Extending analyzed frequency range in interpretation of frequency responses measured on a distribution transformer

In the field of diagnosis of mechanical failures in power transformer’s active part, i.e., windings, leads and the core, the technique of Frequency Response Analysis (FRA) has been recently approved as the main application tool. Mechanical failures in transformer windings reflect changes on measured terminal frequency responses normally in medium frequency range, from several to hundreds of kHz, which is in fact not easy to interpret for diagnosis. The authors proposed a new method based on simulation of a lumped three-phase equivalent circuit of power transformers to interpret frequency responses effectively, but only within low frequency range. This limitation is due to the fact that, the circuit cannot reflect well physical phenomena at medium and higher frequencies. To improve the FRA performance of the proposed method at medium frequencies for transformer failure diagnosis purpose, the paper introduces an investigation on a distributed three-phase equivalent circuit of a 200 kVA 10.4/0.46 kV Yy6 distribution transformer. Result of the investigation is a simplified procedure in determination of electrical parameters associated with the distributed circuit for better simulation based FRA interpretation at medium frequencies.


INTRODUCTION
To understand what happens in transformer's active part after a suspected through fault or during transportation for diagnostic purpose, measurement of terminal frequency responses of voltage ratios (end-to-end, inductive and capacitive interwinding) in broad frequency range, e.g., from 20 Hz to 2 MHz, are often made and then compared with those performed when transformers were in good condition.However, there is no guide from current relevant CIGRE and IEC standards [1,2] to identify type and level of fault based on the comparison since there are so many factors influencing measured frequency responses such as transformer type (normal/auto), winding type (disc/layer/interleaved/helical), winding number (two/three), winding connection (vector group), winding's terminal condition (open-circuited/short-circuited/floating), measurement set-up etc.There was a national standard, the Chinese DL/ T-2004 [3], proposing a quantative analysis but its effectiveness in supporting the diagnosis is limited as investigated in recent publications [4][5][6].For illustration, Figure 1 shows a comparison between end-to-end frequency responses measured before and after a clear partial axial collapse and inter-turn shortcircuit of a tap winding of a large power transformer [2].In reality, when the deviation between compared frequency responses is small, it is difficult to diagnose fault type and level due to the above mentioned influencing factors.The authors proposed a new method for supporting the interpretation of frequency responses in such a way that changes between frequency responses at certain frequencies would be transformed into changes of distinct electrical parameters of power transfomers as this could help to figure out fault location and somewhat level [4][5][6].The proposed method was based on simulation of a lumped three-phase equivalent circuit shown in Figure 2, which has been the state-of-the-art in transformer modeling for transient and frequency response analysis so far.
In the dual magnetic-electric circuit (middle part) in Figure 2, R1//L1, Ry//Ly are nonlinear core leg and yoke impedances, respectively; L3 are perphase leakage inductances; R4//L4 are per-phase zero-sequence impedances; all of them are frequency dependent.The high-and low-voltage (HV and LV) winding circuits (outside parts) represent electrical parameters of the whole winding, i.e., equivalent resistances (RH, RL) and capacitances (CsH, CsL: series; CgH, CgL: ground or shunt; Ciw: inter-winding), and winding connection (wye, delta) in accordance with vector group.More details of the lumped circuit and procedures in determination of its components can be found in [4].Although electrical parameters in the equivalent circuit for two-winding power transformers are effective for diagnosis, the circuit simulation based FRA interpretation is valid within low frequency range, from 20 Hz till several or tens of kHz, depending on transformer and winding type.To illustrate the limitation of the lumped circuit, Figure 3 compares a simulated end-to-end frequency response with the corresponding measured one conducted at HV side of a test transformer whose details will be mentioned at the end of this section.
In Figure 3, the simulation curve is valid from 20 Hz (core region) to around 15 kHz (zerosequence inductance influence).At higher frequencies the lumped electrical parameters cannot reflect well the interaction between sectional inductances and capacitances, and therefore it is necessary to analyze the so-called distributed circuit for interpretation of frequency responses at these frequencies.In order to determine electrical parameters in the distributed circuit based on analytical calculation, complete geometrical data and magnetic-electric properties of transformer components (core, windings and insulation system) must be available [8,9].For a contribution to practice application, the authors propose a new parameter identification procedure where less data will be enough with aim to extend the analyzed frequency range for simulation based frequency response interpretation.
The test object in this paper is a 200 kVA 10.4/0.46 kV Yy6 distribution transformer whose measurement is shown in Figure 3.To facilitate the investigation with the distributed circuit development, after all measurements were carried out, the transformer was disassembed to measure its geometrical parameters (structure and dimensions of the core and windings).

Per segment capacitances CgH0, CgL0, Ciw0
Since influence of series capacitances (CsH and CsL) is insignificant from simulation manipulation of the lumped circuit, only ground and inter-winding capacitances need to be determined and are identified from corresponding lumped capacitances derived from the proposed method in [4] by following relations: where n is the segment number to be selected for investigation.
Therefore, geometrical data and electric properties of windings and insulation system of the transformer are not necessary for analytical calculation of the capacitances.

Per segment inductances Li, Lj and Mij
While inductances in the lupmed circuit represent complete fluxes within core, zerosequence and leakage paths, inductances in the distributed circuit 'break' the fluxes into individual parts caused by current in winding segments and are referred as self and mutual components.Below are analytical formulas for calculating self and mutual inductances in the distributed circuit based on geometrical data and magnetic-electrical properties of the core.

Geometrical data
Figure 5 shows geometrical data of two winding segments with presence of the core.For the test transformer, n = 8 segments is selected, which is relatively a compromise between circuit complexity and simulation accuracy for first investigation.
where Lkm0 mutual inductance between kth and mth sections without the core (air core) Z1(km) mutual impedance between kth and mth sections owing to flux confined in core Z2(km) mutual impedance between kth and mth sections owing to leakage field with core presence The resistive component of Zkm represents eddy current loss in the core whereas the inductive one is the total mutual inductance between two sections.Self inductance is a special case of mutual inductance between a section with itself, i.e., Zkk or Zmm.
Following are detailed formulas for determination of Lkm0, Z1(km) and Z2(km).Since Lkm0 is the winding segment inductance when the core material is non-magnetic (air core), only geometrical data are involved.For calculating Z1(km) and Z2(km), together with geometrical data, two input magnetic-electric properties, effective relative permeability rel and resistivity eff of the solid-considered core, must be available.Iron-core impedance (flux in the core) Z1(km)

Air-core inductance Lkm0
where  h1, w1, h2, w2 dimensions of k th and m th segment respectively  a1, a2, r1, r2 inner, outer radii of k th and m th segment respectively  c ratio between relative permeabilities in axial and radial direction [10].

Parameter calculation
To calculate impedances from (3), ( 4) and ( 5), it is required that value of effective relative permeability rel and resistivity eff of the core should be known in advance.
The two parameters rel and eff can be determined if one has measurements of self/mutual impedances of/between winding segments as investigated in [10].However, it is not the case for this investigation and others in practice since all winding segments are in transformer and cannot be broken to measure.Therefore, a new way in identification of rel and eff is proposed as follows.
First, specific values of rel and eff are initially assigned, e.g., the ones in [10], since the investigated subjects in this reference are power transformers (with rated power from 25 kVA to 200 MVA).Then, by comparing simulated and measured frequency responses at low and medium frequencies where inductive components are dominant, deviations between them reveal whether the assigned rel and eff are correct or should be adapted to compensate the deviations.
The procedure of identifying value of rel and eff is based on the fact that, rel influences much analytical inductances at low frequencies whereas eff shows strong effect at medium frequencies, as illustrated in Figures 6 and 7, respectively.

Per segment resistances and conductances
Resistance component in self and mutual impedances of winding segments representing only eddy current losses in the core is calculated using (1).In addition, another component that accounts for skin effect in the winding itself should be taken into account for a more correct equivalence.
On the other hand, determination of conductances parallel with corresponding capacitances (see Figure 4) needs geometrical data and electrical property of insulation system [9].
Nevertheless, influence of resistances and conductances on frequency responses is of minor importance since they contribute only to damping at resonance peaks.For simplified simulation approach, they can be assigned appropriate values so as good agreement between measurement and simulation is achieved.

RESULTS
For first investigation with the distributed transformer circuit, due to limitation of commercial software in simulating mutual effect and frequency dependent inductance simultaneously, constant self and mutual inductances were selected for simulation whereas constant resistances/conductances were adapted based on agreement between measured and simulated frequency responses within range from 10 kHz to 100 kHz.For better representation, inductances and resistances should be frequency dependent.
Figure 8 shows a comparison between measurement and simulation approaches of the end-to-end frequency response.Better agreement proves that, although constant inductances/resistances were selected, the distributed circuit represents well interactions of reasonably calculated inductances and capacitances between winding segments, which is impossible with the lumped circuit at medium frequencies from 10 kHz to 100 kHz.For determination of electrical parameters in the distributed circuit, the paper introduced a simplified procedure based on the proposed method in [4], i.e., values of lumped capacitances, and analytical calculation.This procedure would be beneficial for real applications since it reduces dependence on geometrical and electrical property of transformer insulation system for capacitance calculation and help to find out magnetic-electric properties of the core for inductance dete rmination, which are mostly unavailable in reality.

Figure 1 .
Figure 1.Comparison of frequency responses measured on a winding before and after fault

Figure 2 .
Figure 2. Lumped circuit of a Yy6 transformer

Figure 3 .
Figure 3. Measurement and simulation of an endto-end frequency response Pure mechanical failures in transformer windings normally show changes on frequency responses starting at medium frequencies [7].For theoretical investigations, simulation technique based on the distributed circuit has been exploited[8, 9].Figure4depicts a per-phase distributed circuit with a multi-segment HV and LV winding, from which the complete circuit of three-phase two-winding transformers is derived by combination of three of them, adding their mutual effect and internal terminal connection.

Figure 4 .
Figure 4. Per-phase distributed equivalent circuit In Figure 4, the HV and LV phase winding are divided into a number of segments each of which has equivalent electrical components: self/mutual inductances (Li, Lj/Mij), ground, series, inter-winding capacitances (Cg0, Cs0, Ciw0

Figure 5 .
Figure 5. Illustration of geometrical data of winding segments and the core circuit Analytical formulas Wilcox et.al. proposed an accurate analytical solution based on Maxwell's equations in determination of self and mutual inductances of transformer winding segments [10]: permeability of vacuum  Nk, Nm turn numbers of k th and m th segment respectively  r, a radii of k th and m th segment from core center respectively apparent length of the magnetic circuit  N constant affecting accuracy degree


eff effective resistivity of the solid core  I0, I1 modified Bessel functions  10 magnetic permeability of medium outside the core Iron-core impedance (leakage flux) Z2(km)