INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 9(6):786-799, December, 2014.

Application of Fuzzy Reasoning Spiking Neural P Systems to
Fault Diagnosis

T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

Tao Wang, Gexiang Zhang*, Haina Rong
School of Electrical Engineering, Southwest Jiaotong University
Chengdu, 610031, China
wangtaocdu@gmail.com, zhgxdylan@126.com
ronghaina@126.com
*Corresponding author: wangtaocdu@gmail.com

Mario J. Pérez-Jiménez
Research Group on Natural Computing
Department of Computer Science and Artificial Intelligence
University of Sevilla, Sevilla, 41012, Spain
marper@us.es

Abstract: This paper discusses the application of fuzzy reasoning spiking neural
P systems with trapezoidal fuzzy numbers (tFRSN P systems) to fault diagnosis of
power systems, where a matrix-based fuzzy reasoning algorithm based on the dynamic
firing mechanism of neurons is used to develop the inference ability of tFRSN P
systems from classical reasoning to fuzzy reasoning. Some case studies show the
effectiveness of the presented method. We also briefly draw comparisons between the
presented method and several main fault diagnosis approaches from the perspectives
of knowledge representation and inference process.
Keywords: fuzzy reasoning spiking neural P system with trapezoidal fuzzy number,
fuzzy reasoning, fault diagnosis, trapezoidal fuzzy number, linguistic term.

1 Introduction

Membrane computing, introduced by Gh. Păun in [1], is an attractive research field of com-
puter science aiming at abstracting computing models, called membrane systems or P systems,
from the structures and functioning of living cells, as well as from the way the cells are organized
in tissues or higher order structures. In recent years, much attention is paid to the spiking neural
P systems (SN P systems, for short), an important class of P systems introduced in [2] and inves-
tigated in a series of papers (see [3]- [12]), which can be described as a directed graph. An SN P
system is a kind of distributed and parallel computing model inspired by the neurophysiological
behavior of neurons sending electrical impulses (spikes) along axons from presynaptic neurons to
postsynaptic neurons. The features of SN P systems, such as inherent parallelism, understand-
ability, dynamics, synchronization/asychronization, non-linearity and nondeterminism [3], [4],
are suitable for solving various engineering problems.

Until now, only a few investigations have focused on the use of SN P systems to solve
engineering problems. In [3], a fuzzy reasoning spiking neural P system with real numbers
(rFRSN P system) was presented to perform diagnosis knowledge representation and reasoning.
In [13], an rFRSN P system was used for fault diagnosis of power systems and three examples
were used to verify its effectiveness. The studies in [3, 13], dealing with the fault diagnosis
problem, used certainty factors and truth degree values, which are described by real numbers
obtained from the frequency of occurrences in historical data, It is known how difficult it is to
obtain and process real-time statistical data from power network data and the knowledge of
dispatchers and experts in electrical power systems as they usually contain linguistic terms with

Copyright © 2006-2014 by CCC Publications



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 787

some degree of uncertainty. So, using various models, including rFRSN P systems to solve fault
diagnosis problems with a certain degree of uncertainty, represents a way to tackle this difficult
problem.

This paper discusses the extended version of rFRSN P systems, i.e., fuzzy reasoning spiking
neural P systems with trapezoidal fuzzy numbers (tFRSN P systems), and its application to
fault diagnosis of power systems. To adapt tFRSN P systems to solve fault diagnosis problems,
a matrix-based fuzzy reasoning algorithm (MBFRA) is used inspired by the dynamic firing
mechanism of neurons. Given initial pulse values of all input neurons of a tFRSN P system,
MBFRA can perform fuzzy inference to obtain the pulse values contained in other neurons and
export reasoning results represented by trapezoidal fuzzy numbers. To make MBFRA suitable
for multiple faults diagnosis of power systems, a defuzzification method is applied for processing
the reasoning results in order to obtain crisp numbers corresponding to them. Some case studies
show the effectiveness of the presented method. We also briefly draw comparisons between
tFRSN P systems and several other fault diagnosis approaches.

The remainder of this paper is organized as follows. Section 2 introduces concepts and
notations used in this work. Section 3 provides the definition of tFRSN P systems and MBFRA.
Section 4 discusses the application of tFRSN P systems to fault diagnosis of power systems.
Discussions on several fault diagnosis methods are made in Section 5. Conclusions are finally
drawn in section 6.

2 Preliminaries

A trapezoidal fuzzy number can be characterized as a 4-tuple of real numbers T̃f = (a, b, c, d), a <
b < c < d, shown in Fig. 1, where a and d represents the left hand and right hand width of the
trapezoidal distribution, (b, c) denotes the interval in which the membership value is equal to 1
and H

T̃f
(x) represents the membership function of T̃f defined as follows [14]:

H
T̃f
(x) =




0, x ≤ a
x−a
b−a , a < x ≤ b
1, b < x ≤ c

d−x
d−c , c < x ≤ d
0, x > d

(1)

Let à and B̃ be two trapezoidal fuzzy numbers, à = (a1, b1, c1, d1) and B̃ = (a2, b2, c2, d2).
The arithmetic operations of à and B̃ are defined as follows [15]:

1. Addition ⊕: Ã⊕ B̃ = (a1, b1, c1, d1)⊕ (a2, b2, c2, d2)=(a1 + a2, b1 + b2, c1 + c2, d1 + d2);

2. Subtraction ⊖: Ã⊖ B̃ = (a1, b1, c1, d1)⊖ (a2, b2, c2, d2)=(a1 −a2, b1 − b2, c1 − c2, d1 −d2);

3. Multiplication ⊗: Ã⊗B̃ = (a1, b1, c1, d1)⊗(a2, b2, c2, d2)=(a1×a2, b1×b2, c1×c2, d1×d2);

4. Division ⊘: Ã⊘ B̃ = (a1, b1, c1, d1)⊘ (a2, b2, c2, d2)=(a1/a2, b1/b2, c1/c2, d1/d2).

We define four logic operations, where à and B̃ are trapezoidal fuzzy numbers, and a, b are
real numbers:

1. Minimum operator ∧: a∧ b = min(a, b);

2. Maximum operator ∨: a∨ b = max(a, b);



788 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

Figure 1: A trapezoidal fuzzy number.

3. and ∧○: Ã ∧○ B̃ = (a1, b1, c1, d1) ∧○ (a2, b2, c2, d2) =((a1∧a2), (b1∧b2), (c1∧c2), (d1∧d2));

4. or ∨○: Ã ∨○ B̃ = (a1, b1, c1, d1) ∨○ (a2, b2, c2, d2)=((a1 ∨a2), (b1 ∨ b2), (c1 ∨c2), (d1 ∨d2)).

We define a scalar multiplication operation as follows:

Scalar Multiplication: x · Ã = x · (a1, b1, c1, d1) = (x ·a1, x · b1, x · c1, x ·d1),
where A is a trapezoidal fuzzy number and x is a real number:

The defuzzification method in [16] is chosen to obtain a crisp number tf associated with a
trapezoidal fuzzy number T̃f , it is shown in (2), where e and g are the extreme values of the
whole fuzzy set range. In this study, e and g are equal to 0 and 1, respectively.

tf =
(d−e) + (c−e)

((d−e) + (c−e))− ((a−g) + (b−g))
(2)

3 tFRSN P systems

A tFRSN P system of m ≥ 1 is a construct Π = (O, σ1, . . . , σm, syn, in, out), where:

(1) O = {a} is a singleton alphabet (a is called spike);

(2) σ1, . . . , σm are neurons, of the form σi = (θi, ci, ri), 1 ≤ i ≤ m, where:

(a) θi is a trapezoidal fuzzy number in [0,1] representing the potential value of spikes (i.e.
value of electrical impulses) contained in neuron σi;

(b) ci is a trapezoidal fuzzy number in [0,1] representing the fuzzy truth value correspond-
ing to neuron σi;

(c) ri represents a firing (spiking) rule contained in neuron σi with the form E/aθ → aβ,
where E is the firing condition and its form will be specified below, θ and β are
trapezoidal fuzzy numbers in [0,1].

(3) syn ⊆{1, 2, . . . , m}×{1, 2, . . . , m} with i ̸= j for all (i, j) ∈ syn, 1 ≤ i, j ≤ m, is a directed
graph of synapses between the linked neurons;

(4) in, out ⊆ {1, 2, . . . , m} indicate the input neuron set and the output neuron set of Π,
respectively.



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 789

In tFRSN P systems, the definition of neurons and pulse values can be extended. Specifically,
in tFRSN P systems, the neurons are extended to four types, i.e., proposition neurons and three
kinds of rule neurons: general, and and or, and the pulse value contained in each neuron is
no longer the number of spikes represented by a real value, but a trapezoidal fuzzy number in
[0, 1], which can be interpreted as the potential value of spikes contained in neuron σi. It is
worth pointing out that the number of spikes in each neuron is determined by the problem to be
solved and the pulse value contained in each neuron is different. If neuron σi contains no spike,
then θi = 0; otherwise, if neuron σi contains only one spike, then θi equals to the pulse value of
this spike; in any other case, θi equals to the result of a operation on all pulse values received
from its presynaptic neurons. For different types of neurons, the operations for pulse values are
different. For proposition neurons and and rule neurons, they use operation ∧○ to handle all
the pulse values received from their presynaptic neurons while or rule neurons use operation
∨○, where symbols ∧○ and ∨○ represent the and and or operators of trapezoidal fuzzy numbers,
respectively. The firing condition E = as means that the spiking rule, E/aθ → aβ, contained in
neuron σi, can be applied if and only if neuron σi contains at least s spikes, otherwise, the firing
rule cannot be applied. More details about tFRSN P systems can be found to the preliminary
work [9].

Fuzzy production rules consist of five types:
Type 1 : Ri(ci) : pj(θj) → pk(θk); θk = θj ⊗ ci.
Type 2 : Ri(ci) : p1(θ1) ∧○ . . . ∧○pk−1(θk−1) → pk(θk); θk = (θ1 ∧○ . . . ∧○θk−1)⊗ ci.
Type 3 : Ri(ci) : p1(θ1) → p2(θ2) ∧○ . . . ∧○pk(θk); θ2 = . . . θk = θ1 ⊗ ci.
Type 4 : Ri(ci) : p1(θ1) ∨○ . . . ∨○pk−1(θk−1) → pk(θk); θk = (θ1 ∨○ . . . ∨○θk−1)⊗ ci.
Type 5 : Ri(ci) : p1(θ1) → p2(θ2) ∨○ . . . ∨○pk(θk).

where Ri represents the ith fuzzy production rule; ci is the certainty factor of rule Ri; pi is a
proposition appearing in the antecedent or consequence part of a rule, 1 ≤ i ≤ k (k is the number
of propositions in a rule-based system); pj in Type 1 represents the jth proposition, 1 ≤ j ≤ k−1;
θi represents the fuzzy truth value corresponding to the ith proposition [15]. ci and θi are
trapezoidal fuzzy numbers defined in the universe of discourse [0, 1]. The causality between a
fault on a faulty section in a power system and the status information about protective relays and
circuit breakers (CBs) of this section can be described by the aforementioned fuzzy production
rules. A simplified transmission network shown in Fig. 2 is used to illustrate the notations in a
fuzzy production rule. According to the protection principle, if there is fault on transmission L,
then its main protective relays, i.e., MLR1 and MLR2 and their corresponding CBs, i.e., CB1
and CB2, will operate to protect L, which can be backward described by a fuzzy production rule:
R(1, 1, 1, 1) : MLR1 operates (0.975, 0.98, 1, 1) ∧○ MLR2 operates (0.975, 0.98, 1, 1) ∧○
CB1 trips (0.975, 0.98, 1, 1) ∧○ CB2 trips (0.975, 0.98, 1, 1) → L has a fault. The certainty
factor of this rule is (1, 1, 1, 1) which represents the contribution of this rule to the final diagnosis
result. In this rule, there are four propositions MLR1 operates, MLR2 operates, CB1 trips
and CB2 trips, and they have an equal fuzzy truth value (0.975, 0.98, 1, 1) representing the
contributions of the propositions to the result L has a fault.

Because rules Type 5 are unsuitable for diagnosis, they are not further described in Section
3. tFRSN P system models for rules Type 1 to Type 4 are shown in Fig. 3.

To adapt tFRSN P systems to solve fault diagnosis problems, we describe MBFRA in the
following description.

Given initial truth values of propositions corresponding to all input neurons in an tFRSN P
system, MBFRA can perform fuzzy reasoning to obtain the fuzzy truth values of other neurons
with unknown pulse values and output reasoning results. Let us assume that the tFRSN P
system contains l proposition neurons and n rule neurons, each of which may be general, and or
or rule neurons, m = l + n, where m is the number of all the neurons in this system.



790 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

Figure 2: A simplified transmission network.

Figure 3: tFRSN P system models for fuzzy production rules. (a) Type 1 ; (b) Type 2 ; (c) Type
3 ; (d) Type 4.

In order to clearly present the reasoning algorithm, we first introduce some parameter vectors
and matrices as follows.

1) θ = (θ1, θ2, . . . , θl)T is a fuzzy truth value vector of the l proposition neurons, where θi
represents the pulse value contained in the ith proposition neuron, 1 ≤ i ≤ l, and is expressed by
a trapezoidal fuzzy number in [0, 1]. If there is not any spike contained in a proposition neuron,
its pulse value is “unknown” or (0, 0, 0, 0).

2) δ = (δ1, δ2, . . . , δn)T is a fuzzy truth value vector of the rule neurons, where δj represents
the pulse value contained in the jth rule neuron, 1 ≤ j ≤ n, and it is expressed by a trapezoidal
fuzzy number [0, 1]. If there is not any spike contained in a rule neuron, its pulse value is
“unknown” or (0, 0, 0, 0).

3) C = diag(c1, c2, . . . , cn) is a diagonal matrix, where cj is the certainty factor of the jth
fuzzy production rule, 1 ≤ j ≤ n, and it is expressed by a trapezoidal fuzzy number.

4) D1 = (dij)l×n is a synaptic matrix representing the direct connection between proposition
neurons and general rule neurons. If there is a directed arc (synapse) from the proposition neuron
σi to the general rule neuron σj, then dij = 1, otherwise, dij = 0.



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 791

5) D2 = (dij)l×n is a synaptic matrix representing the direct connection between proposition
neurons and and rule neurons. If there is a directed arc (synapse) from the proposition neuron
σi to the and rule neuron σj, then dij = 1, otherwise, dij = 0.

6) D3 = (dij)l×n is a synaptic matrix representing the direct connection between proposition
neurons and or rule neurons. If there is a directed arc (synapse) from the proposition neuron σi
to the or rule neuron σj, then dij = 1, otherwise, dij = 0.

7) E = (eji)n×l is a synaptic matrix representing the direct connection between rule neurons
and proposition rule neurons. If there is a directed arc (synapse) from the rule neuron σj to the
proposition neuron σi, then eji = 1, otherwise, eji = 0.

Subsequently, we introduce some multiplication operations as follows.
1) ◦○: C ◦○ δ = (c1 ⊗ δ1, c2 ⊗ δ2, . . . , cn ⊗ δn)T ; DT ◦○ θ = (d̄1, d̄2, . . . , d̄n)T , where d̄j

= d1jθ1 +d2jθ2 + . . . +dljθl, j = 1, 2, . . . , n.
2) ⊙: DT ⊙ θ = (d̄1, d̄2, . . . , d̄n)T , where d̄j = d1jθ1 ∧○ d2jθ2 ∧○ . . . ∧○ dljθl, j = 1, 2, . . . , n.
3) ∗○: ET ∗○ δ = (ē1, ē2, . . . , ēl)T , where ēi = e1iδ1 ∨○ e2iδ2 ∨○ . . . ∨○eniδn, i = 1, 2, . . . , l.
Next, we list the pseudocode of MBFRA.

Algorithm MBFRA
Require: D1, D2,D3, E, C, θ0, δ0
1: Set the termination condition 0 = (unknown, unknown, . . . , unknown)Tn
2: Let t = 0, where t represents the reasoning step
3: while δt ̸= 0 do
4: for each input neuron (t = 0) or each proposition neuron (t > 0) do
5: if the firing condition E = as is satisfied then
6: the neuron fires and computes the fuzzy truth value vector δt+1 via δt+1=(DT1 ◦○ θt)⊕

(DT2 ⊙θt)⊕ (D
T
3 ∗○ θt)

7: if there is a postsynaptic rule neuron then
8: the neuron transmits a spike to the next rule neuron
9: else

10: just accumulate the value in the neuron
11: end if
12: end if
13: end for
14: for each rule neuron do
15: if the firing condition E = as is satisfied then
16: the rule neuron fires and computes the fuzzy truth value vector θt+1 via θt+1= ET ∗○

(C ◦○ δt+1) and transmits a spike to the next proposition neuron
17: end if
18: t = t + 1
19: end for
20: end while
Ensure: θt, which represents the final states of pulse values contained in proposition neurons.

4 Application Examples and Results

In this section, a power system with 14-buses, chosen from [17] and as shown in Fig. 4,
is applied as an example to describe how to use tFRSN P systems with MBFRA to solve a
fault diagnosis problem. The system contains 34 system sections, including 14 buses and 20
transmission lines. The buses are marked as Bpq and the transmission lines are represented



792 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

Figure 4: The power system with 14 buses.

as Lpquv, where 0 ≤ p, q, u, v ≤ 9. The protection system of the 14-bus system contains 174
protective devices consisting of 40 circuit breakers (CBs), 40 main transmission line relays, 40
first backup transmission line relays, 40 second backup transmission line relays and 14 bus relays.
A local part, which is composed of a transmission line L1314, its adjoining two buses, B13 and
B14, and its adjoining three transmission lines, L1213, L0613 and L0914, of the protection system
is given to describe its structure and symbols of protection devices. The local system is shown
in Fig. 5. The operational rules of the protective devices are descried as follows [17].

The main transmission line relay MLR1314 protects the entire line L1314 and it will operate
to trip its associated circuit breaker (CB), i.e., CB1314, to clear a fault on the line L1314. The bus
relay BR13 protects the bus B13 and it will operate to trip the three CBs, i.e., CB1312, CB1306
and CB1314, if there is a fault on the bus B13. The first backup transmission line relay BLR1314
is a local backup of the relay MLR1314 and has the same protection zone as MLR1314. The relay
BLR1314 will operate to trip CB1314 to clear a fault if the fault clearance by the relay MLR1314
fails. Secondary backup transmission line relays SLR1213 and SLR0613 are the remote backups
of the relays MLR1314 and BLR1314. They will operate to trip their corresponding CBs, i.e.,
CB1213 and CB0613, respectively, to clear a fault if the fault clearance by both MLR1314 and
BLR1314 fails. The relays SLR1213 and SLR0613 are also two remote backups of the relay BR13
and they will operate to trip CBs, i.e., CB1213 and CB0613, respectively, to clear a fault if the
fault clearance by the relay BR13 fails. The functions of the four relays, MLR1413, BLR1413,
SLR0914 and BR14, and three CBs, CB1413, CB1409 and CB0914, in the process of protecting
the line L1314 and the bus B14 are similar and the protection systems for other sections in this
14-bus power system have the same protection rules, so it is not necessary to repeatedly describe



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 793

Figure 5: A local part of the protection system of the 14-bus power system.

their operation rules.
The protection rules described above show that when a fault occurs on a certain section of a

power system, protection devices will reach certain statuses to protect the section. Meanwhile,
the relay trip signals and CBs status signals, used as inputs of fault diagnosis models of sections,
can be obtained from remote terminal units (RTUs) of supervisory control and data acquisition
(SCADA) systems. The diagnostic strategy in this study is to build one tFRSN P system
diagnosis model for each candidate fault section of a power system and each model performs
MBFRA by using SCADA data, i.e., relay trip signals and CBs status signals, to get a trapezoidal
fuzzy number which represents the fault confidence level of this section. In a single fault case,
the section with the highest fault confidence level is the faulty section. In multiple faults cases,
several sections with fault confidence levels which are greater than a threshold, which is set as
real number 0.5 in this study, are regarded as faulty sections. Thus, to obtain real numbers for
easily comparing the fault confidence levels with the threshold, a defuzzification method shown
in (2) is used to process the reasoning results represented by trapezoidal fuzzy numbers. In
addition, the fault confidence levels of faulty sections in multiple faults cases are ranked from
high to low to help operators to decide a repair order of the sections.

Fig. 6 and Fig. 7 show the tFRSN P system diagnosis models for L1314 and B13, respectively.
It is worth noting that there are several assistant arcs (synapsises) with different arrow endings
in the figures. For illustration purposes, we take arcs, from σ2 to σ25 and from σ2 to σ26,
as examples. The meanings of the two arcs are that if CB1314 opens, the operation of its
corresponding second backup protective devices, including relays (SLR0613 and SLR1213) and
CBs (CB0613 and CB1213), is invalid and then the values of these relays and CBs are set as
(0, 0, 0, 0); otherwise, the operation of the second backup protective devices is valid. In what
follows we take transmission line L1314 as an example to show the fuzzy reasoning process of
MBFRA based on tFRSN P systems.

Case 1: A single fault. Transmission line L1314 has a fault.
Operated relays: MLR1314, MLR1413, BLR1314. Tripped CBs: CB1314, CB1413.
A tFRSN P system for L1314 is Π1 and its corresponding tFRSN P system diagnosis model

is shown in Fig. 6.

Π1 = (O, σ1, σ2, . . . , σ36, syn, in, out)



794 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

Figure 6: Fault diagnosis model of transmission line L1314 based on a tFRSN P system.

Figure 7: Fault diagnosis model of bus B13 based on a tFRSN P system.



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 795

Table 1: Linguistic terms and their corresponding trapezoidal fuzzy numbers
Linguistic Terms Trapezoidal Fuzzy Numbers
absolutely-false (AF) (0, 0, 0, 0)
very-low (VL) (0, 0, 0.02, 0.07)
low (L) (0.04, 0.1, 0.18, 0.23)
medium-low (ML) (0.17, 0.22, 0.36, 0.42)
medium (M) (0.32, 0.41, 0.58, 0.65)
medium-high (MH) (0.58, 0.63, 0.80, 0.86)
high (H) (0.72, 0.78, 0.92, 0.97)
very-high (VH) (0.975, 0.98, 1, 1)
absolutely-high (AH) (1, 1, 1, 1)

where
1) O = {a} is the singleton alphabet (a is called spike).
2) σ1, . . . , σ22 are proposition neurons corresponding to the propositions with fuzzy truth

values θ1, . . . , θ22; that is, l = 22.
3) σ23, . . . , σ36 are rule neurons, where σ23, . . . , σ29 are and rule neurons, σ30 and σ36 are or

rule neurons and σ31, . . . , σ35 are general rule neurons; that is, n = 14.
4) syn = {(1, 23) , (2, 23), (2, 24), (2, 25), (2, 26), (3, 24), (4, 25), (5, 25), (6, 26), (7, 26),

(8, 27), (9, 27), (9, 28), (9, 29), (10, 28),(11, 29),(12, 29),(13, 30),(14, 30),(15, 31), (16, 32), (17, 33),
(18, 34), (19, 35), (20, 36), (21, 36), (23, 13), (24, 14), (25, 15), (26, 16), (27, 17),(28, 18),(29, 19),(30,
20), (31, 20), (32, 20), (33, 21), (34, 21), (35, 21), (36, 22)}.

5) in = {σ1, σ2, . . . , σ12}, out = {σ22}.
The knowledge of dispatchers in power systems may contain linguistic terms and the sta-

tuses of devices may have a certain degree of uncertainty. Table 1 shows an example of linguistic
terms and their corresponding trapezoidal fuzzy numbers. In the tFRSN P system Π1, input neu-
rons σ1, . . . , σ12 are assigned as the empirical values V H, V H, H, AF, AF, AF, AF, V H, V H, L,
AF, AF , respectively. Certainty factors corresponding to rule neurons σ23, . . . , σ36 are given
values V H, V H, V H, H, V H, V H, H, V H, V H, V H, V H, V H, V H, V H, respectively.

According to Table 1, we obtain the trapezoidal fuzzy numbers θ0 and δ0. In order to
succinctly describe the matrices, let us denote Or = (x1, . . . , xr)T , where xi = (0, 0, 0, 0), 1 ≤
i ≤ r.

θ0 =




(0.975, 0.98, 1, 1)

(0.975, 0.98, 1, 1)

(0.72, 0.78, 0.92, 0.97)

O4
(0.975, 0.98, 1, 1)

(0.975, 0.98, 1, 1)

(0.04, 0.1, 0.18, 0.23)

O12



22×1

, δ0 =
[
O
]
14×1

When t = 0, we get the results



796 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

δ1 =




(0.9506, 0.9604, 1, 1)

(0.702, 0.7644, 0.92, 0.97)

O2
(0.9506, 0.9604, 1, 1)

(0.39, 0.098, 0.18, 0.23)

O8



14×1

, θ1 =




O12
(0.9268, 0.9412, 1, 1)

(0.6845, 0.7491, 0.92, 0.97)

O2
(0.9268, 0.9412, 1, 1)

(0.2808, 0.0764, 0.1656, 0.2231)

O4



22×1

When t = 1, we obtain the results

δ2 =




O7
(0.9268, 0.9412, 1, 1)

O2
(0.9268, 0.9412, 1, 1)

(0.2808, 0.0764, 0.1656, 0.2231)

O2



14×1

, θ2 =




O19
(0.9268, 0.9412, 1, 1)

(0.9268, 0.9412, 1, 1)

O1



22×1

When t = 2, we have the results

δ3 =

[
O13

(0.9268, 0.9412, 1, 1)

]
14×1

, θ3 =

[
O21

(0.9036, 0.9224, 1, 1)

]
22×1

When t = 3, we get the results

δ4 =
[
O
]
14×1

.

Thus, the termination condition is satisfied and the reasoning process ends. We obtain the
reasoning results, i.e., the fuzzy truth value (0.9036,0.9224,1,1) of the output neuron σ22. The
transmission line L1314 is a faulty section with a confidence level (0.9036,0.9224,1,1).

tFRSN P systems and MBFRA are also suitable for multiple faults diagnosis problems in
power systems. In what follows we take an example of the power system in Fig. 4 to show the
effectiveness of the method in diagnosing multiple faults.

Case 2: Multiple faults. Transmission line L1314 and bus B13 have faults.
Operated relays: MLR1314, MLR1413, SLR0613, SLR1213. Tripped CBs: CB1314, CB1413,

CB0613, CB1213.
According to the SCADA data, four candidate fault sections, i.e., L1314, B13, L0613 and L1213,

are selected. The tFRSN P systems of the four sections are established to perform MBFRA,
respectively. After the fuzzy reasoning, fault confidence levels, (0.9036, 0.9224, 1, 1), (0.6673,
0.7341, 0.92, 0.97), (0.2165, 0.299, 0.623, 0.7849) and (0.2165, 0.299, 0.623, 0.7849), represented
by trapezoidal fuzzy numbers of sections L1314, B13, L0613 and L1213 are obtained. According to
(2), we obtain their corresponding real numbers, i.e., 0.92, 0.7595, 0.475 and 0.475. Thus, there
are two faulty sections, i.e., L1314 and B13. The results are summarized in Table 2.

The logic analysis about Case 2 is described as follows. In this case, information of protective
relay BR13 is not observed and CB1312 and CB1306 fail to trip. For transmission line L1314, its
main transmission relays, MLR1314 and MLR1413, operate to trip their corresponding CBs,



Application of Fuzzy Reasoning Spiking Neural P Systems to Fault Diagnosis 797

Table 2: Relay trip signals and CBs status signals observed, and diagnosis results
Candidate Confidence Corresponding Ranking Fault

fault section level real number section
L1314 (0.9036,0.9224,1,1) 0.92 1 Yes
B13 (0.6673,0.7341,0.92,0.97) 0.7595 2 Yes
L0613 (0.2165,0.299,0.623,0.7849) 0.475 - No
L1213 (0.2165,0.299,0.623,0.7849) 0.475 - No

CB1314 and CB1413, to clear a fault. So it is a faulty section. Although main protective relay
BR13 of bus B13 fails to clear a fault, its remote backup protective relays, SLR0613 and SLR1213,
operate to trip their corresponding CBs, CB0613 and CB1213, to clear this fault. So B13 is a
faulty section. For transmission lines L0613 and L1213, only their single-ended remote backup
protective relays SLR0613 and SLR1213 operate to trip their corresponding CBs, CB0613 and
CB1213, respectively. Actually, SLR0613 and SLR1213 and their CBs act as remote backup
protections of B13. So, L0613 and L1213 are not faulty sections. Therefore, according to the
logic analysis and Table 2, we can know that the presented method can obtain correct results in
multiple fault situations.

5 Discussions

A tFRSN P system is a novel graphical model for representing fuzzy knowledge and informa-
tion. This study employs it to diagnose the faults of power systems. The fault diagnosis ability of
a method is usually associated with the knowledge availability and the reasoning process. Thus,
in what follows, we make a comparison between tFRSN P systems and several fault diagnosis
approaches regarding the aspects of knowledge representation and inference process.

(1) Expert systems (ESs). Both ES and the fault diagnosis method based on tFRSN P
systems (FDM-tFRSNP) can make full use of experts’ knowledge. The differences are: an ES
needs long response time and the maintenance of its knowledge base is difficult [18]; FDM-
tFRSNP possesses parallel reasoning ability and adopt graphical knowledge representation and
reasoning, which can avoid the main limitation of ES.

(2) Fuzzy set theory (FST). FST is an effective way to represent uncertain information but the
definition of membership function is a hard job [18]. The FST-based method and FDM-tFRSNP
both possess the ability to deal with uncertain information of protective devices. In addition,
linguistic terms used in both methods make them closer to the human thinking compared with
the methods using crisp numbers. The main differences between them are that FDM-tFRSNP
has a fast reasoning speed and the matrix reasoning process is easier to describe diagnostic
process as well as its programming.

(3) Artificial neural networks (ANNs). ANNs can be regarded as opaque black boxes and can
be easily used. The main problems of ANNs lie in the difficult acquisition of a complete sample
set and a tedious training process needing extra time consumption. In addition, premature
convergence is also a problem. FDM-tFRSNP neither needs a training process with a set of
comprehensive training data nor has a premature convergence problem [13]. Besides, FDM-
tFRSNP can intuitively represent the relationships between faults and operations of protection
devices. This feature is very helpful for operators to analyze and summarize failure processes.

(4) Fuzzy Petri nets (FPNs). Both FPNs and tFRSN P systems have graphical knowledge
representation and parallel computing ability. However, the mechanism of tFRSN P systems is
originated from neurophysiological behavior of neurons or/and living cells. Thus, the working
principle of different types of neurons or/and cells may provide new inspirations for extending SN



798 T. Wang, G. Zhang, H. Rong, M.J. Pérez-Jiménez

P systems (or tFRSN P systems), which can increase the ways of knowledge representation and
reasoning to solve new problems in power systems [4]. In addition, tFRSN P systems with trape-
zoidal fuzzy numbers have three rule neurons types, i.e., general, and and or, and one proposition
neuron type, while FPNs only contain same places and transition types. Thus, different types
of neurons make FDM-tFRSNP have better flexibility and trapezoidal fuzzy numbers (linguistic
terms) make tFRSN P systems more understandable to operators of power systems.

6 Conclusions

In this study, tFRSN P systems and a matrix-based fuzzy reasoning algorithm, MBFRA, for
fault diagnosis are discussed to extend the application area of SN P systems in fault diagnosis
of power systems. MBFRA is based on the dynamic firing mechanism of neurons. Given initial
pulse values of all input neurons of a tFRSN P system, MBFRA can obtain the pulse values of
other neurons by performing fuzzy reasoning. To make MBFRA suitable for fault diagnosis in
power systems, a defuzzification method is employed to treat reasoning results represented by
trapezoidal fuzzy numbers. Application examples show that tFRSN P systems with MBFRA is
effective in diagnosing faulty sections of power systems. Besides, a comparison between tFRSN
P systems and different fault diagnosis approaches is made.

The aim of this study is to construct a tFRSN P system diagnosis model for each candidate
fault section. The scale of each diagnosis model depends on protective devices connections of the
candidate fault section rather than the scale of power systems. Thus, the presented method can
be used for large-scale power systems. This study focuses on the effectiveness and correctness
of the fault diagnosis method and the results of application examples are obtained by manual
computation. To test the speed, convergence and accuracy of MBFRA and to explore automatical
generation of tFRSN P systems in diagnosing faulty sections in power systems, our future work
will simulate them on MATLAB, P-Lingua or MeCoSim [19]- [21]. Moreover, how to verify and
realize the parallelism of tFRSN P systems and MBFRA on hardware such as FPGA and CUDA
is also our further task.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (61170016,
61373047, 61170030), the Program for New Century Excellent Talents in University (NCET-11-
0715) and SWJTU supported project (SWJTU12CX008).

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