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Steady-State Solution for Power Networks Modeled at Bus Section Level
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Abstract
This paper extends the conventional power flow
formulation in order to enable the solution of networks modeled
at the bus section level. The proposed extension is centered on a
methodology to represent zero impedance branches successfully
employed in state estimation studies. Accordingly, the active and
reactive power flows through switches and circuit breakers are
treated as new state variables along with the complex voltage at
the network nodes. Information regarding device statuses is included
into the power flow problem as new (and linear) equations,
producing a solvable non-redundant set of algebraic equations.
Applications of the proposed modifications in connection with the
power flow solution via Newton-Raphson’s method are presented
and discussed. The proposed approach provides an efficient tool to
directly determine the power flow distribution over selected substations
of the network, avoiding unreliable artifices and tedious
post-processing procedures required when a conventional power
flow formulation is applied. The IEEE 24-bus and IEEE 30-bus
test systems are employed to illustrate and evaluate the proposed
approach, considering distinct substation layouts.
Index Terms—Circuit-breaker representation, power flow
analysis, power network modeling, steady-state power network
solution.
I. INTRODUCTION
CONVENTIONAL steady-state analysis of power networks
relies on the so-called bus/branch model. In such
models, substation arrangements are predetermined and a “bus”
is defined by merging sets of electrical substation nodes connected
through closed circuit breakers and switches. Therefore,
switching devices are not explicitly represented.
An advantage of the bus/branch representation is that the
basic network matrices employed for steady-state analysis, such
as the bus admittance matrix, bus/branch incidence matrices,
etc., can be readily built from the model. On the other hand, the
lack of information on substation topologies as well as on the
power flow distribution through substation devices can be seen
as a serious drawback in some applications.
This is the case of real-time modeling, where the current trend
is towards the representation of parts of the power network at the
bus-section, or physical level. That trend is driven by the emergence
of topology estimators [1] and the development of new
Manuscript received May 21, 2009; revised July 13, 2009. First published
December 31, 2009; current version published January 20, 2010. The work of
E. M. Lourenço and A. Simões Costa was supported by CNPq, the Brazilian National
Research Council. In addition, the work of E. M. Lourenço was also supported
by Paraná Araucária Foundation, under Grant 9268. Paper no. TPWRS-
00018-2009.
E. M. Lourenço and R. Ribeiro P., Jr. are with the Federal University of
Paraná, Curitiba, Brazil (e-mail: elizete[at]eletrica.ufpr.br).
A. Simões Costa is with the Federal University of Santa Catarina, Florianópolis,
Brazil (e-mail: simoes[at]labspot.ufsc.br).
Digital Object Identifier 10.1109/TPWRS.2009.2036466
topology error identification algorithms relying on the explicit
representation of switching devices [2], [3]. In both cases, the
search of the correct network topology requires the consideration
of distinct substation arrangements for the same problem,
something which is not easily carried out through the use of the
conventional bus/branch model.
Representing substations at the physical level is also relevant
in other areas, such as corrective switching studies. In such
case, one looks for proper switching strategies to alleviate or
eliminate system overloads [4]–[6]. Since the search may involve
bus-splitting, the need then arises to explicitly represent
some selected circuit breakers at bus section level. The selected
breakers are those seen as good candidates to undergo switching
actions. In the past, artifices have been employed to accommodate
that need within the limitations of the bus/branch model.
However, as discussed in this paper, those artifices tend to cause
numerical problems in the course of power flow solutions.
The objective of this paper is to propose an extension of the
conventional power flow (PF) formulation via Newton-Raphson
(N-R) method in order to accommodate the explicit representation
of switching devices. The resulting extended power
flow (XPF) method is numerically robust and yields the power
flow distribution through switching components for selected
substations of the power network. The proposed approach
builds on previous efforts conducted in the real-time power
system modeling area related to the representation of switching
branches in the state estimation formulation [2], [7]. The concepts
introduced in those references provide the foundations of
the so-called generalized state estimation [8], [9] and underlies
a number of recently proposed methodologies for processing
topology errors in substation arrangements [3], [10]–[13].
The extension of network modeling down to the substation
level is accomplished by expanding the conventional mathematical
model via the inclusion of the power flows through circuit
breakers as new state variables and the aggregation of new equations
determined by the status of those devices and the substation
topology. It should be stressed that the detailed representation
is not intended to be used for every substation of the network.
Instead, it is assumed that only a few substations, selected
by the particular power flow application are to be represented at
physical level.
This paper provides details on how the required changes
should be imbedded in the PF formulation in order to generate
XPF, and also discusses the impact of those changes on the
convergence rate of the iterative process. Two IEEE test systems
are employed to both illustrate and validate the results provided
by the proposed methodology.
0885-8950/$26.00 © 2009 IEEE
LOURENÇO et al.: STEADY-STATE SOLUTION FOR POWER NETWORKS MODELED AT BUS SECTION LEVEL 11
This paper is organized as follows. Section II revisits the conventional
PF problem in order to facilitate the description of the
extended formulation. Section III introduces the fundamentals
and mathematical formulation of XPF. Two important particular
cases leading to degenerate circuit conditions and indeterminate
systems are dealt with in Section IV, where solutions for
such problems are also proposed. Section V presents two illustrative
examples of XPF applications to a small network. The
impact of the changes required by XPF on the convergence rate
of the iterative power flow solution are discussed in Section VI.
Section VII presents illustrative results of the application of XPF
to subnetworks derived from the IEEE 24-bus and 30-bus test
systems. Finally, the main conclusions are drawn and discussed
in Section VIII.
II. CONVENTIONAL POWER FLOW FORMULATION
This section reviews the fundamentals of power flow analysis
through Newton-Raphson method. Since power flow concepts
are well-known, the main purpose of the section is actually to
familiarize the reader with the problem formulation and notation
used in this paper, in order to pave the way for the extensions to
be introduced in Section III.
A. Bus-Branch Network Model
Given a set of bus loads and specified voltage magnitudes/
power injections at generation buses, a conventional power flow
study determines the steady state operating condition of a power
system based on the bus/branch network model. Such a model
is produced by merging adjacent substation nodes present at the
actual bus-section level topology. This procedure circumvents
the explicit representation of switches and circuit breakers. On
the other hand, the bus/branch model does not provide any information
internal to the substations, such as the power flow distribution
through switches and circuit breakers.
The basic power flow equations are obtained by applying the
Kirchhoff’s laws to the network represented by the bus/branch
model. The results can be grouped into two categories: nodal
and branch equations. The nodal equations for an N-bus network
(at the bus/branch level) are the active and reactive power
injections at each bus, given by [9], [14]