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

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.


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][2][3][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. erefore, fault prediction and abductive fault diagnosis are of great significance for improving the working reliability and stability of motors [5].
e 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. en, 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. is 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. e 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. e 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. e 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. us, they cannot effectively diagnose multiple faults, not achieving the requirement of performing an overall fault analysis of the whole machine. erefore, 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][17][18][19][20], Petri nets (PNs) [21][22][23], tissue P systems (TPSs) [24][25][26], and spiking neural P systems (SNPSs) [27][28][29][30][31][32][33][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][31][32].
e 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. us, 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. erefore, 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.
erefore, 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 threephase 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. e 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. e 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.
rMFRSNPS is employed to model the rules to propose a universal rMFRSNPS-based fault analysis model.

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 degree m ≥ 1 is a tuple where , and E � a n ∧α ≥ λ i is the firing condition. e firing rule r i of σ i can be applied if and only if σ i receives, at least, n spikes and the potential value of spikes satisfies that α ≥ λ i . By applying rule r i , σ i 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 neuron σ s+j is of the form (δ j , c j , λ j , r j ), 1 ≤ j ≤ t, where (a) δ j is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in σ s+j . (b) c j is a real number in [0, 1] representing the truth value of σ s+j . (c) λ j is a real number in (0, 1) representing the firing threshold of σ s+j . (d) r j represents a firing (spiking) rule of σ s+j with the form E/a δ ⟶ a β , where δ and β are real numbers in [0, 1], and E � a n ∧δ ≥ λ j is the firing condition. e firing rule r j of σ s+j can be applied if and only if σ s+j receives, at least, n spikes and the potential value of spikes satisfies that δ ≥ λ j . By applying rule r j , σ s+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.  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 is where p 1 is an antecedent proposition and p 2 is a consequent proposition (b) Compound AND rule, whose format is where p 1 , . . . , p k−1 are antecedent propositions, p k is a consequent proposition, and k ≥ 3 (c) Compound OR rule, whose format is where p 1 , . . . , p k−1 are antecedent propositions, p k is a consequent proposition, and k ≥ 3 In fact, there exists another type of rule whose format is where p 1 is an antecedent proposition and p 2 , . . . , p k 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 neuron σ i � (α i , λ i , r i ) is an input neuron, then its initial potential value α i represents the information that σ i has received from the environment.
A general rule neuron in an rMFRSNPS consists of only one presynaptic proposition neuron and one or more postsynaptic proposition neurons. erefore, 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. us, 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. ese rule neurons are represented by a rectangle, and they are graphically illustrated in Figure 1.

Fault Fuzzy Production
Rules for Motors. In this paper, the possible failures in a motor include electrical faults and mechanical ones. e 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 Complexity current asymmetry, and the insulation winding burned down. e 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][38][39][40][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

e rMFRSNPS-Based Model for a Motor.
is section models the fault fuzzy production rules proposed in Section 2.2 and builds a universal rMFRSNPS-based fault analysis model for three-phase induction motors, as shown in Figure 3. e designed rMFRSNPS is of degree m � 84 and specifically contains s � 50 proposition neurons and t � 34 rule neurons.

Fault Analysis Method Based on rMFRSNPSs
is section proposes a fault analysis method based on rMFRSNPSs for three-phase induction motors, whose flowchart is shown in Figure 4, where 0 � (0, . . . , 0) T t×1 . e 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. e 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.
Specifically, in this proposed method, the PVRA (Algorithm 1) 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). us, 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).
Next, we describe Algorithms 1-3 in detail as follows.

Pulse Value Reasoning Algorithm.
To explain this algorithm, we introduce its vectors, matrices, and operators as follows (PN denotes proposition neuron and RN denotes rule neuron): . . , s) represents the pulse value of the i-th PN σ i . If a PN has not any pulse, then its pulse value is 0.
(2) δ � (δ 1 , . . . , δ t ) T is a pulse value vector of RNs, where δ j (j � 1, . . . , t) is the pulse value of the j-th RN σ s+j . If an RN has not any pulse, then its pulse value is 0.
s×t is a synaptic matrix, which represents the directed synaptic connections from PNs to general RNs. If there is a synapse from the PN σ i to the general RN σ s+j , then d ij � 1; otherwise, d ij � 0.
(7) D 2 � (d ij ) s×t is a synaptic matrix, which represents the directed synaptic connections from PNs to and RNs. If there is a synapse from the PN σ i to the and RN σ s+j , then d ij � 1; otherwise, d ij � 0.
s×t is a synaptic matrix, which represents the directed synaptic connections from PNs to or RNs. If there is a synapse from the PN σ i to the or RN σ s+j , then d ij � 1; otherwise, d ij � 0.
t×s is a synaptic matrix, which represents the directed synaptic connections from RNs to PNs. If there is a synapse from the RN σ s+j to the PN σ i , then d ji � 1; otherwise, d ji � 0.

Forward Fault Prediction Reasoning Algorithm.
To explain this algorithm, we introduce its vectors, matrices, and operators as follows: (1) N + p 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) N + r 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.
Note that the vectors α, δ, λ p , λ r , and 0, the matrices D 1 , D 2 , D 3 , D 4 , and C, and the operators * , · and ∘ in Algorithm 2 are the same as the ones in Algorithm 1.

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 Definition 2) into the parallel reasoning ability of SNPSs to propose the BAFDRA for the rMFRSNPS, i.e., Algorithm 3. e 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]. ( where w(σ i ) is a weighted value in [0, 1] representing the tightness degree between PN σ i and its postsynaptic neurons. (b) y(Q g ) is a danger degree of the g-th minimum cut set, i.e., Q g , which is defined as (c) λ y is a number in (0, 1) representing the danger degree threshold of an MCS. When the danger  Algorithm 3 is shown as follows: To explain the algorithm, we introduce its vectors, matrices, and operators as follows: (1) N − p 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) N − r 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) θ p � (θ p 1 , . . . , θ p s ) T is a fault pulse value vector of PNs, where θ p i (i � 1, . . . , s) represents the pulse value of the i-th PN σ i . If a PN has not any pulse, then its pulse value is 0. (4) θ r � (θ r 1 , . . . , θ r t ) T is a fault pulse value vector of RNs, where θ r j (j � 1, . . . , t) represents the pulse value of the j-th RN σ s+j . If a RN has not any pulse, then its pulse value is 0. (5) λ y is a dangerous threshold of an MCS.
weight matrix, where the matrix elements represent the tightness degree between adjacent PNs. If the PNs σ i and σ k are connected, then w ik (σ i ) is a weighted value in [0, 1] representing the tightness degree between σ i and σ k ; otherwise, w ik (σ i ) � 0, 1 ≤ i, k ≤ s. Note that the vectors λ p , λ r , α, δ, and 0, the matrices D 1 , D 2 , D 3 , and D 4 , and the operators ⊗, ⊕, and Δ in Algorithm 3 are the same as the ones in the Algorithms 1 and 2. Rotor winding short circuit p 3 e resistance value of a phase winding decreases p 4 Fuse melt fault p 5 Damage of shaft seal ring structure p 6 Oil sealing material overheating p 7 Excessive roughness value of the seal surface shaft p 8 Excessive temperature p 9 Mechanical fault of the rotor winding p 10 e motor centerline is inconsistent with the pump one p 11 Fault of the bearing locking device p 12 Rotor core deformation p 13 Fracture or shedding of magnetic slot wedges p 14 Dewelding at the joint of the winding and lead wire p 15 Connection box joint loosened p 16 Poor contact of the power control loop switch p 17 Decrease in rotational speed p 18 Excessive current in a phase p 19 Excessive excitation current p 20 A phase voltage loss p 21 Foreign matter enters the rotary shaft clearance p 22 e motor oil intake p 23 Oxidation and decomposition of bearing lubricating oil p 24 Bearing expansion by heat p 25 Bearing generates additional load p 26 Rotor axial moves p 27 e iron core of the stator and rotor has an air gap p 28 Rotor winding open circuit p 29 Contact resistance value increases p 30 Motor overheating p 31 Phase-absent operation p 32 Abnormal rotation or the rotor is stuck p 33 Insulation aging p 34 Reduction of lubricant oil p 35 Friction occurs between the crankshaft ring and shaft hole p 36 Excessive vibration of the motor in operation p 37 Excessive bearing noise p 38 Motor sweeping p 39 ree-phase current of the stator increases p 40 Increased pressure drop p 41 ree-phase current asymmetry p 42 Excessive wear of the bearing p 43 Irregular impact load p 44 Abnormal noise

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 Algorithms 1 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.  e 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. en, we can get that the initial pulse value  (1) Let k � 1 (2) while (δ k ≠ 0) (3) if each proposition neuron satisfies its firing condition E � a n ∧α i ≥ λ p i , 1 ≤ i ≤ s then (4) proposition neurons fire and compute δ k via if each rule neuron satisfies its firing condition E � a n ∧δ j ≥ λ r j , 1 ≤ j ≤ t then (6) rule neurons fire and compute α k via   Figure 3.
null matrix, and E is an identity matrix. e pulse value reasoning process is described as follows: us, 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: if each proposition neuron satisfies its firing condition E � a n ∧α i ≥ λ p i , 1 ≤ i ≤ s then (4) proposition neurons fire and compute δ k and N + r k via (6) if each rule neuron satisfies its firing condition E � a n ∧δ j ≥ λ r j , 1 ≤ j ≤ t then (7) rule neurons fire and compute α k and N + p k via

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 N + p 0 of the PNs with fault pulses is obtained: us, the termination condition is satisfied and the reasoning stops. We find that the neurons with fault pulses are shown in Figure 6. erefore, the potential fault paths are obtained; that is, L 1 � (σ 1 , σ 17 , σ 30 , σ 41 , σ 46 ), us, the checking order of the fault paths is L 2 , L 3 , L 5 , L 4 , L 1 . Note that if each rule neurons satisfies its firing condition E � a n ∧θ j ≥ λ r j , 1 ≤ j ≤ t then (4) rule neurons fire and compute N − r k and θ r k via (6) if each proposition neuron satisfies its firing condition E � a n ∧θ i ≥ λ p i , 1 ≤ i ≤ s then (7) proposition neurons fire and compute N − p k and θ p k via Compute danger degree of MCSs for each PN via y(Q i ) � q j�1 y(σ j ), and screen out the pulse of PNs in a PMCS whose danger degree is larger than λ y (10) update the number of propositional neurons N − p k per the selected pulses, and compute pulse value of fault pulse in proposition neurons after position updating via  e tightness degree between PNs is shown in Figure 7, from which the weight matrix W can be obtained. e abductive fault diagnosis reasoning process is described as follows: When k � 1,  14 Complexity To start with the process, the pulse of PNs in the MCS whose danger degree is less than λ y is deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Similarly, the pulse of PNs in the MCS whose danger degree is less than λ y must also be deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., Repeatedly, it must be made sure to delete the pulse of PNs in the MCS whose danger degree is less than λ y . Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, Once more, the pulse is deleted in PNs in the MCS whose danger degree is less than λ y . Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., N − p 4 � (1, 0, 1, 0, 1, O 21 ) T and θ p 4 � (0.8, 0, 0.9, 0, 0.8, O 21 ) T , respectively. When k � 5, N − r 5 � (O 18 ) T . us, the termination condition is satisfied and the reasoning stops. We find that the rMFRSNPS-based abductive reasoning model is shown in Figure 8, where represents the deleted pulse. en, the fault paths can be found in Figure 8, i.e., L 1 � (σ 1 , σ 17 , σ 30 , σ 41 , σ 46 ), L 2 � (σ 3 , σ 19 , σ 30 , σ 41 , σ 46 ), and L 3 � (σ 5 , σ 21 , σ 32 , σ 41 , σ 46 ), where σ 1 , σ 3 , and σ 5 are the fault source of "insulation winding burned down." Besides, the occurrence probability of each fault path is P(L 1 ) � 0.159, P(L 2 ) � 0.217, and P(L 3 ) � 0.186. en, maintenance personnels can check the motor in turn according to the fault sources and paths got by Algorithm 3. e check order of fault sources is σ 3 , σ 1 , σ 5 , and the fault paths are L 2 , L 3 , L 1 .

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. e experimental results are shown in Table 2, 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.

Conclusions
is paper proposes a fault analysis method for three-phase induction motors based on rMFRSNPSs. Firstly, fault fuzzy production rules are proposed, and then, an rMFRSNPSbased fault diagnosis model is established according to them. en, the PVRA (Algorithm 1), 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 BAF-DRA 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. is 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.

ES:
Expert system ANN: Artificial neural network PN: Petri net SNPS: Spiking neural P system rMFRSNPSs: Modified fuzzy reasoning spiking neural P systems with real numbers PVRA: Pulse value reasoning algorithm FFPRA: Forward fault prediction reasoning algorithm BAFDRA: Backward abductive fault diagnosis reasoning algorithm MCS: Minimum cut set PMCS: Precise minimum cut set PN: Proposition neuron RN: Rule neuron MSAF-12: Method of selection of amplitudes of frequencies FPN: Fuzzy petri net FFPN: Fuzzy fault petri net IAAC: Improved artificial ant clustering.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
All authors declare that they have no conflicts of interest.