The Transformer Modelling for VFTOs Calculation - A The Transformer Modelling for VFTOs Calculation - A Comprehensive Review Mohamed Salah Comprehensive Review Mohamed Salah

The transformer is exposed to very fast transient overvoltages originated from the switching operations in the gas insulated switchgears, these overvoltages may damage the transformer insulation and result in catastrophic failures. Consequently, the transformer should be modeled to handle the transient state. In this work, the methods published in the literature to illustrate the very fast transient overvoltages in gas insulated switchgears are reviewed, and then an overview of the transformer modeling techniques is presented with the contribution and shortcomings of each technique. The determination methods of the parameters required for the modelling of the transformer are also reviewed. In addition, the contributions of utilizing the frequency response analysis as a diagnostic tool of the transformer condition are summarized.


Introduction
T he power transformer is considered as one of the most vital pieces in the power system as it is an essential link in the chain of power system components.The transformer failure may result in catastrophic failures to the other system devices, and economic consequences, incorporating proceeds decrease and market backlash (Abbasi, 2022).Hence, the reliability of the power system relies significantly on the reliability of the transformer.A significant amount of transformer failures is due to the fast and very fast transient overvoltage (VFTO) (Taikina-aho, 2010).As this overvoltage results in a dielectric stresses degrading the insulation between the turns and this leads to hazardous short circuit between the turns, therefore the transformed insulation should be designed well to withstand such overvoltages.
Most of the transient overvoltages in the power system result from the switching operations in the gas-insulated switchgears (GIS) (Vinod Kumar et al., 1999;Lu and Zhang, 2005;Yu et al., 2009).When the disconnector in the switchgear is being opened or closed, arc strikes occur between the contacts, so VFTOs are generated and propagate into the power system.These overvoltages may damage the devices connected to GIS, especially the transformer, as the frequency of the VFTO may match one of the resonant frequencies of the transformer winding, consequently this results in an inter-turn fault as the insulation between the turns is vulnerable to the high-frequency oscillations.
The transformer may pass the standard tests, and in the same time the failure occurs due to overvoltage.To reduce the failure, the system is to be modeled to monitor and assess the state of the system various devices, and to clarify the mutual effect between the transformer and the other system components (Yang et al., 2011).
The transformer modeling is not an easy process for many reasons among them: the transformer behavior is non-linear, frequency dependent, and affected by the variations in the winding and the core construction.In addition to the presence of physical attributes which have to be considered.Many achievements for transformer winding have been published in the literature which can be categorized into: multi transmission line (MTL) model, the lumped parameter models, and the hybrid models.The MTL models are utilized for fast and VFTO, the voltage distribution along the winding can be driven from these models.They require the construction details of the transformer winding and are timeconsuming (Shibuya and Fujita, 2002;Popov et al., 2007;Wang et al., 2018).The lumped parameter models depend on the geometrical data of the transformer (Yang et al., 2011;CIGRE, 2014;Colqui et al., 2019).They are used to simulate the lightning impulse and the fast transient overvoltage and to clarify the interaction between the transformer and the neighboring equipment.The hybrid model is a compromise between the aforementioned two models, where the winding is represented by a single transmission line to compute each disc voltage, and then the coil of interest is studied so as to obtain the voltage on each turn (Shintemirov et al., 2009a).
The above models depend significantly on the capacitance and the inductance matrices, consequently, these matrices have to be calculated precisely.These matrices can be calculated either by analytical expressions or by finite element method (FEM).In the analytical method, the turns are treated as parallel plates (de leon et al., 2009) to calculate the capacitance, Analytical expressions are used to compute self and mutual inductance as given in (Miki et al., 1978).The authors in (Portillo et al., 2020) reviewed the basics of parameters computation for high frequency transients.There are three approaches to calculate the capacitance matrix by FEM: forced voltage, fixed charge and energy method (de leon et al., 2009), while the inductance matrix can be calculated by flux linkage method or energy method.
Condition monitoring, based on the transformer modelling, is an efficient method to reduce the failures, and help in detecting the deformation in earlier stages which prevent the occurrence of catastrophic failures (Zhang et al., 2015).The FRA introduced in (Dick and Erven, 1978) is an efficacious approach to detect mechanical deformations such as buckling and tightening.In the FRA, sinusoidal voltages containing different frequencies are applied to one terminal then the magnitude and phase of selected winding terminal are plotted as a function of frequency.The instrumentation required for this method is commercially available and can be set up to give repeatable results easily.This approach is considered as simple and cheap which allows the analysis of the transformer in wide frequency range.
In this work, first the methods employed to depict VFTOs of GIS are examined.Then the transformer modelling techniques are reviewed in section 3 showing the pros and cons of each model.Section 4 shows the determination methods of the parameters required for the modelling of the transformer.The contributions of utilizing the FRA as a diagnostic tool of the transformer condition are summarized in section 5. Finally, the recommended research issues and conclusions are given in sections 6 and 7, respectively.

The VFTOs of GIS
The majority of transient overvoltages in the power system result from the switching operations in the GIS (Vinod Kumar et al., 1999;Lu and Zhang, 2005;Yu et al., 2009).When the disconnector in the switchgear is being opened or closed, arc strikes occur between the contacts, so VFTOs are generated and propagate into the power system.
An early trail to calculate the transient over voltage was introduced by Lovass-Nagy and Szendy in 1964 (Lovass-Nagy and Szendy, 1964).The study outlined a matrix method to derive relatively straight forward expressions for calculating transient-voltage distribution in a system.The differential equations governing the transient voltages were combined to create a unified matrix equation, the solution of which was obtained through the application of the Laplace transform and the matrixinversion technique.An algorithm incorporating also the numerical inversion of the Laplace transform, accounting for frequency-dependent parameters, was introduced in (Ahmad et al., 1992).This algorithm was applied to transient calculations with good accuracy.The numerical results exhibited agreement with the measured values.
The VFTO results of a 420 KV GIS calculated using EMTP were proposed in (Vinod Kumar et al ).The pattern of VFTO during the switching operations in the disconnector and the circuit breaker was studied.The formulae of VFTO waveform were deduced in a theoretical manner in (Yu et al., 2009) using the transmission line model.The authors assumed the structure of GIS illustrated in Fig. 1 and its equivalent circuit as shown in Fig. 1.It was assumed that the bus 1 had a trapped voltage of E 1 and bus 2 had initial voltage of E 2 when the switch was closed the amplitude of k th harmonic G k was deduced as (Yu et al., 2009) And the DC component E 0 as (Yu et al., 2009) Where T 1 ; T 2 are the transmission times through bus 1 and bus 2, z is the surge impedance of the buses and u is the angular frequency of kth harmonic.The analysis involved examining the impact of the length and the characteristic impedance of the bus, and external equivalent capacitance of the transformer on the VFTO waveform.The simulation waveforms obtained by PSCAD software were aligned with the analytical results.
The work in (Wong and Chen, 2009) demonstrated the switching operations in China light and Power Company in Hong Kong.Simulation analysis was performed to evaluate the potential risks of VFTO in GIS substation.The results revealed that the major components of the VFTO cover frequency range from 0.75 MHz to 7 MHz.It was reported that the VFTO surpassed the dielectric strength withstand level of turn-to-turn insulation, particularly for high-frequency components.
Researchers in (Li et al., 2012) discovered a specific category of VFTOs through onsite testing on an 800 kV GIS in commercial operation.The features of this novel VFTO type were examined, encompassing parameters such as the timing of arc restrikes between contacts, rise rate, damping time constant, oscillating frequency, and trapped charge.The results of the work showed that using closing and opening resistors in disconnect switches had the benefit of suppressing VFTO.In (Zhan et al., 2014), a 252 kV GIS circuit and the measurement system for VFTO and very fast transient current (VFTC) were constructed.The relationship between the VFTC and the arc resistance was discussed.Another VFTC model was proposed in (Duan et al., 2015).Tests were conducted on an actual 220 kV GIS.A measurement setup utilizing a Rogowski coil in conjunction with an optical fiber transmission system was employed to measure VFTCs.
A fitting method was introduced in (Almenweer et al., 2018) to derive the mathematical equation describing the SF6 discharge waveform.The aim of this method is to adjust the parameters of a model function to achieve the best fit with a given data set.A typical data set comprises n points (data pairs) (x i ; y i ), where x i represents an independent variable and y i represents a dependent variable obtained through observation.The model function is represented as (x), where the vector b holds m adjustable parameters, reflecting the fact that the function model incorporates m parameters.The fit of a model to a data point is assessed by its residual, calculated as (Almenweer et al., 2018): The VFTO waveform resulting from a single SF6 gas discharge was derived from the collected data in (Xixiu et al., 2017).The frequency characteristics were extracted from the time-domain waveforms using the fast Fourier transform (FFT) technique implemented in Matlab.It was determined that the Squares Method was suitable for fitting the VFTO expression due to its transient characteristics.The computed values matched effectively with the results obtained using ATP-EMTP software and the measurements reported in (Xixiu et al., 2017).Utilizing EMTP, disconnector switching operations were simulated in (Haseeb and Joy Thomas, 2018) at different locations in a GIS rated at 1100 kV.In the examination of the specified substation, the highest calculated overvoltage reached 1.76 p.u not accounting for the charge present on the bus bar.However, when considering a trapped charge of 1 p.u, the maximum overvoltage elevated to 2.54 p.u.The research outlined in (Ametani et al., 2018) concentrated on analyzing the oscillating frequencies of VFTOs produced during disconnector Fig. 1.Structure of GIS system (Yu et al., 2009).
and circuit breaker switching operations within GIS.The study presented simulation instances that were demonstrated in comparison with test results and showed a reasonable concurrence with the EMTP simulation results.
In (Chung, 2004), a statistical investigation was conducted to analyze the variations in overvoltage associated with the step-up transformer during the switching of vacuum circuit breakers.Certain variables, such as the switching angle, wind turbine power current chopping, high-frequency and current quenching capability were considered as random variables in the analysis.The investigation revealed that relying solely on the amplitude of transient overvoltages was insufficient for forming conclusions regarding the safety of the transformer when subjected to frequent switching surges.
Simulations in (Moreiraa et al., 2020) were performed using EMTP with GIS model for high-frequency analysis and multi-strike electric arc modeling.Field test measurements on 550-kV GIS were compared with simulation results and showed an accurate prediction for VFTO, allowing improved failure analysis.
The work in (Zhao et al., 2022) handled the complex frequency-related parameters.The transfer function of the ultra-high voltage GIS voltage transformer was obtained with the impulse test as exciting signal.Simulation was employed to model the VFTO and their associated conducted disturbances at the Wuhu substation.The waveforms of VFTO, both calculated and measured, exhibited notable consistency in terms of shape, amplitude, and duration.This matching substantiates the accuracy of the employed model.A special online sensor to VFTOs based on the optical sensor technology was introduced in (Chatrefou et al., 2000).A fraction of the high voltage was applied by a geometrical capacitive divider near the base of the high voltage bushing.In (Ma et al., 2009), an innovative measuring system featuring surface-mounted devices was introduced, designed to provide widebandwidth capabilities and ensure stability.The measuring technique in (Ma et al., 2011) was based on a transformer bushing sensor.To address the challenge of frequency limitation, a solution was proposed involving the application of both the convolution model and the incremental Wiener filter.Utilizing this methodology, measurements conducted at the 750 kV substation revealed that the disconnector switching off operations generated a VFTO with a peak value of approximately 775 kV.while the predominant oscillating frequencies of the VFTO were identified as 3.5 MHz, 1.6 MHz, and 650 kHz.
A simplified shrinkage ratio criterion was presented in (Yang et al., 2016) to calculate the distribution of lightning impulse voltage within transformer windings based on the relationship between physical quantities.The authors employed finite element (FE) simulation software to model both the reduced-scale and the original electromagnetic field configurations.
In (Florkowski et al., 2016), an extensive analysis of the propagation of overvoltages through distribution transformer windings was presented, focusing on typical medium and low-voltage electrical networks.The authors in (Gongchang et al., 2013) introduced a complete frequency bandwidth measurement system utilizing a porthole capacitive voltage divider.Three measurement circuits were studied and the effect of cable length on the measurements was discussed.The objective of the study presented in (Soloot et al., 2013) was to examine the impact of resonant overvoltage generated in wind turbine transformers across different winding designs: disc, layer, and pancake configurations.
A transient overvoltage monitoring system was introduced in (Filipovi c- Gr ci et al., 2017).This system used an on-line overvoltage transient recorder with the ability to sample, analyze and store transients at transformer terminals in real-time.
The results of overvoltages resulted from the switching of a compensation coil to the tertiary winding of a transformer were proposed in (Wouters et al., 2019).The method provided bandwidth, up to 10 MHz, Enabling the observation of fast transient events within the waveforms.
A statistical correlation technique was introduced in (Garg et al., 2020) to determine the optimal correlation between transient signals to assess the insulation's condition, distinguishing between pass and fail states.The effectiveness of the technique was confirmed through the validation of partial discharge detection in windings subjected to impulse voltage application, utilizing an 11 kV singlelayer winding.
In (Jurisic et al., 2021), the illustration of 123 measured overvoltages in 220 kV air-insulated substations served the purpose of dimensioning transformer insulation to withstand potential overvoltages in the power network.This precaution aims to prevent potentially hazardous situations arising from resonances.The severity factor in the frequency domain was computed for the observed signals.The authors in (Oliveira et al., 2022) successfully demonstrated the achievement of improved time-domain reproduction by emphasizing the significance of extending frequencydomain measurements into higher frequencies, accomplished through the utilization of S-Parameters.Table 1 gives the contributions and shortcomings of the works which deal with the VFTOs of GIS.

The transformer modelling for VFTOs
As mentioned above the transformer modeling is categorized into three categories:

Lumped parameter model
The model involves dividing the winding into segments, each represented by a ladder network comprising lumped resistive, inductive, and capacitive components, as illustrated in Fig. 3.In this model the voltages and currents are related to each other through the following equation (Van Jaarsveld, 2013): A reduced order lumped parameter model of the transformer windings designed for the analysis of high-frequency transients was presented in (de Leon and Semlyen, 1992).This model underwent validation through both frequency and time domain simulations.A coupled circuit FE model was proposed in (Abed and Mohammed, 2010) with an additional high-frequency branch, operating in parallel with the nominal power frequency model.
The models in (Liang et al., 2008;Wang et al., 2008) used the Krylov subspace technique to reduce the order of the transfer function (TF) derived from the lumped circuit modeling of the transformer under VFTOs.In (Liang et al., 2008) Arnoldi algorithm, one of Krylov subspace methods, was used.In (Wang et al., 2008) a fast algorithm utilizing the Precise Integration Method and Krylov subspace technique was presented.A simulation study using P-Spice program was proposed in (Srinivasamurthy and Dixit, 2016).The study was performed for continuous disc-type winding and interleaved winding.
A methodology for applying black-box transformer models to the lumped-parameter representation of transformer windings was proposed in (Theocharis et al., 2016).The amplification factor N tm;k from an external node t to a specific internal node m concerning the node k where an input is applied is determined by (Theocharis et al., 2016) Where e denotes the node-to-reference voltage in the frequency domain.The internal voltage distribution was calculated by utilizing an approximation with a lumped-parameter model.The method was The running time of the analytical methods was larger than the simulation time.Zhan et al. (2014) 2014 The measurement system for VFTO and VFTC in a 252 kV GIS circuit.
Modification was required in VFTO simulation for extra and ultra-high voltage systems.Ametani et al. (2018) 2018 Illustrating the oscillating frequencies of very fast transients.
The radiation losses were not included.

Chatrefou et al. (2000) 2000
Introducing a special online sensor to VFTOs based on the optical sensor technology.
The vibrations and temperature rise affected the accuracy.Ma et al. (2009) 2009 Proposing a measuring system featuring surface-mounted devices designed for wide bandwidth and stability.The measuring system was stable and applicable in the field.
The coaxial cable used in the measuring system would produce overvoltage on the GIS tank.

Yang et al. (2016) 2016
Presenting A simplified shrinkage ratio criterion to quantify nanosecond pulse voltage and assess the distribution of lightning impulse voltage within transformer windings.
The assumption of constant electric field intensity taken in the design of the shrinkage model reduced the accuracy of the model.

Gongchang et al. (2013) 2013
Presenting a measurement methodology with bandwidth range up to 300 MHz.
The circuit has a limitation in its low frequency response band (below 113.7 Hz).Li et al. (2012) 2012 Illustrating a set of measured overvoltages in 220 kV air-insulated substations.
The correlation between different amplitude and front time or front and tail time was not discussed.
based on the transformation matrix utilization of the voltage distribution factors.The simulation was performed via MATLAB software.
A systematic methodology was proposed in (Zhongyuan et al., 2008) for developing a high-frequency two-port circuit model for transformer windings under VFTO by using its frequency responses.The experiment results showed the practicality and the accuracy of the model.Because of the complexity of the model, the simulation time is relatively large.
A refined lumped RLC model was formulated in (Yang et al., 2011) whose parameters were derived as To counteract the reduction in susceptance with increasing frequency, a negatively valued capacitive branch was introduced in parallel with the inductive branch as shown in Figs.3e5.
The work in (Colqui et al., 2019) showed a refined p-circuit cascade with a damping resistor in series with the shunt capacitor as illustrated in Fig. 3, In this topology, the artificial damping resistance R D is given by (Colqui et al., 2019) where K P is an adjustable factor that varies between 2 and 10, L is the longitudinal per-unit value inductance and Dt is the time step size simulation.The authors in (Ding et al., 2020) proposed a modified lumped parameter model with parameters deduced as (Ding et al., 2020) The model minimized the errors impacted by the lumped parameter approach, and the hyperbolic functions of the distributed parameter model.

MTL model
In the MTL model, each disc or turn is represented by transmission line and these discs are connected to each other in a cascaded manner as shown in Fig. 2 the telegrapher equations can be applied to this circuit as (Van Jaarsveld, 2013) Fig. 2. Thevenin Equivalent of GIS system (Yu et al., 2009).A MTL model was used in (Shibuya et al., 1997), where The entirety of the transformer windings was modeled and simulated as a single transmission line, then the first coil was simulated using the MTL model.The voltage for each turn V i was solved in form (Shibuya et al., 1997) E 0 is the voltage supplied to the winding, and the values of A i and b i are determined through the boundary conditions, c, 3 r are the speed velocity and the relative permittivity of the winding insulation.
Further study was necessary to assess the reliability of the applied model to determine the effects of the VFTOs' on transformers in field.The authors in (Shibuya et al., 2001) aimed to reduce the number of unknowns in the MTL methodology by combining multiple turns under common parameters.
A model based on turn-to-turn modelling of the windings using a wavelet filter bank was proposed in (Saleh and Rahman, 2002).The transformer model utilizing the wavelet filter bank can be depicted by stages, which correspond to the number of turns in the transformer.The model considered the active coupling between the windings and the iron core.
The work in (Zhongyuan et al., 2006) adopted a MTL model, where the computing equations of the model were based on FDTD methodology.The voltage recursion relations can be derived by employing second-order central differences (Zhongyuan et al., 2006) n stands for the time point, and k stands for the position point.Fig. 6.Multiconductor transmission line model, each turn in a disk is represented as a transmission line and connected to the subsequent turn in a zig-zag configuration (Fattal, 2017).
The authors in (Liang et al., 2006a) integrated the theory of compact finite difference with the precise integration method within the MTL model.
A high frequency model based on s-parameters was introduced in (Zhang et al., 2008a).The authors derived voltage TF from s-parameters which was measured or calculated from transformer windings.The vector fitting was used to fit the derived voltage TF, and then the voltage TF was order-reduced by the optimal Pade-approximation algorithm.Frequency-dependent parameters were computed using vector fitting and recursive convolution techniques.The work in (Popov et al., 2007) dealt with layer type transformer windings.A step pulse with 50 ns rise time was used as exciting signal of transformer to measure the response.The MTL model was used to calculate the voltage distribution across the transformer layers and turns.The computations were performed by using the inductance matrix which was calculated by inverse capacitance matrix and by making use of Maxwell formulas.The complete iron core losses and the proximity effect between the layers were considered.More work was needed for the analysis of the voltage distribution along the windings in a three-phase transformer.
The high order Radau method was presented in (Wang and Yangm, 2016).The integration scheme of the high-order Radau method and its features were presented, with particular emphasis on the coefficient matrix satisfying V-transformation.The EMT simulations for overvoltages of transmission line were carried out.
In (Venegas et al., 2011), the electric parameters of the transformer were calculated via conventional formulation.Then these parameters were utilized in MTL model.
A single-input and multiple-output model relied on MTL was presented in (Liang et al., 2006b).Vector fitting and recursive convolution (Grivet-Talocia, 2004) were used to represent the response in the time domain, the voltage and current at the beginning and the end of the coil were related as (Liang et al., 2006b) The results showed that the model was adequate to calculate the resonances in transformer winding under very fast transients.
The work in (Shibuya and Fujita, 2002) introduced the anisotropic complex permeability of the lamination core in computing inductances and resistances required for the MTL model.
Modifications were made in (Zhang et al., 2013) to the MTL to consider the inserted capacitance and the duplex winding to illustrate the electromagnetic induction between the transformer windings.The terminal conditions of the MTL models considering screen tapes were predicted.Another model employing alternative modal decoupling technique was introduced in (de Carvalho et al., 2016).The decoupling was performed using the real and constant Clarke matrix and a reduced frequency dependent matrix for decoupling the remaining mutual terms in the admittance and impedance matrices.
The fundamental principle of rotating transformation was combined with the traditional EMT modeling method in (Yao et al., 2018).The low-frequency line models incorporating single-phase lossless, single-phase lossy, and three-phase decoupling models were deduced.A hybrid simulation involving distinct steps and its sequential coordination method was discussed.
In (Mombelloa and Diaz Florez, 2020) a process for determining the model parameters was introduced that enhanced the model stability.The FEM was used to obtain the basic impedance data.Subsequently, the approach for processing these data to derive parameters for a stable and robust model was illustrated, utilizing Vector Fitting in conjunction with Particle Swarm Optimization.The model was verified by the results of a 50 MVA unit.

Hybrid model
To address the complexities of the MTL model, a hybrid model has been developed to efficiently reduce computation time and hardware requirements.This hybrid model computes overvoltage in two steps.First, the winding is modeled using a single transmission line per disk to determine the voltage of each disk.Second, the disks of interest are represented by an MTL model to compute the voltages for each turn.The Lyout of this model is illustrated in Figs. 3 and 7.
A hybrid model was introduced in (Gharehpetian et al., 1998), consisting of a detailed, coil-by-coil model and an inter-coil black box model.The detailed model's parameter determination and the synthesis of the black box model relied on frequency domain measurements.The model was simulated using EMTP.The measurements verified the accuracy of the model in the frequency range from 10 kHz to few MHz.
In (Babaei et al., 2012) the transformer models used for electromagnetic transient studies were described and compared together.The comparison showed that the lumped parameter models have better characteristics in transient phenomena which have a very high frequency such as the overvoltages introduced by lightning.The number of calculations required for single transmission line model (STLM) was less than other models.The MTL model was more accurate than STLM but the amount of calculation required for this model was more.
The applied algorithm in (Popov et al., 2003) used a hybrid model which was a combination of the MTL model and STLM.By means of STLM, the voltages at the end of each coil were computed, and these values were subsequently employed in the MTL model to calculate the voltage distribution within the coils.
In (Shintemirov et al., 2009b) the voltage and the current at the beginning and the end of the disc is calculated as (Shintemirov et al., 2009b) Iðs Where Iðs; 0Þ, Iðs; lÞ, Uðs; 0Þ and Uðs; lÞ are the currents and the voltages in the beginning and end of the disc, ∅ðlÞ was calculated in iterative manner as Where a i denotes the length of the i th turn and p is the number of turns in the disc.
In (Zhang et al., 2008b), a lumped parameter circuit model is utilized to derive the MTL model of the transformer under VFTO.Each winding was modeled as one unit, others were modeled as another unit, transient over-voltage distribution of practical transformer is calculated using FDTD methodology.
Table 2 gives the contributions and shortcomings of the transformer modelling techniques.

The transformer parameter determination
An early trial to compute the mutual inductance of coaxial cable was by Grover in 1933(Grover, 1933) where tables and formulae were introduced to obtain the mutual inductance.To calculate the leakage reluctance of the transformer, equations were presented in (Rabins, 1956).An impedance formula based on the frequency was introduced in (Wilcox et al., 1989).The authors in (De Leon and Semlyen, 1992;Heinemann and Helfrich, 2000) aimed to calculate the capacitance and the inductance for the high frequency transient models.In (De Leon and Semlyen, 1992) the capacitance was computed by the charge simulation method, while the inductance was computed by the method of image.
A method was introduced in (Cornick, 1992) to obtain the proximity losses of the transformer winding based on the reluctance network of the magnetic paths in the insulation of the winding, and the magnetic flux which penetrates the turn's sides facing each other.
A novel method to calculate the inductance of the transformer was introduced in (Yan et al., 2005).This method was on the basis of the magnetic circuit.The calculation of the leakage and mutual inductances were assured by the dynamic analog tests.
A precise equivalent circuit model designed for three winding transformers was proposed in (Hosimin Thilagar and Sridhara Rao, 2002).The genetic algorithm based parameter estimation method was introduced.One measurement under loaded conditions was performed to determine the parameters which excluded all empirical elements inherent in conventional measurement methods.The mutual coupling between secondary windings and the series-branch leakage reactance was detected by this method.
The work in (Alfuhaid, 2001) deduced the frequency characteristics of the input impedance, considering both open and short-circuited conditions, through graphical representations of the resistance, the reactance, and its phase versus frequency.The natural frequencies derived from the proposed s-domain model exhibited a strong correspondence with outcomes obtained through the SPICE circuit simulation program for both opencircuited and short-circuited secondary configurations, but when the winding was divided into more sections.
In (Martinez et al., 2005), a summary of the implementable models for parameter estimation in transient analysis software like EMTP was introduced.These approaches were valid in the low and medium-frequency domains.The nameplate data were used to deduce the value of the circuit parameters.
A methodology to compute the self capacitance of a high-voltage transformer was presented in (Dalessandro et al., 2007).Measurements were Introducing a methodology to reduce the order of high order models.
The iron core was assumed to have infinite permeability which reduced the accuracy of the model.Abed and Mohammed (2010) 2010 Lumped parameter Proposing high frequency variable model of the transformer with good accuracy.
The simulation time of the model was large because of the complex equations Liang et al. (2008) 1997 Lumped parameter The results matched with the experiments on an actual 525 kV transformer.
VFTOs of Transformers in field were not considered.Gharehpetian et al. (1998) 1998 hybrid Proposing a hybrid model using frequency domain measurements to determine the transformer parameters.
The accuracy was limited to few MHz range Wang et al. (2008) 2001 Lumped parameter The high frequency response of the transformer was calculated directly from the transformer geometry.
Dissipation was not represented.The stability and the accuracy were not guaranteed.Restriction to continual transformers.(Liang et al., 2008;Wang et al., 2008) 2008 Lumped parameter Overcoming the problems of stability and accuracy of the Pade-approximation model (Wang et al., 2018).
Restriction to continual transformers.That model was validated using a scaled down prototype of a power transformer not an actual transformer.Liang et al. (2006b) 2006 MTL Proposing a single input multiple output MTL model and calculating the resonances in transformer winding.
Deviation between the calculated and measured data was noticed.Shibuya and Fujita (2002) 2002 MTL Considering the anisotropic complex permeability within the laminated core in the computation of inductances and resistances for the MTL.The accuracy was confirmed by a comparative analysis between the measured and simulated FRA within the frequency range of 10 Hz to 10 MHz.
The simulation outcomes, when comparing scenarios with and without accounting for the frequency-dependent property of the lamination core didn't exhibit considerable differences.
(continued on next page) performed on three transformers with different construction, and the results of the methodology matched well with these measurements.
A two-dimensional FE model of transformer winding was used in (Du et al., 2008) to obtain the distribution parameters.C matrix is obtained by solving static electric field considering hollow winding and iron core.To evaluate the L and R parameters of a transformer with iron core within the frequency range of MHz, by considering the skin effect of conductors and the associated iron area.The authors in (Ouaddi et al., 2010) calculated the impedance of the transformer utilizing measurements performed on different configurations.
A procedure to calculate the series capacitance and inductance was presented in (Li et al., 2011).The two dimensional FEM was used to analyze the electrostatic fields to obtain the capacitance, and the magnetic fields were analyzed to obtain the inductance using the three dimensional FEM.
An algorithm for the incorporating of hysteresis effect in three phase transformer models for electromagnetic transient investigations was proposed in (Prousalidis et al., 1996a).It relied on a simplification of the Preisach model requiring only a limited amount of easily available data.An agreement was noted between measured excitation currents and their harmonics, with the corresponding calculated values.
To model VFTOs, the multi-layer method of the image was used in (Gomez et al., 2011) to obtain the inductance matrix.The results matched well with the results of the FEM.This work was extended in (Gomez et al., 2013) to consider multi-layer transformers.A novel approach was introduced in (Eslamian and Vahidi, 2012) to consider the core effects in calculating the inductance matrix.The results matched well with the results of the FEM.
A modal experiment was implemented in (Geng et al., 2013) on a real 10 kV power transformer to determine the modal parameters of transformer winding.A refined Empirical Mode Decomposition (EMD) algorithm was introduced for the identification of modal parameters associated with transformer windings.
A recursive least square algorithm utilizing the measured voltage and current for the terminal of single phase transformer, to obtain the resistance and the inductance, was introduced in (Araú et al., 2013).A formula to compute the core inductance component was proposed in (Gomez et al., 2014).The formula relied on the complex flux penetration.This method showed good accuracy in the high frequency range.The accuracy of the model was limited to low frequency range.

Ding et al. (2020) 2020 Lumped parameter
Modifying the lumped parameter model to minimize the errors impacted by the lumped parameter approximation in conjunction with the hyperbolic functions of the distributed parameter model.
Minimizing the global error was not guaranteed.
A methodology providing online monitoring for the equivalent circuit parameters of the transformer was introduced in (Dirik et al., 2014).The determination of series winding parameters was achieved through the application of differential equations algorithm to fundamental frequency components of the transformer data that are obtained utilizing the discrete cosine transform.The nonlinear inductance was derived through polynomial curve fitting utilizing the least squares error method.
The work in (Drandi c et al., 2017) proposed employing a solver based on boundary element methods (BEM) for computing the electric field within a transformer.Linear functions served as base functions, and the point-matching technique, along with both analytical and numerical integration, was employed.The problem was solved for real transformer geometry by both BEM and FEM.The results of both models were in good agreement.
An application of artificial intelligence (AI) techniques was introduced in (Abu-Siada et al., 2020) to determine the electrical parameters of a high-frequency model based on the transformer FRA signature.This approach depended on change of the electrical model parameters results instead of a graphical comparison that may result in inconsistent interpretation for the same FRA signature.The genetic algorithm and PSO were employed for the estimation of equivalent circuit parameters.Table 3 gives the contributions and shortcomings of the transformer parameter determination Methods.

The frequency response analysis of the transformer
The work in (Dick and Erven, 1978) introduced a FRA methodology to identify winding deformation.This method is an alternative to the time domain method like the low voltage impulse method (Lech and Tyminiski, 1966).The work in (Satish and Sahoo, 2005) provided quantitative evidence about how the resultant TF of two winding transformer originates.The work in (Zhongdong and Sofian, 2009) addressed the impact of the winding structure on the FRA responses.The winding was classified into windings of either a high or low series capacitance relative to shunt capacitance.In (Abeywickrama et al., 2006), a model was presented to characterize the quality of the insulation using FRA.The model in (Rahimpor and Bigdeli, 2009) calculated several TFs for several winding configurations to detect the type of fault based on which TF was changed.There was a The equations neglected the adjacent windings and the tank, The yoke was considered to extend to infinity so the accuracy was reduced.De Leon and Semlyen (1992) Introducing an impedance formula based on the frequency.
The accuracy was lower in the case of a transformer with inhomogeneous core.Heinemann and Helfrich (2000) 1992 Calculating the capacitance and the inductance for the high frequency transient models.
The results were efficient and matched well with the results of the FEM and mathematical design formulae.
The frequency response deviated from the field tests above the resonance frequency due to the neglected losses in the simulation.Alfuhaid (2001) 2002 Estimating all impedance parameters of three winding transformers utilizing the genetic algorithm.Detecting the mutual coupling between secondary windings and the seriesbranch leakage reactance This methodology was more complex than traditional optimization techniques.

Li et al. (2011) 2010
Calculating the impedance of the transformer utilizing measurements performed on different configurations.
Discrepancy was observed in the results when comparing with the wide frequency range.

Gomez et al. (2013) 2011
Utilizing the multi-layer method of image to obtain the inductance matrix.
The multi-layer transformer was not considered.Araú et al. (2013) 2013 Determining the modal parameters associated with transformer winding.
The model needs to be extended to incorporate the high voltage power transformers.Drandi c et al. (2017) 2014 Providing online monitoring for the transformer equivalent circuit parameters including core losses and saturation effects.
The accuracy of the model was reduced due to measurement errors of current and voltage transformers.Lech and Tyminiski (1966) 2020 Estimating the electrical parameters of a highfrequency model from the transformer based on FRA signature based on AI techniques with easy implementation The model was not verified by investigations on real transformer.The percentage change in the parameters was not correlated to the fault level.
deviation between the results of this model and the measured data.
The work in (Tang and Wu, 2010) used the simplified hybrid model (Shintemirov et al., 2009a) to detect minor axial winding faults and minor radial winding faults.A quantitative analysis was introduced by statistical indicators to analyze the FRA data.The hybrid model (Shintemirov et al., 2009a) was also employed in (Ji and Wu, 2012) to carry out a comprehensive investigation on frequency response characteristics of winding deformation and measurement connection variation.
The contribution in (Kraetge et al., 2009) summarized various aspects of the practical application of sweep FRA.The different sources of reference data and their significance were discussed.The methods used to assess the measured data were classified into time-based, type-based, and phase comparison.The choice of the proper method was discussed.A summary of guidelines to achieve repeatability was given.
The work in (Mitchell and Welsh, 2011) proposed a modeling methodology utilized to simulate FRA tests on three-phase power transformers.To ensure the validity of the FRA models, several of the parameters, whose values were accurately determined through internal inspection, were compared against their estimated counterparts.
The work in (Gomez-Luna et al., 2013) discussed the status and future trends for online FRA.The online FRA was categorized into impulse FRA (IFRA) and sweep FRA (SFRA).The IFRA seemed to provide increased potential for obtaining satisfactory results.A technique was introduced in (Behjat et al., 2011) for online monitoring of the power transformer winding based on the FRA.The signal was injected into the transformer through a simple and economical capacitive sensor implemented within the surface of the transformer bushing.Another online monitoring technique was presented in (Rahinpour et al., 2016).The model employed a wide-frequency voltage excitation signal and high-frequency current transformers, using the bushing test taps.
In (Hashemnia et al., 2015), the three dimensional FE) analysis was employed to simulate the actual operational behavior of a single-phase transformer by incorporating the physical and geometrical dimensions.An investigation into the influence of each fault level on the electrical parameters of the equivalent circuit of the transformer was conducted.A double-ladder circuit model of transformer winding which considered the frequency-dependent effects and core loss was proposed in (Zhang et al., 2013).The FEM-based software COMSOL Multiphysics was employed to obtain the frequency-dependent parameters.The frequencydependent characteristics of inductance proved that the effect of the transformer core was greater at frequencies less than tens of kilohertz.The results indicated that accurately representing the frequency-dependent parameters of the core and insulation materials resulted in an improved circuit model for the transformer winding.
The model in (Zhao et al., 2019) considered the detailed winding structure, mutual inductance, and interturn capacitance.These parameters were calculated and simulation results were validated through experimental measurements.The study outlined in (Zhou1 et al., 2019) introduced a lumped parameter circuit model that took into account intersection capacitances and mutual inductance between windings.The model was implemented for a 220 kV traction transformer, and the analysis included the examination of parameter variations and characteristics in response to axial displacement across various windings.The spectral clustering of transformer FRA signatures was proposed in (Zhao et al., 2021).A total of 360 sets of FRA signatures were acquired, each corresponding to different types, degrees, and locations of faults.These were obtained using a validated transformer model based on FEM simulations.
The authors in (Duvvury and Pramanik, 2021) proposed an advanced FRA measurement technique designed for the identification of the faulty phase in actual three-phase delta-connected HV windings.The method was illustrated using an appropriate terminal configuration to ensure the symmetry in the physical inter-connection between the middle phase and the other two phase windings.
The effort in (Arumugam, 2021) aimed to bridge the gap between the theoretical concepts and the practical application of white noise as an alternative test for identifying FRA.Pertinent magnitude transfer function revealed the natural frequencies with good accuracy.
The work in (Prousalidis et al., 1996b) analyzed the frequency response and voltage ratio response.Transformer frequency response was compared with transient response to understand the impact of the applied voltage amplitude, rise time, and duty cycle, on transient distribution along the transformer winding.The research helped in the design of transformer insulation subjected to varying transient parameters.Transformer was of core-type construction with concentric windings.The consideration of leakage inductance was confined to the axial path between the high and lowvoltage windings.Rahinpour et al. (2016) 2011 Online monitoring of the power transformer winding on the basis the FRA providing sufficient information concerning the occurrence of faults with sensitivity to a few shorted turns.
The apparatus of online monitoring changed some features of the transfer function then reduced the sensitivity of approach to detect winding inter-turn fault.Christian and Xie (2006) 2006 Utilizing a simplified hybrid model (Zhang et al., 2013) to include new scenarios like the impact of the number of deformed discs, the impact of the degree of deformation, the effect of high contact resistance failure, the impact of the high voltage bushing and the influence of lead deformation.
Restricted to single phase transformers Hashemnia et al. (2015) 2016 Online monitoring of the power transformer winding deformation on the basis of the FRA.Detecting slight changes in the FRA when the load was varied.The changes in transformer parameters affected the FRA response Further research is needed to guarantee its robustness in the presence of lightning and switching surges.Further research was required to develop a reliable expert system based on the detailed transformer equivalent electric circuit model to assess the transformer mechanical integrity automatically.Zhao et al. (2021) 2019 Proposing a lumped parameter circuit model incorporating intersection capacitances and mutual inductance between windings.The viability of the model was substantiated through testing on a 10 kV test transformer.
The model ignored the effect of the insulating paper on the calculation of the inductance.
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Research gap issues and recommendations
There are many contributions used to model the VFTOs.However, there are several challenges and recommended trends that need to be considered and carried out.Some of these are summarized as follows: FRA signatures to assess the transformer's mechanical integrity (g) Online monitoring of transformer utilizing FRA in case of lightning and switching surges.(h) identifying effective mitigation strategies becomes crucial for reducing their impact on transformer insulation and associated equipment.This may involve optimizing transformer design, implementing surge protection devices, or modifying switching procedures to minimize VFTO generation (i) Developing accurate measurement techniques and validation procedures is crucial for verifying VFTO models and ensuring their reliability in practical applications.

Conclusions
This work summarized the transformer modeling techniques published in the literature showing the pros and cons of each technique.The methodologies used to determine the parameters required for the modeling are also reviewed.The contributions of utilizing the FRA as a diagnostic tool of the transformer condition are summarized.Also, research gap topics and the recommended research issues of the discussed topics are illustrated and discussed.The data obtained in this model was not valid for a dry type transformer.

Arumugam (2021) 2021
Proposing an advanced FRA measurement technique to detect the faulty phase in actual three-phase delta-connected HV windings.
The simulation results and the measurements assured the feasibility of the method.
The work did not consider any fault severity and assumed that the fault occurred in one winding at a time.Prousalidis et al. (1996b) 2021 The practical application of the white noise as an alternative test to determine frequency response analysis.
Future work was expected to validate the quality of the obtained results through correlation analysis, and to extend the model to an actual high-power transformer.
Where b V , b I are the voltage and current matrices, and b Z and b Y are the impedance and admittance matrices which can be calculated as (Van Jaarsveld, 2013) b

Fig. 6 .
Fig. 6.A lumped parameter model based on state-space equations, and a distributed parameter model based on the multi conductor line theory and Bergeron's model were introduced in (Villanueva-Ramí et al., 2014).The models were based on MATLAB SIMU-LINK.The results indicated that the distributed parameter model was theoretically more accurate where the time required for the lumped parameter model was smaller.A MTL model was used in(Shibuya et al., 1997), where The entirety of the transformer windings was modeled and simulated as a single transmission
(a) The accurate simulation of VFTOs with highfrequency content (above 2 MHz) is still a challenging issue.So it is recommended that more investigations are carried out to present VFTO models with high frequency simulation capability and small simulation time.(b) Including the saturation effects and the inrush current in the modeling of three phase transformer with faster calculations.(c) Providing models to obtain the voltage distribution along the transformer winding with shorted turns under very fast transients.(d) Accurate online monitoring of the transformer equivalent circuit parameters including core losses and saturation effects.(e) Estimating the impedance parameters of three phase transformer based on AI techniques with less complexity.(f) Developing a reliable expert system based on .,

Table 1 .
Contributions and shortcomings of the illustrating techniques of the VFTOs of GIS.

Table 2 .
Contributions and shortcomings of the transformer modeling techniques.

Table 3 .
The contributions and shortcomings of the transformer parameter determination Methods.
Table 4 illustrates the contributions and shortcomings of the FRA analysis techniques used to monitor the transformer condition.

Table 4 .
Contributions and shortcomings of the FRA analysis techniques used to monitor the transformer condition.

Table 4 .
(continued ) . High frequency circuit model of transformer windings based on frequency responses for VFTO studies.In: 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, pp. 1528e1532.