Fast Frequency Characterization of Inductive Voltage Transformers Using Damped Oscillatory Waves

Measurements of harmonics and, in general, of power quality in distribution grids are becoming more and more an essential task, due to the proliferation of switching power devices. Instrument transformers (ITs), which are necessary for these kinds of measurements in distribution grids, in turn, need to be frequency characterized. Although international standards about ITs do not suggest procedures for their frequency characterization, there are some characterization procedures that are acknowledged in the scientific literature. They are based on the application of realistic power system waveforms to the ITs, in order to test their behavior in conditions very similar to the actual operating conditions. A drawback of these procedures is the quite long duration of the tests, in the order of ~1–10 h. Therefore, this article proposes a new waveform for the frequency characterization of inductive voltage transformers (VTs), composed by the superposition of a fundamental component and a periodic transient phenomenon, namely, a damped sine wave. With its use, it is possible, with a single waveform, to evaluate the frequency behavior of an inductive VT, thus with a test duration in the order of ~1 s. The waveform is theoretically analyzed and experimentally applied to two commercial VTs for medium-voltage distribution grids. Results are compared with those obtained with the IT frequency characterization methods most acknowledged in scientific literature.


I. INTRODUCTION
H ARMONICS are recognized as one of the most critical power quality (PQ) phenomena due to their significant increase in recent years and the numerous issues they can cause.The main sources of harmonics are the switching power converters widely used in renewable power plants, energy storage facilities, and high-voltage direct current (HVdc) systems [1].The energy transition is promoting the use of these devices in distribution grids, making harmonics a crucial issue also at medium-voltage (MV) level.
The growing emphasis on monitoring harmonics has led to the publication of several international standards.These standards establish limits for harmonics [2], prescribe methods and indices for measuring them [3], and provide accuracy limits and guidelines for the characterization of PQ instruments (PQIs) [4], [5], [6], [7], [8].PQIs are designed and built with standardized low input voltage levels.Therefore, measuring harmonics at the MV level requires transducers to reduce voltage to levels compatible with the PQIs input stage.This task is very commonly assigned to inductive instrument transformers (ITs), namely, voltage and current transformers (VTs and CTs), already installed in MV grids for metering and/or protection applications.However, due to a lack of international standards dealing with the characterization of ITs at frequencies different from the power frequency (50/60 Hz), typically, the installed VTs and CTs are characterized only at 50/60 Hz, even if they are used to measure PQ phenomena, which can range up to several kilohertz.
In fact, although a number of international standards deal with limits for PQ disturbances in public distribution systems, measurement of PQ, and characterization of PQIs, the standards on ITs (mainly the IEC 61869 standard family [9], [10], [11], [12]) give accuracy specifications and procedures for the characterization only at power frequency.Instead, at higher frequencies, only accuracy limits are given; in particular, the IEC 61869-1 Ed. 2 [9], published very recently, gives accuracy specifications up to 500 kHz.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
For what concerns the scientific literature, in recent years, a number of articles dealt with the characterization of ITs, both inductive and low-power ITs (LPITs), up to several kilohertz [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]; for the sake of brevity, here, only some articles are cited.With reference to inductive ITs, their main issue is the iron core nonlinearity that makes them introduce errors up to some percents in the measurement of harmonics.For this reason, at the moment this article is written, two methods are the most acknowledged for the accurate characterization of inductive ITs [18], [20].Both the methods involve realistic waveforms (that are waveforms very similar to actual power system waveforms) composed by the fundamental component at rated frequency and amplitude, plus other harmonic components with reduced amplitudes.The first method [18] (in the following, named also FH1, that stands for fundamental plus 1 harmonic tone) includes also one sweeping harmonic tone, whereas the second method [20] (in the following, named also FHN, that stands for fundamental plus N harmonic tones) includes also a number of harmonic tones having random amplitudes and phase angles.
Both these methods involve the generation of a quite high number of waveforms, in the order of ∼100-1000, depending on the frequency range of interest as well as the required accuracy.Thus, they require a significant time, in the order of ∼1-10 h, for the characterization of a single IT.Therefore, they can result unsuitable for the industrial laboratories, where the use of time-consuming techniques represents a bottleneck and then, by extension, a cost.
In this framework, the European research project EMPIR 19NRM05 IT4PQ [27], [28] aims at identifying proper characterization methods and test conditions for VTs, employed for PQ measurements, to be performed in primary as well as in the industrial laboratories; the work here presented is part of this project.
The study presented in [26] analyses PQ phenomena that can impact on the performance of IT.The next step, which is the objective of this article, is the investigation of these phenomena of a novel test waveform designed for the fast and accurate frequency characterization of VT.This waveform is properly chosen in order to meet three features: 1) it is a realistic and already well-standardized waveform; 2) it has the same periodicity as the fundamental tone; and 3) it exhibits a uniform spectral content within the specified frequency range of interest.To achieve this goal, an oscillatory transient (OT) composed by a fundamental tone plus a damped oscillatory waveform (FDOW) has been identified.The DOWs, described in [8], are a transient phenomenon with a wide frequency spectrum [29], [30] that originate in power systems, for instance, following a capacitor bank or a line energization.The generated DOW waveforms have a primary component frequency lower than 5 kHz and a duration from 0.3 to 50 ms [8], [30].Due to their frequent presence in power systems, the DOW waveforms are used for the immunity test of MV equipment [28] and tests for relays and relay systems associated with electric power apparatus [32].Another important feature of the DOWs is the fact that they can be quite easily generated through a generator simpler than those used for FH1 and FHN tests; a circuit composed of a number of passive components, namely, resistors, capacitors, and inductors, opportunely sized, one or more switches and a power frequency or a direct current (dc) generator, as shown, for instance, in [33] and [34], can be used.All the cited features make DOWs very promising for a fast and accurate frequency characterization of VTs to be performed in industrial laboratories.
In this article, the use of DOWs for the fast characterization of inductive VTs is analyzed.It is important to underline that, although DOWs can be generated through circuits like those shown in [33] and [34], the scope of this article is not to validate a circuit for the generation of DOWs.The scope of this article is just to analyze the possibility to employ DOWs for the fast-frequency characterization of inductive VTs.Therefore, general purpose generation and measurement setups were used for this scope.To validate the proposed method, the results obtained with the DOW method are compared with those obtained using FH1 [18] and FHN [20], considered as the reference methods.Since FH1 and FHN are considered as the reference methods, in the rest of this article, the word "accurate," referred to the proposed method, means that the proposed method gives results compatible with those obtained with FH1 and FHN.
This article is organized as follows.Section II introduces the DOW test waveform and Section III provides guidelines on the selection of the DOW parameters, following analytical as well as physical considerations.Section IV introduces a slightly modified version of the DOW test waveform.Sections V and VI describe two different generation and measurement setups and the test procedures adopted to prove the feasibility of the proposed method.In Sections VII and VIII, extensive experimental results are provided and discussed, and finally, Section X draws the conclusion.

II. PROPOSED TEST WAVEFORM
The basic test waveform to employ in IT accuracy evaluation, according to international standards [9], [10], [11], [12], is a virtually pure sine waveform having rated frequency (50/60 Hz) and amplitude.However, in order to evaluate the IT accuracy at other frequencies, more complex waveforms are needed [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].All these authors agree on a general requirement: the use of realistic waveforms in the IT characterization.In fact, especially for inductive ITs, the measurement accuracy can be better evaluated only if they are tested in conditions similar to the actual operating conditions, due to their nonlinear behavior.With reference to inductive IT, Kaczmarek and Brodecki [22] show that it is possible to predict the accuracy in the measurement of a transient phenomenon starting from the accuracy at harmonic frequencies.FH1 and FHN tests require the use of a quite high number of waveforms, in the order of ∼100-1000, thus requiring a significant time, in the order of ∼1-10 h, for the characterization of a single IT.Therefore, this article, starting from the requirement of using a realistic waveform, analyses the possibility of using a different waveform with the aim to make the IT characterization process faster.The proposed waveform is composed of a fundamental tone, having rated amplitude and frequency (50/60 Hz), and a DOW, namely, a damped sinusoidal component, having a higher frequency and a reduced amplitude.One period of this waveform is described by the following equation: where In (1), the expression of the DOW is taken from [30] according to the mathematical modeling of many transient phenomena, and the waveform of DOW is expressed as exponentially decaying sinusoids.For the sake of clarity and simplicity, in this section, t 0 is chosen as equal to zero.In Section III-E, some considerations on t 0 will be shown.In the rest of this article, a waveform expressed by (1) will be referred to as FDOW.
The Fourier transform of the FDOW is described by the following equation: where δ(•) is the Dirac delta function.
Unlike the purely sinusoidal case, Y ( f ) has all the frequency components; moreover, considering the portion of the spectrum given by the DOW, the magnitude peak is obtained exactly when f = ± f d .These properties can be seen in the graphical representation of (1) and of the magnitude of (2), as shown, respectively, in Fig. 1 In order to make the FDOW suitable for VT characterization, it should have a periodicity equal to that of the fundamental component.Therefore, if a DOW is added, every period, to the fundamental component, (1) becomes the following equation: where Moreover, in the rest of this article, for the sake of clarity, instead of t 0 , we will refer to the quantity γ , related to t 0 as in the following equation: The quantity γ represents a phase angle referred to the fundamental component.In particular, it represents the fundamental phase angle at which the DOW starts.The use of γ allows to disengage from the fundamental frequency and, moreover, in the authors' opinion, it is a more intuitive quantity with respect to t 0 .With an abuse of language, the quantity γ will be referred to as the start phase angle of the DOW.

III. ANALYSIS OF THE EFFECTS OF THE DOW PARAMETERS CHOICE
The spectrum of the FDOW depends on the choice of the DOW parameters, i.e., damping factor, amplitude, frequency, phase angle, and γ .This section analyses how changes to these parameters affect the FDOW spectrum.

A. Phase Angle of the DOW: Effect on the DC Component
It can be easily demonstrated that the FDOW signal has a dc component since the exponential decay leads to an asymmetry between the positive and negative DOW half waves.Having a dc component is really undesirable since it can lead the transformers to saturate.Therefore, if a measurement setup including a step-up transformer is used, the presence of the dc component at its input will make it introduce significant spurious harmonics, thus introducing undesired distortion of the test waveform.Even if the measurement setup uses an MV amplifier to supply the test waveform to the inductive VT under test, a dc component could lead the VT to saturation and so to invalidate the test.
The dc component of the DOW can be evaluated starting with the following equation: In ( 6), the two terms in brackets are complex conjugate.Thus, their difference is an imaginary number and it is twice their imaginary part, given by the following equation: To provide a signal with no dc component, the phase angle φ d of DOW has to be chosen such that the numerator of ( 7) is equal to zero.Therefore, it is easy to show that the required value is given by the following equation:

B. Start Phase Angle of the DOW
The choice of the start phase angle γ of the DOW could introduce significant changes in the accuracy of an inductive VT evaluated by using an FDOW.In order to better understand its role, in the following, a brief recall of the operation of an inductive VT is done.
According to Faraday's law [35], when an input voltage is supplied to the primary winding of an inductive VT, it produces a proportional magnetic flux inside the iron core.This has a nonlinear behavior that influences the metrological performances of the VT and depends on the amplitude of the magnetic flux, which, in turn, depends on the supplied voltage amplitude.FH1 and FHN waveforms are basically a sum of steady-state sine waves, namely, the fundamental and the harmonic components.Considering one period of this kind of waveforms, the magnetic flux generated by the harmonic components is always present, for every value of the magnetic flux generated by the fundamental tone.Therefore, when the accuracy of the VT is evaluated at a harmonic frequency, by analyzing a time frame equal to an integer number of periods of the fundamental tone, the obtained values represent a kind of a mean value over all the values of the magnetic flux generated by the fundamental tone.
Different from FH1 and FHN waveforms, one period of an FDOW is the sum of a fundamental tone and a transient event, with a duration quite smaller than the fundamental period [see for instance, Fig. 1(a)].Therefore, in this case, when the accuracy of the VT is evaluated at a harmonic frequency, by analyzing a time frame equal to an integer number of periods of the fundamental tone, the obtained values represent a kind of a mean value over only a small portion of values of the magnetic flux generated by the fundamental tone.Thus, standing the nonlinear behavior of the VT, the position of the DOW inside the fundamental period can have a big influence on the obtained accuracy at a harmonic frequency.For the sake of more clarity, consider also the two following cases.
1) When γ is equal to zero, the DOW starts in correspondence with the positive zero crossing of the fundamental tone.In this case, for the entire duration of the DOW, the most important contribution to the total magnetic flux is given by the DOW.From the point of view of the obtained accuracy at harmonic frequencies, this situation produces results very similar to an FH1 or an FHN waveform having a very small fundamental component, far from its rated value.Crotti et al. [36] and [37] show that in this way the obtained results are not accurate.2) When γ is equal to π/2 rad, the DOW waveform starts in correspondence with the peak value of the fundamental tone.In this case, for the entire duration of the DOW, the total flux is very high, as it is given by the peak of the fundamental flux and by the DOW flux.Also, in this case, the obtained results could be inaccurate as the VT under test can reach the saturation.
These considerations have highlighted that the choice of γ is very important in order to obtain accurate results in the VT characterization by using FDOW.It follows that a reasonable value of γ may be around π/4 or 3π/4.In Section VII, the best value of γ will be experimentally determined.

C. Damping Factor of the DOW
The damping factor α of the DOW is the inverse of the decay time τ of the waveform.The possible range of variation of α is established starting from the typical DOW duration outlined in IEEE 1159 [8] and then narrowing it down.The decision to restrict this range is driven by the need to maintain the periodicity of the FDOW equal to those of the sole fundamental tone.To achieve this, it is crucial that at the end of every period of the fundamental tone, the amplitude of the superimposed DOW is negligible compared to the measurement uncertainty.This constraint is verified by considering that after about 20 times τ , the amplitude of the DOW can be considered zero (within 1 µV/V), so the maximum value of the time constant τ is fixed to 1 ms.Consequently, α has a minimum limit value of 1000 s −1 .
Choosing this minimum value of α implies that the amplitudes of the spectral components in the range [ f d − 500, f d + 500] Hz are quite higher than the amplitudes outside this range.In fact, the tones outside this range are quite overlapped with constant amplitudes, independently of the values of the α.It is worth recalling that one of the features of the novel test waveform should be the near uniformity of its amplitude spectrum.This condition is also suggested by the standards about grid voltage characteristics (see [2]), which establish Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
consistent and quite similar limits for harmonics beyond the 13 th order.
Therefore, the desired features of quite uniformity of the magnitude spectrum of the test waveform can be satisfied as the α parameter increases.However, it is not possible to arbitrarily increase this parameter, since, approaching the upper limit, the amplitudes of the produced spectral tones can reach low values implying higher uncertainty contributions.For these reasons, the α value is chosen as 2000 s −1 .

D. Amplitude of the DOW
According to (2), the amplitude A d of the DOW essentially influences the amplitude of the harmonic tones of the FDOW spectrum.Moreover, the FDOW magnitude spectrum has a peak at f = f d , whereas the amplitude rapidly decreases moving away from f d .In particular, the minimum amplitude value is obtained at the limit frequencies (not considering the fundamental component) of the range of interest.Moreover, it has to be considered that no indications about harmonic amplitudes to use in the tests are provided by international standards on ITs.Therefore, a possible criterion to choose the minimum value of A d can come from the minimum desired amplitude of the harmonic components at the limits of the frequency range of interest of the tests.
For what concerns the maximum value of A d , the limit can be imposed by the used instrumentation.In particular, according to [11], the accuracy of VTs at fundamental frequency must be evaluated by varying the amplitude of the fundamental component between 80% and 120% of the rated value.Typically, calibration laboratories have the capabilities to generate at least 130% of the rated fundamental amplitude.According to Section III-B, γ should be chosen in such a way that the peak amplitude of the FDOW is significantly lower than the sum of the peak amplitudes of the fundamental tone and of the DOW.
Following these considerations, a reasonable range of variation for A d is between 5% and 50% of the fundamental amplitude.

E. Frequency of the DOW
The frequency f d of the DOW [see (2)] is another parameter that influences the amplitude of the FDOW spectrum at the limits of the frequency range of interest of the tests and, therefore, determines the width of the range itself.
According to [4], the basic time frame to use for harmonic analysis is equal to ten cycles of fundamental waveform.Considering the case of a 50 Hz power system, this time frame is equal to 200 ms, giving a frequency resolution of 5 Hz.Theoretically, it is possible to choose the f d value equal to any integer multiple of 5 Hz, including a harmonic frequency of the fundamental tone.In this case, the DOW spectrum has all the harmonic components, including a component at the fundamental frequency, superimposed to the fundamental tone of the test.Moreover, since the DOW magnitude spectrum exponentially decreases, the first harmonics up to the tenth harmonic order can have an amplitude comparable with the spurious harmonics introduced by the fundamental tone, which are a source of errors.Therefore, in order to reduce the errors, a compensation technique, like SINDICOMP [18] or HD [20], should be used.However, in these cases, the number of tests increases.
To summarize, a possible option for the value of f d involves the selection of a harmonic or interharmonic frequency that halves the selected characterization bandwidth of the VT.In such a way, the average amplitude of the harmonic tones, inside the frequency range of interest, is maximized.

F. Ranges of Variation of the DOW Parameters
Following the considerations given in this section, Table I gives the ranges of variation of the DOW parameters.These values are given, without loss of generality, with reference to a 50 Hz power system.

IV. DOW INTRODUCING INTERHARMONIC COMPONENTS
Consider a particular FDOW including a DOW with the frequency f d equal to a harmonic component of the fundamental tone and with an arbitrary value of the phase angle φ d , provided that it increases π at each period of the fundamental component, as in the following equation: where h is an integer number at least ten times higher than 1.
In this case, the periodicity of the FDOW is equal to two cycles of the fundamental component.Therefore, theoretically, considering a 50 Hz power system, the spectrum of this waveform, obtained by analyzing a time frame of 200 ms, should contain tones every 25 Hz, thus harmonics but also interharmonics of the fundamental component.However, due to the particular symmetry of the DOWs in two consecutive periods, the even harmonics of the 25 Hz tone, including the dc component, are zero; therefore, other than the 25 Hz tone and the fundamental tone at 50 Hz, the spectrum contains only the odd harmonics of the 25 Hz component, which are interharmonics for the fundamental tone at 50 Hz.
A great advantage of this waveform is the fact that the introduced interharmonic components do not superimpose to the spurious harmonics due to the fundamental tone, and therefore, no compensation techniques or other supplementary tests are required for their accurate evaluation.Moreover, we can assume a very similar behavior of the VT between two adjacent harmonic frequencies.Therefore, if the VT is characterized by using this particular FDOW that introduces all the interharmonic components multiple of 25 Hz, the VT behavior at harmonic frequencies can be obtained by interpolation between two adjacent interharmonic tones.
In conclusion, the use of the DOW that introduces interharmonic tones seems the best candidate for the accuracy evaluation of VTs since: 1) structurally, no dc component is introduced (whatever the value of φ d ) and 2) no compensation technique nor supplementary tests are required since no harmonic components are introduced.
In the rest of this article, these kinds of waveforms will be referred to as fundamental plus interharmonic DOW (FIDOW).In Sections VI-VIII, both FDOW as well as FIDOW will be extensively analyzed and the related results compared to those obtained with FH1 and FHN tests.The AWG generates a 4 MHz clock that is used to derive the sampling clock; this allows obtaining coherent sampling, thus avoiding spectral leakage.Acquisition of the primary and secondary waveforms of the VT under test has been performed with a multifunction I/O module PXIe-6124 (± 10 V, 16 bit, and maximum sampling rate of 4 MHz).Waveforms have been sampled with a 1 MHz rate.The output of the AWG is connected to an arbitrary four-quadrant voltage and current amplifier Bolab (± 75 V, 40 A, 1 kW, and dc-1 MHz) feeding the VT under test through a 100 V/24 kV step-up transformer.
Primary voltages are scaled with a commercial divider OhmLab KV-10A (high-voltage divider, HVD) having a ratio of 1000 V/V and uncertainties on ratio and phase errors of, respectively, 180 µV/V and 190 µrad (level of confidence 95%) from dc up to 4 kHz.A low-voltage divider (LVD), having a ratio of about 18.5 V/V and uncertainties on ratio and phase errors of, respectively, 100 µV/V and 110 µrad (level of confidence 95%) from dc up to 4 kHz, has been designed and built for measuring the secondary voltage of the VT.Calibration of HVD and the LVD was performed at INRIM.The overall uncertainty (level of confidence 95%) of the measurement setup is lower than 300 µV/V and 300 µrad for the measurement of the ratio error and the phase error, respectively, from dc up to 4 kHz (see Table II).
The reference generation and measurement setup of INRIM is depicted in Fig. 2(b).

TABLE III VTS UNDER TEST PARAMETERS
a reference resistive-capacitive voltage divider (30 kV, dc to 12 kHz) designed, built, and characterized at INRIM [25].
The acquisition system is composed of an NI compact data acquisition system (cDAQ) chassis with various input modules having a 24 bit resolution, 50 kHz maximum sampling rate, and input range from ± 500 mV up to ± 425 V.The uncertainty (level of confidence 95%) of the ratio and phase errors up to 9 kHz is 250 µV/V and 300 µrad, respectively (see Table II).
It is worth noting that an alternative method to analyze and compare the outputs of the reference and the under test sensors is based on the use of a differential measuring system, as shown in [38].

VI. DESCRIPTION OF THE TESTS
In all the performed tests, the waveforms have been analyzed by performing the discrete Fourier transform (DFT) over a time frame of ten cycles of the fundamental component, as suggested by [3] for the measurement of harmonics.
The accuracy of the VTs under test at harmonic frequencies is evaluated by calculating the harmonic ratio and phase errors, through the following equations: where k r = V p,r /V s,r rated transformation ratio (V p,r and V s,r are the rated primary and secondary voltages, respectively); V p,h and V s,h rms values of the primary and secondary h-order harmonic voltages; ϕ p,h and ϕ s,h phase angles of the primary and secondary h-order harmonic voltages.
(10) and ( 11) are a frequency extension of ratio and phase errors defined for fundamental frequency [23].Two commercial inductive VTs for 50 Hz MV power systems, whose main features are shown in Table III, were tested.In particular, VT A is tested at SUN, whereas VT B is tested at INRIM.

A. Reference Tests
As highlighted in Section I, FH1 [18] and FHN [20] tests are assumed as the reference methods for the accuracy evaluation of VTs at the harmonic frequencies.That is, the values of ratio and phase errors at the harmonic frequencies of a VT  IV.Moreover, for each test, the fundamental amplitude A 1 is fixed equal to the rated primary voltage of the VTs under test.
It is worth noting that the VTs under test are preliminarily characterized with a sinusoidal waveform having the rated frequency and amplitude.This preliminary test is necessary to apply the SINDICOMP technique [18] to the results obtained with FH1 and FHN tests, in order to mitigate the nonlinearity errors introduced by the inductive VTs due to the presence of the fundamental component.
Basically, SINDICOMP consists in: 1) measure the VT secondary voltage harmonic phasors in the preliminary sinusoidal test and 2) use them as a correction for the spurious harmonic tones produced in the output voltage in the tests with distorted waveforms.
A detailed description of SINDICOMP can be found in [18] whereas [25] deepens FH1 and FHN tests, their comparison, and the application of SINDICOMP to both.

B. FDOW and FIDOW Tests
In all the FDOW and FIDOW tests, the fundamental amplitude is equal to the rated primary voltage of the VTs under test and the fundamental frequency is 50 Hz.Several tests were performed to analyze the effects of the variation of FDOW and FIDOW parameters on the VT accuracy evaluation through the new proposed procedure.All the obtained results are compared with those obtained with FH1 and FHN tests.Sections VII and VIII deeply discuss the experimental results.

VII. EXPERIMENTAL RESULTS: IMPACT OF THE DOW PARAMETERS
First of all, the impact of the DOW parameters (γ , f d , and A d ) on the evaluation of the VT frequency behavior is analyzed, in Sections VII-A-VII-C.They are chosen as indicated in Table V.Then, a comparison among the responses obtained with FDOW and FIDOW is provided in Section VII-D, highlighting the advantages and disadvantages of both approaches.For the sake of brevity and clarity, for these analyses, reference will be made only to ratio error.

A. Effects of the Start Phase Angle of the DOW
Fig. 3 shows various ratio errors [see (10)] of VT A , from the 20 th to the 80 th harmonic order, obtained with FDOW, by varying γ [fixing A d and f d as in Table V and φ d as in ( 8)], and with FHN.For the FHN test, the ± 95% ratio error variation range, along with the uncertainty (level Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE V DOW TEST PARAMETERS
Fig. 3. VT A ratio error at harmonic order h, obtained with: 1) FDOW with amplitude equal to 50%, frequency equal to 1.5 kHz, and various γ (dot, up-triangle, and down-triangle markers) and 2) FHN (mean value with diamond markers and ± 95 limits with square markers). of confidence 95%), is shown.This range of variation is obtained for an intrinsic reason.Each FHN [25] waveform has all the harmonic components but with random amplitudes and phase angles.The inductive VTs, due to the intrinsic nonlinear effects, have a harmonic behavior that depends on the particular waveform.In other words, by stimulating a VT with M different waveforms (see Table IV), M different frequency behaviors will be obtained.Fig. 3 shows, thus, the ratio error mean value curve and the two curves equal to ± 95% of the ratio error variation range plus the uncertainty (level of confidence 95%).The bars on the FDOW curves in Fig. 3 represent the uncertainty (level of confidence 95%).
In the rest of this article, the uncertainty (level of confidence 95%) is included both in the ± 95% limits as well as in the error bars.
Looking at Fig. 3, when γ is zero, the harmonic ratio error exceeds the FHN ± 95% range, whereas when γ is close to π/4 rad, the harmonic ratio error curve is close to the reference FHN curve.In this respect, Fig. 3 graphically represents what is theoretically explained in Section III-B.To better explain this effect, consider Fig. 4. Fig. 4(a) [Fig.4(b)] reports the ratio error of VT A (VT B ) at the 40 th (60 th ) harmonic order, obtained by varying γ .Fig. 4(a) and (b) also reports the FHN ratio error, both the mean value as well as the 95% limits, at the related harmonic frequency.
First, it can be observed that there is a periodicity of π rad of the ratio error versus γ : this result confirms that γ is strictly related to the flux operating conditions of the VT.
For both the tested VTs, selecting a γ value near to the zero crossing of the fundamental tone (i.e., 0 or π rad) provides results not compatible with the FHN results.Also, the selection of γ near to π/2 rad, that is near the maximum values of the fundamental components, can lead to results not compatible with FHN results.
Another interesting phenomenon to observe is the fact that, for both the VTs, when γ is equal to kπ rad, the obtained curves are always lower than the lower limit of the FHN range.This behavior is always observed when the VT harmonic ratio errors are evaluated with a frequency sweep of sine waves having amplitudes quite lower than the rated VT voltage.This phenomenon is highlighted also in [36] and [37].
These considerations highlight two important aspects of the choice of γ .The first is that, thanks to the periodicity of γ , there are four intervals, contained in one period [0, 2π ] rad of the fundamental tone, which ensure accurate results: they are the intervals around π/4 + kπ/2, with k being an integer number, that is, [π/6 + kπ/2, π/3 + kπ/2] rad.However, due to the considerations shown in Section III-C, related to the duration of the DOW, only the intervals in the first half period of the fundamental tone should be chosen.The second aspect is that it is not important to choose a specific γ for each VT under test since every value in the cited ranges give accurate results.This aspect, in particular, gives flexibility and versatility to the presented characterization method.

B. Effects of the Amplitude of the DOW
The choice of the DOW amplitude A d directly impacts on the amplitude of the produced harmonic tones.The lower is the DOW amplitude, the lower are the harmonic tones amplitudes, especially at harmonic frequencies far from f d .Fig. 5 reports VT A ratio errors measured at different harmonic frequencies by varying A d [ f d and γ are chosen as in Table V and φ d as in (8)].It can be observed that reducing A d produces results affected by a higher uncertainty, in the order of some percents.In particular, the higher uncertainty is observed for A d equal to 5% and for h = 10 and h = 70 that are, in Fig. 5, the harmonic tones farther from f d .This is due to measurement problems related to the necessity of measuring simultaneously harmonic components with amplitudes in the order of ∼1 mV and the fundamental tone with an amplitude of 3 kV.On the contrary, for h = 30, that is the frequency of the DOW, also the result obtained for A d = 5% is affected by quite a low uncertainty, compared to the uncertainty obtained for A d = 5% at the other harmonic orders, and the mean values measured under the three A d conditions are well-overlapped within a deviation of 0.1%.Fig. 6(a) [Fig.6(b)] shows the ratio error of VT B at the 50 th (60 th ) harmonic order, with f d = 3 kHz, γ = π /6 rad, and φ d as in (8) and three different values of A d , 10%, 20%, and 30%; FHN results are also reported.Similar to what was found for VT A , increasing the A d value, the uncertainty of the ratio error decreases.For h = 60, that is equal to f d , the results obtained with the three A d are within the ± 95% FHN range.For h = 50, even if all the FDOW mean values are well-overlapped with the FHN mean value, the uncertainty obtained for A d = 10% is higher than the FHN range limits.
Hence, as expected, increasing the A d values ensures more accurate measurements over the entire considered frequency range.Looking at Fig. 7(a), it can be observed that, for all the cases, ε 40 is compatible with the reference value.In particular, for f d = 2 kHz, that is h = 40, the mean value of the ratio errors obtained with FDOW and with FHN is perfectly overlapped and the deviations are within ± 0.1%.For f d values that differ most from h = 40, both the difference between the FDOW and FHN mean values, as well as FDOW deviations, increase.This result is explained considering that the 40 th harmonic tone has a low amplitude when f d moves away from 2 kHz.

C. Effects of the Frequency of the DOW
Very similar considerations can be made for VT B , looking at Fig. 7(b).This result suggests that f d should be chosen as the half of the upper limit of the frequency range in which the VT has to be characterized.

D. Comparison of FDOW and FIDOW Results
As said in Section IV, the FIDOW is a very promising waveform since comparison with the FDOW does not introduce the dc component and does not require a nonlinearity compensation technique.However, it introduces interharmonics instead of harmonics.Therefore, in order to use FIDOW to evaluate the VT accuracy at harmonic frequencies, we have to prove that the information on the VT behavior obtained with FDOW and FIDOW are equivalent.
For this analysis, both the FDOW as well as the FIDOW waveforms have the same values for the parameters f d , A d , γ , and φ d .The ratio errors of VT A are shown in Fig. 8, where f d = 1.5 kHz, A d = 50% of A 1 , γ = 4π /5 rad, and φ d is chosen as in (8).Instead, the ratio errors VT B are shown in Fig. 9(a), where f d = 3 kHz, A d = 30% of A 1 , γ = π/4 rad, and φ d is chosen as in (8).
Very similar considerations can be made for the two VTs.Excluding the ratio error values until 500 Hz, that is until the tenth harmonic order, the FDOW and FIDOW approaches yield overlapping results with maximum deviations of ± 0.1% Fig. 8.
VT A ratio errors at harmonic order h, obtained with FDOW (square markers) and FIDOW (diamond markers), both having f d = 1.5 kHz, A d = 50% of A 1 , and γ = 4π /5 rad.for VT A and ± 0.05% for VT B .Fig. 9(b) shows the difference between the FDOW and FIDOW ratio errors of VT B from the 20 th up to the 100 th harmonic order.
However, including the lower frequencies, deviations are higher than some percents.These differences are due to nonlinear effects at harmonic frequencies, which are highlighted by the FDOW test but not by the FIDOW test.In particular, FDOW waveforms with f d quite higher than 1 kHz provided to the VT under test primary harmonics with low amplitudes (<0.1%), which can be directly compared to the spurious tones generated by the VT due to the presence of the fundamental component [18].These spurious tones can be compensated by applying nonlinearity compensation techniques known in the literature, such as SINDICOMP [18] or HD [20].The application of SINDICOMP leads to a quite smoother behavior until the first harmonic frequencies, with ratio and phase error values close to those at 50 Hz.Looking at the experimental results, it is evident that a response exhibiting these characteristics (quite flat up to h = 10) is obtained with the FIDOW test, which, by its mathematical definition, does not introduce harmonics but rather interharmonics, which are not affected by the same nonlinearities exhibited by the harmonics.
Although the FIDOW test introduces a slight complication in the test waveform (the phase changes every fundamental period by an amount equal to π ) and requires to be analyzed over time frames integer multiple of two fundamental periods, it does not require the use of a nonlinearity compensation technique.Therefore, it provides results that can be directly compared with FH1 and FHN, provided that FH1 and FHN, as highlighted in Section VI-A, results have been compensated with a suitable nonlinearity compensation technique.
Another advantage of both FDOW and FIDOW tests is the possibility to measure the first resonance frequency of an inductive VT, as it can be seen in Fig. 9(a), with a simple waveform.It is worth noting that this feature is, in principle, obtainable with a waveform containing all the harmonic components in the range of interest.However, with FDOW and FIDOW, it is obtained with a waveform represented by a simple equation.

VIII. COMPARISON OF FIDOW WITH
THE REFERENCE METHODS This section presents a comparison among the VTs ratio and phase error responses obtained with the proposed FIDOW approach and with the reference methods, among the entire frequency range, that is up to 4 kHz for VT A and 9 kHz for VT B .11(a) up to h = 60.For both the VTs, and for both ratio as well as phase errors, we can see that the FH1, FHN, and FIDOW curves are well-overlapped.The maximum deviations between the mean value curves of the FIDOW results and those obtained with FH1 and FHN are 0.2% and 650 µV/V for ratio errors, respectively, for VT A and VT B , whereas they are 500 and 600 µrad for phase errors, respectively, for VT A and VT B .
In particular, Fig. 11(c) shows the deviation between the FIDOW and the FHN ratio errors up to the 110 th harmonic order: the deviation is contained in the range of ± 650 µV/V.

IX. DISCUSSION ON THE MAIN RESULTS
The main outcomes of this article can be summarized as follows.
A new realistic power system waveform, made by the superposition of a fundamental component and a periodic realistic transient (defined in the international standards about power systems [8]), to use for the frequency characterization of inductive VTs, has been defined.
It has a wide spectrum, since contains all the harmonic, in the case of FDOW, or the interharmonic (odd multiples of the half of the fundamental frequency), in the case of FIDOW, components up to some kilohertz.In this way, with a single waveform, it is possible to evaluate the frequency behavior of a VT.
Since a VT can be characterized with a single waveform, the duration of the tests is strongly reduced to the order of ∼1 s, whereas the most acknowledged procedures for frequency characterization of VTs, namely, FH1 [18] and FHN [20], require a test duration in the order of ∼1-10 h.A deep theoretical and experimental analysis has been conducted in order to choose the best values for the damped sine wave parameters.
The version of the waveform that produces only interharmonic components does not need the application of a nonlinearity compensation technique, whereas FH1 and FHN require it.
Two different commercial MV VTs have been characterized with two different measurement setups.The frequency behaviors obtained with the proposed procedures have been compared with those obtained with the reference methods, which are FH1 and FHN, resulting in compatibility and showing maximum deviations among the mean values of 0.2% and 600 µrad, respectively, for ratio and phase errors, up to 9 kHz.

X. CONCLUSION
This article has presented a new waveform that allows performing a fast frequency characterization of inductive VTs.It is a periodic waveform composed of a fundamental tone and a damped sine wave, which begins every period at the same fundamental tone phase angle and ends within the same period.The main outcomes of this article are summarized in Section IX.
It is important to underline that these waveforms can be generated and measured by using the same measurement setups able to implement FH1 and FHN tests.In this respect, it does not require additional features of the measurement setups.In addition, another advantage of the proposed waveform is the fact that it can be quite easily generated with a generator setup quite simpler than those required by FH1 and FHN tests.As shown, for instance, in [33] and [34], a DOW can be generated through a circuit composed of a number of passive components, namely, resistors, capacitors, and inductors, opportunely sized, one or more switches and a power frequency or a dc generator.All these features make the proposed procedure attractive for the frequency characterization of inductive VTs.
V. MEASUREMENT SETUP To experimentally evaluate and demonstrate the validity of the proposed characterization technique, two measurement setups are developed at the Università degli Studi della Campania "Luigi Vanvitelli" (SUN), Aversa, Italy, and Istituto Nazionale di Ricerca Metrologica (INRIM), Turin, Italy.The generation and measurement setup of SUN is shown in Fig. 2(a).The reference voltage signal to be applied to the VTs under test is provided by an arbitrary waveform generator (AWG) National Instrument (NI) PCI eXtension Instrumentation (PXI) 5422 (16 bit, variable output gain, ± 12 V output range, 200 MHz maximum sampling rate, and 256 MB of onboard memory).

Fig. 5 .
Fig. 5. VT A ratio error at harmonic order h, obtained with FDOW with f d and γ equal to 1.5 kHz and 4π /5 rad, respectively, and various A d values.

Fig. 6 .
Fig. 6.VTB ratio error at harmonic order: (a) h = 50 and (b) h = 60, obtained with FDOW with f d and γ equal to 3 kHz and π/6 rad, respectively, and various A d values.

Fig. 7 (
Fig. 7(a) [Fig.7(b)] shows the ratio errors of VT A (VT B ) measured at a fixed harmonic tone, h = 40 (h = 60), obtained with FDOW, by varying f d [fixing A d and γ as in Table V and φ d as in (8)], and with FHN.Looking at Fig.7(a), it can be observed that, for all the cases, ε 40 is compatible with the reference value.In particular, for f d = 2 kHz, that is h = 40, the mean value of the ratio errors obtained with FDOW and with FHN is perfectly overlapped and the deviations are within ± 0.1%.For f d values that differ most from h = 40, both the difference between the FDOW and FHN mean values, as well as FDOW deviations, increase.This result is explained considering that the 40 th harmonic tone has a low amplitude when f d moves away from 2 kHz.Very similar considerations can be made for VT B , looking at Fig.7(b).

Fig. 9 .
Fig. 9. (a) VT B ratio errors at harmonic order h, obtained with FDOW (square markers) and FIDOW (diamond markers), both having f d = 3 kHz, A d = 30% of A 1 , and γ = π/4 rad.(b) VT B ratio error deviations among the mean values of FIDOW and FDOW harmonic ratio errors.

Fig. 10 (
a) [Fig.10(b)] shows the ratio (phase) errors of VT A obtained with FH1, FHN, and FIDOW.Instead, Fig. 11(a) [Fig.11(d)] shows the ratio (phase) errors of VT B obtained with FH1, FHN, and FIDOW.Both for VT A as well as for VT B , FIDOW parameters have the same values of the test described in Section VII-D.Fig. 11(b) shows a zoomed-in view of Fig.

TABLE I SUGGESTED
RANGES OF VARIATION OF DOW PARAMETERS

TABLE II EXPANDED
UNCERTAINTY (LEVEL OF CONFIDENCE 95%) ASSOCIATED WITH THE MEASUREMENT OF RATIO AND PHASE ERROR USING THE TWO EXPERIMENTAL SETUPS The voltage signals are generated by the AWG NI PXI 5421, with 16 bit, variable output gain, ± 12 V output range, 100 MHz maximum sampling rate, and 256 MB of onboard memory.The AWG is housed in a PXI chassis and its 10 MHz PXI clock is used as a reference clock for the phase-locked loop (PLL) circuitry.Another AWG is employed to generate a 12.8 MHz clock, which is provided as a time base clock to the acquisition system.
The low-voltage waveform generated by the AWG is amplified by a Trek high-voltage power amplifier (± 30 kV, ± 20 mA, and 20 kHz).The applied MV values are scaled by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IV REFERENCE
TEST PARAMETERS obtained with these tests are considered the reference values.The parameters of reference tests are shown in Table