DIAGNOSTIC METHODS FOR ELECTRICAL CABLES UTILIZING AXIAL TOMOGRAPHY

Abstract

Cable diagnostic test methods, systems and apparatus are disclosed that utilize “standing wave” principles to facilitate identification and location of insulation defect(s) along a power cable. The methods/systems measure dissipation factors and dielectric constants associated with the power cable insulation and the impedance of the power cable conductor at any number of points or sections along the axial length of the cable. In an exemplary embodiment, the disclosed method involves (i) connecting an alternating voltage source to a cable at a “sending end” thereof; (ii) applying a voltage to the cable at a first frequency to set up a traveling wave along the cable that is reflected at the “receiving end” thereof; (iii) permitting a standing wave pattern to be established along the cable by the traveling wave and the reflection thereof; (iv) measuring the total complex power loss (Sin) at the sending end of the cable; (v) calculating the standing wave voltage at any point/section of the cable based on the load impedance (ZL) connected at the receiving end of the cable, and the characteristic impedance (ZO) of the cable, or the measured/calculated cable parameters for the first frequency of the voltage source, (vi) repeating the foregoing steps while one of: (1) varying at least one of: the load impedance (ZL) connected at the receiving end of the cable, the first frequency of the voltage source; the output impedance of the voltage source, a combination of the load impedance (ZL), the output impedance of the voltage source and the first frequency of the voltage source, and combinations thereof; (2) interchanging sending and receiving cable ends; and (3) a combination thereof, and (vii) determining a dissipation factor (tan δ) and a dielectric constant (∈′), for the insulation, and an impedance, for the conductor at predetermined points/sections along the axis of the cable.

Description

BRIEF DESCRIPTION OF THE FIGURES

To assist those of ordinary skill in the art in making and using the disclosed systems, methods and apparatus, reference is made to the accompanying figures, wherein:

FIG. 1 is a cross-sectional view of a conventional power cable;

FIGS. 2( a), 2(b) and 2(c) are circuit equivalents of a shielded power cable insulation;

FIG. 3( a) is a diagram showing an exemplary cable divided into “n” equal sections along its axial length, wherein the cable is energized by a voltage “V_in” and terminated with a load impedance Z_L.

FIG. 3( b) is the electric circuit representation of each cable section in FIG. 3(a).

FIG. 4 is a schematic diagram of an exemplary system for implementing the disclosed methods/techniques;

FIGS. 5-7 are voltage plots for an exemplary homogenous cable according to the present disclosure;

FIG. 8 is a schematic diagram of an exemplary mixed cable configuration according to the present disclosure;

FIGS. 9-10 are voltage plots for an exemplary mixed cable configuration according to the present disclosure;

FIG. 11 is a schematic diagram of an exemplary branched cable according to the present disclosure; and

FIGS. 12-14 are voltage amplitude plots for the exemplary branched cable at three distinct frequencies.

FIGS. 15-16 are exemplary “tomograms” of a cable having two defective areas with elevated tan δ and dielectric constant.

DESCRIPTION OF EXEMPLARY EMBODIMENT(S)

The disclosed cable diagnostic test methods, systems and apparatus utilize “standing wave” principles to identify and locate defect(s) along a power cable. As described herein, the disclosed methods/systems are effective in measuring dissipation factor (tan δ) and dielectric constant (∈′) associated with the insulation of a power cable, as well as the resistance (R_c) and inductance (L_c) associated with the conductor system, at discrete points along the cable's axial length. The disclosed methods/systems offer significant advantages for cable testing and related defect identification/location.

With reference to FIG. 2(a), the schematic circuit diagram represents a shielded power cable insulation subjected to an alternating voltage source, V=V_m sin(ωt), where V_m is the amplitude of the voltage and ω=2πf is the angular frequency, f being the frequency in Hz. The cable insulation is shown as a pure capacitance, C, across which is connected a resistance, R, representing the dielectric losses. C is expressed in Farads and R in Ohms. FIG. 2(b) is a “phasor” diagram showing the current, I_C, drawn by the ideal capacitor, the current, I_R, drawn by the resistor, and the total current I, which is the sum of the two previously mentioned currents. The angle, δ, between I and I_C is called the dissipation factor angle and represents the effect of the dielectric loss occurring in the cable insulation. The dissipation factor, defined as tan δ, can be shown to be equal to 1/ωRC.

An alternative equivalent notation, often used by physicists (as opposed to engineers), is as follows. The cable is assumed to be represented by the following complex capacitance rather than by a combination of a resistance in parallel with a capacitance: C*=C₀(∈′−j∈″), where ∈′ is called the dielectric constant, ∈″ the dissipation index, and C₀ the geometric capacitance (capacitance of the same cable construction having air as its insulating medium) of the cable insulation. The quantity “j” is the square root of the negative quantity “−1”. The current, I, drawn by this complex capacitance is, therefore: I=V×jωC=VωC₀(∈″+j∈′). The phasor diagram is redrawn in FIG. 2(c) on the basis of this new form of the current, I. The trigonometric tangent of the angle δ, tan δ, is now expressed as the ratio ∈″/∈′. The advantage of this notation is that the axial tomography test would allow ∈′ and ∈″, as well as tan δ, to be determined. The two previous representations are equivalent to each other in accordance with the following relationships:

C=∈′C₀ and 1/R=ωC₀∈″

The cable conductor is represented as a series combination of a frequency dependent resistance, R_c, and an inductive reactance, ωL_c, as shown in FIG. 3(b).

Turning to FIG. 3(a), the voltage source at the sending end of the cable supplies a complex power, S_in, which is the sum of the real Power, P_in, and the reactive power, Q_in, related to each other by the following relationship:

S _in =V _in ×I _in *=P _in +jQ _in  [1]

The quantity I_in* denotes the complex conjugate of the input current, I_in. A similar relationship exists at the receiving, or load, end of the cable:

S _L =V _L ×I _L *=P _L +jQ _L  [2]

The complex power dissipated in the cable consists of two components, the first is the complex power, S_c, dissipated in the cable conductor, and the second is the complex power, S_d, dissipated in the dielectric material, or the insulation, of the cable. As a cable ages, except in few exceptional cases, the complex power dissipated in the conductor remains constant, while that dissipated in the cable insulation tends to increase. All existing global diagnostic tests are intended to determine the changes in the power dissipation of the cable insulation. In the present disclosure, the conductor impedance, R_c+jωL_c, is allowed to vary, as well, along the cable axis. Applying the principle of conservation of energy to this situation, the complex power, S_d, dissipated in the insulation, is found by the relationship:

S _d =S _in −S _L −S _c  [3]

Splitting equation [3] into its real and imaginary components, the following relationships are obtained:

$\begin{array}{cc}{P}_d=\sum _iV_i^2\omega C_i\tan {\delta }_i=\sum _iV_i^2\omega C_o{\varepsilon }_i^{″}=P_{in}-P_L-\sum _iI_i^2R_{ci}& \left[4\right]\\ Q_d=\sum _iV_i^2\omega C_i=\sum _iV_i^2\omega C_o{\varepsilon }_i^{\prime }=Q_{in}-Q_L-\sum _iI_i^2\omega L_{ci}& \left[5\right]\end{array}$

Equations similar to [4] and [5] can be written for various values of load impedance, Z_L, source impedance, Z_s and frequency, ω. If the cable is divided into n sections (not necessarily of equal dimension), a minimum of n equations are needed to solve these equations. The solution will determine the values of tan δ and C, or ∈′ and ∈″ for each section along the cable axial length, thus providing an axial tomogram of the cable insulation. The foregoing process is typically implemented in two distinct steps: first, the conductor is assumed to be homogeneous and the variations of insulation parameters are computed. On the basis of these values, the voltage and current profiles along the cable are re-calculated. In a second step, equations [4] and [5] are solved, assuming the calculated values of the insulation parameters are known, while the conductor parameters, R_c and L_c, are the unknown variables. Through an iterative process, all conductor and insulation parameters are calculated, such that four (4) different axial tomograms can be generated.

With this mathematical context, the systems and methods of the present disclosure establish the voltage and current profiles along the cable using the principle of “standing waves.” For purposes of the disclosed methods/systems, standing wave principles are utilized by establishing an alternating voltage at relatively high frequency, generally on the order of 10-1000 kilohertz (kHz) on the power cable to be tested. The voltage is connected across the cable at the “sending” end (see schematic depiction in FIG. 3(a)). This voltage sets up a traveling wave pattern, which is reflected at the opposite “receiving” end of the cable. The combination of the forward traveling wave and its reflection sets up a “standing wave” pattern within the cable. Of note, each cable section is subjected to an alternating voltage of frequency ω, but with different amplitude. More particularly, a voltage wave travels from the voltage source toward the load at a velocity which is influenced by such parameters as the cable materials and cable construction. The forward moving voltage may be denoted as V⁺. At the load, a portion of the wave is reflected, establishing a voltage V⁻. The reflected portion of the voltage wave is determined by the value of the load impedance and the cable characteristic impedance. The reflected portion of the traveling wave can be determined mathematically if the load impedance, Z_L, connected at the receiving end of the cable and the characteristic impedance, Z₀, of the cable are known. Once the reflected portion of the traveling wave has been determined/calculated, it is possible to calculate the “standing wave” voltage and current at any point or section along the axial length of the cable. Indeed, at any point/section along the cable, the instantaneous voltage (or current) is the sum V⁺+V⁻ (or I⁺+I⁻) based on the waves traveling in the forward and reflected directions, respectively. Inasmuch as both waves are sinusoidal, the summation of the two waves is also sinusoidal.

The voltage and current patterns can be determined either through the application of conventional standing wave equations, or from the solution of the discrete circuit representation of FIG. 3(a), where the sections need not have the same length. Initially, the values of voltage and current are based on the global values of the cable parameters R_c, L_c, C and R, obtained for known load conditions, including short and open-circuit conditions, by direct measurements of voltage and current quantities at the sending and receiving cable ends. These initial values of the cable parameters can be modified, iteratively, as the solutions of equations such as [4] and [5] are obtained.

Absent changing conditions, the combination forms a wave that appears to be stationary (not moving or “standing” still). In reality, each point along the cable is subjected to a sinusoidal voltage of the same frequency as the source frequency; however, at each point/section, the amplitude of the resulting voltage is different. Each standing voltage wave is also accompanied by a standing current wave. With reference to FIGS. 5-7, experimental results are plotted for an exemplary homogeneous cable having the following characteristics/parameters and test conditions:

Length (L)=200 m

Wave velocity (u)=160 m/μs

Frequency (f)=200 kHz (FIG. 5); 400 kHz (FIGS. 6-7)

Wavelength (λ)=800 m; 400 m

Cable characteristic impedance (Z₀)=20Ω

As shown in the foregoing plots, the voltage amplitude at various axial positions along the cable are calculated based on the test conditions according to the present disclosure.

With reference to FIG. 8, an exemplary mixed cable configuration is schematically depicted. The mixed cable includes a first portion/component that is fabricated from crosslinked polyethylene (XLPE) and a second portion/component that is fabricated with an oil-impregnated paper-insulated lead-covered (PILC) cable. With reference to FIGS. 9-10, experimental results are plotted for the noted mixed cable configuration having the following characteristics/parameters and test conditions:

PILC

Cable characteristic impedance (Z₀)=15Ω

Length (L)=600 m

Wave velocity (u)=140 m/μs

XLPE

Cable characteristic impedance (Z₀)=18Ω

Length (L)=200 m

Wave velocity (u)=160 m/μs

Source Impedance

Rs=0 or 15Ω

As shown in FIGS. 9-10, voltage amplitudes are measured/derived for various points/sections along the length of the exemplary mixed cable configuration described herein at frequencies of 200 kHz and 90 kHz according to the present disclosure.

Turning to FIG. 11, a schematic diagram of an exemplary branched cable is provided. Incorporated into the schematic depiction of FIG. 11 are characteristics/parameters associated with an exemplary embodiment thereof. Voltage amplitudes for the exemplary branched cable are provided in the plots of FIGS. 12-14. As shown therein, voltage amplitudes are measured/derived for axial locations along the respective cable branches at frequencies of 80 kHz, 120 kHz, and 160 kHz.

By varying the load impedance, the source impedance or the frequency of the voltage source, or all three, a number of times (e.g., “n” times or more), the needed equations to solve for tan δ_i can be advantageously established. Of note, the dissipation factor at these high frequencies should not change significantly with a modest variation in frequency, and therefore may be assumed to be constant. However, this assumption is not a necessary condition to apply the disclosed method and, in further exemplary embodiments of the present disclosure, potential dissipation factor variations due to frequency variations may be included in the mathematical equations associated with the disclosed systems/methods. If necessary or desired, the same operation can be conducted by exchanging the sending end and the receiving end of the cable, generating half of the equations needed in each of these operations.

FIG. 15 and FIG. 16 are exemplary axial tomograms obtained on a 240 m long cable model in which two areas with elevated tan δ=∈″/∈′ and one area with elevated dielectric constant, ∈′, were simulated.

The foregoing method allows the determination of a dissipation factor profile along the cable length using a broad range of excitation frequencies, e.g., frequencies of 10-1000 kHz. In order to perform the disclosed axial tomography technique using significantly lower frequencies, e.g., frequencies of 50/60 Hz, 1 Hz or 0.1 Hz, or other, the amplitude V_m of the excitation voltage is generally modulated with the lower desired frequency. The result is an “amplitude-modulated” excitation source described mathematically as:

V=V _m sin(ω₁t)sin(ωt)  [6]

where ω₁ is the lower modulating frequency (e.g., 60 Hz or 0.1 Hz). With appropriate circuitry and/or software, the power associated with each of the two frequencies, ω and ω₁, can be separated. Thus, the dissipation factor profile at each of these frequencies can be calculated according to the mathematical techniques described above. The use of this amplitude-modulated technique allows “dielectric spectroscopy” to be performed by means of axial tomography (spectro-tomography).

Thus, in an exemplary embodiment, the disclosed cable detection method/technique involves the following steps:

• • (a) connecting an alternating voltage source to a cable at a “sending end” thereof;
  • (b) applying a voltage to the cable at a first frequency to set up a traveling wave along the cable that is reflected at the “receiving end” thereof;
  • (c) permitting a standing wave pattern to be established along the cable by the traveling wave and the reflection thereof;
  • (d) measuring the total complex power loss (S_in) at the sending end of the cable;
  • (e) measuring or calculating the complex power, (S_L), dissipated in the load impedance (Z_L);
  • (f) repeating the foregoing steps while one of: (1) varying at least one of (i) the load impedance (Z_L) connected at the receiving end of the cable, (ii) the first frequency of the voltage source, (iii) the output impedance of the voltage source, (iv) a combination of the load impedance (Z_L), the output impedance of the voltage source, and the first frequency of the voltage source, and (v) combinations thereof, (2) interchanging sending and receiving cable ends and (3) a combination thereof;
  • (g) calculating the standing wave voltage at any point/section of the cable based on the load impedance (Z_L) connected at the receiving end of the cable, and the characteristic impedance (Z_O) of the cable, or by solving the discrete circuit model of the cable as shown in FIG. 13(a), the global cable parameters, Rc, Lc, C and R, having been determined through measurements of voltage and current at the sending and receiving cable ends under specific load conditions;
  • (h) calculating the complex power loss, S_c, in the conductor system;
  • (i) determining the dissipation factor, tan δ, and the dielectric constant, ∈′, at predetermined points/sections along the axis of the cable;
  • (j) re-calculating the voltage and current profiles according to the new values of cable parameters;
  • (k) determining the values of conductor resistance (Rc) and inductance (Lc)
  • (l) if warranted, repeating (g) through (k) with corrected cable parameters.

The characteristic impedance, Z₀, of the cable can be obtained through theoretical calculations or by direct measurement, according to one of the following exemplary methods:

• • (1) Measuring the voltage, V_in, across the cable and the current, I_in, into the cable at the voltage source end while the load impedance is zero (short circuit) or infinite (open circuit). The short circuit impedance is defined as Z_sc=V_in/I_in when the load is zero. The open circuit impedance is defined as Z_oc=V_in/I_in when the load is infinite. The cable characteristic impedance is then equal to the square root of the product of Z_sc and Z_oc.
  • (2) Changing the load impedance Z_L until the voltage V_in at the source end of the cable and the voltage V_L across the load are equal. This occurs when Z_L is equal to Z₀. The ratio of Z_sc and Z_oc can also be used to determine the propagation constant of the cable (phase constant and attenuation) as well as velocity of propagation, provided the cable length is known.

In a further exemplary embodiment of the present disclosure, the results obtained by partial discharge and axial tomography may be combined, thereby providing a powerful diagnostic tool that is vastly superior to all presently existing tools.

Mixed Cable Systems

The methods, systems and apparatus of the present disclosure also have advantageous applicability to mixed cable systems, i.e., cable systems where two or more cable types, such as extruded polymer and oil-impregnated laminated insulation, are interconnected to each other. These cables have different characteristic impedance, Z₀, and velocity. Equations can be written based on each cable's “input impedance”, Z_in, its length, the velocity of the electromagnetic wave associated with it, and the frequency of the voltage source. These equations will allow the determination of the voltage at each point along the mixed cable system. The velocity/cable length can be measured from a time-domain reflectometry (TDR) test, as is known from prior art partial discharge tests performed by the assignee of the present application, or can be estimated by calculation.

Branched Cable Systems

Cable systems with multiple branches can be successfully tested according to the present disclosure, provided a variable load impedance is connected to the end of each branch. The voltage calculations can, as well, be accomplished by means of the input impedance formulas described herein.

Hardware Configuration

An exemplary system configuration for practicing the methods/techniques of the present disclosure is illustrated in FIG. 4. In the schematic illustration of FIG. 4, the noted components are as follows:

• • Element 1: Remotely controllable, amplitude modulated variable frequency voltage source;
  • Element 2: Current measuring device;
  • Element 3: Voltage measuring device;
  • Element 4: Digitizer;
  • Element 5: Microcomputer with means to communicate with digitizer/receiver/transmitter (element 6) by wireless or through power or optical cable;
  • Element 6: Digitizer/receiver/transmitter communicating with microcomputer (element 5) and/or control console (element 9) by wireless or through power/optical cable;
  • Element 7: Wireless communication system;
  • Element 8: Remotely controllable variable impedance;
  • Element 9: Operator remote control console;
  • Element 10: Fiber-optic communication system.Thus, as schematically depicted in FIG. 4, at the near-end (or sending-end) of the cable, the system may include:
  • A controllable variable frequency or amplitude-modulated variable frequency (for the spectro-tomography test) source with a variable series impedance, Zs;
  • One or more devices for measuring the instantaneous voltage and current;
  • A digitizer to digitize the measurements;
  • A microcomputer to compute power and the voltage profile along the cable; and
  • A control console for the operator with means to communicate remotely with the load-end by wireless or through the power cable under test;
  • Analog or digital filters, as needed, to separate the modulating frequencies from the high “carrier” frequencies.

At the remote-end, the following hardware is provided according to the exemplary embodiment depicted in FIG. 4:

• • A remotely controllable load impedance,
  • A voltage/current measuring device; and
  • A digitizer and means to communicate with the near-end microcomputer by a wireless system or through the power cable itself

A computer/processor is generally provided to solve the set of “n” (or more) equations with “n” unknown quantities to yield the dissipation factor values at each of the “n” sections of the cable. The computer/processor is generally adapted to run system software that is adapted to calculate the voltage profile along the cable, the net power dissipated by the cable insulation for each setting of the load impedance, and the solution of the system of “n” (or more) equations with “n” unknown tan δ and ∈′ values. In addition, the software will include means to display the tomograms (tan δ versus cable position) and to perform filtering operations on the data. The programming of such software is well within the skill of persons of ordinary skill in the art, based on the disclosure set forth herein as to the “n” (or more) equations to be simultaneously solved according to the present disclosure.

Although the methods, system and apparatus for diagnostic testing of cables has been described herein with reference to exemplary embodiments thereof, the present disclosure is not limited to such exemplary embodiments. Rather, the disclosed methods, systems and apparatus are subject to variations, modifications and/or enhancements without departing from the spirit or scope of the present disclosure. Such variations, modifications and enhancements are expressly encompassed within the scope of the present disclosure.

Claims

1. A method for cable testing, comprising: calculating a dissipation factor (tan δ) and a dielectric constant (∈′) at a predetermined point or section along the axis of the cable based, at least in part, on a standing wave established on such cable; andtaking at least one action with respect to the cable based on the calculated dissipation factor (tan δ) and dielectric constant (∈′) for such predetermined point or section of the cable.

2. The method according to claim 1, wherein the at least one action includes identifying or locating a cable defect based on the calculated dissipation factor.

3. The method according to claim 1, wherein the cable is a shielded power cable.

4. The method according to claim 1, wherein dissipation factors and dielectric constants are calculated at a plurality of points or sections along the cable.

5. The method according to claim 4, wherein cable characteristic impedances are calculated at the plurality of points or sections along the cable.

6. The method according to claim 4, wherein the plurality of calculated dissipation factors and dielectric constants are effective to functionally establish at least one of an axial tomographic rendering and a spectro-tomographic rendering of cable condition.

7. A method according to claim 1, wherein said cable is selected from the group consisting of a linear cable, a mixed cable, a branched cable, and combinations thereof.

8. A method according to claim 1, wherein at least one of the dissipation factor (tan δ) and the dielectric constant (∈′) for a given point or section of the cable is used to identify a cable defect.

9. A system for cable testing, comprising: a. a controllable variable frequency source and its series impedance, Zs;b. at least one device for measuring at least one of instantaneous voltage and instantaneous current at a first end of a cable;c. a filter for separating modulated frequencies from carrier frequencies at the first end of the cable;d. a processing unit that is adapted to calculate a dissipation factor (tan δ) and dielectric constant (∈′) at a predetermined point or section along the axis of the cable based, at least in part, on a standing wave established on such cable at the first end of the cable;e. a controllable load impedance at a second end of the cable;f. a measuring device adapted to measure at least one of voltage and current at the second end of the cable; andg. communication means for transmission of data between the controllable load impedance, the measuring device at the second end of the cable, and the processing unit.

10. The system according to claim 9, wherein the controllable variable frequency source is adapted to modulate with a lower frequency at a first cable end.

11. The system according to claim 9, further comprising digitizing units that are adapted to digitize at least one of voltage and current measurements.

12. The system according to claim 9, further comprising a control console in communication with at least one system component.

13. The system according to claim 12, wherein the control console communicates with one or more system components through wireless signals or through the test cable itself.

14. The system according to claim 9, wherein the processing unit is adapted to solve “n” or more equations with “n” unknown quantities to yield dissipation factor values at each of “n” sections of the cable.

15. A method for performing diagnostic testing of a cable, comprising: a. connecting an alternating voltage source to a cable at a “sending end” thereof;b. applying a voltage to the cable at a first frequency to set up a traveling wave along the cable that is reflected at a “receiving end” thereof;c. permitting a standing wave pattern to be established along the cable by the traveling wave and the reflection thereof;d. measuring the total complex power loss (Sin) at the sending end of the cable;e. measuring or calculating the complex power (SL) dissipated in the load impedance (ZL) and (Sc) in the cable conductor;f. repeating the foregoing steps while one of: (1) varying at least one of (i) the load impedance (ZL) connected at the receiving end of the cable, (ii) the first frequency of the voltage source, (iii) the output impedance of the voltage source, (iv) a combination of the load impedance (ZL), the output impedance of the voltage source and the first frequency of the voltage source, and (v) combinations thereof, (2) interchanging the sending and receiving cable ends and (3) a combination thereof;g. calculating the standing wave voltage at least one point or section of the cable based on the load impedance (ZL) connected at the receiving end of the cable, and the characteristic impedance (ZO) of the cable, or the solution of the discrete cable representation whose global parameters have been determined by measurement and calculation; andh. determining a dissipation factor (tan δ) and a dielectric constant (∈′) at a predetermined point/section along the axis of the cable.

16. The method according to claim 15, wherein at least one of the dissipation factor (tan δ) and the dielectric constant (∈′) for a given point or section of the cable is used to identify a cable defect.

17. The method according to claim 15, wherein the first frequency is between about 10 kHz and 100 kHz.

18. The method according to claim 15, wherein an amplitude of the first frequency is modulated at a relatively low frequency.