Power transformer is the most important unit in the power system both economically and functionally. Therefore it is highly necessary to know the real conditions of transformer insulation to increase the reliability. On the other hand, the function and life span of transformer insulation is highly affected by partial discharge. Proper recognition and also PD localization help us to prevent terrible incidents in power system. Methods of PD localization and calculating can be classified in to two groups of electrical and non-electrical methods based on the effects that PD can cause in an insulator. . Non-electrical methods are also categorized into three methods: Audio, chemical and optical by which the presence or absence of partial discharge in the insulator can only be realized. Electrical methods are the most successful and practical methods in localization and calculation of PD. Among electrical methods, methods which are based on transformer modeling are used for PD localization in transformers. In these methods, transformer winding is modeled in high frequencies based on the available information from designing of transformer .
According to the studies done in this respect, we can mention three models: the traveling wave model  multi-conductor transmission line model (MTL)  and RLC ladder network .
After winding modeling, electrical PD pulses are injected in different parts of the winding and the created signals are recorded in terminals. Localization is then done after analyzing the recorded signals. PD localization is done in different ways such as: Transfer Function based on partial discharge localization , PD localization based on zeros and poles of frequency spectrum  and PD localization based on pattern recognition . In most of the presented methods, PD localization is full of errors due to the weakness in precise winding modeling and invalid processing algorithm in high frequencies. Hence, in this paper, a valid algorithm is presented using the wavelet transform (WT) in order to find a suitable location for PD in the transformer winding in offline state by precise modeling of transformer winding and calculation of parameters with high accuracy using finite elements method (FEM) in Ansoft Maxwell software.
This paper consists of six parts. In the second part, precisely introduce the RLC and multi-conductor transmission line models, and in the third part, wavelet transforms and Parseval’s theorem are described. Modeling methods and suggested algorithm simulation on 20 kV winding are also illustrated in the fourth part. In the fifth section, the laboratory results of the above method are implemented on the practical model and a comparison is made between the results of localization obtained from the RLC model and those of multi-conductor transmission line model, and the conclusion is presented in section six.
2. High-frequency Models of Transformer Winding
To study the manner of partial discharge behavior in the transformer winding, there is a need for an accurate modeling of the transformer winding. There are different methods to model the transformer winding. So far, the RLC models and multi-conductor transmission lines (MTL) by which the set of transformer windings can be physically studied and evaluated have been introduced. The RLC model is a circuit with integrated elements and a limited number of nodes. As a result, the frequency validity limit is limited up to about one MHz. Precision in calculating the numeric value of the parameters of the model that is self and mutual inductance and capacitance has a great effect in reducing the errors of simulation and PD localization. In this model each unit of transformer is modeled with an R, L, and C network. Fig. 1 shows the transformer winding RLC model.
Fig. 1.Winding equivalent circuit inform R, L, C Ladder network.
In Fig. 1, Li is self-inductance of i unit, Mij is mutual inductance of i and j units. Ki is capacitance for i unit, Ci is the capacitance between i unit and earth potential (core or tank), Gs is conductance which is representing the dielectric loss of the i unit, Gg is conductance which is representing the dielectric loss of the i unit and earth and r is the ohmic resistance indicating ohmic losses of the i unit.
The multi-conductor transmission line model enjoys higher frequency validity limit contrary to the RLC and can predict the wavy behavior of high-frequency effects. In this model, the winding parameters are considered extensively and the winding behavior is described by transmission line equations; also, each turn of the transformer winding is modeled as one single-conductor transmission line with an equivalent circuit similar to Fig. 2. The equations of transmission lines in time domain can be written as follows:
Fig. 2.Multi-conductor transmission line model
Where in Eqs. (1) and (2), L is the inductance, R is the resistance, C is the capacitor capacitance, and G is the parallel resistance of a small part of the transmission line . Also in Fig. 2, US and IS are the voltage vectors and the current transmitted from the beginning of the transmission line, respectively. UR and IR are the voltage and current vectors, respectively received at the end of the transmission line.
Applying analytical formula to calculate these parameters especially self and mutual inductance is a complex and time-consuming process. Therefore in most of the studies done so far the modeling for each unit of winding, R, L, C values are calculated; these values are then considered for other units. Mutual inductances that exist between different turns of each winding are also ignored which cause errors in modeling and PD localization. One of the best methods to calculate the parameters of detailed models is applying the finite element method (FEM) by Ansoft Maxwell software [10, 11]. This paper has made use of this method to calculate the exact amount of parameters such as mutual inductances.
Modeling of winding, precision of the parameters obtained, and the manner of the propagation of partial discharge signal by using RLC and MTL models are fully presented in references [10, 11]. In reference , a comparison is made among the detailed models of the transformer for the purpose of studying the transient state, and it has been shown that the frequency credit range of RLC model, 10kHz < f < 1MHz and the multi-conductor transmission line model is 1MHz ≤ f ≤ 5 MHz. To localize the partial discharge in the transformer winding, the detailed models are employed in this paper, and by using wavelet transform techniques, the location of PD is estimated in two models. Ultimately, the frequency validity limit of the transformer detailed models has been studied by using the proposed algorithm in localizing PD.
3. Wavelet Transform
Wavelet transform is a scale-time transform saving the data of the two domains of time and frequency. The transform of wavelet is of two types of continuous (CWT) and discrete (DWT). Here also due to the time-consuming calculation of CWT coefficients in computer applications, the DWT is used. Therefore, the discrete wavelet transform for X(t) signal is defined as follows:
In equation (3), x(t) and Ѱ are the initial signal and wavelet function, respectively. Also, kτ0 sj0 is the transform parameter, sj0 is the scale parameter and j shows level of wavelet transform. Among various types of wavelet discretization, binary discretization with S0=2 and τ0=1 is widely used. Discrete wavelet transform can be achieved well with a pair of low-pass and high-pass filter which is shown as L(K) and H(K) respectively. By using these filters, x(n) discrete signal decomposes in to high frequency and low frequency. When the components of high and low frequency are decomposed, the low frequency wavelet is only used in the next step. CA1 and CD1 are approximation coefficient and detail coefficient of x(n) signal in first level respectively and are calculated based on (4) and (5):
It is worth noting that the number of needed levels for discrete wavelet transform depends on the frequency characteristics of the signals under analysis. Finally discrete wavelet transform of the signal is calculated by gathering the output filters. Fig. 3, shows how to calculate the 3 level discrete wavelet transform by using the idea of filter bank for a x(n) desired signal.
Fig. 3.Decomposition of the signal in DWT
3.1 Parseval’s Theorem
In order to make the energy vectors small, energy of various levels of wavelet transform is used instead of wavelet transform coefficients. For this reason, we’ve made use of Parseval’s theorem. Parseval’s relation is:
Where N is the sampling period, and ak is the spectrum coefficients of the Fourier transform.
The signal energy can be obtained from relation (7):
In this relation, CAj,k equals approximation coefficients, CDj,k equals detail coefficients in j scale, N is the number of samples in the input signal, Nj is the number of samples at any analysis level and J is the number of analysis levels. In the right hand side of this relation, the energy is related to the high frequency and low frequency elements. Also, the left side of this relation indicates the whole signal energy.
With regard to the application of Parseval’s theorem in the analysis of discrete wavelet, the signals can be classified with regard to how energy is distributed in the high frequency and low frequency elements. Pa the energy distribution in low frequency elements and Pd the distribution of energy in high frequency elements are calculated from relations (8) and (9):
4. PD Localization Algorithm
The winding under study in this paper is related to the 20 kV transformers the details of whose dimensions have been mentioned in Table 1. The winding experimented in this paper was modeled and simulated by using RLC ladder network and MTL model along with the parameters obtained from FEM with the help of Matlab software. After modeling the transformer winding, it is necessary to enforce the partial discharge pulse to the winding. Hence, it is necessary to obtain the mathematical model of partial discharge pulse. Real PD pulses can be illustrated by combining two exponential functions as follows:
Table 1.The basic characteristics of the 20 kV transformer winding
Where, p and q are first and second time constants, respectively, so that p≥q. By changing the p and q quantities, the rise time and pulse width can be changed.
Based on this, with the help of Matlab software, signals with varied ranges were produced and enforced in various points of the winding in two models. Having the quantities of transfer functions and impulse supposition for the partial discharge current using relations (11) and (12), we can obtain the voltage amounts in the beginning and in the end of the winding for various quantities of injection locations of partial discharge pulse in the two models of RLC and MTL [5,11].
Where, The transfer function from PD source to the line-end (TFL), the transfer function from PD source to the neutral-end (TFN). V and V' are beginning and end of winding voltages, respectively and IPD is the PD pulse current.
Then, for PD localization based on the proposed algorithm, the first step is the application of PD pulse on each round of the windings. Fig. 4 shows the manner of the application of the PD pulse and storage of the voltage signal.
Fig. 4.The circuit of the application of a partial discharge signal on the sample winding.
Consider a reference PD pulse with 20ns rise time and 1μs width like Fig. 5. This pulse is similar to the specifications and results obtained from laboratory tests.
Fig. 5.PD pulse with 20ns rise time and 1μs width.
This pulse is injected to the first turn of winding and its corresponding response i.e. V1 (t) is measured. In this way, this is repeated for all the 38 turn of winding and their corresponding voltages V2 (t) to V38 (t) are recorded.
By doing this, a database of the winding response to the injection of the PD pulse in different locations and in two models of RLC and multi-conductor transmission line is obtained. For example, Figs. 6 to 9 show the V10(t) and V14 (t) signals for two models of RLC and multi-conductor transmission line, respectively.
Fig. 6.Representation of V10(t) signal in RLC model.
Fig. 7.Representation of V10(t) signal in MTL model.
Fig. 8.Representation of V14(t) signal in RLC model.
Fig. 9.Representation of V14(t) signal in MTL model.
In the third step, it is assumed that PD is occurred in an unknown location of the winding, for this purpose, another PD pulse with different specification like 0.35μs rise time and 1.15μs width is injected to one of the winding turn just like the turn number mth and Um (t) is recorded.
For example, Figs. 10 and 11 show the U14(t) voltage, which is the response of the winding after the application of the above PD pulse to the 14th turn in two models.
Fig. 10.Representation of U14(t) signal in RLC model.
Fig. 11.Representation of U14(t) signal in MTL model.
Comparing Um(t) and database in the suggested method, we are looking for a way to determine in which turns PD occurs.
Since the frequency content of the recorded voltages varies with time, in the fourth step of V1(t) to V38(t) signals and also Um(t) signal, the discrete wavelet transform is considered separately and at the first level.
Mother wavelet function which is used in this step is (Daubechies 1) db1. This is for similarity of recorded signals by db1 mother wavelet function. Getting a wavelet transform, each signal is divided in to two components. One of them is detailed component which includes high frequency; the other one is approximate component which includes low frequency. So, for PD identification, detailed component which includes high frequencies must be analyzed. For example Figs. 12 to 15, show the Detail coefficient of first level for V10 (t) and V14 (t) signals. Also Figs. 16 and 17, display the Detail coefficient of first level for U14 (t) signal.
Fig. 12.Representation of Detail coefficients of first level for V10(t) signal in RLC model.
Fig. 13.Representation of Detail coefficients of first level for V10(t) signal in MTL model.
Fig. 14.Representation of Detail coefficients of first level for V14(t) signal in RLC model.
Fig. 15.Representation of Detail coefficients of first level for V14(t) signal in MTL model.
Fig. 16.Representation of Detail coefficients of first level for U14(t) signal in RLC model.
Fig. 17.Representation of Detail coefficients of first level for U14(t) signal in MTL model.
In the fifth step, the correlation between detail Coefficient vector of Um (t) signal and detail Coefficient vector of V1 (t) signal is calculated. The energy of the vector resulting from this correlation is then calculated by the use of Parseval’s theorem. This process has also been repeated for wavelet transform output of V2 (t) to V38 (t). For PD localization, we are looking for a Vn(t) which maximum energy has happened for that. In this way, n is the turn in which PD is occurred and by doing this, PD is localized. The proposed algorithm is shown briefly in the Fig. 20. Tables 2 and 3 represents the results of simulation for U14(t) signal in two models of RLC and MTL. The numbers shown in Tables 2 and 3 express the value of energy. As it is specified in the tables, the maximum amount of energy has occurred for V14(t) Therefore, PD occurs in the 14th turn of winding.
Fig. 20.Different steps of proposed algorithm for localization of PD.
Table 2.Result of calculation of energy obtaining from correlation of detail coefficients between DWT (U14(t)), and DWT (V1(t)) to DWT (V38(t)) signals in RLC model. These results are also repeated for U20(t) signals. (Numbers are based on Micro Joule)
Table 3.Result of calculation of energy obtaining from correlation of detail coefficients between DWT (U14(t)), and DWT (V1(t)) to DWT (V38(t)) signals in MTL model. These results are also repeated for U20 (t) signals. (Numbers are based on Micro Joule)
All the mentioned steps are also repeated for 20th turn, and the results in Tables 2 and 3 indicate the success of the proposed algorithm in the two models. Also, in Figs. 18 and 19, frequency spectrum of V14(t) and U14(t) signals has been shown.
Fig.18.Frequency spectrum of V14(t) signal.
Fig. 19.Frequency spectrum of U14(t) signal.
5. Laboratory Results
The proposed algorithm was implemented on the practical sample in laboratory environment. Fig. 21 shows 20 kV winding which is under examination.
Fig. 21.The tested distribution 20 kV transformer winding
Since the above winding has 38 turns, the partial discharge pulse was injected into different points of the winding by PD Calibrator. The partial discharge signals were measured and the frequency range, discharge time duration as well as the amplitude value of this signal was recorded. These signals were injected into the winding under experimentation within the frequency range of kHz to MHz. First, we injected a PD pulse with specifications in Fig. 5 into the first to thirty-eighth turns, and the response signals of V1(t) to V38(t) were measured and stored by an oscilloscope. For example, Fig. 22 shows the voltage signal V1(t) resulting from the application of PD pulse to the first turn, and Fig. 23 shows the detail coefficients of wavelet transform of this signal in the first level.
Fig. 22.Representation of V1 (t) signal relevant to the practical example recorded in lab setting.
Fig. 23.Representation of Detail coefficients of first level for V1(t) signal relevant to the practical example recorded in lab setting.
In the next section, another PD signal with different amplitude was injected into one of the winding turns e.g. the 11th turn and the response signal was stored again. The results of PD localization by using the proposed algorithm on the 20 kV winding are presented in Table 4. Localization based on RLC model and the injected PD frequency components are within the range of kHz.
Table 4.Result of the calculation of the energy obtaining from the correlation of detail coefficients between DWT (U11 (t)), and DWT (V1 (t)) to DWT (V38(t)) signals in RLC model within the frequency range of 100 kHz < f < 1MHz. (Numbers are based on Micro Joule)
As it is observed in Table 4, the maximum value is devoted to V11(t). As a result, PD has occurred in the 11th turn and the PD location has been correctly estimated.
Next, the algorithm was also repeated for the injected PD frequency components within the Megahertz limit and the localization results were presented in Tables 5 and 6. As observed in Table 5, localization has estimated the location of PD with a minor error by using RLC model, which is due to the limitation of frequency range of RLC model in the range of megahertz. In order to solve this problem, and to obtain more precise results, localization has been studied based on multi-conductor transmission line (MTL). The results are presented in Table 6. As it is observed, since the maximum value is devoted to V11(t), PD is estimated in the 11th turn. The evaluation of the results obtained from the Tables shows that the PD localization by using the proposed algorithm determines the real location of PD along the length of the winding through appropriate estimation. Comparison of the results in Tables 4 to 6 showed better functioning of RLC model within the frequency range of kilohertz, and the MTL model within the frequency range of megahertz by using the proposed algorithm.
Table 5.Result of the calculation of the energy obtaining from the correlation of detail coefficients between DWT (U11 (t)), and DWT (V1 (t)) to DWT (V38(t)) signals in RLC model within the frequency range of f ≤ 5MHz. (Numbers are based on Micro Joule)
Table 6.Result of the calculation of the energy obtaining from the correlation of detail coefficients between DWT (U11 (t)), and DWT (V1 (t)) to DWT (V38(t)) signals in MTL model within the frequency range of f ≤ 5MHz. (Numbers are based on Micro Joule)
As it was observed, partial discharge is the most important source of error in transformer insulation. Therefore, the study of this phenomenon is very important. But, with regard to the frequency ranges of the partial discharge phenomenon, it should be said that the base of the correct and acceptable localization of partial discharge pulses in transformers is modeling the winding at high frequencies. It has also been shown in this paper that if the model employed within the frequency interval of the bandwidth of the PD measuring system is valid, good results will be obtained for PD localization.
With regard to the importance of localization of the partial discharge in transformers due to economic issues, and its greater longevity, proposing a fast, suitable, and secure method for the recognition of the PD location in the transformer winding seems necessary. Almost in all previous papers the both two terminal currents of the transformer winding are analyzed. For example in  the signals of both ends of the winding are registered and then using them, the two transfer functions between the PD location and the both terminals of the winding are obtained. It is shown that the zeros of these transfer functions differ when the location of the PD varies. But the proposed method in this paper is really simple and is also easily applicable because the wavelet packets toolbox is accessible in Matlab software and it has been shown that by taking voltage samples from the beginning of the winding and studying and analyzing the detail component obtained from the wavelet transform of these samples which include the partial discharge pulses, the location of PD is identified.
Therefore, in this paper, the 20 kV winding of the distribution transformer was first simulated and modeled with Matlab software by using RLC and the multiconductor transmission line models; and then, a new method for the localization of PD in the transformer winding was presented by using discrete wavelet transmission. It was also shown that if an unidentified PD signal were measured in the outlet ends of the winding, we could specify the location for the PD pulse injection by using wavelet transform and the energy resulting from the correlation of wavelet coefficients. This method was implemented for PD pulses with different widths to include the random nature of the PD. Hence, this method is considered a suitable and offline one in order to determine the location for the partial discharge in the power transformer. The proposed algorithm was studied through both simulation and experiment on a real 20 kV winding. The results showed the proposed method success and better functioning of the RLC model in frequency ranges under 1 MHz (f < 1MHz) and the MTL model in frequency range of MHz (f ≤ 5MHz) by using the proposed algorithm.