Frequency response signature analysis for winding mechanical fault detection of power transformer using sensitivity method

Abstract

Frequency Response Analysis (FRA) is an effective diagnosis approach to detect the winding mechanical faults in power transformer. For understanding the FRA signatures of transformer windings quantitatively, this paper uses the sensitivity method to study the characteristics of frequency response function (FRF) in equivalent circuit of transformer windings. The sensitivity expressions of FRF can be determined by adjoint network method and Tellegen theorem, which reflects the influence of electric parameter changes on FRA signature under winding faults and within a wide-frequency band (10 Hz to 2 MHz). To verify the sensitivity calculation, a two-winding transformer model with seven discs was established and taken as an example. The results indicate that FRF sensitivity data can provide the valuable insights about how the frequency responses are affected quantitatively with respect to the electric parameter changes or different winding faults. The method and sensitivity analysis discussed in this paper may be useful for diagnosing and predicting different mechanical deformation faults of transformer windings.

Keywords

Transformer winding equivalent circuit frequency response sensitivity analysis

1. Introduction

Power transformer is one of the most essential equipment in electric power systems. Once the transformer failure happens in service, there may have significant safety, economic or societal impacts. So, it is necessary to find reliable and effective methods to detect the healthy or faulty condition in transformer at an early stage. As Fig. 1 presents, it is well known that the winding mechanical deformation produced by huge and dynamical short-circuit electromagnetic forces is one of the most serious problems to make the catastrophic failure in power transformer [1–3]. To detect and diagnose any minor winding fault as soon as possible before transformer failure is very important.

Recently, the method of FRA is used widely to detect the winding mechanical fault, such as winding deformation or displacement [4–6]. In earlier days, Dick and Erven introduced and utilized the FRA idea to diagnose the transformer fault [7]. Until now, FRA measurement has been proved to be as an advanced and effective diagnosis approach in winding mechanical fault detection [8,9]. However, the detection of winding displacement or deformation using FRA normally relies on comparison between two measurement data or with the fingerprint during transformer factory tests. Besides, the current FRA diagnosis can only provide a nonphysical assessment based on explanation of waveform change and evaluation of statistical coefficients calculated from comparisons of the standard frequency responses. So, how to obtain the hidden and quantitative data relevant to winding mechanical faults is a hot and crucial research topic.

To deeply understand the relationship between the states of winding mechanical faults and frequency response signatures, our work tries to quantify this above corresponding relationship and find the method based on sensitivity analysis of FRF for quantizing and interpreting the FRA signatures. In this paper, the sensitivity of frequency responses to each electric parameter change in the equivalent circuit of transformer winding was constructed and determined by using adjoint network method and Tellegen theorem. In the end, a two-winding transformer model with seven discs is taken as an example for verifying the proposed method.

2. Transformer modeling and its frequency response analysis

It is worthwhile to convert the transformer and windings into an equivalent circuit with number of winding discs to investigate the detection method of winding mechanical faults and FRA signature interpretation. Figure 2 shows the 3-D transformer model with High Voltage winding (HV) and Low Voltage winding (LV), where the FRA measurement connection is given. The transformer components, such as windings, core, and insulation, can be represented by equivalent circuits with lumped parameters, such as series or shunt resistances, capacitances and inductances.

Fig. 1.

Typical mechanical fault of transformer winding.

Fig. 2.

Transformer model with two windings.

Thanks to each transformer winding has a unique frequency characteristic, which is sensitive to electrical parameter changes or winding mechanical fault in transformer. The winding mechanical faults usually include deformation and displacement caused by external events, such as tap changer failure and short-circuit or accidents in transport, which may be detected by using FRA measurement [10]. Generally, there are three kinds of terminal connection methods for transfer function (TF) measurements (End-to-end voltage ratio measurement, input admittance measurement and transfer voltage ratio measurements) in FRA of power transformer can be employed. In our work, to investigate the sensitivity of shunt capacitance (Clh) on the frequency responses of the transformer windings, the FRA test connection scheme namely transfer voltage ratio (TVR) measurements is selected as Fig. 2 shows. In this connection method, it is much more sensitive on frequency responses for detecting radial deformation and axial displacement compared with other connection methods, as [11] discusses. Besides, the terminal of voltages of HV and LV windings of transformer can be measured conveniently. Also, the radial buckling deformation of transformer windings may be reflected through the variations of capacitance parameter Clh between HV and LV windings. According to FRA connection of transfer voltage ratio measurement, the FRF is given by $\begin{eqnarray}\displaystyle \text{FRF}(L,C,w,\ldots )=\left|\frac{\dot{U}_{2}}{\dot{U}_{1}}\right| & & \displaystyle\end{eqnarray}$ (1) Where L, C are the electric parameters of this model, which denote the inductances and capacitances, respectively; w denotes the frequency of external excitation voltage $\dot{U}_{1}$ .

In Eq. (1), different electric parameters and different discs may have different influence on the TVR. Then, the typical mechanical faults may be simulated by changing the relevant electric parameters in the equivalent circuit model of transformer [12]. It is believed that if the winding is mechanically damaged, for example the mechanical deformation is produced by electromagnetic forces due to short-circuit fault, the physical position or shape of a winding or its discs will change. Then, the deformation of displacement of the winding or its discs will alter the electric parameter value of the real windings, thus making its FRA trace deviation. To analyze and provide valuable insights about how quantitative the frequency responses are with respect to the changes of electric parameter. Also, to find which parameters are the key factors that affect the FRA curves or signatures in transformer winding during deformation detection, the sensitivity analysis of FRF is discussed in this paper.

3. Sensitivity analysis based on adjoint technique

Generally, for determining the sensitivity, there are two methods including direct differentiation method and adjoint variable/network method. It is well known the adjoint variable/network method is more effectively than the former at the parameter influence quantities analysis. In this paper, the adjoint network method is employed.

3.1. Sensitivity definition of FRF

The sensitivity S of response function is defined as the rate of network response (FRF) to a certain elements or parameters changes expressed as follows [13]: $\begin{eqnarray}\displaystyle S_{p_{i}}^{\text{FRF}}=\frac{p_{i}}{\text{FRF}}\frac{\partial \text{FRF}}{\partial p_{i}} & & \displaystyle\end{eqnarray}$ (2) where p _i is the ith model parameters, such as the resistance R, conductance G, capacitance C and inductance L in equivalent network or circuit of the transformer winding.

3.2. Adjoint network method with Tellegen theorem

Director and Rohrer derived a sensitivity expression based on Tellegen’s Theorem [13]. The adjoint-based sensitivity analysis with multiple design parameters involves as little as a matrix transposition and a few matrix multiplications in addition to the original network analysis. To consider a linear network consisting of R, G, C, L and assume that one component parameter in this circuit has small changes, the relationship between increment of element parameters and increment of FRF approximately may be expressed as: $\begin{eqnarray}\displaystyle {\rm\Delta}\text{FRF}\approx \mathop{\sum }_{R}\frac{\partial \text{FRF}}{\partial R_{i}}{\rm\Delta}R_{i}+\mathop{\sum }_{G}\frac{\partial \text{FRF}}{\partial G_{i}}{\rm\Delta}G_{i}+\mathop{\sum }_{L}\frac{\partial \text{FRF}}{\partial L_{i}}{\rm\Delta}L_{i}+\mathop{\sum }_{C}\frac{\partial \text{FRF}}{\partial C_{i}}{\rm\Delta}C_{i}. & & \displaystyle\end{eqnarray}$ (3)

For adjoint network strategy, an adjoint network ${\tilde{N}}$ is firstly needed to be constructed by referencing the original network N, as Fig. 3 shows. It is noted that, the adjoint network ${\tilde{N}}$ has the same topology as the original one. Then, the voltage and current at each frequency can be determined by solving the original network and the adjoint network easily.

Fig. 3.

Original network (a) and adjoint network (b).

According to Tellegen’s Theorem, the partial derivative of response function to resistance, namely absolute sensitivity, may be obtained as $\begin{eqnarray}\displaystyle \frac{\partial \text{FRF}}{\partial R_{i}}=-\tilde{{\dot{I}}}_{Ri}{\dot{I}}_{Ri}. & & \displaystyle\end{eqnarray}$ (4)

Similarly, the sensitivity of FRF with respect to other electric parameters can be derived as $\begin{eqnarray}\displaystyle S_{R}^{\text{FRF}}=-{\dot{I}}_{R}\tilde{{\dot{I}}}_{R}\frac{R}{|\dot{U}_{2}|},\quad S_{G}^{\text{FRF}}=\dot{U}_{G}\tilde{\dot{U}}_{G}\frac{G}{|\dot{U}_{2}|}\quad S_{L}^{\text{FRF}}=-jw{\dot{I}}_{L}\tilde{{\dot{I}}}_{L}\frac{L}{|\dot{U}_{2}|}\quad S_{C}^{\text{FRF}}=jw\dot{U}_{C}\tilde{\dot{U}}_{C}\frac{C}{|\dot{U}_{2}|}. & & \displaystyle\end{eqnarray}$ (5)

In fact, each transformer winding has a unique frequency characteristic, namely FRA signature which is sensitive to electrical parameter changes. It is certain that some parameters data, such as the resistance R, capacitance C or inductance must be altered once the mechanical faults happens within a transformer winding. Also, parameter changes will lead to a frequency response trace deviation. Then, the knowledge of sensitivity of each electrical parameter to FRF is essential to understand the FRA signature changes [14].

4. Example of a two-winding transformer model

4.1. A 7-Discs transformer model

Take a two-winding transformer with continuous winding as an example, the continuous winding can be visualized as being mapped onto a series of discrete nodes, starting from line to neutral end, as Fig. 4 gives. In the HV and LV windings of this figure, each disc comprises a series capacitance (Csh and Csl), grounded capacitor (Cgh and Cgl), and shunt capacitance (Clh) between HV and LV winding, self- and mutual-inductance (Lhs, Lls and Mij, Mi*j*), where the resistance and conductance are neglected. It is noted that the dielectric insulation (such as transformer oil) between LV winding and core, HV winding and tank is simulated by the grounded capacitance (Cgh and Cgl), respectively. All the electric parameters include inductance and the capacitance can be calculated by using FEM or analytical method on the basis of winding geometry and material properties [14].

Fig. 4.

Transformer model with 7 discs winding as the original network.

Using TVR connection as shown in Fig. 2, the FRA measurements are performed for 800 frequency points. The input and output impedances of measurement cables are selected as 50 Ω, which indicate the R _i and R ₂ in Fig. 4. Then, using the adjoint network method obtains the sensitivity data of frequency response with respect to each electric parameter.

4.2. Sensitivity results

To evaluate the FRA measurement data, the whole frequency band (10 Hz ∼ 2 MHz) may be conveniently divided into three bands, namely low- (<20 kHz), mid- (20–400 kHz) and high-frequency (>400 kHz) band. Figure 5 illustrates the detailed sensitivity calculation results to each kind of electric parameters, respectively.

Fig. 5.

Sensitivity of FRF with respect to each electric parameter at a wide-frequency band (1 Hz–2 MHz). A, B, C, D, E, F and G are the plots of FRF sensitivity to Clh, Cgh, Cgl, Csh, Csl, Lhs and Lls, respectivly.

Fig. 5 (Continued).

Generally, the division of frequency domain into sub-bands depends on the geometrical size of the transformer. However, the division of frequency domain into sub-bands in this paper just to distinguish which kind of parameters plays the key role at the given sub-bands, such as the inductance dominate the transformer winding response within the low frequency range (<20 kHz), while within the medium frequency range (20–400 kHz), the combination of inductive and capacitive components results in multiple resonances, as Fig. 5(F) and (G) plot.

In Fig. 5, it can be seen that there are different sensitivity magnitudes at a fixed frequency for different electric parameters. It is noted that the FRF sensitivity for the shunt capacitance and grounded capacitance are larger than that of the series capacitances. Besides, the maximum magnitude of the FRF sensitivity appears around 15.5 kHz for most electric parameters. Obviously, it can be seen that there are fairly smaller sensitivity magnitudes at high-frequency band compared with the sensitivity at low- and mid-frequency band. At the high-frequency sub-band, there is a fairly larger sensitivity for the first component in each type of electric parameter.

5. Verifying for the sensitivity results

In this work, for verifying the sensitivity results and understanding the FRA signature effectively, different types of winding deformations and displacements are reflected as shifts of the resonant frequencies when being compared with the free-fault response. The magnitude of the frequency response in Bode diagram is derived as $\begin{eqnarray}\displaystyle \text{FRA}_{\text{Magnitude}}=20\log _{10}\left|\frac{\dot{U}_{2}}{\dot{U}_{1}}\right|=20\log _{10}(\text{FRF}). & & \displaystyle\end{eqnarray}$ (6)

In this section, the FRA simulation is carried out to verify the sensitivity results. Different winding deformation forms, such as the shorted turns fault, axial displacement, radial deformation or hoop buckling, may be simulated by modifying the relevant electrical parameters in the transformer model [15].

5.1. Shorted turns fault

For evaluating the calculated sensitivity data conveniently, the FRA measured from a healthy transformer is taken as the fingerprint. The shorted turns fault may be simulated by short-circuiting the series resistance and the series inductance [4]. In this paper, the self-inductance Lh5 is decreased to 30% for No. 5 disc to simulate the shorted turns fault. Figure 6 illustrates the comparison data between the fingerprint and fault results. It is seen that the FRA signature changes slightly at frequency range <30 kHz or >600 kHz. The above result indicates that there is lager sensitivity during mid-frequency band that that of during low- and high-frequency bands, which is agreement with the sensitivity results in Fig. 5.

For further verification, we use both Ll3 and Ll5 changes to simulate the large shorted faults of the transformer windings. Figure 7 gives the comparison results of FRA simulation between the fingerprint and frequency response, which are simulated by decreasing both Ll3 and Ll5 30% in LV windings. It can be seen that there is no significant variation for the simulated FRA at frequency lager than 600 kHz. However, for the FRA variations, as shown in Fig. 7(b), there is a lager change of Lls3 than that of Ll5. While during mid-frequency band, as described in Fig. 7(c), there is a lager change of Lls5 than that of Ll3. The FRA signature variations show the agreement between the FRA simulation and sensitivity calculation as shown in Fig. 5(G).

Fig. 6.

FRA signature with the self-inductance Lh5 decreasing 30%.

Fig. 7.

Comparison of shorted turns fault on the frequency response analysis signature with different inductance parameters. (a) Full- frequency response. (b) Low- frequency response. (c) Mid- frequency response.

5.2. Axial displacement and radial buckling

Axial displacement fault in the transformer windings can be simulated by changing the series capacitance (Csh and Csl) and mutual inductance (Mij) between two discs. The radial or hoop buckling fault may be simulated by changing the parameter of shunt capacitance (Clh) between HV and LV winding.

Fig. 8.

Comparison of axial displacement and radial/hoop buckling effect of on the FRA signature.

In this study, to understand the different sensitivity response in different deformation types the FRA results are compared between changing series capacitance and shunt capacitance, as Fig. 8 describes. The influence of shunt capacitances Clh2-8 which are increasing 10% (increase from 5 nF to 5.5 nF) between HV and LV winding on FRA signatures have been shown in Fig. 8. This figure also gives the influence of series capacitances Csh1-7 increasing 10% in HV winding (from 0.013 nF to 0.0143 nF). It is noted that FRA traces almost has no variation after changing the parameters of Csh compared with the fingerprint over the entire frequency range. These results also indicate that the axial displacement fault have less effect on the FRA signature.

6. Conclusion

Through the transformer modeling, sensitivity calculation and FRA simulation studies, it is found that most of the electric parameters have a higher sensitivity to FRF at the mid-frequency sub-bands than that of at low- or high-frequency sub-bands. Different winding deformation types, location and extent have different effects on the FRA trace, which may be analyzed by using quantitative sensitivity data. The sensitivity analysis of FRF can provide valuable insights about how quantitatively the frequency responses affect the FRA signatures with respect to electric parameters change. Besides, which kind of the parameter are key factors that affect the FRA curve is also may be obtained. The sensitivity analysis may aid in establishing a standard code for FRA signature interpretation for locating the deformation position, as well as identifying the extent of changes in transformer fault diagnosis. The method proposed in this paper is applicable to other kind of transformer and its windings as long as an appropriate circuit model is established.

Footnotes

Acknowledgements

This research was supported by Hubei Provincial Natural Science Foundation of China (2019CFC893), Hubei Superior and Distinctive Discipline Group of “Mechatronics and Automobiles” (XKQ2019015) and Excellent Young, Middle-aged Science, Technology Innovation Team Project in Higher Education Institutions of Hubei Province (T201815).

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