# Fault Diagnosis of Capacitance Aging in DC Link Capacitors of Voltage Source Inverters Using Evidence Reasoning Rule

Authors: Linhao Liao; Haibo Gao; Yelan He; Xiaobin Xu; Zhiguo Lin; Yajie Chen; Fubing You
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Capacitance aging of DC link capacitors in voltage source inverters (VSIs) is a common fault which can lead to instability of the DC voltage. In such a failure state, although the VSI can still work, its performance gradually deteriorates, resulting in a shorter service life of the equipment. Here, an online monitoring and fault diagnosis method for capacitance aging based on evidence reasoning (ER) rule is presented. Features from the DC link voltage data with different levels of capacitance aging are extracted, and data features are generated as pieces of diagnostic evidence, which are then combined according to the ER rule. Finally, capacitance aging fault levels were estimated using the combined results. This method has better diagnostic performance compared to a backpropagation (BP) neural network approach and can be used to flexibly define the relative weighting of each evidence parameter depending on the application. This approach can therefore be widely used for fault diagnosis of an array of different devices.


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## Body

## 1. Introduction

Voltage source inverters (VSIs) are generally reliable, accurate, and effective and have become the most common inverter type used in industry [1]. A capacitor is a key component of VSIs, as they supply the input current to the inverter, compensate for the difference between the power requirement of the inverter and the output power of the rectifier, suppress the current harmonics, absorb feedback energy, meet instantaneous power demand, and stabilize voltage fluctuations [2]. There are three main types of capacitor: electrolytic, ceramic, and film capacitors. Of these, the electrolytic capacitor is most commonly used in VSIs. Electrolytic capacitors generally have a much shorter life expectancy compared with the service life of the inverter. Capacitor faults generally include short circuits, electric leakages, and capacitance attenuation. As inverter operation proceeds, evaporation of electrolyte will cause gradual capacitance attenuation. And although the inverter can still function as capacitance aging increases, overcurrent and under voltage situations increasingly arise when the inverter runs under load, increasing potential safety hazards [3]. This fault is classified as an incipient fault and does not generally trigger the protection system. However, if left undetected and untreated, this situation can significantly reduce the service life of the equipment. It is therefore increasingly important to diagnose capacitance aging in order to evaluate and carry out preventative maintenance in advance.Historically, equivalent series resistance (ESR) and capacitance C are two important parameters used to evaluate capacitance aging. A fault situation is defined if the ESR is more than the double of its initial value. When capacitance attenuation exceeds 20% of the initial value, the capacitor is classified as failed [4]. The majority of methods used to detect capacitor faults are based on monitoring the ESR value, including offline and online modes. Offline detection methods are gradually becoming obsolete as they do not allow dynamic management [5]. On the other hand, for online detection methods, the most direct and effective method is to estimate the ESR value using the correlation between current and voltage values [6, 7]. The ratio of capacitor voltage to supply voltage can also be used to calculate the ESR by reference resistor [8]. However, the accuracy of the technique depends largely on the difference of the reference resistors. The influence of temperature on ESR has also been widely investigated [9, 10]. Improvements in the measuring terminal have been used to obtain more accurate ESR values [11]. In addition, a built-in self-test (BIST) method was used to monitor the attenuation of capacitance C [12]. However, there are few research studies which consider the capacitor fault state jointly determined by ESR value and capacitance C. Oukaour et al. [13] monitored both parameters, and the aging index was estimated using the least squares method, which is proved to be effective.However, all above methods are based on the principle of using formulae to fit the relationship between the acquisition signal source and the capacitor ESR value and capacitanceC which is not ideal. The monitoring parameters are various and complex. Moreover, it is difficult to identify incipient faults and predict serious faults using this approach. At present, data-driven fault monitoring and identification methods are increasingly used due to their broad applicability and high diagnostic accuracy [14, 15]. Data-driven methods using artificial neural networks (ANNs) have been used to estimate capacitance capacity [16]; however, the accuracy of ANN method is limited by the quantity of training samples. Extended neural network (ENN) algorithms and chaotic synchronization detection methods have also been used to identify capacitor faults [17]. An online track circuit compensation capacitor fault diagnosis method based on K- fault diagnosis method was reported [18]. Unfortunately, these methods only consider the identification of serious faults that have occurred and do not carry out further subdivision on the degree of capacitor faults which would allow much earlier intervention and fault correction.Here, we propose a fault diagnosis method for capacitance aging in DC link capacitors used in inverters which is based on the evidence reasoning (ER) rule [19]. Based on the principle of data-driven decision making, this method does not assume any relationship between capacitance and other parameters [20, 21], but only generates types of diagnostic evidence and combines them by measuring the DC link voltage value under different capacitance attenuation degrees and then estimates the fault level of capacitance aging based on the combined results. The advantage of using the ER rule method is that it has the ability to deal with the variable information provided by incipient fault situations and it can also define the relative reliability of different information sources. Moreover, for multiple types of evidence, the relative weighting or importance between different types of evidences can be allocated. This weighted information can then be considered comprehensively to provide more correct data for final decision making. Different fault thresholds can be set according to expert user experience in different practical applications. The method also has the advantages of using fewer parameters but providing higher monitoring sensitivity.


## 2. Simulation Model and Experiment

### 2.1. A Simulation Model for a Marine Electric Propulsion System

In marine electric propulsion systems, the propeller is driven by an electric motor. This propulsion method abandons the traditional way in which the shaft system and propeller are driven by the prime mover and is efficient and energy saving and allows excellent maneuverability and flexible arrangements [22, 23]. The most common electric propulsion system is generally powered by diesel generator sets, which provide electric power to propulsion motors through switchboard and frequency converters and which then drives the propellers to generate thrust. The ship “Jiangsu ferry 3011” was used in this work as a case study and is a ferry ship which is in use and which runs across the Yangtze River between Zhenjiang and Yangzhou in Jiangsu province in China (Figure 1). The ship is 90 meters long and 15.4 meters wide and has a tonnage of 720 tons and is the first civil ship adopting an electric propulsion system (Figure 2) using DC power in China.Figure 1
Jiangsu Ferry Boat 3011 (a) and the DC electric propulsion unit (b).
(a)
(b)Figure 2
Structure diagram of the Jiangsu Ferry 3011 electric propulsion system.A DC power system is used on board, which means the electric power is generated from an alternating current (AC) generator and transformed into DC power by a rectifier to feed the DC bus. Power is then transmitted to inverters to drive propulsion motors and other auxiliary loads. The basic parameters of this marine power system are shown in Table1.Table 1
Basic parameters of the marine power system.
Item
Device
Type
Main parameters
1
Diesel engine
Yuchai Group YC6K560L-C20
6-cylinder, in-line, four-stroke, turbo-charged, displacement 12.939 l
2
Generator
Leroy-Somer FLSMV355LB
Marine brushless three-phase AC asynchronous motor, rated power 380 kW, rated voltage 440 V, rated frequency 60 Hz
3
Rectifier
Vacon NXI_0730
Input voltage 440 V AC, output voltage 750 V DC
4
Inverter
Vacon NXI_0650
Input voltage 750 V DC, output voltage 400 V AC
5
Propulsion motor
CSIEM TYT350-8-AT
Rated voltage 400 V, rated power 350 kW, air-cooled, variable frequency motorThe overall life cycle of electric propulsion ships is characterized by working in highly variable environments with multiple types of interference, strong maneuvering requirements, and complex river conditions for 24 hours per day, always accompanied by potential incipient faults in the electrical system. The inverter is the core component of the ship’s electric propulsion system. The failure of the DC link capacitor is one of the most common faults of the inverter [24]. Due to the different operating environments and different types of electric propulsion systems, the degree and characteristics of capacitor attenuation are irregular and uncertain. In addition, with current monitoring technology, it is difficult to track, collect, and extract data which can identify fault features. The accumulated data are often incomplete and invalid, which frustrates careful research-based and data-driven solutions. However, by building a computer model of this system, we can simulate the life cycle characteristics of the ship’s electric propulsion system and track the fault characteristics of the capacitor with time, laying the foundations for a new and rigorous research-based approach to incipient fault diagnosis which can be developed for use on board these electric ferries.Based on the main parameters in Table1, we developed a simulation model of this marine electric propulsion system using a Matlab/Simulink environment [25] as shown in Figure 3.Figure 3
Simulation model of marine electric propulsion system.In the simulation model, the Jiangsu ferry 3011’s electric propulsion system consists of generator, rectifier, inverter, and propulsion motors. The DC link capacitor is made up of two identical capacitors in series, and N is the midpoint between the two capacitors. The propulsion system consists of two propulsion motors of the same type. Tm1 and Tm2 are the load torques of the two propulsion motors.


### 2.2. Fault Simulation

In this paper, the most frequent working condition was selected for research, and the load torque was set at 2228 Nm. In the DC link, three different levels of capacitance attenuation were set in the simulation model. The initial capacitanceC1 is 7500e − 6 F, and 90∼100% of the initial capacitance is classified as normal. With the gradual deterioration of capacitance, if the capacitance is set to C2 (80∼90% of initial capacitance), it is defined as an incipient fault. If the capacitance value is set to C3 (less than 80% of initial capacitance), it is no longer an incipient fault, but a serious fault. At the same time, the data sampling frequency was set to 50 kHz to collect the instantaneous DC link voltage VDC for 24 seconds, including three levels of capacitance with 1.2 million data points. Among them, the data of C1 are generated in the1∼8th second, the data of C2 are generated in the 8∼16th second, and the data of C3 are generated in the 16∼24th second. In order to make the diagnosis process closer to the actual situation, the measured VDC is added with noise, and the awgn function in Matlab was used to add 30 dB white Gaussian noise to VDC as shown in Figure 4.Figure 4
V
DC after adding noise.In this figure, The fluctuation range of the upper and lower limits of the DC voltage under the capacitanceC1 is from −0.79% to +0.49%; for capacitance C2, the fluctuation range is from −0.87% to +0.56%, and for capacitance C3, the fluctuation range is from −1.09% to +0.65%. So, the instantaneous value of the DC bus voltage under different fault levels has no obvious feature distinction, and it is difficult to effectively diagnose using the general threshold method.We further propose an inference model based on an ER rule to estimate the fault level of capacitance attenuation by collecting and analyzing instantaneous voltage dataVDC from the DC link. The definition of fault levels and their thresholds can be set according to different capacitor types and different application systems. This method does not need to make assumptions about the relationship between ESR values, capacitance values, and voltage values from the DC link as it directly learns the fault characteristics and can then be used to diagnose a predetermined fault level using a data-driven approach.


## 2.1. A Simulation Model for a Marine Electric Propulsion System

In marine electric propulsion systems, the propeller is driven by an electric motor. This propulsion method abandons the traditional way in which the shaft system and propeller are driven by the prime mover and is efficient and energy saving and allows excellent maneuverability and flexible arrangements [22, 23]. The most common electric propulsion system is generally powered by diesel generator sets, which provide electric power to propulsion motors through switchboard and frequency converters and which then drives the propellers to generate thrust. The ship “Jiangsu ferry 3011” was used in this work as a case study and is a ferry ship which is in use and which runs across the Yangtze River between Zhenjiang and Yangzhou in Jiangsu province in China (Figure 1). The ship is 90 meters long and 15.4 meters wide and has a tonnage of 720 tons and is the first civil ship adopting an electric propulsion system (Figure 2) using DC power in China.Figure 1
Jiangsu Ferry Boat 3011 (a) and the DC electric propulsion unit (b).
(a)
(b)Figure 2
Structure diagram of the Jiangsu Ferry 3011 electric propulsion system.A DC power system is used on board, which means the electric power is generated from an alternating current (AC) generator and transformed into DC power by a rectifier to feed the DC bus. Power is then transmitted to inverters to drive propulsion motors and other auxiliary loads. The basic parameters of this marine power system are shown in Table1.Table 1
Basic parameters of the marine power system.
Item
Device
Type
Main parameters
1
Diesel engine
Yuchai Group YC6K560L-C20
6-cylinder, in-line, four-stroke, turbo-charged, displacement 12.939 l
2
Generator
Leroy-Somer FLSMV355LB
Marine brushless three-phase AC asynchronous motor, rated power 380 kW, rated voltage 440 V, rated frequency 60 Hz
3
Rectifier
Vacon NXI_0730
Input voltage 440 V AC, output voltage 750 V DC
4
Inverter
Vacon NXI_0650
Input voltage 750 V DC, output voltage 400 V AC
5
Propulsion motor
CSIEM TYT350-8-AT
Rated voltage 400 V, rated power 350 kW, air-cooled, variable frequency motorThe overall life cycle of electric propulsion ships is characterized by working in highly variable environments with multiple types of interference, strong maneuvering requirements, and complex river conditions for 24 hours per day, always accompanied by potential incipient faults in the electrical system. The inverter is the core component of the ship’s electric propulsion system. The failure of the DC link capacitor is one of the most common faults of the inverter [24]. Due to the different operating environments and different types of electric propulsion systems, the degree and characteristics of capacitor attenuation are irregular and uncertain. In addition, with current monitoring technology, it is difficult to track, collect, and extract data which can identify fault features. The accumulated data are often incomplete and invalid, which frustrates careful research-based and data-driven solutions. However, by building a computer model of this system, we can simulate the life cycle characteristics of the ship’s electric propulsion system and track the fault characteristics of the capacitor with time, laying the foundations for a new and rigorous research-based approach to incipient fault diagnosis which can be developed for use on board these electric ferries.Based on the main parameters in Table1, we developed a simulation model of this marine electric propulsion system using a Matlab/Simulink environment [25] as shown in Figure 3.Figure 3
Simulation model of marine electric propulsion system.In the simulation model, the Jiangsu ferry 3011’s electric propulsion system consists of generator, rectifier, inverter, and propulsion motors. The DC link capacitor is made up of two identical capacitors in series, and N is the midpoint between the two capacitors. The propulsion system consists of two propulsion motors of the same type. Tm1 and Tm2 are the load torques of the two propulsion motors.


## 2.2. Fault Simulation

In this paper, the most frequent working condition was selected for research, and the load torque was set at 2228 Nm. In the DC link, three different levels of capacitance attenuation were set in the simulation model. The initial capacitanceC1 is 7500e − 6 F, and 90∼100% of the initial capacitance is classified as normal. With the gradual deterioration of capacitance, if the capacitance is set to C2 (80∼90% of initial capacitance), it is defined as an incipient fault. If the capacitance value is set to C3 (less than 80% of initial capacitance), it is no longer an incipient fault, but a serious fault. At the same time, the data sampling frequency was set to 50 kHz to collect the instantaneous DC link voltage VDC for 24 seconds, including three levels of capacitance with 1.2 million data points. Among them, the data of C1 are generated in the1∼8th second, the data of C2 are generated in the 8∼16th second, and the data of C3 are generated in the 16∼24th second. In order to make the diagnosis process closer to the actual situation, the measured VDC is added with noise, and the awgn function in Matlab was used to add 30 dB white Gaussian noise to VDC as shown in Figure 4.Figure 4
V
DC after adding noise.In this figure, The fluctuation range of the upper and lower limits of the DC voltage under the capacitanceC1 is from −0.79% to +0.49%; for capacitance C2, the fluctuation range is from −0.87% to +0.56%, and for capacitance C3, the fluctuation range is from −1.09% to +0.65%. So, the instantaneous value of the DC bus voltage under different fault levels has no obvious feature distinction, and it is difficult to effectively diagnose using the general threshold method.We further propose an inference model based on an ER rule to estimate the fault level of capacitance attenuation by collecting and analyzing instantaneous voltage dataVDC from the DC link. The definition of fault levels and their thresholds can be set according to different capacitor types and different application systems. This method does not need to make assumptions about the relationship between ESR values, capacitance values, and voltage values from the DC link as it directly learns the fault characteristics and can then be used to diagnose a predetermined fault level using a data-driven approach.


## 3. Fault Diagnosis Based on Evidence Reasoning Rule

### 3.1. Outline of ER Rule

The evidence reasoning (ER) rule has been proposed by Yang et al. [26, 27], and the evidence can be defined here as follows:(1)ej=θ,pθ,j∀θ⊆Θ,∑θ⊆Θpθ,j=1.Here, defineΘ=h1,h2,…,hN as a frame of discernment; it contains mutually exclusive and collectively exhaustive hypotheses. PΘ and 2Θ represent all the subsets of the power set Θ. θ,pθ,j is an element of evidence ej which indicates the degree of ej approaching θ (not an empty set) is pθ,j, when pθ,j>0, θ,pθ,j is the focal element of ej.The concepts of reliabilityrj and weight wj of ej are introduced in the ER rule. rj is the ability of the information source, which characterizes credibility of this ej for the fault diagnosis system. On the other hand, for wj, it is different from rj, which can be defined by the person who uses this diagnostic method according to subjective conditions such as different equipment or different environments. wj represents the importance degree of different types of evidence. The weighted belief distribution with reliability factor and weight factor is defined as follows:(2)mj=θ,m∼θ,j∀θ⊆Θ;PΘ,m∼PΘ,j,where m∼θ,j represents the degree of ej approaching θ after combining rj and wj, which is defined as follows:(3)m∼θ,j=0,θ=∅,crw,jmθ,j,θ⊆Θ,θ≠∅,crw,j1−rj,θ=Pθ,where mθ,j=wjpθ,j, crw,j=1/1+wj−rj are normalization factors; it makes ∑θ⊆Θm∼θ,j+m∼PΘ,j=1, wherein ∑θ⊆Θpθ,j=1.If the two pieces of evidence are independent of each other, they can be combined with the ER rule as follows:(4a)pθ,e2=0,θ=∅,m^θ,e2∑D⊆Θm^D,e2,θ⊆Θ,θ≠∅,(4b)m^θ,e2=1−r2mθ,1+1−r1mθ,2+∑B∩C=θmB,1mC,2,∀θ⊆Θ,where pθ,e2 represents the degree of how the two pieces of evidence jointly support θ, and the fusion process includes the bounded sum and the orthogonal sum. The bounded sum is reflected in 1−rii=1,2, which means the sum of the two sets of evidence e1 and e2 independently approaches θ, and it includes the interaction between e1 and e2. For example, when e1 is considered to be completely reliable 1−r1=0, then 1−r1mθ,2=0, which means that e2 has no effect on the combined results. Conversely, e1 has no effect on the combined results. In addition, the orthogonal sum indicates the degree of all evidence conjointly approaching toθ. This method can also combine multiple pieces of evidence without order restriction while satisfying the exchange law and the combination law [28, 29].


### 3.2. Diagnostic Process

Figure5 outlines the fault diagnosis process using the ER rule.Figure 5
Incipient fault diagnosis flowchart based on the ER rule.First, the instantaneous DC voltage valueVDC is obtained as shown in Section 2.2 so that the peak-to-peak voltage value VPP can be calculated from VDC. Here, 0.02 seconds, including 1000 data points, is taken as a window length to obtain the average value of VPP of 400 data windows named as VP(t). VR(t) is obtained by calculating the root mean square value of the VDC with same window length, and then by normalization. VPt∈s11,s21 and VRt∈s12,s22, where s11, s21, s12, and s22 are, respectively, the minimum and maximum values of input characteristic signals VP(t) and VR(t). The capacitor aging fault level is then defined as C(t), C(t) = [1, 2, 3]. And VP(t), VR(t), and C(t) are denoted together as T = {[VP(t), VR(t), C(t)] | t = 1, 2, …, TS}, TS = 1200, where [VP(t), VR(t), C(t)] is a sample vector.Secondly, the data [VP(t), VR(t), C(t)] in T are converted to the form of similarity degree to the reference value. The reference value set of the capacitor fault level is D = {Dn | n = 1, …, N}, where N is the number of reference values of the capacitor fault level. The input reference values set of the voltage signal Vi is Ai = {Aji | j = 1, …, Ji}, i = P, R, where Ji is the number of reference values of Vi. The similarity distribution of Vi(t) to reference values Aji is calculated as follows:(5a)TIVit=Aji,αi,jj=1,...,Ji;i=1,2,(5b)αi,j=Aj+1i−VitAj+1i−Aji,αi,j+1=1−αi,j,Aji≤Vit≤Aj+1i,(5c)αi,j′=0,j′=1,…,Ji,j′≠j,j+1,where αi,j represents the similarity of Vi(t) to the reference value Aji.Similarly, the calculation process for the similarity distribution of the fault levelC(t) to the reference value Dn is as follows:(6a)TOCt=Dn,γnn=1,…,N,(6b)γn=Dn+1−CtDn+1−Dn,γn+1=1−γn,Dn≤Ct≤Dn+1,(6c)γn′=0n′=1,…,N,n′≠n,n+1,where γn denotes the similarity of C(t) to DnThe parameter form similar toαi,jγn,αi,j+1γn,αi,jγn+1,αi,j+1γn+1 can be obtained after completing the calculation of all the sample sets in T, where αi,jγn denotes the comprehensive similarity of the input value in sample pair (Vi(t), C(t)) matching the reference value Aji, Dn. Table 2 shows the cast statistics result of all sample pairs in T, and ∑n=1Nδn=∑j=1Jiηj=TS.Table 2
Casting result tables of sample pairs (Vi(t), C(t)).
V
i(t)
C(t)
A
1
i
…
A
j
i
…
A
J
i
i
Total
D
1
a
1,1
…
oa1,j
…
oa1,Ji
δ
1
⋮
⋮
⋮
⋮
⋮
⋮
⋮
D
n
a
n
,
1
…
a
n
,
j
…
a
n
,
J
i
δ
n
⋮
⋮
⋮
⋮
⋮
⋮
⋮
D
N
a
N
,
1
…
a
N
,
j
…
a
N
,
J
i
δ
N
Total
η
1
…
η
j
…
η
J
i
T
SAccording to the casting statistics shown in Table2, the degree of belief βn,ji of ej can be calculated as follows:(7)βn,ji=an,j/δn∑k=1Nak,j/δk,where ∑n=1Nβn,ji=1. Then, the evidence corresponding to the reference value can be defined as follows:(8)eji=β1,ji,β2,ji,…,βN,ji.Therefore, an evidence matrix, as shown in Table3 can be constructed to describe the relationship between Vi(t) and C(t).Table 3
Evidence matrix of (Vi(t), C(t)).
V
i(t)
C(t)
e
1
i
…
e
j
i
…
e
J
i
i
A
1
i
…
A
j
i
…
A
J
i
i
D
1
…
β
1
,
j
i
…
β
1
,
J
i
i
⋮
⋮
⋮
⋮
⋮
⋮
D
n
β
n
,
1
i
…
β
n
,
j
i
…
β
n
,
J
i
i
⋮
⋮
⋮
⋮
⋮
⋮
D
N
β
N
,
1
i
…
β
N
,
j
i
…
β
N
,
J
i
iNow we define reliabilityrj and weight wj of evidence ej, after obtaining the evidence of the input Vi. ri defined in this paper is determined by the Spearman rank correlation coefficient for ranked data [30]. The Spearman rank correlation coefficient expresses the closeness of the relationship between the two sets of variables and it is suitable for describing the reliability between Vi(t) and C(t) in this case. The calculation procedures are as follows:(9a)di=Vit−Ct,(9b)ri=1−6∑i=1Ndi2TSTS2−1,where di is the difference between Vi(t) and C(t) and ri is the reliability of the evidence.For all of the input valuesVi(t), they must be within the interval Aji,Aj+1i formed by the two adjacent reference values, and evidence eji and ej+1i corresponding to the two reference values will be activated. The final evidence of the input values Vi(t) can be weighted by eji and ej+1i with the calculated reliability ri.(10a)ei=Dn,pn,i,n=1,…,N,(10b)pn,i=αi,jβn,ji+αi,j+1βn,j+1i.The initial evidence weight is set aswi = ri, and e1 and e2 are combined by ER rule. The fusion process is as follows:(11a)pn,e2=m^n,e2∑k⊆Dm^k,e2,(11b)m^n,e2=1−r2mn,1+1−r1mn,2+∑B∩C=nmB,1mC,2,(11c)OVt=Dn,pn,e2,n=1,…,N.After the fusion process is completed, the combined result is obtained. Furthermore, the fault level of the capacitor can then be estimated based on the combined results.(12)C^t=Dnmaxpn,e2.


## 3.1. Outline of ER Rule

The evidence reasoning (ER) rule has been proposed by Yang et al. [26, 27], and the evidence can be defined here as follows:(1)ej=θ,pθ,j∀θ⊆Θ,∑θ⊆Θpθ,j=1.Here, defineΘ=h1,h2,…,hN as a frame of discernment; it contains mutually exclusive and collectively exhaustive hypotheses. PΘ and 2Θ represent all the subsets of the power set Θ. θ,pθ,j is an element of evidence ej which indicates the degree of ej approaching θ (not an empty set) is pθ,j, when pθ,j>0, θ,pθ,j is the focal element of ej.The concepts of reliabilityrj and weight wj of ej are introduced in the ER rule. rj is the ability of the information source, which characterizes credibility of this ej for the fault diagnosis system. On the other hand, for wj, it is different from rj, which can be defined by the person who uses this diagnostic method according to subjective conditions such as different equipment or different environments. wj represents the importance degree of different types of evidence. The weighted belief distribution with reliability factor and weight factor is defined as follows:(2)mj=θ,m∼θ,j∀θ⊆Θ;PΘ,m∼PΘ,j,where m∼θ,j represents the degree of ej approaching θ after combining rj and wj, which is defined as follows:(3)m∼θ,j=0,θ=∅,crw,jmθ,j,θ⊆Θ,θ≠∅,crw,j1−rj,θ=Pθ,where mθ,j=wjpθ,j, crw,j=1/1+wj−rj are normalization factors; it makes ∑θ⊆Θm∼θ,j+m∼PΘ,j=1, wherein ∑θ⊆Θpθ,j=1.If the two pieces of evidence are independent of each other, they can be combined with the ER rule as follows:(4a)pθ,e2=0,θ=∅,m^θ,e2∑D⊆Θm^D,e2,θ⊆Θ,θ≠∅,(4b)m^θ,e2=1−r2mθ,1+1−r1mθ,2+∑B∩C=θmB,1mC,2,∀θ⊆Θ,where pθ,e2 represents the degree of how the two pieces of evidence jointly support θ, and the fusion process includes the bounded sum and the orthogonal sum. The bounded sum is reflected in 1−rii=1,2, which means the sum of the two sets of evidence e1 and e2 independently approaches θ, and it includes the interaction between e1 and e2. For example, when e1 is considered to be completely reliable 1−r1=0, then 1−r1mθ,2=0, which means that e2 has no effect on the combined results. Conversely, e1 has no effect on the combined results. In addition, the orthogonal sum indicates the degree of all evidence conjointly approaching toθ. This method can also combine multiple pieces of evidence without order restriction while satisfying the exchange law and the combination law [28, 29].


## 3.2. Diagnostic Process

Figure5 outlines the fault diagnosis process using the ER rule.Figure 5
Incipient fault diagnosis flowchart based on the ER rule.First, the instantaneous DC voltage valueVDC is obtained as shown in Section 2.2 so that the peak-to-peak voltage value VPP can be calculated from VDC. Here, 0.02 seconds, including 1000 data points, is taken as a window length to obtain the average value of VPP of 400 data windows named as VP(t). VR(t) is obtained by calculating the root mean square value of the VDC with same window length, and then by normalization. VPt∈s11,s21 and VRt∈s12,s22, where s11, s21, s12, and s22 are, respectively, the minimum and maximum values of input characteristic signals VP(t) and VR(t). The capacitor aging fault level is then defined as C(t), C(t) = [1, 2, 3]. And VP(t), VR(t), and C(t) are denoted together as T = {[VP(t), VR(t), C(t)] | t = 1, 2, …, TS}, TS = 1200, where [VP(t), VR(t), C(t)] is a sample vector.Secondly, the data [VP(t), VR(t), C(t)] in T are converted to the form of similarity degree to the reference value. The reference value set of the capacitor fault level is D = {Dn | n = 1, …, N}, where N is the number of reference values of the capacitor fault level. The input reference values set of the voltage signal Vi is Ai = {Aji | j = 1, …, Ji}, i = P, R, where Ji is the number of reference values of Vi. The similarity distribution of Vi(t) to reference values Aji is calculated as follows:(5a)TIVit=Aji,αi,jj=1,...,Ji;i=1,2,(5b)αi,j=Aj+1i−VitAj+1i−Aji,αi,j+1=1−αi,j,Aji≤Vit≤Aj+1i,(5c)αi,j′=0,j′=1,…,Ji,j′≠j,j+1,where αi,j represents the similarity of Vi(t) to the reference value Aji.Similarly, the calculation process for the similarity distribution of the fault levelC(t) to the reference value Dn is as follows:(6a)TOCt=Dn,γnn=1,…,N,(6b)γn=Dn+1−CtDn+1−Dn,γn+1=1−γn,Dn≤Ct≤Dn+1,(6c)γn′=0n′=1,…,N,n′≠n,n+1,where γn denotes the similarity of C(t) to DnThe parameter form similar toαi,jγn,αi,j+1γn,αi,jγn+1,αi,j+1γn+1 can be obtained after completing the calculation of all the sample sets in T, where αi,jγn denotes the comprehensive similarity of the input value in sample pair (Vi(t), C(t)) matching the reference value Aji, Dn. Table 2 shows the cast statistics result of all sample pairs in T, and ∑n=1Nδn=∑j=1Jiηj=TS.Table 2
Casting result tables of sample pairs (Vi(t), C(t)).
V
i(t)
C(t)
A
1
i
…
A
j
i
…
A
J
i
i
Total
D
1
a
1,1
…
oa1,j
…
oa1,Ji
δ
1
⋮
⋮
⋮
⋮
⋮
⋮
⋮
D
n
a
n
,
1
…
a
n
,
j
…
a
n
,
J
i
δ
n
⋮
⋮
⋮
⋮
⋮
⋮
⋮
D
N
a
N
,
1
…
a
N
,
j
…
a
N
,
J
i
δ
N
Total
η
1
…
η
j
…
η
J
i
T
SAccording to the casting statistics shown in Table2, the degree of belief βn,ji of ej can be calculated as follows:(7)βn,ji=an,j/δn∑k=1Nak,j/δk,where ∑n=1Nβn,ji=1. Then, the evidence corresponding to the reference value can be defined as follows:(8)eji=β1,ji,β2,ji,…,βN,ji.Therefore, an evidence matrix, as shown in Table3 can be constructed to describe the relationship between Vi(t) and C(t).Table 3
Evidence matrix of (Vi(t), C(t)).
V
i(t)
C(t)
e
1
i
…
e
j
i
…
e
J
i
i
A
1
i
…
A
j
i
…
A
J
i
i
D
1
…
β
1
,
j
i
…
β
1
,
J
i
i
⋮
⋮
⋮
⋮
⋮
⋮
D
n
β
n
,
1
i
…
β
n
,
j
i
…
β
n
,
J
i
i
⋮
⋮
⋮
⋮
⋮
⋮
D
N
β
N
,
1
i
…
β
N
,
j
i
…
β
N
,
J
i
iNow we define reliabilityrj and weight wj of evidence ej, after obtaining the evidence of the input Vi. ri defined in this paper is determined by the Spearman rank correlation coefficient for ranked data [30]. The Spearman rank correlation coefficient expresses the closeness of the relationship between the two sets of variables and it is suitable for describing the reliability between Vi(t) and C(t) in this case. The calculation procedures are as follows:(9a)di=Vit−Ct,(9b)ri=1−6∑i=1Ndi2TSTS2−1,where di is the difference between Vi(t) and C(t) and ri is the reliability of the evidence.For all of the input valuesVi(t), they must be within the interval Aji,Aj+1i formed by the two adjacent reference values, and evidence eji and ej+1i corresponding to the two reference values will be activated. The final evidence of the input values Vi(t) can be weighted by eji and ej+1i with the calculated reliability ri.(10a)ei=Dn,pn,i,n=1,…,N,(10b)pn,i=αi,jβn,ji+αi,j+1βn,j+1i.The initial evidence weight is set aswi = ri, and e1 and e2 are combined by ER rule. The fusion process is as follows:(11a)pn,e2=m^n,e2∑k⊆Dm^k,e2,(11b)m^n,e2=1−r2mn,1+1−r1mn,2+∑B∩C=nmB,1mC,2,(11c)OVt=Dn,pn,e2,n=1,…,N.After the fusion process is completed, the combined result is obtained. Furthermore, the fault level of the capacitor can then be estimated based on the combined results.(12)C^t=Dnmaxpn,e2.


## 4. Analysis of Simulation Examples

### 4.1. Training Model

In the experiment, 80% of the total sample setT (sample size TS = 1200) was randomly selected as the training sample TR = {[VP(t), VR(t), C(t)] | t = 1, 2, …, TSR}, sample size TSR = 960, and the remaining 20% was selected as the test sample TE = {[VP(t), VR(t), C(t)]| t = 1, 2, …, TSE}, sample size TSE = 240. Based on experience, the reference value set for the initial parameter VP(t) is A1 = {6.7877, 7.8030, 8.1517, 8.4575, 8.8254, 9.1441, 9.5400, 9.9043, 10.3445, 10.7758}, including 10 reference values in total. The reference value set for the initial parameter VR(t) is A2 = {0, 0.0679, 0.1245, 0.1733, 0.2177, 0.2759, 0.3375, 0.4224, 0.5122, 0.6417}, including 10 reference values in total. The reference value set D of C(t) is D = {1, 2, 3}, including 3 reference values in total. The training sample set TR is processed through the steps described in Section 4 to obtain casting result tables of sample pairs (VP(t), C(t)), which are presented in Tables 4 and 5.Table 4
Casting result tables of sample pairs (VP(t), C(t)).
C
V
P
Total
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
25.8775
107.6622
83.6619
55.4520
33.0722
11.6113
2.3414
0.3214
0
0
320.0000
D
2
2
0
4.3032
13.6444
43.2252
65.3831
72.6308
73.3652
36.9081
10.3351
0.2047
320.0000
D
3
3
0
0
0
0
0.1070
4.4748
31.4539
58.4470
86.0086
139.5087
320.0000
Total
25.8775
111.9654
97.3063
98.6772
98.5623
88.7169
107.1605
95.6765
96.3437
139.7134
960.0000Table 5
Casting result tables of sample pairs (VR(t), C(t)).
C
V
R
Total
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
43.7841
58.7483
39.0959
29.8323
25.1073
28.5133
29.5964
28.3416
27.0418
9.9390
320.0000
D
2
2
5.3529
35.4618
43.8672
31.0040
37.8508
34.7105
29.9841
30.3652
34.8532
36.5502
320.0000
D
3
3
0
1.3512
10.3416
29.3254
36.0224
37.1118
37.3102
40.4432
38.6807
89.4135
320.0000
Total
49.1370
95.5613
93.3047
90.1617
98.9805
100.3356
96.8907
99.1500
100.5757
135.9027
960.0000After obtaining the casting result tables, the evidence matrix table ofVi is obtained by normalization of likelihoods as shown in Tables 6 and 7.Table 6
Evidence matrix table for inputVP.
C
V
P
e
1
P
e
2
P
e
3
P
e
4
P
e
5
P
e
6
P
e
7
P
e
8
P
e
9
P
e
10
P
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
1
0.9616
0.8598
0.5620
0.3355
0.1309
0.0218
0.0034
0
0
D
2
2
0
0.0384
0.1402
0.4380
0.6634
0.8187
0.6846
0.3858
0.1073
0.0015
D
3
3
0
0
0
0
0.0011
0.0504
0.2935
0.6109
0.8927
0.9985Table 7
Evidence matrix table for inputVR.
C
V
R
e
1
R
e
2
R
e
3
R
e
4
R
e
5
R
e
6
R
e
7
R
e
8
R
e
9
R
e
10
R
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
0.8911
0.6148
0.4190
0.3309
0.2537
0.2842
0.3055
0.2825
0.2689
0.0731
D
2
2
0.1089
0.3711
0.4701
0.3439
0.3842
0.3459
0.3905
0.3063
0.3465
0.2689
D
3
3
0
0.0141
0.1108
0.3253
0.3639
0.3699
0.3851
0.4079
0.3846
0.6579After obtaining the evidence matrix tables, the reliability factorsr1 = 0.8897 and r2 = 0.4117 of the evidence parameters e1 and e2 are calculated with the sample dataset T. The initial fault diagnosis result on the sample training set TR can be obtained by using the ER rule, and the results are presented in the form of a confusion matrix as shown in Table 8.Table 8
Initial diagnostic results of ER.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
267
53
0
320
83.44
C
2
47
238
35
320
74.38
C
3
0
45
179
320
55.94It can be seen from the initial diagnostic results that the accuracy is not satisfactory because the initial set of reference valuesAi, reliability factor rj, and weight wj are not the optimal ER rule parameters, so the ideal diagnostic accuracy cannot be achieved. Therefore, it is necessary to establish a parameter optimization model for the ER rule process.


### 4.2. Parameter Optimization

The mean square error between the estimated value and the real value of the capacitor fault level is set as the objective function, and the set of optimization parametersP is determined at the same time. P = {Aji,wi | i = P, R; j = 2,..., Ji − 1}, wi represents the weight of the evidence, and the other parameters are set to A11=s11, AJ11=s21, A12=s12, AJ22=s22.(13a)minPξP=1N∑t=1TsC^t−Ct2,(13b)s.t.0≤wi≤1,i=1,2(13c)Aj−1i<Aji<Aj+1i,j=2,…,Ji−1.The objective function is optimized by using thefmincon function in Matlab, and the optimal parameter set P is obtained after training. At the same time, the optimized casting result tables, optimized evidence matrix table, and optimized capacitance aging fault diagnosis results can be obtained.


### 4.3. Analysis of Diagnostic Results

As can be seen from Tables4–13, the optimized diagnostic results are more accurate. Especially for capacitor fault levels 2 and 3, the judgment is more accurate, and misjudgment is greatly reduced. In order to further verify the reliability of the ER fault diagnosis model, test set TE was used for diagnosis. In the meanwhile, a classical three-layer BP neural network approach was selected for comparison. The learning rate was set as Ir = 0.05, and the maximum number of iterations was set to 100. The diagnostic results of TE are shown in Table 14 and Figures 6 and 7.Table 9
Optimized casting result table of sample pairs (VP(t), C(t)).
C
V
P
Total
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
14.7558
111.6242
99.3861
56.5894
33.5273
3.6031
0.5140
0
0
0
320.0000
D
2
2
0
3.9400
16.0655
59.1711
98.3794
88.4319
45.0296
8.9332
0.0493
0
320.0000
D
3
3
0
0
0
0
1.9566
30.8117
67.5026
130.7243
82.5786
6.4262
320.0000
Total
14.7558
115.5642
115.4516
115.7605
133.8633
122.8467
113.0462
139.6575
82.6279
6.4262
960.0000Table 10
Optimized casting result table of sample pairs (VR(t), C(t)).
C
V
R
Total
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
42.4839
80.1372
58.1664
70.9277
44.7212
19.6971
3.8077
0.0588
0
0
320.0000
D
2
2
8.5817
57.5337
64.5437
85.1186
48.6665
27.4821
17.0460
10.2378
0.7898
0
320.0000
D
3
3
0.1654
4.3327
50.3746
94.0281
61.7715
28.8358
25.8347
20.3626
22.4779
11.8168
320.0000
Total
51.2310
142.0036
173.0847
250.0744
155.1592
76.0150
46.6884
30.6592
23.2677
11.8168
960.0000Table 11
Optimized evidence matrix table for inputVP.
C
V
P
e
1
P
e
2
P
e
3
P
e
4
P
e
5
P
e
6
P
e
7
P
e
8
P
e
9
P
e
10
P
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
1
0.9659
0.8608
0.4888
0.2505
0.0293
0.0045
0
0
0
D
2
2
0
0.0341
0.1392
0.5112
0.7349
0.7119
0.3983
0.0640
0.0006
0
D
3
3
0
0
0
0
0.0146
0.2508
0.5971
0.9360
0.9994
1Table 12
Optimized evidence matrix table for inputVR.
C
V
R
e
1
R
e
2
R
e
3
R
e
4
R
e
5
R
e
6
R
e
7
R
e
8
R
e
9
R
e
10
R
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
0.8293
0.5643
0.3361
0.2836
0.2882
0.2591
0.0816
0.0019
0
0
D
2
2
0.1675
0.4052
0.3729
0.3404
0.3137
0.3615
0.3651
0.3339
0.0339
0
D
3
3
0.0032
0.0305
0.2910
0.3760
0.3981
0.3793
0.5533
0.6642
0.9661
1Table 13
ER optimized diagnosis results.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
256
64
0
320
80.00
C
2
31
254
35
320
79.38
C
3
0
40
280
320
87.50Table 14
Diagnostic results of the ER/BP test set.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
65/75
15/5
0
80
81.25/93.75
C
2
6/30
70/45
4/5
80
87.50/56.25
C
3
0
5/14
75/66
80
93.75/82.50Figure 6
Results comparison between ER (blue) and BP (red).Figure 7
Diagnostic results of ER/BP test set.
(a)
(b)The difference between the true and estimated levels of capacitance at different level capacitance attenuation in test set is as shown in Figure7. In this figure, the ordinate value of 1 indicates that the capacitance is normal, the value 2 indicates that the capacitance is in incipient fault, and the value 3 indicates that the capacitance is in serious fault. On the time axis of the abscissa, the true level of the capacitance is normal for 0 to 8 seconds, an incipient fault for 8 to 16, and a serious fault for 16 to 24.Figure8 shows the estimated levels of the test set calculated by the trained ER and BP neural network, respectively.Figure 8
Test samples and output of ER/BP.
(a)
(b)From the results presented, the diagnostic accuracy of the BP neural network approach is lower than that obtained using the ER rule model, which is mainly seen in the diagnosis of capacitor fault level 2. This fault level 2 represents an incipient fault. For the case ofC2, the change of DC bus voltage is not as obvious as case of C3, which is a very small change. Therefore, it is difficult to distinguish the change from C1 to C2 in the specific diagnosis process. In this case, there are two reasons for the low accuracy of BP neural network. Firstly, BP neural network is sensitive to initial weights and bias, and these two values are randomly selected in each BP training, so every time, the training tends to converge to different local minimum, resulting in low diagnostic accuracy in some diagnosis experiments. Secondly, the choice of structure of BP neural network has not a unified and complete theoretical guidance, normally only selected by experience. The selected structure may not the most suitable, and there may be a better network structure, but it takes more time to select and adjust. However, the above problem does not exist in ER rule; its initial reliability and weights are derived from sample data, and their calculating method has clear physical meanings. The evidences generated method is based on data statistics and likelihood normalization method. It is strictly based on the principle of sample statistical reasoning, thus reducing the randomness and uncertainty in process of diagnosis.In the process of actual dynamic monitoring, fault trend prediction is of great importance to marine electric propulsion systems. Nowadays, what the marine electric propulsion system operators need urgently is to be able to predict the state of the equipment before faults occur to allow the faults to be corrected prior to more serious damage or economic loss. Therefore, the ER rule method is a flexible and applicable approach which serves as an early warning and management assistance system for fault diagnosis in marine electric propulsion systems.


## 4.1. Training Model

In the experiment, 80% of the total sample setT (sample size TS = 1200) was randomly selected as the training sample TR = {[VP(t), VR(t), C(t)] | t = 1, 2, …, TSR}, sample size TSR = 960, and the remaining 20% was selected as the test sample TE = {[VP(t), VR(t), C(t)]| t = 1, 2, …, TSE}, sample size TSE = 240. Based on experience, the reference value set for the initial parameter VP(t) is A1 = {6.7877, 7.8030, 8.1517, 8.4575, 8.8254, 9.1441, 9.5400, 9.9043, 10.3445, 10.7758}, including 10 reference values in total. The reference value set for the initial parameter VR(t) is A2 = {0, 0.0679, 0.1245, 0.1733, 0.2177, 0.2759, 0.3375, 0.4224, 0.5122, 0.6417}, including 10 reference values in total. The reference value set D of C(t) is D = {1, 2, 3}, including 3 reference values in total. The training sample set TR is processed through the steps described in Section 4 to obtain casting result tables of sample pairs (VP(t), C(t)), which are presented in Tables 4 and 5.Table 4
Casting result tables of sample pairs (VP(t), C(t)).
C
V
P
Total
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
25.8775
107.6622
83.6619
55.4520
33.0722
11.6113
2.3414
0.3214
0
0
320.0000
D
2
2
0
4.3032
13.6444
43.2252
65.3831
72.6308
73.3652
36.9081
10.3351
0.2047
320.0000
D
3
3
0
0
0
0
0.1070
4.4748
31.4539
58.4470
86.0086
139.5087
320.0000
Total
25.8775
111.9654
97.3063
98.6772
98.5623
88.7169
107.1605
95.6765
96.3437
139.7134
960.0000Table 5
Casting result tables of sample pairs (VR(t), C(t)).
C
V
R
Total
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
43.7841
58.7483
39.0959
29.8323
25.1073
28.5133
29.5964
28.3416
27.0418
9.9390
320.0000
D
2
2
5.3529
35.4618
43.8672
31.0040
37.8508
34.7105
29.9841
30.3652
34.8532
36.5502
320.0000
D
3
3
0
1.3512
10.3416
29.3254
36.0224
37.1118
37.3102
40.4432
38.6807
89.4135
320.0000
Total
49.1370
95.5613
93.3047
90.1617
98.9805
100.3356
96.8907
99.1500
100.5757
135.9027
960.0000After obtaining the casting result tables, the evidence matrix table ofVi is obtained by normalization of likelihoods as shown in Tables 6 and 7.Table 6
Evidence matrix table for inputVP.
C
V
P
e
1
P
e
2
P
e
3
P
e
4
P
e
5
P
e
6
P
e
7
P
e
8
P
e
9
P
e
10
P
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
1
0.9616
0.8598
0.5620
0.3355
0.1309
0.0218
0.0034
0
0
D
2
2
0
0.0384
0.1402
0.4380
0.6634
0.8187
0.6846
0.3858
0.1073
0.0015
D
3
3
0
0
0
0
0.0011
0.0504
0.2935
0.6109
0.8927
0.9985Table 7
Evidence matrix table for inputVR.
C
V
R
e
1
R
e
2
R
e
3
R
e
4
R
e
5
R
e
6
R
e
7
R
e
8
R
e
9
R
e
10
R
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
0.8911
0.6148
0.4190
0.3309
0.2537
0.2842
0.3055
0.2825
0.2689
0.0731
D
2
2
0.1089
0.3711
0.4701
0.3439
0.3842
0.3459
0.3905
0.3063
0.3465
0.2689
D
3
3
0
0.0141
0.1108
0.3253
0.3639
0.3699
0.3851
0.4079
0.3846
0.6579After obtaining the evidence matrix tables, the reliability factorsr1 = 0.8897 and r2 = 0.4117 of the evidence parameters e1 and e2 are calculated with the sample dataset T. The initial fault diagnosis result on the sample training set TR can be obtained by using the ER rule, and the results are presented in the form of a confusion matrix as shown in Table 8.Table 8
Initial diagnostic results of ER.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
267
53
0
320
83.44
C
2
47
238
35
320
74.38
C
3
0
45
179
320
55.94It can be seen from the initial diagnostic results that the accuracy is not satisfactory because the initial set of reference valuesAi, reliability factor rj, and weight wj are not the optimal ER rule parameters, so the ideal diagnostic accuracy cannot be achieved. Therefore, it is necessary to establish a parameter optimization model for the ER rule process.


## 4.2. Parameter Optimization

The mean square error between the estimated value and the real value of the capacitor fault level is set as the objective function, and the set of optimization parametersP is determined at the same time. P = {Aji,wi | i = P, R; j = 2,..., Ji − 1}, wi represents the weight of the evidence, and the other parameters are set to A11=s11, AJ11=s21, A12=s12, AJ22=s22.(13a)minPξP=1N∑t=1TsC^t−Ct2,(13b)s.t.0≤wi≤1,i=1,2(13c)Aj−1i<Aji<Aj+1i,j=2,…,Ji−1.The objective function is optimized by using thefmincon function in Matlab, and the optimal parameter set P is obtained after training. At the same time, the optimized casting result tables, optimized evidence matrix table, and optimized capacitance aging fault diagnosis results can be obtained.


## 4.3. Analysis of Diagnostic Results

As can be seen from Tables4–13, the optimized diagnostic results are more accurate. Especially for capacitor fault levels 2 and 3, the judgment is more accurate, and misjudgment is greatly reduced. In order to further verify the reliability of the ER fault diagnosis model, test set TE was used for diagnosis. In the meanwhile, a classical three-layer BP neural network approach was selected for comparison. The learning rate was set as Ir = 0.05, and the maximum number of iterations was set to 100. The diagnostic results of TE are shown in Table 14 and Figures 6 and 7.Table 9
Optimized casting result table of sample pairs (VP(t), C(t)).
C
V
P
Total
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
14.7558
111.6242
99.3861
56.5894
33.5273
3.6031
0.5140
0
0
0
320.0000
D
2
2
0
3.9400
16.0655
59.1711
98.3794
88.4319
45.0296
8.9332
0.0493
0
320.0000
D
3
3
0
0
0
0
1.9566
30.8117
67.5026
130.7243
82.5786
6.4262
320.0000
Total
14.7558
115.5642
115.4516
115.7605
133.8633
122.8467
113.0462
139.6575
82.6279
6.4262
960.0000Table 10
Optimized casting result table of sample pairs (VR(t), C(t)).
C
V
R
Total
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
42.4839
80.1372
58.1664
70.9277
44.7212
19.6971
3.8077
0.0588
0
0
320.0000
D
2
2
8.5817
57.5337
64.5437
85.1186
48.6665
27.4821
17.0460
10.2378
0.7898
0
320.0000
D
3
3
0.1654
4.3327
50.3746
94.0281
61.7715
28.8358
25.8347
20.3626
22.4779
11.8168
320.0000
Total
51.2310
142.0036
173.0847
250.0744
155.1592
76.0150
46.6884
30.6592
23.2677
11.8168
960.0000Table 11
Optimized evidence matrix table for inputVP.
C
V
P
e
1
P
e
2
P
e
3
P
e
4
P
e
5
P
e
6
P
e
7
P
e
8
P
e
9
P
e
10
P
A
1
P
A
2
P
A
3
P
A
4
P
A
5
P
A
6
P
A
7
P
A
8
P
A
9
P
A
10
P
6.7877
7.8030
8.1517
8.4575
8.8254
9.1441
9.5400
9.9043
10.3445
10.7758
D
1
1
1
0.9659
0.8608
0.4888
0.2505
0.0293
0.0045
0
0
0
D
2
2
0
0.0341
0.1392
0.5112
0.7349
0.7119
0.3983
0.0640
0.0006
0
D
3
3
0
0
0
0
0.0146
0.2508
0.5971
0.9360
0.9994
1Table 12
Optimized evidence matrix table for inputVR.
C
V
R
e
1
R
e
2
R
e
3
R
e
4
R
e
5
R
e
6
R
e
7
R
e
8
R
e
9
R
e
10
R
A
1
R
A
2
R
A
3
R
A
4
R
A
5
R
A
6
R
A
7
R
A
8
R
A
9
R
A
10
R
0
0.0679
0.1245
0.1733
0.2177
0.2759
0.3375
0.4244
0.5122
0.6417
D
1
1
0.8293
0.5643
0.3361
0.2836
0.2882
0.2591
0.0816
0.0019
0
0
D
2
2
0.1675
0.4052
0.3729
0.3404
0.3137
0.3615
0.3651
0.3339
0.0339
0
D
3
3
0.0032
0.0305
0.2910
0.3760
0.3981
0.3793
0.5533
0.6642
0.9661
1Table 13
ER optimized diagnosis results.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
256
64
0
320
80.00
C
2
31
254
35
320
79.38
C
3
0
40
280
320
87.50Table 14
Diagnostic results of the ER/BP test set.
Capacitor fault level
Total
Accuracy (%)
C
1
C
2
C
3
Capacitor fault level
C
1
65/75
15/5
0
80
81.25/93.75
C
2
6/30
70/45
4/5
80
87.50/56.25
C
3
0
5/14
75/66
80
93.75/82.50Figure 6
Results comparison between ER (blue) and BP (red).Figure 7
Diagnostic results of ER/BP test set.
(a)
(b)The difference between the true and estimated levels of capacitance at different level capacitance attenuation in test set is as shown in Figure7. In this figure, the ordinate value of 1 indicates that the capacitance is normal, the value 2 indicates that the capacitance is in incipient fault, and the value 3 indicates that the capacitance is in serious fault. On the time axis of the abscissa, the true level of the capacitance is normal for 0 to 8 seconds, an incipient fault for 8 to 16, and a serious fault for 16 to 24.Figure8 shows the estimated levels of the test set calculated by the trained ER and BP neural network, respectively.Figure 8
Test samples and output of ER/BP.
(a)
(b)From the results presented, the diagnostic accuracy of the BP neural network approach is lower than that obtained using the ER rule model, which is mainly seen in the diagnosis of capacitor fault level 2. This fault level 2 represents an incipient fault. For the case ofC2, the change of DC bus voltage is not as obvious as case of C3, which is a very small change. Therefore, it is difficult to distinguish the change from C1 to C2 in the specific diagnosis process. In this case, there are two reasons for the low accuracy of BP neural network. Firstly, BP neural network is sensitive to initial weights and bias, and these two values are randomly selected in each BP training, so every time, the training tends to converge to different local minimum, resulting in low diagnostic accuracy in some diagnosis experiments. Secondly, the choice of structure of BP neural network has not a unified and complete theoretical guidance, normally only selected by experience. The selected structure may not the most suitable, and there may be a better network structure, but it takes more time to select and adjust. However, the above problem does not exist in ER rule; its initial reliability and weights are derived from sample data, and their calculating method has clear physical meanings. The evidences generated method is based on data statistics and likelihood normalization method. It is strictly based on the principle of sample statistical reasoning, thus reducing the randomness and uncertainty in process of diagnosis.In the process of actual dynamic monitoring, fault trend prediction is of great importance to marine electric propulsion systems. Nowadays, what the marine electric propulsion system operators need urgently is to be able to predict the state of the equipment before faults occur to allow the faults to be corrected prior to more serious damage or economic loss. Therefore, the ER rule method is a flexible and applicable approach which serves as an early warning and management assistance system for fault diagnosis in marine electric propulsion systems.


## 5. Conclusions

In this paper, an online monitoring and fault diagnosis method for capacitance aging of DC link capacitors of VSI based on ER rule is proposed. Diagnostic evidence is generated from information sources, and then reliability factors and weighting of each parameter are determined by evaluating the correlation between fault features and fault levels. The ER rule was then used to combine the evidence, and each fault level was estimated based on the combined results. Finally, we defined the objective function to optimize the ER rule parameters to make the diagnostic results more accurate. Compared with most classical data-driven method, BP neural networks, the ER rule method also has higher diagnostic accuracy and applicability in the field of incipient fault diagnosis. Unlike other types of fault diagnosis methods, this data-driven method has the advantage that it does not need to assume any mapping relationships between fault features and fault models. In addition, this method can be extended not only to serious fault diagnosis but also incipient fault diagnosis due to its ability to predefine different weighting to each evidence parameter depending on the application field. When using this method on ships, it is flexible enough to allow changes to the parameters in the ER rule to meet the needs of the different electric propulsion systems in use on board diverse types of vessel. The Jiangsu ferry 3011 is a new ship. However, the aging process of capacitor will take years. Therefore, the actual aging data cannot be obtained for research. The method described in this paper will be a promising diagnostic method to detect incipient fault for Jiangsu ferry 3011; further analysis and discussion will be carried out after collecting overall life cycle’s physical data.



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*Source: 5724019-2020-03-31.xml*