1
Thevenin Equivalents, Adjoint Equivalents and
Power Flow Equations
Daniel F. C. M. S. Dias
Abstract — This work starts by presenting a theoretical study
about power flow, voltage stability, Thevenin’s theorem and
Tellegen’s theorem applied to adjoint networks where some
existing models are also included. ext follows the approach of
the methods of analysis used in this dissertation, with special
attention to the two models of Thevenin equivalent, where each
one represents an operating situation (situation of short circuit
and normal operating conditions of an electrical power system),
and corresponding calculation of voltage at its terminals. In
addition to these, it is also shown how to calculate the power flow
(software MATPOWER) and the method chosen to indicate the
voltage collapse (
). As the simulation part, two types
of disturbances are carried out: at a load bus and at a branch.
The first is tested in three electrical power systems (3, 6 and 39
buses), in order to compare the two operation situations and
their respective equivalent Thevenin models with the solution of
the chosen power flow and obtain conclusions about them. The
second is analyzed only in the 39 bus system, because it is the
most complex one. These disturbances occur on the critical buses
and branches of each system, in order to obtain more accurate
results.
Index Terms — Electrical Power Systems, Power Flow,
Tellegen’s Theorem, Thevenin’s Theorem, Voltage Stability.
I. INTRODUCTION
T
HE electrical power systems (EPS) are normally studied
using the power flow equations, knowing that their
interest is not usually in the solution of all currents and
voltages, but the current and voltage of a small part of the
network. So, given the need to improve the speed of
calculations of these systems, it is convenient to be able to
replace much of the network by a simple equivalent circuit.
It is in this context that, while looking for alternatives to
the solution of the power flow equations for a small part of the
system, the Thevenin’s theorem and the Tellegen’s Theorem
together with the adjoint networks appear as possible valid
equivalent models.
Thevenin’s theorem plays an important role in the study of
linear systems. Knowing that the EPS are essentially
nonlinear, the question lies in the proper function of
Thevenin’s theorem in such systems beyond the usual
calculation of short circuits (already used with guarantees).
Among the circuit analysis theorems, Tellegen’s theorem
is unusual because it depends solely on the Kirchhoff’s laws
and the network topology. This theorem can therefore be
applied to all electrical systems that comply with Kirchhoff’s
laws, be they linear/nonlinear and variants/invariants in time.
Tellegen’s theorem applied to the adjoint network
technology is already considered as a valid and fast solution to
this kind of problems. Its implementation performance in
these systems has been extremely accurate after different types
of disturbances (variations in voltage, impedance, power,
changes in the network structure, etc.).
With increasing demand on the quality, safety and
operating conditions of the electrical power grids, as well as
economic benefits and environmental constraints, the EPS are
operated closer to their limits. Knowing this, it is
understandable that the voltage stability is an issue with
significant importance in the electrical industry of the last
years, which makes it convenient to study it in this work.
Thus, it is quite interesting to study and develop models of
Thevenin equivalents and adjoint equivalents (using
Tellegen’s theorem) in electrical power systems, testing them
over a number of perturbations, particularly on the system’s
voltage stability limits.
In this work a theoretical study is presented of its
important principles, including some already existing models.
Next follows the approach of the analysis models used,
including the chosen method of power flow (Newton’s
method), the voltage collapse proximity indicator and two
models of Thevenin’s equivalent for the two different
operating situations (situation of short circuit and normal
operating conditions of an electrical power system), called
respectively Model 1 and 2. Both these models (especially
Model 2, because it works with the normal operation situation
of an EPS), along with their corresponding calculation of the
voltage at its terminals are the original contributions of this
work.
In the end, there are some case studies conducted with
three different power systems (3, 6 and 39 bust test systems),
in order to compare those two Thevenin equivalents for both
different situations with the power flow equations. For this,
some disturbances in the load buses of all the three systems
and disturbances in a branch of the 39 bus system are
considered, starting always from an initial case also studied.
II. THEORETICAL PRINCIPLES
A. Power Flow
In electrical power systems, power flow (1) is the steady
state solution including the network, generators and loads. It is
a fundamental problem in these kinds of systems.
The power flow and its solution consist of the following
steps:
- Formulation of the mathematical model representative of the
system;
- Specification of the bus type and quantities for each one;
- Numerical solution of the power flow equations, obtaining
the magnitude and argument of the voltages of each bus;
- Solution of the power transfer in each branch.
2
B. Voltage Stability
Voltage stability (2) refers to the ability of a power system
to maintain steady voltages at all buses in the system after
being subjected to a disturbance from a given initial operating
condition. It depends on the ability to maintain/restore
equilibrium between load demand and load supply from the
power system.
Studies have been performed to predict voltage collapse
(lower limit of the voltage from which the system becomes
unstable) with both static and dynamic approaches. Since the
objective of this work does not require the precision of
dynamic analysis methods, mostly static analysis methods are
referred in the following paragraphs.
The P-V and Q-V curve methods (3) are classical
techniques to predict voltage collapse. These curves are
produced by running a serious of load flow cases. The critical
point is where the curve makes an inversion from the upper to
the lower level. Sterling et. al. (4) studied voltage collapse at
load buses of the network using the concept of maximum
power transfer between two buses. Using Thevenin’s
equivalent for the relevant bus, the open circuit voltage and
Thevenin’s equivalent impedance are calculated and utilized
in deriving the stability index. This index is the ratio of
Thevenin’s impedance to load impedance and is at a
maximum of 1 when both are equal, indicating the system’s
maximum stable loading at that bus. If the impedance ratio is
lower than 1, the system is said to be stable. The system looses
stability when the impedance ratio exceeds 1.
Jasmon et. al. (5) formulates a voltage stability index, L,
from the voltage equation derived from a two bus network and
it is computed using Thevenin equivalent circuit of the power
system referred to a load bus. To maintain a stable system at a
specific load bus, the index has to be lower than 1. The system
starts to be close to its voltage stability limit when the index
approaches 1.
Gubina et. al. (6) used Tellegen’s theorem associated with
Thevenin’s theorem to calculate an index of voltage stability,
obtaining satisfactory results. Tellegen’s theorem is used to
simplify determination of the Thevenin parameters, which are
calculated in a different way than adaptive curve-fitting
techniques. To evaluate the system’s voltage stability at a
given bus, it requires the voltage and current phasors
measurements only.
C. Thevenin’s Theorem
Thevenin’s theorem (7) states that a linear circuit ended in
two points a and b, containing any number of voltage
generators, can be replaced by a single voltage generator in
series with an impedance connected between those two points,
called respectively VTH and ZTH (Fig. 1). The voltage generator
VTH is equal to the open circuit voltage measured between a
and b; the impedance ZTH is the impedance of the circuit
measured between a and b when the voltage generators are
short circuited.
Fig. 1 – Thevenin’s equivalent circuit.
This theorem is widely used for the calculation of short
circuits, used with the superposition theorem. Short circuit (1)
means a low impedance path, resulting from a defect, through
which the current (generally very high) is closed. This is an
abnormal situation in EPS that requires immediate action due
to the damage it may do. Thus, the calculation of the short
circuit currents is a fundamental task in this type of systems.
Like said in the section before, Thevenin’s theorem is
recently being used to predict the voltage collapse. Besides the
already referred (3), (4), (5) and (6), others like Wang et. al.
(8) and Yu et. al. (9) proposed Thevenin based models to
observe and assess voltage stability. The first converts the
whole system into an equivalent two bus system by multiple
measured synchronized phasors, using those phasors of the
target bus and the buses connected to it which are measured in
WAMS (wide area measurement system). The relationship of
the load impedance magnitude and the Thevenin’s equivalent
impedance magnitude is analyzed when voltage stability
problem occurs. Yu et. al. (9) monitors the weak voltage load
bus groups with Thevenin equivalent parameters, presenting a
power flow data based simulation method to track those
parameters.
Rao et. al. (10) investigates the validity of his Thevenin
model in obtaining Q-V characteristics of P-Q loads, the main
consideration for analysis of the problem of voltage instability
and reactive power management. The model works with a
very reasonable accuracy in non-linear systems, and cannot be
the same as the one that is determined for fault analysis. To
find the Thevenin model, the load buses were approximated to
constant voltage sources, as well as the generator buses. The
Thevenin impedance is computed at the load bus. The error
increases with the system size and also with the wide
variations in Q.
Bahadornejad et. al. (11) develops a method to estimate
power system Thevenin impedance, that is based on signal
processing on the measured data in the load bus. It is shown
that the cross-correlations of the changes in the load voltage
and current with respect to the changes in the load admittance
can be used to estimate the system Thevenin impedance. The
required steps are clearly shown, being the method validated
by simulation, also on real data.
D. Tellegen’s Theorem and Adjoint !etworks
Tellegen’s theorem (12) expresses a relationship between
magnitudes of the branches of two adjoint networks, i.e.
networks with the same graph. In its strong form, and
as being adjoint networks, expresses the
considering and 3
orthogonality between the voltages of a network and the
currents of the other.
This theorem and the concept of adjoint networks have
been applied to a variety of functions and in several fields, but
it’s their application to power systems that interests to this
work. Examples of some of those applications to power
systems are the voltage stability index calculated by Gubina
et.al. (6), some works from Ferreira ( (13) and (14)) and from
both Ferreira and Jesus ( (15) and (16)). Both of them
confirmed Tellegen’s theorem and adjoint networks as a solid
alternative to the normal power flow equations, using them to
calculate voltages and currents of EPS after disturbances in
that system.
III. ANALYSIS MODELS
A. Power Flow
The computer program chosen to solve power flows is the
MATPOWER (17), which is a package of MatLab files
developed by PSERC (Power Systems Engineering Research
Center) researchers. In this work, the power flow solver is the
one based on a standard Newton’s method (18) using a full
Jacobian, updated at each iteration.
In order to return the solution of the power flows from
MATPOWER in output arguments and to be able to use them,
a MatLab file was created. This file also calculates the
currents injected at each bus and transferred in each branch.
B. Voltage Collapse Proximity Indicator
Despite all the methods to predict the phenomenon of
voltage collapse referred before, the one chosen to use in this
work was the one from Moghavvemi et. al. (19), due to its
simplicity and computational mathematics (compared to other
methods, the solution is faster to find) combined with its
proven capabilities in predicting the voltage collapse.
Considering a typical transmission line of a power system
network, Fig. 2 shows its representation through an equivalent
Thevenin network, modeled with its parameters.
Fig. 2 – Transmission line modeled with its parameters.
In this method, the load of the line is treated as the power
that is transferred at the receiving end through that particular
line only, instead of the total load at the bus. Therefore, ∠
is the line impedance, ∠ф is the corresponding load
impedance and ф = ⁄ .
It is considered the most frequent case, where only the
modulus of the load impedance is varied while ф remains
constant. With the increase of demand in load, decreases
and current I increases, which leads to a voltage drop at the
receiving end. The maximum real power that can be
transferred to the receiving end , and the maximum real
power loss in the line , can then be obtained using the
boundary condition ! ⁄! = 0, which leads to ⁄# = 1.
Base on these maximum permissible quantities, the
following Voltage Collapse Proximity Indicators were
proposed by the author (note that the values of and are
obtained from conventional power flow calculations):
%&' ()* =
= .
%
012ф
[1]
+,-
#
∙
#
− ф .
4 ∙ 012 4
7
2
%&'8(## =
= .
012
[2]
+,- %#
∙
#
− ф .
4 ∙ 012 4
7
2
As the loading condition moves closer to the critical
operation point, both %&' ()* and %&'8(## approach the
value 1. Close to the voltage collapse point, %&' ()* is
less sensitive to further loading, while %&'8(## is found to
be highly sensitive. Therefore, values of %&' ()* are
adequate to indicate for indicating the voltage collapse in the
line, but evaluation of %&'8(## helps to locate the voltage
collapse point exactly.
Because MATPOWER does not save the values of the
power loss in the line from the power flow’s solution (needed
to calculate the indicator %&'8(## ), and knowing that exact
precision is not needed, in this work only the %&' ()* is
used as a voltage collapse proximity indicator. Thus, a
MatLab file was created to calculate the value of this indicator
for all branches of a system, indicating the critical branch and
critical bus, which is the one in the receiving end.
C. Model 1 – Thevenin’s Equivalent for the short circuit
situation
Under the assumption of the short circuit theory, and to
verify the proper function of Thevenin’s equivalent in short
circuits, a model was developed to calculate its equivalent for
this situation that does not depend on computer programs like
MatLab but in the software OrCAD PSpice together with
PSpice Schematics (very important in an electrical engineer
student’s life).
Thus, the following steps are used in the construction of
the electrical circuits for the calculation of ZTH:
- Replace the impedances between the system’s branches by
resistors and capacitors/coils (in their respective metric
system);
- Replace the load buses by their equivalent load impedance,
through:
||.
[3]
9: = ∗
<:
- Place the existing generators in short circuit (instead of
using its transient reactance, like in a normal short circuit
calculation);
4
- Place the AC voltage generator (VAC device) on the bus
where the short circuit happens, with the same value as the
pre-fault voltage on that bus;
@ A:*
@ A:*
- Find the values of %>?
and 'B
through two separate
simulations (20), and thus the value of ZTH with [4]. Note that
@ A:*
the value of %>?
is measured by using VPRINT1 (from
PSpice Schematics), changing the AC, MAG and PHASE
cells (they have to be selected and entered OK). After
@ A:*
removing the VPRINT1 device, the value of 'B
is
measured by the insertion of an IPRINT device (from PSpice
Schematics) between the terminals (between the bus and the
ground or between the two buses), that also has to have the
AC, MAG and PHASE cells selected and entered OK. The
pertinent data from both simulations (set the analysis for AC
Sweep/Noise with a linear sweep beginning and ending at a
frequency of 50 Hz) are found in the output files of PSpice.
@ A:*
%
[4]
9 >? = >?
@ A:*
'B
The value VTH of the equivalent circuit is the electrical
potential difference between the specific bus and the ground
or between buses i and j in the pre-fault situation, depending
on the case.
D. Model 2 – Thevenin’s Equivalent for the normal
operating conditions
Ohm’s law (21) states that the voltage across conducting
materials is directly proportional to the current flowing
through the material. Thus, the impedance is defined as the
ratio of the electrical potential difference between two points
and the current through these two points:
[5]
9=
Model 2 was then created based on these assumptions and,
unlike Model 1, is applied in normal operating conditions of
an EPS. With the aim of injecting power on the bus where the
equivalent should be found, a unitary current source is inserted
between the bus and the ground or between the buses i and j,
as shown in Fig. 3. Thus, the value of 1+j0 A is added to the
previous injected current in bus i. That same value is removed
from the previous injected current in bus j, if there is such a
bus, i.e., -1+j0 A added.
Solving the power flows before and after the current
injection, obtaining both voltage values, it is possible to find
Thevenin’s impedance for both cases:
DAE,8
− AEA:A,8
[6]
A
9 >? = A
∆
DAE,8
DAE,8
GA
− H
I − GAEA:A,8
− HAEA:A,8 I
[7]
A
9 >? =
∆
The value VTH of the equivalent circuit is, like for Model 1,
the electrical potential difference between the specific bus and
the ground or between buses i and j in the pre-fault situation,
depending on the case.
Two MatLab files were created for this model, in order to
be able to find the correspondent power injected in the buses
for both cases, after the insertion of the current source in the
circuit. These files run numerous iterations, during each are
held several cases of small changes (increase/decrease) in the
values of the load power involved. At the end of each
iteration, the best option to achieve the unitary value of
injected current is chosen.
E. Calculating the Voltage between the Thevenin’s
Equivalent Terminals
After calculating Thevenin’s equivalent parameters VTH
and ZTH (via the two described models), it is necessary to
calculate the voltage Vab (voltage between the terminals a and
b of Fig. 1). Depending from the case, these terminals can be
the bus i and the ground (Vab applied to a bus) or the buses i
and j (Vab applied to a branch).
In order to find the impedance between terminals a and b
(Zab) when looking to find Vab applied to a bus, one has to
pay attention to the load before and after the disturbance.
Considering Zo as the initial load impedance and Zc as the
final load impedance, it is concluded that Zc equals to the
parallel between Zo and Zab, like in Fig. 4. Thus, Zab is
calculated by:
9K
9,J =
[8]
9
1 − L KN
9M
Fig. 4 – The parallel between the impedances Zo and Zab equals to Zc.
Fig. 3 – Current injection of 1 A: (a) in bus i; (b) between buses i and j.
Knowing all parameters, the voltage Vab between the bus j
and the ground is found by:
9,J
[9]
,J = >? ∙
9 >? + 9,J
5
When it comes to Thevenin’s equivalent circuit between
buses i and j (branch), the objective is not to calculate the
voltage at its terminals depending on the load disturbances but
to make changes in its network. Therefore, in order to disturb
the system, Zab should lean to zero (short circuit) or to infinity
M
(open circuit). Considering ,J
as the initial impedance of the
branch, the way to put Zab as a short circuit is given by [10]
and to put it as open circuit is given by [11].
9M,J
[10]
9E,J =
[11]
9E,J = ∗ 9M,J
For this section, two MatLab files were also created, that
have the similarity of calculating Thevenin’s equivalent
parameters for the two previously described models. Besides
that, one of them calculates Vab between the bus i and the
ground after load disturbances in that bus and the other one
calculates the effects of the branch disturbances, which are
observed by graphs from MatLab itself.
IV. DISTURBANCE CASE STUDIES
A. Initial Analysis
In order to compare both models of Thevenin’s equivalent
for the two different operating situations with the power flow
by Newton’s method, three test systems (3-bus, 6-bus and 39bus) are used. Both initial data from the 3-bus and 6-bus
systems are given in the Appendix, and were taken from (10)
and (15), respectively. The initial data from the 39-bus system
was taken from (17).
In this section, MATPOWER was used to make an initial
analysis of each of those three systems, indicating all the
power flow results including all the values of the currents
(injected at each bus and transferred in each branch).
B. Critical buses and branches
In order to truly evaluate the two models, the critical
branch and corresponding critical bus of each system is found,
using the voltage collapse proximity indicator studied before
(%&' ()* ).
In the 3-bus system, the critical branch is the one between
the buses 1 and 3 (branch number 2), being the bus number 3
the critical one. The 6-bus system showed that the critical
branch is the one between the buses 1 and 5 (branch number
3), being the bus number 5 the critical one. For the 39 New
England bus system, because the calculated critical bus was a
generator bus (the bus 31 maintains a constant voltage with
power variations), it was necessary to resort to some of the
references ((22) and (23)) to find the critical bus of this well
known system. Therefore, the critical branch used was the one
between the buses 12 and 11 (branch number 35) and the
critical bus of this system is the bus number 12.
C. Disturbances on a load bus
Since the power load is complex, the voltage depends on
the combination of active and reactive power of the concerned
bus. The point of voltage collapse will then be affected by
how these two powers vary independently and/or the
combination of the two, i.e., the apparent power.
Thus, for each test system there are three variations of
power load (only active power, only reactive power and both
powers) only on the critical bus, so that it could tend to
voltage instability. Through these disturbances, both the
magnitude and the argument of the voltage are recorded by the
two Thevenin’s equivalents models and by the power flow
equations from MATPOWER (Newton’s method).
In the 3-bus system, Model 1 presented excellent results,
having a maximum voltage magnitude error of 1,50% near
voltage collapse when changing only the active power. Model
2 produced satisfactory results near voltage collapse, having a
maximum voltage magnitude error of 14,99% for the reactive
power variations.
When studying the 6-bus system, Model 1 again showed
excellent results. The maximum voltage magnitude error was
with the active power disturbance, with a percentage error of
1,69%. On the other hand, Model 2 produced the worst results
of all the three systems. Near voltage collapse, this method
had voltage magnitude errors of 23,11%, 33,81% and 29,80%
when variations of active, reactive and apparent powers
occurred, respectively.
In the 39-bus system, Model 1 presented the worst
maximum voltage magnitude error (active power variations
only), with 2,78%. Still, with an error so low, it is possible to
say that the results were also excellent. Model 2 had again
satisfactory results (although worst than in the 3-bus system),
with maximum voltage magnitude errors near voltage collapse
of 16,61%, 15,41% and 15,53% when variations of active,
reactive and apparent powers occurred, respectively.
Through the obtained results of this study, it is possible to
conclude that Model 1 for the short circuit situation, like
expected, presents good results for the voltage’s magnitude
calculation. On the other hand, Model 2 presents considerable
errors when the voltage in the critical bus is near its collapse.
Note that, in both cases, the errors in the models tend to grow
according to the proximity of the voltage collapse point, as
expected.
D. Disturbances on a branch
Because the possible obtained results for all systems had
the same conclusions for this kind of disturbances, only the
39-bus system was selected to be studied (since it is the
biggest one from all the three test systems).
After the calculation of Thevenin’s equivalent parameters,
with equations [4] and [7], and in order to make the critical
branch tend to short circuit or open circuit, various branches
(n=1000) are placed in parallel with the terminals a and b
(critical branch of the system) or the load is increased
progressively, respectively. The curves of voltage and current
between these terminals are then analyzed and verified that the
branch do tend to those situations. In the short circuit situation
the voltage Vab tends to zero and the current Iab tends to its
P
maximum value (',J = QR ). On the other hand, in the open
SQR
circuit situation the voltage Vab tends to its maximum value
(Vab=VTH) and the current Iab to zero.
6
V. CONCLUSION
This work was initiated with a study on the fundamental
theoretical principles. Thus, a literature review was performed
on the themes outlined in the work, including some of its
existing methods. After it, an approach of the analysis models
used was made, including the two models of Thevenin’s
equivalent for the two operating situations (situation of short
circuit and normal operating conditions of an electrical power
system) and their corresponding calculation of the voltage at
its terminals.
Model 1 uses the software OrCAD PSpice together with
PSpice Schematics to create electrical circuits based on the
short circuits theory and correspondent Thevenin’s equivalent.
This actually proved to be impractical, due to the need to build
a new circuit with every new system or network
reconfiguration. Model 2 is based on Ohm’s law and on the
definition of impedance, and works on the normal conditions
of nonlinear systems, like the EPS. For this model, because
MATPOWER (program used to calculate power flows) does
not let you insert a unitary current source directly in a bus of
the system, some MatLab algorithms had to be created. These
algorithms make it possible to inject unitary current sources in
one or 2 buses of any kind of system, an essential requirement
in this Thevenin’s equivalent model.
In the simulation part of this work two different kinds of
disturbances were made (after the calculation of the critical
buses and branches of all testing systems, starting from an
initial case study). Disturbances in the critical load bus were
tested in three different systems (3-bus, 6-bus and 39-bus),
while disturbances on a critical branch was only made in the
39-bus system.
Model 1, because it uses the situation of short circuits, was
expected to have good results on the load bus disturbance
studies, and it did. On the other hand, because Model 2 works
with the nonlinearity of the electrical power systems, it was
expected to not have good results. That assumption was
correct, since this model usual presented considerable errors
when the bus was near voltage collapse (maximum error of
33,81%, for the 6-bus system). In both models, its errors
tended to grow according to the proximity of the voltage
collapse point.
When concerning the disturbances on the critical branch,
both Thevenin’s equivalent models had similar results. When
the branch tended for short circuit and open circuit, both Vab
and Iab presented expected results.
As future work, and since in this work it was concluded
that Thevenin’s equivalent models for nonlinear electrical
power systems have problems, it seems interesting to find
other kind of equivalent models to use in those nonlinear
systems. Thinking already in that objective, a theoretical study
about Tellegen’s theorem applied to adjoint networks was
made, given its validity in this type of problems.
It is expected that these adjoint equivalents will obtain
accurate results when compared to the Thevenin’s equivalents
developed in this work and even with the power flow
equations solution.
APPENDIX
Fig. 5 – Representation of the 3 bus test system.
TABLE I
BRANCH DATA FOR THE 3 BUS TEST SYSTEM (p.u.)
From
To
Bus
Bus
1
1
2
Resistance
Reactance
Susceptance
2
0,02
0,08
0,02
3
0,02
0,08
0,02
3
0,02
0,08
0,02
TABLE II
BUS DATA FOR THE 3 BUS TEST SYSTEM (p.u.)
Bus
Type
Magnitude
of Voltage
Angle
Pg
Qg
Pc
Qc
1
Slack
1,02
0,00
-
-
-
-
2
Gen
1,01
-
0,50
-
-
-
3
Load
-
-
-
-
1,40
0,50
No.
Fig. 6 – Representation of the 6 bus test system.
7
TABLE III
BRANCH DATA FOR THE 6 BUS TEST SYSTEM (p.u.)
From
To
Bus
Bus
1
Resistance
Reactance
Susceptance
2
0,10
0,20
0,00
1
4
0,05
0,20
0,00
1
5
0,08
0,30
0,00
2
3
0,05
0,25
0,00
2
4
0,05
0,10
0,00
2
5
0,10
0,30
0,00
2
6
0,07
0,20
0,00
3
5
0,12
0,26
0,00
3
6
0,02
0,10
0,00
4
5
0,20
0,40
0,00
5
6
0,10
0,30
0,00
TABLE IV
BUS DATA FOR THE 6 BUS TEST SYSTEM (p.u.)
Bus
Type
Magnitude
of Voltage
Angle
Pg
Qg
Pc
Qc
1
Slack
1,05
0,00
-
-
-
-
2
Gen
1,05
-
0,50
-
-
-
3
Gen
1,07
-
0,60
-
-
-
4
Load
-
-
-
-
0,70
0,70
5
Load
-
-
-
-
0,70
0,70
6
Load
-
-
-
-
0,70
0,70
No.
Fig. 7 – Representation of the 39 bus test system.
REFERENCES
1. Paiva, J. P. Sucena. Redes de Energia Eléctrica – Uma Análise Sistémica.
Instituto Superior Técnico : IST Press, 2005.
2. Definition and Classification of Power System Stability. IEEE/CIGRE
Joint Task Force. Maio 2004, IEEE Trans.Power Systems, Vols. 19, no.3,
pp. 1387 - 1401.
3. An analysis of voltage collapse in electric power systems. Bhat, R. and
Gross, C.A. Outubro 2005, Proceedings of the 37th Annual North American,
pp. 325 - 331.
4. Voltage collapse proximity indicator: behaviour and implications. Chebbo,
A.M.; Irving, M.R.; Sterling, M.J.H. Maio 1992, IEE Proceedings C, Vols.
139, no.3, pp. 241 - 252.
5. A new technique for voltage stability analysis in a power system and
improved loadflow algorithm for distribution network. Rahman, T.K.A.;
Jasmon, G.B. Novembro 1995, Proceedings of EMPD'95, Vols. 2, 21 - 23,
pp. 714 - 719.
6. Local Voltage-Stability Index Using Tellegen's Theorem. Smon, I., Verbic,
G. and Gubina, F. Agosto 2006, IEEE Transactions on Power Systems, Vols.
21, no.3, pp. 1267 - 1275.
7. Stevenson, William D. Elementos de Análise de Sistemas de Potência. s.l. :
McGraw-Hill.
8. A Method for Voltage Stability Assessment Based on Wide Area
Measurement System. Su, Yongchun and Wang, Xiaoming. Março 2009,
APPEEC 2009, pp. 1 - 4.
9. Tracking of Thevenin Equivalent Parameters on Weak Voltage Load Bus
Groups. An, Tianyu, et al. Outubro/Novembro 2006, IEEE PES PSCE '06,
pp. 1570 - 1576.
10. Model Validation Studies in Obtaining Q-V Characteristics of P-Q Loads
in Respect of Reactive Power Management and Voltage Stability. Govinda
Rao, G. and Ramachandra Murthy, K.V.S. Dezembro 2006, International
Conference on PEDES '06, pp. 1 - 5.
11. System thevenin impedance estimation using signal processing on load
bus data. Bahadornejad, M. and Ledwich, G. Novembro 2003, Sixth
International Conference on ASDCOM, Vol. 1, pp. 274 - 279.
12. Jesus, C.M.S.C. Aplicação do Teorema de Tellegen e de Redes Adjuntas
ao Cálculo de Tensões e Perdas em Sistemas de Energia Eléctrica. Instituto
Superior Técnico. Dezembro, 2004. Dissertação de Doutoramento.
13. Tellegen’s Theorem and Power Systems - !ew Load Flow Equations, !ew
Solution Methods. Ferreira, L.A.F.M. Abril 1990, IEEE Transactions on
Circuits and Systems, Vols. 37, no. 4, pp. 519 - 526.
14. Accuracy and Complexity in the Modelling of Power Systems Subject to
Large !etwork Disturbances in a Framework given by Tellegen’s Theorem
and Adjoint !etworks. Ferreira, L.A.F.M. Singapura : s.n., Junho 1991,
IEEE International Symposium on Circuits and Systems, Vol. 2, pp. 1228 1232.
15. Power Flow by Adjoint !etworks. Jesus, C.M.S.C. and Ferreira,
L.A.F.M. Setembro 2003, IEEE PES Transmission and Distribution
Conference and Exposition, Vol. 1, pp. 196 - 200.
16. Local !etwork Power Flow Analysis: an Accuracy Level Comparison for
Two Sets of Equations. Ferreira, L.A.F.M., Jesus, C.M.S.C. Novembro
2006, IEEE Transactions on Power Systems, Vols. 21, no. 4, pp. 1616 - 1623.
17. Zimmerman, R.D., Murillo-Sánchez, C.E. MATPOWER: A MATLAB
Power System Simulation Package. [Online] Setembro 2007.
http://www.pserc.cornell.edu/matpower/.
18. Power Flow Solution by !ewton's Method. Tinney, W.F. and Hart, C.E.
Novembro 1967, IEEE Transactions on Power Apparatus and Systems, Vols.
PAS-86, no. 11, pp. 1449 - 1460 .
19. Real-time contingency evaluation and ranking technique. Moghavvemi,
M. and Faruque, O. Setembro 1998, IEE Proceedings, Vols. 145, no.5, pp.
517 - 524.
20. Robbins, Allan and Miller, Wilhelm C. Circuit Analysis with Devices:
Theory and Practice. 2. s.l. : Cengage Learning, 2003. pp. 722 - 725.
21. William H. Hayt, Jr., Kemmerly, Jack E. and Durbin, Steven M.
Engineering Circuit Analysis. sexta edição. s.l. : McGraw-Hill, 2002.
22. Calculation of the Extreme Loading Condition of a Power System for the
Assessment of Voltage Stability. Semlyen, A., Gao, B. and Janischewskyj,
W. Fevereiro 1991, IEEE Transactions on Power Systems, Vols. 6, no.1, pp.
307 - 315.
23. Method of Combined Static and Dynamic Analysis of Voltage Collapse in
Voltage Stability Assessment. Hasani, M. and Parniani, M. 2005, IEEE/PES
Transmission and Distribution Conference and Exhibition: Asia and Pacific,
pp. 1 - 6.