IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
1271
A Real-Time Voltage Instability Identification
Algorithm Based on Local Phasor Measurements
Sandro Corsi, Member, IEEE, and Glauco N. Taranto, Senior Member, IEEE
Abstract—This paper proposes a new voltage instability risk indicator based on local phasor measurements at fast sampling rate.
The effectiveness of the indicator is analyzed at EHV load and
“transit” buses. The risk criterion is based on the real-time computation of the Thevenin equivalent impedance of the classic electrical
circuit given by an equivalent generator connected to an equivalent
load impedance through an equivalent connecting impedance. The
main contribution of the paper is the innovating algorithm utilized
on the real-time adaptive identification of the Thevenin voltage
and impedance equivalents. The algorithm performance is shown
through a detailed sensitivity analysis. The paper presents important numerical results from the actual Italian EHV network.
Index Terms—Dynamic simulation, long-term voltage stability,
online VSA, phasor measurement unit (PMU), Thevenin equivalent, voltage stability.
I. INTRODUCTION
V
OLTAGE instability in the transmission networks has
led or significantly contributed to some major blackouts
around the world. Its timely recognition is very crucial to allow
effective control and protection interventions. On this concern,
a worldwide interest in defining an effective real-time voltage
stability indicator, under the assumption of very fast measurements of system electrical variables (use of PMU—phasor
measurement unit [1]—or devices computing phasors in comparable speed), is growing. By measuring the local voltage and
current phasors at an EHV bus, the voltage instability analysis
can be performed by considering the classic Thevenin equivalent “seen” from the bus [2]–[5]. As generally understood, the
instability is linked to the condition of equality between the
absolute value of two equivalent impedances—the load and the
Thevenin impedances—with this equality corresponding to the
maximal power transfer.
One of the first works on the subject [2] refers to the two-bus
Thevenin equivalent system, as depicted in Fig. 1. The identifiand
is based on a recursive least-square idencation of
tification method applied to the local voltage and current phasor
measurements. Results are presented for -constant loads as a
sequence of power flow solutions, resembling a continuation
power flow.
Manuscript received January 5, 2007; revised February 15, 2008. The work
of G. N. Taranto was supported in part by CNPq and and in part by CESI during
his sabbatical leave of absence. Paper no. TPWRS-00910-2006.
S. Corsi is with CESI Spa., Milano, Italy (e-mail: corsi@cesi.it ).
G. N. Taranto is with the Federal University of Rio de Janeiro, COPPE/UFRJ,
21945-970 Rio de Janeiro, Brazil (e-mail: tarang@coep.ufrj.br ).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2008.922586
Fig. 1. Two-bus Thevenin equivalent circuit.
The works [5], [6] differ in the identification method utilized,
which is simplified to achieve a high-speed voltage instability
risk evaluation. The work [5] uses the concept of insensitivity
of the apparent power at the receiving end of the transmission
line to infer the voltage instability proximity, whereas the work
[6] uses the same concept of Thevenin equivalent and relies on
the Tellegen’s theorem to identify the Thevenin parameters. In
[3], the authors extend the previous analyses to ZIP loads and
present a mechanism to include the -constant and -constant
portions of the load into the equivalent Thevenin impedance,
allowing to conclude that maximum loadability and voltage instability occur at the same point. They also propose to monitor
the status of the over-excitation limiters (OELs) of nearby generators and use the information for voltage instability proximity
indication.
In practice, the main critical aspects of the methodologies
based on real-time measurements at a given bus are:
• computing uncertainty of the equivalent grid parameters
depending on the identification method, and their high sensitivity to small changes of local measurements at fast sampling rate;
• computing time which often does not allow enough speed
for real-time applications;
• significant performance differences of the real system with
respect to the simple electrical model of the considered
equivalent circuit, when approaching the voltage instability
condition;
• unknown parameters of the ZIP load required in [3] when
applied in the field;
• absence of EHV “transit” buses from these analyses.
“Transit bus” is a bus with no load directly connected to it (in
the sense of a one-line diagram). It normally represents important EHV buses in bulk transmission corridors or nearby load
centers, having weak or no parallel paths.
0885-8950/$25.00 © IEEE
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
This paper has the objective to overcome the above-mentioned criticisms by defining a real-time indicator, introduced in
[7], and able to effectively support practical applications. The
paper presents a new algorithm for fast-tracking the Thevenin
parameters (voltage and reactance) based on the local voltage
and current phasor measurements. Contrary to the least-squaretype identification methods, that generally need a large data
window to suppress oscillations, the proposed algorithm has a
good feature to have these oscillations filtered without significantly delaying the identification process.
For voltage stability studies, some basic and fundamental
considerations should be put in evidence [8]. The first point
is that the Thevenin equivalent has to represent a detailed
dynamic model. Therefore, its parameters change continuously
and quickly while approaching the maximum loadability. The
second refers to the local area OELs and on-load tap changer
(OLTC) transformers whose dynamics strongly impact on the
bus maximum loadability and voltage instability [9]. The third
refers to the P-constant load analysis, which is not adequate
for a correct identification in real systems, where a mixed ZIP
load is usually adopted. The fourth point distinguishes the
maximum loadability point (PV “nose” curve) from the real
instability point, which in general occurs at very low voltage.
The last basic consideration refers to the high speed the PV
curve equilibrium point moves from the PV curve nose tip to
the first unstable point at lower voltage. This fact confirms the
practical importance of the nose tip identification to prevent
against voltage instability, as well as the need of a very fast
identification procedure when approaching the voltage instability limit.
According to these basic considerations the proposed new
way is capable to identify in real-time the Thevenin equivalent, with higher precision in proximity of the voltage instability where the OELs and OLTCs operate. This faster-change
dynamic period also corresponds to the relevant simultaneous
and in opposite direction variations of the load and the Thevenin
equivalent impedances. The proposed algorithm is able to distinguish these critical variations from those happening during
the normal operating conditions, and therefore, to send a timely
alarm at the beginning of the voltage degradation process. We
remark, however, that the proposed identification algorithm is
not suited for short-term voltage instability occurring straight
after large perturbations.
The proposed algorithm is tested through time-domain simulations performed in a large and realistic representation of the
EHV Italian network. Results on both load and “transit” buses
are presented.
II. BASICS OF THE PROPOSED ALGORITHM
Considering the electrical circuit in Fig. 1, the computation
and
, based on the phasors
and
measured at
of
the load bus, is the objective to be pursued.
From Kirchoff’s law
(1)
with
.
Fig. 2. Phasor diagram of the two-bus equivalent circuit.
This equation gives evidence that the considered problem has
infinite solutions. In principle, two subsequent phasor measureand
can be used to compute
and
ments of the pair
, under the assumption that they do not change during the
time interval between the two subsequent measurements. According to that, the corresponding matrix equation takes a high
risk of being singular when considering the required very short
sampling-time interval. Therefore, computations as such, are
very risky due to the numerical difficulties, and the identification procedures based on this approach usually requires time (a
large data window) to converge. Only in the presence of significant system variations between two subsequent measurements
it is possible to achieve an acceptable result. This happens when
the system is collapsing; therefore, it could arrive too late for activating preventive controls and special protections. Because of
that, it is proposed a new algorithm to speed up the identificaand
.
tion of the Thevenin equivalent parameters
III. DESCRIPTION OF THE PROPOSED ALGORITHM
The maximum power transfer to the load in the electric circuit
shown in Fig. 1 occurs when
(2)
.
with
This circuit represents the entire network “seen” from the
considered bus in an equivalent way. According to the phasor
diagram show in Fig. 2 the following relationship holds:
(3)
(4)
,
and
.
with
Separating (4) into real and imaginary parts, yields
(5)
(6)
For the equivalent Thevenin impedance “seen” from an EHV
, and the assumption of
is
bus, we have
very reasonable. Therefore, an initial estimation for is given
CORSI AND TARANTO: REAL-TIME VOLTAGE INSTABILITY IDENTIFICATION ALGORITHM
by
(7)
and are measured quantities taken from the PMUs,
Since
the initial estimation of still depends on
. The admissible
must be in agreement with the electric circuit
range for
laws. Up to the maximum power transfer (MPT) and consid) corresponds
ering an inductive load its minimum value (
to the load voltage, and its maximum value (
) corresponds
(with
). In normal opto the voltage when
erating conditions the load impedance is much higher than the
equivalent Thevenin impedance: a good start estimation value
is the arithmetic average of its extreme values given by
for
(8)
and
, with obwhere
tained from:
Even being inside of its admissible range,
as the free
variable of the problem, should be updated towards its correct
physical value. The appendix shows the proposed theorem and
a simple numerical example that elucidates to which direction
should be updated at each sampling time in order to speed
identification convergence. The proposed algorithm
up the
and
are constant in the brief “
assumes that
” interval of their identification, which requires a very short
sampling time. In brief, the proposed adaptive algorithm will
when the variations of
and the estimated
reduce
have the same direction, otherwise it will increase
. This is
evident from the following (see the Appendix (A8)) where the
has to be reduced to avoid the error on increases
error
when the current reduces:
(9)
The simple numerical example given in the Appendix helps
to further clarify this concept.
Knowing the direction
should be updated at each sampling time, we need now to establish how large this variation
should be. This quantity is calculated as follows:
with
(10)
(11)
(12)
(13)
with being a pre-specified parameter chosen in such way to
constrain the identification error within narrower bounds, and
being the corresponding time step. Most of the times
drives
the identification process, so its specification has a major impact
and
are active only when
in the process. The quantities
the estimated
is close to the edges of its feasible range. In
the following frame the adaptive algorithm that tracks the corto identify
is presented. One possible
rect value of
.
variant of the algorithm is with respect to the calculation of
At Step 4, the calculation of and
can be done by using
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, calculated over a window of approa moving average of
at
priate size ( ), instead of the instantaneous value of
iteration . This variant has the advantage to filter the identified
variables paying the price of a slower identification process. In
value
synthesis, the proposed algorithm will identify the
identification is reached, by imposing an osas soon as the
cillation with small amplitude around their correct values with
frequency half of the sampling rate.
Algorithm to identify
Step 1) Estimate initial values for
according to (7) already considering
according to (8) and
.
Step 2) Calculate
from (6).
Step 3) Calculate
according to the conditions:
If load impedance variation is negative do
If
then
If
then
If load impedance variation is positive do
If
then
If
then
If load impedance is constant
Step 4) Calculate
and
from (7) and (6), respectively.
Step 5) Increment and go to Step 3.
OBS:
is an intermediate evaluation of
that takes into
account the present values of the voltage and current phasor
and .
measurements and the previous values of
IV. SENSITIVITY ANALYSIS OF THE PROPOSED METHOD
Generally the degree of success of identification methods
relies on their tuning parameters. The proposed method has
only one key parameter that has a major impact on the identification process, the parameter . This section starts showing
a sensitivity analysis on the parameter . The results were
obtained from dynamic simulation tests performed on a large
power system considering a sampling rate of 20 ms. The test
system will be further explained in the next section dedicated to
the assessment of voltage instability risk based on the proposed
for different
method. Figs. 3 and 4 show the estimated
values of the parameter . It can be noted that for larger values
of the estimation of
is faster (Fig. 3) but more oscillatory
(Fig. 4). Our experience after many runs of the algorithm says
that good results are obtained when is in the range from
0.01% to 0.1% of the nominal bus voltage.
One good practice is to adopt larger values of during the initialization phase (first seconds or minutes) of the identification
process, and then switch it to a lower value as the value of
approaches to a steady value. Fig. 5 shows the corresponding
estimated
as a function of the parameter . Fig. 6 shows
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Fig. 3. Identification of E
Fig. 4. Identification of E
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
Fig. 5. Identification of X
as a function of k .
Fig. 6. Convergence of E
as a function of its initial value.
Fig. 7. Identification of X
as a function of m.
as a function of k .
as a function of k .
the convergence of the estimated
parameter by increasing
the difference of the initial values of
with respect to the
correct estimation. Similar convergence curves are obtained for
. The results show a good performance of the proposed alwas utilized in
gorithm during the initial phase. A
the results of Fig. 6. Fig. 7 shows the sensitivity with respect
, for the cases with
to the parameter . Fig. 7(a) shows
, 4, 10, and 100, in a period of time where there is a
, and Fig. 7(b) shows
change in the identified value of
at the end of the simulation when reaching a voltage instability.
In both cases, it is clear to see higher delays in the identificaas gets larger. Fig. 8 shows the convergence of
tion of
as a function of the signal-to-noise ratio (SNR) when white
noise is added to the voltage and current phasor measurements.
In the absence of a due filter, observe that when the power of
the noise increases with respect to the power of the phasors, the
algorithm starts to drift from the correct value. Therefore, the
required filter on the phasor signals has to be checked not only
on the cut-off frequency tuning but also on the SNR value effects.
The maximum frequency of the white noise added to the
signal in Fig. 8 coincides with the frequency of the phasor measurement sampling, i.e., 50 Hz. Fig. 9 shows the identification
of the equivalent Thevenin voltage, for the case when SNR
dB, as a function of the cut-off frequency of a low-pass
filter. It can be noted that the performance of the proposed algorithm increases as the cut-off frequency of the low-pass filter
decreases. When the cut-off frequency is half of the measurement frequency the algorithm is already able to track the correct
parameters.
Fig. 8.
E
Convergence as a function of SNR without signal conditioning.
The choice of the filter cut-off frequency is not critical, as
long as the system dynamics under analysis stay lower than 25
Hz (half of the sampling-rate frequency). Thus, Fig. 10 proceeds with a sensitivity analysis of the identification algorithm,
CORSI AND TARANTO: REAL-TIME VOLTAGE INSTABILITY IDENTIFICATION ALGORITHM
1275
Fig. 9. Convergence of E as a function of the cut-off frequency of the lowpass filter, for the case when the SNR = 80 dB.
Fig. 12. The 380-kV Italian network.
is not small enough to track a problem occurring, taking the simulation of Fig. 11 as an example, in a timeframe of 1200 s.
V. SIMULATION RESULTS ON A LARGE-SIZE SYSTEM
Fig. 10. Convergence of E
as a function of SNR with signal conditioning.
This section shows the application of the proposed method
to the Italian system. The Italian grid analyzed contains the
380-kV and 220-kV networks (Fig. 12 depicts the 380-kV network only). The system configuration has 2549 buses, 2258
transmission lines, 134 groups of thermal generators, and 191
groups of hydro generators. Dynamic models for the generator,
AVR, OEL, governor and OLTC are considered.
The system load is approximately 50 GW, represented as a
and
in (14). The system is
static model with
under primary voltage and frequency control only
(14)
Fig. 11. Identification of X
as a function of the sampling rate (s.r.).
for varying SNR, when the phasor signals filtered by low-pass
filter with a cut-off frequency of 5 Hz. The results show that
the algorithm has an acceptable performance for SNR equals or
as a
higher than 40 dB. Fig. 11 shows the identification of
function of the sampling rate (s.r.) of the phasor measurements.
It can be noted that when the algorithm is utilized with a s.r.
is tracked with more precision, although being
of 20 ms,
more corrupted with high frequency noise. With a s.r. of 2.56 s
is sluggish, and it can be noted that before
the tracking of
, the system undergoes voltage
finding a steady value for
instability. This is a very interesting and counter intuitive fact,
because it is difficult to say that a sampling rate of few seconds
All the simulations assume a sampling rate of 20 ms, with
the identification updated at every 20 ms. The identified value
20 ms (for tests on
is based on the sampled data of the last
the Italian system was set equal to 4). The parameter was
set within the range of 0.01% to 0.1% of the nominal voltage.
Two sets of tests were performed, one at a load bus and the other
at a “transit” bus.
A. Application to a Load Bus
The tests performed on a load bus consisted of increasing
the load, while maintaining the power factor constant, in the
neighboring buses at a given rate and in the considered load bus
at a higher rate. Our objective in testing the proposed method
was not to find which bus would first exhibit a voltage instability
problem, instead, how the method would behave as the voltage
instability point was reached at the bus most prone to instability.
The tests were performed in Brugherio 380-kV bus at the Milano
Area (Region A marked in Fig. 12).
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Fig. 13. Identified Thevenin reactance.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
Fig. 15. OEL indicators of generators near Brugherio bus.
Fig. 16. Voltage instability risk indicator for Brugherio load bus.
Fig. 14. Tap evolution of OLTCs near Brugherio bus.
The analysis performed in the Milano Area consisted of increasing the local area load by a rate of 10%/min maintaining
constant the power factor. The load at the Brugherio bus was
increased by a rate of 40%/min. Fig. 13 shows the identified
Thevenin reactance and the load impedance.
The curves in all the figures of this section are drawn up to
the voltage instability point (considered as the point where numerical difficulties start to occur in the simulation program).
Fig. 14 shows the tap variation of the OLTCs located electrically close to Brugherio bus. The heavy solid lines represent the
three OLTCs that feed the local loads at the Brugherio bus. A
continuous tap model represents the OLTCs. Tests with a discrete model were performed without any relevant impact on the
identification process. Fig. 15 shows the OEL indicators of some
important group of generators. The heavy lines correspond to
three groups of generators (Turbigo and Tavazzano 1–2) electrically close to the Brugherio bus. When the indicator reaches
the value zero, the OEL starts to operate. It can be noted that
when these three OEL start to operate at around 550 s, the identified Thevenin reactance, as shown in Fig. 13, increases substantially. Fig. 16 shows the proposed voltage instability risk
indicator defined as the ratio between the identified Thevenin
) and the load impedance ( ). Since the loads
reactance (
are realistically represented with a mixed ZIP model, the voltage
instability occurs [9] a little later than the MPT (when the impedances are equal).
Fig. 17. Equivalent Thevenin voltage seen from Baggio bus.
B. Application to a “Transit” Bus
Not only due to practical interest, but also for being the most
onerous for testing the identification procedure, the proposed
method was checked on a “transit” bus. The “transit” bus is modeled as a load bus according to the assumptions made in [7],
where the transmission lines with power leaving the bus represent the “equivalent load,” and those with power entering the
bus represent the Thevenin equivalent. The algorithm was tested
in the Baggio 380-kV “transit” bus also located at the Milano
Area. Figs. 17 and 18 show the time simulation of the equivalent Thevenin voltage and reactance, respectively.
Fig. 17 also shows the theoretical limit values for the equiva,
) and Fig. 18 also shows the
lent Thevenin voltage (
equivalent load impedance seen from Baggio bus. The discontinuities seen in the curves of the Thevenin voltage, Thevenin
CORSI AND TARANTO: REAL-TIME VOLTAGE INSTABILITY IDENTIFICATION ALGORITHM
Fig. 18. Equivalent Thevenin reactance seen from Baggio bus and equivalent
load impedance.
Fig. 21. Identification of
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X
for load steps having 20% slope in its flat part.
The searching for a best-performance indicator is out of the
focus of this paper.
C. Application in the Case of Large Step Sequence of Load
Perturbations With 20% Slope in the Flat Part
Fig. 19. The voltage instability risk indicator for Baggio “transit” bus.
Here it is shown a very difficult condition to the algorithm
where we combine high frequency large load stepping (50 s-duration pulses) and ramps with small slope. Fig. 21 shows the
identification of the equivalent Thevenin voltage and reactance,
respectively, for a load variation at P. A. Caiano, consisting of
steps having a 20% slope in its flat part. The performance of the
algorithm is satisfactory also in this case.
VI. CONCLUSIONS
Fig. 20. Voltages at Brugherio load bus and Baggio “transit” bus.
reactance, and indicator (Fig. 19) represent the start of operation of nearby OELs. According to the simulation results for
Baggio bus, closer attention should be paid after 400 s. It can be
noted that before 400 s the indicator is reasonably flat, changing
to a steeper increase afterwards. Fig. 20 shows the voltage magnitudes of Brugherio and Baggio buses. At round 580 s (100 s
before the instability) the voltages at both buses are at 0.9 p.u.
Thus, solely relying the voltage instability proximity on voltage
magnitudes may be too late.
Figs. 16 and 19 show a better and earlier insurgence view of
the voltage instability process.
We think that other more sophisticated indicators could be deand
.
fined by having the real time updated values of
The paper proposes a real-time voltage instability algorithm
to recognize in a very fast way the raising of the voltage instability phenomenon seen by an EHV bus provided with PMU.
The first contribution is the description of the intrinsic difficulties to identify in real time the Thevenin’s equivalent circuit
seen by the considered EHV bus. The proposed algorithm is described and deeply checked through a complete sensitivity analysis, based on large power system data with ZIP load characteristics at the sampling rate of 20 ms, with very satisfactory results in terms of reliability, robustness and speed. The effectiveness of the algorithm was also confirmed by the dynamic tests
performed on the Italian large power system through a detailed
simulation model, both on load buses and on “transit” buses.
The results are very clean and precise, notwithstanding the continuous changing of the real system data (50-Hz band) and the
) imposed. The deidentification high speed (
scribed method allows computing a variety of real-time reliable
indicators to be used in practice, very simple and at low comand
putation cost, mainly based on the distance between
or
and
, as well as on the slope of the variables
under identification [10].
APPENDIX
The identification problem covered in this paper is basically a
case of an underdetermined system, where a free variable is arbitrarily selected and the system is solved based on the selection
and then
of the free variable. The free variable chosen is
is calculated. The here proposed theorem is based on the
circuit shown in Fig. 1, and it establishes to which direction (i.e.,
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 3, AUGUST 2008
higher or smaller value) the selection of the free variable should
be updated in order to approach its correct physical value. The
mechanics of the theorem are based on the variation of the load
and
are conimpedance, and in the assumption that
” interval of their identification.
stant in the brief “
Therefore, the correct application of the Theorem does require
a very short sampling time to assure its thesis. At the end of this
appendix the reader can find a simple numerical example that
clarifies the basic concept behind the theorem.
Theorem: In a two-bus Thevenin equivalent circuit in Fig. 1
subject to load variation, the link between the estimation errors
(
and
) of the Thevenin parameters modules is always given, under the following assumptions, by the here after
thesis:
Definitions:
the variation on the load impedance;
•
the initial identification error on
;
•
•
•
•
•
•
•
•
the complex error on the phasor
;
the given identification error on
.
Assumptions:
, that is
;
and
the correct values of the Thevenin parameters, be constant;
is an inductive impedance;
at each step of load variation.
Thesis:
,
If the load increases:
with
then:
with
then:
or analogously:
• If the load reduces:
,
with
then:
with
then:
.
Proof: With reference to Figs. 1 and 2, the circuit current
measured in the load bus is associated to the last estimated
system model that generally includes parameters errors. On
the other hand, that current is coherent with the correct model
(without errors on the parameters). Therefore the following two
equations are consistent:
(A1)
(A2)
Now, without loss of generality
in the real axis. According to that
Note that, with the load held constant, there is no way to identify
the two errors.
Conversely, considering a load change during two subsequent
) and ( ) and maintaining
constant
sampling times (
between the two, it is possible to write
(A6)
(A7)
Subtracting (A6) from (A7) yields
(A8)
(A9)
Equation (A9) confirms the thesis statement.
,
is also overIn words, in case have overestimated
estimated (A4) and increasing the load,
changes, assuming lower values. On the contrary, in case have underesti,
is also underestimated and increasing the load,
mated
assumes higher values. Analogous conclusions are inferred from the case of reducing the load.
Results presented in [7] showed that the simplification of conas a fixed value during load build-up, resulted on
sidering
critically dependent on the value fixed
an identification of
for
and able to achieve the correct value only at the MPT
(we always start from
). In agreement with the results presented in [7], the following corollary is given to show
that at the MPT the errors on both
and
cancel out, and
a precise identification is possible.
and
, that is
Corollary: If
, or
if
and
, that is
,
then there will be a sampling interval
where:
•
;
• this condition corresponds to Thevenin’s equivalent maximum loadability;
•
is the value of the load bus impedance at the
.
reached maximum loadability point, as if
Proof: From (A5), giving evidence to the load impedance
change:
is being considered to lie
(A10)
(A3)
(A4)
A4 links the identification errors at any sampling time and
shows that the two errors have the same sign. Moreover, from
the equality of the current in the first two equations
(A5)
In Fig. 1, the MPT is reached (with
) when
. At this limit condition, reachable under the
given hypotheses when
, the following result is obtained
from (A10) that refers to the analysis with the estimation errors:
(A11)
CORSI AND TARANTO: REAL-TIME VOLTAGE INSTABILITY IDENTIFICATION ALGORITHM
1279
ation of the impedance load and the variation of the Thevenin
impedance have the same direction, the value of the Thevenin
voltage should be reduced, otherwise, it should be augmented.
A similar analysis could have been done for the case when the
load impedance increases. In this case, the conclusions would
be the opposite, i.e., when both impedance variations have the
same direction, the value of the Thevenin voltage should be
augmented, otherwise, it should be reduced.
TABLE I
CIRCUIT VARIABLES WHEN OVER-UNDER ESTIMATING E
The correspondent circuit current is
ACKNOWLEDGMENT
The authors would like to thank F. Parma, M. Salvetti, and
M.Pozzi for their contributions regarding the SICRE simulator.
(A12)
This confirms the thesis that the MPT happens when
, even in the presence of estimation errors.
In words the corollary states that at the sampling time ,
reaches a condition that combined with
is equivalent to say that
. In this condition, it is also
. Therefore,
verified the MPT corresponding
the approaching of the MPT corresponds to cancel the errors on
and
. At , the load bus phasor measurements allow
. That also means
cannot be
the correct estimation of
maintained constant during the identification process otherwise
the correct parameters identification can be achieved at the MPT
only.
A Simple Numerical Example: A very simple numerical
example shows the basis of the parameters adaptive identification. Referring to Fig. 1, we assume that the correct values
for the Thevenin voltage and impedance are 20 V and 1 , respectively. To explain the core feature of the proposed method,
we suppose that these values are not known. To actually estiand
be
mate them we need to assume the values of
constant between two subsequent measurements (Considering
a sampling rate of 20 ms this assumption is very reasonable).
Starting the analysis with load impedance equal to 9 the circuit current is 2 A, and the voltage across the load impedance
is of 18 V. In the next step the load impedance decreases to a
value equal to 8 . Without changing the logic of our analysis,
we could also assume that we had measured the load voltage
and current, and then calculated the load impedance. Now we
separate the analysis in two ways—one for the case when we
overestimate the generator voltage value, and one when we underestimate it.
Overestimation—Table I, when estimating
1)
shows the values of the circuit
variables (columns 5 and 6). Note that when we over, a decrease in
is accompanied by a
estimate
decrease in
.
—Table I, when estimating
2) Underestimation of
shows the values of the circuit variables
,a
(columns 7 and 8). Note that, underestimating
decrease in
is accompanied by an increase in
.
In conclusion, when the load impedance is decreasing, it
can be inferred from this simple analysis, that when the vari-
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Sandro Corsi (M’89) received the Doctorate degree in electronics from the
Polytechnic of Milan, Milan, Italy, in 1973.
In 1975, he joined ENEL’s Automation Research Center, where his main interests were power system voltage control, generator control, power electronics
devices, advanced control technology, and power system automation. He was a
Research Manager at ENEL S.p.A. Ricerca beginning in 1997, in charge of the
Power System Control and Regulation Office. After the ENEL reorganization,
he was a Research Manager at CESI, in charge of the Network and Plant Automation Office from 1999 and of the Electronics and Communications Office
in the Automation Business Unit from 2000. Since 2001, he has been with CESI
R&D and Sales. He is author of more than 80 papers on system control.
Dr. Corsi is a Member of CIGRE, IEEE-PES, CEI, and the IREP Board.
Glauco N. Taranto (S’92–M’96–SM’04) was born in Rio de Janeiro, Brazil,
in 1965. He received the B.Sc. degree in 1988 from the State University of Rio
de Janeiro, the M.Sc. degree in 1991 from the Catholic University of Rio de
Janeiro, and the Ph.D. degree in 1994 from Rensselaer Polytechnic Institute,
Troy, NY, all in electrical engineering.
In 2006, he was on sabbatical leave as a Visiting Fellow at CESI, Milan, Italy.
Since 1995, he has been with the Electrical Engineering Department, Federal
University of Rio de Janeiro /COPPE, Brazil, where he is currently an Associate
Professor. His research interests include power system dynamics and controls,
intelligent control, and robust control design.
Dr. Taranto is a member of the IEEE PES, CSS, PSDP Committee, and
CIGRE.