Transmission systems congestion management by using modal participation factors

Abstract

Power system congestion management is a hard task, and the electricity industry restructuring process tends to difficult it even more. The competitive electric power market arises new congestion management perspectives and difficulties. This work demonstrates that modal analysis technique can be a powerful tool for defining corrective actions to overcome congestion problems. Extensive voltage stability margin assessment, for a test system, shows that modal participation factors allow the identification of congested areas, and also the most adequate active and reactive power based control actions to relieve congestion.

Full text

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I. INTRODUCTION
RANSMISSION network congestion is the main
constraint to the optimum exploitation of energy sources.
Congestion can be caused by transmission line and generator
outages, changes in energy demand and uncoordinated
transactions, which can lead to network congestion when the
system is not able to respect security requirements due to line
overload, transient and voltage stability [1]-[2].
Looking for increased competition on electric power
markets, the industry restructuring process tends to deepen the
congestion problem [3].
Therefore, there is a tradeoff between electricity business
and physical system operation. From the point of view of
business management, generators dispatches could be defined
just by the shortest prices. However, these dispatches could
lead to system congestion due to transmission constraints.
The problem associated with the schemes of congestion
management is how to internalize in the dispatch process the
externalities, such as congestion, without increasing the
electricity prices unreasonably and keeping the motivation for
market players to make investments on system expansion.
II. C
ONGESTION MANAGEMENT
Many congestion management schemes have being recently
studied and implemented. Each one takes into account some
CNPq and FAPESP are acknowledged by the financial support provided to
this work.
All authors are with the Scholl of Electrical and Computer Engineering,
State University of Campinas-UNICAMP, CP 6101, Campinas/SP, Brazil,
CEP 13081-970 (e-mail: lui@dsce.fee.unicamp.br).
Phone: 55-19-3788-3739, Fax: 55-19-3289-1395.
peculiarities of the electric power system. Even with different
ways to overcome the congestion problem, all the schemes
intend to maintain the benefits of open accessed markets.
Thus, ancillary services, transmission tariffs, interruptible
load incentives, bid compensation (such as uplifts), reserves
requirements and other means can contribute to the congestion
management scheme definition [1]-[3].
In any case, to define the best scheme for a particular
system, it is necessary to identify the congested areas and
which deficiencies of the system are producing congested
situations. We propose in this work the use of modal analysis
technique for solving this part of the problem.
Providing the active and reactive power impact of each bus
on system security, the expanded modal analysis technique
allows the best choice of control actions to get over
congestion problems, such as the identification of key areas
for interruptible load incentives, the best generator re-dispatch
scheme for eliminating congestion, critical areas for
investment on reactive power ancillary services and others.
III. E
XPANDED MODAL ANALYSIS TECHNIQUE
The linearized power flow equations can be written as follow:






=












=






9
-
9
--
--
4
3
494
393
∆
θ∆
∆
θ∆
∆
∆
θ
θ
(1)
where J
P
θ
, J
PV
, J
V
θ
and J
QV
are Jacobian sub-matrices
representing the sensitivities of active and reactive power with
respect to voltage angles and magnitudes. J is the standard
load flow Jacobian matrix.
Two reduced Jacobian matrices can be defined as:
,QJV
1-
RQV
∆=∆
by assuming
∆P = 0
(2)
,PJ
1-
RP
∆=θ∆
θ
by assuming
∆Q = 0
(3)
where,
393449549
-----
1−
−=
θθ
(4)
θθθ
44939353
-----
1−
−=
(5)
The reduced matrices J
RQV
and J
RP
θ
represent the
sensitivities of system equilibrium with respect to reactive and
active power incremental variations. Information about system
voltage stability can be obtained from these matrices in both
Transmission Systems Congestion Management
by Using Modal Participation Factors
Igor Kopcak, Luiz C.P. da Silva, Vivaldo F. da Costa, and Jim S. Naturesa
T
0-7803-7967-5/03/$17.00 ©2003 IEEE
Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy

perspectives: reactive and active power conditions.
However, matrices J, J
RQV
and J
RP
θ
are singular at the same
point, the same modal information obtained from J, at its
singularity point (the saddle-node bifurcation point), can also
be obtained from J
RQV
and J
RP
θ
[4]. The reduced matrices
provide didactic advantages by decoupling active and reactive
power impacts on system voltage stability. It does not imply in
further approximation since modal analysis is a linear
technique. It should be emphasized, though, that the
unreduced Jacobian, J, shows computational advantages,
mainly related to its higher sparsity degree.
Modal analysis applied to reduced or unreduced Jacobian
matrices results in [4]-[6]:
Γ
Λ
Φ
- =
(6)
where:
Φ - Contain the right eigenvectors of matrix J;
Γ - Contain the left eigenvectors of matrix J;
Λ - Contain the eigenvalues of matrix J.
From the analysis of the critical eigenvalue of J near the
Saddle-Node bifurcation point with associated left and right
eigenvectors, voltage stability critical areas of a power system
can be identified.
Gao [5] defines the UHDFWLYH(power) SDUWLFLSDWLRQIDFWRU
(53)) from the reactive power reduced matrix J
RQV,
and Da
Silva [4,6] defines the DFWLYH (power) SDUWLFLSDWLRQ IDFWRU
($3)) from active power reduced matrix J
RP
θ
. With this, the
conventional modal analysis technique is expanded to the
active portion of the Jacobian matrix.
Similarly to the RPF [5], the APF is defined as the element-
by-element product of the left and right eigenvectors of the
J
RP
θ
matrix. If λ
i
is the i
th
eigenvalue of J
RP
θ
,
and µ
i
and ν
i
its
right and left eigenvectors related to
λ
i
, the participation factor
of bus k to mode i is defined as:
NLLNNL
$3)
ν
µ
=
(7)
The APF reveals those buses where active power changes
are more detrimental to system voltage stability. They
represent the best locations for planning or operation active
power based control actions, such as load shedding and
generator rescheduling, for improving system power transfer
capability. On the other hand, the RPF is related to the
reactive power demand at load (PQ) buses, and indicates the
best locations for reactive power compensation based control
actions.
The New England test system (10 generators and 29 load
buses) is used in this work. Initially, it is compared the
information provided by left and right eigenvectors for the
critical mode, and also their combination as participation
factors. Figure 1 shows that there are just little differences
between the modal shape provided by left and right
eigenvectors and by the participation factors. It should be
remembered that for identifying critical buses or areas the
most important information is the modal shape, and not the
eigenvector values. It can be seen from Figure 1 that nearly
the same critical buses are indicated by the three indices. This
conclusion is confirmed in Table I, which shows 10 critical
buses ranked by left and right eigenvectors, and participation
factors, respectively. Almost the same set of critical buses is
obtained by the three methods. We have confirmed this
characteristic of left and right eigenvectors for many power
systems, and believe that the participation factor is still the
best option, since it combines the two eigenvectors.
Fig. 1. Active participation factor (APF), left and right eigenvectors for the
New England test system.
TABLE I
C
RITICAL BUSES FROM LEFT AND RIGHT EIGENVECTORS AND FROM APF
Critical
Buses
Left
Eigenvector
Right
Eigenvector
Participation
Factor
1 18 17 17
2 9 18 18
3 10 16 9
4 17 9 10
5 36 13 12
6 27 12 16
7 12 10 13
8 35 27 27
9 34 26 36
10 26 24 26
IV. RESULTS AND DISCUSSION
Congestion due to static security is the main transmission
constraint analyzed so far [1]. But, voltage stability margins
contribute significantly to the system $7& ($YDLODEOH
7UDQVIHU &DSDELOLW\) definition. For this reason, this work
focuses on the congestion assessment due to system inability
to assure minimum voltage stability margin criteria.
In this work voltage stability margin is considered as been
the maximum load increase that the system could supply from
the base case loading until it reaches the voltage stability limit.
Voltage stability margin is obtained by using PV-curve
methods [7], [10]. PV curves are obtained in this work by
considering load increases for all load buses in a proportional
way to the base case loading (keeping constant power factor).

System generation level is also increased (in proportion to the
base case injections) in order to match the load increases
during the PV curve construction process. It should be
emphasized that all generators respond for an increase in
demand, and not only the slack bus. Generators reactive
power and tap limits are also properly considered. For each
load increase it is solved a load flow problem, and the set of
obtained equilibrium points defines the PV curve. The
stability margin represents the distance, in MW or percentage,
from the base case operation point to the maximum power
transfer capability point of the system (PV curve nose point).
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The New England test system is used to demonstrate the
adequacy of participation factors for identifying congested
areas. Initially, it is assessed control actions associated with
reactive power variations (
∆Q), looking for voltage stability
margin enlargement.
Figure 2 shows the RPF for load buses. Buses with larger
RPF represent the best places for applying reactive power
based control actions for improving the security level. Large
RPF’s indicate congested areas, where there is undersupplied
reactive power support.
Fig. 2. RPF of New England system load buses, for the critical mode.
Fig. 3. System voltage stability margin, with and without SVC at bus 16.
The reactive power support of each bus is changed with the
inclusion of 69&¶V (6WDWLF 9DU &RPSHQVDWRU). Figure 3
shows the voltage stability margin with and without the
inclusion of the SVC at bus 16. It can be seen that the SVC
provides a significant gain at the margin. In other words, the
SVC enlarges systems transfer capability. In order to assess
the correlation between the RPF ranking and the voltage
stability margin, the SVC is connected to each load bus one by
one, and the margin gain is checked out. Figure 4 shows the
results of this test. It can be noted that the SVC allocation at
buses with larger RPF provides the largest margin gains. This
test validates the idea of using the RPF for identifying
congested areas from a point of view of reactive power
support deficiencies.
Fig. 4. Normalized voltage stability margin gain with SVC allocation at the
load buses.
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This section investigates control actions associated with
active power variations (
∆P) at load buses. Figure 5 shows the
APF for load buses. Similarly to the RPF, buses with large
APF’s represent the best places for applying active power
based control actions. Adequate modifications on the active
power injections (load shedding) for those places would
produce maximum voltage stability margin gain.
Fig. 5. APF of 1HZ(QJODQG system load buses, for the critical mode.

The test consists of applying a 5MW load shedding for
each load bus, and verifying the provided voltage stability
margin gain. It can be seen, from Figure 6, that the margin
gain can be considered closely correlated to the APF. This test
shows that the APF can easily identify congested areas from a
point of view of extreme active power demand.
Fig. 6. Normalized voltage stability margin gain provided by 5MW load
shedding at each load bus.
From the modal information a congestion cost could be
attributed to the active power demand of each bus, which
allows the identification of areas in which the increase of
demand are acceptable (buses with small APF’s), and also
areas where interruptible load should receive incentives (buses
with large APF’s). The APF could also be used for defining
preventive load shedding schemes for relieving congestion.
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This section evaluates control actions associated with
active power variations (
∆P) at generator buses, such as
generator active power rescheduling.
Generator with large APF’s can provide additional active
power to the system without severely depleting system
reactive reserves [6]. The ones with small APF’s are more
contributing for system congestion. It means that system
reactive reserves are relieved if these units inject less active
power. In other words, by increasing the active power
generation in large APF’s units, and reducing it in buses with
small APF’s, a congested situation could be relieved. It has an
impact, of course, on system dispatch cost, and also on the
energy price.
Figure 7 shows the APF for generator buses. Generator 2 is
the slack bus and has no APF. Generator 3 has the smallest
APF and must be encouraged to generate less active power for
improving voltage stability margin. The test consists of a
20MW generation increase at the buses 1 and 4 to 10
individually, while reducing the same 20MW generation at
Bus 3.
Fig. 7. APF of 1HZ(QJODQG system generator buses, at the critical mode.
The test results are shown at Figure 8. It can be seen that
the generation increases at buses 1, 8 and 10 provide larger
voltage stability margin gain. It should be emphasized that the
voltage stability margin gains are related to the generation
reduction at Bus 3, and because of this there is not a gain for
the Generator 3.
This test demonstrates that the generators APF’s could be
used for active power re-scheduling purposes with the
objective of relieving congested situations, since it indicates
which generators should inject more, and which generators
should inject less active power.
Fig. 8. Normalized voltage Stability Margin gain provided by 20MW
rescheduling.
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Sections A, B and C demonstrates that the modal
participation factors can provide the best candidate buses for
control actions related to active and reactive power changes at
generator and load buses for improving voltage stability
margin. In this section we introduce the participation factor
information at the optimal power flow problem. The idea is
iteratively manage active and reactive power for generators
and loads following the participation factors optimal direction,

in order to improve system margin, and relieve congested
situations.
The modal analysis technique provides voltage stability
critical areas and gives information about the best actions for
improving system stability margins. Modal participation
factors indicate which generators should inject more active (or
reactive) power to improve the voltage stability margin, and
which generators should inject less [6]. This information is
added to the system dispatch problem, so that the final
solution leads to an optimized reactive power injection for
each generator and synchronous condensers, from a
perspective of improving voltage stability margin.
Figure 9 illustrates the voltage stability margin against the
iterations of such a MVAR optimization methodology. It can
be noted that the method leads to a margin improvement of
7.5%. It is significant, since there is no modification
on
generators active power injections. In other words, the
stability margin is improved with no cost deviation compared
with the economical optimal solution. The OPF just redirects
generator reactive injections following the participation
factors information.
Figure 8 – Voltage stability margin improvement with MVar optimization.
Figure 8 also indicates that the margin saturates as the
program perform subsequent iterations. It means that the
optimal MVAR injections from the point of view of margin
improvement have been found, and the process
should be
interrupted.
This idea is also applied for a generator active power re-
dispatching test looking for improved voltage stability margin.
The participation factors are used to define security costs,
which attribute low costs for generators with high APFs, and
high costs for generators with low APF’s. By solving the OPF
with the objective of minimizing system generation cost we
can drive the solution to improved voltage stability margin.
Figure 9 shows the results. It can be seen that there is a margin
improvement of 10.8%. The initial PV curve and final PV
curve, after ten iterations of the active power rescheduling
process, illustrates the margin enlargement.
Figure 9 – Voltage stability margin improvement with MW re-scheduling
It is also shown at Figure 10 that the participation factors
tend to equalize as the process runs. The APF’s for iteration
one shows different values for each generator, it means that
the margin can be improved from active power re-scheduling.
For iteration ten we have almost equal APFs for all
generators, indicating that the optimum dispatch from a
perspective of maximizing voltage stability margin has been
found.
Figure 10 – Generators active participation factor.
The participation factors are also tested for defining
minimum load shedding schemes for improving voltage
stability margin, and consequently for relieving congestion.
APF’s are used to select the best candidate buses for load
shedding, and also used to define load-shedding costs for
fictitious generators installed at the candidate buses. Large
APF loads provide the best margin gain for a given load shed,
so it receives the lowest costs. On the contrary, low APF loads
receive high costs. By running the OPF with the objective of
minimizing the fictitious generators cost we can redirect the
load shedding in the sense of maximizing voltage stability
margin. The result is that it can be obtained a given voltage
stability criteria with minimum load shedding for emergency
situations.
Figure 11 shows the results for this test. It can be observed
that the margin continually grows as load is shed. It means

that the load shedding has been applied at the right places with
adequate amounts. This is the guarantee that minimum load
shedding has been performed. It can also be noted, that a pre-
defined voltage stability margin criteria could be used to stop
the process. In this case 11% of margin gain is obtained with
load shedding.
Figure 11 – Voltage stability margin improvement with minimum load
shedding
V. C
ONCLUSION
This work proposes that the modal analysis technique can be
used for identifying power system congested areas, and for
defining corrective/preventive control actions, based on active
and/or reactive power, for relieving system congestion.
Voltage stability limit is the main congestion constraint
discussed in this work.
From our results it can be concluded that the APF and RPF
indices can provide a complete picture of system critical areas,
and also the most adequate actions to improve system security
from a voltage stability perspective.
The tests results have shown that the modal analysis is a
powerful tool for defining congestion management schemes,
and also for maximizing system power transfer capability. It is
demonstrated that modal participation factors have a close
correlation with the actual results of applied control actions.
The introduction of the participation factors on the optimal
power flow has been demonstrated as very efficient for
generators and synchronous condensers optimum dispatch of
reactive power, for optimum re-scheduling of generators
active power, and for minimum load shedding determination,
always looking for increased voltage stability margin, and so
on relieving congested situations.
VI. REFERENCES
[1] Vournas, C. D., “Interruptible Load as a Competitor to Local Generation
for Preventing Voltage Security”, 3RZHU (QJLQHHULQJ 6RFLHW\ :LQWHU
0HHWLQJ, Vol.1, pp. 236 –240, 2001.
[2] Shahidehpour, M. & Alomoush, M., 5HVWUXFWXUHG (OHFWULFDO 3RZHU
6\VWHPV2SHUDWLRQ7UDGLQJDQG9RODWLOLW\, New York: Marcel Deekker,
2001.
[3] Doorman, G. L., “Optimal System Security Under Capacity Constrained
Conditions”, ,(((3RUWR3RZHU7HFK3URFHHGLQJ, Vol.2, 6 pp, 2001.
[4] Da Silva, L. C. P., da Costa, V. F. e Xu, W., “Preliminary Results on
Improving the Modal Analysis Technique for Voltage Stability
Assessment”, 3URFHHGLQJV RI WKH ,((( 3(6 6XPPHU 0HHWLQJ, Seattle,
USA, 2000.
[5] Gao, B., Morison, G. K. and Kundur, P., “Voltage Stability Evaluation
Using Modal Analysis”, ,(((7UDQVDFWLRQVRQ3RZHU6\VWHPV, vol. 7, no.
4, pp.1529-1542, 1992.
[6] Da Silva, L.C.P., Wang, Y., Da Costa, V.F., Xu, W., "Assessment of
generator impact on system power transfer capability using modal
participation factors", in Proc. of IEE Generation, Transmission, and
Distribution, Vol. 149, No. 5, 2002.
[7] WSCC Reactive Power Reserve Work Group. (1998). Final Report:
Voltage Stability Criteria, Undervoltage Load Shedding Strategy, and
Reactive Power Reserve Monitoring Methodology. Availabe: http://
www.wscc.com
[8] IEEE Working Group on Voltage Stability, Suggested Techniques for
voltage stability analysis, IEEE Power Eng. Society Report, 1993,
98TH0620-5PWR.
[9] C. Taylor, Power System Voltage Stability, New York: McGraw-Hill,
1994, p. 273.
[10] Almeida P. et al., “Criteria and methodologies established in the ambit of
GTAD/SCEL/GCOI Voltage Collapse Task Force for studies on voltage
stability in the Brazilian North/Northeast, South/Southeast and
North/South interconnected systems”, IEEE-PES Summer Meeting, 2000,
pp.531 –536.
VII. BIOGRAPHIES
,JRU.RSFDN, received his B.S. degree in Electrical Engineering from UFMT,
Brazil, in 1999, and his Master degree from UNICAMP-State University of
Campinas-Brazil, in 2003. At present, he is working towards a Ph.D. degree at
UNICAMP. His research interests are on power system stability, and
distributed generation. He can be reached at kopcak@dsce.fee.unicamp.br.

/XL]&3GD6LOYD, received his B.S. degree from UFG, Brazil, in 1995, and
his M.S. and Ph.D. degrees from UNICAMP in 1997 and 2001, respectively.
His research interests are on power systems stability analysis, and distributed
generation. He is with UNICAMP since 2002, where he is currently an
assistant professor. He can be reached at lui@dsce.fee.unicamp.br.
9LYDOGR ) GD &RVWD, received his B.S., M.S. and Ph.D. degrees from
UNICAMP in 1976, 1981 and 1992, respectively. He is with UNICAMP since
1977, where he is currently an associate professor. His research interests
include power system stability analysis and simulation. He can be reached at
vivaldo@dsce.fee.unicamp.br.
-LP 6 1DWXUHVD received his B.S. degree in Electrical Engineering from
UNESP, Brazil, in 1996, and his Master degree from UNICAMP-State
University of Campinas-Brazil, in 2001. At present, he is working towards a
Ph.D. degree at UNICAMP. His research interests are on power system
stability, and FACTS devices. He can be reached at jim@dsce.fee.unicamp.br.