Patent Publication Number: US-2013234731-A1

Title: Systems and methods for determining trapped transmission line charge

Description:
TECHNICAL FIELD 
     This disclosure relates to determining trapped charge on transmission lines. More particularly, this disclosure relates to systems and methods for determining trapped charge on uncompensated phase lines fitted with capacitance-coupled voltage transformers. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Non-limiting and non-exhaustive embodiments of the disclosure are described, including various embodiments of the disclosure, with reference to the figures, in which: 
         FIG. 1  illustrates a simplified circuit diagram of a phase line connected to a capacitance-coupled voltage transformer (CCVT). 
         FIG. 2  illustrates a circuit diagram of a phase line connected to a CCVT, including various tuning and protection circuits. 
         FIG. 3  illustrates an embodiment of a coupling-capacitor voltage divider that may be used to couple a phase line to a CCVT. 
         FIG. 4  illustrates an embodiment of a capacitance-bushing voltage divider that may be used to couple a phase line to a CCVT. 
         FIG. 5  illustrates an oscillographic comparison of an actual phase line voltage and a phase line voltage derived from measurements taken at the output of a CCVT. 
         FIG. 6  illustrates an oscillographic comparison of an actual phase line voltage and a phase line voltage derived from measurements taken at the output of a CCVT following a phase line de-energizing event. 
         FIG. 7  illustrates a method for determining the trapped charge on a phase line coupled to a CCVT. 
         FIG. 8  illustrates a method for determining the trapped charge on a phase line coupled to a CCVT, including adjusting capacitance parameters based on a measured temperature. 
         FIG. 9A  illustrates a circuit diagram of one embodiment of a phase line coupled to a CCVT, including two current sensors useful for determining the voltage on the phase line. 
         FIG. 9B  illustrates a circuit diagram of another embodiment of a phase line coupled to a CCVT, including two current sensors useful for determining the voltage on the phase line. 
         FIG. 9C  illustrates a circuit diagram of another embodiment of a phase line coupled to a CCVT, including two current sensors useful for determining the voltage on the phase line. 
         FIG. 10  illustrates a circuit diagram of a phase line with a trapped charge coupled to a de-energized CCVT. 
         FIG. 11  illustrates an oscillographic comparison of the actual phase line voltage and the phase line voltage derived from CCVT current sensors. 
     
    
    
     In the following description, numerous specific details are provided for a thorough understanding of the various embodiments disclosed herein. The systems and methods disclosed herein can be practiced without one or more of the specific details, or with other methods, components, materials, etc. In addition, in some cases, well-known structures, materials, or operations may not be shown or described in detail in order to avoid obscuring aspects of the disclosure. Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more alternative embodiments. 
     DETAILED DESCRIPTION 
     Intelligent electronic devices (IEDs) may be used for monitoring, protecting, and/or controlling industrial and utility equipment, such as in electric power delivery systems. IEDs may be configured to obtain measurement information from current sensors and/or voltage sensors, such as current transformers (CTs) and/or voltage potential transformers (PTs). IEDs may be configured to obtain measurement information from a variety of other sources, such as optical current transducers, Rogowski coils, light sensors, relays, temperature sensors, and similar devices, as well as from measurements, signals, or data provided by other IEDs. IEDs within a power system may be configured to perform metering, control, switching, and protection functions based on measured data. In some embodiments, an IED may be configured to monitor, protect, and/or control the de-energization and/or re-energization of power distribution lines. 
     Power distribution systems may include various transmission and distribution lines. In many instances power is transmitted as three-phase power, with each phase of power carried over a single phase line. In other embodiments, any number of phase lines may be used to transmit power from one point to another. Phase lines may be switched on and off (de-energized) with relative frequency in some configurations, and rarely de-energized in other situations. For example, a critical transmission line may only be disconnected as the result of a breaker tripping under fault conditions or during scheduled maintenance. When a phase line is initially connected to a power source, the phase line is energized with an initial voltage potential. In a three-phase power system, the phase line may be energized with an alternating current at approximately 60 Hz. 
     When the phase line is disconnected from the power source (e.g., a breaker trips), the phase line looses the 60 Hz voltage signal. Excess charge on the phase line dissipates quickly if the phase line is terminated with a magnetic voltage transformer or other component that allows for DC voltage discharge. However, if the phase line terminates with a capacitive coupled voltage transformer (CCVT), then a DC voltage may remain on the de-energized phase line as “trapped charge.” 
     Re-energization of a phase line with a trapped charge can result in severe switching transients. For example, reclosing a transmission line when trapped charge is present on one of the three phase lines in a three-phase power system may result in severe transient overvoltages, and/or other undesirable conditions. A significant factor in the design of extra-high voltage (EHV) lines is the expected level of switching transients. Accordingly, the ability to limit switching transients to lower levels with controlled closing of de-energized phase lines could provide significant benefits. The benefits may include a reduction in the cost of phase line design and a reduction in temporary overvoltages (TOVs), which may task surge arresters and expose equipment to overvoltages exceeding their voltage ratings. 
     In some embodiments, pre-insertion resistors, surge arresters, and current-limiting reactors may be employed to reduce the magnitude and impact of switching transients. In other embodiments, controlled re-energization can, in many cases, provide an effective means of mitigating transients due to reclosing phase lines with trapped charge. In order to perform an optimized re-energization of a phase line, it is necessary to know the magnitude and polarity of the trapped charge on the phase line. The phase line may be optimally re-energized by matching the prospective re-energizing voltage with the trapped charge. In some embodiments, resistive dividers and/or inductive voltage transformers (IVTs) may be used to accurately determine the voltage on a high-voltage phase line. 
     Due to cost, size, and other considerations, CCVTs are commonly employed in high-voltage systems. However, the secondary output of CCVTs becomes distorted and decays rapidly to zero following the loss of a 60 HZ voltage signal. Accordingly, it is difficult or impossible to determine the trapped charge on a de-energized phase line using the output voltage of a CCVT. Specifically, using the output voltage (the secondary windings) of a step-down transformer in a CCVT configuration is unsuitable for calculating the trapped charge on a phase line on the side of the primary windings of the transformer. This is due, at least in part, to the fact that the CCVT acts as a band-pass filter suppressing low frequency components of the input signal. 
     According to various embodiments of the present disclosure, a transformer is configured in a CCVT configuration using a coupling-capacitor voltage divider or a capacitance-bushing voltage divider. In various configurations, the primary winding of the transformer “taps” a point in between a primary capacitive assembly and an auxiliary capacitive assembly. The primary capacitive assembly may couple the primary winding of the transformer to a high voltage phase line, and the auxiliary capacitive assembly may couple the same side of the primary winding to a neutral point (or other reference point). 
     An IED may be used to measure the current through the primary capacitive assembly using a first current sensor, such as a CT or Rogowski coil. The IED may also measure the current through the auxiliary capacitive assembly using a second current sensor. In some embodiments, the current through the primary capacitive assembly and the auxiliary capacitive assembly may be directly measured. In some embodiments, the current sensors may be positioned at neutral, ground, or low voltage locations in order to reduce the difficulty in obtaining accurate current measurements. 
     For example, the current through the auxiliary capacitive assembly may be measured via a current sensor positioned between the auxiliary capacitive assembly and ground. The current through the primary capacitive assembly may be deduced using the current through the auxiliary capacitive assembly and a current measured between the grounded side of the primary winding and ground. Accordingly, the current through the primary and auxiliary capacitive assemblies may be determined using current sensors positioned at zero-voltage points. 
     The IED may determine a de-energization time corresponding to the instant the phase line is de-energized. The IED may calculate the voltage on the phase line using the current through the primary capacitive assembly, the current through the auxiliary capacitive assembly, and the capacitances of the primary and auxiliary capacitive assemblies. The IED may use the calculated voltage to determine a trapped charge on a transmission line at the de-energization time when the phase line was de-energized. 
     During re-energization, the IED may communicate with one or more additional IEDs, breakers, relays, and/or other power system components in order to ensure that the re-energizing voltage applied to the phase line is matched with the trapped charge on the phase line. As previously described, by matching the re-energizing voltage with the trapped charge, unwanted transients can be minimized or eliminated. 
     The phrases “connected to” and “in communication with” refer to any form of interaction between two or more components, including mechanical, electrical, magnetic, and electromagnetic interaction. Two components may be connected to each other, even though they are not in direct contact with each other, and even though there may be intermediary devices between the two components. 
     As used herein, the term IED may refer to any microprocessor-based device that monitors, controls, automates, and/or protects monitored equipment within a system. Such devices may include, for example, remote terminal units, differential relays, distance relays, directional relays, feeder relays, overcurrent relays, voltage regulator controls, voltage relays, breaker failure relays, generator relays, motor relays, automation controllers, bay controllers, meters, recloser controls, communications processors, computing platforms, programmable logic controllers (PLCs), programmable automation controllers, input and output modules, motor drives, and the like. IEDs may be connected to a network, and communication on the network may be facilitated by networking devices, including, but not limited to, multiplexers, routers, hubs, gateways, firewalls, and switches. Furthermore, networking and communication devices may be incorporated in an IED or be in communication with an IED. The term IED may be used interchangeably to describe an individual IED or a system comprising multiple IEDs. 
     Aspects of certain embodiments described 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 or on a computer-readable storage medium. A software module 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. 
     The embodiments of the disclosure will be best understood by reference to the drawings, wherein like parts are designated by like numerals throughout. The components of the disclosed embodiments, as generally described and illustrated in the figures herein, could be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the systems and methods of the disclosure is not intended to limit the scope of the disclosure, as claimed, but is merely representative of possible embodiments. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of this disclosure. In addition, the steps of a method do not necessarily need to be executed in any specific order, or even sequentially, nor need the steps be executed only once, unless otherwise specified. 
       FIG. 1  illustrates a simplified circuit diagram of a phase line  110  connected to a capacitance coupled voltage transformer (CCVT)  100 . Phase line  110  may be terminated on either end with a breaker  111  and  112 . According to some embodiments, breaker  111  may connect an AC power source to phase line  110 . Breaker  112  may connect phase line  110  to additional components and/or transmission lines. Alternatively, breaker  112  may be omitted and phase line  110  may terminate with CCVT  100 . 
     As illustrated, simplified CCVT  100  includes a transformer  150  including a primary winding  155  and a secondary (output) winding  157 . The output voltage of CCVT  100  is measured across terminals X 1   161  and X 2   162  on secondary winding  157 . In various embodiments, transformer  150  may be a step-down transformer (i.e. the voltage potential across primary winding  155  is higher than that across the secondary winding  157 ). As illustrated, transformer  150  is considered a CCVT because of primary capacitive assembly (C 1 )  120  and auxiliary capacitive assembly (C 2 )  130 . Additionally, an inductor (L 1 )  135  may be configured to tune CCVT  100  in order to improve accuracy. 
     Primary winding  155  may be said to “tap” the junction of C 1   120  and C 2   130 . C 1   120  may couple primary winding  155  to phase line  110 . C 2   130  may couple the high-voltage side of primary winding  155  to ground  140 . Ground  140  may be a physical ground, a neutral point, a neutral phase line, or another phase line in a three-phase power system. CCVT  100  may be used to step down high-voltage phase line  110  to a lower voltage across X 1   161  and X 2   162 . For example, phase line  110  may have a potential of 230 KV and the outputs X 1   161  and X 2   162  of CCVT  100  may have a potential of 115 V. CCVT  100 , including capacitive assemblies C 1   120  and C 2   130 , may be smaller and/or less costly to manufacture than an equivalent inductive transformer. 
     As previously stated,  FIG. 1  illustrates a simplified diagram of a CCVT  100 . The present systems and methods are applicable to both passive CCVTs and active CCVTs.  FIG. 2  illustrates a circuit diagram of a phase line  210  connected to an active single-phase CCVT  200 , including various tuning and protection circuits. Phase line  210  may be selectively disconnected from an AC power source via breaker  211 . Breaker  212  may connect phase line  210  to additional components or phase lines in a power distribution system. CCVT  200  includes primary capacitive assembly (comprising capacitor  220  and capacitor  225 ) and auxiliary capacitive assembly  230 . A compensating reactor  233  may include inductive, capacitive, and/or resistive elements. A ferroresonant suppression circuit (FSC)  270  may be connected to output terminals X 1   261  and X 3   263  across secondary windings  257  and  259 . FSC  270  may reduce or eliminate ferroresonant conditions within the CCVT  200  that might otherwise cause damaging overvoltages and/or overcurrents. 
     Primary capacitive assembly  220  and  225  may couple the high-voltage side of primary winding  255  to high-voltage phase line  210 . Auxiliary capacitive assembly  230  may couple the high voltage side of primary winding  255  to a neutral point  240 , such as ground. Accordingly, primary capacitive assembly  220  and  225  and auxiliary capacitive assembly  230  may be part of a coupling-capacitor voltage divider or a capacitance-bushing voltage divider with the high-voltage side of the primary winding coupled to the “tap” of such devices. Transformer  250  may be a step-down transformer and include one or more secondary windings  257  and  259 . Various desired output voltages may be achieved using any number of secondary windings and associated terminals, such as terminals X 1   261 , X 2   262 , and X 3   263 . 
     The interaction of the various capacitive and reactive elements in CCVT  200  results in transient errors in the secondary voltage output during switching and faults. As previously described, the poor transient response of CCVTs makes it difficult or impossible to accurately determine the trapped charge on a phase line after it has been disconnected from the AC power source via breaker  211  and/or  212 . For example, the voltage measured at outputs X 1   261  and X 3   263  is unsuitable to determine a DC charge on phase line  210  once it is de-energized. 
       FIG. 3  illustrates an embodiment of a coupling-capacitor voltage divider  300  that may be used in conjunction with a transformer to form a CCVT. A primary capacitive assembly, comprising capacitive elements  321 ,  322 ,  323 , and  324  may couple a high-voltage phase line  310  to a “tap”  350 . Since the voltage on both sides of capacitive elements  321 ,  322 ,  323 , and  324  (forming the primary capacitive assembly) is relatively high, primary capacitive elements  321 ,  322 ,  323 , and  324  may be housed within an insulating bushing  315 . “Tap”  350  may be coupled to a neutral point  340  via an auxiliary capacitive assembly  330 . Accordingly, high-voltage phase line  310  may be coupled to neutral via primary capacitive elements  321 ,  322 ,  323 , and  324  and auxiliary capacitive assembly  330 . The primary winding of a transformer may be connected to “tap”  350  positioned between primary capacitive elements  321 ,  322 ,  323 , and  324  and auxiliary capacitive assembly  330 . 
     Capacitive elements  321 ,  322 ,  323 , and  324  allow a 60 Hz AC power signal to flow, but may not allow DC charge to flow. Accordingly, if phase line  310  is disconnected from an AC power source, via breakers  311  and/or  312 , a DC trapped charge may remain on phase line  310 . Phase line  310  may have high-voltage trapped charge, while tap  350  is at zero voltage. 
       FIG. 4  illustrates an embodiment of a capacitance-bushing voltage divider  400  that may be used in conjunction with a bushing potential device. Capacitance-bushing voltage divider  400  may include a center conductor  412  attached to a high-voltage phase line  410 . An insulating bushing  415  may surround center conductor  412 . Insulating bushing  415  may include one or more layers of capacitive and dielectric materials. A first capacitive layer  420  may be used to couple a tap  450  to high-voltage phase line  410 . Accordingly, center conductor  412  and first capacitive layer  420  may form a primary capacitive assembly that couples tap  450  to phase line  410 . A second capacitive layer  430  in conjunction with first capacitive layer  420  and center conductor  412  may form an auxiliary capacitive assembly coupling tap  450  and phase line  410  to neutral point  440 . The primary winding of a transformer may be connected to tap  450  in order to form a bushing potential device. Accordingly, a CCVT, as used herein, may comprise a step-down transformer connected to a phase line via a bushing capacitance-bushing voltage divider  400 , such that first capacitive layer  420  serves as a primary capacitive assembly and second capacitive layer  430  serves as an auxiliary capacitive assembly. 
     Again, if breakers  411  and/or  412  are opened, phase line  410  may be disconnected from an AC power signal. Remaining AC voltages would quickly dissipate on phase line  410 , but a DC trapped charge would remain on phase line  410 . The DC trapped charge would remain on phase line  410  since capacitive layers  420  and  430  may prevent the DC charge from dissipating. 
     As described above, while CCVTs may be smaller and/or cheaper than an equivalent inductive transformer, the poor transient response of CCVTs makes it difficult or impossible to accurately determine the trapped charge on a de-energized phase line using the output of the CCVT. Accordingly, the voltage measured at the output of the CCVT is unsuitable to determine the trapped charge on a de-energized phase line.  FIGS. 5 and 6  illustrate various oscillographic reports that demonstrate the limitations of CCVTs. Specifically, that while the output voltage of a CCVT corresponds to the actual voltage across the primary winding of a transformer, the output voltage of the CCVT cannot be used to accurately determine trapped charge on a de-energized phase line. The oscillographic reports of  FIGS. 5 and 6  correspond to a CCVT, such as CCVT  200  illustrated in  FIG. 2 , which includes primary and auxiliary capacitive assemblies, such as those shown in  FIGS. 3 and 4 . 
       FIG. 5  illustrates an oscillographic comparison  500  of the actual phase line voltage  510  and the phase line voltage derived using the voltage measured at the output of a CCVT (derived phase line voltage  520 ). At a time zero (along the X-axis) the actual phase line voltage  510  alternates between approximately −200 KV and 200 KV (along the Y-axis). At approximately 240 ms, at  530 , the phase line is de-energized and the actual phase line voltage  510  drops to approximately zero for duration  550 . At approximately 320 ms, at  540 , the phase line is re-energized and the actual phase line voltage  510  alternates between approximately −100 KV and 100 KV. 
     The transformer may be a step-down transformer having a known winding ratio. Accordingly, the phase line voltage may be derived using the voltage measured at output of the CCVT. As illustrated in  FIG. 5 , the derived phase line voltage  520  based on the measured voltage at the output of the CCVT is relatively accurate, though imperfect due to the poor transient response of CCVTs. 
       FIG. 6  illustrates an oscillographic comparison  600  of the actual phase line voltage  620  and the phase line voltage derived using the output of the CCVT (derived phase line voltage  610 ). As illustrated, when the phase line is de-energized, at  630 , a DC trapped charge of about −400,000 KV remains on the phase line, at  650 . The derived phase line voltage  610  erroneously indicates that the phase line has a zero voltage, at  650 , during de-energization. Accordingly,  FIGS. 5 and 6  illustrate that while the voltage measured at the output of a CCVT corresponds to the actual phase line voltage (see  FIG. 5 ), a DC trapped charge on the phase line is not calculable using the voltage measured at the output of a CCVT. 
       FIG. 7  illustrates a method  700  for accurately determining the trapped charge on a phase line connected to a CCVT. The method  700  could be repeated for each phase of a multi-phase power system utilizing independent transformers for each phase. Similarly, the method  700  could be adapted to accommodate a multi-phase power system in which one or more phase lines are connected to one or more transformer cores. 
     With regards to one phase in a three-phase power system, an IED may determine a first current through a primary capacitive assembly in a CCVT, at  710 . The IED may determine a second current through an auxiliary capacitive assembly in the CCVT, at  720 . The IED may then determine a de-energization time corresponding to the instant the phase line is de-energized, at  730 . The IED may calculate the voltage on the phase line using the first and second currents, the capacitance (or associated reactance) of the primary capacitive assembly, and the capacitance (or associated reactance) of the auxiliary capacitive assembly, at  740 . The IED may determine the trapped charge on the phase line at the de-energization time, at  750 . 
     According to various embodiments, a local IED may utilize distributed or cloud computing developments to reduce the data storage or processing demands. For example, the local IED may receive signals from current sensors associated with the primary and auxiliary capacitive assemblies and transmit the current signals to a remote IED configured to store and/or process the data. The local IED (or other IED configured to monitor, protect, and/or control aspects of the power system associated with the phase line) may then receive instruction from the remote IED with regards to breaker switching, or other related events, in order to ensure that a phase line is optimally re-energized. 
     In some embodiments, the current through the primary capacitive assembly and the auxiliary capacitive assembly may be directly measured. In some embodiments, current sensors may be positioned at neutral, ground, or low voltage locations in order to reduce the difficulty in obtaining accurate current measurements. For example, the current through the auxiliary capacitive assembly may be measured via a current sensor positioned between the auxiliary capacitive assembly and ground. The current through the primary capacitive assembly may be deduced using the current through the auxiliary capacitive assembly and a current measured between the grounded side of the primary winding and ground. Accordingly, the current through the primary and auxiliary capacitive assemblies may be determined using current sensors positioned at zero-voltage points. 
       FIG. 8  illustrates a related method  800  for accurately determining the trapped charge on a phase line connected to a CCVT. The methods of  FIGS. 7 and 8  are described as being performed by an IED, however, various machines, apparatuses, and/or persons could alternatively perform method  800 . Moreover, the systems and methods described herein could be implemented as machine instructions executable by a processor in an IED. Initially, an IED determines the current through the primary capacitive assembly in the CCVT, at  810 . The IED determines the current through the auxiliary capacitive assembly in the CCVT, at  820 . For purposes of subsequent calculations, the known capacitances of the primary and auxiliary capacitive assemblies may be adjusted to compensate for an associated measured temperature, at  830 . For example, one or more temperature sensors may be used to measure the ambient temperature near a capacitive component in the primary or auxiliary capacitive assemblies. Alternatively, one or more temperature sensors may be used to directly measure the temperature of the primary and/or auxiliary capacitive assemblies. Adjusting the capacitance value of the primary and/or auxiliary capacitive assemblies based on the temperature may provide increased accuracy for subsequent calculations. 
     The IED may then determine a de-energization time corresponding to the instant the phase line is de-energized, at  840 . The IED calculates the voltage on the phase line using the first and second currents and the adjusted capacitances of the primary and auxiliary capacitive assemblies, at  850 . The IED may then determine the trapped charge on the phase line at the de-energization time, at  860 . 
     Again, the current through the primary capacitive assembly and the auxiliary capacitive assembly may be directly measured. Alternatively, current sensors may be positioned at neutral, ground, or low voltage locations in order to reduce the difficulty in obtaining accurate current measurements. Accordingly, the current through the auxiliary capacitive assembly may be measured via a current sensor positioned between the auxiliary capacitive assembly and ground. The current through the primary capacitive assembly may be deduced using the current through the auxiliary capacitive assembly and a current measured between the grounded side of the primary winding and ground. 
       FIG. 9A  illustrates a circuit diagram of one embodiment of a CCVT  900 , including two current sensors  980  and  985  useful for determining the trapped charge on phase line  910 . The illustrated CCVT configuration is similar to that described in conjunction with  FIG. 2 , and may utilize a coupling-capacitor voltage divider or a capacitance-bushing voltage divider, as illustrated in  FIGS. 3 and 4  respectively.  FIG. 9A  illustrates a circuit diagram including the main components of an active single-phase CCVT  900 , including various tuning and protection circuits. CCVT  900  includes primary capacitive assembly (comprising capacitive elements  920  and  925 ) and auxiliary capacitive assembly  930 . A compensating reactor  933  may include inductive, capacitive, and/or resistive elements. A ferroresonant suppression circuit (FSC)  970  may be connected to output terminals X 1   961  and X 3   963  across secondary windings  957  and  959 . FSC  970  may reduce or eliminate ferroresonant conditions within the CCVT that might otherwise cause damaging overvoltages and/or overcurrents. 
     Primary capacitive assembly  920  and  925  may couple the high-voltage side of primary winding  955  to high-voltage phase line  910 . Auxiliary capacitive assembly  930  may couple the high voltage side of primary winding  955  to a neutral point  940 , such as ground. Transformer  950  may be a step-down transformer and include one or more secondary winding  957  and  959 . Various desired output voltages may be achieved using any number of secondary windings and associated terminals, such as terminals X 1   961 , X 2   962 , and X 3   963 . 
     The interaction of the various capacitive and reactive elements in CCVT  900  results in transient errors in the secondary voltage output during switching and faults. As previously described, the poor transient response of CCVTs makes it difficult or impossible to accurately determine the trapped charge on phase line  910  when breaker  911  and/or breaker  912  are opened. For example, the voltage measured at outputs X 1   961  and X 3   963  is unsuitable to determine the DC trapped charge on phase line  910  because capacitive element  920  and/or capacitive element  925  filter DC voltages. 
     IED  905  may be in communication with current sensors  980  and  985  configured to measure the current I C2  through auxiliary capacitive assembly  930  and the current I C1  through primary capacitive assembly  920  and  925 , respectively. Measuring currents I C1  and I C2  and having knowledge of the capacitive values of primary capacitive assembly  920  and  925  and auxiliary capacitive assembly  930  allows for the reconstruction of the voltage of phase line  910 . Knowing the voltage of phase line  910  and detecting a de-energization time corresponding to when breaker  911  is opened (disconnecting phase line  910  from an AC power source), allows for the calculation of the trapped charge on phase line  910 . As previously described, knowledge of the trapped charge on a phase line can be used to considerably reduce undesirable transients during the subsequent re-energization of the phase line. 
     To calculate the voltage of phase line  910 , the following equation can be used: 
         V=jI   C1   X   C1   +jI   C2   X   C2    Equation 1
 
     In equation 1 above, V is the voltage of phase line  910 , I C1  is the current in primary capacitive assembly  920  and  925 , I C2  is the current in auxiliary capacitive assembly  930 , X C1  is the capacitive reactance of primary capacitive assembly  920  and  925 , and X C2  is the capacitive reactance of auxiliary capacitive assembly  930 . In the time domain, equation 1 can be expressed as: 
     
       
         
           
             
               
                 
                   
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     In equation 2 above, V(t) is the instantaneous voltage of phase line  910 , I C1  is the instantaneous current in primary capacitive assembly  920  and  925 , I C2  is the instantaneous current in auxiliary capacitive assembly  930 , C 1  is the capacitance of primary capacitive assembly  920  and  925 , and C 2  is the capacitance of auxiliary capacitive assembly  930 . Using equation 2 above, the DC trapped charge may be calculated by determining the voltage at the instant the phase line underwent a de-energization event (i.e. the de-energization time). 
       FIG. 9A  illustrates one possible configuration of a CCVT  900  and one possible location for positioning current sensors  980  and  985 . It should be apparent to one of skill in the art that two or more current sensors may be placed at varying locations within the circuit diagram of  FIG. 9A  and still allow for the calculation of the currents through the primary and auxiliary capacitive assemblies. Specifically, using the electrical principles described in Kirchhoff&#39;s first and second laws, currents sensors  980  and  985  may be placed in a wide variety of locations. Kirchhoff&#39;s laws may be adapted to account for Faraday&#39;s law of induction related to the inductors associated with CCVT  900  by associating a potential drop or electromotive force with each inductor in the circuit (e.g., primary winding  955 ). 
       FIG. 9B  illustrates a circuit diagram of another embodiment of a CCVT  990 , in which a neutral connection  941  of primary winding  955  of transformer  950  is not shared with neutral connection  940  of auxiliary capacitive assembly  930 . The other components are similar to those described in conjunction with  FIG. 9A  having similar reference numbers. According to the embodiment illustrated in  FIG. 9B , IED  905  may receive current measurements I C2  and I C3 . I C2  corresponds to the current through the auxiliary capacitive assembly. IED  905  may derive the current, I C1 , through primary capacitive assembly  920  and  925  using I C2  and I C3 . 
     Measuring or deriving currents I C1  and I C2  and having knowledge of the capacitive values of primary capacitive assembly  920  and  925  and auxiliary capacitive assembly  930  allows for the reconstruction of the voltage of phase line  910  using equations 1 and/or 2 above. The instantaneous voltage at the de-energization time may be calculated in order to determine the trapped charge on phase line  910  following a de-energization event. As previously described, knowledge of the trapped charge on a phase line can be used to considerably reduce undesirable transients during the subsequent re-energization of the phase line. 
       FIG. 9C  illustrates a circuit diagram of another embodiment of a CCVT  995 , including two high-voltage current sensors  981  and  986  configured to directly measure the current through primary capacitive assembly  920  and  925  and auxiliary capacitive assembly  930 . The other components of CCVT  995  are similar to those described in conjunction with  FIG. 9A  having similar reference numbers. IED  905  may receive the current measurement made by high-voltage current sensors  981  and  986 . The IED  905  may then reconstruct the voltage of phase line  910  using equations 1 and/or 2 above. IED  905  may then determine the instantaneous voltage at the de-energization time in order to calculate the trapped charge on phase line  910 . As previously described, knowledge of the trapped charge on a phase line can be used to considerably reduce undesirable transients during the subsequent re-energization of the phase line. 
     A primary advantage of the embodiments illustrated in  FIGS. 9A and 9B  is that current sensors  980  and  985  are located at a zero-voltage location in the respective CCVTs  900  and  990 . Accordingly, the cost and size of current sensors  980  and  985  may be significantly lower than high-voltage current sensors  981  and  986 . However, any of the configurations shown in  FIGS. 9A-9C  may be used in conjunction with presently described systems and methods. Additionally, current sensors may be positioned in any of a wide variety of locations in a CCVT, so long as they provide sufficient information for an IED to calculate or derive the current through each of the primary and auxiliary capacitive assemblies. 
       FIG. 10  illustrates a circuit diagram of a phase line  1010  with a trapped charge coupled to a de-energized CCVT  1000 . As illustrated, phase line  1010  is coupled to transformer  1050  via a primary capacitive assembly  1020 , and to ground via an additional auxiliary capacitive assembly  1030 . Transformer  1050  is illustrated as de-energized completely as dashed lines, while phase line  1010  is shown has having a DC trapped charge that cannot be discharged because of primary capacitive assembly  1020  of CCVT  1000 . IED  1005  may have used current measurements obtained via current sensors  1080  and  1085  to reconstruct the voltage of phase line  1010 . By detecting the de-energization time, such as when breaker  1011  (or possibly breaker  1012 ) was opened, IED  1005  may determine the magnitude and polarity of the trapped charge on phase line  1010 . 
       FIG. 11  illustrates an oscillographic comparison  1100  of an actual phase line voltage  1110  and a phase line voltage derived (derived phase line voltage  1120 ) using measurements taken from two current sensors. For example, derived phase line voltage  1120  may be determined using currents sensors  980  and  985  as illustrated in one of  FIGS. 9A  or  9 B. As illustrated, actual phase line voltage  1110  and derived phase line voltage  1120  are nearly identical prior to a de-energizing event, at  1130 . At de-energizing event  1130 , the voltage on the primary winding of a CCVT would become zero due to the filtering effects of the primary capacitive assembly. Accordingly, a voltage derived from the output (secondary winding) of the CCVT would indicate that the phase line had a zero-voltage trapped charge following a de-energization event. In contrast, derived phase line voltage  1120 , derived using measurements taken from two current sensors, accurately illustrates a non-zero-voltage trapped charge following the de-energizing event, at  1130 . 
     By contrasting  FIG. 6  with  FIG. 11 , it can be seen that while the output voltage of the CCVT cannot be used to accurately derive the trapped charge on a phase line ( 1110  in  FIG. 6 ), using the current measured (or derived) through the primary and auxiliary capacitive assemblies of the CCVT can be used to accurately calculate the trapped charge on a phase line. Specifically, using the output voltage of the CCVT is unsuitable because of the poor transient response of CCVTs, due in part to the primary coupling capacitor acting as a DC filter. In contrast, and as illustrated in  FIG. 11 , the voltage calculated using the current measured (or derived) through the primary and auxiliary capacitive assemblies of the CCVT can be used to accurately find the trapped charge on a de-energized phase line. 
     The above description provides numerous specific details for a thorough understanding of the embodiments described herein. However, those of skill in the art will recognize that one or more of the specific details may be omitted, modified, and/or replaced by a similar process or system.