# Patent

## Title
Corrective device protection

## Abstract

A control system for an electrical power system includes an electrical corrective device, a voltage measuring device coupled to each phase of the electric power system, a current measuring device connected between each phase of the electric power system and the electrical corrective device, and a protective device connected to outputs of the voltage measuring device and the current measuring device. The protective device includes a controller configured to detect an imbalance using the measured voltages and currents output from the voltage measuring device and the current measuring device without using or independently of other recent voltage or current measurements for other points in the electrical corrective device and independently of a neutral-ground measurement at the electrical corrective device.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims priority to U.S. Application No. 60/619,032, filed Oct. 18, 2004, which is incorporated herein by reference in its entirety.


TECHNICAL FIELD
This document relates to protecting corrective devices in electric power systems.
BACKGROUND
Utilities often connect corrective devices such as reactor or capacitor banks to an electric power system to reduce system losses, neutralize inductive reactance, or improve voltage regulation.
SUMMARY
In one general aspect, a method is performed for detecting imbalances in an electrical corrective device connected to an electric power system. The method includes measuring a voltage of each phase of the electric power system, measuring a current between each phase of the electric power system and the electrical corrective device, and detecting an imbalance in the electrical corrective device. The imbalance is detected using the measured voltages and currents independently of other recent voltage or current measurements for other points in the electrical corrective device and independently of a neutral-ground measurement at the electrical corrective device.
Implementations may include one or more of the following features. For example, the electrical corrective device may include a reactor bank. The electrical corrective device may include a capacitor bank.
Detecting an imbalance may include determining a negative-sequence current using the measured values and comparing the negative-sequence current with a predetermined negative-sequence current calculated for an undamaged electrical corrective device.
The electrical corrective device may be shunt connected. The electrical corrective device may be ungrounded. The electrical corrective device may include a bank of elements that are arranged in a wye configuration. The electrical corrective device may be ungrounded or grounded.
The method may also include activating a switching device to isolate the electrical corrective device from the electric power system when an imbalance is detected.
Detecting the imbalance may include compensating for errors in the measured voltages and currents due to ambient temperature changes.
The method may include determining in which phase the imbalance occurs.
In another general aspect, a control system for an electrical power system includes an electrical corrective device, a voltage measuring device, a current measuring device, and a protective device. The voltage measuring device is connected to measure voltages of each phase of the electric power system. The current measuring device is connected to measure current between each phase of the electric power system and the electrical corrective device. The protective device is connected to outputs of the voltage measuring device and the current measuring device. The protective device includes a controller configured to detect an imbalance using the measured voltages and currents output from the voltage measuring device and the current measuring device independently of other recent voltage or current measurements for other points in the electrical corrective device and independently of a neutral-ground measurement at the electrical corrective device.
Implementations may include one or more of the following features. For example, the electrical corrective device may include a capacitor bank or a reactor bank. The electrical corrective device may be shunt connected. The electrical corrective device may be ungrounded. The electrical corrective device may include a bank of elements that are arranged in a wye configuration. The electrical corrective device may be grounded.
The control system may include a switching device in each phase of the electric power system, and the electrical corrective device may be connected to the switching device to activate the switching device to isolate the electrical corrective device from the electric power system when an imbalance is detected.
In another general aspect, a system for detecting imbalances in an electrical corrective device connected to an electric power system includes a means for measuring a voltage of each phase of the electric power system, a means for measuring a current between each phase of the electric power system and the electrical corrective device, and a means for detecting an imbalance. The means for detecting the imbalance uses the measured voltages and currents independently of other recent voltage or current measurements for other points in the electrical corrective device and independently of a neutral-ground measurement at the electrical corrective device.
In another general aspect, a method is performed for detecting imbalances in an electrical corrective device connected to an electric power system. The method includes measuring a voltage VA, VB, and VC of each phase A, B, C of the electric power system; measuring a current IA, IB, and IC between each phase of the electric power system and the electrical corrective device; and calculating positive-sequence, negative-sequence, and zero-sequence device currents I1, I2, and I0, respectively, and positive-sequence, negative-sequence, and zero-sequence voltages V1, V2, and V0, respectively, on the basis of the currents IA, IB, and IC and the voltages VA, VB, and VC. The method further includes determining a predetermined negative-sequence current I2un of an undamaged electrical corrective device; calculating a negative-sequence current difference by comparing the calculated negative-sequence current I2 with the predetermined negative-sequence current I2un; and de-energizing the electrical corrective device if I2−I2un exceeds a predetermined threshold.
Implementations can include one or more of the following features. For example, the predetermined negative-sequence current I2un may be calculated based on the positive-sequence and negative-sequence voltages V1 and V2, respectively, at the electrical corrective device and the positive-sequence current I1 at the electrical corrective device for an ungrounded electrical corrective device.
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and where V1C, I1C, V2C, and I2C are, respectively, a positive-sequence voltage, a positive-sequence current, a negative-sequence voltage, and a negative-sequence current that were measured when the electrical corrective device was commissioned; I1rC, I1iC are, respectively, real and imaginary components of I1C; V2rC, V2iC are, respectively, real and imaginary components of V2C; and I2rC, I2iC are, respectively, real and imaginary components of I2C.

The predetermined negative-sequence current I2un may be calculated based on the positive, negative, and zero-sequence voltages V1, V2, V0, respectively, at the electrical corrective device, and the positive-sequence current I1 at the electrical corrective device for a grounded electrical corrective device.
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for a grounded electrical corrective device.
In another general aspect, a method is performed for detecting imbalances in an electrical corrective device connected to an electric power system. The method includes measuring a current IA, IB, and IC between each phase of the electric power system and the electrical corrective device; calculating a negative-sequence current I2 on the basis of the currents IA, IB, and IC; determining a predetermined negative-sequence current I2un of an undamaged electrical corrective device; calculating a negative-sequence current difference by comparing the calculated negative-sequence current I2 with the predetermined negative-sequence current I2un; and de-energizing the electrical corrective device if I2−I2un exceeds a predetermined threshold.
For many applications, a double wye form of bank construction is employed. However, this form of bank construction may require more space to install and may be more expensive to fabricate than other configurations, such as a single wye configuration. The described control system and method allows the bank to be configured in a single wye without losing benefits of the double wye construction, and thereby reduces the mounting space in the substation and further lowers the overall cost of the bank.
For single wye systems, the inclusion of a neutral-ground measuring device such as a voltage transducer (for ungrounded banks) or a neutral-ground current transducer (for grounded banks) may be undesirable from a cost and reliability standpoint. The control system and method described below provide comparable degrees of bank protection, but eliminate the need for the neutral-ground measuring devices.
The described control system and method also provides a way of determining an internal imbalance in the bank without making a measurement internal to the bank. Such an internal imbalance determination is not possible by just measuring an output of the bank.
Other features will be apparent from the description, the drawings, and the claims.



DESCRIPTION OF DRAWINGS
FIG. 1 is a diagram of a control system including an ungrounded corrective device;
FIG. 2 is a diagram of a control system including a grounded corrective device.



Like reference symbols in the various drawings may indicate like elements.
DETAILED DESCRIPTION
Corrective devices such as reactor and capacitor banks can be connected in a “shunt” configuration (i.e., a line-neutral arrangement) rather than in a series, or in-line, connection. As a shunt-connected device, the bank may be grounded (i.e., such that the bank neutral point is connected to the power system ground with an intentionally low impedance tie) or ungrounded (i.e., such that there is no intentional connection between the bank neutral point and system ground).
Once the bank is energized, external protective systems are employed to ensure the voltages within the bank are acceptable so as not to overly stress individual reactor or capacitor components of the bank. This is of particular concern in capacitor banks that include dozens of interconnected individual components. If an individual component within the bank fails, internal bank voltages would be altered and increase across the remaining functioning units. If this elevated voltage stress is not detected, and, when necessary, the entire bank is not de-energized in response, internal cascading and often catastrophic failures may result. One way to determine whether potentially damaging voltage levels exist is to measure the voltage stress across each internal component within the bank. However, this may be impractical due to the large number of components. Existing protection systems therefore employ indirect measurements, using voltages external to the bank and possibly current flow through the bank, in conjunction with well-understood equations, to estimate internal voltage levels.
For ungrounded banks, the failure of an individual component creates an imbalance that leads to elevated voltage stress within the bank, as mentioned above. This imbalance manifests as a shift in the neutral-ground voltage. By employing a voltage transducer at the neutral-to-ground location, the neutral voltage can be measured, and a protective system can be used to estimate the magnitude of internal voltage stress.
For grounded banks, the failure of an individual component also creates an imbalance that leads to elevated voltage stress within the bank. This imbalance manifests as a shift in the neutral-ground current. By employing a current transducer at the neutral-to-ground location, the neutral current can be measured, and a protective system can be used to estimate the magnitude of internal voltage stress. Regardless of the grounding configuration, the magnitude of estimated internal voltage stress is compared to thresholds, with time delays, to determine if and when the bank must be switched off from the power system.
Referring to FIGS. 1 and 2, a control system 100 or 200 detects abnormalities such as imbalances within ungrounded (FIG. 1) or grounded (FIG. 2) reactor or capacitor banks that are shunt-connected to an electric power system. The control systems 100 or 200 determine whether potentially damaging voltage levels exist without having to measure the voltage stress across one or more internal components within the bank, that is, independently of the voltage stress across one or more internal components within the bank and without use of a neutral-ground measuring device at the bank.
As noted above, in existing systems, when an ungrounded bank is arranged in a single electrical wye, the voltage between the ungrounded neutral of the bank and the power system ground is monitored. If the bank is arranged in a double wye, then either the current flowing between the neutral points of the two wyes (the “neutral-to-neutral current”) or the neutral-ground voltage is monitored. Typically, these signals are measured with a separate voltage or current transducer, which is an additional component of the power system that raises both procurement and operational costs, and may reduce reliability. Once quantified, the neutral-ground voltage or the neutral-to-neutral current is used to detect when potentially damaging imbalance conditions occur within the bank. Referring in particular to FIG. 1, in the control system 100, electrical signals external to the bank are still employed. However, these voltage and current signals are typically available for other metering and backup protection functions.
The control system 100 is designed for a three-phase (A, B, C) power system and includes a three-phase, high-voltage bus 105 shown at the connection point to a three-phase, ungrounded shunt reactor or capacitor bank 110. The system 100 also includes three current measuring devices such as current transducers 115 that provide currents IA, IB, and IC flowing from each phase of the bus 105 to the shunt bank 110. The current measurements provided by the transducers 115, in addition to being used by the control system 100, also may be used by metering equipment and backup protection in other parts of the power system.
The control system 100 also includes voltage measuring devices such as voltage transducers 120 that provide voltages VA, VB, and VC taken from each phase to ground for each phase of the bus 105. The voltage measurements provided by the voltage transducers 120, in addition to being used by the control system 100, also may be used by metering equipment and backup protection in other parts of the power system.
The control system 100 includes a switching device 125 that is used to connect the bank 110 to the power system as needed as well as to remove the bank 110 from service when damaging imbalances are detected within the system 100. The control system 100 also includes a computerized protective device 130 that performs calculations based on the measured values VA, VB, VC, IA, IB, and IC. The device 130 outputs a decision in the form of a control signal that is sent to the switching device 125 such that the bank 110 may be de-energized when damaging imbalances are detected. In some implementations, the protective device 130 may be a component of the switching device 125.
Referring to FIG. 2, a control system 200 for a three-phase (A, B, C) power system includes a three-phase, high-voltage bus 205 shown at the connection point to a three-phase, grounded shunt reactor or capacitor bank 210. Like the system 100, the system 200 includes three current measuring devices, such as current transducers 215 that provide currents IA, IB, and IC flowing from each phase of the bus 205 to the shunt bank 210. The control system 200 also includes voltage measuring devices, such as voltage transducers 220, that provide voltages VA, VB, and VC taken from each phase to ground for each phase of the bus 205.
The control system 200 includes a switching device 225 that is similar to the switching device 125 and is also used to connect the shunt bank 210 to the power system as needed as well as to remove the bank 210 from service when damaging imbalances are detected within the system 200. The control system 200 includes a computerized protective device 230 that performs calculations based on the measured values VA, VB, VC, IA, IB, and IC in a similar manner as the device 130.
In both of systems 100 and 200, the protective device 130 or 230 calculates positive-sequence, negative-sequence, and zero-sequence device currents I1, I2, and I0, respectively, and positive-sequence, negative-sequence, and zero-sequence voltages V1, V2, and V0, respectively, on the basis of the phase currents IA, IB, and IC and the phase-to-ground voltages VA, VB, and VC of the bank 110 or 210 using the following relationships:

I1=1⁄3×(IA+aIB+a2IC)

I2=1⁄3×(IA+a2IB+aIC)

I0=1⁄3×(IA+IB+IC)

V1=1⁄3×(VA+aVB+a2VC)

V2=1⁄3×(VA+a2VB+aVC)

V0=1⁄3×(VA+VB+VC)

where:





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is the Fortescue operator.

The protective device 130 or 230 performs a negative-sequence current difference calculation for imbalance protection of three-phase reactive devices connected in a single-wye configuration. The calculation is based on a comparison of the bank calculated negative-sequence current I2 (determined above based on the bank phase currents) with a predetermined negative-sequence current of the undamaged bank, I2un. The negative-sequence current I2un is calculated as if asymmetry exists only because of an external (system) voltage imbalance and/or because of the inherent phase imbalance due to manufacturing tolerance (i.e., not all phases of the bank include identical amounts of reactance). The vector difference IUN between the calculated signal I2 and the predetermined negative-sequence current I2un is proportional to the internal asymmetry of a protected bank due to internal faults (unit failures):

IUN=I2−I2un 

An internal fault that requires action such as de-energizing the entire bank exists when the vector difference IUN exceeds a predetermined threshold. As seen in the above equations, I2 is calculated based on measured bank phase currents IA, IB, and IC. The measured signal I2 may include the effects of false imbalance due to either imbalance bus voltages, inherent imbalances in the bank due to manufacturing tolerances, or variations in the bank's electrical properties (such as admittance) due to changes in ambient temperature. The negative-sequence current term I2un removes any such error signals from the calculated signal I2 and is calculated as shown below based on the bank being either ungrounded or grounded.
Calculation of I2un for an Ungrounded Bank
The predetermined negative-sequence current I2un is calculated on the basis of measured positive-sequence and negative-sequence voltages at the bank terminals V1 and V2, as well as on the basis of the measured positive-sequence bank current I1:




  
    
      I
      
        2
        ⁢
        un
      
    
    =
    
    ⁢
    
      
        [
        
          
            
              (
              
                
                  G
                  
                    21
                    ⁢
                    C
                  
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    
                      21
                      ⁢
                      C
                    
                  
                
              
              )
            
            ⁢
            
              V
              1
            
          
          +
          
            
              (
              
                
                  G
                  
                    11
                    ⁢
                    C
                  
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    
                      11
                      ⁢
                      C
                    
                  
                
              
              )
            
            ⁢
            
              V
              2
            
          
        
        ]
      
      ×
      
      ⁢
      
        
        
          
            I
            1
          
          
            
              V
              1
            
            ⁢
            
              
                
                  G
                  
                    11
                    ⁢
                    C
                  
                  2
                
                +
                
                  B
                  
                    11
                    ⁢
                    C
                  
                  2
                
              
            
          
        
        
      
    
  
  ,




where G11C+jB11C and G21C+jB21C are elements of the sequence admittance matrix of the bank 110 or 210 existing when the bank was being commissioned. These values are kept in non-volatile memory of the protective device 130 or 230. In the notation above, as well as in that to follow, the subscript “C” denotes measured or derived quantities available when the bank is commissioned or initially placed in service.

The matrix elements can be calculated on the basis of a positive-sequence voltage V1C, a negative-sequence voltage V2C, a positive-sequence current I1C, and a negative-sequence current I2C that were measured when the bank 110 or 210 was commissioned. Assuming that the phasor V1C coincides with the real axis, the measured symmetrical components can be presented in algebraic form as follows:

V1C=V1C, V2C=V2rC+jV2iC,

I1C=I1rC+jI1iC, I2C=I2rC+jI2iC,

The subscripts “i” and “r” denote, respectively, the imaginary and real parts of the measured components, and V1C, V2rC, V2iC, I1rC, I1iC, I2rC, and I2iC are real numbers.

The parameters G11C, B11C, G21C and B21C can be derived from the following equations:




  
    
      G
      
        11
        ⁢
        C
      
    
    =
    
      
        
          
            V
            
              1
              ⁢
              C
            
          
          ⁢
          
            I
            
              1
              ⁢
              rC
            
          
        
        +
        
          
            V
            
              2
              ⁢
              rC
            
          
          ⁢
          
            I
            
              2
              ⁢
              rC
            
          
        
        +
        
          
            V
            
              2
              ⁢
              iC
            
          
          ⁢
          
            I
            
              2
              ⁢
              iC
            
          
        
      
      
        
          V
          
            1
            ⁢
            C
          
          2
        
        +
        
          (
          
            
              V
              
                2
                ⁢
                rC
              
              2
            
            +
            
              V
              
                2
                ⁢
                iC
              
              2
            
          
          )
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      B
      
        11
        ⁢
        C
      
    
    =
    
      
        
          
            V
            
              1
              ⁢
              C
            
          
          ⁢
          
            I
            
              1
              ⁢
              iC
            
          
        
        +
        
          
            V
            
              2
              ⁢
              iC
            
          
          ⁢
          
            I
            
              2
              ⁢
              rC
            
          
        
        -
        
          
            V
            
              2
              ⁢
              rC
            
          
          ⁢
          
            I
            
              2
              ⁢
              iC
            
          
        
      
      
        
          V
          
            1
            ⁢
            C
          
          2
        
        -
        
          (
          
            
              V
              
                2
                ⁢
                rC
              
              2
            
            +
            
              V
              
                2
                ⁢
                iC
              
              2
            
          
          )
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      G
      
        21
        ⁢
        C
      
    
    =
    
      
        
          I
          
            2
            ⁢
            rC
          
        
        -
        
          
            G
            
              11
              ⁢
              C
            
          
          ⁢
          
            V
            
              2
              ⁢
              rC
            
          
        
        +
        
          
            B
            
              11
              ⁢
              C
            
          
          ⁢
          
            V
            
              2
              ⁢
              iC
            
          
        
      
      
        V
        
          1
          ⁢
          C
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      B
      
        21
        ⁢
        C
      
    
    =
    
      
        
          
            I
            
              2
              ⁢
              iC
            
          
          -
          
            
              G
              
                11
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                iC
              
            
          
          -
          
            
              B
              
                11
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                rC
              
            
          
        
        
          V
          
            1
            ⁢
            C
          
        
      
      .
    
  




As an additional benefit of these techniques, the negative-sequence current of the undamaged bank I2un takes into account changes of the bank reactance with temperature. In particular, the




  
  
    
      I
      1
    
    
      
        V
        1
      
      ⁢
      
        
          
            G
            
              11
              ⁢
              C
            
            2
          
          +
          
            B
            
              11
              ⁢
              C
            
            2
          
        
      
    
  
  




term of the equation used to determine I2un performs automatic temperature compensation. This benefit is most desired when protecting shunt capacitor banks that exhibit a pronounced temperature dependency. Without the temperature compensation, the techniques could yield false imbalance signals, possibly leading to undesired de-energizing of the bank, occurring under large variations of ambient temperature.

Calculation of I2un for a Grounded Bank

The predetermined negative-sequence current I2un is calculated on the basis of measured positive, negative, and zero-sequence voltages at the bank terminals (V1, V2, and V0) as well as on the basis of measured positive-sequence bank current I1:




  
    I
    
      2
      ⁢
      un
    
  
  =
  
    
      I
      
        2
        ⁢
        un
      
    
    =
    
      
        [
        
          
            
              Y
              
                10
                ⁢
                C
              
            
            ⁢
            
              V
              1
            
          
          +
          
            
              Y
              
                11
                ⁢
                C
              
            
            ⁢
            
              V
              2
            
          
          +
          
            
              Y
              
                12
                ⁢
                C
              
            
            ⁢
            
              V
              0
            
          
        
        ]
      
      ×
      
        
        
          
            I
            1
          
          
            
              V
              1
            
            ⁢
            
              Y
              
                11
                ⁢
                C
              
            
          
        
        
      
    
  




The “C” subscript indicates quantities measured or calculated when the bank is initially placed into service. This equation accounts for deviations of matrix elements from the measured ones during the bank commissioning that may take place due to temperature variations by including the factor




  
  
    
      I
      1
    
    
      
        V
        1
      
      ⁢
      
        Y
        
          11
          ⁢
          C
        
      
    
  
  




. The Ynn elements appearing above are calculated using the expression

Y11C=g11I1C+g12I2C+g13I0C,

Y12C=g12I1C+g13I2C+g11I0C,

Y10C=g13I1C+g11I2C+g12I0C.

The gnn elements appearing above are calculated using the expression





  
    
      g
      11
    
    =
    
      
        
          
            (
            
              V
              
                1
                ⁢
                C
              
            
            )
          
          2
        
        -
        
          
            V
            
              2
              ⁢
              C
            
          
          ⁢
          
            V
            
              0
              ⁢
              C
            
          
        
      
      
        
          
            (
            
              V
              
                1
                ⁢
                C
              
            
            )
          
          3
        
        +
        
          
            (
            
              V
              
                2
                ⁢
                C
              
            
            )
          
          3
        
        +
        
          
            (
            
              V
              
                0
                ⁢
                C
              
            
            )
          
          3
        
        -
        
          3
          ⁢
          
            V
            
              1
              ⁢
              C
            
          
          ⁢
          
            V
            
              2
              ⁢
              C
            
          
          ⁢
          
            V
            
              0
              ⁢
              C
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      g
      12
    
    =
    
      
        
          
            (
            
              V
              
                2
                ⁢
                C
              
            
            )
          
          2
        
        -
        
          
            V
            
              1
              ⁢
              C
            
          
          ⁢
          
            V
            
              0
              ⁢
              C
            
          
        
      
      
        
          
            (
            
              V
              
                1
                ⁢
                C
              
            
            )
          
          3
        
        +
        
          
            (
            
              V
              
                2
                ⁢
                C
              
            
            )
          
          3
        
        +
        
          
            (
            
              V
              
                0
                ⁢
                C
              
            
            )
          
          3
        
        -
        
          3
          ⁢
          
            V
            
              1
              ⁢
              C
            
          
          ⁢
          
            V
            
              2
              ⁢
              C
            
          
          ⁢
          
            V
            
              0
              ⁢
              C
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      g
      13
    
    =
    
      
        
          
            
              (
              
                V
                
                  0
                  ⁢
                  C
                
              
              )
            
            2
          
          -
          
            
              V
              
                1
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                C
              
            
          
        
        
          
            
              (
              
                V
                
                  1
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  2
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  0
                  ⁢
                  C
                
              
              )
            
            3
          
          -
          
            3
            ⁢
            
              V
              
                1
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                C
              
            
            ⁢
            
              V
              
                0
                ⁢
                C
              
            
          
        
      
      .
    
  




Determination of Faulted Phase

When elevated levels of IUN indicate a unit failure internal to the bank, the protective device 130 or 230 may command the switching device 125 or 225 to de-energize the bank to prevent voltage stress on the bank's remaining healthy elements. To aid in identifying the failed unit(s) for replacement, it is helpful to not merely indicate the existence of an imbalance problem but also to isolate the failure to the involved phase, that is, A, B, or C. Referring to the table below, for example, the angular relationship between IUN and V1 complex values for a capacitor bank shows in which phase the imbalance has occurred as detailed in the table below.











Phase with failed
Phase angle φ of the imbalance


capacitor elements or
current IUN related to the voltage V1 [Degrees]









capacitor units
Fused Capacitors
Fuseless Capacitors





A
 −80° < φ < −100°
 80° < φ < 100°


B
 20° < φ < .40°
−140° < φ < −160°


C
140° < φ < 160°
−20° < φ < −40°









These relationships can be determined for reactor banks also.

Other implementations are within the scope of the following claims.

## Claims

1. A method of detecting imbalances in an electrical corrective device, connected to an electric power system, the method comprising:
measuring a voltage VA, VB, and VC of each phase A, B, C of an electric power system;
measuring a current IA, IB, and IC between each phase A, B, C of the electric power system and an electrical corrective device connected to the electric power system;
calculating positive-sequence, negative-sequence, and zero-sequence device currents I1, I2, and I0, respectively, and positive-sequence, negative-sequence, and zero-sequence voltages V1, V2, and V0, respectively, on the basis of the currents IA, IB, and IC and the voltages VA, VB, and VC;
determining a predetermined negative-sequence current I2un of an undamaged electrical corrective device;
calculating a negative-sequence current difference by comparing the calculated negative-sequence current I2 with the predetermined negative-sequence current I2un;
comparing the negative-sequence current difference to a predetermined threshold;
if the negative-sequence current difference exceeds the predetermined threshold, determining that an imbalance exists in the electrical corrective device; and
de-energizing the electrical corrective device when an imbalance exists in the electrical corrective device.

2. The method of claim 1, wherein the predetermined negative-sequence current I2un is calculated based on the positive-sequence and negative-sequence voltages V1, and V2, respectively, at the electrical corrective device and the positive-sequence current I1 at the electrical corrective device for an ungrounded electrical corrective device.

3. The method of claim 1, wherein:




  
    
      I
      
        2
        ⁢
        un
      
    
    =
    
    ⁢
    
      
        [
        
          
            
              (
              
                
                  G
                  
                    21
                    ⁢
                    C
                  
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    
                      21
                      ⁢
                      C
                    
                  
                
              
              )
            
            ⁢
            
              V
              1
            
          
          +
          
            
              (
              
                
                  G
                  
                    11
                    ⁢
                    C
                  
                
                +
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  
                    B
                    
                      11
                      ⁢
                      C
                    
                  
                
              
              )
            
            ⁢
            
              V
              2
            
          
        
        ]
      
      ×
      
      ⁢
      
        
        
          
            I
            1
          
          
            
              V
              1
            
            ⁢
            
              
                
                  G
                  
                    11
                    ⁢
                    C
                  
                  2
                
                +
                
                  B
                  
                    11
                    ⁢
                    C
                  
                  2
                
              
            
          
        
        
      
    
  
  ,




for an ungrounded electrical corrective device, wherein:



  
    
      
        G
        
          11
          ⁢
          C
        
      
      =
      
        
          
            
              V
              
                1
                ⁢
                C
              
            
            ⁢
            
              I
              
                1
                ⁢
                rC
              
            
          
          +
          
            
              V
              
                2
                ⁢
                rC
              
            
            ⁢
            
              I
              
                2
                ⁢
                
                  
                
                ⁢
                rC
              
            
          
          +
          
            
              V
              
                2
                ⁢
                
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              I
              
                2
                ⁢
                
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                C
              
            
          
        
        
          
            V
            
              1
              ⁢
              
                
              
              ⁢
              C
            
            2
          
          +
          
            (
            
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
                2
              
              +
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
                2
              
            
            )
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        B
        
          11
          ⁢
          C
        
      
      =
      
        
          
            
              V
              
                1
                ⁢
                C
              
            
            ⁢
            
              I
              
                1
                ⁢
                iC
              
            
          
          +
          
            
              V
              
                2
                ⁢
                iC
              
            
            ⁢
            
              I
              
                2
                ⁢
                
                  
                
                ⁢
                rC
              
            
          
          +
          
            
              V
              
                2
                ⁢
                
                  
                
                ⁢
                r
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              I
              
                2
                ⁢
                
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                C
              
            
          
        
        
          
            V
            
              1
              ⁢
              
                
              
              ⁢
              C
            
            2
          
          -
          
            (
            
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
                2
              
              +
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
                2
              
            
            )
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        G
        
          21
          ⁢
          C
        
      
      =
      
        
          
            I
            
              2
              ⁢
              rC
            
          
          -
          
            
              G
              
                11
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                
                  
                
                ⁢
                rC
              
            
          
          +
          
            
              B
              
                11
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                
                  
                
                ⁢
                iC
              
            
          
        
        
          V
          
            1
            ⁢
            
              
            
            ⁢
            C
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        B
        
          21
          ⁢
          C
        
      
      =
      
        
          
            I
            
              2
              ⁢
              iC
            
          
          -
          
            
              G
              
                11
                ⁢
                C
              
            
            ⁢
            
              
                V
                
                  2
                  ⁢
                  iC
                
              
              
                
              
            
          
          -
          
            
              B
              
                11
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                r
                ⁢
                
                  
                
                ⁢
                C
              
            
          
        
        
          
            
              V
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
            
              
            
          
          
            
          
        
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      
        
          
            
              V
              
                1
                ⁢
                C
              
            
            =
            
              V
              
                1
                ⁢
                C
              
            
          
          ,
        
      
      
        
          
            
              V
              
                2
                ⁢
                C
              
            
            =
            
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  rC
                
              
              +
              
                j
                ⁢
                
                  
                
                ⁢
                
                  V
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
              
            
          
          ,
        
      
    
    
      
        
          
            
              I
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
            =
            
              
                I
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  rC
                
              
              +
              
                jI
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  iC
                
              
            
          
          ,
        
      
      
        
          
            
              I
              
                2
                ⁢
                C
              
            
            =
            
              
                I
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  rC
                
              
              +
              
                j
                ⁢
                
                  
                
                ⁢
                
                  I
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
              
            
          
          ,
        
      
    
  



where V1C, I1C, V2C, and I2C are, respectively, a positive-sequence voltage, a positive-sequence current, a negative-sequence voltage, and a negative-sequence current that were measured when the electrical corrective device was commissioned; I1rC, I1iC are, respectively, real and imaginary components of I1C; V2rC, V2iC are, respectively, real and imaginary components of V2C; and I2rC, I2iC are, respectively, real and imaginary components of I2C.

4. The method of claim 1, wherein the predetermined negative-sequence current I2un is calculated based on the positive, negative, and zero-sequence voltages V1, V2, V0, respectively, at the electrical corrective device, and the positive-sequence current I1 at the electrical corrective device for a grounded electrical corrective device.

5. The method of claim 1, wherein:



  
    
      
        I
        
          2
          ⁢
          
            
          
          ⁢
          un
        
      
      =
      
        
          I
          
            2
            ⁢
            
              
            
            ⁢
            un
          
        
        =
        
          
            [
            
              
                
                  Y
                  
                    10
                    ⁢
                    C
                  
                
                ⁢
                
                  V
                  1
                
              
              +
              
                
                  Y
                  
                    11
                    ⁢
                    C
                  
                
                ⁢
                
                  V
                  2
                
              
              +
              
                
                  Y
                  
                    12
                    ⁢
                    C
                  
                
                ⁢
                
                  V
                  0
                
              
            
            ]
          
          ×
          
            
            
              
                I
                1
              
              
                
                  V
                  1
                
                ⁢
                
                  Y
                  
                    11
                    ⁢
                    C
                  
                
              
            
            
          
        
      
    
    ;
  




  
    
      
        Y
        
          11
          ⁢
          C
        
      
      =
      
        
          
            g
            11
          
          ⁢
          
            I
            
              1
              ⁢
              
                
              
              ⁢
              C
            
          
        
        +
        
          
            g
            12
          
          ⁢
          
            I
            
              2
              ⁢
              
                
              
              ⁢
              C
            
          
        
        +
        
          
            g
            13
          
          ⁢
          
            I
            
              0
              ⁢
              C
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        Y
        
          12
          ⁢
          C
        
      
      =
      
        
          
            g
            12
          
          ⁢
          
            I
            
              1
              ⁢
              
                
              
              ⁢
              C
            
          
        
        +
        
          
            g
            13
          
          ⁢
          
            I
            
              2
              ⁢
              
                
              
              ⁢
              C
            
          
        
        +
        
          
            g
            11
          
          ⁢
          
            I
            
              0
              ⁢
              C
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        
          Y
          
            10
            ⁢
            C
          
        
        =
        
          
            
              g
              13
            
            ⁢
            
              I
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
          
          +
          
            
              g
              11
            
            ⁢
            
              I
              
                2
                ⁢
                
                  
                
                ⁢
                C
              
            
          
          +
          
            
              g
              12
            
            ⁢
            
              I
              
                0
                ⁢
                C
              
            
          
        
      
      ;
    
  




  and




  
    
      
        g
        11
      
      =
      
        
          
            
              (
              
                V
                
                  1
                  ⁢
                  C
                
              
              )
            
            2
          
          -
          
            
              V
              
                2
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              V
              
                0
                ⁢
                C
              
            
          
        
        
          
            
              (
              
                V
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  0
                  ⁢
                  C
                
              
              )
            
            3
          
          -
          
            3
            ⁢
            
              V
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                C
              
            
            ⁢
            
              V
              
                0
                ⁢
                C
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        g
        12
      
      =
      
        
          
            
              (
              
                V
                
                  2
                  ⁢
                  C
                
              
              )
            
            2
          
          -
          
            
              V
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              V
              
                0
                ⁢
                C
              
            
          
        
        
          
            
              (
              
                V
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              )
            
            3
          
          +
          
            
              (
              
                V
                
                  0
                  ⁢
                  C
                
              
              )
            
            3
          
          -
          
            3
            ⁢
            
              V
              
                1
                ⁢
                
                  
                
                ⁢
                C
              
            
            ⁢
            
              V
              
                2
                ⁢
                C
              
            
            ⁢
            
              V
              
                0
                ⁢
                C
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        
          g
          13
        
        =
        
          
            
              
                (
                
                  V
                  
                    0
                    ⁢
                    C
                  
                
                )
              
              2
            
            -
            
              
                V
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              ⁢
              
                V
                
                  2
                  ⁢
                  C
                
              
            
          
          
            
              
                (
                
                  V
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
                )
              
              3
            
            +
            
              
                (
                
                  V
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    C
                  
                
                )
              
              3
            
            +
            
              
                (
                
                  V
                  
                    0
                    ⁢
                    C
                  
                
                )
              
              3
            
            -
            
              3
              ⁢
              
                V
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  C
                
              
              ⁢
              
                V
                
                  2
                  ⁢
                  C
                
              
              ⁢
              
                V
                
                  0
                  ⁢
                  C
                
              
            
          
        
      
      ;
    
  



for a grounded electrical corrective device.

6. The method of claim 1, wherein the negative-sequence current difference is I2−I2un.

7. The method of claim 6, wherein I2−I2un is proportional to an internal asymmetry of the electrical corrective device caused by a failure of a discrete component included in the electrical corrective device.

8. The method of claim 1, wherein the imbalance in the electrical corrective device exists due to a failure of a discrete corrective component included in the electrical corrective device.

9. The method of claim 1, wherein the electrical corrective device includes a capacitor bank.

10. The method of claim 1, wherein the electrical corrective device includes a reactor bank.

11. The method of claim 1, wherein the electrical corrective device is shunt connected.

12. The method of claim 1, further comprising:
if the predetermined threshold is exceeded, determining in which phase A, B, C the imbalance occurred.

13. The method of claim 12, wherein determining in which phase A, B, C the imbalance occurred comprises accessing data that defines a correlation between a phase angle of the imbalance and the positive-sequence voltage.

14. A method of detecting imbalances in an electrical corrective device connected to an electric power system, the method comprising:
measuring a current IA, IB, and IC between each phase of an electric power system and an electrical corrective device connected to the electric power system;
calculating a negative-sequence current I2 on the basis of the currents IA, IB, and IC;
determining a predetermined negative-sequence current I2un of an undamaged electrical corrective device;
calculating a negative-sequence current difference by comparing the calculated negative-sequence current I2 with the predetermined negative-sequence current I2un;
comparing the negative-sequence current difference to a predetermined threshold;
if the negative-sequence current difference exceeds the predetermined threshold, determining that an imbalance exists in the electrical corrective device; and
de-energizing the electrical corrective device when an imbalance exists in the electrical corrective device.

15. The method of claim 14, wherein the electrical corrective device includes a capacitor bank.

16. The method of claim 14, wherein the electrical corrective device includes a reactor bank.

17. The method of claim 14, wherein the electrical corrective device is shunt connected.

18. The method of claim 14, wherein the negative-sequence current difference is I2−I2un.

19. The method of claim 18, wherein I2−I2un is proportional to an internal asymmetry of the electrical corrective device caused by a failure of a discrete component included in the electrical corrective device.

20. The method of claim 14, wherein the imbalance in the electrical corrective device exists due to a failure of a discrete component included in the electrical corrective device.