Traveling wave analysis and fault locating for electric power systems

Systems may include a first data acquisition subsystem, a second data acquisition subsystem, and a traveling wave analysis subsystem. The traveling wave analysis subsystem may be configured to analyze measured traveling wave data from the first data acquisition subsystem and the second data acquisition subsystem. Additionally, methods of analyzing traveling wave data resulting from a fault on an electric power delivery system may involve analyzing measured electrical properties of at least one traveling wave.

TECHNICAL FIELD

This disclosure relates to systems, devices, and methods for utilizing traveling wave analysis for determining fault locations in electric power systems. More particularly, but not exclusively, this disclosure relates to analyzing one or more traveling wave forms with Bewley lattice diagrams for determining fault locations.

DETAILED DESCRIPTION

Traveling waves (TWs) are surges of electricity resulting from sudden changes in voltage that propagate at nearly the speed of light on overhead power transmission lines and at approximately half the speed of light on underground cables. When launched by a line fault, the step change in voltage launches current and voltage TWs that initiate at the fault and travel towards each end of the line. As they travel along the line, coupling may occur between the three phases of the power system, with a relationship that is dependent on the fault type. These TWs may arrive at line terminals within a short span of time (e.g., a few milliseconds depending on the line length and fault location). When the initial TWs arrive at a line terminal, some of the energy is reflected back towards the fault as a reflected TW. When the reflected TW arrives at the fault, some of the energy continues in the same direction and some of the energy is reflected again and returns to the line terminal. Relative arrival times, polarities, and magnitudes of TWs may allow for identifying the fault type and locating the fault with accuracy on the order of a single tower span.

Embodiments consistent of the present disclosure provide significant flexibility in analyzing TWs by allowing an operator to adjust a plurality of parameters to match electric parameters associated with a TW. For example, multiple TW arrival times resulting from reflections may be fixed to provide additional control over the solution space. In some embodiments, the TW may be represented by a system of linear equations (e.g., one each per TW arrival time) that include variables that represent a variety of parameters (e.g., fault location, line length, impedances, resistances, fault resistances, propagation times, propagation velocities, capacitances, inductances, reactances, lengths, and/or the like) may be solved.

In some cases, well-known features, structures, or operations are not shown or described in detail. Furthermore, the described features, structures, or operations may be combined in any suitable manner in one or more embodiments. It will also be readily understood that the components of the embodiments, as generally described and illustrated in the figures herein, could be arranged and designed in a wide variety of different configurations. For example, throughout this specification, any reference to “one embodiment,” “an embodiment,” or “the embodiment” means that a particular feature, structure, or characteristic described in connection with that embodiment is included in at least one embodiment. Thus, the quoted phrases, or variations thereof, as recited throughout this specification are not necessarily all referring to the same embodiment.

Several aspects of the embodiments disclosed herein may be implemented as software modules or components. As used herein, a software module or component may include any type of computer instruction or computer-executable code located within a memory device that is operable in conjunction with appropriate hardware to implement the programmed instructions. A software module or component may, for instance, comprise one or more physical or logical blocks of computer instructions, which may be organized as a routine, program, object, component, data structure, etc., that performs one or more tasks or implements particular abstract data types.

An event, such as a fault, that initiates TWs may occur at a specific time, which is referred to as “T0” herein. When an event occurs mid-line (e.g., distant from a terminal where measurements may be taken), T0is not directly measurable.

When a power line is energized by closing a circuit breaker at one end, TWs are initiated that are observed nearly immediately at the terminal where the circuit breaker was closed and observed at the other line terminal after the TWs propagate along the entire length of the line. The time from the initiation of a TW at one terminal and the TW arrival at the other terminal is the TW line propagation time, T, which may be calculated by using Eq. 1,

T=(L⁢Lv*c)⁢(1+eps)Eq.1
wherein LL represents the line length, v represents the propagation velocity as a fraction of the speed of light (c), and eps is a dispersion factor. The LL may be difficult to precisely determine due to line sag and the topography of the area, but the LL may be estimated to a sufficient degree of accuracy. Additionally, the LL may change due to environmental conditions. For example, the line sag may increase in relatively high temperatures. Likewise, the propagation velocity depends on the inductance and capacitance of the line; therefore, it may be affected by environmental conditions, such as humidity, temperature, air pressure, etc.

Faults that occur on the line are located a distance m (expressed as a per-unitized fraction of line length) from the local terminal and a distance 1−m from the remote terminal where measurements may be taken. The per-unitized fault location, m, may be converted to a linear distance by multiplying it by the line length (LL) or converted to a time by multiplying by the TW line propagation time (T). The propagation time from the fault to the local and remote terminals may be determined using Eqs. 2 and 3, respectively.
Propagation Time to Local Terminal=m*TEq. 2
Propagation Time to Remote Terminal=(1−m)*TEq. 3

For a fault initiated at time T0, the initial TW will arrive at the local terminal at a time determined using Eq. 4 and will arrive at the remote terminal at a time determined using Eq. 5.
Local Terminal Arrival Time=T0+mTEq. 4
Remote Terminal Arrival Time=T0+(1−m)TEq. 5

Additionally, it may take further time for the TW to travel down current transformer (CT) or potential transformer (PT) cables to the device taking measurements. CTs and PTs are collectively known as instrument transformers. The delay associated with the instrument transformer cables at the local terminal, dL, and remote terminal, dR, may be included in the arrival times at the recording device using Eqs. 6 and 7.
Local Terminal Arrival Time=T0+mT+dLEq. 6
Remote Terminal Arrival Time=T0+(1−m)T+dREq. 7

The difference in the arrival times for the initial TW at the local terminal versus the remote terminal may be used to estimate the fault location. Accordingly, both TW arrival times may be shifted by dLsuch that the TW arrival times are determined using Eq. 4 for the local terminal and Eq. 8 for the remote terminal.
Remote Terminal Arrival Time=T0+(1−m)T+(dR−dL)  Eq. 8

The impact of the instrument transformer cable delay on the fault location estimation is therefore dependent on the difference between dLand dR, d=dR−dL. A TW system may further account for a time skew parameter. In some embodiments, a time skew may represent a difference between clock synchronization at the remote terminal and the local terminal. Further, the skew may account for differences in TW shape or different group delays for instrument transformers that may impact time of arrival determinations. Therefore, the instrument transformer cable delay, d, may be modified to include the time skew, dSkew, and may be determined using Eq. 9.
d=(dR−dL)+dSkewEq. 9

FIG.1illustrates a TW Bewley lattice diagram10for correlating TW arrival times with actual TW paths through time on a transmission line and consistent with embodiments of the present disclosure. The subscript L and R correspond to local and remote respectively but could also be interpreted as left and right as the local terminal is shown on the left side12and remote terminal on the right side14inFIG.1.

When the TWs are launched, the propagation time to the local terminal may be determined using Eq. 2, and the propagation time to the remote terminal may be determined using Eq. 3. The first right TW arrival time is labeled TR. Subsequent TW arrival times are labeled based on the terminals from which it reflected last. Therefore, the TW arrival time that is a reflection from TLto a fault location16and back to the left side12is TLL. A TW arrival that reflects from TLthen passes through the fault location16to arrive at the right side14is labeled TLR.

Two full traversals are illustrated for reflections between the local terminal12and the fault location16. Each of the TW arrival times can be represented in terms of T0, mT, T, and d. Eq. 10 shows a matrix representation for the reflection at TL, and Eq. 11 represents the reflection at TR.

The form of the reflections may be generalized as shown in Eq. 13.

The parameters other than T0may be measured by a TW detection system and used to determine T0using Eqs. 14-16.

FIG.2shows a simplified schematic diagram of a system20for analyzing TW data and for determining a fault location in an electric power delivery system, according to an embodiment of the present disclosure. The system20may include a first data acquisition subsystem22, a second data acquisition subsystem24, and a traveling wave analysis subsystem26. The first data acquisition subsystem22may be located and configured to receive at least one of a measured voltage and a measured current of a power transmission line30at a first location32(e.g., a local terminal), and the second data acquisition subsystem24may be located and configured to receive at least one of a measured voltage and a measured current of the power transmission line30at a second location34(e.g., a remote terminal). The voltage and/or current measurements of the power transmission line30by the first data acquisition subsystem22and the second data acquisition subsystem24may include TW data, such as TWs originating from a fault location36.

The traveling wave analysis subsystem26is configured to receive and analyze measured TW data from the first data acquisition subsystem22and the second data acquisition subsystem24. The traveling wave analysis subsystem26may compensate for a time skew between actual TW arrival times and measured TW arrival times from the first data acquisition subsystem22and the second data acquisition subsystem24and determine a fault location36on the power transmission line30between the first location32and the second location34.

To facilitate the TW analysis, the traveling wave analysis subsystem26may include a graphic user interface40configured to display a Bewley lattice diagram including TW data received from each of the first data acquisition subsystem22and the second data acquisition subsystem24. The traveling wave analysis subsystem26may additionally include a processor42and a computer-readable storage medium44.

The processor42may be configured to process data received from the first data acquisition subsystem22and the second data acquisition subsystem24. The processor42may operate using any number of processing rates and architectures, and may be configured to perform various algorithms and calculations described herein. The processor42may be embodied as a general-purpose integrated circuit, an application specific integrated circuit, a field-programmable gate array, and/or any other suitable programmable logic device.

The computer-readable storage medium44may be the repository of a database containing properties for the power transmission line30and/or each section of the power transmission line30, such as impedances, resistances, propagation times, reactances, lengths, and/or the like. The computer-readable storage medium44may also be the repository of various software modules configured to perform any of the methods described herein. Computer-readable storage medium44may comprise volatile storage (e.g., random access memory) and non-volatile storage.

FIG.3shows a graphic user interface of the traveling wave analysis subsystem ofFIG.2, and shows a cursor located at each TW arrival time on a Bewley lattice diagram, consistent with embodiments of the present disclosure. A cursor50a-50lis shown located at each TW arrival time on the lattice. The time-aligned TW data is plotted next to each side of the lattice after passing through sufficient signal processing steps to make the TW arrival pulses visible. The signal processing may include any or all of transformation, low frequency removal, smoothing, differentiation, edge detection, matched filter squaring, and absolute value. For example, some transformations, such as conversion from ABC phases to Clarke components, may take advantage of cross coupling between phases. Further, once a TW has been identified, the first arrival (which tends to be of the largest magnitude) may be captured and applied as a matched filter to the entire signal at that terminal. In another example, once the signals are sufficiently processed, an interpolation algorithm may be applied to provide additional precision beyond the raw sample rate of a TW recorder. One method of interpolation is to apply a least-squared-error polynomial (e.g., parabolic for a second-order polynomial) fit to a range of points surrounding a TW arrival peak and assigning the timestamp of the TW arrival time to be the peak (or trough for signals in the negative direction) of the polynomial function.

The TW arrival times may be skewed (time delayed) by different amounts because different filters may be applied at each of the first data acquisition subsystem and the second data acquisition subsystem, and each filter may have a different delaying effect on the peak times. A traveling wave analysis subsystem can compensate via adjustments to the d parameter to account for this delay.

Referring again toFIG.3, each cursor50a-50lmay consist of three parts. First, a circular handle52may allow the operator to move a cursor50a-50lup (e.g., earlier in time) or down (e.g., later in time). Second, a triangular pin54may allow the operator to indicate that this is a fixed time value and movement of other cursors will not affect it. Note that a pinned cursor itself may still be moved. Pinning prevents the cursor from being indirectly moved due to the manipulation of other cursors. Third, a line, such as a dashed line56, that may move with a cursor50a-50land may be overlaid on the TW plot may aid in aligning a cursor50a-50lto a TW arrival time.

An operator may select any cursor50a-50lto move. Since any row of the [A] matrix is of rank 1, moving a cursor50a-50l, with no other cursors pinned, has the effect of shifting the entire lattice up/down. Once a cursor50a-50lis aligned with a TW arrival time, it can be pinned.

Pinning a cursor may modify the ability to pin and/or manipulate other cursors50a-50l. For any other cursor50a-50l, if the two rows of the [A] matrix corresponding to a pinned cursor50a-50land a cursor50a-50lof interest are examined, an additional cursor50a-50lmay be visible if the left two columns form a rank 2 matrix. If the resulting 2×2 matrix is of rank 1, certain cursors50a-50lmay not be displayed and may not be selectable.

In an additional embodiment, cursor50a-50lvisibility may be determined when one cursor50a-50lis pinned by checking the rank of the first two columns (for manipulating the fault location, mT) or by forming the 2×2 matrix from the first and third column (for manipulating the propagation time, T). In this embodiment, the cursors50a-50lthat manipulate T may be modified to distinguish them (e.g., shape, color, label) from the cursors50a-50lthat manipulate mT.

As shown inFIG.4, cursor50a(corresponding to TW arrival time TL) may be aligned and pinned. The pinning of cursor50ais indicated by the triangle cursor being solid. Pinning cursor50amay cause subsequent reflections (i.e., cursors50h,50ecorresponding to TW arrival times TLRand TRLL) to be hidden.

At this point, manipulating the cursor50a(TL) may still have the effect of shifting the lattice up or down as there is only a single cursor of interest. If one of the other visible cursors50b,50c,50d,50f,50g,50i-50lis manipulated, and thus including the row corresponding to that cursor in the equation, a rank 2 matrix may be formed, thus any manipulation results in modification to both T0and mT (the fault location).

As shown inFIG.5, cursor50aand cursor50c(corresponding to TW arrival times TLand TLLL, respectively) may be pinned. Again, the triangle cursors of50aand50care solid. Cursor50b(corresponding to TW arrival time TLL) may be hidden, as its row of the [A] matrix is linearly dependent on TLand TLLL.

At this point, manipulating cursor50aand cursor50cmay still have the effect of modifying T0and mT as the rank 2 matrix is maintained. If one of the other visible cursors50d-50lis manipulated, a rank 3 matrix may be formed, thus any manipulation may result in modification to T0, mT, and T (the TW propagation time).

As shown inFIG.6, cursor50a, cursor50c, and cursor50e(corresponding to TL, TLLL, and TRLL, respectively) may be pinned. Cursors50b,50d, and50fmay be hidden, as their rows of the [A] matrix are linearly dependent on the pinned cursors50a,50c, and50e. Manipulating any of the pinned cursors50a,50c, and/or50emay similarly modify the values of T0, mT, and T.

If one of the visible non-pinned cursors50g-50lare manipulated, their combined rows of the [A] matrix may form a rank 4 matrix, allowing the manipulation of d in addition to T0, mT, and T.

As shown inFIG.7, a nonzero value for d may be shown as an offset (or as an arrow) between the cursor50g-50llocations and the corresponding TW arrival times (TR, TLR, TLLR, TRR, TLRR, and TRRR) on the Bewley lattice diagram.

If a fourth cursor50jis pinned, as shown inFIG.8, all remaining cursors50b,50d,50f-50i,50k,50lmay be hidden as a full rank (rank 4) for the system may be reached. Any manipulation of a fifth cursor would be redundant on this fully determined system of equations.

As described above with reference toFIGS.3-8, the degrees of freedom available for manipulation may be based on the number of cursors50a-50lthat are pinned. An additional implementation provides another method to select which degrees of freedom are available by pinning mT, T, T0, and/or d, as discussed below.

Since all of the columns are linearly independent, holding them constant may be accomplished by extracting columns of the [A] matrix as follows.

To hold T0constant, the T0column may be extracted.

And the resulting equation may be solved.

This may be especially useful when an operator has confidence in a value for T, but not in d. By holding T constant, the cursors50a-50lcan be manipulated to adjust T0, mT, and d.

An additional method is to swap the third and fourth columns of A and T and d in the right-hand vector. This results in modification to d when two cursors are pinned and a third cursor is moved. Then, pinning the third cursor and moving a fourth cursor may modify T as desired. The modified matrix equation is shown below.

Note that modification of the columns of the [A] matrix and associated vector of T0, mT, d, and T while applying the standard algorithm described above allows the order of variable manipulation to be set arbitrarily.

If the fault location, m, is known, an operator may want to hold m constant instead of mT. Since mT and T both include T, it may be desirable to isolate m from mT. To do this the [A] matrix may be modified to include a fixed m value. To perform the modification, one may begin with the original equations:

Then m may be moved into the [A] matrix.

Next, the second and third columns may be consolidated, and the resulting equation may be solved.

[TLTL⁢LTL⁢RTRTR⁢RTR⁢L]=[1m013⁢m011+m111-m113-3⁢m112-m0][T0Td]
Because of the interdependence of T and mT, a similar procedure may be used to solve for T0, m, and d while holding T constant.

Up to four TW arrival times may be assigned cursors to provide an exact match between the Bewley Lattice and the recorded TW signals. A best-fit result is also possible if, instead of providing a direct solution to the Bewley Lattice equations, a Moore-Penrose pseudo-inverse is applied to the constructed [A] matrix. The pseudo-inverse may provide a best-fit solution in the least-squared-error sense to a system of linear equations. Thus, more than four cursors can be pinned to provide an approximate solution.

Furthermore, a weighted-pseudo-inverse also exists by which different rows of the [A] matrix may have a stronger influence on the result than others. In some embodiments, this may be applied by assigning a weight to a TW arrival time proportional to the absolute magnitude of the TW arrival pulse for that row of the [A] matrix. In some embodiments, the weight can be manually assigned.

Other best-fit solutions may also be applied. For example, a minimum-squared-error method uses a 2-norm to minimize cost function. Other weights may equally be applied such as 1-norm of the absolute value, 4-norm, etc.

In view of the foregoing, methods of analyzing TW data resulting from a fault on an electric power delivery system according to embodiments of the present disclosure may comprise measuring electrical properties of a first TW on a power transmission line at a first location, measuring electrical properties of a second TW on the power transmission line at a second location, and analyzing the measured electrical properties of the first TW and the second TW to simultaneously determine the value of four parameters related to the first TW and the second TW.

The four parameters may be selected from a time (To) of a TW initiation at a fault location, a length of time (T) for a TW to traverse the entire line length, a time (mT) for the first TW arrival at the first location, a time skew (d) between the actual relative measured arrival times and the actual arrival times at the first location and the second location, and a location (m) of the initiation of the first TW. The value of any of the four parameters may be fixed and any of the four parameters that are not fixed may be calculated.

Optionally, signal processing, such as transformation, low frequency removal, smoothing, differentiation, edge detection, matched filter, squaring, and/or absolute value, may be performed on the measured electrical properties of the first TW and the second TW, and a time skew caused by signal processing between measured TW arrival times and actual TW arrival times at the first location and the second location may be compensated for.

To facilitate the analysis, a plurality of cursors may be arranged on a Bewley lattice diagram of a graphic user interface. An operator may pin up to four of the cursors to fix one or more of the four parameters. Additionally, a time skew (d) between the relative measured arrival times and the actual arrival times of the TWs at the first location and the second location may be displayed as an offset, arrow, or other offset indicator between the cursor locations and corresponding reflections on the Bewley lattice diagram.

FIG.9shows an additional embodiment of the graphic user interface40of a traveling wave analysis subsystem including an additional embodiment of a Bewley lattice diagram and additional parameter displays and inputs consistent with embodiments of the present disclosure. In this embodiment a line, which may be a solid and/or dashed line, associated with each TW reflection may function as a cursor60(e.g., a handle) that may be utilized by a user to move the location of the line and the associated TW reflection on the Bewley lattice diagram.

As shown, each cursor60may be positioned overlying a graphical representation of a respective measured traveling wave signal62,64on a timescale66. Each timescale66may include the time T0located near the top of the timescale66with time progressing forward in a downward direction. In some embodiments the diagram may be drawn vertically as shown, with time progressing from top to bottom. In additional embodiments, the diagram may be drawn horizontally with time progressing from left to right. The graphic user interface40may include an input70to allow an operator to select the time to be displayed relative to the TW fault time (T0), as shown inFIG.9, or to be displayed using the time recorded by the data acquisition subsystem. Once a cursor60and the associated TW reflection on the Bewley lattice diagram is placed in a desired location, an operator may pin a cursor60, which may be indicated by a symbol72(e.g., a symbol of a thumbtack).

Additionally, an operator may choose to display reflections propagated through a fault76(as indicated by the lines inFIG.9) or choose to have them hidden by selecting the input78.

In addition to manipulating parameters with the cursors60on the Bewley lattice diagram, an operator may manipulate parameters in the fields provided on the left-hand side of the graphic user interface40. A line length field80may import a value for a line length (LL) from a database where a known line length is stored and/or a value for the line length may be input directly or changed by an operator or calculated from other system parameters.

A fault location may be displayed in a left pane fault location field82and a right pane fault location field84, each indicating a respective distance between the fault location and end of the line. These values may be calculated and displayed based on other measured and entered parameters. These fault location values may additionally be entered by an operator, should the operator know the fault location and/or desire to examine other parameters related to the measured traveling wave signals62,64, such as to evaluate potential delays or discrepancies in the system.

Likewise, first TW arrival times may be displayed in a left pane first TW arrival time field86and a right pane first TW arrival time field88. These TW arrival times may be determined by positioning the cursors60on the Bewley lattice diagram and/or by entering a time directly into the left pane first TW arrival time field86and/or the right pane first TW arrival time field88.

Additional fields may also be provided that may include values that may be calculated, retrieved from a database, and/or entered by an operator, such as a TW line propagation time field90, a TW propagation velocity field92, a left pane CT cable delay compensation field94, a right pane CT cable delay compensation field96, and a time skew field, such as a right pane time skew field98. Directly entering some values may affect other values (for example, how propagation time, propagation velocity, and line length are related by equation 1). A reset input100may also be provided to enable an operator to reset the Bewley lattice diagram to its prior and/or default state. These parameter fields are provided as non-limiting examples, and additional or fewer parameter fields may be provided according to various embodiments of the present disclosure.

A time skew99is shown on the right pane of the Bewley lattice diagram and in the right pane time skew field98, indicating a time difference between peaks visible in the measured traveling wave signal64and the arrival of each TW according to the Bewley lattice diagram in the right pane. There may be many causes for such a time skew, such as a difference in the measured times between a local terminal and a remote terminal, CT or PT group delay, or clock synchronization error, for example. The system may allow the time skew to be removed by the operator, which may result in improved alignment between the measured TW signals64and the Bewley lattice diagram, as will be described further with reference toFIGS.10and11.

FIG.10shows the graphic user interface40ofFIG.9after a time shift has been applied to the measured traveling wave signal64. As shown, an operator may shift the graph of the measured traveling wave signal64on the timescale66in the negative direction by an amount equal to the magnitude of the time skew. Thus, the arrival of TWs according to the Bewley lattice diagram in the right pane and the peaks visible in the measured traveling wave signal64may be better aligned on the Bewley lattice diagram.

FIG.11shows the graphic user interface ofFIG.10after the time skew has been removed to account for the time shift applied to the measured traveling wave signal64. Accordingly, the Bewley lattice diagram may be modified by an operator to shift the graph of a measured traveling wave signal on a timescale, such as to compensate for a time skew, for improved TW analysis, and for a clear and precise understanding of the Bewley lattice diagram.

While specific embodiments and applications of the disclosure have been illustrated and described, it is to be understood that the disclosure is not limited to the precise configurations and components disclosed herein. Accordingly, many changes may be made to the details of the above-described embodiments without departing from the underlying principles of this disclosure. The scope of the present invention should, therefore, be determined only by the following claims.