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Volt–time characteristics of small airgaps with Hyperbolic model
Electric Power Systems Research 80 (2010) 739–742
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Short communication
Volt–time characteristics of small airgaps with Hyperbolic model S. Venkatesan a,∗,1 , S. Usa a,b,2 a b
Cardiff University, School of Engineering, Queens Building, The Parade, Cardiff, Wales CF24 3AA, United Kingdom Division of High Voltage Engineering, Anna University, Chennai - 600025, India
a r t i c l e
i n f o
Article history: Received 13 January 2008 Received in revised form 14 November 2008 Accepted 13 December 2009 Available online 6 January 2010
a b s t r a c t An experimental investigation of the volt–time characteristics of small airgaps is performed. A Hyperbolic model is proposed to account for the results. The constants of the model have a direct bearing on parameters of the Disruptive Effect model for breakdown with non-standard Lightning Impulses (LIs). Analyses with uniform and non-uniform electrodes show their effect on the Hyperbolic model parameters. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
Keywords: Disruptive Effect Nonstandard impulse Transformer Unconditionally sequential approach
1. Introduction The major proportion of failures in transformers are due to inter-turn and inter-disc winding insulation (oil and OIP) failures, where clearances are in the order of millimeters. Small voids in dry type resin cast transformers may lead to permanent failure in the near future. During transient conditions, these insulations can be stressed with non-standard impulse voltages of both unidirectional and bi-directional oscillating waveshapes. The inter-turn and inter-disc winding insulation failures are attributed to these oscillating voltages. There exists an analytical prediction procedure (Modified Disruptive Effect method) [2] to estimate the insulation strength of these non-standard waveshapes. This method serves as a measure of insulation strength under oscillating voltages and estimate the breakdown strength irrespective of the medium of the insulation and of the physical processes involved [1,2]. It is important to note that this method utilise volt–time characteristics to obtain the Disruptive Effect (DE) model parameters. To estimate the strength under non-standard waveshape, one should first estimate the experimental volt–time characteristics using the standard 1.2/50 s impulse waveshape. If there exist an analytical prediction model to obtain the volt–time characteristics, then the efforts
in doing experimentation could be minimized. One such model namely a Hyperbolic model based on the DE model parameters is proposed here. A basic study with this model on small airgaps of 1–6 mm gap distances is dealt here. It is well known that the Disruptive Effect model [3] is an effective tool to estimate the strength under non-standard waveshape. It is stated that when the area under the impulse curve above the onset voltage is greater than critical value (DE* ) the breakdown takes place. This phenomenon is explained in the DE model [3], which states that the condition for sparkover at time ‘tb ’ is given by Eq. (1).
DE∗ =
tb
(U − U0 ) dt
(1)
0
where U0 is the onset voltage, U is the applied voltage, DE* is the critical Disruptive Effect area above the onset voltage U0 and tb is the time to breakdown. When this area above the onset voltage (U0 ) exceeds a critical value, the breakdown takes place. Based on this model, parameters (DE* , U0 ) are extracted from the standard 1.2/50 s impulse volt–time characteristics. 2. Experimental volt–time characteristics
∗ Corresponding author. Tel.: +44 2920870673. E-mail address: [email protected] (S. Venkatesan). 1 S. Venkatesan is currently Research Associate in School of Engineering, Cardiff University, Cardiff, UK. 2 S. Usa was his research supervisor and is Professor in High Voltage Engineering Division, Anna University, Chennai, India.
A MWB (Messwandler-Bau), 100 kV, 250 kJ impulse generator with fine voltage control facility is utilised for generating standard 1.2/50 s impulse voltages. The test cell consists of uniform plane electrodes with the provision for minute gap adjustment. The uniformity in electrical field distributions is verified with the help of
0378-7796/$ – see front matter. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.12.002
740
S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
Table 1 Mean (t ) and SD ( t ) values of tb (4 mm GAP). Voltage (kV)
t (s)
t (s)
12.17 12.28 12.38 12.49 12.60 12.71 12.82 12.93 13.10 13.43
7.94 7.2 6.08 5.53 4.71 4.6 3.36 3 3.26 2.69
0.94 0.76 0.4 0.27 0.54 0.95 0.94 0.78 0.9 0.50
Finite Element Method (FEM) software. From the field plots, maximum electric field (Emax ) and utilization factor () are determined for each gap distance and the variation in is found to be in the range of 0.92–0.85 [1]. This clearly shows that the field is more uniform in plane–plane configuration while compared to other geometries. The measuring system consists of a Tektronix TDS 420 oscilloscope, which is connected to the computer via GPIB (IEEE 488) interface, by which digitized waves can be acquired to the computer for further analysis. Due to variations in tb , it is essential to plot the mean statistical impulse volt–time curve. It is observed that while performing constant voltage test, the time to breakdown, where breakdown occurs in the tail of the impulse, follows Normal distribution. Hence, for each gap setting, constant voltage tests are performed at different voltage levels, so that breakdown occurs in tail of the impulse. A number of ni impulses resulting in breakdown are applied at every voltage level and the time to breakdown tb is measured every time. A sample case for 4 mm gap distance is considered and the mean (t ) and standard deviation ( t ) of tb for the plane–plane configuration is as shown in Table 1. With the mean time to breakdown and the voltage applied, a curve is plotted, which is denoted as the mean volt–time curve. As the time to breakdown has a Normal distribution, the probability of a voltage to breakdown at a time ‘tb ’ shall be expressed and based on which the 5% and 95% probability breakdown curves were also constructed. These curves are utilized in extracting the parameters of the DE model. 2.1. Extraction of DE parameters
Fig. 1. Onset voltage (U0 ) vs distance.
this onset voltage is the factor responsible for the time lag. Let a constant proportional to this area be defined as ‘B’ and is given by
tb
(U − U0 ) dt
B∝
(2)
0
The breakdown voltage (Ub ) can be defined with the above said constants as Ub = A +
B t
(3)
This hyperbolic function defines the breakdown voltage in terms of the onset voltage (U0 ), Critical Disruptive area (DE* ) and the time to breakdown. In order to verify this model, with the voltage and mean values of time to breakdown (U, t ) obtained from the statistical mean volt–time curve, a regression analysis is carried out to obtain the best fit. With this model, the volt–time curves are reconstructed and the errors in reconstruction are found to be within 5% limit. 3.1. Hyperbolic and DE model parameters Since the Hyperbolic model has been derived from the parameters of the DE model, a regression analysis is carried out to find
The DE parameters satisfying the equal area criterion are U0 and DE* . From the statistical volt–time curve, digitized waves corresponding to the points that lie on the mean (50%) volt–time curve are considered. By using the equal area criterion [3], the DE parameters are extracted. These values correspond to the 50% probability volt–time curve. Similar procedure is followed in estimating the parameters for the 5% and the 95% probability volt–time curves. Figs. 1 and 2 shows the values of U0 and DE* as a function of gap distance. The procedure to obtain the DE parameters involves a major experimental part in obtaining the volt–time characteristics. In the next section, the proposed Hyperbolic model is introduced. 3. Hyperbolic volt–time curve model During impulse voltage application, a minimum voltage is required for the initiation of breakdown processes. This voltage is dependent on gap distance, dielectric and also on electrode configuration. Let a constant proportional to this voltage be defined as ‘A’. Above this onset voltage the breakdown takes place at different times depending on the amount of over voltage and other factors. As explained in the Disruptive Effect model, the area above
Fig. 2. DE* vs distance.
S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
741
Fig. 5. B vs distance for plane–plane and plane–cone electrodes.
Fig. 3. A vs U0 and B vs DE* .
3.2. Relation between Emax and Hyperbolic model parameters
out the relationship, and from Fig. 3, it can be concluded that there exist a linear relationship between A and U0 and also between B and DE* . This serves as a direct proof for the derived model. The effect of electrode geometry on the Hyperbolic model parameters could be verified by obtaining the parameters for a non-uniform electrode configuration. A plane–cone electrode configuration as detailed in [1] is chosen for this purpose. Hyperbolic model parameters are derived using the similar procedure, as stated for plane–plane electrode configuration. The parameters ‘A’ and ‘B’ are as shown in Figs. 4 and 5 respectively. As the plane–cone electrode has a lower onset voltage (U0 ) than the uniform plane–plane configuration, a similar trend is observed in ‘A’ values. The variation in ‘B’ with respect to distance is less significant in plane–cone configuration, while compared to the plane–plane electrode configuration. This clearly explains the influence of electrode geometry in both the parameters (‘A’ and ‘B’). Also the proposed model is verified with standard (1.2/50 s), non-standard (with different front and tail times) and also with unidirectional oscillating voltages of frequency of oscillation 0.1, 0.2 and 0.3 MHz.
As shown in the previous section, the electrode geometry has a significant influence in values of the Hyperbolic model parameters. To perform an in-depth analysis, Emax values calculated using the FEM software are utilised. Figs. 4 and 5 show a definite relationship between the maximum field Emax and the Hyperbolic model parameters, which will be useful in obtaining the volt–time characteristics in terms of computed Emax . The Fig. 6 indicate a similar trend of variation in ‘A’ with respect to Emax for both plane–plane and plane–cone electrode configurations. This indicates a possibility that A can be expressed as a function of Emax and distance. In Fig. 7, it can be observed that B values are higher in the case of plane–plane electrode configuration and much lower in case of plane–cone electrode configuration. This again indicates the possibility that B could be expressed as a function of electrode configuration and gap distance. Once the volt–time characteristic is expressed with the Hyperbolic model, Disruptive Effect model parameters can easily be arrived. The proposed model denotes a simplified procedure to estimate the insulation strength under non-standard waveshapes. The Hyperbolic model serves as a better representation of the volt–time characteristics in terms of the DE model parameters. Hence, by
Fig. 4. A vs distance for plane–plane and plane–cone electrodes.
Fig. 6. ‘A’ as a function of ‘Emax ’.
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S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
4. Conclusion The behavior of small airgaps under impulse voltages is studied. A Hyperbolical model, in terms of the Disruptive Effect model parameters is proposed. The model constant A represents the Onset voltage and B represents the area above the onset voltage. With this model the volt–time curve can be predicted for any gap distance. The utility of the model lies in a simple prediction of the behavior of insulation under non-standard impulse waveshapes. Acknowledgments The Authors wish to thank Dr. V. Jayashankar, Associate Professor, IITM, Chennai, for his fruitful discussions and moral support throughout this work. Also, the authors wish to thank AICTE, New Delhi for funding this project. References Fig. 7. ‘B’ as a function of ‘Emax ’.
expressing the volt–time characteristics with this model, one can determine the DE model parameters. This will eventually serve as the base to determine insulation strength under non-standard waveshape. As an analytical model which is independent of the physical processes involved, this method can also be expanded to other insulations like oil and OIP.
[1] S. Venkatesan, S. Usa, Impulse strength of transformer insulation with nonstandard waveshapes, IEEE Trans. Power Deliv. 22 (October (4)) (2007) 2214–2221. [2] S. Usa, V. Jayashankar, K. Udayakumar, Modified Disruptive Effect method as a measure of insulation strength for non-standard lightning waveforms, IEEE Trans. Power Deliv. 17 (April (2)) (2002) 510–515. [3] M. Darveniza, A.E. Vlastos, The generalized integration method for predicting volt–time characteristics for non-standard waveshapes—a theoretical basis, IEEE Trans. Electr. Insul. 23 (33) (1988) 373–381.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Short communication
Volt–time characteristics of small airgaps with Hyperbolic model S. Venkatesan a,∗,1 , S. Usa a,b,2 a b
Cardiff University, School of Engineering, Queens Building, The Parade, Cardiff, Wales CF24 3AA, United Kingdom Division of High Voltage Engineering, Anna University, Chennai - 600025, India
a r t i c l e
i n f o
Article history: Received 13 January 2008 Received in revised form 14 November 2008 Accepted 13 December 2009 Available online 6 January 2010
a b s t r a c t An experimental investigation of the volt–time characteristics of small airgaps is performed. A Hyperbolic model is proposed to account for the results. The constants of the model have a direct bearing on parameters of the Disruptive Effect model for breakdown with non-standard Lightning Impulses (LIs). Analyses with uniform and non-uniform electrodes show their effect on the Hyperbolic model parameters. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
Keywords: Disruptive Effect Nonstandard impulse Transformer Unconditionally sequential approach
1. Introduction The major proportion of failures in transformers are due to inter-turn and inter-disc winding insulation (oil and OIP) failures, where clearances are in the order of millimeters. Small voids in dry type resin cast transformers may lead to permanent failure in the near future. During transient conditions, these insulations can be stressed with non-standard impulse voltages of both unidirectional and bi-directional oscillating waveshapes. The inter-turn and inter-disc winding insulation failures are attributed to these oscillating voltages. There exists an analytical prediction procedure (Modified Disruptive Effect method) [2] to estimate the insulation strength of these non-standard waveshapes. This method serves as a measure of insulation strength under oscillating voltages and estimate the breakdown strength irrespective of the medium of the insulation and of the physical processes involved [1,2]. It is important to note that this method utilise volt–time characteristics to obtain the Disruptive Effect (DE) model parameters. To estimate the strength under non-standard waveshape, one should first estimate the experimental volt–time characteristics using the standard 1.2/50 s impulse waveshape. If there exist an analytical prediction model to obtain the volt–time characteristics, then the efforts
in doing experimentation could be minimized. One such model namely a Hyperbolic model based on the DE model parameters is proposed here. A basic study with this model on small airgaps of 1–6 mm gap distances is dealt here. It is well known that the Disruptive Effect model [3] is an effective tool to estimate the strength under non-standard waveshape. It is stated that when the area under the impulse curve above the onset voltage is greater than critical value (DE* ) the breakdown takes place. This phenomenon is explained in the DE model [3], which states that the condition for sparkover at time ‘tb ’ is given by Eq. (1).
DE∗ =
tb
(U − U0 ) dt
(1)
0
where U0 is the onset voltage, U is the applied voltage, DE* is the critical Disruptive Effect area above the onset voltage U0 and tb is the time to breakdown. When this area above the onset voltage (U0 ) exceeds a critical value, the breakdown takes place. Based on this model, parameters (DE* , U0 ) are extracted from the standard 1.2/50 s impulse volt–time characteristics. 2. Experimental volt–time characteristics
∗ Corresponding author. Tel.: +44 2920870673. E-mail address: [email protected] (S. Venkatesan). 1 S. Venkatesan is currently Research Associate in School of Engineering, Cardiff University, Cardiff, UK. 2 S. Usa was his research supervisor and is Professor in High Voltage Engineering Division, Anna University, Chennai, India.
A MWB (Messwandler-Bau), 100 kV, 250 kJ impulse generator with fine voltage control facility is utilised for generating standard 1.2/50 s impulse voltages. The test cell consists of uniform plane electrodes with the provision for minute gap adjustment. The uniformity in electrical field distributions is verified with the help of
0378-7796/$ – see front matter. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.12.002
740
S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
Table 1 Mean (t ) and SD ( t ) values of tb (4 mm GAP). Voltage (kV)
t (s)
t (s)
12.17 12.28 12.38 12.49 12.60 12.71 12.82 12.93 13.10 13.43
7.94 7.2 6.08 5.53 4.71 4.6 3.36 3 3.26 2.69
0.94 0.76 0.4 0.27 0.54 0.95 0.94 0.78 0.9 0.50
Finite Element Method (FEM) software. From the field plots, maximum electric field (Emax ) and utilization factor () are determined for each gap distance and the variation in is found to be in the range of 0.92–0.85 [1]. This clearly shows that the field is more uniform in plane–plane configuration while compared to other geometries. The measuring system consists of a Tektronix TDS 420 oscilloscope, which is connected to the computer via GPIB (IEEE 488) interface, by which digitized waves can be acquired to the computer for further analysis. Due to variations in tb , it is essential to plot the mean statistical impulse volt–time curve. It is observed that while performing constant voltage test, the time to breakdown, where breakdown occurs in the tail of the impulse, follows Normal distribution. Hence, for each gap setting, constant voltage tests are performed at different voltage levels, so that breakdown occurs in tail of the impulse. A number of ni impulses resulting in breakdown are applied at every voltage level and the time to breakdown tb is measured every time. A sample case for 4 mm gap distance is considered and the mean (t ) and standard deviation ( t ) of tb for the plane–plane configuration is as shown in Table 1. With the mean time to breakdown and the voltage applied, a curve is plotted, which is denoted as the mean volt–time curve. As the time to breakdown has a Normal distribution, the probability of a voltage to breakdown at a time ‘tb ’ shall be expressed and based on which the 5% and 95% probability breakdown curves were also constructed. These curves are utilized in extracting the parameters of the DE model. 2.1. Extraction of DE parameters
Fig. 1. Onset voltage (U0 ) vs distance.
this onset voltage is the factor responsible for the time lag. Let a constant proportional to this area be defined as ‘B’ and is given by
tb
(U − U0 ) dt
B∝
(2)
0
The breakdown voltage (Ub ) can be defined with the above said constants as Ub = A +
B t
(3)
This hyperbolic function defines the breakdown voltage in terms of the onset voltage (U0 ), Critical Disruptive area (DE* ) and the time to breakdown. In order to verify this model, with the voltage and mean values of time to breakdown (U, t ) obtained from the statistical mean volt–time curve, a regression analysis is carried out to obtain the best fit. With this model, the volt–time curves are reconstructed and the errors in reconstruction are found to be within 5% limit. 3.1. Hyperbolic and DE model parameters Since the Hyperbolic model has been derived from the parameters of the DE model, a regression analysis is carried out to find
The DE parameters satisfying the equal area criterion are U0 and DE* . From the statistical volt–time curve, digitized waves corresponding to the points that lie on the mean (50%) volt–time curve are considered. By using the equal area criterion [3], the DE parameters are extracted. These values correspond to the 50% probability volt–time curve. Similar procedure is followed in estimating the parameters for the 5% and the 95% probability volt–time curves. Figs. 1 and 2 shows the values of U0 and DE* as a function of gap distance. The procedure to obtain the DE parameters involves a major experimental part in obtaining the volt–time characteristics. In the next section, the proposed Hyperbolic model is introduced. 3. Hyperbolic volt–time curve model During impulse voltage application, a minimum voltage is required for the initiation of breakdown processes. This voltage is dependent on gap distance, dielectric and also on electrode configuration. Let a constant proportional to this voltage be defined as ‘A’. Above this onset voltage the breakdown takes place at different times depending on the amount of over voltage and other factors. As explained in the Disruptive Effect model, the area above
Fig. 2. DE* vs distance.
S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
741
Fig. 5. B vs distance for plane–plane and plane–cone electrodes.
Fig. 3. A vs U0 and B vs DE* .
3.2. Relation between Emax and Hyperbolic model parameters
out the relationship, and from Fig. 3, it can be concluded that there exist a linear relationship between A and U0 and also between B and DE* . This serves as a direct proof for the derived model. The effect of electrode geometry on the Hyperbolic model parameters could be verified by obtaining the parameters for a non-uniform electrode configuration. A plane–cone electrode configuration as detailed in [1] is chosen for this purpose. Hyperbolic model parameters are derived using the similar procedure, as stated for plane–plane electrode configuration. The parameters ‘A’ and ‘B’ are as shown in Figs. 4 and 5 respectively. As the plane–cone electrode has a lower onset voltage (U0 ) than the uniform plane–plane configuration, a similar trend is observed in ‘A’ values. The variation in ‘B’ with respect to distance is less significant in plane–cone configuration, while compared to the plane–plane electrode configuration. This clearly explains the influence of electrode geometry in both the parameters (‘A’ and ‘B’). Also the proposed model is verified with standard (1.2/50 s), non-standard (with different front and tail times) and also with unidirectional oscillating voltages of frequency of oscillation 0.1, 0.2 and 0.3 MHz.
As shown in the previous section, the electrode geometry has a significant influence in values of the Hyperbolic model parameters. To perform an in-depth analysis, Emax values calculated using the FEM software are utilised. Figs. 4 and 5 show a definite relationship between the maximum field Emax and the Hyperbolic model parameters, which will be useful in obtaining the volt–time characteristics in terms of computed Emax . The Fig. 6 indicate a similar trend of variation in ‘A’ with respect to Emax for both plane–plane and plane–cone electrode configurations. This indicates a possibility that A can be expressed as a function of Emax and distance. In Fig. 7, it can be observed that B values are higher in the case of plane–plane electrode configuration and much lower in case of plane–cone electrode configuration. This again indicates the possibility that B could be expressed as a function of electrode configuration and gap distance. Once the volt–time characteristic is expressed with the Hyperbolic model, Disruptive Effect model parameters can easily be arrived. The proposed model denotes a simplified procedure to estimate the insulation strength under non-standard waveshapes. The Hyperbolic model serves as a better representation of the volt–time characteristics in terms of the DE model parameters. Hence, by
Fig. 4. A vs distance for plane–plane and plane–cone electrodes.
Fig. 6. ‘A’ as a function of ‘Emax ’.
742
S. Venkatesan, S. Usa / Electric Power Systems Research 80 (2010) 739–742
4. Conclusion The behavior of small airgaps under impulse voltages is studied. A Hyperbolical model, in terms of the Disruptive Effect model parameters is proposed. The model constant A represents the Onset voltage and B represents the area above the onset voltage. With this model the volt–time curve can be predicted for any gap distance. The utility of the model lies in a simple prediction of the behavior of insulation under non-standard impulse waveshapes. Acknowledgments The Authors wish to thank Dr. V. Jayashankar, Associate Professor, IITM, Chennai, for his fruitful discussions and moral support throughout this work. Also, the authors wish to thank AICTE, New Delhi for funding this project. References Fig. 7. ‘B’ as a function of ‘Emax ’.
expressing the volt–time characteristics with this model, one can determine the DE model parameters. This will eventually serve as the base to determine insulation strength under non-standard waveshape. As an analytical model which is independent of the physical processes involved, this method can also be expanded to other insulations like oil and OIP.
[1] S. Venkatesan, S. Usa, Impulse strength of transformer insulation with nonstandard waveshapes, IEEE Trans. Power Deliv. 22 (October (4)) (2007) 2214–2221. [2] S. Usa, V. Jayashankar, K. Udayakumar, Modified Disruptive Effect method as a measure of insulation strength for non-standard lightning waveforms, IEEE Trans. Power Deliv. 17 (April (2)) (2002) 510–515. [3] M. Darveniza, A.E. Vlastos, The generalized integration method for predicting volt–time characteristics for non-standard waveshapes—a theoretical basis, IEEE Trans. Electr. Insul. 23 (33) (1988) 373–381.