomplete information reconstruction of distribution networks in ship integrated power systems. The algorithm is used to optimize the weights and thresholds of a BPNN, achieving accurate completion of incomplete information in distribution networks. Fully verified through multiple groups of comparative simulation experiments, this method is comprehensively superior to the BPNN and SH5N1-BPNN comparison methods in four core dimensions: convergence performance, reconstruction accuracy, real-time response capability and scenario robustness. It exhibits strong adaptability and stable reconstruction performance in the complex operating scenarios of ship integrated power systems with multiple power systems, multiple operating conditions and multiple damage levels. This method can provide a reliable technical solution for the accurate reconstruction of state information of ship integrated power systems under incomplete data and possesses high engineering application value.

In subsequent research, we will further expand the scope of the study. On the one hand, this method will be integrated with the multi-physical domain model of the overall ship giant system to carry out research on the full-ship and full-system information reconstruction covering the internal characteristics of generators and the internal characteristics of propulsion loads, further exploring the application potential of this method in ship-wide state awareness and cooperative control. On the other hand, in view of the topology change scenarios of ship power distribution networks, we will study incremental model update methods and explore how to complete the rapid adaptation of the model with minimal retraining costs when incremental topology changes occur in the power distribution network, further improving the engineering adaptability of this method.