Improved Temporal Fuzzy Reasoning Spiking Neural P Systems for Power System Fault Diagnosis

Abstract

Fuzzy and temporal reasoning can effectively improve the accuracy of fault diagnosis methods. However, there are challenges in practical applications, such as missing alarm messages, temporal reasoning with complex calculations, and complex modeling processes. Therefore, this study proposes an improved temporal fuzzy reasoning spiking neural P (ITFRSNP) system for power system fault diagnosis. First, the ITFRSNP system and its reasoning method are proposed to perform association reasoning between confidence degrees and temporal constraints. Second, a general fault diagnosis model and process are developed based on the ITFRSNP system to diagnose various faulty components and simplify the modeling process. In addition, a search method is provided for identifying suspected faulty components, considering the missing alarm message of the circuit breaker. Simulation results of fault cases demonstrate that the proposed method exhibits high accuracy and fault tolerance. It can precisely identify faulty components despite incorrect operations or inaccurate alarm messages of protective relays and circuit breakers. Moreover, the search method effectively narrows down the diagnostic scope without missing suspected faulty components in scenarios where alarms from boundary circuit breakers are missing, thereby enhancing the fault diagnosis efficiency. The fault diagnosis model features a straightforward structure and reasoning process with minimal computational complexity, making it suitable for real-time diagnosis of complex faults within power systems.

1. Introduction

Faults are inevitable during the operation of a power system. Therefore, power systems are equipped with many protective relays (PRs) and circuit breakers (CBs), which quickly, reliably, and selectively isolate faults, limit the extent of power outages, and reduce economic loss. After a fault occurs, alarm messages from the PRs and CBs are reported to the dispatch center, providing important decision-making support for regulating and operating power systems. Rapid and accurate diagnosis of faulty components is essential for the rapid restoration of power supply, reducing the influence scope of the accident, and ensuring the safety and stability of the power system. Achieving this goal not only effectively reduces the duration of power outages and ensures normal social production and residential life but also reduces the economic losses caused by power outages. In addition, efficient fault diagnosis methods provide important decision support for the maintenance and improvement of the power system, thereby improving the efficiency and accuracy of fault resolution and establishing a solid foundation for the long-term stable operation of the power system.

Recently, experts and scholars have proposed various fault diagnosis methods based on alarm messages, including expert systems [1,2], analytical models [3,4,5], artificial neural networks [6,7], Bayesian networks [8,9], Petri nets [10,11,12,13], and spiking neural P systems [14,15]. Expert systems match alarm messages with a knowledge base to identify faulty components. The knowledge base of an expert system is difficult to build and maintain for a large-scale power system, and its generalizability and fault tolerance must be improved. The analytical model constructs a mathematical model that expresses the logical relationship between components, PRs, and CBs, according to the protection configuration and operation rules. It represents the fault diagnosis problem as an integer programming problem, and then applies an intelligent optimization algorithm to find the fault hypothesis that best explains the alarm messages. Artificial neural networks represent the causal relationships between alarm information and faulty components using high-order nonlinear equations. However, artificial neural networks require many training samples, and their portability is poor. Bayesian networks involve considerable effort to determine prior probabilities and face difficulties in online applications without automated modeling techniques. Petri nets graphically explain the topology of the grid and use mathematical methods to reason and analyze the logical relationships between components, PRs, and CBs implied in fault messages. Petri nets require reconstruction or adjustment of their fault diagnosis models to accommodate topological changes, complicating the modeling process.

The concept of a fault diagnosis method utilizing a spiking neural P (SNP) system was first proposed in 2013. The SNP system employs synapses to connect propositions and rule neurons, thereby delineating the relationships between faulty components, PRs, and CBs. The confidence degrees of PRs and CBs follow the spikes passing along the synapses between neurons to simulate the dynamic process of fault isolation. Notably, SNP systems do not require training and can explicitly describe the causal logic among components, PRs, and CBs, demonstrating commendable fault tolerance, flexibility, and scalability. Nonetheless, the potential for PRs and CBs to malfunction or cease functioning, coupled with the possibility of false alarms and omissions in their alarm messages, complicates fault diagnosis due to the uncertainty of operational states and alarm messages. In response, enhanced models and schemes for SNP systems have been proposed to address problems faced in practical fault diagnosis applications. Owing to the uncertainty of alarm messages, scholars have adopted trapezoidal fuzzy numbers [16], intuitionistic fuzzy numbers [17], and interval-valued fuzzy numbers [18] to characterize the operational states of PRs and CBs, thereby improving the accuracy of the diagnosis results. Moreover, a combination of biological apoptosis mechanisms and rough sets has been applied [19] to simplify the fault diagnosis model and manage the uncertainty and incompleteness of alarm messages.

The study referenced in [20] employs the temporal constraints between fault occurrence and operation times of PRs or CBs to distinguish between valid and invalid alarm messages. It assigns confidence degrees to valid alarm messages based on their temporal constraints and correspondence. Subsequently, ref. [21] introduced causal networks to identify PRs and CBs that do not conform to these temporal constraints, enabling adjustments to their confidence degrees. The research in [22] developed the time sequence spiking neural P system with real numbers (rTSSNPS), which facilitates the logical association of confidence levels with temporal constraints. The temporal features of the alarm message are introduced in [20,21,22] to prevent non-faulty components from being misclassified as faulty components and enhance the accuracy of the fault diagnosis results. However, the above methods encounter limitations in practical applications: (1) The modeling process is complicated by traversing all the suspected faulty components to construct fault diagnosis models, and these models need to be reconstructed when adapting to topology changes. (2) The missing alarm message of the CB reduces the accuracy and fault tolerance of fault diagnosis results. Moreover, the outage area may not be recognized, and all components in the power grid need to be diagnosed, which seriously affects the efficiency of fault diagnosis.

To solve these problems, this study proposes a fault diagnosis method based on an improved temporal fuzzy reasoning spiking neural P (ITFRSNP) system. The main contributions of this study are as follows:

• The ITFRSNP system and its associated reasoning algorithm are proposed. The ITFRSNP system contains new rule neurons and reasoning methods, simplifying the correlation reasoning of confidence degrees and temporal features. The ITFRSNP system can effectively improve the accuracy and fault tolerance of fault diagnosis. This lays a theoretical foundation for constructing a general fault diagnosis model and its diagnosis reasoning method.
• The CB subnet is defined based on the configuration of the CBs and corresponding protections. A general fault diagnosis model and its fault diagnosis process are developed based on the ITFRSNP system for CB subnets. The proposed model simplifies the modeling process and can be directly applied to fault diagnosis of various components with good versatility and adaptability.
• A search method for suspected faulty components is proposed. The search method does not need to traverse the entire power system and does not miss suspected faulty components. This method narrows the scope of topology search and fault diagnosis and can effectively improve the efficiency of fault diagnosis.

The remainder of this study is organized as follows. Section 2 defines the ITFRSNP system and its reasoning algorithms. Section 3 describes the fault diagnosis method based on the ITFRSNP system. Section 4 presents the fault case validation and algorithm comparison, and Section 5 concludes this study.

2. ITFRSNP System

This section proposes the ITFRSNP system with new rule neurons and a reasoning algorithm. This section provides a theoretical foundation for a general fault diagnosis model and its diagnosis process.

2.1. Temporal Constraints and Temporal Reasonings

The temporal constraints are defined as follows [10]:

• The time-point constraint (T(a) = [t_a⁻, t_a⁺]) indicates that event a occurs in the time interval [t_a⁻, t_a⁺]. If t_a⁻ = t_a⁺, event a occurs at a specific time.
• The time-distance constraint (D(a, b) = [Δt_ab⁻, Δt_ab⁺]) indicates that the time distance between the occurrence times of events a and b is within the time interval [Δt_ab⁻, Δt_ab⁺].

According to the above definitions, the following are temporal reasonings that utilize the interval calculation rules.

• Reverse reasoning: The time-point constraint T(b) and the time-distance constraint D(a, b) are known. The following definition can be employed to infer the time-point constraint T(a):

  $$T(\mathrm{a})=T(\mathrm{b})-D(\mathrm{a},\mathrm{b})=[t_{\mathrm{b}}^{-}-\Delta t_{\mathrm{ab}}^{+},t_{\mathrm{b}}^{+}-\Delta t_{\mathrm{ab}}^{-}]$$

  (1)
• Merge reasoning: If event a is known to satisfy multiple time-point constraints T₁(a), T₂(a), T₃(a), …, T_k(a) simultaneously, the following definition can be employed to infer the time-point constraint T(a).

  $$T(\mathrm{a})=T_1(\mathrm{a})\cap T_2(\mathrm{a})\cap \cdots \cap T_k(\mathrm{a})$$

  (2)

2.2. The TFRSNP System

The ITFRSNP system is defined as follows:

$$\mathsf{\Pi }=(A,N_p,N_r,syn,I,O)$$

(3)

where the following are defined:

• A = {a} is a singleton alphabet (a is called spike).
• N_p = {σ_p1, σ_p2, …, σ_pm} denotes the proposition neuron set, where σ_pi (1 ≤ i ≤ m) is associated with a fuzzy proposition. Each proposition neuron σ_pi has the form σ_pi = (α_pi, T_pi, θ_pi, λ_pi, r_pi), where the following are defined:
  • α_pi (α_pi ∈ [0, 1]) is the spike value in σ_pi;
  • T_pi (T_pi = [t_pi⁻, t_pi⁺]) is the time-point constraint in σ_pi;
  • θ_pi is an integer representing the number of spikes received by σ_pi;
  • λ_pi is an integer that represents the spike number threshold required for firing σ_pi;
  • r_pi is a firing/spiking rule of σ_pi with the form E/a^θ → a^β, where θ and β are real numbers in [0, 1].
• N_r = {σ_r1, σ_r2, …, σ_rn} is the rule neuron set, where σ_rj (1 ≤ j ≤ n) is associated with a fuzzy production rule. N_r contains two types of rule neurons: TIME-type and OR-type. Each rule neuron σ_rj has the form σ_rj = (α_rj, T_rj, D_rj, θ_rj, λ_rj, r_rj), where:
  • α_rj (α_rj ∈ [0, 1]) is the spike value in σ_rj;
  • T_rj (T_rj = [t_rj⁻, t_rj⁺]) is the time-point constraint in σ_rj;
  • D_rj (D_rj = [Δt_rj⁻, Δt_rj⁺]) is the time-distance constraint in σ_rj, which denotes the time-distance constraint between the propositions of the input and output proposition neurons associated with σ_rj;
  • θ_rj is an integer representing the number of spikes received by σ_rj;
  • λ_rj is an integer that represents the spike number threshold required for firing σ_rj;
  • r_rj is a firing/spiking rule of σ_rj, with the form E/a^θ→ a^β, where θ and β are real numbers in [0, 1].
• $syn\subseteq \{N_p\times N_r\}\cup \{N_p\times N_r\}$ is a directed graph that indicates the synapses between propositions and rule neurons. Input matrices U₁ = {u_{1(i, j)}}_m×n and U₂ = {u_{2(i, j)}}_m×n and output matrix V = {v_{(j, i)}}_n×m, where u_{1(i, j)}, u_{2(i, j)}, and v_{(i, j)} ∈ [0, 1], are used to represent syn. If there is a synapse from proposition neuron σ_pi to TIME-type neuron σ_rj, then u_{1(i, j)} = 1; otherwise, u_{1(i, j)} = 0. If there is a synapse from proposition neuron σ_pi to OR-type neuron σ_rj, then u_{2(i, j)} = 1; otherwise, u_{2(i, j)} = 0. If there is a directed arc (synapse) from rule neuron σ_rj to proposition neuron σ_pi, then v_{(j, i)} = 1; otherwise, v_{(j, i)} = 0.
• I and O are input and output neuron sets, respectively.

Graphical models of the proposition neuron, TIME-type rule neuron, and OR-type rule neuron are shown in Figure 1. Their respective interpretations are provided below:

• The proposition neuron (Figure 1a) describes the operational events of PRs and CBs. When a PR alarm message is received, the spike value in the proposition neuron corresponds to its confidence degree, and the time-point constraint corresponds to its operation time. The proposition neuron can delineate the state and time of the PR and CB operations, which are then inputted into the fault diagnosis model for inference. Moreover, this neuron can record the intermediate and output results of the fault diagnosis.
• The TIME-type rule neuron (Figure 1b) receives spikes from the proposition neuron. It then transmits the spike value to the next proposition neuron, and deduces the time-point constraints backward according to Equation (1) before forwarding them. This rule neuron is responsible for characterizing the temporal distance constraints between propositions that correspond to their associated propositional neurons. The TIME-type rule neurons employ reverse reasoning to deduce the time-point constraints for the PRs from those of the corresponding CBs and their time-distance constraints. Similarly, they apply to determine the time-point constraints of the faults by leveraging the time-point constraints and time-distance constraints of the PRs.
• The OR-type rule neuron (Figure 1c) receives spikes from the proposition neurons and takes the maximum spike value α_l and the corresponding time-point constraint T_l to transmit to the next proposition neuron. Each component is equipped with multiple sets of protections, including primary, near backup, and remote backup protections. The OR-type rule neuron determines the protection type for isolating the faulty component by selecting the PR and CB with the highest confidence degree.

2.3. The Reasoning Algorithm of the ITFRSNP System

Before describing the reasoning algorithm for the ITFRSNP system, the definitions of the matrices and vectors used in the reasoning process are first introduced.

• α_p = (α_p1, α_p2, …, α_pm)^T and α_r = (α_r1, α_r2, …, α_rn)^T are vectors consisting of spike values in m proposition and n rule neurons, respectively. α_pi (1 ≤ i ≤ m) and α_rj (1 ≤ j ≤ n) represent the spike values of the i-th proposition neuron and j-th rule neuron, respectively. This relationship between the vector elements and neurons applies to the subsequent vector definitions as well.
• T_p = (T_p1, T_p2, …, T_pm)^T and T_r = (T_r1, T_r2, …, T_rn)^T are vectors consisting of time-point constraints in m proposition and n rule neurons, respectively.
• D_r = (D_r1, D_r2, …, D_rn)^T is a vector of the time-distance constraints in n rule neurons.
• θ_p = (θ_p1, θ_p2, …, θ_pm)^T and θ_r = (θ_r1, θ_r2, …, θ_rn)^T are vectors consisting of spike numbers in m proposition and n rule neurons, respectively.
• λ_p = (λ_p1, λ_p2, …, λ_pm)^T and λ_r = (λ_r1, λ_r2, …, λ_rn)^T are vectors consisting of spike number thresholds in m proposition and n rule neurons, respectively.
• β_p = (β_p1, β_p2, …, β_pm)^T and β_r = (β_r1, β_r2, …, β_rn)^T are vectors consisting of spike values output from m proposition neurons and n rule neurons, respectively.
• τ_p = (τ_p1, τ_p2, …, τ_pm)^T and τ_r = (τ_r1, τ_r2, …, τ_rn)^T are vectors consisting of time-point constraints output from m proposition and n rule neurons, respectively, after firing.
• b_p = (b_p1, b_p2, …, b_pm)^T and b_r = (b_r1, b_r2, …, b_rn)^T are vectors consisting of spike numbers output from m proposition neurons and n rule neurons, respectively, after firing.
• B_p = diag(b_p) and B_r = diag(b_r) are diagonal matrices, where diag( ) is the diagonal function.

The following matrix calculation rules are defined by incorporating temporal reasoning into fuzzy reasoning.

• H = C$\ast$D:

  $$H_i=\underset{C_{ik}\ne 0,1\le k\le t}{\cap }{D}_k$$

  (4)

  where D and H are t × 1-and s × 1-dimensional vectors of time-distance constraints, respectively, and C is an s × t-dimensional binary matrix.
• J = G$-$H:

  $$J_i=G_i-H_i$$

  (5)

  where G and J are s × 1-dimensional vectors of time-point constraints.
• {P, Q} = {C, D}$\oplus${F, G}:

  $$\{\begin{array}{c}P=C^{\mathrm{T}}\times F\\ Q_i=\underset{C_{ik}^{\mathrm{T}}\ne 0,1\le k\le s}{\cap }{(G-C\ast D)}_k\end{array}$$

  (6)

  where F and P are s × 1 and t × 1-dimensional vectors of spike values, and Q is a t × 1-dimensional vector of time-point constraints.
• {P, Q} = {C,D} $\otimes$ {F,G}:

  $$\{\begin{array}{l}{P}_i=\underset{C_{ik}^{\mathrm{T}}\ne 0,1\le k\le s}{\max }(F_k)\\ \begin{array}{cc}{Q}_i=G_j,& 1\le j\le s,F_j=P_i\end{array}\end{array}$$

  (7)
• {P, Q} = {$P^{\prime }$,$Q^{\prime }$}$\stackrel{⌢}{\times }${$P^{″}$,$Q^{″}$}:

  $$\{\begin{array}{l}{P}_i=\max ({P^{\prime }}_i,{P^{″}}_i)\\ Q_i={Q^{\prime }}_i\cup {Q^{″}}_i\end{array}$$

  (8)

  where ${\mathit{P}}^{\prime }$ and ${\mathit{P}}^{″}$ are t × 1-dimensional vectors of the spike values, and $Q^{\prime }$ and $Q^{″}$ are t × 1-dimensional vectors of the time-point constraints.
• {P, Q} = {${\mathit{P}}^{\prime }$,${\mathit{Q}}^{\prime }$}$\stackrel{⌢}{+}${${\mathit{P}}^{″}$,${\mathit{Q}}^{″}$}:

  $$P_i=\{\begin{array}{cc}\mathrm{avg}({P^{\prime }}_i,{P^{″}}_i),& {P^{\prime }}_i\ne 0\\ {P^{″}}_i,& {P^{\prime }}_i=0\end{array}$$

  (9)

  $$Q_i=\{\begin{array}{cc}{Q^{\prime }}_i\cap {Q^{″}}_i,& {Q^{\prime }}_i\cap {Q^{″}}_i\ne \varnothing \\ {Q^{\prime }}_i\cup {Q^{″}}_i,& {Q^{\prime }}_i\cap {Q^{″}}_i=\varnothing \end{array}$$

  (10)
• {β,τ} = fire (α, T, θ, λ):

  $${\beta }_i=\{\begin{array}{cc}{\alpha }_i,& {\theta }_i={\lambda }_i\\ 0,& {\theta }_i<{\lambda }_i\end{array}$$

  (11)

  $${\tau }_i=\{\begin{array}{cc}{T}_i,& {\theta }_i={\lambda }_i\\ \varnothing ,& {\theta }_i<{\lambda }_i\end{array}$$

  (12)
• {α,T} = update (α, T, θ, λ):

  $${\alpha }_i=\{\begin{array}{cc}{\alpha }_i,& 0<{\theta }_i<{\lambda }_i\\ 0,& {\theta }_i=0 \mathrm{or} {\theta }_i={\lambda }_i\end{array}$$

  (13)

  $$T_i=\{\begin{array}{cc}{T}_i,& 0<{\theta }_i<{\lambda }_i\\ \varnothing ,& {\theta }_i=0 \mathrm{or} {\theta }_i={\lambda }_i\end{array}$$

  (14)
• b = fire (1, θ, λ):

  $${\beta }_i=\{\begin{array}{cc}1,& {\theta }_i={\lambda }_i\\ 0,& {\theta }_i<{\lambda }_i\end{array}$$

  (15)
• θ = update (θ, θ, λ):

  $${\theta }_i=\{\begin{array}{cc}0,& {\theta }_i={\lambda }_i\\ {\theta }_i,& {\theta }_i<{\lambda }_i\end{array}$$

  (16)

Based on the matrix calculation rules, the reasoning algorithm for the ITFRSNP system is as follows.

• Input parameter matrices and vectors U₁, U₂, V, λ_p, λ_r, D_r, θ_p, and θ_r. The number of interactive inferences is g, and the vectors of the spike values and time-point constraints in the g-th iteration are ${\alpha }_p^g$, $T_p^g$, ${\alpha }_r^g$, and $T_r^g$. Initialize the number of iterations g = 0 and input ${\alpha }_p^0$, $T_p^0$, ${\alpha }_r^0$, and $T_r^0$.
• Process the firing of proposition neurons:

  $$\{\begin{array}{c}\{{\mathit{\beta }}_p^g,{\mathit{\tau }}_p^g\}=\mathrm{fire}({\mathit{\alpha }}_p^g,{\mathit{T}}_p^g,{\mathit{\theta }}_p^g,{\mathit{\lambda }}_p)\\ {\mathit{b}}_p^g=\mathrm{fire}(1,{\mathit{\theta }}_p^g,{\mathit{\lambda }}_p)\\ {\mathit{B}}_p^g=\mathrm{diag}({\mathit{b}}_p^g)\\ \{{\mathit{\alpha }}_p^g,{\mathit{T}}_p^g\}=\mathrm{update}({\mathit{\alpha }}_p^g,{\mathit{T}}_p^g,{\mathit{\theta }}_p^g,{\mathit{\lambda }}_p)\\ {\mathit{\theta }}_p^g=\mathrm{update}({\mathit{\theta }}_p^g,{\mathit{\theta }}_p^g,{\mathit{\lambda }}_p)\end{array}$$

  (17)
• Compute ${\alpha }_r^{g+1}$, $T_r^{g+1}$ and ${\theta }_r^{g+1}$:

  $$\{\begin{array}{c}\{{\mathit{\alpha }}_r^{g+1},{\mathit{T}}_r^{g+1}\}=\left(\{({\mathit{B}}_p^g\cdot {\mathit{U}}_1),{\mathit{D}}_r\}\oplus \{{\mathit{\beta }}_p^g,{\mathit{\tau }}_p^g\}\right)\stackrel{⌢}{\times }\left(\{({\mathit{B}}_p^g\cdot {\mathit{U}}_2),{\mathit{D}}_r\}\otimes \{{\mathit{\beta }}_p^g,{\mathit{\tau }}_p^g\}\right)\\ {\mathit{\theta }}_r^{g+1}={\mathit{\theta }}_r^g+\{{[{\mathit{B}}_p^g\cdot ({\mathit{U}}_1+U_2)]}^{\mathrm{T}}\times {\mathit{b}}_p^g\}\end{array}$$

  (18)
• Process the firing of rule neurons:

  $$\{\begin{array}{c}\{{\mathit{\beta }}_r^{g+1},{\mathit{\tau }}_r^{g+1}\}=\mathrm{fire}({\mathit{\alpha }}_r^{g+1},{\mathit{{\rm T}}}_r^{g+1},{\mathit{\theta }}_r^{g+1},{\mathit{\lambda }}_r)\\ {\mathit{b}}_r^{g+1}=\mathrm{fire}(1,{\mathit{\theta }}_r^{g+1},{\mathit{\lambda }}_r)\\ {\mathit{B}}_r^{g+1}=\mathrm{diag}({\mathit{b}}_r^{g+1})\\ \{{\mathit{\alpha }}_r^{g+1},{\mathit{{\rm T}}}_r^{g+1}\}=\mathrm{update}({\mathit{\alpha }}_r^{g+1},{\mathit{{\rm T}}}_r^{g+1},{\mathit{\theta }}_r^{g+1},{\mathit{\lambda }}_r)\\ {\mathit{\theta }}_r^{g+1}=\mathrm{update}({\mathit{\theta }}_r^{g+1},{\mathit{\theta }}_r^{g+1},{\mathit{\lambda }}_r)\end{array}$$

  (19)
• Compute ${\alpha }_p^{g+1}$, $T_p^{g+1}$ and ${\theta }_p^{g+1}$:

  $$\{\begin{array}{c}\{{\alpha }_p^{g+1},{{\rm T}}_p^{g+1}\}=\{{\alpha }_p^g,{{\rm T}}_p^g\}\stackrel{⌢}{+}[\{(V\cdot B_r^{g+1}),D_r\}\otimes \{{\beta }_r^g,{\tau }_r^g\}]\\ {\theta }_p^{g+1}={\theta }_p^g+[{(V\cdot B_r^{g+1})}^{\mathrm{T}}\times b_r^g]\end{array}$$

  (20)
• If ${\theta }_p^{g+1}$ = (0, 0, …, 0)^T and ${\theta }_r^{g+1}$ = (0, 0, …, 0)^T, then halt and export reasoning results (${\alpha }_p^g$ and $T_p^g$); otherwise, g = g + 1 and return to Step 2.

2.4. Comparison between ITFRSNP and RTSSNPS Systems

In Section 2.2 and Section 2.3, the ITFRSNP system includes only two rule neurons (TIME-type and OR-type) to express the reasoning rules required by the subsequent general fault diagnosis model. This study formulates a set of reasoning algorithms based on matrix calculations to realize iterative reasoning of the fault diagnosis model. The rTSSNPS system requires three types of rule neurons (general neurons, “and” neurons, and “or” neurons) to express the reverse reasoning, comparison, and fusion of temporal constraints and confidence degrees. Additionally, rTSSNPS requires two sets of reasoning algorithms. The first algorithm realizes the reverse reasoning of temporal constraints and comparison of confidence, and the second algorithm fuses the confidence degrees and temporal constraints obtained by the first algorithm. By contrast, the ITFRSNP system uses fewer rule neurons to realize correlation reasoning with confidence and temporal constraints, simplifying the reasoning process and reducing its complexity.

3. ITFRSNP System-Based General Fault Diagnosis Model and Its Fault Diagnosis Process

This section proposes a general fault diagnosis model and its fault diagnosis process based on the ITFRSNP system, to simplify the modeling process of the fault diagnosis. Moreover, a new search method for suspected faulty components is proposed to improve the efficiency of fault diagnosis.

3.1. ITFRSNP System-Based General Fault Diagnosis Model and Fault Diagnosis Principle

The fault diagnosis components include the lines, buses, and transformers. The CBs and relay protection configurations are shown in Figure 2. The bus (B2) is configured with CBs (CB_bm1 and CB_bm2), bus PR (Bm), remote backup PRs (Bs1 and Bs2), and their corresponding CBs (CB_bs1 and CB_bs2). The sending (s) and receiving (r) ends of the line (L2) are equipped with CBs (CB_lsm and CB_lrm), their associated main PRs (Lsm and Lrm), and near backup PRs (Lsp and Lrp), as well as remote backup PRs (Lss and Lrs) and corresponding CBs (CB_lss and CB_lrs), respectively. The transformer (T) is equipped with CBs (CB_tsm and CB_trm), main PR (Tm), near backup PR (Tp), remote backup PRs (Lss and Lrs), and corresponding CBs (CB_tss and CB_trs).

As shown in Figure 2, the main and near backup PRs of each CB and the remote backup PRs and their corresponding CBs are configured similarly. Therefore, to conveniently describe the construction principle of the fault diagnosis model, the net consisting of each CB and its main and near backup PRs, as well as the CBs and their remote backup PRs corresponding to the remote end of the adjacent line, can be regarded as a subnet (the subnet is named after the CB is connected to the component, as shown in the dotted box in Figure 2). The PRs and CBs configured for each component can be divided into a corresponding number of subnets, based on the number of connected CBs. For example, the PRs and CBs configured for bus B2 are divided into two subnets of CB_bm1 and CB_bm2.

Each CB subnet consists of a main PR, near-backup PR, remote backup PRs, and their corresponding CBs. The operation logic of the PRs and CBs in each subnet is consistent. The difference between each subnet is the number of remote backup PRs, corresponding CBs, and protection types in the subnet. Based on the commonality and characteristics of subnets, this study constructs an ITFRSNP system-based general fault diagnosis model for the subnet according to the PRs, CBs, and their operation logic. The number of remote backup PRs and corresponding CBs for each component varies owing to the number of neighboring lines. The remote backup PRs and corresponding CBs are preprocessed, and a proposition neuron is used to represent the operational states of all remote backup PRs (corresponding CBs) in the subnet.

The ITFRSNP system-based general fault diagnosis model is shown in Figure 3. σ_p1, σ_p2, and σ_p3 record the confidence degrees and time-point constraints of the CBs in the subnet, σ_p4, σ_p5, and σ_p6 record the confidence degrees and time-point constraints of the PRs in the subnet. The corresponding devices are marked below the proposition neurons in the CB and PR layers, respectively. CB_mPR and CB_nPR represent the CB connected to the components in a subnet. Because the CB has different confidence degrees when corresponding to the main PR or near backup PR (as shown in

Table 1. Spike values of PRs and CBs with alarm messages.

Components		Line	Bus	Transformer
Main	PR	0.9913	0.8564	0.7756
	CB	0.9833	0.9833	0.9833
Near backup	PR	0.80	0	0.75
	CB	0.85	0	0.8
Remote backup	PR	0.70	0.70	0.70
	CB	0.75	0.75	0.75

), σ_p1 and σ_p2 record the confidence degrees when the CB corresponds to the main PR and near the backup PR, respectively. CB_rPR and rPR denote the remote CBs of the adjacent lines and the corresponding remote backup PRs in the subnet, respectively. σ_r1, …, σ_r6 are TIME-type rule neurons that perform reverse reasoning on the time-point constraints of the PRs and CBs. σ_p7, σ_p8, and σ_p9 in the middle layer recorded the fusion results of CBs and PRs. σ_r7 is an OR-type rule neuron that selects the protection type for the fault isolation. σ_p10 outputs diagnosis results for the subnet.

The pretreatment process of remote backup PRs and their corresponding CBs is as follows:

$$\{\begin{array}{l}{\alpha }_{p3}=\frac{n_{\mathrm{CBrPR}.\mathrm{o}}\times {\phi }_{\mathrm{CBCBrPR}.\mathrm{o}}+n_{\mathrm{CBrPR}.\mathrm{c}}\times {\phi }_{\mathrm{CBrPR}.\mathrm{c}}}{n_{\mathrm{CBrPR}.\mathrm{o}}+n_{\mathrm{CBrPR}.\mathrm{c}}}\\ {\alpha }_{p6}=\frac{n_{\mathrm{rPR}.\mathrm{o}}\times {\phi }_{\mathrm{rPR}.\mathrm{o}}+n_{\mathrm{rPR}.\mathrm{c}}\times {\phi }_{\mathrm{rPR}.\mathrm{c}}}{n_{\mathrm{rPR}.\mathrm{o}}+n_{\mathrm{rPR}.\mathrm{c}}}\\ T_{p3}=\underset{i=1}{\overset{\mathrm{CBrPR}.\mathrm{o}}{\cup }}{T}_{\mathrm{CBrPR}.\mathrm{o}i}\\ T_{p6}=\underset{i=1}{\overset{n\mathrm{rPR}.\mathrm{o}}{\cup }}{T}_{\mathrm{rPR}.\mathrm{o}i}\end{array}$$

(21)

where α_p3 and T_p3 are the spike value and time-point constraint of the CB corresponding to the remote backup PR in the subnet, respectively, α_p6 and T_p6 are the spike value and time-point constraint of the remote backup PR in the subnet. n_CBrPR.o (n_CBrPR.c) is the number of operated (non-operated) CBs corresponding to the remote backup PRs and n_rPR.o (n_rPR.c) is the number of operated (non-operated) remote backup PRs. φ_CBrPR.o (φ_CBrPR.c) is the spike value of the operated (non-operated) CB corresponding to the remote backup PR; φ_rPR.o (φ_rPR.c) is the spike value of the operated (non-operated) remote backup PR; T_rPR.oi and T_CBrPR.oi are the time-point constraints for the i-th operated remote backup PR and its corresponding operated CB, respectively.

Combined with Section 2.3, the diagnosis process of the general fault diagnosis model is as follows.

• σ_p1, σ_p2, and σ_p3 fire, and the confidence degrees and time-point constraints of the CBs are input into σ_r1, σ_r2, and σ_r3.
• σ_r1, σ_r2, and σ_r3 infer the time-point constraints of the PRs according to the time-point constraints and time-distance constraints of the CBs. Then, σ_r1, σ_r2, and σ_r3 fire, and the confidence degrees of the CBs and the temporal reasoning results of the PRs are input into σ_p4, σ_p5, and σ_p6.
• σ_p4, σ_p5, and σ_p6 fuse the confidence degrees of the PRs and CBs, and the temporal reasoning results from σ_p4, σ_p5, and σ_p6 are matched with the time-point constraints of the PRs in σ_p4, σ_p5, and σ_p6. Then, σ_p4, σ_p5, and σ_p6 fire, and the confidence fusion results and temporal matching results are input into σ_r4, σ_r5, and σ_r6.
• σ_r4, σ_r5, and σ_r6 infer the fault time based on the temporal matching results and the time-distance constraints of the PRs. Then, σ_r4, σ_r5, and σ_r6 fire, and the confidence fusion results and temporal reasoning results of the fault time are input into σ_p7, σ_p8, and σ_p9. Finally, σ_p7, σ_p8, and σ_p9 fire, these results are input into σ_r7.
• σ_r7 is responsible for selecting the highest confidence degree among σ_p7, σ_p8, and σ_p9 and its time-point constraints as the diagnosis result of the subnet output into σ_p10.

The fault diagnosis results for each subnet can be obtained using a general fault diagnosis model. These diagnosis results of the suspected faulty component can be fused to determine whether the component is faulty. The fault diagnosis principle of the general fault diagnosis model is shown in Figure 4.

Fault diagnosis models are built based on PRs, CBs, and their operational logic in the existing methods. However, the number of adjacent lines may differ for each component and the numbers of PRs and CBs configured for each component are also different. Therefore, fault diagnosis models for components have different scales and structures, and must be modeled online. Additionally, when the power grid topology changes, the number of PRs and CBs configured for the component changes accordingly. Therefore, it is necessary to adjust or reconstruct a fault diagnosis model to adapt it to a new topology. Existing methods do not propose efficient automatic modeling methods, and their process of constructing and reconstructing fault diagnosis models is complicated for online applications, thereby reducing the efficiency of fault diagnosis. In this study, a general fault diagnosis model is constructed for a CB subnet. Its structure and scale are not affected by the component types, number of remote backup PRs and their corresponding CBs, or topological changes. It is suitable for fault diagnosis of various components with good universality and adaptability. The fault diagnosis principle simplifies the modeling process and improves the efficiency of fault diagnosis.

3.2. Model Parameters of the General Fault Diagnosis Model

The parameters of the general fault diagnosis model according to the propositions and reasoning rules of the neurons are as follows.

• The input matrices U₁ and U₂, and the output matrix V of the general fault diagnosis model are as follows:

  $${\mathit{U}}_1=\left[\begin{array}{cc}{I}_{6\times 6}& O_{6\times 1}\\ O_{4\times 6}& O_{4\times 1}\end{array}\right]$$

  (22)

  $$U_2=\left[\begin{array}{cc}{O}_{6\times 6}& O_{6\times 1}\\ O_{3\times 6}& E_{3\times 1}\\ O_{1\times 6}& O_{1\times 1}\end{array}\right]$$

  (23)

  $$\mathit{V}=\left[\begin{array}{cc}{\mathit{O}}_{7\times 3}& {\mathit{I}}_{7\times 7}\end{array}\right]$$

  (24)

  where I is the unit matrix, O is the zero matrix, and E is the matrix with all the elements equal to 1.
• Considering the uncertainty of the alarm messages, the spike values of PRs and CBs with alarm messages are listed in Table 1, and the spike values of PRs and CBs without alarm messages are 0.2 [21]. The spike values of the proposition and rule neurons in the other layers are 0.
• If the PR/CB operation time is t₀, the time-point constraint of the related proposition neuron is [t₀, t₀]. For PR/CB without an alarm message, the time-point constraint of the related proposition neuron is $\varnothing$.
• The time-distance constraints for the bus PR, main PR, near backup PR, and remote backup PR are [10, 40] ms, [10, 40] ms, [310, 340] ms, and [510, 540] ms relative to the fault time, respectively. The time-distance constraint for the CB is [20, 40] ms relative to the protection operation time. Therefore, the vector D_r of the general fault diagnosis model is as follows:

  $$D_r=[[20,40],[20,40],[20,40],[10,40],{[310,340],[510,540],\varnothing ]}^{\mathrm{T}}$$

  (25)
• The spike numbers of the proposition neurons in the PR and CB layers are set to 1, and the spike numbers of the other neurons are set to zero to ensure that spikes from proposition neurons can be delivered to rule neurons. Therefore, the vectors θ_p and θ_r of the general fault diagnosis model are as follows:

  $$\{\begin{array}{l}{\theta }_p={[1,1,1,1,1,1,0,0,0,0]}^T\\ {\theta }_r={[0,0,0,0,0,0,0]}^T\end{array}$$

  (26)
• Because the PR layer neurons need to obtain spikes from the CB layer neurons to fire, the spike number thresholds of all proposition neurons in the PR layer are set to two, and the spike number thresholds of other proposition neurons are set to one. Rule neurons must obtain spikes from all input proposition neurons to fire; therefore, their spike number thresholds are equal to the number of their input proposition neurons. Therefore, the vectors λ_p and λ_r of the general fault diagnosis model are as follows:

  $$\{\begin{array}{l}{\lambda }_p={[1,1,1,2,2,2,1,1,1,1]}^{\mathrm{T}}\\ {\lambda }_r={[1,1,1,1,1,1,3]}^{\mathrm{T}}\end{array}$$

  (27)

3.3. ITFRSNP System-Based General Fault Diagnosis Process

3.3.1. Suspected Faulty Component Search Method

As mentioned in the introduction, the missing alarm message from the CB at the boundary of the outage area (referred to as the boundary CB) significantly affects the diagnosis efficiency. A missing boundary CB alarm can result in outage areas not being recognized. All components in the power system are diagnosed to identify faulty components. In practical applications, it is difficult to traverse the entire power system topology and construct fault diagnosis models for all the components, and the workload is large. Therefore, this study proposes a new search method for suspected faulty components. According to the relay protection configuration, when a fault occurs in a component, the main PR trips the CBs to isolate the faulty component; if the CB refuses to trip, the backup protection at the remote end of the neighboring component trips the corresponding CB. It illustrates that the faults of the components connected and adjacent to the CB cause the CB to trip. Therefore, this study searches for components connected and adjacent to the tripped CBs as their isolation ranges; the isolation ranges of different CBs take the intersection of two to determine the suspected faulty components, thus narrowing down the scope of fault diagnosis.

The specific steps to search for suspected faulty components are as follows.

• Alarm messages are received from the W CBs. These CBs form the CB set (CBS). Initialize w = 1 and begin the search.
• Select the w-th CB (CB_w) in CBS.
• Buses, lines, and transformers connected to CB_w are searched to form the w-th subset of suspected faulty components (COM_w).
• Adjacent buses, lines and transformers are searched in the direction of the line or transformer connected to the CB_w and placed into COM_w.
• If w = W, proceed to the next step; otherwise, w = w + 1 and return to Step 3.
• The suspected faulty component subsets intersect in pairs. If the intersection of two suspected faulty components is not empty, then the intersection results are combined into a total set of suspected faulty components (COMS). If the intersection of a certain suspected faulty component subset with other subsets is empty, all the components in this subset are input into COMS. The components in COMS are suspected to be faulty components.

3.3.2. Fault Diagnosis Process

The suspected fault component search method is used to determine the diagnosis range, and then the fault diagnosis is performed individually for the subnets of the suspected faulty components. Finally, the diagnosis results of each subnet are fused to determine the faulty components. The specific fault diagnosis process used in this study is as follows.

• After receiving the alarm messages, start the search method for suspected faulty components and set the total number of suspected faulty components as X. Initialize x = 1.
• Select the x-th suspected faulty component and set the number of CBs connected to it as Y. Initialize y = 1.
• Initialize the general fault diagnosis model with neuron spike values of zero and time-point constraints of $\varnothing$. The y-th CB is selected for search. If the x-th suspected faulty component is a line or transformer, search for the main PR and near-backup PR corresponding to the CB. If the x-th suspected faulty component is a bus, the bus PR corresponding to the CB is searched. According to the alarm messages, σ_p1, σ_p2, σ_p4, and σ_p5 in the general fault diagnosis model are assigned spike values and time-point constraints concerning Section 3.1.
• The number of adjacent lines of the y-th CB is set as Z. If Z = 0, skip to Step 8; otherwise, initialize z = 1.
• The initial values of n_CBrPR.o, n_CBrPR.c, n_rPR.o, and n_rPR.c are 0; the initial values of T_CBrPR.oi and T_rPR.oi are $\varnothing$. Search for CBs at the far end of the z-th adjacent line. If the CB has an alarm message at t_CBrPRz, then n_CBrPR.o = n_CBrPR.o + 1, and T_CBrPR = T_CBrPR ∪ [t_CBrPRz, t_CBrPRz]; otherwise, n_CBrPR.c = n_CBrPR.c + 1. Search for the remote backup PR corresponding to the CB at the far end of the z-th adjacent line, and if this remote backup PR has an alarm message at t_rPRz, then n_rPR.o = n_rPR.o + 1 and T_rPR = T_rPR ∪ [t_rPRz, t_rPRz]; otherwise, n_rPR.c = n_rPR.c + 1.
• If z < Z, z = z + 1, and return to Step 5; otherwise, substitute n_CBrPR.o, n_CBrPR.c, n_rPR.o, n_rPR.c, T_CBrPR, and T_rPR into (21) to yield α_p3, T_p3, α_p6, and T_p6.
• Based on the parameter definitions in Section 2.3, the inputs (${\alpha }_{p(xy)}^0$ and $T_{p(xy)}^0$) are determined for the general fault diagnosis model corresponding to the y-th CB subnet of the x-th suspected faulty component. The reasoning algorithm in Section 2.3 is introduced to obtain the diagnosis result (α_p10(xy) and T_p10(xy) of σ_p10) for the y-th CB subnet.
• If y < Y, then y = y + 1 and return to Step 3; otherwise, the diagnosis results (α_(x) and T_(x)) of the x-th suspected faulty component are obtained by substituting the diagnosis results of the subnets of this component into the following equations:

  $$\{\begin{array}{c}\alpha (x)=\underset{y=1}{\overset{Y}{avg}}(\alpha p10(xy))\\ T(x)=\underset{y=1}{\overset{Y}{\cap }}(\alpha p10(xy))\end{array}$$

  (28)
• The measured value η_(x) of the x-th suspected faulty component is obtained by substituting α_(x) and T_(x) into the measurement function, as shown in Equation (29). The threshold of the measured value was set to 0.6, considering the uncertainty and timestamp errors of the alarm messages. A component is faulty if η_(x) is greater than 0.6.

  $$\{\begin{array}{c}R\left(T(x)\right)=\{\begin{array}{cc}1,& T(a)\ne \varnothing \\ 0,& other\end{array}\\ \eta (x)=0.7\alpha (x)+0.3R(T(x))\end{array}$$

  (29)
• If x < X, then x = x + 1 and return to Step 2; otherwise, complete the fault diagnosis for all suspected faulty components and end the fault diagnosis process.

4. Case Studies

In this section, the IEEE-39 bus power system (as shown in Figure 5) is used to simulate the fault cases and verify the effectiveness and reliability of the proposed method. These cases include single faults, double faults, PR/CB rejections, missing alarm messages, and error timestamps. They prove that the proposed method can ensure the accuracy and fault tolerance of the diagnosis results for complex faults.

In Figure 5, B denotes buses, L denotes lines, T denotes transformers, Lp denotes line PR, Bp denotes bus PR, Tp denotes transformer PR, and the subscripts m, p, and s indicate the main, near backup, and remote backup PRs, respectively. For example, the line connecting B₀₃ and B₁₈ is L₀₃₁₈; the transformer connecting B₀₂ and B₃₀ is T₀₂₃₀; the CB of L₀₃₁₈ near B₀₃ is CB₀₃₁₈; the CB near B₁₈ is CB₁₈₀₃; Lp₀₃₁₈m, Lp₀₃₁₈p, and Lp₀₃₁₈s indicate the main, near backup, and remote backup PRs configured by CB₀₃₁₈; Bp₀₃ is the bus PR configured by B₀₃; and Tp₀₂₃₀m and Tp₀₂₃₀p are the main and near backup PRs configured by T₀₂₃₀.

4.1. Fault Case Analysis

4.1.1. Case 1 (Bus Fault)

Considering Figure 5 as an example, a fault occurs at B₁₈. Bp₁₈ operates, and then CB₁₈₁₇ and CB₁₈₀₃ trip. The alarm messages are briefly described as Bp₁₈(0), CB₁₈₁₇(35), and CB₁₈₀₃(40), where the brackets are the operation times, the first alarm message is the zero moment, and the unit is ms. The fault diagnosis process is as follows:

• CBS = {CB₁₈₁₇, CB₁₈₀₃} can be obtained from the alarm message. Traverse CBS and search for suspected faulty components. There are a subset COM₁₈₁₇ = {B₁₈, L₁₈₁₇, B₁₇, L₁₇₂₇, L₁₆₁₇}, a subset COM₁₈₀₃ = {B₁₈, L₀₃₁₈, B₀₃, L₀₂₀₃, L₀₃₀₄}, and a total set COMS = {B₁₈}.
• B₁₈ is selected for fault diagnosis. CB₁₈₀₃ connected to B₁₈ is selected for the search, and its corresponding bus PR is Bp₁₈, respectively. According to the alarm messages, the spike values of σ_p1, σ_p2, σ_p4, and σ_p5 are 0.9833, 0.85, 0.8564, and 0, respectively, and their time-point constraints are [40, 40], [40, 40], [0, 0], and $\varnothing$, respectively.
• The adjacent line corresponding to CB₀₃₁₈ is L₀₃₁₈, the CB at the far end is CB₀₃₁₈, and the remote backup PR is Lp₀₃₁₈s. Because the alarm information is not received from the above PRs and CBs, n_CBrPR.o = 0, n_CBrPR.c = 1, T_CBrPR = $\varnothing$, n_rPR.o = 0, n_rPR.c = 1, and T_rPR = $\varnothing$. Substituting them into (21) to obtain the spike values and time-point constraints for σ_p3 and σ_p6 are 0.2 and $\varnothing$, respectively.
• ${\alpha }_{p(11)}^0$ and $T_{p(11)}^0$ of the fault diagnosis model for the CB₁₈₀₃ subnet are as follows:

  $$\{\begin{array}{l}{\alpha }_{p(11)}^0={[0.9833,0.85,0.2,0.8564,0,0.2,0,0,0,0]}^{\mathrm{T}}\\ T_{p(11)}^0={[[40,40],[40,40],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (30)
• ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$ and model parameters are imported into the reasoning algorithm, the diagnosis results of the CB₁₈₀₃ subnet are α_p10(11) = 0.9199, T_p10(11) = [−40, −10].
• Similarly, ${\mathit{\alpha }}_{p(12)}^0$ and ${\mathit{T}}_{p(11)}^0$ of the fault diagnosis model for the CB₁₈₁₇ subnet are as follows:

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(12)}^0={[0.9833,0.85,0.2,0.8564,0,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(12)}^0={[[35,35],[35,35],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (31)
• Similarly, the diagnosis results for the CB₁₈₁₇ subnet are α_p10(12) = 0.9199 and T_p10(12) = [−40, −10].
• α_p10(11), T_p10(11), α_p10(12) and T_p10(12) are substituted into (28), and the diagnosis results α₍₁₎ = 0.9199 and T₍₁₎ = [−40, −10] for B₁₈. α₍₁₎ and T₍₁₎ are substituted into (29), and the measured value of B₁₈ is η₍₁₎ = 0.9439, which is greater than 0.6. Therefore, B₁₈ is a faulty component.

This case demonstrates that the proposed method can diagnose bus faults and their corresponding time-point constraints.

4.1.2. Case 2 (Line Fault with the CB Rejection)

Considering Figure 5 as an example, a fault occurs at L₀₃₁₈. Lp₀₃₁₈m and Lp₁₈₀₃m operate, and then CB₀₃₁₈ trips, but CB₁₈₀₃ fails to operate; thus, Lp₁₇₁₈s operates and CB₁₇₁₈ trips. The alarm messages are briefly described as Lp₁₈₀₃m(0), Lp₀₃₁₈m(2), CB₀₃₁₈(30), Lp₁₇₁₈s(520), and CB₁₇₁₈(541). The fault diagnosis process is as follows.

• Based on the alarm messages, COMS = {L₀₃₁₈, L₁₇₁₈, B₁₈} is obtained.
• L₀₃₁₈ is selected for fault diagnosis. The ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$, ${\mathit{\alpha }}_{p(12)}^0$, and ${\mathit{T}}_{p(12)}^0$ of the fault diagnosis model for the CB₀₃₁₈ and CB₁₈₀₃ subnets are as follows.

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(11)}^0={[0.9833,0.85,0.2,0.9913,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(11)}^0={[[30,30],[30,30],\varnothing ,[2,2],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (32)

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(12)}^0={[0.2,0.2,0.75,0.9913,0.2,0.7,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(12)}^0={[\varnothing ,\varnothing ,[541,541],[0,0],\varnothing ,[520,520],\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (33)
• The diagnosis results for the CB₀₃₁₈ subnet are α_p10(11) = 0.9873 and T_p10(11) = [−38, −8]. The diagnosis results of the CB₁₈₀₃ subnet are α_p10(12) = 0.725 and T_p10(12) = [−20, 10]. The diagnosis results of L₀₃₁₈ are α₍₁₎ = 0.8562 and T₍₁₎ = [−20, 10], and its measured value is η₍₁₎ = 0.8993, which is greater than 0.6. Thus, L₀₃₁₈ is a faulty component.
• Similarly, the measured values of L₁₇₁₈ and B₁₈ are 0.3733 and 0.42, respectively, and L₁₇₁₈ and B₁₈ are non-faulty components.

CB₁₈₀₃ refused to operate in this case, and the suspected faulty component extended to L₁₇₁₈ and B₁₈. However, the proposed method is not misdiagnosed, and its diagnosis results are accurate. This case proves that the method in this study can diagnose a faulty line with CB rejection and non-faulty components.

4.1.3. Case 3 (Transformer Fault with Missing CB Alarm Message)

Considering Figure 5 as an example, a fault occurs at T₁₂₁₃. Tp₁₂₁₃m operates, and then CB₁₂₁₃ and CB₁₃₁₂ trip. However, the alarm message for CB₁₃₁₂ is lost. The alarm messages are briefly described as Tp₁₂₁₃m(0) and CB₁₂₁₃(32). The fault diagnosis process is as follows:

• Based on the alarm messages, COMS = {T₁₂₁₃, B₁₂, B₁₃, L₁₃₁₄, L₁₀₁₃} is obtained.
• T₁₂₁₃ is selected for fault diagnosis. The ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$, ${\mathit{\alpha }}_{p(12)}^0$, and ${\mathit{T}}_{p(12)}^0$ of the fault diagnosis model for the CB₁₃₁₂ and CB₁₂₁₃ subnets are as follows.

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(11)}^0={[0.2,0.2,0.2,0.7756,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(11)}^0={[\varnothing ,\varnothing ,\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (34)

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(12)}^0={[0.9833,0.8,0.2,0.7756,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(12)}^0={[[32,32],[32,32],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (35)
• The diagnosis results for the CB₁₃₁₂ subnet are α_p10(11) = 0.4878 and T_p10(11) = [−40, −10]. The diagnosis results of the CB₁₂₁₃ subnet are α_p10(12) = 0.8795 and T_p10(12) = [−40, −10]. The diagnosis results of T₁₂₁₃ are α₍₁₎ = 0.6836 and T₍₁₎ = [−40, −10], and its measured value η₍₁₎ = 0.7785, which is greater than 0.6. Thus, T₁₂₁₃ is a faulty component.
• Similarly, the measured values of B₁₂, B₁₃, L₁₃₁₄, and L₁₀₁₃ are 0.2771, 0.2042, 0.1881, and 0.1881, respectively, and they are non-faulty components.

In this case, CB₁₃₁₂ is a boundary CB, and its alarm message is lost. However, it was not possible to confirm the outage area. The suspected fault component search method searches for T₁₂₁₃, B₁₂, B₁₃, L₁₃₁₄, and L₁₀₁₃ within the protection range of CB₁₂₁₃. This search method avoids fault diagnosis in the entire power system. This improves the fault diagnosis efficiency when an alarm message from the boundary CB is lost.

4.1.4. More Cases

More fault cases are presented to demonstrate the accuracy and fault tolerance of the proposed method, and the fault diagnosis results are presented in

Table 2. Fault diagnosis results of more cases.

No.	Fault Descriptions	Brief Descriptions	Suspected Faulty Components	Spike Values	Time-Point Constraints	Measured Values	Faulty Components
4	The fault occurs on B18, and the alarm message of CB0318 is missing.	Bp18(0), CB1817(29)	B18	0.7240	[−40, −20]	0.8068	B18
			B17	0.2917	$\varnothing$	0.2042
			L1718	0.3958	$\varnothing$	0.2771
			L1727	0.2688	$\varnothing$	0.1881
			L1617	0.2688	$\varnothing$	0.1881
5	A fault occurs on B18, CB1803 fails to trip, Lp0318s operates, and CB0318 trips.	Bp18(0), CB1718(29), Lp0318s(520), CB0318(540)	B18	0.8549	[−20, −10]	0.8757	B18
			L1718	0.5625	$\varnothing$	0.3938
6	A fault occurs on B03, and the alarm on Bp03 is missing.	CB0302(29), CB0304(31), CB0318(32)	B03	0.5917	[−48, −1]	0.7142	B03
7	A fault occurs on B03, Bp03 fails to operate, and remote backup PRs operate.	Lp1803s(0), Lp0403s(2), Lp0203s(6), CB1803(32), CB0403(35), CB0203(36)	B03	0.725	[−534, −510]	0.8075	B03
			L0203	0.6583	$\varnothing$	0.4608
			L0304	0.6583	$\varnothing$	0.4608
			L0318	0.6583	$\varnothing$	0.4608
8	A fault occurs on L0318, Lp1803m fails to operate, and Lp1803p operates, but the timestamp of Lp1803p is incorrect.	Lp0318m(0), CB0318(30), Lp1803p(200), CB1803(325)	L0318	0.9062	[−40, −5]	0.9343	L0318
9	A fault occurs on L0318, CB1803 fails to trip, and the alarm message of Lp0318m is missing.	Lp1803m(0), CB0318(25), Lp1718s(500), CB1718(515)	L0318	0.6583	[−40, −10]	0.7608	L0318
			L1718	0.5333	$\varnothing$	0.3733
			B18	0.6	$\varnothing$	0.42
10	A fault occurs on L0318, CB1803 fails to trip, and the alarm message of CB0318 is missing.	Lp0318m(0), Lp1803m(5), Lp1718s(500), CB1718(515)	L0318	0.6603	[−35, −10]	0.7622	L0318
			L1718	0.3958	$\varnothing$	0.2771
			B18	0.4625	$\varnothing$	0.3238
			B17	0.3958	$\varnothing$	0.2771
11	A fault occurs on T1213 and the alarm message of CB1312 is missing.	Tp1213m(0), CB1213(5)	T1213	0.6836	[−40, −10]	0.7785	T1213
			B12	0.3958	$\varnothing$	0.2771
			B13	0.2917	$\varnothing$	0.2042
			L1314	0.2688	$\varnothing$	0.1881
			L1013	0.2688	$\varnothing$	0.1881
12	A fault occurs on T1213, CB1803 fails to trip, Lp1413 and Lp1013s operate, and CB1413 and CB1013 trip.	Tp1213m(0), CB1213(32), Lp1413s(525), Lp1013s(530), CB1413(540), CB1013(545)	T1213	0.8224	[−15, −10]	0.8757	T1213
			B13	0.475	$\varnothing$	0.3325
			L1314	0.5333	$\varnothing$	0.3733
			L1013	0.5333	$\varnothing$	0.3733
13	A fault occurs on L0318, the alarm message of CB0318 is missing. A fault occurs on L1718.	Lp0318m(0), Lp1803m(5), CB1803(35), Lp1817m(120), Lp1718m(128), CB1817(145), CB1718(153)	L0318	0.7915	[−35, −10]	0.8540	L0318 L1718
			L1718	0.9873	[88, 110]	0.9911
			B17	0.3306	$\varnothing$	0.2314
			B18	0.5917	$\varnothing$	0.4142
14	A fault occurs on L0318, and CB1803 fails to trip. A fault occurs on B17.	Lp1803m(0), Lp0318m(2), CB0318(30), Lp1718s(520), CB1718(541), Bp17(100), CB1727(135), CB1716(140),	L0318	0.8562	[−20, 10]	0.8993	L0318 B17
			L1718	0.5333	$\varnothing$	0.3733
			B17	0.9199	[60, 80]	0.9493
			B18	0.6	$\varnothing$	0.42
15	A fault occurs on B03, the alarm message of CB0318 is missing. A fault occurs on B03, but the timestamp of CB1415 is incorrect.	Bp03(0), CB0302(29), CB0304(31), Bp14(250), CB1404(275), CB1415(284), CB1413(385)	B03	0.8549	[−27, −10]	0.8984	B03 B14
			B14	0.9199	[210, 240]	0.9493
16	A fault occurs on B18, Bp18 fails to operate, and remote backup PRs operate. A fault occurs on T1213, and the alarm on Bp11 is missing; a fault occurs on L0204.	Lp0318s(0), Lp1718s(5), CB0318(32), CB1718(35) CB1312(50), CB1213(60)	B18	0.725	[−534, −510]	0.8075	B18 T1213
			L0318	0.6583	$\varnothing$	0.4608
			L1718	0.6583	$\varnothing$	0.4608
			T1213	0.5917	[−20, 20]	0.7142
			B12	0.3958	$\varnothing$	0.2771
			B13	0.3306	$\varnothing$	0.2314

. Since the reasoning processes in these cases are consistent with the above cases, their specific reasoning processes are not provided.

Cases 4 and 5 are based on Case 1, simulating the missing alarm message and rejection of CBs. Although the number of suspected faulty components and spike values of non-faulty components increased, the proposed method still accurately diagnoses the faulty bus.

Cases 6 and 7 are bus faults with missing alarm messages and rejection of the bus PR. If the spike value is the diagnosis basis and its threshold is set to 0.6, B₀₃ will be diagnosed as a non-faulty component in Case 6, and the lines adjacent to B₀₃ will be misdiagnosed as fault components in Case 7. Because the bus PR is related to multiple CBs and remote backup PRs, the missing alarm message and maloperation of the bus PR significantly affect the diagnosis results of the faulty bus and adjacent non-faulty components. In this case, the proposed method can still diagnose a faulty bus without missed diagnosis and misdiagnosis.

In Case 9, the CB refused to operate, the main PR alarm message was lost, and the spike values of L₁₇₁₈ and B₁₈ are 0.5333 and 0.6, respectively, which are close to the threshold. Because the fault times of L₁₇₁₈ and B₁₈ cannot be inferred, L₁₇₁₈ and B₁₈ are independent of the operating PR and CB and are not faulty components. The main PR configuration of the transformer is similar to that of the bus, whereas that of the remote backup PR configuration is similar to that of the line. Therefore, when the CB alarm message is lost (Case 11), the diagnosis results of the suspected faulty components are similar to those in Case 4 (bus fault with a missing CB alarm message). When the CB refuses to operate, and the backup PR is initiated (Case 12), the diagnosis results of the suspected faulty components are similar to those in Case 12 (line fault with CB rejection). Cases 8–12 prove that the proposed method can accurately diagnose faulty lines and transformers in complex fault scenarios.

Compared with the single faults in the above fault cases, the adjacent faulty components in the double faults (Cases 13–16) have a greater impact on the non-faulty components. For example, the confidence degrees of B₁₈ in Case 13, L₁₇₁₈ in Case 14, and L₀₃₁₈ and L₁₇₁₈ in Case 16 are close to or even exceeded the threshold. In Case 16, because the main PR of the transformer is associated with multiple CBs, the loss of the main PR alarm message causes the spike value of the faulty transformer to be lower than 0.6. The diagnosis results of Cases 12–16 are still accurate because of the measured values as the diagnosis basis, which proves that the proposed method still has high accuracy and fault tolerance for double faults.

It is found that the timestamp error of alarm messages (Cases 8 and 15) and adjacent component faults (Cases 13 and 14) affect temporal reasoning. The former is because the wrong time-point constraint cannot satisfy the time-distance constraint. The latter is because the CB time-point constraint cannot satisfy the time-distance constraints for all corresponding PRs. The proposed method can complete temporal reasoning by simultaneously performing reverse reasoning on the CBs and PRs. When the PR or CB in a subnet satisfies the temporal constraint, the temporal reasoning proceeds normally. Moreover, the merged reasoning between subnets can mutually verify whether the temporal reasoning results of the subnets are correct. Therefore, PR/CB maloperation/rejection, timestamp errors, and complex faults have little effect on the temporal reasoning of this method, and the accuracy and fault tolerance of the diagnosis results are not reduced.

The fault diagnosis method based on the ITFRSNP system simultaneously accomplishes temporal and confidence reasoning. The results of temporal and confidence reasoning are quantized using the measurement function, and the results of confidence inference are corrected using temporal reasoning. The proposed method can diagnose faulty components and fault times with high accuracy and fault tolerance in complex fault scenarios, such as single faults, multiple faults, PR/CB rejection, missing alarm messages, and timestamp errors. The fault diagnosis model and reasoning method have good generality and can be applied to the fault diagnosis of buses, lines, and transformers.

4.2. Actual Case Analysis

This study verifies the effectiveness of the proposed fault diagnosis model and diagnosis process with actual fault cases from a city power system in Shandong Province, 2020. The wiring diagram of the power system associated with the fault case is shown in Figure 6. It contains seven substations, nine buses, twenty-one lines, and two transformers.

4.2.1. Case 1

A fault occurs at L_SW. Lp₇₁m and Lp₃₃m operate, and then CB₇₁ and CB₃₃ trip. However, the alarm message for CB₃₃ is lost. The alarm messages are briefly described as Lp₇₁m(0), Lp₃₃m(5) and CB₇₁(25). The fault diagnosis process is as follows:

• CBS = {CB₇₁} can be obtained from the alarm message. Traverse CBS and search for suspected faulty components. There are a subset COM₇₁ = {L_SW, L_XQ, B₃, B₇}, and a total set COMS = {L_SW, L_XQ, B₃, B₇}.
• L_SW is selected for fault diagnosis. The ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$, ${\mathit{\alpha }}_{p(12)}^0$, and ${\mathit{T}}_{p(12)}^0$ of the fault diagnosis model for the CB₇₁and CB₃₃ subnets are as follows.

  $$\{\begin{array}{l}{\alpha }_{p(11)}^0={[0.9833,0.2,0.2,0.9913,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ T_{p(11)}^0={[[25,25],[25,25],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (36)

  $$\{\begin{array}{l}{\alpha }_{\mathrm{p}(12)}^0={[0.2,0.2,0.2,0.9913,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ T_{p(12)}^0={[\varnothing ,\varnothing ,\varnothing ,[5,5],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (37)
• The diagnosis results for the CB₇₁ subnet are α_p10(11) = 0.9873 and T_p10(11) = [−40, −10]. The diagnosis results of the CB₃₃ subnet are α_p10(12) = 0.5567 and T_p10(12) = [−35, −5]. The diagnosis results of L_SW are α₍₁₎ = 0.7915 and T₍₁₎ = [−35, −10], and its measured value η₍₁₎ = 0.8540, which is greater than 0.6. Thus, T₁₂₁₃ is a faulty component.
• Similarly, the measured values of L_XQ, B₃ and B₇ are 0.2363, 0.14, and 0.2085, respectively, and they are non-faulty components.

4.2.2. Case 2

A fault occurs at B₅. Bp₅ operates, and then CB₅₁ trips, but CB₅₃ fails to operate; thus, Lp₉₁s operates and CB₉₁ trips. The alarm messages are briefly described as Bp₅(0), CB₅₁(29), Lp₉₁s(525), and CB₉₁(550). The fault diagnosis process is as follows:

• CBS = {CB₅₁, CB₉₁} can be obtained from the alarm message. Traverse CBS and search for suspected faulty components. There are a subset COM₅₁ = {B₅, T₁, B₄, L_CXI, B_CXII, L_CM, L_SCI, L_SCII}, a subset COM₉₁ = {B₉, L_CL, B₅, T₁}, and a total set COMS = {B₅, T₁}.
• B₅ is selected for fault diagnosis. The ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$, ${\mathit{\alpha }}_{p(12)}^0$, and ${\mathit{T}}_{p(12)}^0$ of the fault diagnosis model for the CB₅₁ and CB₅₂ subnets are as follows.

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(11)}^0={[0.9833,0.2,0.2,0.8564,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(11)}^0={[[29,29],[29,29],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (38)

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(12)}^0={[0.2,0.2,0.75,0.9913,0.2,0.7,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(12)}^0={[\varnothing ,\varnothing ,[550,550],[0,0],\varnothing ,[525,525],\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (39)
• The diagnosis results for the CB₅₁ subnet are α_p10(11) = 0.9873 and T_p10(11) = [−40, −10]. The diagnosis results of the CB₅₂ subnet are α_p10(12) = 0.5567 and T_p10(12) = [−15, 15]. The diagnosis results of B₅ are α₍₁₎ = 0.7915 and T₍₁₎ = [−15, −10], and its measured value η₍₁₎ = 0.8540, which is greater than 0.6. Thus, B₅ is a faulty component.
• Similarly, the measured value of T₁ is 0.2363, and T₁ is a non-faulty component.

4.2.3. Case 3

A fault occurs at T₁. Tp₁m operates, and then CB₄₄ and CB₅₁ trip. However, the timestamp of the CB₅₁ alarm message is incorrect due to transmission delays. The alarm messages are briefly described as Tp₁m(0), CB₄₄(25), and CB₅₁(70). The fault diagnosis process is as follows:

• CBS = {CB₄₄, CB₅₁} can be obtained from the alarm message. Traverse CBS and search for suspected faulty components. There is a subset COM₄₄ = {B₄, T₁, B₅, L_CL}, a subset COM₅₁ = {B₅, T₁, L_CXI, B₄, L_CXII, L_CM, L_SCI, L_SCII}, and a total set COMS = {T₁, B₄, B₅}.
• T₁ is selected for fault diagnosis. The ${\mathit{\alpha }}_{p(11)}^0$, ${\mathit{T}}_{p(11)}^0$, ${\mathit{\alpha }}_{p(12)}^0$, and ${\mathit{T}}_{p(12)}^0$ of the fault diagnosis model for the CB₄₄ and CB₅₁ subnets are as follows.

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(11)}^0={[0.9833,0.2,0.2,0.7756,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(11)}^0={[[25,25],[25,25],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (40)

  $$\{\begin{array}{l}{\mathit{\alpha }}_{p(12)}^0={[0.9833,0.2,0.2,0.7756,0.2,0.2,0,0,0,0]}^{\mathrm{T}}\\ {\mathit{T}}_{p(12)}^0={[[70,70],[70,70],\varnothing ,[0,0],\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ,\varnothing ]}^{\mathrm{T}}\end{array}$$

  (41)
• The diagnosis results for the CB₄₄ subnet are α_p10(11) = 0.8797 and T_p10(11) = [−40, −10]. The diagnosis results of the CB₅₁ subnet are α_p10(12) = 0.8795 and T_p10(12) = [−40, 40]. The diagnosis results of T₁₂₁₃ are α₍₁₎ = 0.8795 and T₍₁₎ = [−40, −10], and its measured value η₍₁₎ = 0.9156, which is greater than 0.6. Thus, T₁ is a faulty component.
• Similarly, the measured values of B₄ and B₅ are 0.1857 and 0.1881, respectively, and they are non-faulty components.

4.3. Comparison Analysis with Other Methods

4.3.1. Case Result Comparisons

The proposed method and other existing methods are used for fault diagnosis in Cases 1–16, and the diagnosis results are shown in

Table 3. Fault diagnosis results of the proposed method and existing methods for cases 1–16.

No.	Suspected Faulty Components		Fault Diagnosis Results								Faulty Components
	[16,21,22]	This Study	[13]	[23]	[17]	[18]	[20]	[21]	[22]	This Study
1	B18	B18	B18	B18	B18	B18	B18	B18	B18	B18	B18
2	L0318, L1718, B18	L0318, L1718, B18	L0318, B18	L0318	L0318, B18	L0318, B18	L0318	L0318, B18	L0318	L0318	L0318
3	-	T1213, B12, B13, L1314, L1013	T1213	-	T1213	-	-	T1213	-	T1213	T1213
4	-	B18, B17, L1718, L1727, L1617	B18	-	B18	B18	B18	B18	B18	B18	B18
5	L0318, B03	B18, L1718	B18	B18	B18	B18	B18	B18	B18	B18	B18
6	B03	B03	B03	-	-	-	-	-	B03	B03	B03
7	B03, L0203, L0304, L0318	B03, L0203, L0304, L0318	B03, L0203, L0304, L0318	B03	B03, L0203, L0304, L0318	B03, L0203, L0304, L0318	B03	B03	B03	B03	B03
8	L0318	L0318	L0318	L0318	L0318	L0318	L0318	-	L0318	L0318	L0318
9	L0318, L1718, B18	L0318, L1718, B18	L0318, B18	-	L0318, B18	L0318, B18	-	B18	L0318	L0318	L0318
10	-	L0318, L1718, B18, B17	L0318, B18, L1718	-	L0318, B18	L0318, B18	-	B18	L0318	L0318	L0318
11	-	T1213, B12, B13, L1314, L1013	T1213	-	T1213	-	-	T1213	-	T1213	T1213
12	T1213, B13, L1314, L1013	T1213, B13, L1314, L1013	T1213, B13, L1314, L1013	-	T1213, B13, L1314, L1013	B13, L1314, L1013	-	T1213	-	T1213	T1213
13	B18, L1718	L0318, L1718, B17, B18	L1718	L1718	L1718	L1718	L1718	L1718	L1718	L0318, L1718	L0318, L1718
14	L0318, L1718, B17, B18	L0318, L1718, B17, B18	L0318, B17, B18	L0318, B17	L0318, B17	L0318, B17	L0318, B17	L0318, B17	L0318, B17	L0318, B17	L0318, B17
15	B14	B03, B14	B14	B14	B14	B14	B14	B14	B14	B03, B14	B03, B14
16	B18, L0318, L1718, T1213	B18, L0318, L1718, T1213, B12, B13	B18, L0318, L1718, T1213	B18	B18, L0318, L1718	B18, L0318, L1718	B18	B18	B18	B18, T1213	B18, T1213

. The methods in [13,17,18,23] do not consider the temporal features of alarm messages, whereas the methods in [20,21,22] consider temporal features.

Existing methods must identify the outage area to reduce suspected faulty components and improve the efficiency of the fault diagnosis. Outage and active areas are separated using tripped CBs. Once the alarm message of boundary CB is lost (Cases 3 and 4), the outage area cannot be determined. Existing methods must diagnose all the components in the power system. In complex faults (Cases 13 and 15), the existing methods only search for the outage area with complete CB alarm messages. They failed to identify the faulty components (L₀₃₁₈ in Case 13 and B₀₃ in Case 15) in the outage area without an alarm message regarding the boundary CBs. Therefore, this study proposes a search method for suspected faulty components by considering the missing alarm messages of boundary CBs. Suspected faulty components corresponding to tripped CBs are searched for to form subsets. These subsets complement and verify each other to avoid missing faulty components in case of missing CB alarms. Moreover, this method does not need to traverse the entire power system topology, reduce the scope of the topology search, or improve the fault diagnosis efficiency.

In [13], the confidence degrees are corrected according to the quantitative relationship between PRs and CBs. The method in [13] can avoid the misdiagnosis problem when the key alarm message is missing (Cases 6 and 16). However, the adjacent components are misdiagnosed when the remote backup PR and its CBs operate. For example, Lp₁₇₁₈s and CB₁₇₁₈ were operational in Case 2. Because they are also the remote backup PR and corresponding CB of B₁₈, the B₁₈ diagnosis result exceeded the threshold in [13], and B₁₈ was misdiagnosed as a faulty component.

The method in [23] has a high diagnosis accuracy when the alarm message is complete. However, this method considers the minimum confidence degree in PRs and CBs as the diagnosis result for suspected faulty components. Therefore, when an alarm message is missing (Cases 10–12), this method cannot diagnose the fault component with the missing alarm message, and the fault tolerance is poor.

The methods in [17,18] can diagnose faulty components in most cases. However, misdiagnosis and missed diagnosis occur during remote backup PR operation (Cases 10 and 16) and missing alarm messages (Cases 6 and 16), respectively. The cause of the misdiagnosis is the same as that in [13], without considering the remote backup PR and CB of adjacent components. The main reason for the missed diagnosis is that the bus PR and transformer PR are associated with multiple CBs simultaneously, and the missing alarm message of the bus PR and transformer PR will seriously affect the fault diagnosis results of the faulty bus and transformer.

The accuracy of fault diagnosis is improved using temporal features and reasoning in [20,21]. However, only the PR and CB confidence degrees that do not satisfy the temporal constraints are punished, and there are no corresponding compensation measures. When the alarm message is missing, or the timestamp is incorrect, there is a problem of missed diagnosis, and the fault tolerance must be improved [20,21].

In reference [20], alarm messages are categorized as valid or invalid based on their relationship and temporal constraints with PRs and CBs, with valid alarms being assigned confidence degrees. However, the method in [20] classifies the alarm message of CB₀₃₁₈ as invalid (with a confidence degree of 0.2) in Case 9, because CB₀₃₁₈ has no corresponding PR alarm message. The confidence degree of L₀₃₁₈ is 0.3836 (lower than the threshold in [20]), and L₀₃₁₈ is diagnosed as a non-faulty component. The method in [21] corrects the confidence degrees of PRs and CBs that do not meet the temporal constraint in the sequence of “Fault time—PR operation time—CB trip time”. If the timestamp of a PR alarm message is incorrect, this method determines that the PR and the corresponding CB do not meet their temporal constraints, resulting in a missed diagnosis. For example, in Case 8, because the timestamp of Lp₁₈₀₃p is incorrect, Lp₁₈₀₃p and CB₁₈₀₃ do not meet the temporal constraints, their confidence degrees are corrected to 0, and L₀₃₁₈ is diagnosed as a non-faulty component. In addition, the temporal reasoning in [21] defaults the fault time to zero, but the alarm message does not include the fault time; thus, determining the fault time is a problem that this method must solve.

Combining the above case comparisons, the accuracy and fault tolerance comparison of the proposed method with other methods is shown in

Table 4. Comparison of the proposed method with the existing methods.

	[13]	[23]	[17]	[18]	[20]	[21]	[22]	This Study
Consideration of temporal features	No	No	No	No	Yes	Yes	Yes	Yes
Consideration of neighboring components configured with the backup protection	No	Yes	No	No	Yes	Yes	Yes	Yes
Consideration of incorrect operations of the PRs and CBs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Consideration of missing alarms of the boundary CBs	No	No	No	No	No	No	No	Yes
Consideration of the uncertainty of alarm messages	Yes	No	Not considering the main protection of the bus and transformer	Not considering the main protection of the bus and transformer	Not considering the main protection of the bus and transformer	Not considering the main protection of the bus and transformer	Yes	Yes
Consideration of timing errors	No	No	Yes	No	Yes	No	Yes	No
Accuracy and fault tolerance	General	Poor	General	General	Good	Good	Excellent	Excellent

. The proposed method deduces the fault time by the operation times of PRs and CBs, adjusting the confidence degrees of the suspected faulty components by using a measure function and fault time. Furthermore, this study addresses the misdiagnosis problem in the case of neighboring components configured with the same backup protection and incorrect operations of the PRs and CBs. It also maintains diagnostic precision even when alarm messages of the boundary CB or main PRs of the bus and transformer are absent. Temporal errors do not affect the temporal reasoning of PRs and CBs. In contrast, the temporal results of different CB subnets are cross-verified, enhancing the reliability of temporal reasoning and adjustments made by the measure function. The comparison of fault results demonstrates that this study makes full use of temporal features and reasoning in complex fault scenarios, effectively enhancing the accuracy and fault tolerance of fault diagnosis.

4.3.2. Performance Comparison

Table 5. Performance comparison between the proposed method and existing methods.

Method	Fault Diagnosis Models	Search Method for Suspected Faulty Components	Calculation Tasks	Calculation Methods and Calculation Amounts	Performance
This study	No need to build separate models No need to adjust diagnosis models for topology changes	Search for suspected faulty components configured the tripped CBs	Correlation reasoning between confidence degrees and temporal constraints	Matrix operation with a calculation amount of μν	Simple reasoning and calculation principles Strong generality and adaptability of the fault diagnosis model
[24]	Construction of the objective function based on the alarm message	Search the outage area to identify suspected faulty components	Fault hypothesis with the minimum loss value	Look-up table and optimization algorithm with a calculation amount of $2^{\mu }v$	Relatively slow convergence speed High calculation complexity
[10]	Individual construction of fault diagnosis models for suspected faulty components. Need for reconstruction of models based on topology changes for fault diagnosis.		Temporal matching of alarm messages Fuzzy reasoning of confidence degrees	Event correlation, look-up table, and matrix operation with a calculation amount of μ2ν	Independence of confidence reasoning and temporal reasoning processes High complexity in the alarm message association and temporal matching process
[20]			Temporal matching and classification of alarm messages Fuzzy reasoning of confidence degrees	Event correlation, look-up table, and matrix operation with a calculation amount of 2μν
[21]			Temporal matching of alarm messages Fuzzy reasoning of confidence degrees	Event correlation, look-up table, and matrix operation with a calculation amount of 2μν
[22]			Correlation reasoning between confidence degrees and temporal constraints	Matrix operation with a calculation amount of μν	Relatively complex reasoning and calculation process in fault diagnosis High efficiency in diagnosis Lack of consideration for transformer fault diagnosis

Note: μ denotes the number of received alarm messages, and ν denotes the number of suspected faulty components.

compares the performance of the proposed method with the existing methods in [10,20,21,22,24] that consider temporal features.

In [24], temporal constraints are added to the loss function of the analytical model to solve the multi-solution problem. However, more variables increase the number of calculations and reduce the convergence speed of the optimization algorithm. The action event coding and the temporal rule bases of PRs and CBs are required in [10,20,21]. After a fault occurs, the alarm message is matched with the rule base to determine an effective alarm message and modify its confidence degree. The temporal matching and reasoning processes of the fault diagnosis model are independent of each other [10,20,21]. The fault diagnosis process is complex, and the number of calculations is large, which restricts online applications. The rTSSNPS system also implements correlation reasoning between confidence degrees and temporal constraints; however, its reasoning method is more complex and does not realize fault diagnosis of transformer components [22].

Compared to the above methods, the ITFRSNP system and its reasoning method do not require the construction of corresponding event codes and temporal rule libraries for each suspected fault component. The proposed method avoids complex temporal matching and confidence correction processes in temporal reasoning. Simultaneously, the fault diagnosis principle based on the general fault diagnosis model simplifies the association reasoning steps of the confidence degrees and temporal constraints, thus reducing the complexity of the reasoning algorithm and improving fault diagnosis efficiency.

In addition, the above methods do not fully consider the impact of the missing boundary CB alarm message on fault diagnosis results and efficiency. Once the boundary CB alarm message is lost, it is not possible to form an outage area when diagnosing a single fault. It is necessary to diagnose all the components in the power grid with a significant amount of computation and reduced fault efficiency. Moreover, when diagnosing multiple faults, only the outage area with complete boundary CB alarm messages can be correctly identified, and faulty components located in the unrecognized outage area are missed, thereby reducing the fault diagnosis accuracy.

The proposed method suspected faulty components within the isolation range of tripped CBs based on the operation logic of the PRs and CBs. Compared with other methods, the search algorithm for suspected fault components in this study does not need to traverse the whole grid topology, thus enhancing search efficiency. Meanwhile, it can narrow down the diagnosis scope of a single fault without missing faulty components in multiple faults in the case of missing alarm messages of boundary CBs. In the case of rejections and missing/false alarm messages, this study combines confidence degrees and temporal constraints for fault diagnosis to reduce misdiagnoses and missed diagnoses.

Owing to the different types and topologies of the components, it is necessary for [10,20,21,22,24] to build the corresponding fault diagnosis model for each component separately. The structure and parameters of the fault diagnosis model are reconstructed or adjusted according to topology changes. In contrast to the above methods, this study can directly use a general fault diagnosis model to diagnose the subnets of suspected faulty components and fuse the diagnosis results of each subnet to diagnose the faulty components. For example, as shown in Figure 7, L₀₃₀₄ is out of operation because of maintenance, the number of adjacent lines of B₀₃ and B₀₄ is reduced, and the number of remote backup PRs of L₀₂₀₃ and L₀₃₁₈ is reduced. If a fault occurs in B₀₃, the proposed method diagnoses the CB₀₃₀₂ and CB₀₃₁₈ subnets of B₀₃. If L₀₂₀₃ (L₀₃₁₈) fails, the fault diagnosis process in this study automatically searches for remote backup PRs and CBs of L₀₂₀₃ (L₀₃₁₈) and their alarm messages. The input of the general fault diagnosis model of the corresponding subnet is then adjusted according to (21).

Taking B₀₃ and L₀₃₁₈ in Figure 7 as examples, the fault diagnosis models are constructed using the methods in [20,21,22] and compared with the proposed method. The numbers of proposition neurons α_p and rule neurons α_r in the fault diagnosis models are listed in

Table 6. The number of neurons in the fault diagnosis models.

Components		This Study		[20]		[21]		[22]
		αp	αr	αp	αr	αp	αr	αp	αr
Before L0304 outage	B03	10	7	21	9	25	10	25	16
	L0318	10	7	20	8	35	11	23	16
After L0304 outage	B03	10	7	14	6	16	7	17	11
	L0318	10	7	18	8	29	9	21	15

. Hierarchical modeling is employed in [20,21,22], which can flexibly adjust the hierarchical structure of the fault model according to topology changes; however, more neurons are needed to link the sub-models of different hierarchies for fault diagnosis. As more neurons require more synapses for association, the size of the fault diagnosis model increases, and the structure becomes more complex. After the L₀₃₀₄ outage, the remote backup protection of adjacent components is reduced, and the fault diagnosis models of the above methods must delete the corresponding neurons. If L₀₃₀₄ restarts operation, it is necessary to add the corresponding neurons in the fault diagnosis models. The structure, input matrix, and output matrix of the fault diagnosis model are changed. This illustrates that topology changes have an impact on the fault diagnosis models of the existing methods; in particular, the changes are most pronounced for the fault diagnosis models of the bus. Therefore, existing methods need to adjust the parameters of the fault diagnosis model according to topology changes to ensure the accuracy of the fault diagnosis.

This study defines a CB subnet and proposes a general fault diagnosis model. The PRs and CBs of faulty components are decomposed into multiple CB subnets with the same relay protection configurations, and the general fault diagnosis model is invoked through the diagnosis process to diagnose the CB subnets. This study does not need to build a diagnosis model for a single component online, and the general fault diagnosis model has a simple structure that can be applied to the fault diagnosis of various components. Moreover, the general fault diagnosis model and its reasoning process can automatically adapt to topological changes and simplify the fault diagnosis modeling process.

5. Conclusions

This study proposes a fault diagnosis method based on the ITFRSNP system and a fault diagnosis principle based on a general fault diagnosis model. First, this study proposes the ITFRSNP system, which contains new rule neurons and reasoning methods to realize correlation reasoning of confidence degrees and temporal constraints. This provides a theoretical basis for a general fault diagnosis model. Then, the concept of the CB subnet is defined according to the configuration of the CBs and corresponding PRs. A general fault diagnosis model based on the ITFRSNP system is constructed for the CB subnet. The subnets of the suspected faulty components are diagnosed using the general fault diagnosis model. The diagnosis results of the subnets are fused to determine faulty components. This study also proposes a new search method for suspected faulty components, which avoids the expanded diagnosis scope and incorrect diagnosis. Finally, the effectiveness and reliability of this method are verified and analyzed using a case simulation.

The proposed method can diagnose fault components and time in complex fault scenarios with high accuracy and tolerance. Additionally, the general fault diagnosis model effectively accommodates a variety of components and topological alterations, demonstrating broad applicability and flexibility while streamlining the modeling process. The search algorithm for suspected faulty components reduces the scope of the topology search without missing the faulty components. Owing to its reliability, simplicity, and computational efficiency, the method fulfills the demands of online fault diagnosis, showcasing promising application potential.

The proposed method encounters challenges in diagnosing faults in complex wiring connections, such as double bus and 3/2 wiring connections. In double bus connections, CBs are interconnected between buses, whereas in 3/2 wiring connections, CBs are linked directly. The relay protection configurations in these complex wiring connections are different from those in the CB subnet in this study. Therefore, the proposed method is not applicable to the fault diagnosis of double bus and 3/2 wiring connections. Future research will concentrate on analyzing relay protection configuration and operation logic within complex wiring connections, developing a topology search method, a general fault diagnosis model and a fault diagnosis process to realize the fault diagnosis of complex wiring.