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Application of Cole–Cole model to transformer oil‐paper insulation considering distributed dielectric relaxation

2019; Institution of Engineering and Technology; Volume: 4; Issue: 1 Linguagem: Inglês

10.1049/hve.2018.5079

2096-9813

S. K. Ojha, P. Purkait, Biswendu Chatterjee, S. Chakravorti,

Magnetic Properties and Applications

High VoltageVolume 4, Issue 1 p. 72-79 Research ArticleOpen Access Application of Cole–Cole model to transformer oil-paper insulation considering distributed dielectric relaxation Sandip Kumar Ojha, Sandip Kumar Ojha Department of Electrical Engineering, Haldia Institute of Technology, Haldia, 721657 West Bengal, IndiaSearch for more papers by this authorPrithwiraj Purkait, Corresponding Author Prithwiraj Purkait praj@ieee.org Department of Electrical Engineering, St. Thomas' College of Engineering & Technology, Kolkata, 700023 West Bengal, IndiaSearch for more papers by this authorBiswendu Chatterjee, Biswendu Chatterjee Electrical Engineering Department, Jadavpur University, Kolkata, 700032 West Bengal, IndiaSearch for more papers by this authorSivaji Chakravorti, Sivaji Chakravorti Electrical Engineering Department, Currently at NIT Calicut on lien from Jadavpur University, Kolkata, 700032 West Bengal, IndiaSearch for more papers by this author Sandip Kumar Ojha, Sandip Kumar Ojha Department of Electrical Engineering, Haldia Institute of Technology, Haldia, 721657 West Bengal, IndiaSearch for more papers by this authorPrithwiraj Purkait, Corresponding Author Prithwiraj Purkait praj@ieee.org Department of Electrical Engineering, St. Thomas' College of Engineering & Technology, Kolkata, 700023 West Bengal, IndiaSearch for more papers by this authorBiswendu Chatterjee, Biswendu Chatterjee Electrical Engineering Department, Jadavpur University, Kolkata, 700032 West Bengal, IndiaSearch for more papers by this authorSivaji Chakravorti, Sivaji Chakravorti Electrical Engineering Department, Currently at NIT Calicut on lien from Jadavpur University, Kolkata, 700032 West Bengal, IndiaSearch for more papers by this author First published: 04 February 2019 https://doi.org/10.1049/hve.2018.5079Citations: 23AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Researchers have been exploring dielectric testing techniques both in time and frequency domain for insulation condition assessment of oil-paper insulated transformers. In a practical dielectric system, dipoles are found to behave according to a distribution of elementary Debye relaxation properties. Suitable distribution density functions have been proposed to characterise such many-body interaction processes. Cole–Cole diagrams can be one of the methods for studying the nature of frequency dependency of dielectric materials of complex structure. Cole–Cole plots are commonly used for characterising different materials such as dielectric mixtures, ionic liquids, cable insulating oil, polar liquids etc. The scope of its application for assessing transformer oil-paper insulation considering distributed relaxation process has not been explored yet. The present contribution discusses mathematical formulations used for transforming the experimentally obtained time domain dielectric response test data to distribution domain and further to frequency domain for obtaining the Cole–Cole plots. Findings about the influence of various operating conditions and insulation status on the Cole–Cole diagram have been reported in this contribution. Results of tests on field transformers are also presented. This paper attempts to employ the features of Cole–Cole diagrams as potential indicators for analysing condition of the oil-paper insulation considering distributed relaxation process. 1 Introduction INTERPRETATION of the measured dielectric response test data on transformer oil-paper insulation has been of great interest to researchers in the recent past [1-3]. Analysis outcome in such cases may be accused of being superficial unless the results could be correlated to the physical processes involved during dielectric response measurements. In an insulation system, two types of interaction forces are prevalent during dielectric relaxation process. In the short range, the force due to chemical bonds, Van der Waals attraction, and repulsion forces are prevalent, while in the longer range, dipolar interaction forces are to be considered [4]. The relaxation behaviour is influenced by the fact that dipoles in a complex insulation structure is expected to have several equilibrium positions, each occupied with a probability that is influenced by multi-factor dependency of the dielectric relaxation process [5, 6]. A polarised dielectric in such cases can be considered as combination of small regions each having a certain dipole moment distributed over a wide range of relaxation frequencies [7-12]. In a recent publication [11], the authors have proposed the use of suitable distribution density functions for physical understanding of such multi-body interaction processes. Whereas dielectric responses on transformer oil-paper insulation in both time as well as frequency domain have been independently used as tolls for assessment, correlation between the two is often used by researchers for better understanding of polarisation process [12-15]. Hitherto, evaluation of dielectric test data has been based on the assumption of single (average) relaxation process [7, 8] or at most a limited number of discrete relaxation processes [3, 6]. In order to include the influence of distributed dipole relaxation phenomena observed in practical insulation systems, the present contribution adopts the theoretical guidelines provided in [10, 16-18] for analytically modelling the behaviour of transformer oil-paper insulation system considering distributed relaxation process. The so called Cole–Cole diagram was introduced by K. S. Cole and R. H. Cole [19, 20] as one potential tool for studying frequency dependency of complex permittivity of dielectric materials. Such a diagram is the plot between real and imaginary components of complex permittivity (ε′ and ε″) with frequency as the running parameter. Cole–Cole plots have thenceforth been used by researchers for characterising different materials and composites including dielectric mixtures [21], ferrite films [22], ionic liquids [23], cable insulating oil [24], polar liquids [25], dielectric liquids [26] etc. Initial attempts for analysing dielectric response measurement results of transformer oil-paper insulation with Cole–Cole model were carried out by S. Wolny et al. [27-29]. They highlighted the fact that real dielectrics rarely have a single dominant relaxation frequency and the intermolecular interactions should not be ignored as such while interpreting their dielectric response. The present contribution probes further and attempts to investigate the influence of transformer oil-paper insulation status on Cole–Cole diagrams while considering the relaxation processes involved to be widely distributed in frequency domain. The distributed nature of relaxation phenomena is demonstrated with the help of distribution density functions estimated from measured time domain polarisation (and depolarisation) current data. Data from distribution domain are further transformed numerically to obtain frequency domain dielectric response corresponding to the distributed relaxation processes. Cole–Cole diagrams are finally constructed from frequency domain data and an attempt has been made to demonstrate the impact of condition of the insulation on Cole–Cole parameters. Whereas time domain and frequency domain dielectric responses are considered to be good indicators of general condition of the oil-paper insulation, there has been lack of reliable quantifiable indicators of insulation condition that are universally accepted. In that respect, novelty of the present article is in exploration of Cole–Cole parameters as potential tool for quantitative analysis of oil-paper insulation status considering distributed nature of its relaxation process. 2 Experimental details Paper-wrapped conductors forming coil structures with spacer in between were immersed in mineral oil to imitate a scaled down oil-paper insulation structure of a real transformer. Polarisation/depolarisation current (PDC) measurements were performed on these samples under varying operating states and insulation conditions. The samples M1, M2, M3, P1, P2, and P3 were prepared under controlled laboratory conditions. Copper conductors of rectangular section (10 × 2 mm) wrapped with two layers of 0.35-mm-thick paper insulation have been used for preparing the above samples. These paper-insulated conductors were obtained from standard winding conductors used for transformer manufacturing. The method of moisture conditioning of the paper samples involves measuring the water vapour pressure and the temperature in the conditioning vessel, and then utilising the Piper chart [30] to monitor the paper moisture content level. That is, inside a sealed container under constant water vapour and constant temperature, a certain percentage of moisture content in the paper can be achieved if conditioning is continued for long period of time. Before the moisture conditioning, the samples were first dried at 100°C under vacuum. After this, the temperature inside the vessel was reset to a lower value for conditioning. Then, water vapour at specified pressure and temperature was supplied to the vessel. In this way, three separate batches of paper samples were prepared with low, medium, and high levels of moisture contents. After the desired moisture levels in paper were attained, dry and degassed hydrocarbon oil was injected into the vessel while it was under vacuum. The paper/oil equilibrium was achieved in about one week. After the equilibrium process was complete, moisture contents of both paper and oil were measured using Karl Fischer Titration technique. The entire system was cooled down to ambient temperature before the dielectric tests were done. Further details on the experimental setup are available in [11]. The scaled down transformer model used as samples B1, B2, B3 and A1, A2, A3 are prepared in the laboratory by using double paper insulation covered round copper conductors as used in a real transformer. Inner LV coil is wound over a thick pressboard cylinder. Separation between LV and HV coils are maintained with pressboard spacers of 2, and 5 mm thickness. Geometrical dimensions of the coils and insulation have been so constructed that it resembles capacitance value of a real transformer. Photograph of coil and insulation structure is included in Fig. 1. A metal tank is made to enclose the whole coil and insulation assembly. Tank clearances from the outer coil are also made in proportion to those in a real transformer to obtain realistic values of capacitance. Fig. 1Open in figure viewer Process flow scheme Descriptions of all test samples and operating conditions are summarised in Table 1. Table 1. Description of test samples Sample identifier Description M1 paper moisture = 2.8%, oil moisture = 11.7 ppm M2 paper moisture = 2.6%, oil moisture = 16.1 ppm M3 paper moisture = 2.7%, oil moisture = 25.2 ppm P1 paper moisture = 1.86%, oil moisture = 25.6 ppm P2 paper moisture = 3.05%, oil moisture = 25.4 ppm P3 paper moisture = 3.15%, oil moisture = 23.3 ppm B1 test temperature 40°C B2 test temperature 50°C B3 test temperature 70°C A1 charging voltage 100 V A2 charging voltage 200 V A3 charging voltage 300 V T1 30 MVA transformer before oil filtering T2 same transformer after oil filtering and drying 3 Mathematical formulations The process flow diagram describing the intermediate steps for obtaining Cole–Cole diagram starting from time domain polarisation measurements is shown in Fig. 1. The paper wrapped insulation sample as shown in Fig. 1 has been immersed in dry mineral oil–sealed inside a tank to imitate scaled down transformer model. Numerical transformations that are backbone to the intermediate steps are described in the following sections. 3.1 Conversion from time domain to distribution domain The dielectric response function f(t) according to the classical Debye model [7, 8] can be written as: (1)The time constant τD in (1) is called the 'Debye dielectric relaxation time' or simply the 'relaxation time'. It denotes the time required by dipoles to rearrange themselves after the exciting electric filed is removed. For step input excitation, , the polarisation process can be expressed as: (2)Here, ε0 is vacuum permittivity. Putting (1) in (2) (3)Thus, (4)Noting the monotonically decreasing property of response function f(t), at long times such that from (4) we can derive the value of the integration constant λ as: (5)Putting this value of λ from (5), and combining it with (2), from (4) we have: (6)The value of P(t) at steady state when t→∝ is given by: (7)where is defined as the steady-state susceptibility. Thus, combining (6) and (7), the response of a dielectric with single relaxation time τD can be expressed by a first-order rate equation as: (8)When expressed in terms of the relaxation frequency , the relaxation response (8) of a dielectric with single relaxation time τD can be expressed by a first-order rate equation as [18]: (9)The Debye relaxation theory [7, 8] as followed above, assumes that molecules present in the dielectric structure are spherical in shape and hence its axis of rotation has no influence in the value of its permittivity under external field. In real dielectrics, however, not only that the molecules can have different shapes, they often have linear configurations such that in long chain polymers like paper or pressboard [31]. Further, in the oil-paper insulation configuration, the dipoles invariably interact with each other during relaxation and are thus not independent in their response to external field. The relaxation properties in such composite materials thus are different depending upon the axis of rotation, and degree of interaction and, as a result, the dispersion commonly occurs over a considerably wide frequency range. In order to identify the underlying nature of such a frequency-dispersed relaxation process, the measured PDC test data are converted numerically to distribution domain in which a suitable distribution density function f(ν) has been estimated. Such a distribution density function demonstrates how the relaxation frequencies are distributed, rather than having a single relaxation frequency. The distribution function f(ν) is defined in terms of the rate of change of total polarisation Pt with respect to relaxation frequency ν following [11, 18] as: (10)The distribution density F(ν) is then defined in terms of the distribution function as: (11)The depolarisation current i(t) in continuous time domain can be expressed in terms of the distribution density F(ν) corresponding to the relaxation frequency ν as: (12)where Pot is the total polarisation in the dielectric under equilibrium condition. With Pot normalised to 1, (12) can be utilised in discrete form to calculate (νk, Fk) from the experimentally obtained depolarisation current data (tdi, idi) as: (13)An iterative procedure [11, 18] is to be adopted to compute the relaxation frequency distribution spectra from time domain depolarisation current data following (13). 3.2 Conversion from distribution domain to frequency domain To observe the influence of distributed relaxation process on frequency response characteristics of the insulation structure, data from distribution domain is transformed to frequency domain. Under the influence of an external electric field, the effect of polarisation is observed in a dielectric due to the combined effects of orientation of dipoles and also through induced polarisation of the individual atoms. The polarisation P in such a dielectric material is related to the electric field E by the relation: (14)whereas electric permittivity ε is more commonly used to describe dielectric phenomena in a capacitor, the susceptibility χ is more suitable for complex dielectric structures, such as in transformers. The frequency spectrum of susceptibility is thus more appropriate for representing the frequency domain characteristics or frequency domain spectra (FDS) of complex insulation systems. The polarisation developed in a dielectric material upon application of a sinusoidal electric field is also sinusoidal with initial value P0 and angular frequencyω: (15)With sinusoidal excitation, the relaxation response given in (9) can be re-written in terms of a single relaxation frequency as: (16)From where we obtain: (17)The expression (17) is designated as the complex relaxational susceptibility of the material such that: (18)The real and imaginary parts of the complex susceptibility can hence be formulated as: (19) (20)With the relaxation process being dispersed along frequency, the total susceptibility χt corresponding to all the processes involved at the instant t also follows the distribution function f(ν) in a manner similar to (10) as: (21)Within the small frequency range ν to (ν + dν), (21) can be utilised to express the effect of equilibrium polarisation in terms of total susceptibility of the system as: (22)Thus from (18) and (22): (23)Defining as earlier, superposition of all such elemental relaxation process produces the frequency dispersion of complex susceptibility: (24)Real and imaginary parts of complex susceptibility can now be formulated following (19), (20) and (24) as: (25) (26)Normalising total equilibrium susceptibility , real and complex susceptibilities in discrete form can be obtained numerically from known set of values of discrete frequency distribution function (νk, Fk) using modified forms of (25) and (26) as: (27) (28)where k = (N − i + 1). Frequency spectrum of susceptibility of the material considering distributed relaxation processes can now be obtained by plotting χ′(ωi) and χ″(ωi) against ωi. The real part χ′(ω) of the complex susceptibility is due to contribution of free space, and hence does not indicate any dielectric power loss as such. On the other hand, the FDS ofχ″(ω) can be utilised to describe the dielectric loss phenomena in an insulation, in a manner similar to how dissipation factor (tanδ) is used. 3.2.1 Constructing cole–cole diagram from distributed frequency spectrum The Cole–Cole model can be originated once again from the basic Debye model [7, 8] that presents a normalised form of complex permittivity ε(ω) based on a single average relaxation time τd as: (29)here is the angular frequency, ε(0) is the static dielectric constant, and ε∝ is the dielectric constant at infinite frequency. K. S. Cole and R. H. Cole in [19, 20] developed a method to correlate the dielectric response of real materials with idealised Debye behaviour. They obtained a plot of ε″(ω) of the complex permittivity ε(ω) of (29) versus ε′(ω) over the entire range of frequency and found that it forms a semicircle with its centre at a point on the ε' = 0 axis. It was, however, noticed during experimental observations on real dielectrics that the plot often formed only an arc of a circle, rather than a full semicircle, while its centre lying below the ε″ = 0 axis [19, 20, 25]. Such deviation in the Cole–Cole plot was attributed to the distributed nature of relaxation process taking place in the complex dielectric material. They proposed an equation that is analytical modification of the Debye single relaxation model (29): (30)The distribution parameter α is used to characterise the width of relaxation time distribution within the range 0–1. Withα = 0, the Cole–Cole model reduces to Debye model and α > 0 indicates that the relaxation process is dispersed over a wide range of frequency. The real and imaginary parts of complex permittivity are related to the complex susceptibility counterparts (27)–(28) as [8, 16]: (31) (32)The Cole–Cole diagram, which essentially is the plot of versus , can also be obtained by suitably scaling and shifting the plot axes and then using the values ofχ″(ω)and χ′(ω) from (19) and (20) instead. The general shape of such a scaled and shifted Cole–Cole diagram obtained by plotting χ″(ω) against χ′(ω) is shown in Fig. 2. The two vectors u and v in Fig. 2 are defined in (33) and (34). Fig. 2Open in figure viewer Cole-Cole diagram with centre of the circle lying below the horizontal axis It was reported in [27-29, 32] that the distribution parameter α and the time constant τd are sensitive to operating temperature and moisture level of the insulation. Next sections are thus dedicated to extracting these two parameters from the Cole–Cole plot obtained for all test conditions and observe their interdependence. 3.3 Extracting parameters of the cole–cole plot For determining the time constant τd and coefficient α of Cole–Cole diagram, two vectors u and v are defined as suggested in [33], in the complex plane as: (33) (34)So that: (35)Relating with the modified stretched Cole–Cole expression obtained from (30), we have: (36) (37) (38)Equation (38) when plotted on logarithmic scale produces a straight line as shown in Fig. 3, based on which the parameter values τd and α of Cole–Cole model can be determined. Value of the time constant τd can be calculated from the intersection of the linear approximation of (38) with the vertical axis, while the coefficient α can be obtained from the slope of this linear approximation. Fig. 3Open in figure viewer Determining τd and α from Cole-Cole diagram 4 Results and discussion Following the process flow diagram of Fig. 1, relaxation frequency distribution functions of all samples described in Table 1 are computed using (13). These frequencies dispersed relaxation functions are utilised to obtain values of FDS parameters χ′(ω) and χ″(ω) in discrete for according to (19) and (20). Cole–Cole diagrams of all test samples are then plotted utilising these FDS data. Finally, the Cole–Cole parameters τd and α are determined as per the process outlined by (38). To describe the process, depolarisation current plots for the test samples M1, M2, and M2 that represent insulation with similar paper moisture content, but different oil moisture content values are plotted in Fig. 4. An obvious increase in magnitudes of depolarisation current is observed in Fig. 4 owing to increase in conductivities of oil at higher moisture contents. The difference is primarily in the initial values of depolarisation currents since oil conductivity plays dominant role in influencing the charging and discharging currents at smaller times. Fig. 4Open in figure viewer Depolarisation current plots for samples M1, M2 and M3 at different oil moisture content values Nature of frequency dispersed relaxation properties of the oil-paper samples M1, M2, and M3 are demonstrated by the plots of their distribution density functions shown in Fig. 5. Fig. 5Open in figure viewer Relaxation frequency distribution functions for M1, M2, and M3 Fig. 5 shows growing peaks of relaxation frequency distribution function in the high-frequency zone with increase in oil moisture content values. Higher moisture content in oil due to its high polarity greatly influences spectrum of dipole particle relaxation in the higher frequency (short time) region. Plots of FDS parameters χ′(ω) and χ″(ω) obtained from the distribution density functions of M1, M2, and M3 are shown in Fig. 6. Fig. 6Open in figure viewer FDS for M1, M2, and M3 It is observed in Fig. 6 that FDS plots of χ′ and χ″ are lower for insulation with lower oil moisture contents as compared to the ones with higher oil moisture content. Real part of susceptibility at low frequency χ′(0) is higher for samples with higher moisture content because it takes higher charging current due to higher oil conductivity at longer times (DC). Peak of imaginary part of susceptibility (χ″) that indicates dielectric loss is higher as expected for the sample with higher moisture contents in oil. However, the shape of χ″ peak is not sharp, rather it is reasonably broad, indicating frequency dispersed relaxation process. The plot of χ′ vs. χ″ with ω as the running parameter gives a scatter plot in the shape of arc of a circle. A numerical curve fitting technique is employed to construct the circle that best fits these scatter points. The Cole–Cole, thus, obtained for samples M1, M2, and M3 are shown in Fig. 7. Deviation of the original (scatter) plots from pure semi-circular shape towards a skewed arc shape and also their centres lying well below the horizontal axes are clear indication of distributed relaxation process in the material [29]. It can be seen from Fig. 7 that with increase in oil moisture content, radius of the circle increases. Fig. 7Open in figure viewer Cole-Cole plots for M1, M2, and M3 Similar Cole–Cole plots obtained for the other test samples are shown in Figs. 8, 10, 9. Only portion of the arcs above horizontal axis are shown in these figures. Though all these figures take basic shape of circles, axes limits are different in different figures mainly due to differences in rating and physical size of insulation structures being tested. Fig. 8Open in figure viewer Cole-Cole plots for P1, P2, and P3 at different paper moisture content values Fig. 9Open in figure viewer Cole-Cole plot for B1, B2 and B3 at different operating temperatures Fig. 10Open in figure viewer Cole-Cole plot for A1, A2, and A3 at different charging voltages It could be observed from Fig. 8 that the shape of Cole–Cole diagram deviates much from being pure semicircle as the paper moisture content increases. In each of the samples, presence of two circles could be identified. It has been reported that appearance of more than one circle indicates the presence of multiple dominant relaxation times [34]. In some cases, more than one circle is distinctly identifiable, while in others, such multiple circles may be too small to be clearly identified. If the relaxation times are well separated, then Cole–-Cole plot will exhibit clearly resolved dispersions, separated by plateaus at intermediated frequencies. In case the dominant relaxation times are not clearly separated or are close, then those circles may not be distinct and may overlap. Out of the two circles, the one corresponding to high-frequency range (closer to origin) represents condition of oil, while the one corresponding to lower frequency range represents paper condition. As expected, size of the second circle increases with increase in paper moisture contents. In the series combination of oil and paper, the cellulose acts as a barrier for the ionic charge carriers in the oil and surface charges accumulate on the solid surfaces. Therefore, any variation in paper moisture content also affects the oil conductivity which is visible from the change in sizes of the first circles also in the Cole–Cole plots for samples P1, P2, and P3 in Fig. 8. Fig. 9 shows that with rising temperature, radius of the Cole–Cole circles are found to be bigger. Similar findings on variation of Cole–Cole circle radius with operating temperature were reported in [23]. Increase in temperature is said to affect the dipole ordering, increases chaos, and thereby it results in higher FDS values causing increase in Cole–Cole circle size. The samples A1, A2, and A3 are obtained by performing PDC measurements at three different charging voltages on the same insulation sample. It is found (Fig. 10) that the Cole–Cole plot in each case have two circles. It is interesting to note that diameter of the circles representing oil condition increases with increase in charging voltage, whereas there is no such trend as such in circles representing the paper condition. Cole–Cole plots for two field transformers, T1 and T2 are shown in Fig. 11, where two circles in each plot could be identified. For the extremely aged transformer T1 that was running under fully loaded condition is found to have both its circles of higher diameter than those for T2 which is relatively less aged. Fig. 11Open in figure viewer Cole-Cole plot for T1 and T2 Values of τd and α have been computed from the Cole–Cole plots following (30) and Fig. 3 for all the test samples detailed in Table 1. The results are summarised in Table 2. Table 2. Relaxation time and Distribution parameters obtained from Cole-Cole diagram for all test samples Sample identifier α τd, μs M1 0.3807 94.8 M2 0.3578 83.2 M3 0.3228 69.7 P1 0.3444 68.4 P2 03435 65.5 P3 0.3361 64.3 B1 0.0147 4.39 B2 0.0119 4.35 B3 0.010 4.22 A1 0.029 5.88 A2 0.0188 4.45 A3 0.0131 4.32 T1 0.256 128.02 T2 0.262 201.17 It is found from Table 2 that with increasing temperature (B1, B2, B3) and increasing moisture content (M1, M2, M3 and P1, P2, P3) in insulation, the time constant τd is shortened. With increasing amount of oil moisture contents in samples having similar paper moisture (samples M1, M2, M3), there is noticeable reduction in values of both distribution parameter and relaxation time. It is quite possible that increased oil moisture