A Fault Diagnosis Method for Power Transmission Networks Based on Spiking Neural P Systems with Self-Updating Rules considering Biological Apoptosis Mechanism

School of Electrical Engineering and Electronic Information, Xihua University, Chengdu, China Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu, China College of Electrical Engineering, Sichuan University, Chengdu, China Key Laboratory of Network Assessment Technology, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China Department of Energy, Politecnico di Torino, Turin, Italy School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China Shandong Electrical Engineering & Equipment Group Co., Ltd, Shandong, China


Introduction
Fast and accurate fault diagnosis is very important for power system restoration after a serious blackout [1,2]. is rst requires the identi cation of faulty equipment [3,4]. To identify faulty equipment, dispatchers should rst analyze the fault alarm messages received from the supervisory control and data acquisition (SCADA) system according to ancillary facilities and their operational experience. Fault events in a power system can cause lots of fault alarm messages, and parts of them may be redundant ones which are not important with respect to the fault. Besides, protection devices may also fail leading to incomplete action information. All these cases can increase the uncertainty and incompleteness of fault alarm messages, making fault diagnosis more di cult [4][5][6][7]. erefore, improving the fault information processing ability of fault diagnosis methods is signi cant for faulty equipment identi cation.
Among these fault diagnosis methods, the SNPS is a novel bio-inspired distributed parallel computing model, which has powerful information processing and parallel computing ability (most models have been verified to be Turing equivalent [39,40]). e SNPS has become a hot research topic in Membrane Computing [41] and Natural Computing. In recent years, it has been used to explore the new fault information processing mechanism and SNPSbased fault diagnosis methods with rich achievements. SNPSs are a special kind of neural-like P systems [42], inspired by the mechanism that biological neurons store, transmit, and exchange information by pulses (spikes) along axons from presynaptic neurons to postsynaptic neurons in a distributed and parallel manner [4,43,44]. Due to the similarity between the spike transmission among different neurons through synapses and the fault propagation in power systems, fault diagnosis models based on different variants of spiking neural P systems are proposed to reason fault events to find faulty equipment [4,[36][37][38][45][46][47][48][49].
At present, SNPS-based fault diagnosis methods can be divided into two categories (according to the fault information processing way): fuzzy reasoning with real numbers (FRRN) [36][37][38][45][46][47][48] and fuzzy reasoning with fuzzy numbers (FRFN) [4,[50][51][52]. e FRRN refers to process uncertain and incomplete fault alarm messages using probability numbers which correspond to historical statistics of action information of protection devices including protective relays (PRs) and circuit breakers (CBs). erefore, the FRRN strongly depends on the historical data. However, with the increasing complexity of power systems, it is difficult to accurately obtain the historical statistics and update them in time. For the second kind of method, the FRFN is to deal with the uncertainty and incompleteness by fuzzy numbers. Wang et al. [4], for the first time, introduces trapezoidal fuzzy numbers, fault fuzzy production rules, and fault confidence levels into the framework of SNPSs to propose a fault diagnosis method for power transmission networks, where neurons, spikes, firing rules, and firing conditions are redefined. en, Tao et al., Yu et al., and Peng et al. [50][51][52] continue to explore and introduce triangular fuzzy numbers, interval valued fuzzy numbers, and intuitionistic fuzzy sets, respectively, to diagnose faults of transmission networks. However, for abovementioned types of FRFN, the different kinds of fuzzy numbers are all obtained via expert experience.
erefore, the FRFN has high subjectivity, and its nonsubjectivity is still a difficult problem to be solved. In addition, both the FRRN and the FRFN do not consider the preprocessing of fault alarm messages, that is, the fault alarm messages cannot be effectively utilized by deleting redundant information before modeling the SNPS-based diagnosis models. erefore, when the scale and complexity [53][54][55][56] of a power system increase and the redundancy of fault information is high, the fault tolerance of SNPS-based methods will reduce rapidly. erefore, more attention should be paid to how to effectively overcome the above shortcomings.
On the contrary, the rough set [57,58] is a typical mathematical tool to deal with uncertain and imprecise information in an objective way. However, the rough set generally should be combined with other methods because it can only reason and calculate fault data itself weakly. To address the aforementioned issues, this paper first proposes a spiking neural P system with self-updating rules (srSNPS) considering biological apoptosis mechanism, which integrates the strong objectivity (i.e., no need of any priori or additional information [59]) and good uncertainty handling capacity of rough sets and excellent parallel information processing ability of SNPSs. en, an srSNPS-based fault diagnosis method for transmission networks is further devised. e main contributions of this paper are described as follows: (1) To effectively deal with uncertain and incomplete fault information in an objectivity way, an srSNPS and its apoptosis algorithm (Algorithm 1) for condition neurons are proposed. is is the first time to availably unify the attribute reduction function of rough sets and the apoptosis mechanism of biological neurons in the framework of membrane computing. is way makes the SNPSs handle the uncertainty and incompleteness without historical statistics and expert experience.
(2) Besides, the self-updating matrix reasoning algorithm (Algorithm 2) and self-updating rules for input neurons are proposed. So, the srSNPS can effectively make comprehensive use of fault information including action information, start information, and overlimit signals of protection devices to improve its uncertainty and incomplete processing capacity based on a simple matrix reasoning process with a vivid graphical modelbuilding way.

Complexity
(3) Based on the proposed srSNPS and two algorithms, combined with the depth-first search algorithm (DSA), the weight network segmentation method (WNSM) and the protection device event tree (PDET), an srSNPS-based fault diagnosis method is proposed. is method has good diagnosis result interpretability while maintaining high-fault tolerance and a fast diagnosis speed under the uncertainty and incompleteness of fault alarm messages. In addition, it does not require historical statistics and expertise. ese are not the case for previous SNPSbased methods (even for all the graphical reasoning diagnosis methods). e remainder of this paper is organized as follows. Section 2 proposes the srSNPS and its algorithms. e srSNPS-based fault diagnosis method is proposed in Section 3. In Section 4, the proposed diagnosis method is applied to the IEEE 14-and 118-bus systems with the analysis of their effectiveness and superiority. Conclusions are finally drawn in Section 5.

Spiking Neural P Systems with Self-Updating Rules
is section first proposes the srSNPS and then presents its SMRA.

Spiking Neural P Systems with Self-Updating Rules
Definition 1. A spiking neural P system with self-updating rules (srSNPS) is a tuple: where in M e and represents Input: t condition neurons, s decision-making neurons (1) Calculate h(C) via (2) and let L � C Output: survival condition neuron set L ALGORITHM 1: Apoptosis algorithm for condition neurons.
if input neurons satisfy with the firing conditions of self-updating rules, then (3) correct θ g and the algorithm jumps to Step 7 (4) else (5) the algorithm jumps to Step 7 without pulse value correction (6) end if (7) while (θ g ≠ 0 1 or δ g ≠ 0 2 ) (8) if proposition neurons satisfy their firing conditions then (9) proposition neurons fire and compute δ g+1 via (11) if rule neurons satisfy their firing conditions then (12) rule neurons fire and compute θ g+1 via θ g+1 � E T ∘ δ g+1 (13) end if (14) where g represents the reasoning step of the reasoning algorithm of the srSNPS, i.e., the selfupdating matrix reasoning algorithm proposed in the next subsection. It means that if f i is applied, then spikes in the D i are emptied and the calculation process will be reinitialized (i.e., g � 0). e firing condition is E � θ di/T ′ ≠ θ di/T , representing that the pulse value in D i has changed.
is the j-th condition neuron (CN) in M e and represents a protective relay or circuit breaker in the targeted power transmission network, where (i) θ cj/T equals to 0 or 1 representing the pulse value of the j-th CN at time T. Note that θ cj/T � 1 means the protection device corresponding to C j has acted and vice versa. (ii) T is the sequence time of spikes, and a CN produces a spike at each unit time. us, a pulse value sequence is formed and recorded in the CN after T unit time. (iii) f j is a forgetting rule of the j-th CN with the form E/ a θ cj/T ⟶ λ; g � 0 . It means that if f j is applied, then spikes in C j are emptied and the calculation process will be reinitialized (i.e., g � 0). e firing condition is E � θ cj/T ′ ≠ θ cj/T , representing that the pulse value in C j has changed.
(c) A rj represents an apoptosis rule in M e with the form E/ C j ⟶ Algorithm 1 L { } , which means that if A rj is applied, then C j executes its apoptosis rule to determine whether it lives or dies.
at is, CNs execute Algorithm 1 to output survival condition neuron (containing important information) set L. In the same time, CNs including redundant information die and will not participate in fault reasoning. e firing condition is E � g � 0 ∩ sig j � 0 indicating that the rule A rj can be applied if and only if in the initial configuration [43] (that is, g � 0) with sig j � 0 (please see Definition 2).
(3) σ i � (θ i , r i , e i ), 1 ≤ i ≤ p, is the i-th proposition neuron (PN) corresponding to a protection device or the equipment, σ j � (δ j , r j ), 1 ≤ j ≤ q, is the j-th rule neuron (RN) corresponding to a fault production rule, and p + q � m, where (a) θ i and δ j are pulse values (equal to 0 or 1) in proposition neuron σ i and rule neuron σ j , respectively. (b) r i represents a firing rule of σ i with the form E/a θ ⟶ a θ , where E � a { } is the firing condition, which means that once σ i contains a spike, r i can be applied. en, σ i will consume a spike with pulse value θ, produce a new spike with the same pulse value, and then transmit it to its postsynaptic neurons; r j represents a firing rule of σ j with the form E/a δ ⟶ a β , where E � a { } is the firing condition, which means that once σ j contains a spike, r j can be applied. en, σ j will consume a spike with pulse value δ, produce a new spike with pulse value β (equals to 0 or 1), and then transmit it to its postsynaptic neurons. (c) e i represents a self-updating rule of the form E/a θ ⟶ a θ . e firing condition is E � ε i > 0 , which means that e i can be applied if and only if the self-updating operator ε i > 0. en, σ i will consume a spike with pulse value θ, produce a new spike with pulse value θ (equals to 0 or 1, called the antispike of θ ), and then ε i � ε i − 1. Note that only input proposition neurons contain self-updating rules. Note that if a PN is an input neuron, then it corresponds to a protection device, i.e., a protective relay or a circuit breaker. In this case, the pulse value in σ i represents the action information of its corresponding protection device, where θ i � 1 means that the device has acted, while θ i � 0 means it does not. If a PN is an output neuron, then it corresponds to suspicious faulty equipment. In this case, θ i � 1 indicates that the equipment is faulty, and θ i � 0 means it is not.
where h(C − C j ) is the conditional information entropy (CIE) of the other condition neurons to decision-making neurons after C j dies.

Definition 3. e CIE of a condition neuron C is
where U is called the universe, D is a DN, p(X i ) represents the probability of indicates the probability of the event Y j under the occurrence event X i .

Definition 4. M e of neurons in an srSNPS is a knowledge base
. . , Y m as the partitions of P and Q on U, respectively. e probability distributions of P and Q on the σ algebra composed of subsets of U are where Algorithm 1 used in the apoptosis rule is the apoptosis algorithm for condition neurons, described as follows. Its output is a survival condition neuron set L. e neurons in L will be connected according to the synaptic connection rule in Definition 5. To improve the intelligibility, a sketch map of neurons and the microenvironment are shown in Figure 1, and a diagram of evolutionary process of an srSNPS is shown in Figure 2, where the ODT represents the original decision table of a fault to be diagnosed. In an srSNPS, there are three types of rule neurons, i.e., General-, And-, and Or-rule neurons. All of them can represent fault production rules but with different processing methods of spikes, which will be described in detail in Section 2.2.

Self-Updating Matrix Reasoning Algorithm.
To explain the SMRA, we first introduce its vectors, matrices, and operators as follows: If σ i is an input neuron, then θ i � 1 means the protection device associated with σ i has acted, and θ i � 0 means it has not acted. If σ i is an output neuron, then θ i � 1 means the corresponding suspicious equipment is faulty, and θ i � 0 means it is not faulty.
p×q is a synaptic matrix, which represents the directed synaptic connection from PNs to And RNs. If there is a synapse from PN σ i to And RN σ j , then d ij � 1; otherwise, d ij � 0. (6) D 3 � (d ij ) p×q is a synaptic matrix, which represents the directed synaptic connection from PNs to Or RNs. If there is a synapse from PNs σ i to Or RN σ j , then d ij � 1; otherwise, d ij � 0. (7) E � (e ji ) q×p is a synaptic matrix, which represents the directed synaptic connection from RNs to PNs. If there is a synapse from RN σ j to PN σ i , then e ji � 1; otherwise, e ji � 0.
To parallelly reason and calculate fault alarm messages, we design the SMRA for srSNPSs as follows.

The Proposed Methodology
is section proposes a fault diagnosis method for power transmission networks based on the srSNPS, whose flowchart is shown in Figure 3 and described as follows: Step 1: transmission network partition.   [60,61] to split the DST while ensuring that the computational burden of each subnetwork after network partition is approximately the same.
Step 2: establish an srSNPS-based fault diagnosis model for each subnetwork. e subnetworks perform the following steps (a)-(d) in a parallel way: (a) Select Living Neurons. (i) Transport the ODT of the fault into the microenvironment, i.e., feed   Step 3: correct pulse values of input neurons. Integratedly utilize the start information and action information of protection devices based on the fault information matrix (please see Subsection 3.1) to get the self-updating operator vector of input neurons, i.e., ε. en, each input neuron applies its self-updating rule to correct and update its pulse value. Accordingly, the wrong action information of protection devices is corrected because the pulse value of each input neuron represents the action information of a protective relay or a circuit breaker. e flowchart of this step is shown in Figure 4.
Step 4: compute pulse values of output neurons. Each subnetwork performs the SMRA (i.e., Algorithm 2) of its srSNPS-based fault diagnosis model in a parallel way to reasoning out the pulse values of output neurons.
Step 5: output diagnosis results. If and only if the pulse value of an output neuron is 1, its corresponding equipment is faulty.
Step 6: evaluate uncertain behaviors of protection devices. Search the protection device event tree (PDET, please see Subsection 3.2) to find the false or lost alarm messages of protection devices. e purpose of Step 1 includes (a) simplify decisionmaking samples for srSNPS-based diagnosis models. e typical fault diagnosis methods based on rough sets establish the ODT of a fault event via the fault data of the entire transmission network, while our proposed method concurrently establishes the ODT for each subnetwork; (b) improve the topological adaptability of networks. When the topology of a transmission network changes, it just needs to modify the srSNPS-based diagnosis model of the corresponding subnetwork. Besides, Steps 1 and 2 are finished before faults occur, and Steps 3-6 are executed immediately after failures.
us, the proposed method can save some time.

Correct Pulse Values of Input Neurons in srSNPS-Based Diagnosis Models. Define the fault information matrix of a CB (labeled as A) as
where P (u) stA is the start information of PRs, P (u) acA is the action information of PRs, and u � 1, 2, 3, 4 represents the main protective, first backup protective, second backup protective, and bus protective relays associated with the CB, respectively; R acA is the action information of the CB. Particularly, if there is no bus PR, P (4) stA � P (4) acA � 0. Complexity 7 Define the fault information identification matrix N A as N A � P (1) jA R jA P (2) jA R jA P (3) jA R jA P (4) jA R jA where P (u) jA (P (u) jA � P (u) stA ⊕ P (u) acA , u � 1, 2, 3, 4) represents decision value of status for the corresponding PR and ⊕ is the xor operation. When P (u) jA � 1, read overlimit signals of protection devices corresponding to the suspicious equipment.
ε � (ε 1 , . . . , ε l ) T is the self-updating operator vector of input neurons. Define ε i (1 ≤ i ≤ l) as where the symbol "-" is not the fraction line. e symbols above "-" indicate self-updating operators of lines where protective devices associated with input neurons are located, while the ones under "-" represent self-updating operators of buses.

Search the Protection Device Event
Tree. According to start and action messages of PRs, action messages of CBs, and overlimit signals of faulty equipment, the PDETfor a fault event is generated as shown in Figure 5, where P sti and P aci are the start and action information of the i-th PR, respectively, R aci is the action information of the corresponding CB, and OL i is the overlimit signal of the faulty equipment we have found; the start and action messages are represented by , the overlimit signals are indicated by , and the behaviors of protection devices are represented by ; the pink in above symbol represents the corresponding protection device has acted, while the gray color means it does not. When faulty equipment is got in Section 3-Step 5, then we search the PDET to evaluate uncertain behaviors of protection devices.

Case Study
e proposed method is applied to the IEEE 14-bus and IEEE 118-bus systems to demonstrate its effectiveness and superiority.

Comparative
Tests. Ten cases including uncertainty and incompleteness, such as maloperation and misinformation, are used to do the comparisons between the proposed method and three typical fault diagnosis methods, i.e., the cause-effect network (CEN) in [2], the fuzzy Petri net (FPN) in [18], and the fuzzy reasoning spiking neural P system (FRSNPS) in [36]. e reason choosing the three methods is that their performance over many other approaches has been demonstrated. e diagnosis results are shown in Table 1.
For cases 1 and 2, four methods can find the right faulty equipment. erefore, they are all effective without uncertain or incomplete fault information. For case 3, only the proposed method and the FRSNPS are successful, while both CEN and FPN are failed. For cases 4-7, only our method can accurately diagnose the faults. Cases 8-10 are three extreme examples, i.e., all the action information of protective relays or circuit breakers are lost. e CEN, FPN, and FRSNPS are all failed for cases 8-10, while our method is successful for cases 8 and 9 and failed for the case 10. is is because the lost information in cases 8 and 9 is the redundant information in our method, and the one in case 10 is the core attribute information. erefore, we can see that if the uncertainty and incompleteness occur in the redundant information, it has no effect on the diagnosis results of the proposed method. However, when it happens in the core attribute information, it will influence the results. Fortunately, due to the stability of the relay protection system, the case of complete loss of all core attribute information is very rare under normal operation of power systems. erefore, Table 1 shows that the proposed method can obtain satisfying results in the situations with incomplete or uncertain alarm information for both single and multiple faults.
Besides, the computational complexities of the CEN, FPN, and FRSNPS are analyzed. Note that the computational complexities of the four methods are all in the linear order. Since the proposed method reduces redundant information by the apoptosis algorithm of condition neurons, the number of neurons used for establishing srSNPS-based fault diagnosis models is smaller than that of the CEN, FPN, and FRSNPS. e more the redundant information, the smaller the computational complexity of the proposed method. In a fault diagnosis problem, there is typically much redundant information. So, the upper limit of algorithm computational complexity for the proposed method is far less than that of the other three methods. erefore, we can see that, although our method needs more information (such as start information and overlimit signals) to improve its diagnostic accuracy, the time complexity does not increase.

Disturbed Tests.
To verify the fault-tolerant ability of our method, we take the disturbed test under different uncertain information ratios (UIRs). e test results are shown in Table 2.
Define the UIR as where N fault is the number of fault alarm messages from the SCADA system and N uncertain is the number of the uncertain fault messages which are caused by the refuse operation, unwanted operation, and information loss of protection devices (including PRs and CBs). In Table 2, each diagnostic accuracy for different κ is got by computing the mean value under 1000 random tests. Note that the headers "Uncertainty is in the RAI" and "Uncertainty is in the CAAI" represent that uncertain fault alarm messages appear only in the redundant alarm information (RAI) and the core attribute alarm information (CAAI), respectively. Moreover, the "Uncertainty is in the RAAI" represents that uncertain fault alarm messages appear randomly both in RAI and CAAI, which is called random attribute alarm information (RAAI). Table 2 shows that when uncertain alarm messages contained in κ are all RAI, the diagnostic accuracy is the highest compared to the other two situations. is is because the RAI is deleted in Step 2 (a) and is not used in the fault diagnosis process. When there is too much RAI, it will interfere with the judgment of the CAAI which should be used in the diagnosis process. erefore, when the uncertainty is in the RAI, the diagnostic accuracy is less than 100%, and it will decrease with the increasing UIRs of the RAI.
When the uncertain fault messages contained in κ are CAAI and RAAI, the diagnostic accuracy is also high because the start information and overlimit signals of protection devices will guarantee the diagnostic accuracy. Table 2 shows that the diagnostic accuracy of our proposed method is very high when κ is less than 5%, and the diagnosis results are still acceptable when κ is between 5% and 10%. e diagnostic accuracy drops quickly when κ is more than 10%. Fortunately, this situation will not happen unless there are attacks [62,63]. erefore, we can get from Table 2 that the diagnostic accuracy of the proposed method is high for the three cases (uncertainty is in the RAI, CAAI, and RAAI, respectively) when κ is less than 5%, and the RAI has a smaller impact on fault diagnostic results.

Working Details of the Proposed Fault Diagnosis Method.
is section takes the IEEE 14-bus system as an example to show how the proposed fault diagnosis method works.
(1) Network Partition. e system is first abstracted as a DST by the depth-first search algorithm, as shown in Figure 6(a). en, the weight network segmentation method is employed to divide the DST, as shown in Figure 6(b). e relationship of subnetworks after division is shown in Figure 7. Note that the information on each line in Figure 7 will be used by its linked subnetworks. Finally, we get the network partition result, as shown in Figure 8.
(2) Establish srSNPS-Based Models for Subnetworks. Subnetwork S6 is considered to show how to establish an srSNPS-based fault diagnosis model for a subnetwork. e protection configuration of S6 is shown in Figure 9.
First, the ODT for S6 is established by using the protection configuration information and causality between PRs and CBs. en, we obtain living neurons by executing Step 2 (a) in Section 3, where L C � BR 13 Table 3) and accordingly the FPRS of S6 is created, as shown in Table 4. e srSNPS-based diagnosis model for S6 is built, as shown in Figure 10, by connecting synapses based on the FPRS shown in Table 4 and the synaptic connection rule given in Definition 5.
(3) Computational Process. In this section, case 3 in Table 1 is considered as an example to show how an srSNPS-based model works.
Case 3. L 1213 and B 12 fault. Operated protective relays: MLR 1213 , MLR 1213 , and SLR 0612 . Tripped CBs: CB 1206 . Lost information: CB 1312 and CB 1213 . e fault alarm messages obtained from the SCADA system are shown in Table 5.
According to Table 5, we can get the fault information matrices of CB 1213 , CB 1312 , and CB 1206 which are M CB 1213 , M CB 1312 , and M CB 1206 , respectively. According to their fault information, the identification matrices N CB 1213 , N CB 1312 , and N CB 1206 are obtained.
14 (1) 13 (2) 12 (3) 6 (4) 11 (5) 10 ( e pulse value vector of proposition neurons before correction is e pulse value vector of proposition neurons after correction is When the SMRA stops, we get that Now, both pulse values of output neurons σ 9 and σ 11 are 1. erefore, the equipment corresponding to them is faulty, i.e., L 1213 and B 12 have failed. en, we search the PDET in Figure 5 for the fault event that "B 12 is faulty," and the searching path is P stMLR1213 � 1 ⟶ P acMLR1213 � 1 ⟶ R stCB1213 � * ⟶ 10 , i.e., the action information of CB 1213 is lost. e action information of protection devices of L 1213 can be got in a similar way. Finally, we find that the action information of CB 1206        line main protective relays, 360 line first backup protective relays, 360 line second backup protective relays, and 236 bus protective relays. e system is divided into 12 subnetworks, as shown in Figure 11. For this system, ten typical cases are considered to do the comparison tests for the proposed method, CEN, FPN, and FRSNPS. e results are shown in Table 6.   Figure 11: e partition result of the IEEE 118-bus system. 14 Complexity In Table 6, cases 1-4 have faults with less information loss, while cases 5-10 have faults with serious information loss or many information errors. For cases 1 and 2, all the four methods can diagnose the right faulty equipment. For case 3, the FPN is failed, while the other three methods are successful. For case 4, only our method can diagnose the right faults and all the three other methods have missed diagnosis of faulty equipment. For case 5, CEN and FPN are failed, while the FRSNPS and the proposed method are successful. For case 6, theCEN, FPN, and FRSNPS have missed diagnosis, while the proposed method can diagnose all the right faults. When the lost information increases, such as cases 7-9, the missed diagnosis of faulty equipment for the CEN, FPN, and FRSNPS becomes more serious, while the proposed method is still successful. Furthermore, for case 10, not only many fault alarm messages are lost but also the false information is involved. e results show that, for this case, our method is still valid, while all the other three methods misdiagnose the faults. is is because that the redundant fault alarm information reduction ability of the apoptosis algorithm (Algorithm 1) and the false information correction ability of self-updating rules guarantee the high fault tolerance of the proposed method. erefore, we can see from Table 6 that the proposed method can also obtain satisfying results for a complex system with fault information loss and information errors. Besides, to further demonstrate the effectiveness and superiority of our proposed method, the IEEE 118-bus system is employed to do disturbed tests for different κ. e disturbed test method is the same as that of the IEEE 14-bus system, and test results are shown in Table 7. Table 7 shows that when uncertain alarm messages appear only in the RAI, the diagnostic accuracy is the highest. Besides, when the uncertain fault messages are contained in the CAAI and RAAI, the diagnostic accuracy is also high. Comparing the data in Tables 2 and 7, we find that, for the same case, the diagnostic accuracy of the IEEE 118-bus system is only 0.01 lower than that of the IEEE 14-bus system on average. erefore, Table 7 shows that the proposed method is still feasible and effective with high diagnostic accuracy and fault tolerance for different kinds of faults for complex systems.

Conclusions
To reduce the error caused by historical statistic, expertise, and redundant fault information, this paper proposes a fault diagnosis method of power transmission lines based on an srSNPS, considering the biological apoptosis mechanism. e attribute reduction capacity of rough sets and the apoptosis mechanism of neurons are integrated in a system in the framework of membrane computing for the first time.
e srSNPS can deal with uncertain and incomplete fault alarm messages without historical statistic and expert experience, while its apoptosis algorithm for CNs can delete the redundant fault information before modeling. is simplifies the problem complexity. Besides, the transmission network partition improves the topological adaptive ability. Case studies show that the proposed method has high diagnostic accuracy and fault tolerance with good diagnosis result interpretability and fast speed. Owing to the complexity of fault diagnosis for power systems, future work will focus on different applications, such as power plants, substations, distribution networks, integrated energy system, and cyber-physical power system considering network attacks.

Nomenclature
SCADA: Supervisory control and data acquisition ES: Expert system ANN: Artificial neural network BN: Bayesian network PN: Petri net CEN: Cause-effect network OM: Optimization method FL: Fuzzy logic RS: Rough set SNPS: Spiking neural P system FRRN: Fuzzy reasoning with real numbers FRFN: Fuzzy reasoning with fuzzy numbers srSNPS: Spiking neural P system with self-updating rules SMRA: Self-updating matrix reasoning algorithm DSA: Depth-first search algorithm WNSM: Weight network segmentation method PDET: Protection device event tree DN: Decision-making neuron CN: Condition neuron PR: Protective relay CB: Circuit breaker PN: Proposition neuron RN: Rule neuron CIE: Conditional information entropy MRDT: Minimum reduction decision table FPRS: Fault production rule set ODT: Original decision table FPN: Fuzzy Petri net FRSNPS: Fuzzy reasoning spiking neural P system UIA: Uncertain information ratio RAI: Redundant alarm information CAAI: Core attribute alarm information RAAI: Random attribute alarm information.

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.