A MLP Neural Network-Based Unified Intelligent Model for Predicting Short-Gap Breakdown Voltages in Air

Electrical insulation is an elementary characteristic when designing and implementing electrical systems, whose aim is to guarantee the separation of conductor or electrical constituents. This will prevent, any unwanted or unexpected contact that can lead to electric shocks, short circuits, or other unappreciated phenomena resulting from electricity circulation. The breakdown voltage represents the maximum voltage limit that an insulating material can withstand before suddenly losing its insulating properties and becoming conductive. This phenomenon's consequences are multiple: irreversible damage to the insulation compromising its insulation capacity, short circuits between conductors leading to electrical equipment malfunctions, etc. This damage can lead to a power supply interruption, which can have significant repercussions in critical areas [1-3].

Like any gaseous medium, atmospheric air is considered an insulator. In an aerial network, the air is subjected to sufficient electrical voltage. In the presence of impurities and severe atmospheric conditions, a current of electrically charged particles becomes possible through the medium partial ionization [4-6]. Consequently, the air becomes conductive, which leads to breakdown. Alternatives and models for estimating electrical breakdown voltages are in high demand for areas involving electrical insulation, more particularly, in energy transportation and distribution networks; The models allow, in fact, to predict with certainty the electrical equipment sizing, to develop efficient technologies for preventing atmospheric discharges, and to accurately deduce safety distances.

The study of breakdown in air gaps has been the subject of several research in recent years. However, the airflow mechanism still needs to be fully understood. Methods for measuring breakdown voltage are subdivided into two approach categories: conventional methods (experimentally and theoretical) [7-10] and methods based on Artificial Intelligence (AI). Traditional methods of measuring breakdown voltage have certain disadvantages. On the one hand, these methods involve sophisticated equipment and require complex installations (such as special measuring chambers), thus making their implementation very expensive and limiting their accessibility and use. On the other hand, the results obtained are hazardous and of limited use due to the atmospheric conditions influence and the discharge energy influence on the samples studied. These drawbacks push the scientific community to intensify their research in order to propose more ingenious, efficient and economical breakdown measurement approaches. AI methods, inspired by living things' reasoning mechanisms, are seen as a critical solution due to the multitude of significant advantages they present. Indeed, AI promises to save time through the automation of the estimation process. AI ensures accurate, reliable results and reduces computing costs by eliminating expensive devices. Finally, AI has the particularity of learning from experience and adapting to situations never seen before, which makes it very suitable for the problem of approximating the electrical insulators' breakdown voltages, particularly air. With this in mind, several ingenious research in the field of breakdown voltage estimation based on intelligent regression approaches have been proposed, including those based on neural networks [11-15], those based on Support Vector Regression (SVR) [16-20], those based on Support Vector Machine (SVM) [11, 21, 22], those based on fuzzy logic [23, 24] and those based on extremely randomized trees algorithm [11]. In what follows, we will detail some studies that are pretty similar to ours:

• Yang et al. and Yao et al. [16, 17] proposed an intelligent system based on SVR optimized by Cuckoo Search (CS) and weighted by grey relation analysis (CS-ω-SVR) in order to calculate the 50% breakdown voltage of the long rod-rod gap under different atmospheric conditions and different distances. The proposed method was validated on 70 samples. The authors selected two error indices to quantify the accuracy of the predicted results: mean absolute percentage error, which is on the order 8.8%, respectively. The researchers also highlighted that the proposed optimization algorithm was found to be more accurate than other algorithms such as GA and PSO.
• Kalenderli et al. [12] proposed a study to predict the breakdown voltage and gap types of a point-barrier air gap by training a Multi-Layer Perceptron neural network (MLP). The authors also present a comparison of the implemented method obtained results with those obtained by the methods of: maximum likelihood, graphical techniques and the least squares method. The results show that the breakdown voltages obtained by the MLP are practically identical to the reference values compared to the four other methods.
• Mokhnache and Boubakeur [15] implemented a Radial Basis Function (RBF) improved by the random optimization method (ROM) to predict the breakdown voltage in a barrier-point-plane air space for different voltage pulse forms (lightning pulse and switching pulse). In conclusion, the authors underline the method's effectiveness and even underline the possibility of extending their approach to the other parameters' prediction.
• The research by Qiu et al. [21] presents a new method for predicting breakdown voltages of typical air gaps (considering several configurations, namely, sphere-sphere, rod-plane, sphere-plane and sphere-plane-sphere), based on the Electric Field (EF) characteristics (in agreement with the results of finite element calculation) and an SVM (translating the regression problem into another dichotomous discrimination problem).
• Qiu et al. [22] presented a model for predicting the breakdown voltage of short air gaps (three configurations are considered in this study, namely, sphere-sphere, rod-plane, and rod-gaps between rods) based on the static EF distribution characteristics, an Orthogonal Design (OD) feature reduction approach and SVM. The obtained results showed a satisfactory correlation with the experimental data, with an average error percentage of around 3.9%.
• Yao et al. [14] proposed a new approach for determining the precise breakdown voltages of the rod-plane space and took into consideration two critical factors (temperature and humidity). The authors use an approach based on a regularization Invariant Risk Minimization Neural Network (IRM-NN), and they obtained an average error percentage of around 2.59%.

A thorough analysis of the intelligent systems proposed previously allowed us to raise the following limits: (1) Unlike MLP, RBF is generally favorable for studies whose training sample numbers are quite limited; however, it is necessary to improve them by optimization techniques and vector quantification otherwise. (2) SVR remains favorable in both cases. Nevertheless, their rather complex mathematical foundation and the need to adjust their hyper parameters make their implementation very difficult and expensive in terms of implementation time. (3) The SVM use based on the translation of a regression problem to a classification problem is a bad idea because the obtained results will be non-precise and bounded in an interval. (4) Fuzzy logic is a compelling alternative. However, it remains based on the living beings' logical reasoning, which can make it ineffective in the face of unforeseen situations. In this study, we opted for using MLP because, on the one hand, we have a reasonably rich database which will allow us to train them well; and on the other hand, MLP-type neural networks offer a multitude of advantages, including: adaptability, non-linear data management, ability to extract characteristics from noisy data, robustness to noisy data, effectiveness for regression problems and performance improvement with the data quantity augmentation, etc.

Furthermore, the state of the art allowed us to conclude that despite the existence of a multitude of approaches to estimate the breakdown voltage of different air gap configurations and despite the existence of models which will enable switching between one configuration to another by taking only one configuration at a time. It isn't easy to find models that can guarantee the ability to easily adapt to additions of configurations (the possibility of extending the model) in a relatively simple way (natural, flexible and intuitive) without having to resort to neglecting configurations previously implemented in the same model. The only researches found are that of the studies [21, 22], which proposes a system that manages several configurations at the same time. All the same, this system remains primarily based on breakdown voltage estimation techniques based on mathematical and experimental calculation models (the disadvantages of which are cited above in the introduction); the SVM are only used to refine the tensions already obtained with precision. Furthermore, despite the good results obtained, the implemented systems remain quite complicated and require a fairly long execution time.

In this sense, the present study first presents a unique configuration based on MLP (in adequacy with existing models), with the aim of separately approaching the breakdown voltages of three distinct air gap configurations: Electrode-Sphere-Sphere-Earth (ESSE), Electrode-Sphere-plane (ESP), Electrode-Symetry-Sphere (ESS). Subsequently, the novelty of this research lies in the proposition of a new 100% intelligent unifier architecture is proposed to support all the air gap configurations studied. Thus, the proposed unified model has as first objective to propose a simple and flexible architecture 100% intelligent, which allows a fluid extension of the model to manage other configuration types. The model has as second objective to propose an efficient estimator with the speed and the precision of calculation, which will allow to consider an integration in real time/processes.

This paper is organized as follows: Section 2 will be devoted to the used database presentation. Section 3 will be dedicated to a brief description of the MLP mathematical foundation, followed by a well-established description of the proposed unified model. Section 4 will be reserved for the listing and discussion of the obtained results. Section 5 of this paper will serve as a conclusion.

MLP is a supervised learning model with high computational capacity. Its structure consists of an “input layer”, an “output layer” (interpreted as the response of the network) and one or more intermediate layers called “hidden layers”. A neuron in the bottom layer can only be connected to neurons in subsequent layers. In this study, the MLP is considered for the breakdown voltages calculation in short air gap configurations; it is, in fact, a regression problem.

The regression model hyperparameters selection is a crucial and essential step for an intelligent mathematical model design, capable of behaving effectively in the face of test examples never before seen. For an MLP type approximator, the optimized parameters are those relating to: hidden layer number, the hidden layer neuron number, the activation functions, the metric chosen for the learning error evaluation and the learning algorithm.

• The hidden layer neuron number is usually proportional to the input vector size. Our test and training examples are samples that each has three parameters, therefore, a single hidden layer is sufficient to guarantee good prediction results (a higher number of hidden layers can cause the estimator overfitting).
• The hidden layer neurons number is fixed empirically by testing several configurations whose neurons number is included in the range (8-25); each configuration was repeated ten times due to the synaptic weights random initialization (at the beginning of training). The keeping configuration is the one that gives the best result (Table 2).
• As activation functions, we used a “Tansig-type” activation function for the hidden layer neurons and a “Purelin-type” activation function for the output layer neurons.
• The learning algorithm and the error evaluation metric used are respectively, the backpropagation learning rule and Mean Square Error (MSE); due to their effectiveness which has been proven in several applications.

3.1 Gradient back propagation algorithm

In the multilayer network structure, the gradient back propagation method is most frequently used to minimize the Mean Square Error (MSE) between the network output and the desired output. This algorithm back-propagates the error gradient from the output to the input. This operation is repeated until the difference between the network output and the desired output becomes acceptable (Figure 5).

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Figure 5. Synaptic weights modification in supervised learning

Learning by backpropagation is carried out in successive steps, which are described above [26, 27]:

1) The synaptic coefficients are initialized with small random values (NB: large values tend to cause the activation functions to become saturated).

2) Presentation of the characteristic vectors as input to the classifiers.

3) Iteratively calculate the neuron states in the following layers using Eq. (2).

$y_j^k=f\left(\sum_{i=1}^N w_{j i}^k \cdot y_i^{k-1}\right)$      (4)

with:

$f$: activation function.

$N$: neurons number in layer k-1.

$k$: current layer number.

$y_j^k$: output of neuron j from layer k.

$w_{j i}^k$: connection between neuron i of layer k-1 and neuron j of layer k.

$y_i^{k-1}$: output of neuron $i$ from layer k-1.

4) Calculation of the error gradient made on neuron i in the output layer:

$\Delta_j^k=2 f^{\prime}\left(\sum w_{j k}^k \cdot y_j^{k-1}\right) \cdot\left(d_j^k-y_i^k\right)$    (5)

with:

$f^{\prime}$: the derivative of the function f.

$d_j^k$: the desired output for neuron i in layer k.

5) Back-propagated error gradients calculation from layer k to layer 1:

$\Delta_j^{k-1}=2 f^{\prime}\left(\sum_{i=1}^N w_{j k}^k \cdot y_j^{k-1}\right) \cdot \sum_{i=1}^N \Delta_j^k w_{j i}^k$          (6)

6) Synaptic coefficients update:

$w_{j i}^k(t+1)=w_{j i}^k(t)+\Delta \cdot \delta_j^k \cdot y_j^k$          (7)

with:

$\Delta$:  is the gradient step.

Repeat all steps, except the first, each time for a new characteristic vector, until the stop test is activated.

3.2 MLP for a separate configuration breakdown voltage estimating

This sub-section will be devoted to presenting the methodology adopted to implement our unique intelligent estimators, each of which is responsible for interpolating the breakdown voltages of a single electrode geometry configuration. Each selected architecture is a three-layer configuration (Figure 6):

- An input layer: made up of two neurons, the first receiving the distance, and the second receiving the radius.

- A single hidden layer: the number of which is fixed empirically by testing several values and keeping the one that gave the best result (Table 2).

- An output layer is made up of a single neuron that returns the estimated breakdown voltage.

The MSE evaluation is shown in Figures 7-9. From the three figures, we can see a perfect convergence with quadratic errors, which are of the order of ESSE: 6.295, ESP: 6.966 and ESS: 2.0598. These findings prove the effectiveness of the unique configurations proposed.

3.3 MLP unified model

The goal of this section is to propose a compact architecture that makes it possible to address the problem in its entirety by supporting all three configurations (ESSE, ESP, and ESS). The training and testing data are the concatenation of the sets used to train and test the networks of the three configurations (ESSE, ESP and ESS) individually. The development of this unique network (see Figure 10) requires:

- There are three input neurons: one that receives the distance from which we want to estimate the breakdown voltage, one that receives the radius, and one that receives the configuration type (1 for SST, 2 for ESP, and 3 for ESS).

- One hidden layer whose neuron number was fixed empirically by testing several values and keeping the one that gave the best result (Table 3).

- A single output neuron that gives the estimated breakdown voltage.

The squared error during the iterations is schematized in Figure 11. We see a very satisfactory convergence (with an MSE of around 11.5835), which once again proves the effectiveness of the proposed configuration.

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Figure 6. Graphical representation of the single model's configuration

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Figure 7. MSE error evolution during the iteration (ESSE configuration)

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Figure 8. MSE error evolution during the iteration (ESP configuration)

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Figure 9. MSE error evolution during the iteration (ESS configuration)

10.png

Figure 10. Graphical representation of the unique model configuration

Table 2. MSE evolution according to the hidden layer neurons number

Table 3. MSE evolution according to the hidden layer neurons number

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Figure 11. MSE error evolution during the iteration (unique configuration)

Through this research, a new unified intelligent model for predicting short-gap breakdown voltages based on the MLP neural network is proposed. To do this, we proceeded in a progressive manner:

- We first proposed specific automatic networks to estimate breakdown voltages separately for each of the following three configurations: ESSE, ESP and ESS.

- We have introduced a global estimator based on an MLP, able to support all configurations, thus allowing an integrated approach for the breakdown voltage estimation.

- We also proposed a learning process to determine the neuron's numbers in the hidden layer for the selected configurations, which contributes to minimizes the MSE as much as possible and optimizes the networks estimation.

The obtained estimates and desired outputs are approximately identical in all cases, and the error is practically zero for all the geometries considered in this study (the unified architecture and the specific architectures). Comparisons between the obtained results with different approaches reported in the literature [12, 14-17, 21, 22] also indicate the effectiveness, validity and significant impact of the proposed unified model despite its simplicity.

These results are very encouraging and confirm the effectiveness of the adopted approach and its potential to improve the breakdown voltage estimation performance in areas involving electrical insulation, bringing precision and safety in the electrical equipment sizing. However, this research could be improved by considering the following future work: studying the environmental factors (temperature, humidity, pressure) effect, testing other geometries type, and implementing other automatic regressors type and provide the merger for greater credibility.