# A Fault Analysis Method for Three-Phase Induction Motors Based on Spiking Neural P Systems

Authors: Zhu Huang; Tao Wang; Wei Liu; Luis Valencia-Cabrera; Mario J. Pérez-Jiménez; Pengpeng Li
Journal: Complexity
 (2021)

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## Abstract

The fault prediction and abductive fault diagnosis of three-phase induction motors are of great importance for improving their working safety, reliability, and economy; however, it is difficult to succeed in solving these issues. This paper proposes a fault analysis method of motors based on modified fuzzy reasoning spiking neural P systems with real numbers (rMFRSNPSs) for fault prediction and abductive fault diagnosis. To achieve this goal, fault fuzzy production rules of three-phase induction motors are first proposed. Then, the rMFRSNPS is presented to model the rules, which provides an intuitive way for modelling the motors. Moreover, to realize the parallel data computing and information reasoning in the fault prediction and diagnosis process, three reasoning algorithms for the rMFRSNPS are proposed: the pulse value reasoning algorithm, the forward fault prediction reasoning algorithm, and the backward abductive fault diagnosis reasoning algorithm. Finally, some case studies are given, in order to verify the feasibility and effectiveness of the proposed method.


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## Body

## 1. Introduction

As an important part of industrial and agricultural productions, the normal operation of three-phase induction motors plays a pivotal role in economic benefits and security risks. For a motor, any potential failure that cannot be predicted or detected in time may produce damage on it, resulting in downtime with potentially huge economic losses [1–4]. In addition, when a motor has faults and is shut down, the first task is to perform abductive fault diagnosis to find its failure causes, which can effectively help the operation and maintenance personnels to locate faulty parts quickly. Therefore, fault prediction and abductive fault diagnosis are of great significance for improving the working reliability and stability of motors [5].The fault prediction of a motor is usually realized based on an online monitoring system to detect the early failure symptoms and trend parameters that can reflect hidden troubles. Then, these symptoms and parameters are processed by prediction algorithms to obtain early-warning information and integrated decision making [6] to prevent motor failures. For example, [7] diagnosed mechanical faults of motors by vibration analysis, which was carried out through a noncontact approach based on an optical computer mouse and a digital signal processing device. Reference [8] proposed a two-stage machine learning analysis architecture, where a recurrent neural network-based variational autoencoder was proposed in the first stage, and principal components analysis and linear discriminant analysis techniques were applied in the second stage. This architecture was useful to accurately predict the motor fault modes only by using motor vibration time-domain signals. In [9], a hybrid technique for bearing prognostics was proposed, which utilized a regression-based adaptive prediction model to find the evolution trend of bearing health indices. However, so far, most fault prediction methods require a huge number of historical data to perform the training and learning processes of their predictive models.The abductive fault diagnosis of a motor consists in finding failure causes from its fault phenomena and operation data, so that a motor can be effectively repaired, thus reducing economic losses [10]. In [11], an instantaneous frequency analysis method based on abnormal sounds was proposed. However, when the acoustic signals of a motor were mixed by other acoustic signals (such as reﬂected signals and overlapped signals), it was difficult to extract the features of bearing fault information. In [12], a new current signature analysis-based fault detector for motors based on a matched subspace technique was proposed. However, it was only effective for detecting eccentricity faults, bearing faults, and broken rotor bars. Reference [13] proposed a technique based on vibration information to identify and classify different bearing failure conditions. The setting and testing of parameters was strict and difficult; for example, the accelerometer needs to be very close to the motor, and the setting of accelerometer and data logger should be the same. However, this method needed much historical data with a complex computing process. In [14], an intelligent fault diagnosis of three-phase induction motors using a signal-based method was proposed and tested in different situations, in order to verify its availability in diagnosing failures, even when the operating mode data were limited. However, the experimental results showed that it was only suitable for the diagnosis of broken bars and bearing failure.The aforementioned methods have their own advantages with the same disadvantages implying that they mainly focus on the diagnosis of a single fault, such as the rotor bar breaking or the stator short circuit. Thus, they cannot effectively diagnose multiple faults, not achieving the requirement of performing an overall fault analysis of the whole machine.Therefore, how to improve the abovementioned fault prediction and abductive fault diagnosis methods or put forward new ones is the main issue in the corresponding engineering domain for the motors. On the other hand, with the rapid development of artificial intelligence technology, intelligent analysis and diagnosis methods are gradually developed, such as expert systems (ESs) [15], artificial neural networks (ANNs) [16–20], Petri nets (PNs) [21–23], tissue P systems (TPSs) [24–26], and spiking neural P systems (SNPSs) [27–34]. Specifically, SNPS is a novel high-performance bioinspired distributed parallel computing model with powerful information processing ability. It is a special kind of neural-like P system [29] inspired by the topological structure of biological neural networks and the way that biological neurons store, transmit, and exchange messages, i.e., by sending electrical impulses (spikes) along axons in a distributed and parallel manner [30–32].The SNPS-based fault diagnosis methods (for example, the ones for power systems) are derived from the similarities between the pulse transmission between neurons via synapses and the fault propagation in power systems. Accordingly, the basic mechanism to address fault diagnosis based on SNPSs is to find faulty sections by dealing with the uncertainty [35] of fault alarm information. In general, the input neurons of an SNPS correspond to protective devices (including protective relays and circuit breakers), and the output neurons are associated with suspicious fault sections. Thus, the pulse values of input neurons represent the action information of protective devices, that is, the actual tripping information from the supervisory control and data acquisition system or the action confidence levels represented by fuzzy numbers [36]. On the other hand, the pulse values of output neurons express the trip information o fault confidence levels of the suspicious sections. When the fault reasoning is finished, faulty sections are finally determined based on the fault confidence levels according to criterion rules.Because of the high requirement of fault diagnosis methods for processing fault information, the SNPS-based diagnosis methods have become a hot research topic with rich research results [27–29, 33, 34]. However, up to now, the relevant research work is mainly focused on the fault diagnosis of power systems. Besides, the existing work mainly studies the postevent diagnosis problems. Therefore, to give full play to the excellent information processing ability and computing power of SNPSs, it is of great importance to expand their scope to different application fields, as well as extend the applications from the postante ones to new ex-ante analysis and prediction frameworks.Therefore, this paper moves forward in this widening of the scope of SNPSs. More specifically, the work proposes a fault analysis method based on modified fuzzy reasoning spiking neural P systems with real numbers (rMFRSNPSs) for three-phase induction motors. As an important part of this new method presented here, the forward fault prediction reasoning algorithm (FFPRA) and the backward abductive fault diagnosis reasoning algorithm (BAFDRA) are proposed. The main contributions of this paper are described as follows:(1)
Based on the existing variants of SNPSs, we propose a modified fuzzy reasoning spiking neural P system with real numbers by simplifying previously existing ones. In order to enable the rMFRSNPSs to achieve fault prediction and abductive diagnosis, three algorithms are proposed,i.e., the pulse value reasoning algorithm (PVRA), the FFPRA, and the BAFDRA, respectively.(2)
Fault fuzzy production rules for motors are presented to obtain the relationships between failure symptoms and different faults. Moreover, the rMFRSNPS-based model for a motor is built via modelling the production rules, which is the basis for the fault analysis from the point of view of a whole machine.(3)
Firstly, the SNPS is introduced to solve the fault diagnosis of motors, including forward fault prediction and backward abductive fault diagnosis. In addition, we also extend its application from the postante diagnosis to a new ex-ante prediction framework. The new framework not only can take full advantages of the SNPS for the fault prediction with potential fault paths and their occurrence probabilities in an ex-ante prediction problem but also can effectively find failure causes with abductive reasoning paths and their probabilities in a postante fault diagnosis problem.


## 2. The rMFRSNPS-Based Fault Model for Motors

In this section, we first present the rMFRSNPS and then propose fault fuzzy production rules of motors. Finally, the rMFRSNPS is employed to model the rules to propose a universal rMFRSNPS-based fault analysis model.


### 2.1. Modified Fuzzy Reasoning Spiking Neural P Systems with Real Numbers

Definition 1.
A modified fuzzy reasoning spiking neural P system with real numbers (rMFRSNPS, for short) of degreem≥1 is a tuple(1)∏=O,σ1,…,σm,syn,in,out,where(1)
O=a is a singleton alphabet (a is called a spike, O is a set of spikes).(2)
Q=Qp∪Qr is a neuron set, where Qp=σ1,…,σs is the proposition neuron set and Qr=σs+1,…,σs+t is the rule neuron set, being s+t=m. Each proposition neuron σi is of the form αi,λi,ri, 1≤i≤s, where(a)
αi is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in σi.(b)
λi is a real number in (0, 1) representing the firing threshold of σi.(c)
ri represents a firing (spiking) rule of proposition neuron σi with the form E/aα⟶aθ, where α and θ are real numbers in [0, 1], and E=an∧α≥λi is the firing condition. The firing rule ri of σi can be applied if and only if σi receives, at least, n spikes and the potential value of spikes satisfies that α≥λi. By applying rule ri, σi will consume (remove) a spike with pulse value α and then not only produce (emit) a new spike with pulse value θ but also transmit it to its postsynaptic neurons.Each rule neuronσs+j is of the form δj,cj,λj,rj, 1≤j≤t, where(a)
δj is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in σs+j.(b)
cj is a real number in [0, 1] representing the truth value of σs+j.(c)
λj is a real number in (0, 1) representing the firing threshold of σs+j.(d)
rj represents a firing (spiking) rule of σs+j with the form E/aδ⟶aβ, where δ and β are real numbers in [0, 1], and E=an∧δ≥λj is the firing condition. The firing rule rj of σs+j can be applied if and only if σs+j receives, at least, n spikes and the potential value of spikes satisfies that δ≥λj. By applying rule rj, σs+j will consume (remove) a spike with pulse value δ and then not only produce (emit) a new spike with pulse value β but also transmit it to its postsynaptic neurons.(3)
syn⊆1,…,m×1,…,m with i,i∉syn for 1≤i≤m, is a directed graph of synapses between linked neurons.(4)
in⊆1,…,m and out⊆1,…,m represent the sets of input and output neurons of ∏, respectively.
Fuzzy production rules can be modelled in the framework of rMFRSNPSs. Let us recall that there are, basically, three types of fuzzy production rules [33].(a)
GENERAL rule, whose format isRiCF=ci: IF p1α1 THEN p2α2, where p1 is an antecedent proposition and p2 is a consequent proposition(b)
Compound AND rule, whose format isRiCF=ci: IF p1α1 AND…AND pk−1αk−1 THEN pkαk, where p1,…,pk−1 are antecedent propositions, pk is a consequent proposition, and k≥3(c)
Compound OR rule, whose format isRiCF=ci: IF p1α1 OR…OR pk−1αk−1 THEN pkαk, where p1,…,pk−1 are antecedent propositions, pk is a consequent proposition, and k≥3
In fact, there exists another type of rule whose format isRiCF=ci: IF p1α1 THEN p2α2 AND…AND pkαk, where p1 is an antecedent proposition and p2,…,pk are consequent propositions, with k≥3. However, this kind of rules can be considered as a particular case of a composition of k−1 GENERAL rules.
In order to model fuzzy production rules by means of rMFRSNPSs, a proposition neuron in an rMFRSNPS is associated with a proposition in the fuzzy production rules. Such neurons will be represented by a circle. If a proposition neuronσi=αi,λi,ri is an input neuron, then its initial potential value αi represents the information that σi has received from the environment.
Ageneral rule neuron in an rMFRSNPS consists of only one presynaptic proposition neuron and one or more postsynaptic proposition neurons. Therefore, in a natural manner, a general rule neuron can be associated with a general rule, that is, with a fuzzy production rule which has only one proposition on its antecedent part. An and rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an AND relationship among them and only one postsynaptic proposition neuron. Thus, in a straightforward way, an and rule neuron can be associated with each compound AND fuzzy production rule. Finally, an or rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an OR relationship among them and only one postsynaptic proposition neuron. According to the previous comments, an or rule neuron can be associated with each compound OR fuzzy production rule. These rule neurons are represented by a rectangle, and they are graphically illustrated in Figure 1.Figure 1
rMFRSNPS-based models for fuzzy fault production rules. (a) GENERAL rule; (b) compound AND rule; and (c) compound OR rule.
(a)(b)(c)


### 2.2. Fault Fuzzy Production Rules for Motors

In this paper, the possible failures in a motor include electrical faults and mechanical ones. The first class includes failures such as the excessive current in a phase, the excessive excitation current, a phase voltage loss, the phase-absent operation, the three-phase current asymmetry, and the insulation winding burned down. The second class contains failures such as the bearing expansion by heat, the excessive wear of bearing, the excessive vibration of motor in operation, the abnormal noise, the rotor stuck or stopped rotating, and the motor sweeping. According to the principle of motor failures [23, 37–41] and the fault simulation model in Figure 2, fault fuzzy production rules of motors are obtained as follows, where events corresponding to the propositions in the rules are shown in Table 1.Rule 1 (c1): IF p1 OR p2 occurs, THEN p17 occursRule 2 (c2): IF p2 AND p3 occur, THEN p18 occursRule 3 (c3): IF p3 occurs, THEN p19 occursRule 4 (c4): IF p4 occurs, THEN p20 occursRule 5 (c5): IF p5 occurs, THEN p21 occursRule 6 (c6): IF p6 OR p7 occurs, THEN p22 occursRule 7 (c7): IF p8 occurs, THEN p23 occursRule 8 (c8): IF p8 occurs, THEN p24 occursRule 9 (c9): IF p9 OR p10 occurs, THEN p25 occursRule 10 (c10): IF p10 OR p11 occurs, THEN p26 occursRule 11 (c11): IF p12 OR p13 occurs, THEN p27 occursRule 12 (c12): IF p14 occurs, THEN p28 occursRule 13 (c13): IF p15 AND p16 occur, THEN p29 occursRule 14 (c14): IF p17 OR p18 OR p19 occurs, THEN p30 occursRule 15 (c15): IF p20 occurs, THEN p31 occursRule 16 (c16): IF p21 occurs, THEN p32 occursRule 17 (c17): IF p22 occurs, THEN p33 occursRule 18 (c18): IF p22 OR p23 occurs, THEN p34 occursRule 19 (c19): IF p24 occurs, THEN p35 occursRule 20 (c20): IF p25 occurs, THEN p36 occursRule 21 (c21): IF p26 occurs, THEN p37 occursRule 22 (c22): IF p27 occurs, THEN p38 occursRule 23 (c23): IF p28 occurs, THEN p39 occursRule 24 (c24): IF p29 occurs, THEN p40 occursRule 25 (c25): IF p30 OR p31 OR p32 OR σ33 occurs, THEN p41 occursRule 26 (c26): IF p34 OR p35 occurs, THEN p42 occursRule 27 (c27): IF p36 occurs, THEN p43 occursRule 28 (c28): IF p37 OR p38 occurs, THEN p44 occursRule 29 (c29): IF p39 OR p40 occurs, THEN p45 occursRule 30 (c30): IF p41 occurs, THEN p46 occursRule 31 (c31): IF p42 OR p43 occurs, THEN p47 occursRule 32 (c32): IF p44 occurs, THEN p48 occursRule 33 (c33): IF p45 occurs, THEN p49 occursRule 34 (c34): IF p46 OR p47 OR p48 OR p49 occurs, THEN p50 occursFigure 2
Fault simulation model of a three-phase induction motor.Table 1
The meaning of propositions in fuzzy fault production rules.
PropositionsEventsp1Overloadp2Rotor winding short circuitp3The resistance value of a phase winding decreasesp4Fuse melt faultp5Damage of shaft seal ring structurep6Oil sealing material overheatingp7Excessive roughness value of the seal surface shaftp8Excessive temperaturep9Mechanical fault of the rotor windingp10The motor centerline is inconsistent with the pump onep11Fault of the bearing locking devicep12Rotor core deformationp13Fracture or shedding of magnetic slot wedgesp14Dewelding at the joint of the winding and lead wirep15Connection box joint loosenedp16Poor contact of the power control loop switchp17Decrease in rotational speedp18Excessive current in a phasep19Excessive excitation currentp20A phase voltage lossp21Foreign matter enters the rotary shaft clearancep22The motor oil intakep23Oxidation and decomposition of bearing lubricating oilp24Bearing expansion by heatp25Bearing generates additional loadp26Rotor axial movesp27The iron core of the stator and rotor has an air gapp28Rotor winding open circuitp29Contact resistance value increasesp30Motor overheatingp31Phase-absent operationp32Abnormal rotation or the rotor is stuckp33Insulation agingp34Reduction of lubricant oilp35Friction occurs between the crankshaft ring and shaft holep36Excessive vibration of the motor in operationp37Excessive bearing noisep38Motor sweepingp39Three-phase current of the stator increasesp40Increased pressure dropp41Three-phase current asymmetryp42Excessive wear of the bearingp43Irregular impact loadp44Abnormal noisep45Rotor rotation is weak or the rotor does not turn and hump46Insulation winding burned downp47Motor axle holderp48Rotor stuck or stopped rotatingP49The motor appears local high heatp50The motor cannot work——


### 2.3. The rMFRSNPS-Based Model for a Motor

This section models the fault fuzzy production rules proposed in Section2.2 and builds a universal rMFRSNPS-based fault analysis model for three-phase induction motors, as shown in Figure 3. The designed rMFRSNPS is of degree m=84 and specifically contains s=50 proposition neurons and t=34 rule neurons.Figure 3
An rMFRSNPS-based model for three-phase induction motors.


## 2.1. Modified Fuzzy Reasoning Spiking Neural P Systems with Real Numbers

Definition 1.
A modified fuzzy reasoning spiking neural P system with real numbers (rMFRSNPS, for short) of degreem≥1 is a tuple(1)∏=O,σ1,…,σm,syn,in,out,where(1)
O=a is a singleton alphabet (a is called a spike, O is a set of spikes).(2)
Q=Qp∪Qr is a neuron set, where Qp=σ1,…,σs is the proposition neuron set and Qr=σs+1,…,σs+t is the rule neuron set, being s+t=m. Each proposition neuron σi is of the form αi,λi,ri, 1≤i≤s, where(a)
αi is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in σi.(b)
λi is a real number in (0, 1) representing the firing threshold of σi.(c)
ri represents a firing (spiking) rule of proposition neuron σi with the form E/aα⟶aθ, where α and θ are real numbers in [0, 1], and E=an∧α≥λi is the firing condition. The firing rule ri of σi can be applied if and only if σi receives, at least, n spikes and the potential value of spikes satisfies that α≥λi. By applying rule ri, σi will consume (remove) a spike with pulse value α and then not only produce (emit) a new spike with pulse value θ but also transmit it to its postsynaptic neurons.Each rule neuronσs+j is of the form δj,cj,λj,rj, 1≤j≤t, where(a)
δj is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in σs+j.(b)
cj is a real number in [0, 1] representing the truth value of σs+j.(c)
λj is a real number in (0, 1) representing the firing threshold of σs+j.(d)
rj represents a firing (spiking) rule of σs+j with the form E/aδ⟶aβ, where δ and β are real numbers in [0, 1], and E=an∧δ≥λj is the firing condition. The firing rule rj of σs+j can be applied if and only if σs+j receives, at least, n spikes and the potential value of spikes satisfies that δ≥λj. By applying rule rj, σs+j will consume (remove) a spike with pulse value δ and then not only produce (emit) a new spike with pulse value β but also transmit it to its postsynaptic neurons.(3)
syn⊆1,…,m×1,…,m with i,i∉syn for 1≤i≤m, is a directed graph of synapses between linked neurons.(4)
in⊆1,…,m and out⊆1,…,m represent the sets of input and output neurons of ∏, respectively.
Fuzzy production rules can be modelled in the framework of rMFRSNPSs. Let us recall that there are, basically, three types of fuzzy production rules [33].(a)
GENERAL rule, whose format isRiCF=ci: IF p1α1 THEN p2α2, where p1 is an antecedent proposition and p2 is a consequent proposition(b)
Compound AND rule, whose format isRiCF=ci: IF p1α1 AND…AND pk−1αk−1 THEN pkαk, where p1,…,pk−1 are antecedent propositions, pk is a consequent proposition, and k≥3(c)
Compound OR rule, whose format isRiCF=ci: IF p1α1 OR…OR pk−1αk−1 THEN pkαk, where p1,…,pk−1 are antecedent propositions, pk is a consequent proposition, and k≥3
In fact, there exists another type of rule whose format isRiCF=ci: IF p1α1 THEN p2α2 AND…AND pkαk, where p1 is an antecedent proposition and p2,…,pk are consequent propositions, with k≥3. However, this kind of rules can be considered as a particular case of a composition of k−1 GENERAL rules.
In order to model fuzzy production rules by means of rMFRSNPSs, a proposition neuron in an rMFRSNPS is associated with a proposition in the fuzzy production rules. Such neurons will be represented by a circle. If a proposition neuronσi=αi,λi,ri is an input neuron, then its initial potential value αi represents the information that σi has received from the environment.
Ageneral rule neuron in an rMFRSNPS consists of only one presynaptic proposition neuron and one or more postsynaptic proposition neurons. Therefore, in a natural manner, a general rule neuron can be associated with a general rule, that is, with a fuzzy production rule which has only one proposition on its antecedent part. An and rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an AND relationship among them and only one postsynaptic proposition neuron. Thus, in a straightforward way, an and rule neuron can be associated with each compound AND fuzzy production rule. Finally, an or rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an OR relationship among them and only one postsynaptic proposition neuron. According to the previous comments, an or rule neuron can be associated with each compound OR fuzzy production rule. These rule neurons are represented by a rectangle, and they are graphically illustrated in Figure 1.Figure 1
rMFRSNPS-based models for fuzzy fault production rules. (a) GENERAL rule; (b) compound AND rule; and (c) compound OR rule.
(a)(b)(c)


## 2.2. Fault Fuzzy Production Rules for Motors

In this paper, the possible failures in a motor include electrical faults and mechanical ones. The first class includes failures such as the excessive current in a phase, the excessive excitation current, a phase voltage loss, the phase-absent operation, the three-phase current asymmetry, and the insulation winding burned down. The second class contains failures such as the bearing expansion by heat, the excessive wear of bearing, the excessive vibration of motor in operation, the abnormal noise, the rotor stuck or stopped rotating, and the motor sweeping. According to the principle of motor failures [23, 37–41] and the fault simulation model in Figure 2, fault fuzzy production rules of motors are obtained as follows, where events corresponding to the propositions in the rules are shown in Table 1.Rule 1 (c1): IF p1 OR p2 occurs, THEN p17 occursRule 2 (c2): IF p2 AND p3 occur, THEN p18 occursRule 3 (c3): IF p3 occurs, THEN p19 occursRule 4 (c4): IF p4 occurs, THEN p20 occursRule 5 (c5): IF p5 occurs, THEN p21 occursRule 6 (c6): IF p6 OR p7 occurs, THEN p22 occursRule 7 (c7): IF p8 occurs, THEN p23 occursRule 8 (c8): IF p8 occurs, THEN p24 occursRule 9 (c9): IF p9 OR p10 occurs, THEN p25 occursRule 10 (c10): IF p10 OR p11 occurs, THEN p26 occursRule 11 (c11): IF p12 OR p13 occurs, THEN p27 occursRule 12 (c12): IF p14 occurs, THEN p28 occursRule 13 (c13): IF p15 AND p16 occur, THEN p29 occursRule 14 (c14): IF p17 OR p18 OR p19 occurs, THEN p30 occursRule 15 (c15): IF p20 occurs, THEN p31 occursRule 16 (c16): IF p21 occurs, THEN p32 occursRule 17 (c17): IF p22 occurs, THEN p33 occursRule 18 (c18): IF p22 OR p23 occurs, THEN p34 occursRule 19 (c19): IF p24 occurs, THEN p35 occursRule 20 (c20): IF p25 occurs, THEN p36 occursRule 21 (c21): IF p26 occurs, THEN p37 occursRule 22 (c22): IF p27 occurs, THEN p38 occursRule 23 (c23): IF p28 occurs, THEN p39 occursRule 24 (c24): IF p29 occurs, THEN p40 occursRule 25 (c25): IF p30 OR p31 OR p32 OR σ33 occurs, THEN p41 occursRule 26 (c26): IF p34 OR p35 occurs, THEN p42 occursRule 27 (c27): IF p36 occurs, THEN p43 occursRule 28 (c28): IF p37 OR p38 occurs, THEN p44 occursRule 29 (c29): IF p39 OR p40 occurs, THEN p45 occursRule 30 (c30): IF p41 occurs, THEN p46 occursRule 31 (c31): IF p42 OR p43 occurs, THEN p47 occursRule 32 (c32): IF p44 occurs, THEN p48 occursRule 33 (c33): IF p45 occurs, THEN p49 occursRule 34 (c34): IF p46 OR p47 OR p48 OR p49 occurs, THEN p50 occursFigure 2
Fault simulation model of a three-phase induction motor.Table 1
The meaning of propositions in fuzzy fault production rules.
PropositionsEventsp1Overloadp2Rotor winding short circuitp3The resistance value of a phase winding decreasesp4Fuse melt faultp5Damage of shaft seal ring structurep6Oil sealing material overheatingp7Excessive roughness value of the seal surface shaftp8Excessive temperaturep9Mechanical fault of the rotor windingp10The motor centerline is inconsistent with the pump onep11Fault of the bearing locking devicep12Rotor core deformationp13Fracture or shedding of magnetic slot wedgesp14Dewelding at the joint of the winding and lead wirep15Connection box joint loosenedp16Poor contact of the power control loop switchp17Decrease in rotational speedp18Excessive current in a phasep19Excessive excitation currentp20A phase voltage lossp21Foreign matter enters the rotary shaft clearancep22The motor oil intakep23Oxidation and decomposition of bearing lubricating oilp24Bearing expansion by heatp25Bearing generates additional loadp26Rotor axial movesp27The iron core of the stator and rotor has an air gapp28Rotor winding open circuitp29Contact resistance value increasesp30Motor overheatingp31Phase-absent operationp32Abnormal rotation or the rotor is stuckp33Insulation agingp34Reduction of lubricant oilp35Friction occurs between the crankshaft ring and shaft holep36Excessive vibration of the motor in operationp37Excessive bearing noisep38Motor sweepingp39Three-phase current of the stator increasesp40Increased pressure dropp41Three-phase current asymmetryp42Excessive wear of the bearingp43Irregular impact loadp44Abnormal noisep45Rotor rotation is weak or the rotor does not turn and hump46Insulation winding burned downp47Motor axle holderp48Rotor stuck or stopped rotatingP49The motor appears local high heatp50The motor cannot work——


## 2.3. The rMFRSNPS-Based Model for a Motor

This section models the fault fuzzy production rules proposed in Section2.2 and builds a universal rMFRSNPS-based fault analysis model for three-phase induction motors, as shown in Figure 3. The designed rMFRSNPS is of degree m=84 and specifically contains s=50 proposition neurons and t=34 rule neurons.Figure 3
An rMFRSNPS-based model for three-phase induction motors.


## 3. Fault Analysis Method Based on rMFRSNPSs

This section proposes a fault analysis method based on rMFRSNPSs for three-phase induction motors, whose flowchart is shown in Figure4, where 0=0,…,0t×1T. The proposed method includes two parts, one is for fault prediction before fault occurrence while the other one is for abductive diagnosis reasoning after failures. Moreover, a diagrammatic sketch of the application scenario for the proposed method is shown in Figure 5, where red circles represent the already happened events while blue circles express the not occurred ones. The status of a motor is monitored in real time. When the motor has fault symptoms or faults, relevant state data will be transmitted to the fault analysis center, where our method will be used to handle the events.Figure 4
Fault reasoning flow chart based on rMFRSNPSs.Figure 5
A diagrammatic sketch of the application scenario for the proposed method.Specifically, in this proposed method, the PVRA (Algorithm1) is proposed to get the potential value of spikes in neurons using historical data and expertise. When a motor has no faults, but is accompanied by fault symptoms, the FFPRA (Algorithm 2) is employed to predict propagation paths with occurrence probabilities. When a motor fails, the fault positions (corresponding to neurons with fault pulses) are found according to failure phenomena, and then, failure causes with probabilities are obtained according to the BAFDRA (Algorithm 3). Thus, the maintenance efficiency can be improved accordingly to check the motor on the basis of results got by the prediction reasoning or abductive reasoning. Note that the historical data include fault probabilities of fault events (Algorithms 1–3), certainty factors of fault production rules (Algorithms 1 and 2), and the tightness degree between related fault events (Algorithm 3).Algorithm 1: Pulse value reasoning algorithm.
Input: α0,δ0,D1,D2,D3,D4,C,λp,λr,0=0,…,0tT(1)
Letk = 1(2)
while (δk≠0)(3)
if each proposition neuron satisfies its firing conditionE=an∧αi≥λpi,1≤i≤sthen(4)
proposition neurons fire and computeδk viaδk=D1T∗αk−1+D2T⋅αk−1+D3T∘αk−1(5)
if each rule neuron satisfies its firing conditionE=an∧δj≥λrj,1≤j≤tthen(6)
rule neurons fire and computeαk viaαk=D4T∘C∗δk(7)
end if(8)
k = k + 1(9)
end whileOutput: The pulse value of all neurons.Algorithm 2: Forward fault prediction reasoning algorithm.
Input: Np0+,Nr0+,α0,δ0,λp,λr,C,0=0,…,0tT,D1,D2,D3,D4(1)
Letk = 1(2)
while (Nrk+≠0)(3)
if each proposition neuron satisfies its firing conditionE=an∧αi≥λpi,1≤i≤sthen(4)
proposition neurons fire and computeδk and Nrk+ viaδk=D1T∗αk−1+D2T⋅αk−1+D3T∘αk−1,Nrk+=D1⊕D3TΔNpk−1+⊗αk−1Θλp⊕D2T∇Npk−1+⊗αk−1Θλp(5)
end if(6)
if each rule neuron satisfies its firing conditionE=an∧δj≥λrj,1≤j≤tthen(7)
rule neurons fire and computeαk and Npk+ viaαk=D4T∘C∗δk,Npk+=D4TΔNrk+⊗δkΘλr(8)
end if(9)
k = k + 1(10)
end whileOutput: Potential fault paths and their occurrence probabilities.Algorithm 3: Backward abductive fault diagnosis reasoning algorithm.
Input: Np0−,Nr0−,θp0,θr0,α,δ,λp,λr,λy,0=0,…,0tT,D1,D2,D3,D4,W(1)
Letk = 1(2)
while (Nrk−≠0)(3)
if each rule neurons satisfies its firing conditionE=an∧θj≥λrj,1≤j≤tthen(4)
rule neurons fire and computeNrk− and θrk viaNrk−=D4ΔNpk−1−θrk=Nrk−⊗δ(5)
end if(6)
if each proposition neuron satisfies its firing conditionE=an∧θi≥λpi,1≤i≤sthen(7)
proposition neurons fire and computeNpk− and θpk viaNpk−=D1⊕D2⊕D3ΔNrk−θpk=Npk−⊗α(8)
computeYk− viaYk−=W⊗θpk(9)
determine the MCSsQ1,…,Ql of each PN in Npk−1−, where Qi=σ1,…,σq,1≤q,i<s. Compute danger degree of MCSs for each PN via yQi=∏j=1qyσj, and screen out the pulse of PNs in a PMCS whose danger degree is larger than λy(10)
update the number of propositional neuronsNpk− per the selected pulses, and compute pulse value of fault pulse in proposition neurons after position updating viaθpk=Npk−⊗α(11)
end if(12)
k = k + 1(13)
end whileOutput: Failure causes, paths of abductive reasoning and their probabilities.Next, we describe Algorithms1–3 in detail as follows.


### 3.1. Pulse Value Reasoning Algorithm

To explain this algorithm, we introduce its vectors, matrices, and operators as follows (PN denotesproposition neuron and RN denotes rule neuron):(1)
α=α1,…,αsT is a pulse value vector of PNs, where αii=1,…,s represents the pulse value of the i-th PN σi. If a PN has not any pulse, then its pulse value is 0.(2)
δ=δ1,…,δtT is a pulse value vector of RNs, where δjj=1,…,t is the pulse value of the j-th RN σs+j. If an RN has not any pulse, then its pulse value is 0.(3)
λp=λp1,…,λpsT is a firing threshold vector of PNs.(4)
λr=λr1,…,λrtT is a firing threshold vector of RNs.(5)
C=diagc1,…,ct is a diagonal matrix of truth values of RNs, where cjj=1,…,t is the truth value of the j-th RN σs+j.(6)
D1=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to general RNs. If there is a synapse from the PN σi to the general RN σs+j, then dij = 1; otherwise, dij = 0.(7)
D2=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to and RNs. If there is a synapse from the PN σi to the and RN σs+j, then dij = 1; otherwise, dij = 0.(8)
D3=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to or RNs. If there is a synapse from the PN σi to the or RN σs+j, then dij = 1; otherwise, dij = 0.(9)
D4=djit×s is a synaptic matrix, which represents the directed synaptic connections from RNs to PNs. If there is a synapse from the RN σs+j to the PN σi, then dji = 1; otherwise, dji = 0.(10)
0=0,…,0t×1T is null vector.(11)
DT∗α=d¯1,…,d¯tT, where d¯j=d1j×α1+⋯+dsj×αs,1≤j≤t.(12)
DT⋅α=d¯1,…,d¯tT, where d¯j=mind1j×α1+⋯+dsj×αs,1≤j≤t.(13)
DT∘α=d¯1,…,d¯tT, where d¯j=maxd1j×α1+⋯+dsj×αs,1≤j≤t.


### 3.2. Forward Fault Prediction Reasoning Algorithm

To explain this algorithm, we introduce its vectors, matrices, and operators as follows:(1)
Np+ is the number vector of PNs where pulses are located. If a PN contains a pulse, then the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(2)
Nr+ is the number vector of RNs where pulses are located. If an RN contains a pulse, the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(3)
AΔB=ci1x×1, where A=aikx×y,B=bk1y×1, and ci1=maxaikbk1,1≤i≤x,1≤k≤y.(4)
A∇B=ci1x×1, where A=aikx×y,B=bk1y×1. If ∑k=1yaikbk1<2, then ci1=0, otherwise ci1=1,1≤i≤x,1≤k≤y.(5)
A⊗B=cijx×y, where A=aijx×y,B=bi1x×1, and cij=aijbi1,1≤i≤x,1≤j≤y.(6)
A⊕B=cijx×y, where A=aijx×y,B=bijx×y, and cij=maxaij,bij,1≤i≤x,1≤j≤y.(7)
AΘB=cijx×y, where A=aijx×y,B=bijx×y. If aij≥bij, then cij=1, otherwise, cij=0, 1≤i≤x,1≤j≤y.Note that the vectorsα,δ,λp,λr, and 0, the matrices D1,D2,D3,D4, and C, and the operators ∗,⋅ and ∘ in Algorithm 2 are the same as the ones in Algorithm 1.


### 3.3. Backward Abductive Fault Diagnosis Reasoning Algorithm

To improve the accuracy of backward abductive reasoning, this paper integrates a fault screening mechanism of the precise minimum cut set (please see Definition2) into the parallel reasoning ability of SNPSs to propose the BAFDRA for the rMFRSNPS, i.e., Algorithm 3. The precise minimum cut set effectively combines the abductive principle of top events in minimum cut sets [42] with the screening mechanism, where, in two adjacent fault events, a bottom event corresponds to a fault or a fault symptom and a top event corresponds to a fault. Moreover, the screening mechanism is used to improve the abductive reasoning accuracy by eliminating pulses contained in the minimum cut set whose danger degree is lower than the dangerous threshold, where the danger degree is used to access the fault risk of motors [43].Definition 2.
A precise minimum cut set (PMCS) is a tuple(2)Q=Q1,…,Ql,1≤l≤s,where(1)
Q1,…,Ql are l minimum cut sets (MCSs), where(a)
Ageneral rule neuron has one presynaptic proposition neuron (corresponding to a bottom event) and one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered only by the bottom event; thus, the MCS of the top event consists of the presynaptic neuron.(b)
Anor rule neuron has more than one presynaptic proposition neurons (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered by any bottom event; thus, the MCS of the top event consists of any presynaptic neuron. That is, if there are l bottom events that can trigger the top event, then the top event will have l MCSs with each MCS consisting of one presynaptic neuron corresponding to one of the l bottom events.(c)
Anand rule neuron has more than one presynaptic proposition neuron (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can only be triggered by all bottom events at the same time, so that the MCS of the top event consists of all the presynaptic neurons. That is, if there are l bottom events that can trigger the top event, then the top event will have only one MCS and it should consist of all the presynaptic neurons corresponding to the l bottom events.(2)
Qg=yσi,yQg,λy,1≤g≤l,1≤i≤s is the g-th MCS, where(a)
yσi is the danger degree of the i-th PN σi, defined as(3)yσi=wσi×αi,wherewσi is a weighted value in [0, 1] representing the tightness degree between PN σi and its postsynaptic neurons.(b)
yQg is a danger degree of the g-th minimum cut set, i.e., Qg, which is defined as(4)yQg=∏j=1qyσj,1≤q≤s.(c)
λy is a number in (0, 1) representing the danger degree threshold of an MCS. When the danger degree of an MCS is greater than λy, then the MCS is called a PMCS and the PNs with pulse in the PMCS form the fault paths, with the first PN in each path being called the fault source.
Algorithm3 is shown as follows:
To explain the algorithm, we introduce its vectors, matrices, and operators as follows:(1)
Np− is the number vector of PNs where fault pulses are located. If a PN contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(2)
Nr− is the number vector of RNs where fault pulses are located. If a rule neuron contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(3)
θp=θp1,…,θpsT is a fault pulse value vector of PNs, where θpii=1,…,s represents the pulse value of the i-th PN σi. If a PN has not any pulse, then its pulse value is 0.(4)
θr=θr1,…,θrtT is a fault pulse value vector of RNs, where θrjj=1,…,t represents the pulse value of the j-th RN σs+j. If a RN has not any pulse, then its pulse value is 0.(5)
λy is a dangerous threshold of an MCS.(6)
W=w11σ1…w1kσ1…w1sσ1⋱⋮wikσi⋮⋱ws1σs…wskσs…wssσss×s is a weight matrix, where the matrix elements represent the tightness degree between adjacent PNs. If the PNs σi and σk are connected, then wikσi is a weighted value in [0, 1] representing the tightness degree between σi and σk; otherwise, wikσi=0, 1≤i,k≤s.(7)
Yk−=y11σ1…y1kσ1…y1sσ1⋱⋮yikσi⋮⋱ys1σs…yskσs…yssσss×s is a danger degree matrix of PNs, where yikσi is a number in [0, 1] representing the danger degree of the event corresponding to the PN σi triggers the one associated with the PN σk, 1≤i,k≤s. If σi can emit a spike to σk, then yikσi is obtained via (3); otherwise, yikσi=0.
Note that the vectorsλp,λr,α,δ, and 0, the matrices D1,D2,D3, and D4, and the operators ⊗,⊕, and Δ in Algorithm 3 are the same as the ones in the Algorithms 1 and 2.


## 3.1. Pulse Value Reasoning Algorithm

To explain this algorithm, we introduce its vectors, matrices, and operators as follows (PN denotesproposition neuron and RN denotes rule neuron):(1)
α=α1,…,αsT is a pulse value vector of PNs, where αii=1,…,s represents the pulse value of the i-th PN σi. If a PN has not any pulse, then its pulse value is 0.(2)
δ=δ1,…,δtT is a pulse value vector of RNs, where δjj=1,…,t is the pulse value of the j-th RN σs+j. If an RN has not any pulse, then its pulse value is 0.(3)
λp=λp1,…,λpsT is a firing threshold vector of PNs.(4)
λr=λr1,…,λrtT is a firing threshold vector of RNs.(5)
C=diagc1,…,ct is a diagonal matrix of truth values of RNs, where cjj=1,…,t is the truth value of the j-th RN σs+j.(6)
D1=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to general RNs. If there is a synapse from the PN σi to the general RN σs+j, then dij = 1; otherwise, dij = 0.(7)
D2=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to and RNs. If there is a synapse from the PN σi to the and RN σs+j, then dij = 1; otherwise, dij = 0.(8)
D3=dijs×t is a synaptic matrix, which represents the directed synaptic connections from PNs to or RNs. If there is a synapse from the PN σi to the or RN σs+j, then dij = 1; otherwise, dij = 0.(9)
D4=djit×s is a synaptic matrix, which represents the directed synaptic connections from RNs to PNs. If there is a synapse from the RN σs+j to the PN σi, then dji = 1; otherwise, dji = 0.(10)
0=0,…,0t×1T is null vector.(11)
DT∗α=d¯1,…,d¯tT, where d¯j=d1j×α1+⋯+dsj×αs,1≤j≤t.(12)
DT⋅α=d¯1,…,d¯tT, where d¯j=mind1j×α1+⋯+dsj×αs,1≤j≤t.(13)
DT∘α=d¯1,…,d¯tT, where d¯j=maxd1j×α1+⋯+dsj×αs,1≤j≤t.


## 3.2. Forward Fault Prediction Reasoning Algorithm

To explain this algorithm, we introduce its vectors, matrices, and operators as follows:(1)
Np+ is the number vector of PNs where pulses are located. If a PN contains a pulse, then the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(2)
Nr+ is the number vector of RNs where pulses are located. If an RN contains a pulse, the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(3)
AΔB=ci1x×1, where A=aikx×y,B=bk1y×1, and ci1=maxaikbk1,1≤i≤x,1≤k≤y.(4)
A∇B=ci1x×1, where A=aikx×y,B=bk1y×1. If ∑k=1yaikbk1<2, then ci1=0, otherwise ci1=1,1≤i≤x,1≤k≤y.(5)
A⊗B=cijx×y, where A=aijx×y,B=bi1x×1, and cij=aijbi1,1≤i≤x,1≤j≤y.(6)
A⊕B=cijx×y, where A=aijx×y,B=bijx×y, and cij=maxaij,bij,1≤i≤x,1≤j≤y.(7)
AΘB=cijx×y, where A=aijx×y,B=bijx×y. If aij≥bij, then cij=1, otherwise, cij=0, 1≤i≤x,1≤j≤y.Note that the vectorsα,δ,λp,λr, and 0, the matrices D1,D2,D3,D4, and C, and the operators ∗,⋅ and ∘ in Algorithm 2 are the same as the ones in Algorithm 1.


## 3.3. Backward Abductive Fault Diagnosis Reasoning Algorithm

To improve the accuracy of backward abductive reasoning, this paper integrates a fault screening mechanism of the precise minimum cut set (please see Definition2) into the parallel reasoning ability of SNPSs to propose the BAFDRA for the rMFRSNPS, i.e., Algorithm 3. The precise minimum cut set effectively combines the abductive principle of top events in minimum cut sets [42] with the screening mechanism, where, in two adjacent fault events, a bottom event corresponds to a fault or a fault symptom and a top event corresponds to a fault. Moreover, the screening mechanism is used to improve the abductive reasoning accuracy by eliminating pulses contained in the minimum cut set whose danger degree is lower than the dangerous threshold, where the danger degree is used to access the fault risk of motors [43].Definition 2.
A precise minimum cut set (PMCS) is a tuple(2)Q=Q1,…,Ql,1≤l≤s,where(1)
Q1,…,Ql are l minimum cut sets (MCSs), where(a)
Ageneral rule neuron has one presynaptic proposition neuron (corresponding to a bottom event) and one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered only by the bottom event; thus, the MCS of the top event consists of the presynaptic neuron.(b)
Anor rule neuron has more than one presynaptic proposition neurons (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered by any bottom event; thus, the MCS of the top event consists of any presynaptic neuron. That is, if there are l bottom events that can trigger the top event, then the top event will have l MCSs with each MCS consisting of one presynaptic neuron corresponding to one of the l bottom events.(c)
Anand rule neuron has more than one presynaptic proposition neuron (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can only be triggered by all bottom events at the same time, so that the MCS of the top event consists of all the presynaptic neurons. That is, if there are l bottom events that can trigger the top event, then the top event will have only one MCS and it should consist of all the presynaptic neurons corresponding to the l bottom events.(2)
Qg=yσi,yQg,λy,1≤g≤l,1≤i≤s is the g-th MCS, where(a)
yσi is the danger degree of the i-th PN σi, defined as(3)yσi=wσi×αi,wherewσi is a weighted value in [0, 1] representing the tightness degree between PN σi and its postsynaptic neurons.(b)
yQg is a danger degree of the g-th minimum cut set, i.e., Qg, which is defined as(4)yQg=∏j=1qyσj,1≤q≤s.(c)
λy is a number in (0, 1) representing the danger degree threshold of an MCS. When the danger degree of an MCS is greater than λy, then the MCS is called a PMCS and the PNs with pulse in the PMCS form the fault paths, with the first PN in each path being called the fault source.
Algorithm3 is shown as follows:
To explain the algorithm, we introduce its vectors, matrices, and operators as follows:(1)
Np− is the number vector of PNs where fault pulses are located. If a PN contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(2)
Nr− is the number vector of RNs where fault pulses are located. If a rule neuron contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(3)
θp=θp1,…,θpsT is a fault pulse value vector of PNs, where θpii=1,…,s represents the pulse value of the i-th PN σi. If a PN has not any pulse, then its pulse value is 0.(4)
θr=θr1,…,θrtT is a fault pulse value vector of RNs, where θrjj=1,…,t represents the pulse value of the j-th RN σs+j. If a RN has not any pulse, then its pulse value is 0.(5)
λy is a dangerous threshold of an MCS.(6)
W=w11σ1…w1kσ1…w1sσ1⋱⋮wikσi⋮⋱ws1σs…wskσs…wssσss×s is a weight matrix, where the matrix elements represent the tightness degree between adjacent PNs. If the PNs σi and σk are connected, then wikσi is a weighted value in [0, 1] representing the tightness degree between σi and σk; otherwise, wikσi=0, 1≤i,k≤s.(7)
Yk−=y11σ1…y1kσ1…y1sσ1⋱⋮yikσi⋮⋱ys1σs…yskσs…yssσss×s is a danger degree matrix of PNs, where yikσi is a number in [0, 1] representing the danger degree of the event corresponding to the PN σi triggers the one associated with the PN σk, 1≤i,k≤s. If σi can emit a spike to σk, then yikσi is obtained via (3); otherwise, yikσi=0.
Note that the vectorsλp,λr,α,δ, and 0, the matrices D1,D2,D3, and D4, and the operators ⊗,⊕, and Δ in Algorithm 3 are the same as the ones in the Algorithms 1 and 2.


## 4. Case Studies

In this section, several cases about possible faults on a motor are considered, in order to show the feasibility and validity of our proposed method. Note that the initial pulse values of input neurons in Algorithms1 and 2 are the occurrence probabilities of fault symptoms obtained based on historical data and expertise. Since Algorithm 3 is used to find fault causes and fault sources after a motor fails, its initial pulse values are the event probabilities obtained by Algorithm 1, including the occurrence probabilities of both the fault symptoms and failures.


### 4.1. Pulse Value Reasoning of Neurons

The initial pulse value of input neurons and truth value of rule neurons are obtained via historical data and expert experience [23].Here, we take the “insulation winding burned down” as an example. Then, we can get that the initial pulse value vectors of proposition neurons and rule neurons areα0=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O18T,δ0=O18T, respectively.The truth value diagonal matrix of RNs isC=diag0.8,0.88,0.87,0.8,0.92,0.89,0.89,0.89,0.89,0.92,0.94,0.9,0.92,0.91,0.94,0.93,0.97,0.93.The firing threshold vectors for PNs and RNs areλp=0.5,…,0.5s×1T and λr=0.5,…,0.5t×1T, respectively.The synaptic matricesD1,D2,D3, and D4 are obtained via the rMFRSNPS-based fault analysis model, as shown in Figure 3.(5)D1=V7×6O7×2O7×5O7×1O7×4O1×6E1×2O1×5O1×1O1×4O7×6O7×2V7×5O7×1O7×4O1×6O1×2O1×5E1×1O1×4O10×6O10×2O10×5O10×1V10×4,D2=O1×1O1×1O1×16O2×1E2×1O2×16O23×1O23×1O23×16,D3=V8×6O8×3O8×3O8×3O8×3O3×6V3×3O3×3O3×3O3×3O2×6O2×3O2×3O2×3O2×3O4×6O4×3O4×3V4×3O4×3O3×6O3×3O3×3V3×3O3×3O4×6O4×3O4×3O4×3V4×3O2×6O2×3O2×3O2×3O2×3,D4=O18×8E18×18,where V7×6=O2×2O2×3O2×2O3×2E3×3O3×2O1×2O1×3O1×2T,V7×5=O1×3O1×3O1×1O3×3E3×3O3×1O1×3O1×3O1×1T,V10×4=O2×6O2×1O2×3O1×6E1×1O1×3O1×6O1×1O1×3T,V4×3=E1×2O1×1O1×1O1×2O1×1O1×1O1×2O1×1E1×1T,V3×3=O2×3E1×3T,V8×6=E1×2O1×3O1×2O1×1O4×2O4×3O4×2O4×1O1×2O1×3E1×2O1×1T,O is a null matrix, and E is an identity matrix.The pulse value reasoning process is described as follows:Whenk = 1, δ1=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O10T, α1=O8,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,O10TWhenk = 2, δ2=O8,0.78,0.74,0.74,0.8,0.8,0.73,O4T, α2=O16,0.69,0.68,0.7,0.72,0.74,0.66,O4TWhenk = 3, δ3=O14,0.72,0.74,O2T, α3=O22,0.68,0.69,O2TWhenk = 4, δ4=O16,0.68,0.69T, α4=O24,0.66,0.64TWhenk = 5, δ5=O18TThus, the termination condition is satisfied and the reasoning stops. We obtain the reasoning results,i.e., the pulse value of all neurons, shown as follows:(6)α=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,0.69,0.68,0.7,0.72,0.74,0.66,0.68,0.69,0.66,0.64T,δ=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.78,0.74,0.74,0.8,0.8,0.73,0.72,0.74,0.68,0.69T.


### 4.2. Forward Fault Prediction Reasoning

Let us assume that the following fault symptoms of a motor are monitored online: overload (p1), resistance value of a phase winding decreases (p3), damage of shaft seal ring structure (p5), and excessive roughness value of seal surface shaft (p7). Accordingly, the initial number vector Np0+ of the PNs with fault pulses is obtained: Np0+=1,0,1,0,1,0,1,O19T.The synaptic matricesD1,D2,D3, and D4 are the same as the ones in Section 4.1.The initial pulse value vectors of PNs and RNs areα0=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O18T,δ0=O18T, respectively.The truth value diagonal matrix of RNs isC=diag0.8,0.88,0.87,0.8,0.92,0.89,0.89,0.89,0.89,0.92,0.94,0.9,0.92,0.91,0.94,0.93,0.97,0.93.The fault prediction reasoning process is described as follows:Whenk = 1, δ1=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O10T, Nr1+=1,0,1,0,1,1,O12T. α1=O8,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,O10T, Np1+=O81,0,1,0,1,1,O12T.Whenk = 2, δ2=O8,0.78,0.74,0.74,0.8,0.8,0.73,O4T, Nr2+=O8,1,0,1,1,1,O5T. α2=O16,0.69,0.68,0.7,0.72,0.74,0.66,O4T, Np2+=O161,0,1,0,1,11,O5T.Whenk = 3, δ3=O14,0.72,0.74,O2T, Nr3+=O14,1,1,O2T; α3=O22,0.68,0.69,O2T, Np3+=O22,1,1,O2T.Whenk = 4, δ4=O16,0.68,0.69T, Nr4+=O15,1,1T. α4=O24,0.66,0.64T, Np4+=O241,1T.Whenk = 5, δ5=O18T, Nr5+=O18T.Thus, the termination condition is satisfied and the reasoning stops. We find that the neurons with fault pulses are shown in Figure6. Therefore, the potential fault paths are obtained; that is, L1=σ1,σ17,σ30,σ41,σ46, L2=σ3,σ19,σ30,σ41,σ46, L3=σ5,σ21,σ32,σ41,σ46, L4=σ7,σ22,σ33,σ41,σ46, and L5=σ7,σ22,σ34,σ42,σ47, where σ17,σ19,σ21,σ22,σ30,σ32,σ33,σ34,σ41,σ42,σ46, and σ47 are potential faults. The occurrence probability of each fault path is PL1=0.159, PL2=0.217, PL3=0.186, PL4=0.16, and PL5=0.162, respectively. Thus, the checking order of the fault paths is L2,L3,L5,L4,L1. Note that the probability of a path is obtained by multiplying the pulse value of the neurons in each path.Figure 6
Forward prediction of potential fault paths.


### 4.3. Backward Abductive Fault Diagnosis Reasoning

This section assumes that the motor has failures. Let us take insulation winding burned down (p46) as an example. Accordingly, the number vector of PNs with fault pulse is Np0−=O24,1,O1T.The pulse value vectors of PNs and RNs areα and δ, respectively, given as follows:(7)α=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,0.69,0.68,0.7,0.72,0.74,0.66,0.68,0.69,0.66,0.64T,δ=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.78,0.74,0.74,0.8,0.8,0.73,0.72,0.74,0.68,0.69T.The synaptic matricesD1,D2,D3, and D4 are the same as those in Section 4.1.The tightness degree between PNs is shown in Figure7, from which the weight matrix W can be obtained.Figure 7
The tightness degree among PNs.The abductive fault diagnosis reasoning process is described as follows:Whenk = 1,(8)Nr1−=O16,1,O1T,θr1=O16,0.68,O1T,Np1−=O22,1,O3T,θp1=O22,0.68,O3T,Y1−=O24×22O24×1O24×3O1×220.68O1×3O1×22O1×1O1×3T,Q1=σ41,yQ1=yσ41=0.68.To start with the process, the pulse of PNs in the MCS whose danger degree is less thanλy is deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np1−=O22,1,O3T and θp1=O22,0.68,O3T, respectively.Whenk = 2,(9)Nr2−=O14,1,O3T,θr2=O14,0.72,O3T,Np2−=O16,1,1,1,1,O6T,θp2=O16,0.69,0.68,0.7,0.72,O6T,Y2−=O22×16O22×1O22×1O22×1O22×1O22×6O1×160.610.570.60.58O1×6O3×16O3×1O3×1O3×1O3×1O3×6T,Q1=σ30,Q2=σ31,Q3=σ32,Q4=σ33,yQ1=yσ30=0.61,yQ2=yσ31=0.57,yQ3=yσ32=0.6,yQ4=yσ33=0.58.Similarly, the pulse of PNs in the MCS whose danger degree is less thanλy must also be deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np2−=O16,1,0,1,O7T, θp2=O16,0.69,0,0.7,O7T.Whenk = 3,(10)Nr3−=O8,1,0,1,O7T,θr3=O8,0.78,0,0.74,O7T,Np3−=O8,1,1,1,0,1,O13T,θp3=O8,0.64,0.53,0.78,0,0.74,O13T,Y3−=O16×8O16×1O16×1O16×1O16×1O16×1O16×13O1×80.610.50.72O1×1O1×1O1×13O1×8O1×1O1×1O1×1O1×1O1×1O1×13O1×8O1×1O1×1O1×1O1×10.74O1×13O7×8O7×1O7×1O7×1O7×1O7×1O7×13T,Q1=σ17,Q2=σ18,Q3=σ19,Q4=σ21,yQ1=yσ17=0.61,yQ2=yσ18=0.5,yQ3=yσ19=0.72,yQ4=yσ21=0.74.Repeatedly, it must be made sure to delete the pulse of PNs in the MCS whose danger degree is less thanλy. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np3−=O8,1,0,1,0,1,O13T and θp3=O8,0.64,0,0.78,0,0.74,O13T, respectively.Whenk = 4,(11)Nr4−=1,0,1,0,1,O13T,θr4=0.8,0,0.9,0,0.8,O13T,Np4−=1,1,1,0,1,O21T,θp4=0.8,0.6,0.9,0,0.8,O21T,Y4−=O8×1O8×1O8×1O8×1O8×1O8×210.710.55O1×1O1×1O1×1O1×21O1×10.510.75O1×1O1×1O1×21O1×1O1×10.9O1×1O1×1O1×21O1×1O1×1O1×1O1×1O1×1O1×21O1×1O1×1O1×1O1×10.8O1×21O13×1O13×1O13×1O13×1O13×1O13×21T,Q1=σ1,Q2=σ2,Q3=σ3,Q4=σ5,yQ1=yσ1=0.71,yQ2=yσ2=0.55,yQ3=yσ3=0.9,yQ4=yσ5=0.8.Once more, the pulse is deleted in PNs in the MCS whose danger degree is less thanλy. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np4−=1,0,1,0,1,O21T and θp4=0.8,0,0.9,0,0.8,O21T, respectively.Whenk = 5, Nr5−=O18T.Thus, the termination condition is satisfied and the reasoning stops. We find that the rMFRSNPS-based abductive reasoning model is shown in Figure8, where  represents the deleted pulse. Then, the fault paths can be found in Figure 8, i.e., L1=σ1,σ17,σ30,σ41,σ46, L2=σ3,σ19,σ30,σ41,σ46, and L3=σ5,σ21,σ32,σ41,σ46, where σ1,σ3, and σ5 are the fault source of “insulation winding burned down.” Besides, the occurrence probability of each fault path is PL1=0.159,PL2=0.217, and PL3=0.186.Then, maintenance personnels can check the motor in turn according to the fault sources and paths got by Algorithm3. The check order of fault sources is σ3,σ1,σ5, and the fault paths are L2,L3,L1.Figure 8
Backward fault abductive paths.


### 4.4. Comparisons

In this section, the usefulness of the proposed method is justified by comparison with different approaches: the method of selection of amplitudes of frequencies (MSAF-12) [2], improved artificial ant clustering (IAAC) [14], fuzzy fault Petri net (FFPN) [23], and fuzzy Petri net (FPN) [44] for the abductive fault diagnosis.Historical statistics and expertise [37] show that most faults of three-phase induction motors are related to bearings, windings, and stators. Consequently, five relevant typical cases have been considered, which are “insulation winding burned down,” “excessive wear of bearing,” “motor overheating,” “motor overheating and abnormal rotation or the rotor is stuck,” and “phase-absent operation, abnormal rotation, or the rotor is stuck and excessive wear of bearing,” respectively.The experimental results are shown in Table2, where cases 1–3 are single faults while cases 4-5 are multiple ones. For cases 1–3, the FFPN find more fault causes and fault sources, while the FPN and IAAC diagnose more fault causes and cannot find any fault source. Besides, although the MSAF-12 can obtain right fault causes without redundant ones for case 2, it gets wrong results for cases 1 and 3. For cases 4-5, the FFPN still cannot find the accurate fault causes and fault sources, while the MSAF-12, FPN, and IAAC are unable to find any fault source. In contrast, the rMFRSNPS performs better, finding all the sources and avoiding redundancies. Accordingly, the inspection and repair scope for the motor obtained by our method is smaller than the ones got by the MSAF-12, IAAC, FFPN, and FPN.Table 2
Results comparison of the rMFRSNPS and four other methods.
CasesPreset faultsMethodsResultsFault eventFault locationsFault sourcesFault causesFault sourcesFault causesRedundant fault sourcesRedundant fault causes1Insulation winding burned downσ46σ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21, σ30, σ32, σ41MSAF-12—σ1, σ2, σ3, σ4, σ5, σ17, σ18, σ19, σ20, σ21, σ30, σ31, σ32, σ41—σ2, σ4, σ18, σ20, σ31,IAAC—σ1, σ2, σ3, σ4, σ5, σ17, σ18, σ19, σ20, σ21, σ30, σ31, σ32, σ41—σ2, σ4, σ18, σ20, σ31,FFPNσ1, σ2, σ3, σ4, σ5, σ6, σ7σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ17, σ18, σ19, σ20, σ21, σ22, σ30, σ31, σ32, σ33, σ41σ2, σ4, σ6, σ7σ2, σ4, σ6, σ7, σ18, σ20, σ22, σ31, σ33FPN—σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ17, σ18, σ19, σ20, σ21, σ22, σ30, σ31, σ32, σ33, σ41—σ2, σ4, σ6, σ7, σ18, σ20, σ22, σ31, σ33rMFRSNPSσ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21, σ30, σ32, σ41——2Excessive wear of bearingσ42σ6, σ8σ6, σ8, σ22, σ24, σ34, σ35MSAF-12—σ6, σ8, σ22, σ24, σ34, σ35——IAAC—σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35—σ7, σ23FFPNσ6, σ7, σ8σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35σ7σ7, σ23FPN—σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35—σ7, σ23rMFRSNPSσ6, σ8σ6, σ8, σ22, σ24, σ34, σ35——3Motor overheatingσ30σ1, σ3σ1, σ3, σ17, σ19MSAF-12—σ1, σ2, σ3, σ17, σ18, σ19σ2, σ18IAAC—σ1, σ2, σ3, σ17, σ18, σ19σ2, σ18FFPNσ1, σ2, σ3σ1, σ2, σ3, σ17, σ18, σ19σ2σ2, σ18FPN—σ1, σ2, σ3, σ17, σ18, σ19—σ2, σ18rMFRSNPSσ1, σ3σ1, σ3, σ17, σ19——4Motor overheating and abnormal rotation or the rotor is stuckσ30, σ32σ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21MSAF-12—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2, σ18,IAAC—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2, σ18,FFPNσ1, σ2, σ3, σ5σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2σ2, σ18,FPN—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21—σ2, σ18,rMFRSNPSσ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21——5Phase-absent operation, abnormal rotation, or the rotor is stuck and excessive wear of bearingσ31, σ32, σ42σ4, σ5, σ6, σ8σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35MSAF-12—σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35——IAAC—σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35—σ7, σ23FFPNσ4, σ5, σ6, σ7, σ8σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35σ7σ7, σ23FPN—σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35—σ7, σ23rMFRSNPSσ4, σ5, σ6, σ8σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35——Note: “—” means that there is no such kind of information.


## 4.1. Pulse Value Reasoning of Neurons

The initial pulse value of input neurons and truth value of rule neurons are obtained via historical data and expert experience [23].Here, we take the “insulation winding burned down” as an example. Then, we can get that the initial pulse value vectors of proposition neurons and rule neurons areα0=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O18T,δ0=O18T, respectively.The truth value diagonal matrix of RNs isC=diag0.8,0.88,0.87,0.8,0.92,0.89,0.89,0.89,0.89,0.92,0.94,0.9,0.92,0.91,0.94,0.93,0.97,0.93.The firing threshold vectors for PNs and RNs areλp=0.5,…,0.5s×1T and λr=0.5,…,0.5t×1T, respectively.The synaptic matricesD1,D2,D3, and D4 are obtained via the rMFRSNPS-based fault analysis model, as shown in Figure 3.(5)D1=V7×6O7×2O7×5O7×1O7×4O1×6E1×2O1×5O1×1O1×4O7×6O7×2V7×5O7×1O7×4O1×6O1×2O1×5E1×1O1×4O10×6O10×2O10×5O10×1V10×4,D2=O1×1O1×1O1×16O2×1E2×1O2×16O23×1O23×1O23×16,D3=V8×6O8×3O8×3O8×3O8×3O3×6V3×3O3×3O3×3O3×3O2×6O2×3O2×3O2×3O2×3O4×6O4×3O4×3V4×3O4×3O3×6O3×3O3×3V3×3O3×3O4×6O4×3O4×3O4×3V4×3O2×6O2×3O2×3O2×3O2×3,D4=O18×8E18×18,where V7×6=O2×2O2×3O2×2O3×2E3×3O3×2O1×2O1×3O1×2T,V7×5=O1×3O1×3O1×1O3×3E3×3O3×1O1×3O1×3O1×1T,V10×4=O2×6O2×1O2×3O1×6E1×1O1×3O1×6O1×1O1×3T,V4×3=E1×2O1×1O1×1O1×2O1×1O1×1O1×2O1×1E1×1T,V3×3=O2×3E1×3T,V8×6=E1×2O1×3O1×2O1×1O4×2O4×3O4×2O4×1O1×2O1×3E1×2O1×1T,O is a null matrix, and E is an identity matrix.The pulse value reasoning process is described as follows:Whenk = 1, δ1=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O10T, α1=O8,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,O10TWhenk = 2, δ2=O8,0.78,0.74,0.74,0.8,0.8,0.73,O4T, α2=O16,0.69,0.68,0.7,0.72,0.74,0.66,O4TWhenk = 3, δ3=O14,0.72,0.74,O2T, α3=O22,0.68,0.69,O2TWhenk = 4, δ4=O16,0.68,0.69T, α4=O24,0.66,0.64TWhenk = 5, δ5=O18TThus, the termination condition is satisfied and the reasoning stops. We obtain the reasoning results,i.e., the pulse value of all neurons, shown as follows:(6)α=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,0.69,0.68,0.7,0.72,0.74,0.66,0.68,0.69,0.66,0.64T,δ=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.78,0.74,0.74,0.8,0.8,0.73,0.72,0.74,0.68,0.69T.


## 4.2. Forward Fault Prediction Reasoning

Let us assume that the following fault symptoms of a motor are monitored online: overload (p1), resistance value of a phase winding decreases (p3), damage of shaft seal ring structure (p5), and excessive roughness value of seal surface shaft (p7). Accordingly, the initial number vector Np0+ of the PNs with fault pulses is obtained: Np0+=1,0,1,0,1,0,1,O19T.The synaptic matricesD1,D2,D3, and D4 are the same as the ones in Section 4.1.The initial pulse value vectors of PNs and RNs areα0=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O18T,δ0=O18T, respectively.The truth value diagonal matrix of RNs isC=diag0.8,0.88,0.87,0.8,0.92,0.89,0.89,0.89,0.89,0.92,0.94,0.9,0.92,0.91,0.94,0.93,0.97,0.93.The fault prediction reasoning process is described as follows:Whenk = 1, δ1=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,O10T, Nr1+=1,0,1,0,1,1,O12T. α1=O8,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,O10T, Np1+=O81,0,1,0,1,1,O12T.Whenk = 2, δ2=O8,0.78,0.74,0.74,0.8,0.8,0.73,O4T, Nr2+=O8,1,0,1,1,1,O5T. α2=O16,0.69,0.68,0.7,0.72,0.74,0.66,O4T, Np2+=O161,0,1,0,1,11,O5T.Whenk = 3, δ3=O14,0.72,0.74,O2T, Nr3+=O14,1,1,O2T; α3=O22,0.68,0.69,O2T, Np3+=O22,1,1,O2T.Whenk = 4, δ4=O16,0.68,0.69T, Nr4+=O15,1,1T. α4=O24,0.66,0.64T, Np4+=O241,1T.Whenk = 5, δ5=O18T, Nr5+=O18T.Thus, the termination condition is satisfied and the reasoning stops. We find that the neurons with fault pulses are shown in Figure6. Therefore, the potential fault paths are obtained; that is, L1=σ1,σ17,σ30,σ41,σ46, L2=σ3,σ19,σ30,σ41,σ46, L3=σ5,σ21,σ32,σ41,σ46, L4=σ7,σ22,σ33,σ41,σ46, and L5=σ7,σ22,σ34,σ42,σ47, where σ17,σ19,σ21,σ22,σ30,σ32,σ33,σ34,σ41,σ42,σ46, and σ47 are potential faults. The occurrence probability of each fault path is PL1=0.159, PL2=0.217, PL3=0.186, PL4=0.16, and PL5=0.162, respectively. Thus, the checking order of the fault paths is L2,L3,L5,L4,L1. Note that the probability of a path is obtained by multiplying the pulse value of the neurons in each path.Figure 6
Forward prediction of potential fault paths.


## 4.3. Backward Abductive Fault Diagnosis Reasoning

This section assumes that the motor has failures. Let us take insulation winding burned down (p46) as an example. Accordingly, the number vector of PNs with fault pulse is Np0−=O24,1,O1T.The pulse value vectors of PNs and RNs areα and δ, respectively, given as follows:(7)α=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.64,0.53,0.78,0.74,0.74,0.8,0.73,0.73,0.69,0.68,0.7,0.72,0.74,0.66,0.68,0.69,0.66,0.64T,δ=0.8,0.6,0.9,0.92,0.8,0.9,0.62,0.82,0.78,0.74,0.74,0.8,0.8,0.73,0.72,0.74,0.68,0.69T.The synaptic matricesD1,D2,D3, and D4 are the same as those in Section 4.1.The tightness degree between PNs is shown in Figure7, from which the weight matrix W can be obtained.Figure 7
The tightness degree among PNs.The abductive fault diagnosis reasoning process is described as follows:Whenk = 1,(8)Nr1−=O16,1,O1T,θr1=O16,0.68,O1T,Np1−=O22,1,O3T,θp1=O22,0.68,O3T,Y1−=O24×22O24×1O24×3O1×220.68O1×3O1×22O1×1O1×3T,Q1=σ41,yQ1=yσ41=0.68.To start with the process, the pulse of PNs in the MCS whose danger degree is less thanλy is deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np1−=O22,1,O3T and θp1=O22,0.68,O3T, respectively.Whenk = 2,(9)Nr2−=O14,1,O3T,θr2=O14,0.72,O3T,Np2−=O16,1,1,1,1,O6T,θp2=O16,0.69,0.68,0.7,0.72,O6T,Y2−=O22×16O22×1O22×1O22×1O22×1O22×6O1×160.610.570.60.58O1×6O3×16O3×1O3×1O3×1O3×1O3×6T,Q1=σ30,Q2=σ31,Q3=σ32,Q4=σ33,yQ1=yσ30=0.61,yQ2=yσ31=0.57,yQ3=yσ32=0.6,yQ4=yσ33=0.58.Similarly, the pulse of PNs in the MCS whose danger degree is less thanλy must also be deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np2−=O16,1,0,1,O7T, θp2=O16,0.69,0,0.7,O7T.Whenk = 3,(10)Nr3−=O8,1,0,1,O7T,θr3=O8,0.78,0,0.74,O7T,Np3−=O8,1,1,1,0,1,O13T,θp3=O8,0.64,0.53,0.78,0,0.74,O13T,Y3−=O16×8O16×1O16×1O16×1O16×1O16×1O16×13O1×80.610.50.72O1×1O1×1O1×13O1×8O1×1O1×1O1×1O1×1O1×1O1×13O1×8O1×1O1×1O1×1O1×10.74O1×13O7×8O7×1O7×1O7×1O7×1O7×1O7×13T,Q1=σ17,Q2=σ18,Q3=σ19,Q4=σ21,yQ1=yσ17=0.61,yQ2=yσ18=0.5,yQ3=yσ19=0.72,yQ4=yσ21=0.74.Repeatedly, it must be made sure to delete the pulse of PNs in the MCS whose danger degree is less thanλy. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np3−=O8,1,0,1,0,1,O13T and θp3=O8,0.64,0,0.78,0,0.74,O13T, respectively.Whenk = 4,(11)Nr4−=1,0,1,0,1,O13T,θr4=0.8,0,0.9,0,0.8,O13T,Np4−=1,1,1,0,1,O21T,θp4=0.8,0.6,0.9,0,0.8,O21T,Y4−=O8×1O8×1O8×1O8×1O8×1O8×210.710.55O1×1O1×1O1×1O1×21O1×10.510.75O1×1O1×1O1×21O1×1O1×10.9O1×1O1×1O1×21O1×1O1×1O1×1O1×1O1×1O1×21O1×1O1×1O1×1O1×10.8O1×21O13×1O13×1O13×1O13×1O13×1O13×21T,Q1=σ1,Q2=σ2,Q3=σ3,Q4=σ5,yQ1=yσ1=0.71,yQ2=yσ2=0.55,yQ3=yσ3=0.9,yQ4=yσ5=0.8.Once more, the pulse is deleted in PNs in the MCS whose danger degree is less thanλy. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Np4−=1,0,1,0,1,O21T and θp4=0.8,0,0.9,0,0.8,O21T, respectively.Whenk = 5, Nr5−=O18T.Thus, the termination condition is satisfied and the reasoning stops. We find that the rMFRSNPS-based abductive reasoning model is shown in Figure8, where  represents the deleted pulse. Then, the fault paths can be found in Figure 8, i.e., L1=σ1,σ17,σ30,σ41,σ46, L2=σ3,σ19,σ30,σ41,σ46, and L3=σ5,σ21,σ32,σ41,σ46, where σ1,σ3, and σ5 are the fault source of “insulation winding burned down.” Besides, the occurrence probability of each fault path is PL1=0.159,PL2=0.217, and PL3=0.186.Then, maintenance personnels can check the motor in turn according to the fault sources and paths got by Algorithm3. The check order of fault sources is σ3,σ1,σ5, and the fault paths are L2,L3,L1.Figure 8
Backward fault abductive paths.


## 4.4. Comparisons

In this section, the usefulness of the proposed method is justified by comparison with different approaches: the method of selection of amplitudes of frequencies (MSAF-12) [2], improved artificial ant clustering (IAAC) [14], fuzzy fault Petri net (FFPN) [23], and fuzzy Petri net (FPN) [44] for the abductive fault diagnosis.Historical statistics and expertise [37] show that most faults of three-phase induction motors are related to bearings, windings, and stators. Consequently, five relevant typical cases have been considered, which are “insulation winding burned down,” “excessive wear of bearing,” “motor overheating,” “motor overheating and abnormal rotation or the rotor is stuck,” and “phase-absent operation, abnormal rotation, or the rotor is stuck and excessive wear of bearing,” respectively.The experimental results are shown in Table2, where cases 1–3 are single faults while cases 4-5 are multiple ones. For cases 1–3, the FFPN find more fault causes and fault sources, while the FPN and IAAC diagnose more fault causes and cannot find any fault source. Besides, although the MSAF-12 can obtain right fault causes without redundant ones for case 2, it gets wrong results for cases 1 and 3. For cases 4-5, the FFPN still cannot find the accurate fault causes and fault sources, while the MSAF-12, FPN, and IAAC are unable to find any fault source. In contrast, the rMFRSNPS performs better, finding all the sources and avoiding redundancies. Accordingly, the inspection and repair scope for the motor obtained by our method is smaller than the ones got by the MSAF-12, IAAC, FFPN, and FPN.Table 2
Results comparison of the rMFRSNPS and four other methods.
CasesPreset faultsMethodsResultsFault eventFault locationsFault sourcesFault causesFault sourcesFault causesRedundant fault sourcesRedundant fault causes1Insulation winding burned downσ46σ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21, σ30, σ32, σ41MSAF-12—σ1, σ2, σ3, σ4, σ5, σ17, σ18, σ19, σ20, σ21, σ30, σ31, σ32, σ41—σ2, σ4, σ18, σ20, σ31,IAAC—σ1, σ2, σ3, σ4, σ5, σ17, σ18, σ19, σ20, σ21, σ30, σ31, σ32, σ41—σ2, σ4, σ18, σ20, σ31,FFPNσ1, σ2, σ3, σ4, σ5, σ6, σ7σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ17, σ18, σ19, σ20, σ21, σ22, σ30, σ31, σ32, σ33, σ41σ2, σ4, σ6, σ7σ2, σ4, σ6, σ7, σ18, σ20, σ22, σ31, σ33FPN—σ1, σ2, σ3, σ4, σ5, σ6, σ7, σ17, σ18, σ19, σ20, σ21, σ22, σ30, σ31, σ32, σ33, σ41—σ2, σ4, σ6, σ7, σ18, σ20, σ22, σ31, σ33rMFRSNPSσ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21, σ30, σ32, σ41——2Excessive wear of bearingσ42σ6, σ8σ6, σ8, σ22, σ24, σ34, σ35MSAF-12—σ6, σ8, σ22, σ24, σ34, σ35——IAAC—σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35—σ7, σ23FFPNσ6, σ7, σ8σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35σ7σ7, σ23FPN—σ6, σ7, σ8, σ22, σ23, σ24, σ34, σ35—σ7, σ23rMFRSNPSσ6, σ8σ6, σ8, σ22, σ24, σ34, σ35——3Motor overheatingσ30σ1, σ3σ1, σ3, σ17, σ19MSAF-12—σ1, σ2, σ3, σ17, σ18, σ19σ2, σ18IAAC—σ1, σ2, σ3, σ17, σ18, σ19σ2, σ18FFPNσ1, σ2, σ3σ1, σ2, σ3, σ17, σ18, σ19σ2σ2, σ18FPN—σ1, σ2, σ3, σ17, σ18, σ19—σ2, σ18rMFRSNPSσ1, σ3σ1, σ3, σ17, σ19——4Motor overheating and abnormal rotation or the rotor is stuckσ30, σ32σ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21MSAF-12—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2, σ18,IAAC—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2, σ18,FFPNσ1, σ2, σ3, σ5σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21σ2σ2, σ18,FPN—σ1, σ2, σ3, σ5, σ17, σ18, σ19, σ21—σ2, σ18,rMFRSNPSσ1, σ3, σ5σ1, σ3, σ5, σ17, σ19, σ21——5Phase-absent operation, abnormal rotation, or the rotor is stuck and excessive wear of bearingσ31, σ32, σ42σ4, σ5, σ6, σ8σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35MSAF-12—σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35——IAAC—σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35—σ7, σ23FFPNσ4, σ5, σ6, σ7, σ8σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35σ7σ7, σ23FPN—σ4, σ5, σ6, σ7, σ8, σ20, σ21, σ22, σ23, σ24, σ34, σ35—σ7, σ23rMFRSNPSσ4, σ5, σ6, σ8σ4, σ5, σ6, σ8, σ20, σ21, σ22, σ24, σ34, σ35——Note: “—” means that there is no such kind of information.


## 5. Conclusions

This paper proposes a fault analysis method for three-phase induction motors based on rMFRSNPSs. Firstly, fault fuzzy production rules are proposed, and then, an rMFRSNPS-based fault diagnosis model is established according to them. Then, the PVRA (Algorithm1), the FFPRA (Algorithm 2), and the BAFDRA (Algorithm 3) are designed to realize the fault analysis of motors. Specifically, the pulse value of spikes in neurons predict propagation paths with occurrence probabilities, and failure causes with probabilities are obtained by the abovementioned three algorithms in turn, respectively.Finally, the fault diagnosis method based on rMFRSNPSs is proposed, where the FPRA can effectively predict potential failures of motors to reduce the fault rate, while the BAFDRA can carry out the abductive fault diagnosis of any failure in the proposed model to the detection range of fault sources and failures. In this paper, we extend the spectrum of applications of SNPSs to the fault analysis of motors, which not only expands the application fields of membrane computing but also extends the SNPS-based fault analysis from postante applications to a new ex-ante analysis and prediction framework. Moreover, the proposed method can meet the needs of a motor for its overall fault analysis. Case studies with a detailed reasoning process assess the feasibility and effectiveness of the proposed method. This paper focuses on proposing the fault analysis method and designing related algorithms from a mathematical point of view. Besides, some of our planned lines of future work include the systematic research about the software simulation of these methods, along with the in-depth exploration of practical applications where the proposed method can provide a significant value.



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*Source: 2087027-2021-01-13.xml*