A FRSN P system with degree m ≥ 1 is a construct of the form [23] ;

where:

1. is a spike in the neurons;

2. are proposition, rule neurons and of the form;

where:

A. is spikes potential value of neuron expressed by [0,1];

B. is truth value of neuron expressed by [0, 1];

C. is a firing rule of neuron of the form E/

where:

a) E is a regular expression.

b) & are expressed by [0, 1].

3. syn is a directed graph of synapses between neurons, where:

, with for all,.

4. are input and output neuron respectively.

According to their usage in this study, neurons in FRSN P systems are classified into four types of neurons;

1. Proposition neurons

In this kind of neuron, If neuron as input proposition neuron in P, then; otherwise equals all pulse values received from their presynaptic rule neurons based on logical or operation [23] .

2. General rule neurons

If neuron as general rule neuron in P, then the pulse value equals the pulse value received from their presynaptic proposition neuron [2] , their representation by fuzzy production rules;

As shown in Figure 1(a), The fuzzy truth value of the of proposition is.

3. And rule neurons

If neuron as and rule neuron in P, then the pulse value equals all pulse values received from their presynaptic proposition neurons based on logical and operation [23] , their representation by fuzzy production rules;

As shown in Figure 1(b), The fuzzy truth value of propositions is .

4. Or rule neurons

If neuron as or rule neuron in P, then the pulse value equals all pulse values received from their presynaptic proposition neurons based on logical or operation [23] , their representation by fuzzy production rules;

As shown in Figure 1(c), The fuzzy truth value of the of proposition is.

We defined some matrices, reasoning processes and execution rules as follows [23] .

1) is a fuzzy truth value vector of n proposition neurons.

2) is a fuzzy truth value vector of the u rule neurons.

3) is diagonal matrix confidence factor of rule neurons.

4) , , is directed synaptic matrix from proposition to general , and and or rule neurons respectively.

5) is directed synaptic matrix from rule to proposition neurons.

In order to represent the execution rules of FRSN P systems formally, we introduce some fuzzy matrix operations [23] .

1) :, where A, B and C are all r ´ s matrices,.

2) :, where A, B and C are all r ´ s, s ´ t and r ´ t matrices respectively,.

3) :, where A, B and C are all r ´ s, s ´ t and r ´ t matrices respectively,.

Dissolved gas analysis (DGA) is powerful technique has been used to identify the incipient power oil transformers faults. In this technique can be identified according to the gases concentrations dissolved in oil of transformer, hydrogen (H₂), (CH₄), (C₂H₆), (C₂H₄), (C₂H₂), various interpretative DGA methods has been established, such as Gas key method, IEC ratio method, and the graphical representation method [1] .

In this study we propose adaptive IEC ratio (AIEC ratio) method as first incipient diagnosis of the possible faults of oil transformer, in order to identifying the fault types based incipient possible faults diagnosed by IEC ratio method, we use the ratio of gases as input data to FRSN P systems diagnosis model and the output fuzzy reasoning results as shown in Figure 2.

In IEC ratio method, five gases, H₂, CH₄, C₂H₂, C₂H₄ and C₂H₆, as concentration gases in oil transformer. From these gases produce three ratios [21] ;

, ,

Table 1 presents the transformer DGA faults are classified to six types, low energy discharge, high energy discharge, partial discharge, low thermal, medium thermal and high thermal faults, which is widely used to interpret the DGA [1] .

Table 2 shows the interpreting fault types of IEC 60599 standard with values of three gas ratio [22] .

From the operator expert knowledge, in real word fault diagnosis events, in this study linguistic terms are always used to express the fault types related with gas concentrations ratio, such as (C₂H₂)/(C₂H₄) very low, low, medium, high and very high in the transformer oil.

In this proposed method, we use the linguistic terms to describe a degree of gas concentrations ratio to become more capable to use fuzzy knowledge with fuzzy numbers. We can use adaptive IEC ratio (AIEC ratio) to deal with FRSN P systems and graphically represents with fault diagnosis model from input proposition neurons by reasoning steps to reach the final rezoning results after computation halts in output proposition neurons.

Table 3 shows the classification of gas ratio concentration based on IEC 60599 Gas ratio Limits in Table 2.

FRSN P systems diagnostic model based DGA shown in Table 3, we can constrict the graphical diagnosis model of FRSN P systems with reasoning steps to identify the fault type of oil transformer, see Figure 3.

In this graphical model, IEC ratio with fuzzy representation as linguistic terms can built the FRSN P systems diagnostic model as shown in Figure 4.

Three ratio and, each ratio with five levels very low(VL), low(L), medium (M), high(H) and very high(VH) respectively as input proposition neuron with initial values and after reasoning steps , six fault types identified by confidence levels to give us which one with more confident with linguistic expression. This allowed us to diagnosis the fault with more informative and more correctly decisions.

From the historical database of transformer we can use the confidence level of each fault dissolved gas to use it in the matrix calculations of proposed method based on their experience operator and also we have to certainty factor to represent the degree of confidence fault occurs.

The rule neurons with synapse input neurons, the confidence (0.8) and other rule neurons (1.0)

From the definition (P), we can use FRSN P systems to built fault diagnosis model for transformer based DGA ratio for all possible combinations of gases ratio based in AIEC ratio table, see Figure 4.

, n = 32 proposition neurons, u = 17 rule neurons

1. is the singleton alphabet (a is called spike);

2. are proposition neurons.

3. are rule neurons;

where

a) , and are general rule neurons.

b) , , are or rule neurons.

c) are and rule neurons .

4. Syn. Shown in Figure 4.

5. in =, out =.

Each input proposition neurons will be assigned a truth degree value based on observation of the transformer history data, if the gas ratio limited values of AIEC ratio the of a transformer is actually observed, the input proposition neurons will have a truth degree value (0.9), otherwise truth degree value of non observed gases (0.1).

Each rule neurons with a certainty factor, which describes the confidence level based on experience of operator, in these cases, will be given the same values (0.8) and will be given the same values (1.0).

Case 1#: The observed gases data are listed in Table 4 and gases ratio are listed in Table 5.

From and Figure 4, the input neurons, the initial truth values of proposition neurons are (0.9, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.9, 0.9, 0.1, 0.1, 0.1, 0.1) respectively, and their truth values are (0) ,certainty factors corresponding to the rule neurons are given values (0.8) and (1.0).

The inference procedures are described step by step as follows:

,

where if; otherwise, , (,).

where if (i, j) = {(16, 12), (16, 15), (16, 16), (16, 17), (17, 14), (18, 13), (19, 12), (20, 13), (20, 14), (21, 15), (21, 16), (21, 17), (22, 12), (22, 15), (23, 16), (24, 13), (25, 14), (26, 17)}; Otherwise, , (,).

where if (i, j) = {(3, 2), (4, 2), (4, 3), (5, 3), (7, 5), (8, 5), (9, 6), (10, 6), (11, 7), (12, 7), (13, 8), (13, 9), (14, 8), (14, 9), (14, 10), (15, 9), (15, 10)}; Otherwise, , (,).

where if (j, i) = {(1, 16), (2, 17), (3, 18), (4, 19), (5, 20), (6, 21), (7, 22), (8, 23), (9, 24), (10, 25), (11, 26), (12, 27), (13, 28), (14, 29), (15, 30), (16, 31), (17, 32)}; Otherwise, , (,).

where,.

At t = 0

At t = 1

At t = 2

Thus, the FRSN P system computation halts and the reasoning fault diagnosis results is, The truth values of neurons propositions, , , , and are 0.08, 0.08, 0.08, 0.72, 0.08, 0.08

The reasoning results indicate the PD confidence (0.08), D₁ confidence (0.08), D₂ confidence (0.08), T₁ confidence (0.72), T₂ confidence (0.08) and T₃ confidence (0.08).

So T₁ with highest confidence level and greater than threshold (0.50) is thermal faults T < 300◦C, and results for other cases are listed in Table 6.

In these cases, comparative studies of FRSN P systems with ratio support vector machine method (SVMR) and graphical support vector machine (SVMG), considered the same cases fault situations, the status tested gas data of transformer for eight tested cases are shown in Table 4, and the FRSN P systems diagnosis results are shown in Table 6. From the case studies (1, 2, 3), the fault type is Thermal faults T < 300◦C (T₁) with confidence level (0.72), case studies (4, 5, 6) their isn’t fault with confidence level ( 0.08), case (7) is High energy discharge (D₂) and case (8) is Thermal faults 300 < T < 700◦C (T₂) fault with confidence level ( 0.72).

Table 7 show us, the comparing results proposal method with SVMR and SVMG methods, according to test results in this table, the FRSN P systems is more suitable as dissolved gas signature and solved the problem of conflict between SVMR and SVMG.