Langrangian , Φ with respect to the weights w **, bias** b and α **[13] given by:** 

$$\phi=\int_{2}^{\infty}\left|w\right|^{2}-\sum\alpha_{i}\left(y_{i}\left[\left(w,x_{i}\right)+b\right]-1\right)\right.\tag{6}$$

The multipliers are constrained by 0≤αi,αi
*≥ C **where** 
C **is the misclassification tolerance or capacity. If the** 
value of C **is too large, the kernel function will overfit the training data and will not have good** 
generalization properties. 

## 8 Proposed Frameworks 

![0_Image_0.Png](0_Image_0.Png)

The proposed frameworks for fault diagnosis are a two-level implementation. The first level of the diagnosis identifies if the bushing is faulty or not. If the bushing is faulty, the second level determines the types of faults, which are thermal fault, PD faults and faults caused by an unknown source. Generally, the procedure of fault diagnosis includes three steps, extracting feature and data pre-processing, training the classifiers and identifying transformer fault with the trained classifiers. Fig.4 shows the block diagram of the proposed methodology. 

8.1 Data processing DGA is used to determine the faulty gases in the bushing oil. The content information reflects the states of the transformer and bushing. Ten diagnostic gases mentioned in Section 2 are extracted, which are CH4, C2H6, C2H4, C2H2, H2, CO, CO2, N2, O2 **and total** 
dissolved combustible gases. The total dissolved combustible gas is given by the sum of methane, hydrogen, acetylene, ethane, ethylene and hydrogen. The faulty gases are analysed using the IEEE C57.104 standards [14]. Data pre-processing is an integral part of neural network architecture. Data pre-processing makes it easier for the network to learn. Data are normalized to fall within 0 and 1, using linear normalization. 

## 8.2 **Mlp, Rbf And Svm Classifiers** In This Method Mlp, Rbf And Svm Are Trained. As

the classifier for the first level and second level classifications are trained in a similar way, the training procedure explained below applies to both the two-class and three class problems. The optimal number of hidden layer was found by using exhaustive search. An MLP with optimal number of hidden layer neuron was trained using the scaled conjugate gradient [11]. The centroids of RBF 
are found by using the Gaussian mixture model with circular covariance using the Expectation Maximization (EM) algorithm [10] and the output layer was trained using the scaled conjugate gradient algorithm. Cross-validation is used to ensure that a network with good generalization property is achieved. Cross-validation was used to determine the best kernel function and capacity of the SVM. Then SVM was trained with the optimal hyperplane and the optimal C. The networks are tested with 1000 data points that are randomly selected from the data set. Table 1 shows the results of the networks trained to identify if the transformer is faulty or not which is called second level stage. The methods are tested using specificity and sensitivity. Sensitivity is defined as the probability of the classifier predicting faults correctly and specificity is the probability of a classifier predicting the non-faulty state correctly. Table 1: Comparison of the performance of different frameworks for first level of fault diagnosis 

|                        | MLP    | RBF    | SVM      |
|------------------------|--------|--------|----------|
| Accuracy (%)           | 98.9   | 97.4   | 98.5     |
| Specificity            | 0.796  | 1.000  | 0.996    |
| Sensitivity            | 0.999  | 0.885  | 0.885    |
| Training Time(s)       | 41.236 | 0.625  | 1975.437 |
| Classification Time(s) | 0.0157 | 0.0314 | 104.314  |

The table compares the framework in terms of accuracy, training and testing time. MLP classifier shows classification accuracy of 98.9%, RBF shows 97.4% and SVM gives 98.5% classification accuracy. This table shows that there is no significant difference between SVM and MLP classifiers. Although, RBF 
performs worse than MLP and SVM in terms of 