Transformer Resonance: Reasons, Modeling Approaches, Solutions

Due to the extension of the power grid with many complex and compact pieces of power equipment, transformers will be more exposed to fast transient, resulting from various resonance conditions. Transformer resonance can result in severe overvoltage on internal parts of the winding, leading to insulation failure and, consequently, transformer outage. The main reasons for resonance occurrence, the practical method to measure the resonance of transformers, and the solution for preventing transformer resonances have been discussed in the scientific reports over the past few decades; however, a comprehensive review of these studies is not present in the literature. Hence this paper aims to provide a comprehensive review to categorize the main reasons for transformer resonance, modeling methods, and appropriate solutions to suppress this phenomenon and suggest some prospective protection for future works.


I. INTRODUCTION
As a well-known power system element, the transformer is an electromagnetic device that transforms the voltage level at the same power transfer and frequency in the electrical network [1]. There are two types of transformers: power transformers that supply electric energy and instrument transformers that measure power system signals [2]. The protection of this critical device is essential for the system's operation [3].
Resonance is one of the most significant phenomena affecting transformers' operation [4]. Various types of resonances may influence the operation of transformers [5]. Transformer resonance may occur due to the inductive and capacitive nature of the transformer windings and the connected power system, which can occur at a frequency different than the power frequency [6]. Most of the resonances in the transformer cause a harmful effect that needs to be prevented by protection methods [7]. The Diverse nature of transformers' The associate editor coordinating the review of this manuscript and approving it for publication was Hui Ma . resonance is classified in [8] based on how they affect the system.
Measurement and modeling of transformers in a wide frequency range not only help to optimize the design of the transformer but also provide insight into the risk of resonance during various transformer operations [9], [10]. Several types of solutions have been investigated to protect transformers against resonance faults [11]. These solutions are categorized as parallel damper circuits [12], series components [13], and solid-state protection devices [14]. Parallel dampers are nonlinear resistors known as surge arresters, or R-C circuits, which are mature solutions for transformer overvoltage protection [15]. They can be series components such as fast frequency suppressing circuits (FTSC) [16] and frequency-dependent devices (FDDs) [17]. In addition, the solid-state device includes controllable switches, a capacitor bank, or a reactor that can help to eliminate the overvoltages due to transformer resonance successfully [18], [19].
This paper aims to provide a clear review of significant concerns of resonance occurrence for the most crucial part of the power system (transformers). The main purpose of this paper is to provide a review of the most important reasons for transformer resonance occurrence, transformer modeling approaches, and the industrial methods to suppress transformer resonance overvoltage.
This paper is organized in the following way. Section II deals with the reasons for resonance occurrence in transformers, presented based on recently reported scientific research. In section III, different methods to measure the resonance of transformers are expressed. Next, the methods to suppress the resonance of transformers are debated, and their main features are extracted. Section IV deals with the existing protection topologies. Finally, section V summarizes the results and provides insight for future work in this field.

II. REASONS FOR TRANSFORMERS' RESONANCE
Two main reasons for the resonance occurrence are the high voltage multiple restrikes or the power line switching because of either the fault and protection equipment operation or renewable resources management [20]. When the system's topology changes, the predominantly inductive circuit changes to capacitive and the other way around. Consequently, the system is exposed to oscillations with different frequencies and amplitudes [21], and therefore, the components like transformers may experience resonances during the system transient [22]. The reasons for resonance occurrence are presented in the following subsections.

A. TRANSFORMER RESONANCE OCCURRING DUE TO LIGHTNING EFFECT
Lightning overvoltages are the crucial source of transformer resonance when the lightning impulse occurs in the vicinity of a power transformer [23]. Lightning modeling is explained in [24], [25], [26], and [27]. These models include electromagnetic models, distributed-circuit models, and engineering models. In the case of the electrical system studies engineering model comprises of Duality of Engineering models, lightning striking tall objects, and ground-truth testing are employed [24]. As shown in Fig. 1, there are four types of lightning strikes in power systems: lightning strike to a ground wire, phase conductor, ground near towers [24], as well as to a wind generator tower, which causes a surge that proceeds toward the transformer [28]. Generally, it occurs due to their height and installation location. Under these circumstances, several tens of kilo amperes may flow from the tower, which can induce a high overvoltage nearby the power transformers [28]. The lightning current usually includes a current with a high rate of rise, around 10 kA/µs, and a slower rate of fall, such as 0.6 kA/µs [29], demonstrating a variety of harmonic contents.
This signal can induce a voltage with different harmonics and stimulate the transformer's resonance frequency after the lightning strike takes place [30]. The explained phenomenon can easily affect transformers with resonance overvoltage, partially or totally damaging the transformer's isolation [30]. Therefore, the best approach is to examine the transformer using the lightning test before finalizing the   manufacturing process [31] and protect transformers against lightning effects [32], which this standard is provided in IEEE Std 998™-2012. In this standard, firstly, the detail of charge formation in the cloud is expressed to show various charge distributions in lightning, and then the shield wire design is defined in detail to protect the transformer of the substation from lightning strikes [32]. The standard test of the induced voltage of lightning (transformer lightning test) is shown in Fig. 2 [33], considering that this voltage waveform is similar to a lightning impulse [34].

B. TRANSFORMER RESONANCE OCCURRED DUE TO BREAKER SWITCHING
Circuit breakers are a crucial part of the power system, which generally either operate to protect the power system from fault current damage or disconnect the unloaded transformer [35]. Fig. 3 depicts the configuration of the circuit breaker and the transformer.
By opening the circuit of the unloaded transformer, overvoltage occurs on both the circuit breaker and the transformer VOLUME 11, 2023 58693 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  due to the magnetic energy stored in the transformer [36]. Equations (1) and (2) define the transformer and the circuit breaker voltages, respectively, throughout the breaking process by ignoring the damping effect [36].
Here, U t and U cb are the voltage of the transformer and the circuit breaker, respectively. Ich is the chopping current of the circuit breaker, L m and C t are the magnetization inductance and the transformer capacitor, E kr is the electric field between the contacts, v is the speed of contact separation, and t refers to the time. Fig. 4(a)-(e) depicts the operation sequence of the circuit breaker. It clarifies that during the operation of the circuit breaker, the electric arc will be formed and interrupted several times. Finally, the breaker reaches the end of the cycle, where maximum isolation is applied, and the arc interrupts permanently [37]. The no-load transformer can easily face resonance overvoltage considering harmonically polluted voltage during the breaker's operation [38], [39].  [38] provides analytical and experimental tests of CB prestrike and restrike effects on the winding of the transformer. By performing a test setup of three-phase switching of a CB by applying different cable lengths at different loading conditions, the occurred resonance overvoltage of the transformer was studied. The obtained results clarify the effects of vacuum CB operation on the transformer overvoltage. This operation may cause the transformer resonance.
Another reason that may cause resonance is the no-loaded transformer energization. By closing the breaker to the energized transformer, the fast transient phenomenon may excite resonance in the internal transformer windings, causing high voltage overvoltages [10].

C. TRANSFORMER RESONANCE OCCURRED DUE TO THE OPEN BREAKER EFFECTS
After power line interruption, the no-loaded transformer is always connected in series with the circuit breaker,   considering there is no transient phenomenon in the power system. In this state, the circuit breaker behaves as a series capacitor, and the transformer (power or voltage transformer) is a large reactor [40]. This situation is illustrated in Fig. 6.
Considering the open circuit breaker capacitive model, a series resonance [41] may occur. However, by increasing the transformer current, the core will reach the saturation region, and the ferroresonance may happen [42].
In addition, the harmonic content of voltage during the ferroresonance can cause many harmonic for nearby transformers [43], [44]. An example of the voltage signal of the transformer in the ferroresonance state is presented in Fig. 7 [19].
Observing Fig. 8, the nonlinear ferroresonance voltage of the transformer causes overvoltages higher than 2 p.u. Additionally, in the case of transformer ferroresonance, equation (3) presents the nonlinear characteristic of the transformer as a generic n-order polynomial. The order n is defined by the character of the transformer's core. Then applying the derivative to (3) transformer voltage can be 58694 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  expressed by (4) [19].
Here ϕ refers to the transformer core and L, which are the transformer winding inductance and current, respectively.

D. TRANSFORMER RESONANCE DUE TO NONLINEAR LOADS
The introduction of DC systems and DC nonlinear loads contribute to the increase of power system harmonic content [45]. Some harmonics may be close to the resonance frequency of large transformers [46]. Considering the power quality standard IEC 61000 in the power system, the magnitude of these harmonics should not exceed the values defined in the standard. However, the harmonics' magnitude might rise because of a power system malfunction, faults, or heavy nonlinear load switching [47], [48], resulting in transformer resonance and overvoltage. Due to the injection of harmonics in the transformer [49], as depicted in Fig. 8.
According to IEEE Std 1159-1995, the amplitude of the harmonics and the total harmonic distortion should be below the permissible values [50]. However, in system dynamics and power equipment interactions, the harmonic content can be increased and induce transformer resonance [51]. An example of the resonance possibility due to generated harmonics of the HVDC voltage source converter (VSC) in the High Voltage Direct Current (HVDC) power system is given in Fig. 9 [52]. It is shown that the transformer can be exposed to a risk of resonance caused by nonlinear power electronic devices, mainly triggered by the harmonics of the 30 th -60 th order.

E. TRANSFORMER RESONANCE CAUSED BY CABLE MODEL
In the power system, the high-voltage cable is used for power transmission or distribution between substations around urban areas. Accurate cable modeling is needed to investigate the interaction between the cable and the transformer [53]. The theoretical formulation of the cable, considering bundle-shielded cable as the most common cable, shows the capacitive cable's impedance in various harmonic domains [54], which may interact with the inductive impedance of the transformer.  The cable's analysis conducted in the power system transient studies [55] proves its influence on the overvoltage and resonance [56], as it is illustrated in Fig. 10.
Considering the R-L-C models of the transformer and the cable, many resonance points may appear, which can be stimulated by the harmonic content of fast transients and cause transformer overvoltage. The critical length of the connected cables in both the primary and secondary of the transformer is determined by the travel time or the cable's resonance frequency and the transformer's resonance frequency which can be the main reason for transformer damage by overvoltage [57]. Mostly, cables have been modeled using frequency-dependent models designed using the cable's geometry [58] to study their effects. One of the cases in which resonance overvoltage occurs is where the transformer's secondary side is not loaded, and its primary winding is connected to the grid by a cable [59].

III. METHODS OF TRANSFORMER RESONANCE MEASUREMENT AND MODELING
This section presents the method which is currently applied for the transformer's frequency-dependent modeling and measurement. Data is categorized, including theoretical modeling of transformer resonance and measurement methods.

1) BLACK BOX MODEL
In this model, the transformer is considered a black box, and then the electrical behavior of internal circuits is studied based on measurements [61]. The back box model is the standard and traditional method proposed in 1988, which helps to understand power transformers' behavior within the high-frequency range [61]. As a black box, the three-phase transformer is connected to the source and the measurement device, and its behavior is examined [62]. In this model, the source can be considered as a sinusoidal variable frequency [62] or an impulse generator [63], while the oscilloscope records measured signals from a terminal of the black-boxed transformer [62], [63].
Applying different configurations to this black box (short circuit or open circuit connection), connecting specific terminals with the source, and measuring currents and/or voltages is used to approximate admittance matrix elements [64]. An example of a black box model setup for the single phase and zero phase injection test for the black box model of the transformer is illustrated in Fig. 11 [62].

2) WHITE BOX MODEL
White box models, also known as physical models, are based on either FEM (Finite Element Method) or analytical analysis of R-L-C elements that define the transformer's characteristic (also called a detailed model). The white box is a model considering the geometry of all layers in the transformer, such as conductors, isolators, and core, as well as their interaction. The simulation of the transformer used to extract the results requires a long computational processing time due to the large matrices, with elements that should be computed for each frequency of the selected bandwidth [65].
The geometrical position of winding turns, the cross-section of the core, and the cross-section of the windings' wire are considered in a FEM transformer model [66]. Moreover, the capacitor calculation method between winding disks can be implemented using series-parallel capacitors, which improves the accuracy of the transformer white box model [67]. A 3D design and a 2D scheme can focus on a simplified simulated model of a section of the transformer to increase the accuracy of the transformer FEM model. According to the procedure of the white box model, the self-inductance of each turn or disk can be calculated, and then the mutual inductance of windings (between the disk of windings or between HV-LV) is studied. In addition, the capacitance of each disk to the transformer tank and capacitance between disks and between HV to LV winding is needed to create a comprehensive model [68]. An example of an accurate calculation of the inductance and capacitance of a transformer coil is illustrated based on geometrical features in Fig. 12.

3) GRAY BOX MODEL
The gray box model (so-called hybrid model) comprises a combination of black box and white box models, which is the method to provide faster and more accurate analysis [69].
In this method, firstly, a signal is applied as input of the transformer, and the output as a system response is measured. Then, the transfer function is calculated using the transformer input and output. This model is improved by considering the winding geometry and FEM to calculate self-and mutual inductance [70]. In [71], the R-L-C model of the transformer is described, and it shows that the most critical issue of the gray box model is its ability to identify the model parameters when no information about the geometry of the transformer is available. Therefore, it is a practical method to model transformers with unidentified geometry or an old transformer (referred to as dielectric aging) and to diagnose the problem of the power system-connected transformer [72]. In addition, by simplifying the gray box model, it will be possible to improve the analysis and parameter estimation of the model faster and more accurately [65]. By developing STM (Simplified Transient Model) in [65], the advantages of the method are presented as follows: • The time simulation is adequately low.
• Using STM along with the other models of power network elements for power system studies and analysis is possible.
• The parameter identification of STM does not need any costly and time-consuming measurements.
• Considering the effects of faults on STM is possible.

B. TRANSFORMER RESONANCE STUDY BASED ON TRANSFORMER MODEL
One of the vital goals of studying transformer models in the wideband frequency domain is the detection of the resonance 58696 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  frequencies of the transformer [73]. The resonance frequencies closely depend on the type of transformer and its design conditions, which can be estimated by the nodal admittance measurement [73]. The frequency-dependent impedance of the transformer is a crucial issue that has to be studied to find extreme impedance points that show the resonance frequency points and transformer phase changes [9]. By studying the resonance frequency of the transformer, it is observed that it may cause disk overvoltage. This overvoltage can occur for several resonance frequencies, and the overvoltage magnitude might reach more than 5 p.u [74]. For monitoring power transformer resonance, electromagnetic simulation by EMTP software and using algorithms to diagnose resonance class is suggested in [75]: For the transformer resonance determination and voltage distribution in each coil, it is investigated that a sinusoidal impulse with a 250 ns period is applied to the transformer winding. Then in [76], it is recommended that before applying the single-transmission line model (STLM) method to calculate transformer overvoltage against the transient, actual measurement of the frequency characteristic of the inter-turn voltage distribution, the resonant frequencies and the amplitude of the inter-turn overvoltage should be considered. Fig. 13 depicts an example of the magnitude of the transformer's impedance as a function of frequency and points out the resonance frequencies.

C. TRANSFORMER RESONANCE MEASUREMENT METHODS
In this subsection, the methods of transformer measurement are introduced. These methods are widely used to extract the resonance frequency of transformers in practice. These methods provide appropriate protection of the transformer around the resonance frequency points [10].
The frequency response analysis (FRA) test is based on measuring the transformer response in a wide frequency bandwidth, which is known as a vital method to diagnose resonance frequencies of transformer [6]. This method is presented in IEC 60076-18. The FRA device consists of three channels. The first channel comprises a source connected in series with a resistor. The second channel provides a reference value using a voltage drop on a 50 resistor, and the third  channel is the measurement port [77]. There are two types of FRA, depending on their internal source. The first type is designed based on an impulse source frequency response analyzer (IFRA), and the second one is designed using a sinusoidal sweep frequency response analyzer (SFRA) [77]. SFRA is the most commonly used device.
FRA test is a method to test the terminal or internal resonance of various types of transformers as complex electrical circuits [78]. In addition, it is an appropriate test method to study wideband analysis of transformer admittance and modeling of transformer [79], [80], [81]. Omicron SFRA is a commonly utilized device to test power transformers to diagnose their internal faults or resonance overvoltage [82]. Fig. 14 depicts the interface of the Omicron device in frequency sweep [83].
The second wideband measurement method uses a vector network analyzer (VNA) device [84]. The most crucial difference between VNA and SFRA is the different types of connection to the transformer, and only SFRA passes the requirement of the standard for transformer measurement [84], [85]. In [85], transformer measurements were compared for black-box modeling by employing two different methods and measurement equipment (sFRA and VNA). These two methods had a close agreement in the different ranges of frequencies (up to several MHz). For the frequencies higher than the MHz range, the results obtained by the VNA device were shifted to higher frequencies compared with the results measured with the SFRA. Fig. 15 shows VNA (Bode 100) and its measurement interface.

IV. SOLUTIONS FOR TRANSFORMER RESONANCE
By taking into account the state-of-the-art about transformer protection against resonance overvoltage, several solutions VOLUME 11, 2023  have been introduced to suppress transient effects in power transformers. These structures are series or parallel protection devices that can act as filters to reduce and suppress transient resonance overvoltage [11]. Due to the operation, the content of harmonics that stimulate the transformer's resonance frequency will decrease to the acceptable value. The parallel protection equipment imposes mainly the capacitive-resistive model in parallel with the transformer, and series protection components impose the inductive-resistive model in series with the transformer; their operation is based on magnetic effects [12], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98].

A. THE PARALLEL-CIRCUIT SOLUTION AGAINST TRANSFORMER RESONANCE 1) SURGE ARRESTER PROTECTION
The most practical protection device against resonance (PDAR), which only limits the occurred overvoltage of a transformer, is the use of surge arresters connected in parallel to the transformer terminals. Surge arresters are known as overvoltage protection, developed mainly from Zinc oxide material [86]. Nevertheless, it can just chop overvoltage occurred by each reason. Moreover, it cannot behave as a harmonic filter [12], [87]. Surge arresters are categorized based on the voltage level, particularly between 24-800 kV (RMS) [87]. The configuration of the surge arrester is illustrated in Fig. 16 [88].

2) SURGE CAPACITOR PROTECTION
The solution of transformer resonance through a surge capacitor is very similar to the surge arrester protection. In this method, a surge capacitor is able to tolerate high dU/dt values in parallel with the transformer to suppress the transient voltage of the system as a low-pass filter. In contrast with   the surge arrester, the surge capacitor damps the magnitude of high-frequency overvoltage and decreases the transient's harmonic content, which may stimulate transformer inter-disk or terminal resonance [89]. Fig. 17 [90] shows an in-market surge capacitor and its configuration to protect the transformer. The conventional surge capacitors are designed for a 7.2-36 kV (medium voltage) range, and their capacity is between 0.05-0.8 µF [90].

3) R-C SNUBBER
Using a shunt R-C circuit as a snubber is a standard method to protect transformers from transient and resonance. This method connects a high-impedance R-C in parallel to the transformer [91]. The operation of R-C acts as a low-pass filter and absorbs the energy of the transient signal in the resistor. The voltage range of the conventional R-C snubber of the power system is between 7-34.5 kV, and the range of the resistor is 20-500 (300-750 W). Also, the range of the capacitance is 0.08-0.75 µF [92]. Fig. 18 shows a configuration of RC snubber with a transformer [92].

4) ZNO-RC PROTECTION
A modified structure of the R-C snubber has been developed as a solution to prevent transformer resonance, known as 58698 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.    ZORC. This structure comprises Zinc Oxide (ZnO) surge arresters, resistors, and capacitors, which are very effective devices for suppressing the transient oscillations and protecting the transformer against resonance. Its operation is very similar to RC snubber; however, it includes a nonlinear resistor which can improve the operation of RC snubber [93], [94]. A conventional ZORC is designed for a 3-40 kV voltage level. Fig. 19 illustrates a ZORC to protect the transformer [94].

B. SERIES-CIRCUIT SOLUTION AGAINST TRANSFORMER RESONANCE 1) PRE-INSERTION RESISTOR PROTECTION
A practical way to mitigate the high rate of transient variations in the power system is by using pre-inserted resistors (PIR) before switching [95]. Two types of PIRs exist: series switches PIR or parallel switches connection PIR. Both can protect the transformer against resonance. This method's advantage is that it applies to high voltage transformers up to 400 kV, and its challenge is very high power loss during power switching. Also, this method suppresses only the transients occurred by switching [96]. Fig. 20 shows the configuration of PIR with a transformer [97].

2) FREQUENCY-DEPENDENT PROTECTION DEVICE
The series R-L circuit is a recently presented solution for protecting the transformer against transient phenomenon and resonance. This circuit can be realized by using ferromagnetic shielded wire as a frequency-dependent device (FDD) [98]. This device comprises a multilayer wire using both aluminum and ferromagnetic material, which are wounded as a reactor and connected to input cables of a power substation. FDD provides meager resistance for grid frequency (50 or 60 Hz) and imposes high resistance of nearly 200 for frequencies around 200 kHz [99]. In [99], it is obtained that FDD protects well from short pulses and has little effect on long pulses. The designed device parameters must be developed or integrated with devices like the arrester to protect the transformer properly. The variable resistor of FDD is created based on the skin effect of composite ferromagnetic material [100]. Fig. 21 depicts the equivalent circuit of FDD and in-practice configuration [99].

3) CHOKE PROTECTION
Choke is a series device that protects the medium voltage power transformer against transients and resonance. VOLUME 11, 2023 The R-L choke model can appropriately mitigate transient harmonic content and drastically reduce the transformer's resonance overvoltage [16]. In normal operation, the impedance of this component (power network 50/60Hz frequency) is nearly zero. During high-frequency transient, these applications behave as a parallel R-L and suppress the high-frequency oscillation by using the effect of ferromagnetic core [101], [102]. Fig. 22 shows the choke configuration [11].

4) POWER ELECTRONIC-BASED PROTECTION
The solid-state DC reactor suppressor (DCRS) component is normally used to protect transformers against transient and transformer resonance. This practical method protects the transformer against nonlinear resonance comprised of a solid-state bridge (including four diodes) and parallel R-L. It can appropriately protect the transformer by suppressing nonlinear oscillations of voltage transformer ferroresonance, and mitigating harmonic overvoltage on the transformer terminal [103]. The topology of this power electronic-based transient suppressor device is presented in Fig. 23.
TABLE 1 summarizes the essential data concerning reasons, measurement, and associated protection methods. Considering the transformer resonance issue, it is crucial to determine the reason for transformer resonance and then provide an accurate model for the studied transformer. In the final stage, providing the appropriate compound solution for the resonance occurrence of the studied transformer should be considered.
By observing in the summary TABLE 1, it is clear that there is no comprehensive solution to protect transmission level transformers against grid transients and resonance overvoltage.
Considering the circumstance, the applicability of the available solution in the next section is briefly studied to obtain the next step of the research in this field.

V. STUDY OF APPLICABLE PROTECTION AGAINST TRANSMISSION TRANSFORMER RESONANCE
This section provides a brief overview of the limitations of the already used protection methods for the protection of high-voltage transmission transformers. The evaluation of the available protection solutions at a 200 kV voltage level has been considered.

A. PARALLEL PROTECTION METHODS EVALUATION 1) SURGE ARRESTER
This component is connected in parallel with the transformer to protect it from overvoltage. There are two operating concerns: The transformer model is inductive and dependent on the voltage magnitude. An equivalent model of the surge arrester is illustrated in Fig. 24 [12].
When the transients do not exceed the arrester's continuous operation voltage, the equivalent arrester circuit is a series RLC. The circuit depicted in Fig. 25 (a) is a middle-passed  filter around its resonance frequency. However, the remaining frequency content can cause the transformer's inter-disk resonance while the transformer's terminal voltage never experiences an overvoltage due to the surge arrester suppressing the voltage.
The next concern about surge arrester protection is caused by high voltage and high-frequency transients. In this case, the surge arrester behaves as an R-L circuit, as shown in Fig. 25 (b). This circuit can cause an operational delay against a very fast transient, resulting in transformer resonance before the arrester operates. These two reasons show that surge arresters cannot provide comprehensive protection for transformer resonance: 1) Surge arrester cannot protect transformer against low voltage high-frequency transients. 2) Operational delay considering its inductive circuit against high voltage high-frequency transients.

2) PARALLEL CAPACITIVE PROTECTION
The capacitive transformer protection consists of three types [89], [92], and [93]: surge capacitor, R-C snubber, and ZORC. Considering the industrial datasheets, to protect a 200 kV transformer, using 0.1 µF, 200 (for R-C snubber and ZORC), and at least 360 kV capacitor (considering that transient impulse adds 50-100 kV to the standard voltage of system) is needed. Condition from equation (5) proves that the steady-state impedance of the capacitor is much higher than the resistance. Therefore, its effect can be ignored in calculating the reactive power consumption of the protection branch.
In addition, the nominal voltage of each industrial capacitor module (existing in the market) is 36 kV, 0.1 µF. Hence, to create a 360 kV, a 0.1 µF series-parallel connection of 100 capacitors is needed. This number is estimated under the 58700 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  assumption of linear voltage distribution. The leakage capacitor affects the high-frequency content if it is not distributed linearly. Thus, the number of capacitors should be higher than 100.
In the steady state, the protection capacitor bank will consume reactive power Q as calculated by: Where ω is the angular frequency of the grid, C s is the capacity of the surge capacitor, and V s is grid voltage. For the described example, the value of consumed reactive power will be 1.25 MVAR.
Furthermore, a higher capacitor number can decline the device's reliability. Moreover, the reliability of the capacitor bank will decrease considering the grid's harmonic pollution, which might increase the consumed reactive power in the steady state even more. Therefore, the surge capacitor for high-voltage transformer protection is not applicable.

3) SERIES INDUCTIVE PROTECTION METHOD EVALUATION
The FDD and the choke [99], [102] are the two main methods for protecting the transformer from resonance overvoltage. The design relies on the ferromagnetic power conductor. However, there are some differences between these two schemes. Fig. 26 depicts a 2D illustration of FDD and the choke structure cross-section.
Namely, the choke contains the core isolated from the main conductor, whereas, in the FDD, the ferromagnetic shield is connected electrically to the main conductor. Furthermore, choke has a resistive secondary turn, while FDD does not. Also, FDD is built by wounding the extensive length of the illustrated cross-section, but choke lengths are very shorter. FDD's ferromagnetic thickness is much smaller than the choke.
Both structures modeled as series R-L can protect transformers against resonance because they do not meet power grid voltage during a steady state. They can filter the harmonic content of transients and protect transformers.
However, the concerns are the deep saturation of the choke core, high power losses, and thermal desaturation of FDD during steady state. This concern can also affect the correct transient operation of these devices. Besides, the wideband operation of these series filters needs to increase the inductance, which causes reactive power loss in the steady state. Equations (7) and (8) show that core saturation depends on core cross-section and component inductance. The consumed reactive power depends on the inductance value, as seen in equation (9). Finally, equation (10) shows that the protection component's imposed impedance should be comparable to the transformer impedance.
Here, B, L, and A core are the protection component's magnetic flux density, inductance, and core cross-section. Z T , and I T are the transformer's impedance and current, and ω T is the angular frequency of the transformer's resonance.
It should be mentioned that the consumed active and reactive power can be reduced using the solid-state DC reactor method.
However, the positive aspects of the presented inductive protection methods are as follows: 1) The component is not too bulky, 2) It does not add any electric elements to the power conductor, 3) It is a reliable component.

VI. CONCLUSION
This paper reviews the crucial issue of power transformers' resonance, addressing the reasons of transformer resonance occurrence, transformer wideband frequency model, and protection solution. Eight series and parallel protection strategies are reviewed concerning transformer resonance protection. However, there are limited transformer protection solutions for different inter-disk and terminal resonance frequencies in high-voltage (transmission voltage level) applications. The main shortcoming is expressed by comparing the most significant limitations of the presently used protection devices. The next step of this research field will be the development of a protection device for transmission transformers against transient for both inter-disk and terminal resonance overvoltage.