## Broadband model of the distribution network

Research output: Book/Report › Ph.D. thesis › Research

### Abstract

Due to increased interest in the power quality of the distribution network, it is necessary to have an accurate model of the distribution network. One of the commonly used components in the distribution, is the low-voltage four-wire PEX-M-AL distribution cable. There exists no model of this component, and it is chosen to focus on the development of such a model.

Based on the electromagnetic field equations, the shunt admittance and series impedance parameters of the four-wire cable are derived. The influence of the conductance is considered negligible and is not included in the shunt admittance. Two expressions are derived for the capacitance to ground, one of which is based on the cable forming a plate capacitor to ground, the other is based on the cable and ground forming a coaxial cable. For the modelling of the series impedance, specific attention is given to include the influence of non-ideal ground, skin effect and proximity effect. Non ideal ground is modelled on the basis of Carson expressions for overhead lines. A new method is developed for the calculation of skin effect based on the depth of penetration. The result of this method is compared with the general expressions for circular conductors involving Bessel series. The two methods show equal values of resistance, but there is considerable difference in the values of internal inductance. A method for calculation of proximity effect is derived for a two-conductor configuration. This method is expanded to the use on the fourwire cable.

A number of measurements are performed on two four-wire cables, including frequency sweeps of open ended and shorted cables, as well as square wave measurements. The frequency sweep measurements reveal the values of capacitance and series impedance between the four conductors as a function of frequency up to 200 kHz. The square wave measurements reveal the complete capacitance matrice at a frequency of approximately 12.5 MHz as well as the series inductance between the four conductors. The influence of non-ideal ground could not be measured due to the high impedance of the grounding device.

Comparison between the analytical expressions and the measurements reveal good agreement. The expression for capacitance to ground based on the plate capacitor approximation show the best agreement. The coaxial cable approximation shows poor agreement with the square wave measurement. Analysis of the internal impedance show that the developed method based on the depth of penetration yields a better agreement. The measurements also verify the correctness of the expressions for influence of proximity effect. Both skin effect and proximity effect has a significant influence upon the series impedance of the four-wire cable.

With the help of Dr. Ani Gole, Garth Irwin and the HVDC research centre the four-wire cable model is implemented in the commercial simulation program EMTDC. Comparison between simulation and measurements reveal, that in general EMTDC is inaccurate around the natural frequency of the four-wire cable, but above and below the natural frequency there is good agreement between simulation and measurements. The problem with the natural frequency is not IV related specificly with the four-wire cable model, but is a general problem related with the distributed nature of transmission lines. For cables with a length of less than 100 m, the simulation of the four-wire cable is inaccurate. The Bergeron model yield a good solution of open ended cables, but it is less accurate for shorted cables at frequencies below 1 kHz, above 1 kHz there is good agreement with the measurements. The Phase model is inaccurate for both open ended and shorted cables with a length of less than 100 m. For cables with a length of 100 m or more the Phase model provide a good solution.

The Danish 10 kV test site at Kyndby is used for analysis and further verification of the four-wire cable model. Measurements at this site reveal that the cables in the ground have different values of capacitance than the previous measurements. It is measured that the capacitance between the cables and the capacitance to ground are frequency dependent. It is suggested that the mantle and insulation have absorbed an unknown amount of water. This is possible as water is used in the manufacturing of the cable insulation and mantle. Insertion of new material properties in the EMTDC four-wire cable model, results in good agreement with the measurement once the Bergeron model is used. The new material properties result in poor agreement between measurement and simulation, once the Phase model is used. No explanation is found on why the new material properties cause error in the Phase model.

At the kyndby 10 kV test site a non-linear load is inserted on the secondary side of normal distribution transformer and the phase voltage and current is measured. The measurement are performed with and without the four-wire cable inserted between the transformer and load. The 10 kV test-site is modelled in EMTDC with standard components. Similarly, the non-linear load is modelled as a six-pulse diode bridge loaded with a resistor on the DC-side. The simulations show good agreement with the measurements, including the small oscillation in voltage, which occur after each commutation.

It is concluded that the developed four-wire cable model is correct including the influence of nonideal ground, skin effect and proximity effect. It is also concluded that the implemented EMTDC four-wire cable is correct. Several interesting questions have arised which remain unanswered, such as, why do the generally accepted equations for skin effect show poor agreement with the measurements and it has also been shown, that the general transmission line equations lead to an erronous result in the frequency area around the natural frequency of the cable or overhead line.

### Details

Due to increased interest in the power quality of the distribution network, it is necessary to have an accurate model of the distribution network. One of the commonly used components in the distribution, is the low-voltage four-wire PEX-M-AL distribution cable. There exists no model of this component, and it is chosen to focus on the development of such a model.

Based on the electromagnetic field equations, the shunt admittance and series impedance parameters of the four-wire cable are derived. The influence of the conductance is considered negligible and is not included in the shunt admittance. Two expressions are derived for the capacitance to ground, one of which is based on the cable forming a plate capacitor to ground, the other is based on the cable and ground forming a coaxial cable. For the modelling of the series impedance, specific attention is given to include the influence of non-ideal ground, skin effect and proximity effect. Non ideal ground is modelled on the basis of Carson expressions for overhead lines. A new method is developed for the calculation of skin effect based on the depth of penetration. The result of this method is compared with the general expressions for circular conductors involving Bessel series. The two methods show equal values of resistance, but there is considerable difference in the values of internal inductance. A method for calculation of proximity effect is derived for a two-conductor configuration. This method is expanded to the use on the fourwire cable.

A number of measurements are performed on two four-wire cables, including frequency sweeps of open ended and shorted cables, as well as square wave measurements. The frequency sweep measurements reveal the values of capacitance and series impedance between the four conductors as a function of frequency up to 200 kHz. The square wave measurements reveal the complete capacitance matrice at a frequency of approximately 12.5 MHz as well as the series inductance between the four conductors. The influence of non-ideal ground could not be measured due to the high impedance of the grounding device.

Comparison between the analytical expressions and the measurements reveal good agreement. The expression for capacitance to ground based on the plate capacitor approximation show the best agreement. The coaxial cable approximation shows poor agreement with the square wave measurement. Analysis of the internal impedance show that the developed method based on the depth of penetration yields a better agreement. The measurements also verify the correctness of the expressions for influence of proximity effect. Both skin effect and proximity effect has a significant influence upon the series impedance of the four-wire cable.

With the help of Dr. Ani Gole, Garth Irwin and the HVDC research centre the four-wire cable model is implemented in the commercial simulation program EMTDC. Comparison between simulation and measurements reveal, that in general EMTDC is inaccurate around the natural frequency of the four-wire cable, but above and below the natural frequency there is good agreement between simulation and measurements. The problem with the natural frequency is not IV related specificly with the four-wire cable model, but is a general problem related with the distributed nature of transmission lines. For cables with a length of less than 100 m, the simulation of the four-wire cable is inaccurate. The Bergeron model yield a good solution of open ended cables, but it is less accurate for shorted cables at frequencies below 1 kHz, above 1 kHz there is good agreement with the measurements. The Phase model is inaccurate for both open ended and shorted cables with a length of less than 100 m. For cables with a length of 100 m or more the Phase model provide a good solution.

The Danish 10 kV test site at Kyndby is used for analysis and further verification of the four-wire cable model. Measurements at this site reveal that the cables in the ground have different values of capacitance than the previous measurements. It is measured that the capacitance between the cables and the capacitance to ground are frequency dependent. It is suggested that the mantle and insulation have absorbed an unknown amount of water. This is possible as water is used in the manufacturing of the cable insulation and mantle. Insertion of new material properties in the EMTDC four-wire cable model, results in good agreement with the measurement once the Bergeron model is used. The new material properties result in poor agreement between measurement and simulation, once the Phase model is used. No explanation is found on why the new material properties cause error in the Phase model.

At the kyndby 10 kV test site a non-linear load is inserted on the secondary side of normal distribution transformer and the phase voltage and current is measured. The measurement are performed with and without the four-wire cable inserted between the transformer and load. The 10 kV test-site is modelled in EMTDC with standard components. Similarly, the non-linear load is modelled as a six-pulse diode bridge loaded with a resistor on the DC-side. The simulations show good agreement with the measurements, including the small oscillation in voltage, which occur after each commutation.

It is concluded that the developed four-wire cable model is correct including the influence of nonideal ground, skin effect and proximity effect. It is also concluded that the implemented EMTDC four-wire cable is correct. Several interesting questions have arised which remain unanswered, such as, why do the generally accepted equations for skin effect show poor agreement with the measurements and it has also been shown, that the general transmission line equations lead to an erronous result in the frequency area around the natural frequency of the cable or overhead line.

Original language | English |
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Place of Publication | Aalborg |
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Publisher | Institut for Energiteknik, Aalborg Universitet |

ISBN (Print) | 87-89179-43-9 |

Publication status | Published - 2003 |

Publication category | Research |