# Patent

## Title
Monitoring voltage stability of a transmission corridor

## Abstract

A voltage stability monitoring apparatus monitors the voltage stability of a transmission corridor through which power flows between different parts of a power system. The apparatus monitors an equivalent load impedance at an interface between the transmission corridor and a part of the power system designated as generating the power. This equivalent load impedance at the interface comprises a ratio of a voltage phasor at the interface to a current phasor at the interface. The apparatus tracks a Thevenin equivalent voltage and impedance of the designated part by separately updating that voltage and impedance. Notably, the apparatus updates the imaginary part of the Thevenin equivalent voltage to reflect the magnitude of any changes in the voltage phasor that are associated with large variations in the magnitude of the equivalent load impedance at the interface. The apparatus computes an index indicating the voltage stability as a function of this tracked Thevenin equivalent voltage and impedance.

## Background
RELATED APPLICATIONS
The present application is a continuation of U.S. patent application Ser. No. 14/281,336, filed May 19, 2014, which application is a continuation-in-part of U.S. patent application Ser. No. 14/071,167, filed Nov. 4, 2013, which claims the benefit of U.S. Provisional Application No. 61/825,121, filed May 20, 2013, the disclosures of which are incorporated herein by reference in their entirety.


TECHNICAL FIELD
The present invention relates generally to monitoring the voltage stability of a transmission corridor through which power flows between different parts of a power system, and in particular relates to computing an index indicating such voltage stability.
BACKGROUND
A power system generates electric power at one part of the system and transmits that power via a transmission corridor for use by another part of the system. The transmission corridor is considered to be stable in terms of voltage if the corridor maintains steady acceptable voltages not only under normal operating conditions but also after a disturbance to the system (e.g., a line outage). A voltage stable corridor therefore regains acceptable voltages after a disturbance, rather than oscillating or monotonically decreasing even in response to attempted voltage restoration mechanisms.
Transmission corridor voltage instability causes power blackouts and therefore has huge economic and societal costs. Known approaches to preventing blackouts monitor the corridor's real-time proximity to voltage instability and take appropriate control and protective actions as needed to mitigate system degradation or disturbance propagation. For example, such actions may include load shedding.
Many of these known approaches exploit the relation of the corridor's voltage instability to the power system's maximum loadability. In particular, the approaches identify the corridor's voltage instability as being strongly related to the inability of the combined generation and transmission parts of the system to provide the power requested by the receiving (i.e., load) part of the system. The approaches therefore employ a voltage instability criterion expressed directly or indirectly in terms of the system's maximum deliverable power, which is reached when the magnitude of the Thevenin equivalent impedance of the combined generation and transmission parts of the system equals the magnitude of the equivalent load impedance of the power receiving part: |ZTh|=|Zl|.
At least some of these approaches estimate the Thevenin equivalent impedance of the combined generation and transmission parts of the system in stages. Such multi-stage approaches involve estimating the power generating part's Thevenin equivalent. Some known techniques for “estimating” the power generating part's Thevenin equivalent simply assume that either the Thevenin equivalent voltage or impedance of the power generating part is known. See U.S. Pat. No. 7,200,500 B2, April 2007, which is incorporated by reference herein in its entirety. Other techniques actually identify (i.e., track) the power generating part's Thevenin equivalent in the interest of accuracy, i.e., without making the above assumption. One such tracking technique performs recursive least squares using voltage and current phasor measurements taken at an interface between the power generating part and the transmission corridor. See U.S. Pat. No. 6,219,591 B1, which is incorporated by reference herein in its entirety. To avoid delays associated with the recursive least squares technique, an alternative tracking technique separately updates the power generating part's Thevenin equivalent voltage and impedance. S. Corsi and G. N. Taranto, “A real-time voltage instability identification algorithm based on local phasor measurements,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1271-1279, August 2008.
SUMMARY
One or more embodiments herein track the Thevenin equivalent of a power generating part of a power system with improved accuracy as compared to known tracking techniques. This improved accuracy advantageously prevents or at least mitigates false alarms in terms of prematurely detecting transmission corridor voltage instability.
More particularly, embodiments herein include a method of monitoring voltage stability of a transmission corridor through which power flows between different parts of a power system. The method is implemented by a voltage stability monitoring apparatus. The method includes monitoring an equivalent load impedance at an interface between the transmission corridor and a part of the power system designated as generating the power. The equivalent load impedance at this “power generating part” interface comprises a ratio of a voltage phasor at the interface to a current phasor at the interface.
The method further includes tracking a Thevenin equivalent voltage and impedance of the designated part by separately updating that voltage and impedance. Updating the Thevenin equivalent voltage in this regard comprises updating the voltage to reflect the magnitude of any changes in the voltage phasor that are associated with large variations in the magnitude of the equivalent load impedance at the power generating part interface. Such large variations include variations greater than a threshold-defined variation.
The method finally includes computing an index indicating the voltage stability as a function of the tracked Thevenin equivalent voltage and impedance.
In at least some embodiments, updating the Thevenin equivalent voltage comprises, for each of a plurality of phasor measurement times, determining whether or not variation in the magnitude of the equivalent load impedance at the power generating part interface since a previous phasor measurement time is greater than the threshold-defined variation. If so, the method comprises adjusting the Thevenin equivalent voltage computed for the previous phasor measurement time by the magnitude of the change in the voltage phasor since that previous phasor measurement time.
Thevenin equivalent voltage in some embodiments is also updated responsive to small variations in the magnitude of the equivalent load impedance at the power generating part interface. Specifically, the Thevenin equivalent voltage is decreased or increased by a predefined percentage change when those small variations do or do not have the same polarity as variations in the Thevenin equivalent impedance, respectively. Such small variations include variations less than the threshold-defined variation.
In some embodiments, updating the Thevenin equivalent voltage comprises updating the Thevenin equivalent voltage's complex value in rectangular coordinates. Alternatively or additionally, the method includes dynamically adjusting a threshold defining the threshold-defined variation, as a function of the Thevenin equivalent voltage.
Additionally, in one or more embodiments, updating the Thevenin equivalent impedance comprises solving a set of two linear equations with two unknown variables. These two unknown variables comprise the real and imaginary parts of the Thevenin equivalent impedance. Known variables in the set of two linear equations include the real and imaginary parts of the Thevenin equivalent voltage, as updated to reflect the magnitude of any changes in the voltage phasor at the power generating part interface.
Still further, the method in some embodiments entails dynamically adapting which part of the power system is designated as generating the power. Such dynamic adaptation is performed responsive to detecting a change in direction or magnitude of power flowing through one or both interfaces between the transmission corridor and the parts of the power system.
Finally, in one or more embodiments, the method involves monitoring whether a breaker for each line associated with the power generating part interface is open or closed. In this case, monitoring the equivalent load impedance at that interface comprises dynamically computing the equivalent load impedance at the interface exclusively from phasor measurements taken at lines whose breakers are closed. In some of these embodiments, responsive to detecting the opening or closing of one or more of these breakers, the method entails updating the Thevenin equivalent voltage to reflect the magnitude of the resulting change in the voltage phasor, as dynamically computed, without re-initializing the Thevenin equivalent voltage.
Embodiments herein further include a voltage stability monitoring apparatus configured to perform the above-described method.
Of course, the present invention is not limited to the above features and advantages. Indeed, those skilled in the art will recognize additional features and advantages upon reading the following detailed description, and upon viewing the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram of a power system and a voltage stability monitoring apparatus according to one or more embodiments.
FIG. 2 is a logic flow diagram of a method implemented by a voltage stability monitoring apparatus according to one or more embodiments.
FIG. 3 is a schematic diagram of a circuit model of the power system according to one or more embodiments.
FIG. 4 is a logic flow diagram of a process for tracking the equivalent load impedance at the power generating part interface and for updating the Thevenin equivalent voltage at that interface, according to one or more embodiments.
FIG. 5 is a schematic diagram of a power system circuit model which illustrates the equivalent load impedance at the power generating part interface, according to one or more embodiments.
FIG. 6 is a schematic diagram of a power system circuit model which illustrates modeling the transmission corridor as a T-equivalent circuit, according to one or more embodiments.
FIG. 7 is a schematic diagram of a power system circuit model which illustrates modeling the combined generation-transmission parts of the system as a Thevenin equivalent circuit, according to one or more embodiments.
FIG. 8 shows an exemplary transmission corridor for accounting for structural changes within the corridor, according to one or more embodiments.
FIG. 9 shows exemplary scenarios for dynamically adapting which part of the power system is designated as the power generating part as a function of aggregate active power, according to one or more embodiments.
FIG. 10A illustrates an exemplary corridor.
FIGS. 10B-C illustrate quantitative advantages of accounting for structural changes within the exemplary corridor of FIG. 10A, according to one or more embodiments, with FIG. 10B illustrating these advantages in terms of a ratio of impedance and FIG. 100 illustrating the advantages in terms of reactive power margin.
FIG. 11A illustrates a different exemplary corridor.
FIGS. 11B-C illustrate quantitative advantages of dynamically adapting which part of the exemplary corridor of FIG. 11A is designated as the power generating part, according to one or more embodiments, with FIG. 11B illustrating these advantages in terms of a ratio of impedance and FIG. 110 illustrating the advantages in terms of reactive power margin.



DETAILED DESCRIPTION
FIG. 1 illustrates a power system 2 according to one or more embodiments. The power system 2 generates electric power at one part of the system 2 and transmits that power via one or more transmission lines 4 for use by another part of the system 2. As shown, for example, parts A, B, and C of the power system 2 are each characterized by some degree of power generation (G) and loading (L). But power in the aggregate flows in a direction from part A towards part B via the one or more transmission lines 4. Regardless of the actual power flow direction, though, such characterization of the power system 2 into different parts effectively means that different transfer cuts A and B define a (virtual) transmission corridor 6 between those parts.
The transmission corridor 6 is composed of the one or more physical transmission lines 4 via which power flows between the power system parts. FIG. 1 shows these transmission lines 4 interface parts A and B of the power system 2 at physical buses A1-A2 and B1-B3. This means that the corridor 6 interfaces parts A and B at interfaces IntA and IntB. As shown, interface IntA is a virtual bus comprising a group of physical buses A1-A2. Likewise, interface IntB as shown is a virtual bus comprising a group of physical buses B1-B3.
The transmission corridor 6 is considered to be stable in terms of voltage if the corridor 6 maintains steady acceptable voltages not only under normal operating conditions but also after a disturbance to the system 2 (e.g., a line outage). A voltage stable corridor therefore regains acceptable voltages after a disturbance, rather than oscillating or monotonically decreasing even in response to attempted voltage restoration mechanisms.
A voltage stability monitoring apparatus 8 is configured to monitor the voltage stability of the transmission corridor 6, in order to detect in real-time the proximity of the corridor 6 to voltage instability. The apparatus 8 in this regard comprises one or more communication interface circuits 10 configured to communicatively couple the apparatus 8 to a plurality of time synchronized phasor measurement units (PMUs) 12 deployed at both ends of the corridor 6. Any given PMU 8 deployed at an interface between the corridor 6 and a part of the system 2 measures a voltage phasor and/or a current phasor locally at that interface, and communicates those phasor measurements to the apparatus 8 (e.g., at a rate of 10-120 samples per second). For example, PMUs 12 deployed at IntA measure voltage phasors at and current phasors in transmission lines 4 associated with physical buses A1-A2.
The voltage stability monitoring apparatus 8 also comprises one or more processing circuits 14 configured to use the received phasor measurements to compute an index indicating the corridor's voltage stability. The one or more processing circuits 14 do so according to the processing 100 illustrated in FIG. 2.
As shown in FIG. 2, processing 100 by the one or more processing circuits 14 includes monitoring an equivalent load impedance at an interface Intg between the transmission corridor 6 and a part of the power system designated as generating said power (e.g., IntA in FIG. 1) (Block 110). This interface Intg is also referred to herein as the “power generating part interface” for convenience. The equivalent load impedance at this interface Intg comprises a ratio of a voltage phasor at the interface Intg to a current phasor at the interface Intg. In at least some embodiments, these voltage and current phasors at the interface Intg are computed from the voltage and current phasor measurements received from the one or more PMUs 12 deployed at the one or more physical buses which collectively form that interface Intg. Regardless, the equivalent load impedance at interface Intg as used herein characterizes the load “seen” by interface Intg and therefore characterizes both the transmission corridor 6 and the power receiving part of the system 2.
Processing 100 by the one or more processing circuits 14 further includes tracking a Thevenin equivalent voltage and impedance of the designated power generating part, by separately updating that voltage and impedance (Block 120). As used herein, separately updating the Thevenin equivalent voltage and impedance means updating the Thevenin equivalent voltage separately from updating the Thevenin equivalent impedance, rather than jointly updating both the voltage and impedance by simultaneously solving for them. Notably, updating the Thevenin equivalent voltage comprises updating the voltage to reflect the magnitude of any changes in the voltage phasor at the interface Intg that are associated with large variations in the magnitude of the equivalent load impedance at the interface Intg. Large variations in this regard include variations greater than a threshold-defined variation.
Processing 100 finally includes computing an index indicating the corridor's voltage stability as a function of the tracked Thevenin equivalent voltage and impedance (Block 130). This index as used herein comprises any indicator that quantifies the transmission corridor's proximity to voltage instability. With the corridor's voltage stability quantified in this way, actions can be taken as needed to control the corridor's stability and/or mitigate system degradation or disturbance propagation. In some embodiments, for example, processing 100 further comprises automatically performing a prescribed action based on the index. In other embodiments, processing 100 just comprises displaying the index, e.g., to system operators that initiate control and/or protective actions as they deemed appropriate.
In any event, regardless of the particular form of the index, computing the index as described advantageously prevents or at least mitigates false alarms in terms of prematurely detecting transmission corridor voltage instability. This is because the Thevenin equivalent voltage and impedance of the designated power generating part are tracked more accurately as compared to known approaches. Indeed, rather than assigning any variation in the magnitude of the equivalent load impedance at the interface Intg to the Thevenin equivalent impedance, embodiments herein selectively assign variations in that magnitude which are deemed large to the Thevenin equivalent voltage instead. Such more accurately reflects the fact that these large variations are attributable to otherwise unmeasured changes in the designated power generating part (e.g., switching shunt capacitors on or off, generators hitting their reactive power limits, line outages, etc.), as distinguished from changes in local load.
FIG. 3 illustrates additional details about how the voltage stability monitoring apparatus 8 tracks the Thevenin equivalent voltage and impedance according to one or more embodiments. As shown, the voltage stability monitoring apparatus 8 models the power system 2 as including a power generating part g (e.g., part A in FIG. 1), a power receiving (i.e., load) part/(e.g., part B in FIG. 1), and a transmission corridor 6 that interfaces part g at a power generating part interface Intg and that interfaces part/at a power receiving part interface Intl. The apparatus 8 further models the power generating part g in terms of its Thevenin equivalent voltage Ēg and impedance Zg, and models the power receiving part/in terms of its equivalent load impedance Zl. The apparatus 8 tracks Ēg and Zg based exclusively on a voltage phasor VIntgi and a current phasor ĪIntgi at interface Intg, and tracks Zl based exclusively on a voltage phasor VIntli and a current phasor ĪIntli at interface Intl.
FIG. 4 illustrates processing performed by the apparatus 8 to track Ēg and Zg in this way, according to one or more embodiments (different with respect to U.S. Pat. No. 7,200,500 B2, April 2007, which is incorporated by reference herein in its entirety). FIG. 4's processing is performed for each of a plurality of phasor measurement times i=0, 1, . . . , N; that is, for each time i a synchronized measurement sample is received from the one or more PMUs 12 deployed at interface Intg. A measurement sample received from a given PMU 12 deployed at interface Intg includes voltage phasors VIntgi measured at and current phasors ĪIntgi measured in transmission lines associated with interface Intg. Of course, Ēg be initialized at the start of FIG. 4's processing. By considering Ēg in rectangular coordinates (described by its real and imaginary parts) and taking voltage VIntgi as reference (setting its imaginary part to zero), Ēg is initialized as:




  
    
      
        
          
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where




  
    
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R indicates real and I imaginary part of complex Thevenin voltage Ēg.

Regardless, as shown, the apparatus 8 computes a voltage phasor VIntgi at interface Intg for measurement time i (Block 210). The apparatus computes VIntgi in at least some embodiments as:




  
    
      
        
          
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where Pci is the power transfer through transmission line c at measurement time i. The apparatus 8 also computes an aggregated current phasor ĪIntgi at interface Intg for measurement time i (Block 220). In at least some embodiments, the apparatus 8 computes ĪIntgi as a function of the aggregated active and reactive powers at interface Intg. In this case, the apparatus 8 computes ĪIntgi as:





  
    
      
        
          
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where

PIntgi=ΣcεIntgPci  (4)

QIntgi=ΣcεIntgQci  (5)

Regardless, the apparatus 8 then computes an equivalent load impedance Zsi at interface Intg for measurement time i (Block 230). As modeled in FIG. 5, the equivalent load impedance Zs is the load “seen” by interface Intg (in the direction away from the power generating part g). Accordingly, for the current measurement time i, the apparatus 8 according to one or more embodiments computes the equivalent load impedance Zs at interface Intg as:




  
    
      
        
          
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Having computed Zsi, the apparatus 8 tracks Ēg and Zg by computing Ēgi and Zgi for the current measurement time i in different ways depending on how much the magnitude of Zs has varied since the previous measurement time i−1. Specifically, the apparatus 8 determines the size of the variation in the magnitude of Zs, as |Zsi|−|Zsi−1| (Block 240). The apparatus 8 is configured to deem the size of this variation as “large” if the size of the variation is greater than a threshold-defined variation. In at least some embodiments, this threshold-defined variation is a predefined percentage variation since the previous measurement time i−1, defined as ε1×|Zsi−1|. As an example, the threshold ε1 may have a value in the range from 0.01 to 0.05. On the other hand, the apparatus 8 is configured to deem the size of the variation as “small” if the size of the variation is less than the threshold-defined variation. That said, in least some embodiments, the apparatus 8 deems particularly small variations as “insignificant” variations that do not justify updating Zg. Insignificant variations in this case are deemed to be variations that are less than a second threshold-defined variation. Such second threshold-defined variation may be a second predefined percentage variation since the previous measurement time i−1, defined as ε2×|Zsi−1|. As an example, the threshold ε2 may have a value in the range from 0.00005 to 0.001.
In at least some embodiments, the apparatus 8 dynamically adjusts the threshold defining the first and/or the second threshold-defined variation, as a function of the Thevenin equivalent voltage Ēg. That is, the apparatus 8 dynamically adjusts ε1 and/or ε2 as a function of Ēg In one embodiment, for example, the apparatus 8 dynamically decreases ε1 and ε2 responsive to increases in Ēg.
Irrespective of these details, if the apparatus 8 deems the size of the variation in the magnitude of Zs as “large”, the apparatus 8 adjusts imaginary part of the Thevenin equivalent voltage computed for the previous phasor measurement time (i.e., Eg,Ii−1) by the magnitude of the change in voltage phasor VIntg at interface Intg since that previous phasor measurement time (i.e., by |VIntgi−VIntgi−1)(Block 250) (real part of the Thevenin equivalent voltage is set to Eg,Ri=VIntgi). In at least some embodiments, the apparatus 8 updates Eg in this way by updating the complex value Ēg in rectangular coordinates (real and imaginary parts), as opposed to separately updating the magnitude and angle of the complex value Ēg. In this case, the apparatus 8 computes imaginary part of the Thevenin equivalent voltage Eg,Ii for the current phasor measurement time i as:

Eg,Ii=Eg,Ii−1(1+|VIntgi−VIntgi−1|)  (7)

By contrast, if the apparatus 8 deems the size of the variation in the magnitude of Zs as “small”, the apparatus 8 increases or decreases imaginary part of the Thevenin equivalent voltage computed for the previous phasor measurement time (i.e., Eg,Ii−1) by a predefined percentage change (Block 260). In some embodiments, for instance, the predefined percentage change is |Eg,Ii−1×k|, where k is a pre-specified parameter configured to constrain tracking error within predefined bounds. As an example, k may have a value in the range from 0.01 to 0.0001, with k being set as higher within this range for a certain number of initial measurement sample times (e.g., until i=100) and being set as lower within the range thereafter. Regardless, the apparatus 8 decreases Eg,I−1 by the predefined percentage change when the variation in the magnitude of Zs (i.e., |Zsi|−|Zsi−1|) has the same polarity as estimated variation in the Thevenin equivalent impedance Zg (i.e., |Zgi*|−|Zgi−1|, with Zgi* being an estimate or intermediate evaluation of Zgi that takes into account the present value of VIntg and ĪIntg and the previous value of Ēg). Conversely, the apparatus 8 increases Eg,Ii−1 by the predefined percentage change when |Zsi|−|Zsi−1| does not have the same polarity as |Zgi*|−|Zgi−1|.
Similarly to the case for largely sized variations, the apparatus 8 in some embodiments updates Ēg responsive to small sized variations by updating the complex value Ēg rectangular coordinates (real and imaginary parts), thus directly providing correction for both the magnitude and angle of Ēg. This is contrasted with known approaches that separately update the magnitude and angle of the complex value Ēg. Updating the complex value Ēg in rectangular coordinates advantageously allows the apparatus 8 to better track Ēg in the face of dynamic changes in the system. In one or more embodiments, for instance, the apparatus 8 decreases Eg,Ii−1 by unconditionally computing Eg,Ii for the current phasor measurement time i as:

Eg,Ii=Eg,Ii−1(1−|Eg,Ii−1×k|)  (8)

Likewise, the apparatus 8 increases Eg,Ii−1 by unconditionally computing Eg,Ii for the current phasor measurement time i as:

Eg,Ii=Eg,Ii−1(1+|Eg,Ii−1×k|)  (9)

By updating Ēg according to equations (8) and (9), the Thevenin equivalent tracking technique herein advantageously applies to cases when active and reactive power flows over the corridor 6 have opposite directions. This represents improvement over known Thevenin equivalent tracking approaches where such is not the case. For example, Corsi and Taranto's approach bound updates of Ēg by a lower bound εinf and an upper bound εsup. S. Corsi and G. N. Taranto, “A real-time voltage instability identification algorithm based on local phasor measurements,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1271-1279, August 2008. But the lower bound εinf is valid only if the active and reactive power flows have the same direction. And the upper bound εsup is valid only at the voltage instability point (maximum deliverable power). According to one or more embodiments, therefore, decreasing Eg,Ii−1 by unconditionally computing Eg,Ii according to equation (8) and increasing Eg,Ii−1 by unconditionally computing Eg,Ii according to equation (9) means that the apparatus 8 does not condition the amount by which Eg,Ii−1 is decreased or increased on the value of Corsi and Taranto's lower and upper bounds εinf, εsup.

Finally, if the apparatus 8 deems the size of the variation in the magnitude of Zs as “insignificant”, the apparatus 8 does not adjust the Thevenin equivalent voltage Ēg (Block 270). That is, the apparatus 8 simply computes imaginary part of the Thevenin equivalent voltage Eg,Ii for the current phasor measurement time i as:

Eg,Ii=Eg,Ii−1  (10)

Irrespective of how the apparatus 8 updates the Thevenin equivalent voltage Ēg, the apparatus 8 then updates the Thevenin equivalent impedance Zg separately; that is, rather than jointly updating Ēg and Zg for the current measurement time i by simultaneously solving for Ēg and Zg, the apparatus 8 first updates Ēg without updating Zg and then updates Zg based on the updated Ēg. One or more embodiments herein are indifferent to the method the Thevenin equivalent impedance Zg is updated. In embodiments shown in FIG. 4, though, the apparatus 8 updates the Thevenin equivalent impedance Zg by solving a set of two linear equations for Zg, based on the Thevenin equivalent voltage Ēg (as updated/adjusted for the current measurement time) (Block 280). Specifically, the apparatus 8 solves:




  
    
      
        
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        )
      
    
  




where Rgi+jXgi as the real and imaginary parts of Zg are the two unknown variables for which the apparatus 8 solves the set of equations, and where Eg,Ri and Eg,Ii as the real and imaginary parts of Egi, VIntg,Ri and VIntg,Ii as the real and imaginary parts of VIntgi, and IIntg,Ri and IIntg,Ii as the real and imaginary parts of IIntgi are known variables. Real part of the Thevenin equivalent voltage is set to Eg,Ri=VIntgi. Computing Thevenin equivalent impedance Zg in this way does not involve assuming that equivalent reactive is negligible. Unlike Corsi and Taranto's approach which makes this assumption, therefore, the Thevenin equivalent tracking approach herein advantageously extends to lower voltage levels where this assumption does not hold.

Updating the Thevenin equivalent voltage Ēg and impedance Zg as described above more accurately accounts for the impact that the power generating part g of the system 2 has on the stability conditions of the transmission corridor 6. Indeed, changes in such stability conditions are caused either by (A) changes in the power receiving (i.e., load) part l; or (B) changes in the power generating part g not directly measured by available measurements (e.g., switching shunt capacitors on or off, generators hitting their reactive power limits, line outages, etc.). If changes in the equivalent load impedance Zl of the power receiving part/are only accompanied by changes in the current ĪIntg at the power generating part interface Intg, without large changes in the voltage VIntg at that interface Intg, those changes should be account for with updates to the equivalent load impedance Zl. The Thevenin equivalent voltage Ēg of the power generating part g should only be updated to reflect small changes in the system 2 (i.e., according to equations (8) and (9)). The apparatus 8 detects that this scenario applies when the apparatus 8 detects relatively small variations in the magnitude of the equivalent load impedance Zs at interface Intg. On the other hand, if changes in the equivalent load impedance Zl of the power receiving part/are accompanied by changes in the current ĪIntg at the power generating part interface Intg, as well as by large changes in the voltage VIngg at that interface Intg, those changes should be account for updates to the Thevenin equivalent voltage Ēg of the power generating part g (according to equation (7)). The apparatus 8 detects that this scenario applies when the apparatus 8 detects relatively large variations in the magnitude of the equivalent load impedance Zs at interface Intg.
Regardless of these additional details for tracking the Thevenin equivalent voltage Ēg and impedance Zg of the power generating part g, the apparatus 8 herein of course computes an index indicating the voltage stability of the corridor 6 as a function of that tracked voltage Ēg and impedance Zg (Block 130 of FIG. 2). In at least some embodiments, the apparatus 8 computes this index also as a function of the T-equivalent of the corridor 6. FIG. 6 shows an example of such embodiments.
In FIG. 6, the apparatus 8 computes the T-equivalent of the corridor 6 based exclusively on the voltage phasor VIntg and current phasor ĪIntg at interface Intg and on the voltage phasor VIntl and current phasor ĪIntl at interface Intl. Specifically, the apparatus 8 computes the T-equivalent represented in FIG. 6 by the complex impedances ZT and Zsh as:




  
    
      
        
          
            Z
            _
          
          T
        
        =
        
          2
          ⁢
          
            (
            
              
                
                  
                    V
                    _
                  
                  
                    Int
                    g
                  
                
                -
                
                  
                    V
                    _
                  
                  
                    Int
                    l
                  
                
              
              
                
                  
                    I
                    _
                  
                  
                    Int
                    g
                  
                
                -
                
                  
                    I
                    _
                  
                  
                    Int
                    l
                  
                
              
            
            )
          
        
      
    
    
      
        (
        15
        )
      
    
  
  
    
      
        
          
            Z
            _
          
          sh
        
        =
        
          
            
              
                
                  V
                  _
                
                
                  Int
                  g
                
              
              ⁢
              
                
                  I
                  _
                
                
                  Int
                  l
                
              
            
            -
            
              
                
                  V
                  _
                
                
                  Int
                  l
                
              
              ⁢
              
                
                  I
                  _
                
                
                  Int
                  g
                
              
            
          
          
            
              
                (
                
                  
                    I
                    _
                  
                  
                    Int
                    l
                  
                
                )
              
              2
            
            -
            
              
                (
                
                  
                    I
                    _
                  
                  
                    Int
                    g
                  
                
                )
              
              2
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




Note that the apparatus 8 computes the T-equivalent in this way based exclusively on phasor measurements for a current measurement time i, meaning that the computation is advantageously performed without delay.

Having computed the T-equivalent in this way, the apparatus 8 in these embodiments proceeds by computing the Thevenin equivalent of the combination of the power generating part g and the transmission corridor 6, as shown in FIG. 7. Specifically in this regard, the apparatus 8 computes the Thevenin equivalent impedance Zth of the combined generating part g and corridor 6 as:




  
    
      
        
          
            Z
            _
          
          th
        
        =
        
          
            
              
                Z
                _
              
              T
            
            2
          
          +
          
            1
            
              
                1
                
                  
                    Z
                    _
                  
                  sh
                
              
              +
              
                2
                
                  
                    
                      Z
                      _
                    
                    T
                  
                  +
                  
                    2
                    ⁢
                    
                      
                        Z
                        _
                      
                      g
                    
                  
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




The apparatus 6 then computes the Thevenin equivalent voltage Ēth of the combined generating part g and corridor 6 as:





  
    
      
        
          
            E
            _
          
          th
        
        =
        
          
            
              V
              _
            
            
              Int
              l
            
          
          ⁡
          
            (
            
              
                
                  
                    Z
                    _
                  
                  th
                
                +
                
                  
                    Z
                    _
                  
                  l
                
              
              
                
                  Z
                  _
                
                l
              
            
            )
          
        
      
    
    
      
        (
        18
        )
      
    
  




where the equivalent load impedance Zl of the power receiving (i.e., load) part l is of course:





  
    
      
        
          Z
          l
        
        =
        
          -
          
            
              
                V
                _
              
              
                Int
                l
              
            
            
              
                I
                _
              
              
                Int
                l
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Regardless of the particular technique for calculating the Thevenin equivalent of the combination of the power generating part g and the transmission corridor 6, the apparatus 8 in at least some embodiments uses this Thevenin equivalent in order to express the voltage stability index directly or indirectly in terms of the system's maximum deliverable power. The system's maximum deliverable power in this regard is reached when the magnitude of the Thevenin equivalent impedance Zth of the combined generation and transmission parts of the system 2 equals the magnitude of the equivalent load impedance Zl of the power receiving part: |ZTh=|Zl|. The apparatus 8 is configured to compute any stability index that exploits this relation, such as a ratio of equivalent and load impedances, and reactive power margin (active, reactive, apparent).
One or more embodiments compute this voltage stability index according to the method of FIG. 2 in order to not only realize the advantages described above but to also more accurately account for structural changes within the transmission corridor 6. Specifically, the apparatus 8 in such embodiments monitors whether a breaker for each line 4 associated with the power generating part interface Intg is open or closed. The apparatus 8 exploits this breaker monitoring in order to improve its monitoring of the equivalent load impedance Zs at the power generating part interface Intg. The apparatus 8 in this regard dynamically computes Zs exclusively from phasor measurements taken at lines 4 whose breakers are closed.
Where the apparatus 8 computes Zs according to equations (2)-(6), for example, the apparatus 8 excludes from consideration (and effectively “removes” from the corridor 6) transmission lines cε Intg that are flagged as having open breakers. In at least some embodiments, the apparatus 8 receives signals indicating the status of line breakers from phasor measurement units deployed for those lines and dynamically flags lines that have open breakers and that therefore should be excluded from the corridor 6. Effectively “re-computing” the structure of the corridor 6 responsive to line breaker status in this way advantageously prevents zero-valued phasor measurements at open lines from introducing inaccuracies in computation of Zs (different with respect to U.S. Pat. No. 7,200,500 B2, April 2007, which is incorporated by reference herein in its entirety).
In one or more embodiments, the apparatus 8 not only accounts for structural changes within the corridor 6 that result in the opening of a whole transmission path, but also that result in the opening of a part of a transmission path. Consider the example shown in FIG. 8.
As shown in FIG. 8(a), the structure of the transmission corridor 6 initially comprises the collection of three transmission paths 1, 2, and 3 extending between virtual cuts 1 and 2 adjacent to physical buses 1 and 2. Transmission path 1 consists of line 1A extending between physical buses 1 and 5, line 1B extending between physical buses 4 and 5, and line 1C extending between physical buses 4 and 2. Transmission path 2 just consists of line 2A extending between physical buses 1 and 2. Finally, transmission path 3 consists of line 3A extending between physical buses 1 and 3, and line 3B extending between physical buses 3 and 2. Responsive to detecting that the breaker status for line 3A has changed from closed to open, the apparatus 8 changes from computing PInt1 as P1,1+P1,2−P1,3 to computing PInt1 as P1,1+P1,2. As shown in FIG. 8(b), though, the apparatus 8 advantageously recognizes that the breaker status for line 3A changing to open only results in a partial opening of transmission path 3. The apparatus 8 therefore adds the remaining portion of path 3 to bus 2 as injection (here, as a positive injection given the direction of the flow of P2,3). This means that the apparatus 8 continues to compute PInt2 as P2,1−P2,2−P2,3 both before and after detecting the change of the breaker status for line 3A.
Intuitively, a change in the structure of the corridor 6 requires re-initialization of the Thevenin equivalent voltage Ēg. However, one or more embodiments herein refrain from re-initializing the Thevenin equivalent voltage Ēg in this case and instead rely on updates to the Thevenin equivalent voltage Ēg to accurately account for the corridor structure changes. Specifically, responsive to detecting the opening or closing of one or more breakers for lines associated with the power generating part interface Intg, the apparatus 8 simply updates the Thevenin equivalent voltage Ēg to reflect the magnitude of the resulting change in the voltage phasor VIntg at interface Intg, as dynamically computed (to account for the corridor structure changes), without re-initializing the Thevenin equivalent voltage Ēg.
Those skilled in the art will appreciate that the various embodiments described herein are presented as non-limiting examples. For instance, although FIG. 1 illustrates part A of the system 2 as being designated as the power generating part g, those skilled in the art will appreciate that such need not be the case. In fact, the apparatus 8 according to one or more embodiments dynamically adapts which of the parts of the system 2 is designated as the power generating part g. The apparatus 8 does so responsive to detecting a change in the direction or magnitude of power flowing through one or both interfaces between the corridor 6 and the parts of the power system 2. In some embodiments, the apparatus 8 monitors for such a change at every measurement time i.
The apparatus 8 dynamically detects a power flow direction change in one or more embodiments as a function of the aggregated active power at the corridor interfaces. Aggregated active power at an interface in this sense means the algebraic sum of active power over the transmission lines associated with an interface. Consider the example shown in FIG. 9 (different with respect to U.S. Pat. No. 7,200,500 B2, April 2007, which is incorporated by reference herein in its entirety).
As shown in FIG. 9(a), the apparatus 8 designates virtual bus 1 as the power generating part interface Intg responsive to detecting that the aggregated active power P1agg at virtual bus 1 is positive and the aggregated active power P2agg at virtual bus 2 is negative (where positive active power indicates power flow out of the bus and negative active power indicates power flow into the bus, as is conventional). Conversely, as shown in FIG. 9(b), the apparatus 8 designates virtual bus 2 as the power generating part interface Intg responsive to detecting that the aggregated active power P2agg at virtual bus 2 is positive and the aggregated active power P1agg at virtual bus 1 is negative. Finally, as shown in FIG. 9(c), the apparatus designates whichever virtual bus 1 or 2 has the larger aggregated active power P1agg or P2agg responsive to detecting that both the aggregated active power P1agg and P2agg at buses 1 and 2 are positive. In some embodiments, though, the apparatus 8 proceeds as shown in FIG. 9(c) only upon validating the positive polarity of both aggregated active powers. Specifically, the apparatus validates that the phasor measurements used for computing the aggregated active powers were not performed over a short duration caused by fault conditions, as opposed to multi-terminal corridors with tapped configuration.
Of course, although the above embodiments have been described with reference to the direction and/or magnitude of active power flow, the embodiments apply equally to the direction and/or magnitude of reactive power flow. In fact, in at least some embodiments, the direction of active power flow and the direction of reactive power flow are assumed to be the same.
Regardless, dynamically adapting the power generating part designation in this way advantageously generalizes voltage stability index computation for application to a wide range of power system and transmission corridor types. For instance, in some embodiments the power system 2 comprises an interconnected transmission system where the power flow direction changes based on a series of operational decisions, such as market rules, the schedule of power flow, and the demand-supply chain process. Of course, embodiments are also applicable to a radial system, where power statically flows from one part to another part of the system, but in such cases there is no requirement that one part be preselected as the power generating part. Rather, changes in power flow direction are handled through a series of recursive validation of aggregated active powers at both ends of the corridor 6.
In one or more embodiments, the apparatus 8 re-initializes the Thevenin equivalent voltage Ēg responsive to dynamically adapting which part of the system 2 is designated as the power generating part g. This requires a certain number of measurement samples to be used for Thevenin equivalent voltage identification.
Advantages of improvements in voltage stability monitoring of transmission corridors introduced by one or more embodiments herein are illustrated using phasor measurement units recordings for the exemplary corridor shown in FIG. 10A. Synchronized phasor measurements in this example are collected at the rate of 60 samples/second. The exemplary corridor shown in FIG. 10A includes a line outage between buses 1 and 3 at time t=60 seconds, with the same line re-closed at time t=192 seconds. Advantages are illustrated in terms of two voltage stability indices derived from the concept of Thevenin equivalent: ratio of equivalent and load impedances, as shown in FIG. 10B, and reactive power margin, as shown in FIG. 10C. Time evolutions of these two indices, with and without detection and incorporation of structural change within the corridor, are given in FIGS. 10B and 10C. If the opened line is considered as part of the monitored corridor with measurement equal to zero (i.e., breaker status unknown), it produces errors in corridor stability conditions computation of approximately 0.3 in ratio of impedances and 400 Mvars in reactive power margin as compared to the case when the opened line, together with its counterpart line at Cut 2, is taken out from the consideration in the algorithm (breaker status known).
A more complicated corridor is shown in FIGS. 11A-C as another example, to demonstrate the advantages of dynamically adapting which part of the power system is designated as the power generating part. This case includes consecutive outages of two lines at t=7 second and t=13 seconds, followed by reclosing the lines in opposite order of outages at t=20 seconds and t=25 seconds. Within the monitored corridor, these changes do not present structural changes, since they are not directly related to defined cuts of the corridor. Aggregated active powers on the cuts are such that both are directed toward the corridor, with the power at Cut 1 having the bigger value and therefore Bus 1 being designated as the power generating part. Computations introduced by one or more embodiments herein shown as being performed with correct (Bus 1 as power generating part) and wrong (Bus 2 intentionally set as power generating part) polarity. Time evolutions of two major voltage stability indices are given in FIG. 11B (ratio of impedances) and FIG. 11C (reactive power margin). Wrong polarity results in considerable error: approximately 0.1 in ratio of impedances and 500 Mvars (biggest error) in reactive power margin.
In view of the above, one or more embodiments herein have advantages over U.S. Pat. No. 7,200,500 B2. First, the generation area (sending end) Thevenin equivalent is identified using a specific procedure; it is not assumed as having very strong network on sending and assuming small value for its equivalent impedance that is never updated later. This offers the possibility to reflect changes in the sending area (e.g. some generator hitting the limit, switching on/off capacitor, etc.), which is not possible with the solution in the '500 patent. Second, one or more embodiments herein detect and handle changes within the corridor. In the '500 patent this is not possible and it could lead to wrong assessment of stability conditions. The '500 patent works only if the flow is from sending to receiving and fixed structure of the corridor. This feature extends embodiments herein to a much wider number of possible corridors. Third, one or more embodiments detect changes in polarity (flow direction) and change the position of the sending and receiving ends. This is contrasted, again, with the '500 patent, which works only for one flow direction for both active and reactive powers) for sending to receiving.
Those skilled in the art will appreciate that the one or more “circuits” described herein may refer to a combination of analog and digital circuits, and/or one or more processors configured with software stored in memory and/or firmware stored in memory that, when executed by the one or more processors, perform as described herein. One or more of these processors, as well as the other digital hardware, may be included in a single application-specific integrated circuit (ASIC), or several processors and various digital hardware may be distributed among several separate components, whether individually packaged or assembled into a system-on-a-chip (SoC).
Thus, those skilled in the art will recognize that the present invention may be carried out in other ways than those specifically set forth herein without departing from essential characteristics of the invention. The present embodiments are thus to be considered in all respects as illustrative and not restrictive, and all changes coming within the meaning and equivalency range of the appended claims are intended to be embraced therein.

## Claims

1. A method of monitoring voltage stability of a transmission corridor through which power flows between different parts of a power system, the method comprising the following performed by a voltage stability monitoring apparatus:
monitoring an equivalent load impedance at an interface between the transmission corridor and a part of the power system designated as generating said power, the equivalent load impedance at said interface comprising a ratio of a voltage phasor at said interface to a current phasor at said interface;
tracking a Thevenin equivalent voltage and impedance of said designated part by separately updating that voltage and impedance, wherein updating the Thevenin equivalent voltage comprises updating the imaginary part of the voltage to reflect the magnitude of any changes in said voltage phasor that are associated with large variations in the magnitude of the equivalent load impedance at said interface, said large variations including variations greater than a threshold-defined variation;
computing an index indicating said voltage stability as a function of the tracked Thevenin equivalent voltage and impedance; and
performing one or more actions, based on the computed index, as needed to control the transmission corridor's voltage stability and/or mitigate system degradation or disturbance propagation.

2. The method of claim 1, wherein updating the Thevenin equivalent voltage further comprises setting the real part of the voltage to said voltage phasor for a current phasor measurement time.

3. The method of claim 1, wherein said updating the Thevenin equivalent voltage comprises, for each of a plurality of phasor measurement times, determining whether or not variation in the magnitude of the equivalent load impedance at said interface since a previous phasor measurement time is greater than the threshold-defined variation, and, if so, adjusting the imaginary part of the Thevenin equivalent voltage computed for the previous phasor measurement time by the magnitude of the change in said voltage phasor since that previous phasor measurement time.

4. The method of claim 1, wherein updating the Thevenin equivalent impedance comprises solving a set of two linear equations with two unknown variables that comprise the real and imaginary parts of the Thevenin equivalent impedance, wherein known variables in the set of two linear equations include the imaginary part of the Thevenin equivalent voltage as updated to reflect the magnitude of any changes in said voltage phasor and the real part of the Thevenin equivalent voltage as set to said voltage phasor for a current phasor measurement time.

5. The method of claim 1, further comprising dynamically adjusting a threshold defining the threshold-defined variation, as a function of the Thevenin equivalent voltage.

6. The method of claim 1, further comprising dynamically adapting which of said parts of the power system is designated as generating said power, responsive to detecting a change in direction or magnitude of power flowing through one or both interfaces between the transmission corridor and said parts of the power system.

7. The method of claim 1, further comprising monitoring whether a breaker for each line associated with said interface is open or closed, and wherein monitoring the equivalent load impedance at said interface comprises dynamically computing the equivalent load impedance at said interface exclusively from phasor measurements taken at lines whose breakers are closed.

8. The method of claim 7, wherein, responsive to detecting the opening or closing of one or more of said breakers, said updating comprises updating the imaginary part of the Thevenin equivalent voltage to reflect the magnitude of the resulting change in said voltage phasor, as dynamically computed, without re-initializing the imaginary part of the Thevenin equivalent voltage.

9. The method of claim 1, wherein updating the Thevenin equivalent voltage further comprises, responsive to small variations in the magnitude of the equivalent load impedance at said interface, decreasing or increasing the imaginary part of the Thevenin equivalent voltage by a predefined percentage change when said small variations do or do not have the same polarity as estimated variations in said Thevenin equivalent impedance, respectively, said small variations including variations less than the threshold-defined variation.

10. The method of claim 9, wherein, responsive to small variations in the magnitude of the equivalent load impedance at said interface, increasing the imaginary part of the Thevenin equivalent voltage comprises computing the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1+|Eg,Ii−1×k|) and decreasing the imaginary part of the Thevenin equivalent voltage comprises computing the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1−|Eg,Ii−1×k|), where Eg,Ii is the imaginary part of the Thevenin equivalent voltage for a current phasor measurement time i, Eg,Ii−1 is the imaginary part of the Thevenin equivalent voltage for a previous phasor measurement time i−1, and k is a pre-specified parameter configured to constrain tracking error within predefined bounds, and wherein updating the Thevenin equivalent voltage further comprises setting the real part of the Thevenin equivalent voltage Eg,Ri for the current phasor measurement time i as Eg,Ri=VIntgi.

11. The method of claim 10, wherein, responsive to small variations in the magnitude of the equivalent load impedance at said interface, increasing the imaginary part of the Thevenin equivalent voltage comprises unconditionally computing the imaginary part of Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1+|Eg,Ii−1×k|) and decreasing the imaginary part of the Thevenin equivalent voltage comprises unconditionally computing the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1−|Eg,Ii−1×k|).

12. The method of claim 1, wherein said performing comprises automatically performing a prescribed action based on the computed index.

13. The method of claim 1, wherein said performing comprises displaying the computed index.

14. A voltage stability monitoring apparatus configured to monitor voltage stability of a transmission corridor through which power flows between different parts of a power system, the voltage stability monitoring apparatus comprising one or more processing circuits configured to:
monitor an equivalent load impedance at an interface between the transmission corridor and a part of the power system designated as generating said power, the equivalent load impedance at said interface comprising a ratio of a voltage phasor at said interface to a current phasor at said interface;
track a Thevenin equivalent voltage and impedance of said designated part by separately updating that voltage and impedance, wherein updating the Thevenin equivalent voltage comprises updating imaginary part of the voltage to reflect the magnitude of any changes in said voltage phasor that are associated with large variations in the magnitude of the equivalent load impedance at said interface, said large variations including variations greater than a threshold-defined variation;
compute an index indicating said voltage stability as a function of the tracked Thevenin equivalent voltage and impedance; and
perform one or more actions, based on the computed index, as needed to control the transmission corridor's voltage stability and/or mitigate system degradation or disturbance propagation.

15. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to update the Thevenin equivalent voltage further by setting the real part of the voltage to said voltage phasor for a current phasor measurement time.

16. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to update the Thevenin equivalent voltage by, for each of a plurality of phasor measurement times, determining whether or not variation in the magnitude of the equivalent load impedance at said interface since a previous phasor measurement time is greater than the threshold-defined variation, and, if so, adjusting the imaginary part of the Thevenin equivalent voltage computed for the previous phasor measurement time by the magnitude of the change in said voltage phasor since that previous phasor measurement time.

17. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to update the Thevenin equivalent impedance by solving a set of two linear equations with two unknown variables that comprise the real and imaginary parts of the Thevenin equivalent impedance, wherein known variables in the set of two linear equations include the imaginary part of the Thevenin equivalent voltage as updated to reflect the magnitude of any changes in said voltage phasor and the real part of the Thevenin equivalent voltage as set to said voltage phasor for a current phasor measurement time.

18. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are further configured to dynamically adjust a threshold associated with the threshold-defined variation as a function of the Thevenin equivalent voltage.

19. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are further configured to dynamically adapt which of said parts of the power system is designated as generating said power, responsive to detecting a change in direction or magnitude of power flowing through one or both interfaces between the transmission corridor and said parts of the power system.

20. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are further configured to monitor whether a breaker for each line associated with said interface is open or closed, and are configured to monitor the equivalent load impedance at said interface by dynamically computing the equivalent load impedance at said interface exclusively from phasor measurements taken at lines whose breakers are closed.

21. The voltage stability monitoring apparatus of claim 20, wherein the one or more processing circuits are configured, responsive to detecting the opening or closing of one or more of said breakers, to update the imaginary part of the Thevenin equivalent voltage to reflect the magnitude of the resulting change in said voltage phasor, as dynamically computed, without re-initializing the imaginary part of the Thevenin equivalent voltage.

22. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to update the Thevenin equivalent voltage also by, responsive to small variations in the magnitude of the equivalent load impedance at said interface, decreasing or increasing the imaginary part of Thevenin equivalent voltage by a predefined percentage change when said small variations do or do not have the same polarity as estimated variations in said Thevenin equivalent impedance, respectively, said small variations including variations less than the threshold-defined variation.

23. The voltage stability monitoring apparatus of claim 22, wherein the one or more processing circuits are configured, responsive to small variations in the magnitude of the equivalent load impedance at said interface, to increase the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1+|Eg,Ii−1×k|) and decrease the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1−|Eg,Ii−1×k|), where Eg,Ii is the imaginary part of the Thevenin equivalent voltage for a current phasor measurement time i, Eg,Ii−1 is the imaginary part of the Thevenin equivalent voltage for a previous phasor measurement time i−1, and k is a pre-specified parameter configured to constrain tracking error within predefined bounds, and to set the real part of the Thevenin equivalent voltage Eg,Ri for the current phasor measurement time i as Eg,Ri=VIntgi.

24. The voltage stability monitoring apparatus of claim 23, wherein the one or more processing circuits are configured, responsive to small variations in the magnitude of the equivalent load impedance at said interface, to increase the imaginary part of the Thevenin equivalent voltage by unconditionally computing the imaginary part of the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1+|Eg,Ii−1×k|) and to decrease the imaginary part of the Thevenin equivalent voltage by unconditionally computing the Thevenin equivalent voltage as Eg,Ii=Eg,Ii−1(1−|Eg,Ii−1×k|).

25. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to automatically perform a prescribed action based on the computed index.

26. The voltage stability monitoring apparatus of claim 14, wherein the one or more processing circuits are configured to display the computed index.