12-09-2012, 05:28 PM
Power System Load Flow Analysis using an Excel Workbook
[attachment=33649]
Abstract
These notes describe the features of an MS-Excel Workbook which illustrates four
methods of power system load flow analysis. Iterative techniques are represented by
the Newton-Raphson and Gauss-Seidel methods. The Workbook also includes two
search algorithms: genetic algorithms and simulated annealing.
Introduction
Load flow studies [1,2] are used to ensure that electrical power transfer from
generators to consumers through the grid system is stable, reliable and economic.
Conventional techniques for solving the load flow problem are iterative, using the
Newton-Raphson or the Gauss-Seidel methods. Recently, however, there has been
much interest in the application of stochastic search methods, such as Genetic
Algorithms [3,4,5], to solving power system problems. The increasing presence of
distributed alternative energy sources, often in geographically remote locations,
complicates load flow studies and has triggered a resurgence of interest in the topic.
The principles of power system load flow studies are taught within elective modules
in the later years of undergraduate electrical engineering courses, or as essential
components of specialist masters programmes in electrical power engineering. From
the educational viewpoint, therefore, the topic is important, yet a complete coverage
presents some significant challenges. Pre-requisites include fundamental concepts
from a.c. circuit analysis, such as phasor notation, impedance and admittance, power
and reactive power, three-phase and per-unit systems, all of which are regarded as
‘difficult’ by many students. The load flow solution techniques bring extra
mathematical hurdles, including matrix representation (with complex number
coefficients), iterative methods and probability functions.
Formulation of the Load Flow Problem
Load flow studies are based on a nodal voltage analysis of a power system. As an
example, consider the very simple system represented by the single-line diagram in
Fig. 1. Here two generators (1 and 2) are interconnected by one transmission line
and are separately connected to a load (3) by two other lines. If the phasor currents
injected into the system are I1, I2, and I3, and the lines are modelled by simple series
admittances, then it is possible to draw the equivalent circuit for one representative
phase of the balanced three-phase system, as shown in Fig. 2.
Stochastic Search Techniques
General Principles
Recent developments in load flow analysis have moved attention away from the
iterative methods and towards so-called stochastic search methods. Two such
methods – Genetic Algorithms and Simulated Annealing – are described here and
are implemented in the Excel Workbook. Both approaches use a series of trial
solutions to the problem and develop better solutions in the light of experience
gained from these trials. The computational effort for each trial is kept as low as
possible, so a very large number of trials can be conducted.
For the example problem being considered throughout this paper, there are three
variables: the voltage magnitude V3 and the phase angles 2 3 δ ,δ . In any one trial
some appropriate values are chosen for these variables. The choice may be an
entirely random selection across the entire possible range of values (termed the
‘search space’) or the choice may be informed by previous experience. Once these
trial values are chosen, the phasor voltages at all three nodes are defined, because
all of the other voltage magnitudes and phase angles are fixed.
[attachment=33649]
Abstract
These notes describe the features of an MS-Excel Workbook which illustrates four
methods of power system load flow analysis. Iterative techniques are represented by
the Newton-Raphson and Gauss-Seidel methods. The Workbook also includes two
search algorithms: genetic algorithms and simulated annealing.
Introduction
Load flow studies [1,2] are used to ensure that electrical power transfer from
generators to consumers through the grid system is stable, reliable and economic.
Conventional techniques for solving the load flow problem are iterative, using the
Newton-Raphson or the Gauss-Seidel methods. Recently, however, there has been
much interest in the application of stochastic search methods, such as Genetic
Algorithms [3,4,5], to solving power system problems. The increasing presence of
distributed alternative energy sources, often in geographically remote locations,
complicates load flow studies and has triggered a resurgence of interest in the topic.
The principles of power system load flow studies are taught within elective modules
in the later years of undergraduate electrical engineering courses, or as essential
components of specialist masters programmes in electrical power engineering. From
the educational viewpoint, therefore, the topic is important, yet a complete coverage
presents some significant challenges. Pre-requisites include fundamental concepts
from a.c. circuit analysis, such as phasor notation, impedance and admittance, power
and reactive power, three-phase and per-unit systems, all of which are regarded as
‘difficult’ by many students. The load flow solution techniques bring extra
mathematical hurdles, including matrix representation (with complex number
coefficients), iterative methods and probability functions.
Formulation of the Load Flow Problem
Load flow studies are based on a nodal voltage analysis of a power system. As an
example, consider the very simple system represented by the single-line diagram in
Fig. 1. Here two generators (1 and 2) are interconnected by one transmission line
and are separately connected to a load (3) by two other lines. If the phasor currents
injected into the system are I1, I2, and I3, and the lines are modelled by simple series
admittances, then it is possible to draw the equivalent circuit for one representative
phase of the balanced three-phase system, as shown in Fig. 2.
Stochastic Search Techniques
General Principles
Recent developments in load flow analysis have moved attention away from the
iterative methods and towards so-called stochastic search methods. Two such
methods – Genetic Algorithms and Simulated Annealing – are described here and
are implemented in the Excel Workbook. Both approaches use a series of trial
solutions to the problem and develop better solutions in the light of experience
gained from these trials. The computational effort for each trial is kept as low as
possible, so a very large number of trials can be conducted.
For the example problem being considered throughout this paper, there are three
variables: the voltage magnitude V3 and the phase angles 2 3 δ ,δ . In any one trial
some appropriate values are chosen for these variables. The choice may be an
entirely random selection across the entire possible range of values (termed the
‘search space’) or the choice may be informed by previous experience. Once these
trial values are chosen, the phasor voltages at all three nodes are defined, because
all of the other voltage magnitudes and phase angles are fixed.