Maintenance task affects the reliability and availability of equipment, which is the key factor to minimize failure time and reduce operation cost in the lifecycle of equipment [1]. Currently, the majority of methods of equipment maintenance are planned maintenance [2–10]. Among them, AKYUZ and CELIK designed an enhanced planned maintenance system (E-PMS) for a ship by using A’WOT, and their study had made a great contribution to improving the performance of equipment [2]. In practice, however, the planned maintenance is greatly influenced by the external environment and heavily depends on human effort, which leads to low efficiency and poor accuracy in the following two aspects: one is excessive maintenance, which means the unnecessary maintenance of better equipment, and the other one is insufficient maintenance, and the equipment has broken down before the maintenance period due to various reasons, but restricted by the maintenance plan, it has to operate with faults.

Therefore, fault prediction is the key to realizing the transformation of support mode from planned maintenance to predictive maintenance, which can perform the warning before the failure of equipment occurs. Also, more and more attention has been focused on fault prediction for equipment [11–22], such as fault prediction for the vehicle [11, 12], the watercraft [13–15], the aircraft engine [16–18], the power supply system [19, 20], and the track circuit [21–23]. Due to the complexity of watercraft structure and the diversity of the marine environment, it is challenging and difficult to study the fault prediction for watercraft equipment.

Over the past few years, a large number of methods were explored to predict the failure, such as the grey model [24, 25], the BP neural network [26, 27], the RBF neural network [28, 29], the data-driven model [30–33], deep learning [34], and the grey relational analysis method [35]. Although the grey models in [24, 25] were effective to a certain extent, they only considered the development of a single or several characteristic parameters independently. There were also some results that focused on theoretical research and had made some contributions [36, 37]. But their experimental data was generated from the simulation, which cannot represent the data characteristics of a real physical system. Meanwhile, a data-driven prediction method was introduced into the failure prognosis of marine diesel engines, and a discrete wavelet transform was applied to process the data based on the data characteristics [30]. In order to take multiple characteristic parameters into account comprehensively, the grey neural network model was first introduced into fault prediction for ship machinery in [38]. However, most of the existing works [30] only analyze whether the watercraft equipment will break down; they cannot explain the type of the fault and when the fault will occur. In response to the aforementioned problems, the healthy degree model based on grey relational analysis of watercraft equipment is proposed in this study, and it has been applied to a certain type of watercraft and the improvement of performance has been approved. The highlights of this study are summarized as follows.

∗ A novel healthy degree model is put forward by using grey relational analysis. The healthy state of watercraft equipment can be predicted by the value of the healthy degree. If the healthy degree is greater than 1, then the watercraft equipment will be healthy. If the healthy degree is less than 1, then the watercraft equipment will break down, the fault mode will be identified, and the fault occurrence time will be predicted.

The rest of this article is arranged as follows: at first, we describe the problem to be resolved in Section 2, which consists of the necessary notations and the data generation method. Then, the healthy degree model is introduced in Section 3 to predict whether the watercraft equipment needs to be repaired, what kind of equipment fault it is, and when the fault will occur. In Section 4, an example of fault prediction is provided for the engine equipment in a certain type of watercraft, and the simulation results are obtained to verify the validity and accuracy of the fault prediction. Finally, some concluding remarks are made to end this study.

The traditional grey relational coefficient used in [39] is calculated merely depending on the difference between two sequences. Actually, the area can represent the grey relational coefficient between two sequences; more obviously, the larger the area is, the smaller the grey relational coefficient will be. Then the area between the two sequences labeled x _i and y _j is described as follows: S i j = ∫ 0 t 1 x i − y j d t + ∫ t 1 t 2 x i − y j d t + ⋯ + ∫ t l − 1 t l x i − y j d t , (4) where l is the number of intersections between the two sequences, and intersections are expressed as t k k = 1,2 , … , l . The similarity whose monotonicity is opposite to the area between two sequences is structured as follows: δ i j = 1 n − 1 ∫ 2 n 1 1 + x i − y j d t . (5)

After that, the grey relational coefficient is represented as follows: r i j = r x i , y j = 1 1 + S i j 1 − δ i j , (6) where S _ij and δ _ij are the area and similarity between the two sequences, respectively. Clearly, the grey relational coefficient increases with the increase of similarity, while it decreases when the area is increased.

Finally, the healthy degree is expressed as follows: h i j = e − r i j − r i 0 . (7)

If all of h _ij is greater than 1 for any i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then the watercraft equipment is in a healthy state. If there exists any h _ij smaller than 1 for i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then the watercraft equipment will break down, and the failure mode can be identified as the jth failure mode.

Theorem 1

: For ∀ i ∈ 1,2 , … , n , j k k = 1,2,3 ∈ 1,2 , … , g , if h i j 1 ≤ h i j 2 and h i j 2 ≤ h i j 3 , then h i j 1 ≤ h i j 3 .

Proof: According to Eq. 7 , the following equations can be obtained: h i j k = e − r i j k − r i 0 , k = 1,2,3 . (8)

From the known condition h i j 1 ≤ h i j 2 , we have e − r i j 1 − r i 0 ≤ e − r i j 2 − r i 0 . On account of which the function f x = e − x is a decreasing function, then Eq. 9 can be obtained: r i j 1 ≥ r i j 2 . (9)

Based on Eq. 6 , Eq. 9 can be transformed into 1 1 + S i j 1 1 − δ i j 1 ≥ 1 1 + S i j 2 1 − δ i j 2 . Because the function f x = 1 x is a decreasing function, Eq. 10 can be acquired: S i j 1 1 − δ i j 1 ≤ S i j 2 1 − δ i j 2 . (10)

In the same way, Eq. 11 can be deduced: S i j 2 1 − δ i j 2 ≤ S i j 3 1 − δ i j 3 . (11)

On the basis of Eq. 10 and Eq. 11 , we can obtain S i j 1 1 − δ i j 1 ≤ S i j 3 1 − δ i j 3 . (12)

Then, we have 1 1 + S i j 1 1 − δ i j 1 ≥ 1 1 + S i j 3 1 − δ i j 3 , that is, r i j 1 ≥ r i j 3 . (13)

By virtue of which the function f x = e − x is a decreasing function, then Eq. 14 can be described as follows: e − r i j 1 ≤ e − r i j 3 . (14)

Multiplying both sides of Eq. 14 by the same positive number e r i 0 , we have e − r i j 1 − r i 0 ≤ e − r i j 3 − r i 0 , that is, h i j 1 ≤ h i j 3 . Henceforth, Theorem 1 can be proved.

Theorem 2

: If r _ij < r _i0 for ∀ i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then the equipment will be healthy h i j > 1 . If there exists r _ij > r _i0 for i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then the equipment will break down h i j < 1 .

Proof: Due to r _ij < r _i0 for ∀ i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then we can get r i j − r i 0 < 0 . (15)

On account of which the function f x = e − x is a decreasing function, then Eq. 15 can be changed as follows: e − r i j − r i 0 > 1 . (16)

Based on Eq. 7 , we can obtain h _ij > 1, that is, the equipment will be healthy.

With the same method, if there exists r _ij > r _i0 for i ∈ 1,2 , … , n and j ∈ 1,2 , … , g , then the equipment will be judged to break down. Thus, Theorem 2 can be proved.

A fault prediction example is given for the engine equipment in a certain type of watercraft. Three kinds of common faults are chosen to establish the healthy degree model. Fault 1 is excessive clearance of the crankpin bearing or main bearing, fault 2 is cooling water leakage, and fault 3 is propeller overload. Parameters of cylinder temperature (labeled CT), oil pressure (labeled OP), oil temperature (labeled OT), freshwater pressure (labeled FWP), freshwater temperature (labeled FWT), and exhaust temperature (labeled ET) are selected as the character parameters. According to their values of them in the states of health and three failure modes, the average value of them in each state is considered as the normalized vector which is described in Table 1 where P is the parameter and NV is the normalized vector.

Taking time as the horizontal axis and healthy degree as the vertical axis, three kinds of fault prediction curves are represented to analyze the trend of the curve and predict the occurrence time of the fault included. In this case, the time period is 100 s with a failure point included. The corresponding historical failure record is exhibited in Table 2. If the predicted results obtained by the healthy degree model are consistent with Table 2, then the effectiveness of the healthy degree model can be verified.

In the process of the engine equipment from normal operation to the failure of excessive clearance of the crankpin bearing or main bearing, the time period 100 s including the failure point is studied, and the data are collected every 10 s. The collected data are shown in Table 3, where P and T mean the parameter and time.

The generated data aforementioned are brought into the healthy degree model to calculate the healthy degree exhibited in Table 4, where HD is healthy degree and T is time, and the graph of fault prediction curves is displayed in Figure 1.

It is obvious that only the healthy degree of fault 1 starts to be less than 1 in the time period from 60 to 70 s, and it decreases with time. In other words, the failure of excessive clearance of the crankpin bearing or main bearing will occur in the time period from 60 to 70 s, which is consistent with Table 2.

In the process of engine equipment from normal operation to fault 2, in this case, the time period is 100 s with a failure point included, and the data are collected every 10 s which is described in Table 5.

Based on the collected data and the healthy degree model, the healthy degree is obtained in Table 6, and the graph of fault prediction curves is shown in Figure 2.

According to Figure 2, only the fault prediction curve of the leakage of cooling water whose healthy degree starts to be less than 1 in the time period from 50 to 60 s, and the healthy degree decreases with time. Therefore, the fault of the leakage of cooling water will happen in the time period from 50 to 60 s, which is in line with Table 2.

The time period of 100 s with the failure point included is discussed in the process of engine equipment from normal operation to fault 3. The data are collected every 10 s, as shown in Table 7.

The collected data are used to calculate the healthy degree in Table 8, and the graph of fault prediction curves is displayed in Figure 3.

It is evident that only the fault prediction curve of the propeller overload whose healthy degree starts to be less than 1 when the time is close to 90 s, and the healthy degree decreases with time. In short, the occurrence time of the propeller overload failure will be close to 90 s, which matches the data shown in Table 2.

In summary, the healthy degree model of watercraft equipment is proposed in this article. On the basis of grey relational analysis, the healthy degree model can solve three major problems: whether watercraft equipment needs to be repaired, what kind of fault it is, and when the fault will occur. After that, we analytically derive two theorems related to the healthy degree model, which are conducive to comprehending and applying the healthy degree model. Finally, the real data of the engine equipment in a certain type of watercraft are utilized and the relevant simulation results are provided to verify the effectiveness of the healthy degree model, and there are failure data of three types of common faults which are excessive clearance of the crankpin bearing or main bearing, the leakage of cooling water, and the propeller overload. Obviously, the predicted results of the healthy degree model are consistent with reality. The current analysis of fault prediction will be beneficial to change the support mode of watercraft equipment and realize the transformation of watercraft equipment support from planned maintenance to predictive maintenance.

In addition, there still remains a disadvantage due to the lack of failure data in this study. We will further study the generation of failure data via establishing the simulation model of watercraft equipment, carrying out the fault simulation experiment, and fully utilizing the data science in [40, 41].