Investigating the applicability of the finite integration technique for studying the frequency response of the transformer winding

Electrical Power and Energy Systems 110 (2019) 411–418

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Investigating the applicability of the finite integration technique for studying the frequency response of the transformer winding

T

⁎

Mohammad Hamed Samimia, , Philipp Hillenbrandb, Stefan Tenbohlenb, Amir Abbas Shayegani Akmala, Hossein Mohsenia, Jawad Faiza a b

High Voltage Institute, ECE Department, University of Tehran, Tehran, Iran Institute of Power Transmission and High Voltage Technology (IEH), Stuttgart University, Stuttgart, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Fault diagnosis Finite integration technique Frequency response Indices Power transformers testing Simulation

The frequency response analysis (FRA) is a promising diagnostic method for detecting mechanical faults inside a power transformer. Despite the standardization of the FRA measurement procedure, the interpretation of the results is still a subject of study. The reason lies in the fact that the frequency response is different from case to case. Therefore, for interpreting a transformer FRA result, the effects of various possible mechanical changes on the FRA trace of that particular transformer should be known. Additionally, real mechanical deformations cannot be executed for obtaining the required information due to the destructive nature. Hence, the transformer winding models are utilized instead to study the frequency response. While the circuit model has been extensively discussed in the literature, there is little focus on the numerical methods for the FRA purpose. To take a step forward, this contribution investigates the applicability of the finite integration technique for the FRA studies. The presented model can simulate different fault types for the FRA interpretation while it has also other applications regarding the frequency response. In this contribution, the FRA traces are derived directly from a finite integration model which is an important improvement compared with the previous papers. The agreement between the simulation results and the experimental data indicates that the proposed model has the potential of providing a powerful tool to decipher the transformer frequency behavior.

1. Introduction Power transformers have significant importance in the power system and require careful monitoring and diagnosis to ensure their healthy operation. Accordingly, different diagnostic methods are established for transformers [1–4]. Both electrical and mechanical faults can be fatal for transformers, and the frequency response analysis (FRA) is a method that is mainly utilized to detect the mechanical faults though its diagnosis does not limit to mechanical investigations [5,6]. Early detection of such faults can save the transformer from a consecutive, catastrophic, electrical failure [7,8]. Although the FRA technique has a standardized measurement procedure, its interpretation is still a subject of study [9,10]. In this technique, a fingerprint trace is recorded from the transformer in the intact state which serves as the reference for the comparison. After an accident, another trace is captured and compared with the fingerprint. The difference between these two traces indicates the mechanical change inside the transformer [11]. The main challenge of FRA interpretation is that a general rule ⁎

cannot be settled since each transformer has a special construction and FRA behavior [12]. In other words, the mechanical alterations introduce changes in the FRA traces which can be different between transformers. Consequently, it is important to study these changes in each transformer separately [13]. To study the effects of mechanical changes, the most reliable procedure is the experiment. However, this attitude is not practical since it impairs the transformer and, therefore, modeling approaches should be replaced. The circuit model is one of the main modeling approaches in the FRA studies [14]. In this model, each section of the winding is represented by various circuit elements [15]. To utilize this model, the changes in the transformer structure are converted into the corresponding variations in the circuit elements first. Then, the elements variations are imported into the circuit model to investigate the changes in the FRA trace [16]. However, the conversion process introduces extra error in the model [17]. Moreover, including some mechanical faults in the circuit model is complicated which is the main drawback of the circuit model. Another approach is to estimate the amount of circuit elements from the fields calculated by the finite

Corresponding author. E-mail address: [email protected] (M.H. Samimi).

https://doi.org/10.1016/j.ijepes.2019.03.015 Received 7 November 2018; Received in revised form 1 February 2019; Accepted 11 March 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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element method (FEM) [18,19]. For instance, [20–22] determine the elements based on the energies stored in the calculated fields. It is noteworthy that there are also some procedures for accurate resonance analysis of the transformer which utilize the circuit model [23]. In all previous works, the output of a FEM simulation is turned to the corresponding elements of the circuit model as an intermediary procedure, and the FRA is derived then from the circuit model. The current contribution, by contrast, demonstrates an approach to derive the FRA traces directly from the windings simulation by the finite integration technique (FIT). In the proposed approach, a sinusoidal voltage source is connected to the winding model similar to the FRA measurement setup to sweep the frequency in the corresponding range. Then, the FRA trace is derived by measuring the voltages at the winding terminals, analogous to the sweep frequency response analysis. Different mechanical deformations can be implemented in the windings model to discern their effects on the FRA without any need for the intermediary circuit model. Correspondingly, it is possible to study the mechanical alterations which are difficult to be included in circuit models. The results of this paper show that the model responses are in good agreement with the experimental measurements especially for the purpose of predicting the numerical indices in winding movements. In fact, the main contribution of this work is showing that an FIT model can predict the amounts of numerical indices with sufficient accuracy. Currently, where differences exist between the new FRA trace and the fingerprint, the extent of the mechanical movement is not exactly known since there are no experimental data available. However, the FIT can correlate the FRA differences and the mechanical movements in terms of numerical indices. This is the missing stage in the FRA interpretation procedure, and the novelty of the current contribution is to provide the required data using the proposed FIT model. In the rest of the paper, the experimental setup and the windings utilized for the measurements are introduced first. Then, the FIT model and its characteristics are addressed. Afterwards, the results of the FIT model are compared with the measurements. In the next step, the consistency of the model in predicting the indices behavior in the winding movement is presented. Finally, another application is addressed to establish the efficacy of the proposed model.

Fig. 1. Experimental setup: (a) The windings and axial displacements in the HV winding, (b) the diagram of the windings. The dimensions are in cm.

2. Experimental setup II configurations. In the EE, the signal is injected to the HV terminal, and the voltage at the HV neutral is measured. In the case of the II configuration, the voltage of the LV terminal is measured while the neutrals of both HV and LV windings are grounded. The FRA traces are evaluated in the frequency range 100–1000 kHz because of the following reasons. First, the FRA traces do not show any resonance point below 100 kHz due to the absence of the magnetic core and, therefore, this region is of minor importance. Second, the effect of core magnetization is completely negligible in the chosen range since it is far beyond the frequencies where the core magnetization affects the FRA. This is important because the experimental setup does not contain the core and, thereby, the FRA in the lower frequencies may not be comparable with a real case. Third, this work is focused mainly on the axial displacement, and for this mechanical fault, the lower frequencies do not have significant importance [14]. Finally, the frequencies higher than 1 MHz are skipped since it is influenced mainly by the measurement setup and, accordingly, it is not recommended by technical committees [9,10,26]. In addition to the intact state of the windings, axial displacements are implemented in the experimental setup as one of the common mechanical faults. Two situations are considered regarding the axial displacement. In an exaggerated case, the HV winding is shifted 4 cm vertically in 5 mm steps using the spacers demonstrated in Fig. 1a. Although this amount of axial displacement is not acceptable for a winding, the displacements are exaggerated here since it leads to more

In order to validate and compare the FIT model with real measurements, an experimental setup is employed in this research which consists of two windings. The HV winding is a continuous disc type with 660 turns in 60 discs whereas the LV winding is a helical type winding, 24 turns with 12 parallel conductors in each turn. Fig. 1 shows the windings, their diagrams, and corresponding dimensions. Two grounded aluminum cylinders are used inside and outside of the windings to model the core and tank, respectively. Since the FRA is evaluated above 100 kHz, and the skin depth is very small in this range (tens of micrometer), replacing the core with the hollow metallic cylinder does not impair the results [14,6,16,24]. Furthermore, the same situation with the hollow cylinder is considered in the simulation, i.e. the simulated model also contains no core, and a hollow metallic cylinder is utilized instead. In other words, the model is similar to the experimental setup, and if an effect is somehow missing in the experimental setup, it is also missing in the simulation model. Due to these facts, the results of the simulation are comparable with the experimental data. The FRA traces are recorded with the sweep frequency response method [25]. In this method, the frequency of a sinusoidal source is swept while the voltages of two points are measured, and the ratio of these voltages is considered as the transfer function. Two most used connection schemes recommended by standards are implemented in this research: end-to-end (EE) and inductive inter-winding (II) connections [26,27]. Fig. 2 illustrates the measuring diagram of the EE and 412

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Fig. 2. Connection schemes for measuring the end-to-end (EE) and inductive inter-winding (II) frequency response. The ratio of U2/U1 is considered as the transfer function.

each mesh can contain more than one material and, thereby, the meshes can be larger reducing the amount of computational power needed to solve the problem. As a result, this technique is suitable for complex geometries like a transformer winding, and this is the motivation of employing the FIT as the solution method. This method is implemented in the computer simulation technology (CST) software and, hence, the winding is modeled in this software. The windings are simulated in the circuit and components module of the software CST STUDIO SUITE™ 2016. The modeling is performed in a two-step simulation. First, a geometry model is created which exhibits the key geometric features of the experimental setup. The model comprises the HV and LV windings including the insulation papers, the radial spacers, the inter-winding press board as well as the metallic cylinders. Excitation ports are set at both open ends of the two windings between the grounded cylinder and the winding itself. Based on model order reduction, a broadband frequency-domain electromagnetic field computation is performed, resulting in a multi-port network model which characterizes the electromagnetic properties of the geometric model. In the case implemented here, the output is a two-port module since each of the HV and LV windings has two connection points: the terminal and the neutral. After the two-port module is generated, it is possible to attach normal circuit elements to it in the CST STUDIO. In the second step, a sinusoidal voltage source plus termination resistors are attached to the two-port module to calculate the FRA trace. These elements are attached according to Fig. 2 while the proper points are grounded in the case of inductive inter-winding connection. The same dimensions of the test setup are taken to model the windings in detail, Fig. 3. The HV winding is drawn with 60 discs and 11 turns in each disc. However, the LV winding is simplified to ease the mesh generation and model solution. Since the insulation paper between the parallel conductors of the LV winding is very thin, it obliges the minimum mesh dimensions to be very small. This increases the total mesh number and the computation time exponentially. In order to remove the inner paper layer of the LV winding from the model, each 6 parallel conductors are replaced with a single conductor. Accordingly, the LV winding has 24 turns with two parallel conductors instead of 12 parallel conductors in each turn. The size of the equivalent conductor is defined equal to the sum of the six parallel conductors as displayed in Fig. 4. Finally, two grounded hollow cylinders are also considered inside and outside of the windings analogous to the test setup.

extensive changes in the FRA traces followed by a simpler comparison with the FIT model. In a realistic case, the HV winding is shifted in 0.7 mm steps till 2.8 mm. The results of both cases are reported in the results and discussion section. 3. Finite integration technique The main idea of employing the finite integration technique is to replicate the real procedure of measuring the transformer FRA trace. After modeling the windings, it is possible to connect an imaginary sinusoidal source to the terminal of the HV winding and to sweep its frequency for calculating the transferred voltage at the other end of the winding analogous with the real measurement procedure. Finite integration technique was first introduced by Weiland in 1977 [28]. This technique, which can be used for field calculation in time and frequency domains, utilizes a discrete form of Maxwell equations in their integral notation [29]. The integral form of Maxwell’s equations can be written as follows:

⎯ ⎯→ ⎯ ⎯→ ∂ ∮ E · ds = − ∂t ∂A ⎯ ⎯→ ⎯ ⎯→ ∮ H · ds = ∂A

⎯→ ⎯ ⎯⎯⎯→

∬ B · dA , A

⎯⎯⎯⎯→

∬ ⎡⎢ ∂∂Dt A

⎯→ ⎯ ⎯⎯⎯→ ∯ B · dA = 0, ∂V

⎣

→⎤ ⎯⎯⎯→ + j ⎥· dA , ⎦

(1)

(2)

(3)

⎯→ ⎯ ⎯⎯⎯→ ∮ D · dA = Q.

(4) ⎯→ ⎯ → ⎯→ ⎯ where E and H are the electric and the magnetic field, respectively, j ⎯→ ⎯ is the electric current density, Q is the total electric charge, D is the ⎯→ ⎯ displacement field, and B is the magnetic flux density. In the above equations, t is the time, and V is a volume with a closed boundary ⎯⎯⎯→ surface ∂V where dA denotes the vector element of this surface. ⎯→ ⎯ Moreover, A is a surface with a closed boundary curve ∂A while ds is the infinitesimal element of this curve. In the FIT, the calculation domain is split into several small domains by the meshing system. Then, Maxwells equations are solved while the boundary conditions satisfied in interfaces of the meshes. Using the FIT,

∂V

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skin effect losses changes with the square root of the frequency. Consequently, the surface impedance should be modeled with a f function. By using this estimation, the results show good agreement with the experiment. In addition to the metal losses, the dielectric losses should also be considered in the model. As the dielectric is not ideal, it has the dissipation which is, mainly, due to the dipoles polarization. The parameter which determines the dielectric losses is the dielectric dissipation factor or tanδ defined as follows:

tanδ =

Fig. 4. Simplying the LV winding. The figure shows the cross section of one turn.

The conductor losses due to the skin and proximity effects should be considered in the model as well as the dielectric losses in the insulation paper. Otherwise, the results are neither in accordance with the measurements nor useful. First, the loss due to the skin effect is discussed. In each frequency, the skin depth determines to what extent the electromagnetic field penetrates the metal. The skin depth can be calculated as follows:

ρ , πfμ0 μr

(6)

where ε″ and ε′ are the imaginary and real parts of the dielectric permittivity, respectively. The tanδ can be directly defined for a material in the CST. The dependency of the tanδ of the oil impregnated paper on frequency has been measured and discussed in [33,34]. For the frequency range of the current study, no resonance exists in the imaginary part of the permittivity (ε″ ) and, therefore, the changes of the tanδ versus the frequency can be neglected [14]. As a result, a constant dissipation factor should be defined for the dielectrics in the model. It is noteworthy that by considering a constant dissipation factor, the dielectric loss increases with the frequency to the first power. After defining the two aforementioned parameters, the rest of the model is dependent only on the geometry which has been detailed previously. The geometry has no symmetry and, therefore, the symmetry plane cannot be defined which results in the need for solving the whole geometry. Tetrahedrons are employed to mesh the geometry leading to approximately 2.3 million mesh cells in the final model. The broadband AC solver with the ‘fast reduced order model’ option and the 1st order direct type solver is implemented in the software to sweep the frequency in the range of 100–1000 kHz. It is worthwhile to mention that CST can calculate the broadband frequency behavior from a finite integration model in only one solving step which eliminates the need for calculating multiple frequency samples and greatly speeds up the simulation process. This is the main reason of using CST in this contribution rather than other typical FIT software.

Fig. 3. The geometry of the windings in CST STUDIO.

δ=

ε″ , ε′

(5) 4. Results and discussion

where ρ , f , μr , and μ0 are the resistivity of the conductor, frequency, the relative magnetic permeability of the conductor, and the permeability of the free space, respectively. As the frequency increases, the skin depth declines and, therefore, the field appears in a thinner layer of the metal. In order to obtain enough accuracy in a numerical model, the maximum mesh dimension in the metal has to be one order of magnitude smaller than the skin depth. This leads to highly minuscule meshes which, accordingly, raises the mesh numbers and the computational power needed to solve the problem. In order to overcome this issue, a computational technique is used which is called the surface impedance technique [30]. As it is explained above, when the frequency increases, the field only penetrates to a thin layer of the metal and, thus, the inner part of metal has no contribution and can be omitted from the simulation. In other words, since the penetration depth is small, the volumetric element corresponding to the metal parts can be replaced with sheets having zero width. This attitude is correct since the capacitive role of the metal parts is included. However, in addition to the capacitive effect, the eddy losses in the metal should be considered since these losses change the magnitudes of an FRA trace. Accordingly, the surface impedance technique replaces the volumetric metal parts with thin sheets while including the corresponding losses in the sheet surface impedance [31]. In other words, by defining a surface resistance for the sheet, the equivalent loss of the metal is also involved in the model. The same technique is utilized in this contribution in order to reduce the simulation time. Ref. [32] describes that the resistance which models the

4.1. FRA traces in the intact state Both the measurement and simulation results for the intact state of the windings are demonstrated in Figs. 5 and 6 corresponding to the EE and II connections, respectively. The figures indicate the consistency of the model with real measurements. It should be mentioned that there are some resonance frequency and magnitude mismatches between the simulation and the measurement. One of the reasons that can be stated for the frequency and magnitude mismatches, is the LV winding simplification in the model. In one hand, the simplification results in a different series capacitance in the LV winding which, in turn, affects the resonance frequencies. On the other hand, the FRA magnitude is influenced vastly by the losses, and the simplification of the LV winding leads to differences in both the copper and dielectric losses. The dielectric loss is different since the volume of the paper in the model is not the same as the real case. The copper loss is also distinct from the real case because the proximity and skin effects are different in the model due to less parallel paths for the current. In other words, the effective cross sections of the conductors that carry the current are unequal resulting in altered losses. Nevertheless, the good agreement proves the potential of the proposed method in modeling transformer windings for FRA applications. The next sections provide further proofs to confirm this claim.

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Measurement Simulation

-35 -40 -45 -50

Magnitude (dB)

Phase (degree)

20 mm

30 mm

40 mm

-50

(a)

-30

50 -50 -150 200

400

600

Frequency (kHz)

800

-30

-40

-50

1000 -60

(b) 200

400

600

800

1000

Frequency (kHz)

Fig. 5. Comparing the simulation results of the EE configuration with the measurements.

Magnitude (dB)

10 mm

-40

-60

150

Fig. 7. The II FRA traces corresponding to different axial displacements: (a) Simulation results, (b) measurements.

Measurement Simulation

150

predicting the behavior of numerical indices towards mechanical changes, the axial displacement is implemented in both the experimental setup and the model. For the axial displacements, the two cases similar to the experiments are implemented in the simulation. The traces corresponding to the exaggerated case are shown in Fig. 7. As can be seen, the model behavior towards axial displacements is completely in accordance with the experimental results. Two numerical indices, namely correlation coefficient (1 − CC ) and Euclidean distance (ED) [36], are extracted from each step of the axial displacement and the intact case as follows:

100

1 − CC = 1 −

-35 -40 -45 -50

Phase (degree)

0 mm

-30

Magnitude (dB)

Magnitude (dB)

-30

N

∑i = 1 X (i) Y (i) N

N

∑i = 1 [X (i)]2 ∑i = 1 [Y (i)]2

, (7)

50 0

ED = X − Y = 200

400

600

Frequency (kHz)

800

1000

N

∑i =1 (Y (i) − X (i))2 ,

(8)

where Y and X are the magnitude vectors of the new FRA trace and its fingerprint, respectively, and Y (i) and X (i) are the i-th elements of these vectors. The amounts of these indices for the II configuration, as an instance, are extracted from the simulation and experiment and demonstrated in Fig. 8 versus the axial displacement. The insets of figures demonstrate the results of the realistic case more closely. The results show that despite mismatches between the simulation and measurement in each step, the indices corresponding to the alteration amounts are in excellent agreement with each other. Particularly, the permissible amounts of mechanical change are less than 10 mm where the simulation results adhere to the measurements [37]. This shows the potential of the simulation method to replace destructive tests since the output of the indices match for both cases. Two points should be stated here. First, this method needs to be examined on several real transformers with multiple windings and different mechanical faults to prove its complete functionality. Second, the proposed method predicts the behavior of indices by simulations, but for defining the permissible threshold, a set of mechanical simulation is needed plus the material properties to determine the amounts of critical displacements. Such simulations are normally carried out to calculate the short-circuit withstand capability of transformers [8] and, therefore, the pertinent data are defined in the design stage. There are also rules of thumbs for them. As an example, if the material properties and the mechanical simulation leads to the outcome that 0.3% of the winding height is the critical displacement, the amount of an index

Fig. 6. Comparing the simulation results of the II configuration with the measurements.

4.2. Predicting the numerical indices Implementing numerical indices is among the methods proposed for the FRA interpretation, where an index is calculated from the new FRA trace and its fingerprint [35]. Afterwards, a faulty case is identified if the index exceeds a threshold [2]. The dilemma is that the threshold is different from transformer to transformer, and the experimental data are not available to define the threshold. The reason lies in the fact that the required experiments are destructive and not allowed to be carried out on transformers. Moreover, if a piece of data becomes available due to an accident, it can be only applied to a transformer with a similar construction. Up to now, no promising method is proposed in the literature to predict the numerical indices and set their thresholds. However, the proposed 3D simulation can be beneficial for estimating the behaviors of the indices. Such a simulation can be employed by manufacturers since they access the construction data and, therefore, can define the indices threshold for each transformer using the 3D simulation which is not a costly task. In order to prove the potential of the proposed simulation in 415

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x 10

3

x 10

4

2

3

1

2

0 0

150 10

-5

1

2

3

1 0 0

Euclidean Distance

Correlation Coefficient

-3

5

Measurement Simulation 10

20

30

Axial displacements (mm)

100

5 0 0

2

3

Measurement Simulation

0 0

40

1

50

10

20

30

Axial displacements (mm)

40

Fig. 8. Comparison of the simulation results with the measurement in terms of two indices: correlation coefficient (left) and Euclidean distance (right).

different types and structures of windings to generalize the sensitivity results. It is obvious that a generalized recommendation needs to be examined on different cases beforehand, and this is not conceivable since there is no wide access to different windings in the case of FRA. If a 3D model can provide reliable and precise data, it is possible to simulate different windings and derive a generalized recommendation. This section confirms that the presented model can perform this task satisfactorily. The sensitivity analysis is carried out easily using the presented model. In fact, when the simulation is carried out for the first time, the two-port module is generated, and changing the surrounding circuit elements does not require another simulation. In other words, the software converts the 3D geometry to a two-port module with corresponding scattering parameters. For the next step, the software only deals with the two-port module and not the geometry. Thereby, while the geometry is unchanged, there is no need to run the simulation again. In the study of different connection schemes, only the surrounding circuit elements change not the geometry. Therefore, all possible configurations can be derived in few seconds after the first simulation. In this section, the ten different configurations shown in Fig. 9 are implemented for the FRA measurement. The voltage is injected to the winding where the arrow is depicted, and the ratio of U2/ U1 is considered as the transfer function similar to Fig. 2. Therefore, the configurations (a) and (h) in Fig. 9 correspond to the EE and II configuration of Fig. 2, respectively. All these configurations are implemented in both the experimental setup and the simulation model. After implementing 10 mm of axial displacements as an instance, the amount of change corresponding to each configuration is extracted using the 1 − CC and ED indices. Then, the changes are normalized using the Lever rule by calculating the change ratio (CR) index [7,38]:

Fig. 9. Different configurations implemented for comparing their sensitivity toward axial displacement.

calculated by simulating the intact state and the state of 0.3% axial displacement can be set as the index threshold. The overall procedure for the transformer diagnosis using the proposed method can be summarized as follows: 1. The transformer windings are simulated in their intact situation, and an FRA trace is obtained. 2. The amount of permissible deformation or displacement is defined based on mechanical simulations or from the rules of thumbs given by the manufacturers. 3. The transformer is simulated again while the permissible amount of mechanical change is exerted in the model. A second FRA trace is attained in this step which corresponds to the faulty case. 4. An index is calculated from the both FRA traces. This value is set as the threshold for that index. These steps are normally performed when the transformer is new. 5. When the transformer experiences a severe short circuit current or an accident, an FRA trace is captured. 6. The amount of the index is derived from the new FRA trace captured and the fingerprint. 7. If the index value surpasses the threshold, the transformer is faulty and should undergo a repair procedure. Otherwise, the transformer is sound and ready to continue the service.

CR =

1 ⎛ (1 − CC ) − (1 − CC )min ED − EDmin ⎞ + × 100. 2 ⎝ (1 − CC )max − (1 − CC )min EDmax − EDmin ⎠ ⎜

⎟

(9)

where subscript min and max declare the minimum and maximum of the indices in each set of data. Fig. 10 demonstrates the CR of configurations due to axial displacement in comparison with each other. Because the Lever rule is applied, the least sensitive connection has a 0% while the most sensitive one has a CR equal to 100%. Both experimental and simulation results are included in the figure showing that the configuration (j) has the best sensitivity to the axial displacement. As can be seen, the simulation results follow the experimental case very closely which indicate that the presented model has also the potential to be applicable for the purpose of sensitivity analysis of different configurations in the FRA method. Correspondingly, different transformers can be studied using the presented 3D model to draw a general conclusion about the most sensitive connection in detecting the mechanical changes.

4.3. Comparing different FRA connection schemes It is possible to utilize the presented 3D model for comparing the sensitivity of different connection schemes in the FRA method. Such a study is presented in [7] for one winding using experimental measurements. If the results of the 3D model are in accordance with the experimental results, it is possible to use the model for examining 416

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CR (%)

80

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Experiment Simulation

60 40 20 0 a

b

c

d

e

f

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Different Connections

h

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Fig. 10. Comparing the relative sensitivity of different configurations to axial displacement.

5. Conclusion This paper presented a 3D simulation method for studying the FRA of transformers. The main purpose of this contribution was to implement a model instead of experiments to study the effect of faults on the FRA of a transformer since the corresponding experiments cannot be performed on transformers due to their destructive nature and, thereby, a substitution is needed. The most important characteristic of the proposed model is its ability to extract the FRA directly from the finite integration simulation. Axial displacements were employed as a common mechanical fault to demonstrate that the indices extracted from the simulation agree with the measurement. This stage is missing in the FRA interpretation since the indices are different from case to case, and the experimental data are not available. The proposed 3D simulation, however, can be used for predicting the indices in different cases. Additionally, comparing the sensitivity of different FRA configurations was presented as another application of the model. The promising performance of the model in different applications shows that it has the potential to be utilized in future FRA studies where the real experimental data are rarely available. The next steps of the research, which can be carried out by the proposed model, are: 1. implementing different mechanical faults in the model to investigate their effects on the FRA traces. 2. calculating different indices to determine the best ones by comparison. 3. modeling the maximum permissible mechanical changes and assessing the corresponding variations in the indices to define their thresholds. 4. modeling other types of typical windings such as interleaved and layer windings and implementing the previous steps for them to determine the behavior of indices due to mechanical changes in different cases. It is noteworthy that only after studying different transformers and mechanical changes in them, a proper objective interpretation can be established for the FRA technique, and the proposed model serves as a tool to accomplish this goal. References [1] Theocharis A, Popov M, Seibold R, Voss S, Eiselt M. Analysis of switching effects of vacuum circuit breaker on dry-type foil-winding transformers validated by experiments. IEEE Trans Power Delivery 2015;30(1):351–9. https://doi.org/10.1109/ TPWRD.2014.2327073. [2] Tenbohlen S, et al. Diagnostic measurements for power transformers. Energies 2016;9(5):347. https://doi.org/10.3390/en9050347. [3] Samimi MH, Akmal AAS, Mohseni H, Tenbohlen S. Detection of transformer mechanical deformations by comparing different fra connections. Int J Electr Power Energy Syst 2017;86:53–60. [4] Samimi MH, Bahrami S, Akmal AAS, Mohseni H. Effect of nonideal linear polarizers, stray magnetic field, and vibration on the accuracy of open-core optical current transducers. IEEE Sens J 2014;14(10):3508–15. [5] Gomez-Luna E, Mayor GA, Gonzalez-Garcia C, Guerra JP. Current status and future trends in frequency-response analysis with a transformer in service. IEEE Trans Power Delivery 2013;28(2):1024–31. https://doi.org/10.1109/TPWRD.2012.

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