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A Heuristic Nonlinear Constructive Method for Electric Power
Distribution System Reconfiguration
Thomas E. McDermott
Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and
State University in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Electrical Engineering
Robert P. Broadwater, Chair
Walling Cyre
Terry Herdman
Yilu Liu
Hugh VanLandingham
April 23, 1998
Blacksburg, Virginia
Keywords: Discrete Ascent Optimal Programming, Load Restoration, Losses, Optimization
Methods, Power Distribution Control, Power Distribution Planning, Power System Restoration
Copyright 1998, Thomas E. McDermott
A Heuristic Nonlinear Constructive Method for Electric Power
Distribution System Reconfiguration
Thomas E. McDermott
(ABSTRACT)
The electric power distribution system usually operates in a radial configuration, with tie switches
between circuits to provide alternate feeds. The losses would be minimized if all switches were
closed, but this is not done because it complicates the system’s protection against overcurrents.
Whenever a component fails, some of the switches must be operated to restore power to as many
customers as possible. As loads vary with time, switch operations may reduce losses in the
system. Both of these are applications for reconfiguration.
The problem is combinatorial, which precludes algorithms that guarantee a global optimum. Most
existing reconfiguration algorithms fall into two categories. In the first, branch exchange, the
system operates in a feasible radial configuration and the algorithm opens and closes candidate
switches in pairs. In the second, loop cutting, the system is completely meshed and the algorithm
opens candidate switches to reach a feasible radial configuration. Reconfiguration algorithms
based on linearized transshipment, neural networks, heuristics, genetic algorithms, and simulated
annealing have also been reported, but not widely used. These existing reconfiguration algorithms
work with a simplified model of the power system, and they handle voltage and current
constraints approximately, if at all.
The algorithm described here is a constructive method, using a full nonlinear power system model
that accurately handles constraints. The system starts with all switches open and all failed
components isolated. An optional network power flow provides a lower bound on the losses.
Then the algorithm closes one switch at a time to minimize the increase in a merit figure, which is
the real loss divided by the apparent load served. The merit figure increases with each switch
closing. This principle, called discrete ascent optimal programming (DAOP), has been applied to
other power system problems, including economic dispatch and phase balancing. For
reconfiguration, the DAOP method’s greedy nature is mitigated with a backtracking algorithm.
Approximate screening formulas have also been developed for efficient use with partial load flow
solutions. This method’s main advantage is the accurate treatment of voltage and current
constraints, including the effect of control action. One example taken from the literature shows
how the DAOP-based algorithm can reach an optimal solution, while adjusting line voltage
regulators to satisfy the voltage constraints.
Acknowledgements
I thank the Electric Power Research Institute for supporting development of the DEWorkstation,
which made this research possible.
Special thanks are due to my wife, Kathleen Retcofsky McDermott, for her patience and
understanding during my pursuit of this degree.
iii
Contents
Chapter 1 - Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Effect of Constraints on Reconfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 - Summary of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Branch Exchange Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Loop Cutting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Heuristic Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Transshipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Discrete Ascent Optimal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 3 - Reconfiguration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 DEWorkstation Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Operating Switches in DEWorkstation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Isolating Failed Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Reconfiguration with Full Load Flow Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Reconfiguration with Approximate Loss Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6 Backtracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Example Reconfiguration with Backtracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter 4 - Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Dolloff’s Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Civanlar 2-Feeder Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Civanlar 3-Feeder Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Glamocanin Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Baldick Discussion Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Baran and Wu Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Baran and Wu Circuit with Voltage Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter 5 - Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Appendix A - User Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.1 Introduction to Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.2 Interactive Inputs - Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.3 Control Center - Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.4 Switching Configurations - Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.5 Outputs - Loss Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Appendix B - Derivation of Approximate Loss Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
iv
List of Figures
Figure 1.1 - A simple distribution system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Figure 1.2 - Capacitor switching can alleviate voltage constraints during
reconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Figure 2.1 - A weakly meshed distribution network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Figure 2.2 - Linearized feeder transshipment costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure 2.3 - Baldick’s problem with shunt capacitor banks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Figure 3.1 - Faulted circuit with DEWorkstation component numbering
. . . . . . . . . . . . . . . . . . 13
Figure 3.2 - Algorithm to isolate failed components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Figure 3.3 - Reconfiguration algorithm with full load flow screening . . . . . . . . . . . . . . . . . . . . . 16
Figure 3.4 - Reconfiguration algorithm with approximate loss evaluation . . . . . . . . . . . . . . . . . . 19
Figure 3.5 - Open loop backtracking scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Figure 3.6 - Sample circuit with constant-current loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 4.1 - Civanlar’s 2-Feeder Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 4.2 - Civanlar’s 3-Feeder Test Circuit before Reconfiguration (line flows
are in amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 4.3 - Civanlar’s 3-Feeder Test Circuit after Reconfiguration (line flows are
in amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 4.4 - Glamocanin’s Test Circuit before Reconfiguration (line flows are in
amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Figure 4.5 - Glamocanin’s Test Circuit after Reconfiguration (line flows are in
amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Figure 4.6 - Baldick’s Discussion Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 4.7 - Baran and Wu’s Test Circuit Before Reconfiguration (line flows are
in amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 4.8 - Baran and Wu’s Test Circuit After Reconfiguration (line flows are in
amperes, customer service voltage is italicized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 4.9 - Baran and Wu’s Test Circuit with Voltage Regulators (customer
service voltage before reconfiguartion in red, after reconfiguration in
blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure A.2 - Setup dialog for loss minimization application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure A.3 - Control dialog for loss minimization - before execution
. . . . . . . . . . . . . . . . . . . . . 48
Figure A.4 - Control dialog for loss minimization - after execution . . . . . . . . . . . . . . . . . . . . . . . 49
Figure A.5 - Dialog to manage switching configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Figure A.6 - Switching operation results dialog for loss minimization . . . . . . . . . . . . . . . . . . . . . 50
Figure A.7 - Sample report from loss minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure B.1 - A three-phase lossy device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
v
List of Tables
Table 3.1
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
Table 4.7
Table 4.8
Reconfiguration algorithm steps for Figure 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Test System Size Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Benchmark, networked, and final system losses [kW] . . . . . . . . . . . . . . . . . . . . . . . . . 27
Simulation times on a Pentium 90 [seconds] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Load data for Dolloff’s circuit in DEWorkstation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Impedance and load data for Civanlar 2-Feeder circuit . . . . . . . . . . . . . . . . . . . . . . . . 29
Impedance and load data for Glamocanin circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Impedance and load data for Baran and Wu circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vi
Chapter 1
Introduction
The electric power distribution system delivers power to the customers from a set of distribution
substations. While the transmission and subtransmission lines are configured in a meshed
network, the distribution feeders are configured radially in almost all cases. This radial
configuration simplifies overcurrent protection of the feeder. To help restore power to customers
following a fault, most feeders are provided with tie circuits to neighboring feeders from either the
same or different substations. A number of switching operations are needed to restore power
using these ties.
Loads vary with time of day, day of the week, and season. Each type of load (residential,
commercial, industrial) has a different time profile, and each feeder serves a different mix of loads.
Therefore, the load pattern on each feeder varies constantly, and with a different variation on each
feeder. This creates an opportunity to constantly keep losses at a minimum by reconfiguring the
feeders during the day. Automatic switches and control systems must be installed to perform this
distribution automation, at a cost that must be balanced against the savings in losses.
The distribution system should be operated at minimum cost, subject to a number of constraints:
1. radial configuration
2. all loads are served
3. overcurrent protective devices are coordinated
4. lines, transformers, and other equipment within current capacity limits
5. voltage drop within limits
Reconfiguration is the process of operating switches to change the circuit topology so that
operating costs are reduced, while still meeting the constraints listed above. For the distribution
system, the cost to be minimized is essentially the cost of losses, and the losses are essentially a
quadratic function of the line currents. There may be other costs to consider in the
reconfiguration analysis, such as the cost to operate or install switches.
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Thomas E. McDermott
Chapter 1 - Introduction
Figure 1.1 - A simple distribution system.
Figure 1.1 shows a distribution system with two sources and several switching devices. SW1,
SW5, and RC3 are open so that the system is operating radially. The ending loads in feeder
segments LN2 and LN6 are some distance from their source, SS2. To correct for possible
voltage drop problems at these ending loads, it’s typical to use capacitor banks, as illustrated
between LN4 and SW2, or voltage regulators, as illustrated between LN3 and LN9. The
capacitor banks may be of fixed size, controlled locally by temperature or time of day, or
controlled remotely. Voltage regulators are usually controlled locally. For the system operating
as shown in Figure 1.1, it’s likely that losses could be reduced by transferring some of the load
from source SS2 to source SS1. For example, RC3 might be closed and then SW2 opened,
transferring the loads in LN5 and LN6 from SS2 to SS1. Analyzing these options is the subject of
a reconfiguration algorithm.
Since a typical distribution system would have hundreds of switches, a combinatorial analysis of
the options (2m) could easily take too long. Also, the analysis and application of the constraints
takes more time and complicates the use of classical optimization techniques. However, the
potential cost savings increase dramatically as the number of switches increases [Willis et. al.,
1996], so it is important for the analysis method to handle large systems.
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Thomas E. McDermott
Chapter 1 - Introduction
Reconfiguration may be considered in two different situations. First, in the system planning or
system design stage, the normal operating configuration and switch placement can be adjusted to
achieve a minimum cost system. Capital and maintenance costs of new switches must be
considered along with the cost of losses. Second, in the real-time operation mode, the protective
system may have operated during a fault to reconfigure the system. In this new configuration, it
would be useful to examine switching operations that might reduce losses while not disconnecting
any more customers. Here, the cost and the time involved in operating switches must be
considered along with the cost of losses.
The two situations differ in other ways, too. In the real-time mode, reconfiguration analysis must
be very quick, and it starts with an existing radial system. In the planning or design mode, the
analysis could take longer if it is more thorough, and also more of the system could be considered
a blank slate.
1.1 Effect of Constraints on Reconfiguration
One of the major constraints on reconfiguration is that the network must not have any loops.
Some of the general optimization algorithms have difficulty enforcing this radial constraint
directly. This is a particular problem for reconfiguration, because a meshed network with all
switches closed will have minimum losses, but that violates the radial constraint. A method that
builds the network from scratch can easily enforce the radial constraint, simply by not closing any
switches that form a loop.
The other major constraint on reconfiguration is the limit on line capacity or voltage drop. These
constraints are closely related, because a line carrying current close to its thermal capacity will
also have a larger resistive voltage drop. Heuristically, solutions with a good voltage profile are
more likely to have low losses and line currents within limits. In a given situation either the
voltage or the current constraint may be reached first.
Current constraints can be alleviated by switching, or in the longer term, by re-conductoring the
feeder or installing a larger substation transformer. Voltage constraints can be alleviated by
switching or installing capacitor banks, or by adjusting voltage regulators. Secondarily, capacitor
banks and voltage regulators can reduce line currents and thus may help to alleviate current
constraints.
For example, suppose the last load in Figure 1.2 can be served by closing either switch A or B.
Switch A produces minimum losses, but violates the voltage drop constraint because its feeding
path is already heavily loaded. Therefore switch B is closed to serve the last load. In most design
processes, capacitor banks and voltage regulators will then be adjusted on the reconfigured
network to improve the voltage profile. After the capacitors and regulators have operated,
switching operations that were previously discarded due to constraints may become feasible. For
example, by switching on the capacitor in Figure 1.2, it might be possible to close switch A
instead of B, without violating the voltage constraint.
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Thomas E. McDermott
Chapter 1 - Introduction
Figure 1.2 - Capacitor switching can alleviate voltage constraints during reconfiguration.
1.2 Problem Statement
The objective of this dissertation is to develop an efficient reconfiguration algorithm that directly
constructs an optimal solution for minimum losses, accounting for constraints.
The
reconfiguration should consider capacitor and voltage regulator control along with switching
operations. The algorithm should also isolate any failed components, and restore power to as
many customers as possible.
4
Chapter 2
Summary of Previous Work
The only method known to produce a guaranteed optimal reconfiguration is “branch and bound”
[Boardman, 1985]. This amounts to an exhaustive search, except that a search direction will be
terminated when it becomes obvious that its end result will be less optimal than a previously found
solution. Branch and bound will work better if the initial solution is close to optimal, because
more pruning will occur. Branch and bound also benefits from breaking the problem into
subproblems, each of which can be optimized separately. Even so, branch and bound is a
combinatorial method and hence too slow for practical use.
Therefore, most of the recent work on reconfiguration has used either a branch exchange method
or sequential switch opening method. Heuristics are applied in most cases to reduce the number
of switching options considered. Even so, approximate loss formulas, linearized costs, and other
simplifying assumptions are typically used to avoid repetitive solution of system load flows.
Attempts have also been made to use neural networks, genetic algorithms, and simulated
annealing solutions for the reconfiguration problem.
2.1 Branch Exchange Methods
Branch exchange starts with a feasible solution - the distribution network operating in a radial
configuration. One of the tie switches is closed, and then another switch is opened in the loop
created, which restores a radial configuration. The switch pairs are chosen through heuristics and
approximate formulas for the change in losses. The branch exchange process is stopped when no
more loss reductions are available.
As developed by [Civanlar, 1989], this method requires a full system load flow analysis at the
current operating condition. The change in losses effected by operating a pair of switches is
estimated from:
 

*
∆P = Re 2 ∑ I i (Em − En )  + Rloop
  i∈D 

∑ Ii
2
i ∈D
(2.1)
For the sample system in Figure 1.1, to analyze a possible exchange of RC3 and SW2, the
resistance Rloop would be the series resistance of LN8, 7, 6, 5, 4, 9, 13, 14, 15, and 16. Em and En
are the resistive components of the voltage drop from source to bus on each side of the switch to
be opened, RC3. En is defined on the load-transferred-from side of the switch to be opened, or
the SS1 side of RC3. Therefore, the losses are predicted to decrease when En > Em, or when load
is transferred from the higher voltage drop side of the open switch. The transferred load is
represented by Σ Ii, the sum of the load currents in LN5 and LN6. After the switch operation,
the system load flow is solved again to analyze the next switch exchange.
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Thomas E. McDermott
Chapter 2 - Previous Work
This method is very fast and good for use in real-time operations. Even though it is easily trapped
in a local minimum, it can be used to provide a loss reduction after an emergency reconfiguration.
The method can be used with the total number of switching operations limited to some number
that reflects crew availability and switching time.
A variation on the branch exchange method has been used in Japan, where the density of
automatic switches is evidently rather high. Aoki defines an “arc” as a segment between two
branch points; it usually has several automatic switches along its length. On an open loop, the arc
containing the tie point is moved in one direction or the other. Then, a continuous piecewise
quadratic problem is solved to determine the point on the arc where the break should occur. This
solution is then rounded to the nearest actual switch location.
2.2 Loop Cutting Methods
The loop cutting or sequential switch opening method [Shirmohammadi, 1989] starts with all tie
switches closed. The typical distribution system would be “weakly meshed” in this state,
compared to the transmission system. A load flow of the meshed system will provide a
minimum-loss solution (in the absence of any control action). However, the system must be
brought into a radial configuration. This is done by opening switches that carry the least current,
on the premise that these will least disturb the meshed load flow solution. After each switch
opening, the meshed load flow is solved again before selecting the next switch to open. The
algorithm stops when the system is radial.
For the sample system in Figure 2.1, all three switches A, B, and C would be closed initially. If
switch C were opened first, switch B could not be opened without isolating some of the load, and
vice versa. The number of weakly meshed load flows that must be performed is usually low, at
most two in the case of Figure 2.1.
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Thomas E. McDermott
Chapter 2 - Previous Work
Figure 2.1 - A weakly meshed distribution network.
2.3 Heuristic Search Methods
Search methods involve building the network in stages, or analyzing a decision tree of switching
operations, using a depth-first or best-first algorithm. The best-first algorithm is sometimes called
a “greedy” search. The greedy search cannot ensure a global optimum, though use of an
adjustable number of wrong steps has been investigated for reconfiguration [Carrillo-Caicedo,
1995]. Branch and bound is the broadest search algorithm, but various heuristics have been
applied to make its running time reasonable for reconfiguration [Hsu, Morelato, Taylor].
Heuristics represent the application of engineering judgment without detailed analysis of each
alternative, so they are desirable in any algorithm.
2.4 Transshipment
The linearized transshipment method [Glamocanin 1990] is most closely related to the method
presented in this dissertation. The network starts with just the distribution substations, and the
feeder segments are switched in one at a time. Of the segments available to switch in, the one
with minimum unit transshipment cost is chosen first. Then, of the other segments available to
7
Thomas E. McDermott
Chapter 2 - Previous Work
serve that new load point, the chosen segment is checked to make sure it results in the minimum
increase in losses. The increase in losses for each switching is an approximate quadratic formula.
The algorithm stops when all loads are served; the system is kept radial as the segments are
switched in. It is also a greedy search.
The transshipment cost was linearized to the feeder resistance multiplied by maximum current
capacity. Figure 2.2 shows the I-squared R power loss for two feeders of different capacity,
along with linear approximations through the points of maximum current. The linearized
transshipment cost defines these straight lines. For feeder 1, the linearized transshipment cost is
10 amps multiplied by 1.5 ohms, or 15. For feeder 2 the linearized cost is 15 amps multiplied by
0.7 ohms, or 10.5. Thus feeder 2 has the lowest cost per unit ampere, but its cost relative to
feeder 1 is greater than the simple ratio of resistances. Note that the transshipment cost does not
depend on the actual load current nor on any other operating condition.
Figure 2.2 - Linearized feeder transshipment costs.
The two-stage comparison feature is important. If the segment that results in minimum loss is
chosen at each step, the tendency will be to add the smallest loads first, using the best feeder
segments. Obviously this will not lead to an optimal or even a good solution. In a discussion of
8
Thomas E. McDermott
Chapter 2 - Previous Work
Glamocanin’s paper, Baldick showed this problem for the simple system shown in Figure 2.3.
Adding the minimum loss segment at each step resulted in feeders 1-2-3 and 1-4, with total losses
of 6.25 kW. In his closure, Glamocanin showed that his transshipment method produced the
optimal solution of one feeder, 1-2-3-4, with losses of 5.25 kW. He also recommended that
feeder reconfiguration and capacitor placement be solved as separate problems in any case.
Figure 2.3 - Baldick’s problem with shunt capacitor banks.
The transshipment problem can be solved with linear programming techniques. However, the
linear transshipment cost is an approximation for distribution networks. If there are fixed costs,
these must be ignored. Further, the solution from a general transshipment algorithm may not be
radial [Wall].
2.5 Simulated Annealing
Simulated annealing is not a reconfiguration algorithm by itself, but is a modification to some
other basic algorithm [Chang, Sarfi]. Its purpose is to avoid being trapped in local minima, by not
always taking the best choice at each step. Under simulated annealing, a random choice is
accepted with a probability that decreases exponentially with each iteration. Simulated annealing
has more potential to find the global optimum, though this can't be proven. It is readily applied on
decision tree or branch exchange methods, by adding an extra outer loop to the basic algorithm.
9
Thomas E. McDermott
Chapter 2 - Previous Work
10
Because of the extra analysis time, simulated annealing is more applicable to planning rather than
operations.
2.6 Genetic Algorithms
Genetic algorithms have become very popular as a method of finding global optimums. As
applied to reconfiguration [Nara], the switch states are encoded in strings of 0/1 "chromosomes",
and a population of, for example, 50 topologies is built at random. At each iteration, two parent
topologies are selected at random for crossbreeding, which is a process of combining the
chromosomes according to some defined algorithm. Then mutation, a random alteration of some
chromosomes, may occur with a certain probability. If the resulting child is better, it replaces an
existing topology in the population of 50. This process of crossbreeding continues for a number
of iterations. The population also has to be re-seeded periodically with random strings to avoid
inbreeding. As the population evolves, there will always be a best solution that should steadily
improve.
Genetic algorithms are most attractive for parallel processing environments, and when each child
can be evaluated quickly. The crossbreeding and mutation algorithms must be custom designed
and tested for each application. Parameters such as the number of crossbreedings per generation,
mutation probability, number of generations, population size, and percentage of population
reseeded must all be determined by testing. Applications to reconfiguration have used simplified
network analysis because many thousands of topologies are considered, so the resulting solution
may not be optimal with a more detailed model. The method presented in this dissertation uses
more comprehensive analysis with fewer iterations.
2.7 Neural Networks
Neural networks have been applied to recognize a load pattern from feeder measurements and
other data, then select a pre-analyzed topology and switching strategy to reconfigure the network
for loss reduction [Hayashi, Kim]. It’s necessary to discretize load levels and combine similar
topologies, otherwise the training sets become too large. The neural network serves as a state
estimator but doesn’t analyze the topologies, so this method is not really applicable to the
problem statement.
2.8 Discrete Ascent Optimal Programming
Discrete ascent optimal programming (DAOP) has been applied to optimal load flow and phase
balancing for distribution systems [Dolloff]. In DAOP, the load is picked up in step increments,
with each step added at the ending node that will produce the smallest increase in total losses. At
each step the loss will increase in discrete increments, hence the name “discrete ascent” optimal
programming. DAOP is essentially a greedy algorithm, although it was proved to converge to the
global optimum in the case with 3 substations and no constraints.
Thomas E. McDermott
Chapter 2 - Previous Work
11
In this thesis, the DAOP method is extended to the reconfiguration analysis problem. This
method provides advantages similar to the linearized transshipment algorithm, but with more
accurate load flow analysis. Rather than increment load in steps, the network size is incremented
in steps of one switching operation. At each step, each candidate switching operation is evaluated
from the increase in loss, divided by the switched segment’s apparent load power. This measure
is similar to the linearized transshipment cost, except that it includes nonlinear load flow analysis.
Capacitor switching and voltage regulator operation have to be considered as part of the
reconfiguration problem, and this is another improvement over the transshipment method.
Since DAOP is a greedy search, it cannot be assured of reaching the global optimum except in
simplified or special cases. The minimum loss problem is quadratic or piecewise quadratic, so one
might expect that with no constraints, DAOP would converge to a global optimum. However, as
switches are closed in a DAOP algorithm, certain subsequent switch closings will be ruled out by
the radial constraint. Even when all voltage and current limits remain satisfied, the DAOP
algorithm will encounter constraints. One of the examples used in Dolloff’s thesis shows how a
pure DAOP reconfiguration algorithm can be trapped in a local minimum. A backtracking
scheme, described in sections 3.6 and 3.7, was developed to address this issue.
Chapter 3
Reconfiguration Algorithms
This chapter describes how a reconfiguration algorithm was implemented in the Distribution
Engineering Workstation (DEWorkstation), funded by the Electric Power Research Institute
(EPRI). Sections 3.1 and 3.2 describe the advantages of using DEWorkstation for the
implementation, and some of the platform-specific issues encountered. The algorithm’s features
include:
1. isolating failed components (Section 3.3)
2. loss minimization by a properly selected merit figure (Section 3.4)
3. accelerated solution using approximate loss evaluation of candidate switch closings (Section
3.5)
4. backtracking to avoid local minima (Section 3.6)
Section 3.7 walks through an example of the algorithm’s main features. Chapter 4 describes
several test cases obtained from the literature, and Appendix A contains user instructions for
running the application in DEWorkstation.
3.1 DEWorkstation Platform
The reconfiguration algorithm based on DAOP was implemented and tested in EPRI’s
DEWorkstation [Broadwater, 1995]. The DEWorkstation includes a system model based on
graph edges, rather than separate buses and branches. This model organization affects the design
of applications for DEWorkstation, and the reconfiguration algorithm takes advantage of
component traces built into DEWorkstation:
1. forward and backward component traces
2. feeder path component traces back to the source
3. pointers to components in adjacent circuits
4. brother pointers that identify downstream components fed by this component
The DEWorkstation executive maintains these traces, along with a database and graphical
interface. The reconfiguration algorithm uses other applications in DEWorkstation, in particular
the radial and network power flow calculations. This architecture saves development time and
improves the consistency of results.
12
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
13
Figure 3.1 shows a sample circuit in DEWorkstation, with components numbered in a forward
trace starting from the source. The numbering is not unique, but must follow one rule – whenever
a component is numbered, all downstream components must be numbered before any others.
From the faulted component in Figure 3.1, the feeder path trace would be 4-3-2-1. Some
components have pointers to adjacent circuits. In Figure 3.1, only component 7 would have a
pointer to the adjacent tie switch. A component’s brother pointer jumps to the next forward trace
position that doesn’t have that component in the feeder path. Figure 3.1 lists the brother for each
component in the sample circuit (0 implies a NULL pointer). The brother pointer is useful in
finding branches, and summing loads served by a switch.
1
9
8
2
3
7
4
6
5
fault
component
brother
1
0
2
8
3
8
4
7
5
7
6
7
7
8
8
0
9
0
Figure 3.1 - Faulted circuit with DEWorkstation component numbering
The DEWorkstation includes a comprehensive database and schematic editor for the distribution
system model, with a suite of applications to run against the model. The reconfiguration
algorithm calls the following DEWorkstation applications while executing:
1. radial load flow
2. network load flow
3. load estimation
This architecture makes the full analysis capability of DEWorkstation available to the
reconfiguration algorithm. In particular, unbalanced loads, regulator control action, and capacitor
bank control are accounted for without making approximations in the reconfiguration algorithm.
In other reconfiguration algorithms, these effects are often ignored. Results are consistent
between applications; the user will not run the reconfiguration algorithm, and then find that
DEWorkstation’s load flow program gives a different answer.
The load flow calculations in DEWorkstation may be run on the entire system, or on a selected
circuit. The reconfiguration algorithm runs most of its load flows on just one circuit at a time,
which saves execution time.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
14
3.2 Operating Switches in DEWorkstation
The user may have specified that certain switches may not be operated, either by specifying types
of switches, or selecting individual switches. Before attempting to minimize losses, any failed
components are isolated by opening switches according to the procedure in section 3.3. All of
these switches are marked as inoperable for the reconfiguration algorithm. They are excluded
from the opening and closing sequences described in this section, and the automatic sequences
described in sections 3.4 through 3.6.
The reconfiguration algorithm needs to open and close switches automatically. DEWorkstation
can create loops by closing switches that have one line segment connected to each end, and no
other components connected. This allows DEWorkstation to solve a network power flow, and
the reconfiguration algorithm uses this to provide a lower bound on the system losses. Once a
loop has been created, the switch that was closed to create the loop must be opened before any
other switch in that loop is opened. The reconfiguration algorithm needs to account for these
switching restrictions, in order to successfully open and close all switches in the system.
Before the optimization process begins, all operable switches must be opened in two passes.
First, all of the loop switches are opened. From the DEWorkstation data schema [EPRI], loop
switches are identified by checking the flag pCmp->sCtCmp != 0. Second, all remaining operable
switches are opened.
To perform a network power flow, all operable switches must be closed. As discussed above,
only certain switches may be closed to create loops. The sequence of switch closings is done in
two passes. First, all of the operable switches that could not possibly create a loop are closed.
These ineligible loop switches meet one of the following two conditions:
1. either end has more than one component connected to it
2. one of the ends does not have a line segment connected to it
In the second pass, all remaining operable switches are closed.
This process does not guarantee that all operable switches can be closed. Some system
configurations may not have an eligible loop closing switch for every loop. In those cases, the
user could insert zero-impedance line segments to create valid loop switches, without changing
the model results. DEWorkstation does not allow nested loops; this case does not have a
workaround. The two-pass closing process does tend to maximize the number of loops closed,
and it closed all possible loops in the chapter 4 test cases.
3.3 Isolating Failed Components
One of the first steps in reconfiguration is to isolate faults by opening and locking switches.
Figure 3.2 shows the algorithm used. Referring to the example in Figure 3.1, a feeder path trace
from 4 would find the switch at 2, and lock it open. The forward trace stop point is then set at 8.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
15
The forward trace from 2 will find the switch at 5, lock it open, and then jump to 7. At that point,
the adjacent tie switch would be locked open. The trace stops at 8, without opening the switch at
8.
select a failed component
set the stop to 0, current to failed component
feeder path trace to first operable switch
open and lock
set stop to switch's brother
set current to switch's forward
is current == stop?
Done
if adjacent circuit component is an operable
switch, lock it open
operable switch?
lock it open
set current to switch's brother
set current to switch's forward
Figure 3.2 - Algorithm to isolate failed components
3.4 Reconfiguration with Full Load Flow Evaluation
Figure 3.3 shows the main reconfiguration algorithm based on DAOP, assuming a full load flow
evaluation of candidate switch operations. The user can specify the types of switches to use, and
manually lock selected switches through the interface. The fault isolation process may lock more
switches. At the end of this process, the algorithm will have a list of operable switches for
reconfiguration. The benchmark losses and unserved load are solved with these operable switches
in their current state. It’s also useful to perform a network load flow with all operable switches
closed. This condition provides a lower bound on the losses.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
16
build operable switch list
isolate failed components
benchmark radial load flow
optional: do network load flow for ideal losses
open all operable switches, do base load flow
repeat:
list candidate switches with voltage on just 1 side
set BestMerit = infinity, BestSwitch = not found
for each candidate:
close candidate, run load flow on that circuit
is candidate Merit < BestMerit?
voltage, current constraints okay?
yes: update BestMerit and BestSwitch
no: warning message on component
open candidate, restore base load flow
was a BestSwitch found?
close BestSwitch and update base load flow
optional: check for backtracking
until: no BestSwitch found
report: losses, unserved load, switching operations
Figure 3.3 - Reconfiguration algorithm with full load flow screening
The reconfiguration starts by opening all operable switches, and then building a list of candidate
switches at each step. The candidates must have voltage on just one side of the switch. At each
step, all candidates are closed one at a time and the merit figure is evaluated. The candidate with
minimum merit figure is closed at that step, and the base case is updated. The total losses
increase at each step, according to the discrete ascent nature of the algorithm.
M = DP loss /(DS load + 0.001DQ cap )
(3.1)
∆Ploss is the incremental loss caused by the candidate switch closing, ∆Sload is the load kVA picked
up, and ∆Qcap is the utility capacitor bank kVAR picked up.
The merit figure favors minimum loss increment, normalized by the total apparent load picked up,
at each step. The normalization could be done equally well using load current magnitude, but
normalizing by the real load biased the solution in some cases, as described in section 4.6.
Equation (3.1) includes utility-owned capacitor banks as a low-weighted special case, even
though they provide no direct revenue or customer service benefit. Without this provision, the
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
17
algorithm could get stuck at a switched segment with capacitors but no customer load – closing
that switch increases the losses but not the load served. Once the capacitors are switched on, the
DEWorkstation load flow will operate their controls at each step. Another modification to (3.1)
is that any switching operation with no loss increment has a merit figure of 0, even if the
denominator is also 0. This lets the algorithm switch on zero-impedance branches that don’t
serve any direct load.
The algorithm stops when no suitable candidate can be closed in a given step. This may occur
when:
1. all load has been served – there are no more candidates
2. no more switches can be closed without violating voltage or current constraints
Whenever a component’s voltage or current constraint prevents an “attractive” candidate switch
closing, the warning message on that component serves as a clue that the system might be
upgraded in that area. The user can also modify the voltage and current constraint criteria to
reflect emergency conditions.
3.5 Reconfiguration with Approximate Loss Screening
The algorithm in Figure 3.3 calls for a full nonlinear load flow to evaluate each candidate
switching operation. This load flow runs on just one circuit, which will usually be a small subset
of the total system. Figure 3.4 modifies the algorithm’s main loop to use an approximate loss
screening of each candidate, using (3.2) for the loss in each switched branch. The switched
segment voltage is assumed constant and equal to the existing voltage behind the candidate
switch, and the load current angles are assumed balanced. The voltage and current constraints are
checked approximately, neglecting any decrease in the voltage at the switch. After selecting a
best candidate this way, the algorithm runs a full nonlinear load flow and checks the constraints
again. If this check fails, the algorithm repeats the step and all future evaluations of that candidate
will use a full load flow. At the conclusion of each step in Figure 3.4, the algorithm always has a
full load flow solution that satisfies all constraints, but the total number of load flows run is
greatly reduced.
P loss = I 21 R 11 + I 22 R 22 + I 23 R 33
Â
Â
Â
Â
Â
Â
Â
Â
Â
Â
Â
Â
+ 2R 12 (A re B re + A imB im ) + 2R 13 (A re C re + A im C im ) + 2R 23 (B re C re + B im C im )
(3.2)
I 1 = A 2re + A 2im
(3.3)
I 2 = B 2re + B 2im
(3.4)
I 3 = C 2re + C 2im
(3.5)
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
18
A re = kWflow1 /kV 1
(3.6)
A im = kVARflow1 /kV 1
(3.7)
B re = kWflow2 /kV 2
(3.8)
B im = kVARflow2 /kV 2
(3.9)
C re = kWflow3 /kV 3
(3.10)
C im = kVARflow3 /kV 3
(3.11)
Â
A re = A re
Â
A im = A im
Â
B re = −0.5B re + 0.866B im
Â
B re = −0.5B im − 0.866B re
Â
C re = −0.5C re − 0.866C im
Â
C re = −0.5C im + 0.866C re
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
In (3.2)-(3.11), the subscripts 1, 2, and 3 refer to three phase conductors, R refers to resistive
elements of the branch series impedance matrix, kWflow is the nominal real power flow through
the branch, kVARflow is the nominal reactive power flow through the branch, and kV is the
voltage behind the candidate switch. Subscripts re and im refer to real and imaginary parts of the
current, and primed quantities have been rotated to the same phase angle reference.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
19
repeat:
list candidate switches with voltage on just 1 side
set BestMerit = infinity, BestSwitch = not found
for each candidate:
approximate calculation of candidate Merit
is candidate Merit < BestMerit?
voltage, current constraints approximately okay?
yes: update BestMerit and BestSwitch
no: warning message on component
was a BestSwitch found?
close BestSwitch and update base load flow
actual voltage, current constraints violated?
switch to full load flow evaluation of BestSwitch
warning message on BestSwitch
optional: check for backtracking
until: no BestSwitch found
Figure 3.4 - Reconfiguration algorithm with approximate loss evaluation
3.6 Backtracking
Figures 3.3 and 3.4 show an optional backtracking step at the end of the main loop, further
detailed in Figure 3.5. Whenever a switch closes to complete an open loop, at least one of the
other candidates from that step will now have voltage on both sides. Those alternates are saved
in a list structure, keyed on the switch that was actually selected to close. Subsequent steps in the
algorithm may add load to that circuit, and make one of the alternates more attractive. The
algorithm checks alternate feeds whenever a selected candidate switch has one or more of the
open loop key switches in its feeder path. After opening the key switch, if an alternate feed has a
higher voltage magnitude than the key switch, the algorithm runs a load flow to see if the losses
decrease. If so, the key switch and best alternate are exchanged. An open loop may swap only
once per step, but may swap again in a later step.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
20
any other candidates have voltage on 2 sides?
add OpenLoop structure, with key = BestSwitch
alternates = list of candidates with voltage on 2 sides
mark all OpenLoops unswapped
set BaseLoss from existing load flow solution
repeat:
set BestLoop to not found
for each unswapped OpenLoop with key != BestSwitch
open key switch
set BestAlternate to not found
for each alternate with switch voltage > key voltage
close alternate and run load flow
losses < BaseLoss and constraints okay?
update BaseLoss and BestAlternate
open alternate and restore load flow
found a BestAlternate?
update BestLoop
close key switch and restore load flow
found a BestLoop?
swap BestLoop key and BestAlternate
update load flow
mark BestLoop swapped
until: no BestLoop found
Figure 3.5 - Open loop backtracking scheme
3.7 Example Reconfiguration with Backtracking
Figure 3.6 shows a sample circuit from [Dolloff], used to illustrate how the basic algorithm works
with backtracking. The series branches have been simplified to resistances, and the loads are
modeled as constant current. The system voltage is 13.8 kV, but in this case (3.1) is modified to
normalize by load current rather than load kVA. SWA is a proposed new switch location, not
considered in the first algorithm solution. The base case has switches 3 and 5 open, with losses of
90.14 kW per phase. A network load flow with all switches closed produces losses of 66.36 kW
per phase. Table 3.1 shows the switches closed, candidate switches, merit figures for each
candidate, and best candidate at each step.
Thomas E. McDermott
SW1
0.285 Ω 0.285 Ω SW2
Chapter 3 - Reconfiguration Algorithms
0.527 Ω 0.627 Ω SWA
S1
SW3
0.284 Ω 0.286 Ω SW4
SW5
209.185 Α
21
S2
41.837 Α
209.185 Α
0.570 Ω
1.710 Ω
SW6
83.674 Α
1.710 Ω
0.399 Ω
SW7
209.185 Α
S3
Figure 3.6 - Sample circuit with constant-current loads
Table 3.1 Reconfiguration algorithm steps for Figure 3.6
Step
1
2
3
4
5
Switches Closed
none
1
1, 4
1, 4, 7
1, 2, 4, 7
Candidates
1, 4, 7
2, 4, 7
2, 3, 7
2, 3, 6
5, 6
Merit Figures
59.62, 59.83, 83.46
165.13, 59.83, 83.46
165.13, 169.73, 83.46
165.13, 169.73, 486.48
402.97, 486.48
Best
1
4
7
2
5
To work through the solution step by step, the incremental load served by each candidate switch
is taken from the load currents shown on Figure 3.6. The incremental loss is calculated by
summing the I2R loss in each resistance along the feeder path containing the candidate switch,
then subtracting the previous total loss along that feeder path. The merit figure is the incremental
loss divided by the incremental load.
DAOP Step 1 - no switches closed
The candidate switches in step 1 are SW1, SW4, and SW7. The calculation of each candidate’s
merit figure follows:
SW1: Loss
= 0.285 * 209.185 * 209.185 = 12471.13
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
22
Load = 209.185
Merit = 12471.13 / 209.185 = 59.62
SW4: Loss
= 0.286 * 209.185 * 209.185 = 12514.89
Load = 209.185
Merit = 12514.89 / 209.185 = 59.83
SW7: Loss
= 0.399 * 209.185 * 209.185 = 17459.59
Load = 209.185
Merit = 17459.59 / 209.185 = 83.46
SW1 is closed because it has the lowest merit figure. Three feeder path load flow calculations
were performed in this step.
DAOP Step 2 - SW1 already closed
The candidate switches in step 2 are SW2, SW4, and SW7.
SW2: Loss
= 0.285 * (209.185 + 41.837)2 + (0.285 + 0.527) * 41.8372 - 12471.13
= 19379.70 - 12471.13 = 6908.57
Load = 41.837
Merit = 165.13
SW4: Merit = 59.83 from step 1
SW7: Merit = 83.46 from step 1
SW4 is closed because it has the lowest merit figure. One feeder path load flow calculation was
performed in this step, for SW2. We did not have to repeat the load flow calculations for SW4
and SW7.
DAOP Step 3 - SW1 and SW4 already closed
The candidate switches in step 3 are SW2, SW3, and SW7.
SW2: Merit = 165.13 from step 2
SW3: Loss
= 0.286 * (209.185 + 41.837)2 + (0.284 + 0.627) * 41.8372 - 12514.89
= 19616.00 - 12514.89 = 7101.11
Load = 41.837
Merit = 7101.11 / 41.837 = 169.73
SW7: Merit = 83.46 from step 1
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
23
SW7 is closed because it has the lowest merit figure. One feeder path load flow calculation was
performed in this step.
DAOP Step 4 - SW1, SW4, and SW7 already closed
The candidate switches in step 4 are SW2, SW3, and SW6.
SW2: Merit = 165.13 from step 3
SW3: Merit = 169.73 from step 3
SW6: Loss
= 0.399 * (209.185 + 83.674)2 + (1.71 + 1.71) * 83.6742 - 17459.59
= 58165.37 - 17459.59 = 40705.78
Load = 83.674
Merit = 40705.78 / 83.674 = 486.48
SW2 is closed because it has the lowest merit figure. This operation causes SW3 to have power
supplied on both sides, so SW3 cannot be closed in any subsequent step. For possible
backtracking, we will remember that SW2 is an open loop key switch, and that SW3 is its
corresponding open loop alternate switch.
One feeder path load flow calculation was performed in this step.
DAOP Step 5 - SW1, SW2, SW4, and SW7 already closed
The candidate switches in step 5 are SW5 and SW6.
SW5: Loss
= 0.285 * (209.185 + 41.837 + 83.674)2
+ (0.285 + 0.527) * (41.837 + 83.674)2
+ (0.627 + 0.570) * 83.6742 - 19379.70
= 53098.15 - 19379.70 = 33718.45
Load = 83.674
Merit = 33718.45 / 83.674 = 402.97
SW6: Merit = 486.48 from step 4
SW5 is closed because it has the lowest merit figure. One feeder path load flow calculation was
performed in this step. There have been a total of seven feeder path load flow calculations during
the solution so far; but none of these have needed to solve the whole system.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
24
DAOP Backtracking at Step 5
The solution after step 5 has SW3 and SW6 open, with losses of 83.75 kW per phase. When the
algorithm finishes, the sequence of partial load flows has constructed a full system load flow
solution, and the total system loss will be available without a repeat load flow solution.
When SW5 is closed at step 5, the open loop key switch SW2 is found in SW5’s new feeder path.
Exchanging the key switch SW2 with its alternate SW3, which had been identified in step 4, the
system losses decrease from 83.75 kW per phase to 71.53 kW per phase. Thus, the simple
backtracking procedure in Figure 3.5 improves the solution for this example, and in many of the
other test cases examined.
Referring to Figure 3.6, note that SW2 is electrically closer to the 41.837-ampere load than SW3
is. This is why SW2 is preferred at step 4. In the next step, a larger 83.674-ampere load must be
served, and SW3 is electrically closer to this larger load than is SW2. The alternative SW6 is not
attractive. The DAOP algorithm has been “fooled” by the 41.837-ampere load, and commits to
SW2 too early. The single-step backtracking scheme helps to mitigate this greedy nature of
DAOP.
The entire solution with backtracking has required seven partial load flows for candidate switch
evaluation, plus another load flow on two feeder paths for the backtracking step.
Transshipment Solution
The problem in Figure 3.6 can also be solved using the linearized transshipment method described
in section 2.4. Assuming all line segments have the same current capacity, the segment cost will
be the sum of resistances between the candidate switch and the load it serves. The transshipment
algorithm starts with all switches open.
Transshipment Step 1 - no switches closed
The candidates in step 1 are SW1, SW4, and SW7. The segment costs are:
SW1: Cost = 0.285
SW4: Cost = 0.286
SW7: Cost = 0.399
SW1 has the lowest cost, so its 209.185-ampere load will be picked up first. There are no other
switches that can serve this load, so SW1 is closed. No load flow calculations were performed in
this step.
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
25
Transshipment Step 2 - SW1 already closed
The candidates in step 2 are SW2, SW4, and SW7. The segment costs are:
SW2: Cost = 0.285 + 0.527 = 0.812
SW4: Cost = 0.286
SW7: Cost = 0.399
SW4 has the lowest cost, and no other switch can serve its 209.185-ampere load. SW4 is closed.
Transshipment Step 3 - SW1 and SW4 already closed
The candidates in step 3 are SW2, SW3, and SW7. The segment costs are:
SW2: Cost = 0.285 + 0.527 = 0.812
SW3: Cost = 0.284 + 0.627 = 0.911
SW7: Cost = 0.399
SW7 has the lowest cost, and no other switch can serve its 209.185-ampere load. SW7 is closed.
Transshipment Step 4 - SW1, SW4, and SW7 already closed
The candidates in step 4 are SW2, SW3, and SW6. The segment costs are:
SW2: Cost = 0.285 + 0.527 = 0.812
SW3: Cost = 0.284 + 0.627 = 0.911
SW6: Cost = 1.71 + 1.71 = 3.42
SW2 has the lowest cost, so the 41.837-ampere load will be picked up next. SW3 can also serve
this load, so the transshipment algorithm will check both SW2 and SW3, and close the one that
results in the lowest losses. Glamocanin used an approximate loss formulation to do this, but a
full load flow solution could also be done. Using either method, closing SW2 will result in the
lowest losses, and the algorithm will select SW2. It is the 83.674-ampere load that changes this
selection; since it has not been served yet, the transshipment algorithm does not account for it.
Transshipment Step 5 - SW1, SW2, SW4, and SW7 already closed
The candidates in step 5 are SW5 and SW6. The segment costs are:
SW5: Cost = 0.627 + 0.57 = 1.197
SW6: Cost = 1.71 + 1.71 = 3.42
SW5 has the lowest cost, so the 83.674-ampere load will be picked up next. SW6 can also serve
this load, but it will be found to produce greater losses than SW5. Therefore, SW5 is closed and
Thomas E. McDermott
Chapter 3 - Reconfiguration Algorithms
26
the transshipment algorithm solution is complete. At this point, a full system load flow will
calculate the total losses as 83.75 kW per phase.
Comparison of DAOP and Transshipment Algorithms
The transshipment solution is the same as the DAOP solution without backtracking. The
transshipment algorithm in practice would require one feeder path load flow at the end of each
step to verify the voltage and current constraints, plus two more feeder path load flows in steps 4
and 5, to evaluate the alternate switches SW3 and SW6. This is approximately the same
computational burden as DAOP with no backtracking. The DAOP algorithm can account for
nonlinearities and control action during the candidate switch evaluations, which is an advantage.
With backtracking, DAOP requires more load flow calculations than transshipment, but it reaches
a better solution. This is also an advantage over the linearized algorithm. Use of the full load
flow calculation would not help the transshipment solution in this case. Adding a switch exchange
algorithm at the end of the transshipment solution might be helpful in this case. However,
Glamocanin stated that a prime motivation for the transshipment algorithm was to avoid switch
exchanges.
Chapter 4
Test Cases
Tables 4.1 through 4.3 summarize results for a variety of test systems used in the literature. A
constant-current load model was used for all results in Table 4.2, except for the Baran and Wu
circuit, which used a constant-power load model. Since the algorithm runs load flows on one
circuit at a time, a rough measure of system size would be the number of switches divided by the
number of circuits. By this measure, the Baran and Wu system is much larger than the others.
The algorithm, run with backtracking and approximate loss screening, reached the best solution in
all these cases, including the improved solution reported in [Goswami] for the Baran and Wu
system. The algorithm worked with either constant-current or constant-power loads in all test
systems, and with various capacitor configurations in the two systems that have capacitors.
Each system was tested with a variety of component failures, and the algorithm served as much
load as possible while minimizing losses. In those cases, the losses decrease compared to the
unfaulted state reported in Table 4.2.
The running time is mostly dependent on the number of full load flows run, and Table 4.3 shows
about a six-to-one reduction in load flows by using the approximate loss evaluation. In all tests
examined so far, the approximate loss evaluation produces the same reconfiguration as the full
load flow screening.
Table 4.1 Test System Size Characteristics
Test System
Dolloff
Civanlar 2-Feeder
Civanlar 3-Feeder
Glamocanin
Baldick Discussion
Baran and Wu
Circuits
3
2
3
3
1
1
Loads
5
10
13
9
2
32
Lines
10
11
16
13
4
37
Switches
8
12
17
13
5
37
Capacitors
0
0
7
0
1
0
Table 4.2 Benchmark, networked, and final system losses [kW]
Test System
Dolloff
Civanlar 2-Feeder
Civanlar 3-Feeder
Glamocanin
Baldick Discussion
Baran and Wu
Benchmark
270.41
310.00
488.46
255.65
18.78
202.68
Network
199.08
305.22
412.34
229.53
10.63
124.55
27
Final
202.28
306.04
447.45
247.49
15.75
142.60
Thomas E. McDermott
Chapter 4 - Test Cases
28
Table 4.3 Simulation times on a Pentium 90 [seconds]
Test System
Dolloff
Civanlar 2-Feeder
Civanlar 3-Feeder
Glamocanin
Baldick Discussion
Baran and Wu
Network
1.867
1.373
3.625
2.966
0.494
14.063
Full
2.910
5.384
15.658
6.373
1.373
226.697
Approximate
0.988
1.153
2.964
0.933
0.220
20.451
In DEWorkstation, the test circuits are modeled by drawing the line segments, switches,
substations, and capacitor banks. The branch impedances are entered through editing dialogs for
each line segment. For other components, including substations, capacitor banks, and line voltage
regulators, the user must enter the equipment data into DEWorkstation’s parts library, before
placing these parts on the schematic. Sections 4.1 through 4.7 list all of the required parts, branch
impedance, and load data needed to model the test circuits in DEWorkstation.
Most of the published examples list bus loads, which is the normal practice for networked
transmission and subtransmission systems. DEWorkstation associates loads with line segments,
which is common practice for radial distribution systems. The DEWorkstation assumes all
segment load is lumped at the far end of the line segment, to give conservative estimates of the
voltage drop. If a segment is switched to an alternate feed, its load can move from one end to the
other in DEWorkstation’s model. In order to make valid comparisons, the load locations must be
the same as in the published results. In DEWorkstation, this was accomplished using
zero-impedance stubs at each bus, with all load at that bus represented as a spot load on that stub
segment.
4.1 Dolloff’s Circuit
The system shown in Figure 3.6 was modeled in DEWorkstation to verify the algorithm’s
computer implementation. At a nominal substation voltage of 13.8 kV line-to-line, the
constant-current loads in Figure 3.6 are input as the kW loads specified in Table 4.4. The branch
impedances are specified with the resistance values given in Figure 3.6, and a transformer turns
ratio of 63.5:1 in each line segment. At 13.8 kV, this transformer turns ratio produces a customer
service voltage of 120 volts:
13800/( 3 & 63.51) = 120.0
(4.1)
Table 4.4 Load data for Dolloff’s circuit in DEWorkstation
Current [A]
41.837
83.674
209.185
Power [kW]
1000.0
2000.0
5000.0
Thomas E. McDermott
Chapter 4 - Test Cases
29
The DEWorkstation solutions of Dolloff’s circuit use three-phase losses, which are three times
the per-phase values calculated in section 3.7. In the base case with SW3 and SW5 open, which
looks symmetrical on Figure 3.6, the total losses are 270.41 kW. Running the algorithm without
backtracking produces a configuration of SW3 and SW6 open, with total losses of 251.25 kW.
Repeating the solution with backtracking, the algorithm finds the correct solution of SW2 and
SW6 open, with total losses of 214.59 kW. The network system losses are 190.07 kW, so most
of the available improvement has been achieved with SW2 and SW6 open.
The use of backtracking indicates a possible benefit from installing more switches in that area. If
SWA is installed, the optimal solution has SWA and SW6 open, with losses of 202.27 kW.
Compared to the network solution, very little improvement is possible.
4.2 Civanlar 2-Feeder Circuit
Table 4.5 contains the branch impedance and bus load data for Civanlar’s 2-Feeder test circuit.
The nominal substation voltage is 23 kV line-to-line, which requires a distribution transformer
turns ration of 110.0:1 in each line segment. There are no current constraints given.
Table 4.5 Impedance and load data for Civanlar 2-Feeder circuit
Bus
From
A
5
4
3
2
1
1’
2’
3’
4’
5’
A
5
4
Impedance [ohms]
R
X
0.3968 0.5290
0.4232 0.5819
0.4761 0.6348
0.2116 0.2116
0.1587 0.1587
0.2116 0.0529
0.5290 0.5290
0.5819 0.5819
0.4761 0.6348
0.2910 0.5819
0.5290 0.5290
To
5
4
3
2
1
1’
2’
3’
4’
5’
B
3
2
1
Figure 4.1 - Civanlar’s 2-Feeder Test Circuit
1'
Load at To Bus [kVA]
P
Q
2000.0
600.0
3000.0
1300.0
2000.0
500.0
1500.0
300.0
500.0
100.0
1000.0
200.0
1500.0
200.0
2500.0
600.0
3000.0
400.0
2500.0
900.0
2'
3'
4'
5'
B
Thomas E. McDermott
Chapter 4 - Test Cases
30
Civanlar’s 2-feeder test circuit does not require backtracking. The algorithm was tested with both
constant current and constant power load models, and reached the same configuration (1’-2’
open) in both cases.
4.3 Civanlar 3-Feeder Circuit
Table 4.6 contains the branch impedance and bus load data for the circuit in Figure 4.2. The
nominal substation voltage is 23 kV line-to-line, which requires a distribution transformer turns
ratio of 110.0:1 in each line segment. The capacitor banks are all wye grounded, and comprised
of parallel combinations of units rated 300 kVAR and 100 kVAR at 23 kV.
Figure 4.2 shows the base case, and Figure 4.3 shows the optimal configuration found by the
DAOP algorithm. The solution in Figure 4.3 matches that found by Civanlar. This test case
illustrates that the DAOP algorithm works with capacitor banks. No voltage or current
constraints were encountered, but backtracking was initiated during the solution.
Table 4.6 Impedance and load data for Civanlar 3-Feeder circuit
Bus
From
1
4
4
6
2
8
8
9
9
3
13
13
15
5
10
7
To
4
5
6
7
8
9
10
11
12
13
14
15
16
11
14
16
Impedance [ohms]
R
X
0.3968 0.5290
0.4232 0.5819
0.4761 0.9522
0.2116 0.2116
0.5819 0.5819
0.4232 0.5819
0.5819 0.5819
0.5819 0.5819
0.4232 0.5819
0.5819 0.5819
0.4761 0.6348
0.2432 0.5819
0.2116 0.2116
0.2116 0.2116
0.2116 0.2116
0.4761 0.6348
Load at To Bus [kVA]
P
Q
Cap Bank
2000.0 1600.0
3000.0 1500.0
1100.0
2000.0 800.0
1200.0
1500.0 1200.0
4000.0 2700.0
5000.0 3000.0
1200.0
1000.0 900.0
600.0 100.0
600.0
4500.0 2000.0
3700.0
1000.0 900.0
1000.0 700.0
1800.0
1000.0 900.0
2100.0 1000.0
1800.0
Thomas E. McDermott
1
Chapter 4 - Test Cases
2
120.7
225.0
31
3
120.7
389.4
128.1
118.2
4
119.6
76.0
13
8
119.3
5
120.0
33.8
117.3
11
37.1
10
253.9
14
120.1
118.0
19.3
120.7
117.3
90.2
77.9
9
119.8
6
15
117.1
119.0
12
119.7
48.2
56.3
7
118.9
16
119.7
Figure 4.2 - Civanlar’s 3-Feeder Test Circuit before Reconfiguration (line flows are in amperes,
customer service voltage is italicized)
Thomas E. McDermott
1
Chapter 4 - Test Cases
2
120.7
236.0
32
3
120.7
347.7
154.6
118.5
4
119.6
90.4
13
8
119.3
119.8
119.3
5
120.7
50.4
11
10
238.8
19.5
119.5
14
33.8
119.6
117.6
90.2
77.9
9
119.9
6
15
117.3
119.0
12
119.5
48.2
56.2
7
118.9
16
119.4
Figure 4.3 - Civanlar’s 3-Feeder Test Circuit after Reconfiguration (line flows are in amperes,
customer service voltage is italicized)
4.4 Glamocanin Circuit
Table 4.7 contains the branch impedance and bus load data for the circuit shown in Figure 4.4. In
DEWorkstation, the impedances are entered directly from Table 4.7, but each line segment is
assigned a conductor with a rating of 194 amperes. This conductor assignment matches the
current constraint used in [Glamocanin]. The nominal substation voltage is 10 kV line-to-line,
because the example is based on a European system. The distribution transformer ratio is 48.11:1
in each line segment. The voltage drop constraint is 5%, corresponding to a customer service
voltage of 114 volts.
Figure 4.4 shows the base case load flow solution, and Figure 4.5 shows the load flow solution
after the DAOP algorithm’s reconfiguration. Figure 4.5 matches the configuration found by
Glamocanin. Voltage constraints were encountered and successfully handled during the solution.
Glamocanin pointed out that the branch exchange method would not work on the base case
shown in Figure 4.4, due to voltage and current constraints.
Thomas E. McDermott
Chapter 4 - Test Cases
33
Table 4.7 Impedance and load data for Glamocanin circuit
Bus
From
1
1
1
2
2
3
4
3
2
3
6
8
4
Impedance [ohms]
R
X
0.7820 0.2120
0.7820 0.2120
1.5640 0.4240
0.7820 0.2120
1.1730 0.3180
1.3685 0.3710
1.1730 0.3180
0.7820 0.2120
1.1730 0.3180
1.1730 0.3180
0.7820 0.2120
1.1730 0.3180
1.1730 0.3180
To
2
3
4
5
6
7
8
9
10
5
9
9
10
Load at To Bus [kVA]
P
Q
600.0
400.0
500.0
300.0
100.0
90.0
600.0
400.0
1300.0
1100.0
1300.0
1000.0
100.0
90.0
800.0
600.0
300.0
100.0
1
120.0
157.1
169.8
10
73.2
117.1
18.3
2
117.5
5
116.3
98.3
3
117.3
41.6
4
117.7
94.7
6
7
115.3
114.7
65.5
8
116.2
57.7
114.8
9
Figure 4.4 - Glamocanin’s Test Circuit before Reconfiguration (line flows are in amperes,
customer service voltage is italicized)
Thomas E. McDermott
Chapter 4 - Test Cases
34
1
120.0
181.3
185.9
10
33.1
118.5
18.3
2
117.2
41.6
98.3
5
3
117.1
4
116.5
118.9
94.7
6
7
114.9
114.5
7.7
8
57.7
118.7
116.2
9
Figure 4.5 - Glamocanin’s Test Circuit after Reconfiguration (line flows are in amperes, customer
service voltage is italicized)
4.5 Baldick Discussion Circuit
Figure 4.6 shows the test circuit with a capacitor bank from Baldick’s discussion of [Glamocanin].
The nominal substation voltage is 13.8 kV line-to-line, which requires a distribution transformer
turns ratio of 63.5:1 in each segment. The 50-ampere load is 1195 kW three-phase at 13.8 kV,
and the 100+j100 ampere load is 2390 kW plus 2390 kVAR at 13.8 kV. The 100-ampere
capacitor bank was modeled with a 1951.706 kVAR bank at 12.47 kV.
The pure transshipment algorithm does not find the optimal configuration for this circuit, as
discussed in section 2.4. The DAOP algorithm works, because it assigns a low but non-zero
weight to shunt capacitor banks in the merit figure. The weight is not zero, because in some
configurations that can act as a barrier to the DAOP algorithm. With a capacitor bank weighting
of zero, DAOP would never close a switch that served just a capacitor bank, because that would
increase losses with no corresponding increase in load served. If there were customer loads
beyond the capacitor bank, and no alternate feeds for those customer loads, they would never be
served in the DAOP algorithm.
Thomas E. McDermott
Chapter 4 - Test Cases
35
1
0.1 Ω
0.2 Ω
2
3
0.1 Ω
4
0.2 Ω
50 A
-j100 A
100 + j100 A
Figure 4.6 - Baldick’s Discussion Test Circuit
4.6 Baran and Wu Circuit
Table 4.8 contains the branch impedance and bus load data for the circuit in Figure 4.7. The
substation voltage is 12.66 kV line-to-line, which requires a transformer ratio of 60.91:1 in each
segment.
This test circuit is more severe than the others, because of the low voltage conditions, the large
reactive load at bus 29, and the number of tie switches within a single feeder. To solve this circuit
with the DAOP algorithm, an emergency voltage limit of 110 volts must be specified,
corresponding to the ANSI B range. Otherwise, the DAOP algorithm will stop before serving all
the load, due to voltage constraints. The optimal configuration in Figure 4.8 has lower system
losses, but still does not meet the normal operating voltage criteria of at least 114 volts.
The solution in Figure 4.8, which was obtained using backtracking, matches the one found by
Goswami and Basu. The solution originally reported by Baran and Wu, obtained using their own
algorithm and the loop cutting algorithm, was slightly less optimal than Figure 4.8, with
approximately 600 watts additional loss. In practice, the difference would not be significant, and
this illustrates why it is not always necessary to find the absolute global optimum.
In most cases, the merit figure can be calculated with the customer load’s real power in the
denominator. This has the advantage of corresponding directly to the utility’s revenue-earning
load. For the Baran and Wu test case, the merit figure was altered to use the customer load’s
apparent power. Otherwise, the large reactive load at bus 29 caused the DAOP algorithm to find
a sub-optimal solution. The real and reactive loads at bus 29 must be served together, so they
should be combined as apparent power in the merit figure. Considering only the real power, the
losses increase by a disproportionate amount due to the current drawn by the reactive load, and
this biases the DAOP solution in favor of other bus loads. Most loads have real power greater
Thomas E. McDermott
Chapter 4 - Test Cases
36
than reactive power, so the difference is not important. After making this change in the merit
figure, all test cases were repeated and produced the same results.
The other test cases described in this chapter involve at least two feeders. The test circuit in
Figures 4.7 and 4.8 has just one source, representing a single feeder with parallel branches. Each
load flow in DAOP amounts to a full system load flow in this test case, because there are no other
circuits to exclude from the load flow solution. The DAOP algorithm would be relatively more
efficient when there are multiple sources.
Thomas E. McDermott
Chapter 4 - Test Cases
Table 4.8 Impedance and load data for Baran and Wu circuit
Bus
From
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
18
19
20
2
22
23
5
25
26
27
28
29
30
31
7
8
11
17
24
To
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
20
14
21
32
28
Impedance [ohms]
R
X
0.0922 0.0470
0.4930 0.2511
0.3660 0.1864
0.3811 0.1941
0.8190 0.7070
0.1872 0.6188
0.7114 0.2351
1.0300 0.7400
1.0440 0.7400
0.1966 0.0650
0.3744 0.1238
1.4680 1.1550
0.5416 0.7129
0.5910 0.5260
0.7463 0.5450
1.2890 1.7210
0.7320 0.5740
0.1640 0.1565
1.5042 1.3554
0.4095 0.4784
0.7089 0.9373
0.4512 0.3083
0.8980 0.7091
0.8960 0.7011
0.2030 0.1034
0.2842 0.1447
1.0590 0.9337
0.8042 0.7006
0.5075 0.2585
0.9744 0.9630
0.3105 0.3619
0.3410 0.5302
2.0000 2.0000
2.0000 2.0000
2.0000 2.0000
0.5000 0.5000
0.5000 0.5000
Load at To Bus [kVA]
P
Q
100.0
60.0
90.0
40.0
120.0
80.0
60.0
30.0
60.0
20.0
200.0
100.0
200.0
100.0
60.0
20.0
60.0
20.0
45.0
30.0
60.0
35.0
60.0
35.0
120.0
80.0
60.0
10.0
60.0
20.0
60.0
20.0
90.0
40.0
90.0
40.0
90.0
40.0
90.0
40.0
90.0
40.0
90.0
50.0
420.0
200.0
420.0
200.0
60.0
25.0
60.0
25.0
60.0
20.0
120.0
70.0
200.0
600.0
150.0
70.0
210.0
100.0
60.0
40.0
37
Thomas E. McDermott
Chapter 4 - Test Cases
38
120.0
1
119.6
18
2
22
187.1
119.6
119.2
117.5
118.0
210.4
48.5
116.7
18.1
19
43.7
23
134.6
13.6
3
117.1
21.9
9.1
127.9
114.0
6
20
116.3
4
113.5
119.1
24
116.2
5
58.4
124.8
47.6
28
112.0
111.1
57.0
65.4
7
4.5
27
113.7
113.0
36.8
25
112.2
50.6
8
59.6
62.5
111.5
119.0
9
21
33.4
26
29
110.1
113.4
111.4
110.6
14
30.6
23.4
10
14.2
109.9
11.2
30
15
28.0
110.1
15.1
11
111.2
31
8.1
13
21.2
24.6
110.0
110.2
3.6
16
4.9
109.6
12
110.5
17
109.6
32
110.0
Figure 4.7 - Baran and Wu’s Test Circuit Before Reconfiguration (line flows are in amperes,
customer service voltage is italicized)
Thomas E. McDermott
Chapter 4 - Test Cases
39
120.0
1
119.4
18
2
22
134.6
119.6
117.4
118.0
118.4
207.1
48.3
117.2
67.8
19
43.5
23
82.9
63.3
3
117.9
21.8
58.7
76.3
116.1
6
20
116.8
4
116.0
116.8
24
5
10.6
117.4
73.3
28
30.6
114.3
113.4
52.4
60.5
7
25.5
27
115.9
115.5
20.0
25
115.1
46.2
8
55.0
57.7
115.5
116.4
9
21
26
17.0
29
114.4
115.6
115.5
18.9
113.0
14
3.0
19.4
10
114.2
14.2
30
15
5.5
112.6
11.3
11
115.6
31
11.2
13
6.9
10.2
112.5
115.2
16
113.8
12
115.3
8.2
17
113.7
3.5
32
113.7
Figure 4.8 - Baran and Wu’s Test Circuit After Reconfiguration (line flows are in amperes,
customer service voltage is italicized)
Thomas E. McDermott
Chapter 4 - Test Cases
40
4.7 Baran and Wu Circuit with Voltage Regulator
The Baran and Wu test system is shown in Figure 4.9, with two line voltage regulators added.
Each line regulator is rated 12.66 kV line-to-line and 10 MVA, for a continuous current rating of
1400 amperes. The leakage impedance is 0.5 ohms, the winding losses are 1 watt, and the
magnetizing current is 1 ampere. Each regulator is set for 120 volts, with a deadband of 2 volts,
and no line drop compensation.
In the base case, with the 5 segments shown in dashed lines open, the service voltage is 109.6
volts at bus 17. This would violate an emergency limit of 110 volts. After the optimal
reconfiguration of four switch exchanges shown in Figure 4.9, the service voltage is still 112.5
volts at bus 31. This would violate a normal limit of 114 volts. The utility might use line voltage
regulators to deal with these voltage problems, as shown in Figure 4.9. Note that after the
optimal switch swaps, the two regulators are not in series. With these regulators, the base case
losses are 196.91 kW with a minimum service voltage of 116.3 volts at bus 24. The
reconfiguration algorithm finds the same optimal switch exchanges, with total losses of 137.32
kW and a minimum service voltage of 115.2 volts at bus 13. During the solution, the regulator
controllers move the taps appropriately during each load flow. This example illustrates how the
reconfiguration algorithm can simulate control action, although in practice, the circuit in Figure
4.9 should also have some shunt capacitor banks installed.
Thomas E. McDermott
Chapter 4 - Test Cases
41
118.0
118.4
119.6
119.4
1
18
119.2
117.4
2
117.5
118.0
119.6
119.6
22
19
4
3
116.2
117.4
116.7
117.2
117.1
117.9
23
119.1
116.9
120.7
121.0
20
116.3
116.8
120.9
120.8
6
5
25
24
120.6
120.6
121.1
121.0
119.3
119.4
26
7
27
120.1
115.5
8
120.3
115.5
119.0
116.4
28
118.4
118.5
121.0
119.1
9
21
29
119.0
118.4
118.0
118.1
14
120.2
115.5
10
118.8
118.2
30
15
117.5
117.7
11
120.0
115.6
31
13
117.4
117.7
119.1
115.2
16
12
119.4
115.3
118.6
117.9
17
118.5
117.8
32
117.4
117.7
Figure 4.9 - Baran and Wu’s Test Circuit with Voltage Regulators (customer service voltage
before reconfiguartion in red, after reconfiguration in blue)
Chapter 5
Conclusions
This algorithm differs from most others, by constructing the system from scratch, rather than
performing switch exchanges or sequential switch openings. An approximate loss formula helps
to quickly screen candidate switch closings, but this method still performs more load flow
calculations than other methods. Most of the load flow calculations only work with a subset of
the system. The algorithm can solve either load restoration or loss minimization problems.
Several test cases show that this algorithm reaches the same optimal solutions found by other
investigators. The execution time is longer, but constraints and control actions are handled more
accurately. One of the test cases with line voltage regulators, described in Section 4.7, can not be
solved properly with other published reconfiguration algorithms.
The work reported here builds on the linearized transshipment algorithm [Glamocanin] and
discrete ascent optimal programming [Dolloff]. The original contributions of this work, not
previously reported, are:
1. The reconfiguration algorithm considers control action and other nonlinear effects during the
solution, not after the optimal configuration has been found. This affects both the loss
evaluation and the feasible switching operations, as illustrated by the voltage regulator
example in Section 4.7.
2. Approximate loss formulas were developed in Section 3.5 to screen candidate switching
operations, making use of partial load flow solutions. After a candidate switch has been
selected, the algorithm updates the complete load flow solution. In all test cases evaluated,
screening with the approximate loss formulas produced the same solution as screening with
full load flow solutions.
3. The heuristic backtracking method presented in Section 3.6 was developed to mitigate the
greedy nature of DAOP, solving one of the problems encountered by Dolloff.
4. The DAOP merit figure was adjusted to provide robust solutions with large reactive loads and
capacitor banks in the system, solving one of the problems encountered by Glamocanin.
5. A method to isolate failed components has been developed and customized to the data
structures in EPRI’s DEWorkstation.
6. The network load flow solution provides a lower bound on the losses, and a measure of how
good the algorithm solution is. This would be a useful addition to other methods, since none
can guarantee a global optimum.
42
Thomas E. McDermott
Chapter 5 - Conclusions
43
7. The implementation allows a user to manually operate and lock switches before the algorithm
runs. This would also be a useful addition to other methods.
Since this algorithm has been implemented in EPRI’s DEWorkstation, it is ready for application
by electric utilities in real-world situations. Future work should take further advantage of the
integrated design environment in DEWorkstation:
1. The algorithm could include overcurrent protective coordination constraints, once the
DEWorkstation’s overcurrent protection application has a suitable programming interface.
2. The algorithm could include service reliability indices in the merit figure, once the
DEWorkstation reliability application has been developed.
When the algorithm backtracks, or encounters a voltage or current constraint, this indicates an
area where new equipment may improve the solution. Considerable design judgment is still
required to select and size the new equipment. A new feature planned for DEWorktation will
incorporate the output from a variety of applications, such as reconfiguration, capacitor
placement, overcurrent protection, and reliablity analysis. The reconfiguration algorithm could be
interfaced to this new integrated design tool, as it is developed.
Appendix A
User Manual
This appendix describes the user interface for the DAOP reconfiguration algorithm as
implemented in DEWorkstation. The material is taken from the on-line help developed for the
application, supplemented by screen shots from DEWorkstation and the DAOP reconfiguration
application.
The application is installed by copying lossmin.dll to the DEWorkstation directory, and
lossmin.hlp to the DEWorkstation help directory. The application has been named “loss
minimization” to distinguish it from another DEWorkstation reconfiguration algorithm, which is
based entirely on heuristics. Figure A.1 shows the DAOP reconfiguration algorithm selected to
run, in DEWorkstation’s application setup dialog.
Figure A.1 - Loss minimization algorithm selected to run in DEWorkstation
44
Thomas E. McDermott
Appendix A - User Manual
45
A.1 Introduction to Loss Minimization
This application reconfigures the system switches for minimum losses, using the method of
discrete ascent optimal programming (DAOP). Before the analysis runs, the user may specify
failed components, and the algorithm will determine which switches must be opened to isolate
these failed components. The user may also specify certain switches to lock open or closed, and
the algorithm will not operate these switches.
After these preliminaries, the algorithm performs a power flow solution to determine the losses in
the existing configuration. Then a network power may be performed with all operable switches
closed. This determines the minimum possible loss, and provides a measure of how good the
DAOP solution is compared to the existing configuration.
The DAOP solution commences by opening all operable switches, so that no load is served. Or,
the user might wish to lock many of the existing switches in their present state, running DAOP on
just a portion of the system. In that case the initial load served is not zero, but some base level.
Then DAOP proceeds in steps of closing one switch at a time, increasing the total load served and
the total losses at each step, until all the load possible has been served. During this process the
system is kept radial, and all voltage and current constraints are respected.
The DAOP method is a greedy search with some heuristic modifications. As such it should
provide a good but not necessarily optimum solution. Compared to other methods of optimal
switch reconfiguration, the main advantages of DAOP are more robust handling of constraints,
and full nonlinear system analysis - leading to better accuracy.
A.2 Interactive Inputs - Loss Minimization
The analysis setup dialog for loss minimization, shown in Figure A.2, lets you specify which kinds
of switches to operate, and what the voltage and current constraints will be. Specifying any failed
components or locked switches is done later, using the control center.
Thomas E. McDermott
Appendix A - User Manual
46
Figure A.2 - Setup dialog for loss minimization application
Nonsupervisory Switches To Use - This group of checkboxes allows you to specify that load
break switches, fused cutouts, and/or reclosers are operable. If a box is unchecked, the algorithm
will lock all switches of that class in the existing position. These choices interact with the
Switching Restrictions checkboxes. Also, supervisory switches are always used.
Switching Restrictions - These checkboxes are optional. If three-phase only is checked, all
single phase switches will be locked in their existing positions. If automatic only is checked, only
supervisory switches or those that have an attached communications receiver will be operated in
the algorithm.
The algorithm chooses switches to use as follows:
1.
exclude all switches with Allow Device Movement turned off from the Schematic Control
Center
2.
exclude all switches operated using Lock Switches from the Loss Minimization Control
Center
3.
if three-phase only has been selected, exclude all single-phase and two-phase switches
4.
of those remaining, all supervisory switches are used
5.
nonsupervisory switches are also used if:
a.
their type matches a selected checkbox for load break switches, fused cutouts, or
recloser
Thomas E. McDermott
b.
Appendix A - User Manual
47
if automatic only has been selected, an attached communications receiver is also
required
Do Full Evaluation Load Flows - If selected, each candidate switch closing is evaluated using
the full radial power flow on that circuit. If not selected, short-cut calculations are used to screen
each candidate and the full radial power flow is done only on the apparent best switch closing.
The short-cut calculations run significantly faster and will not allow a switch closing that violates
any constraints (because of the verifying radial power flow calculation). However, there is some
risk that a best switch closing will be screened out due to apparent constraint violations that
would not actually occur in a full radial power flow solution.
Do Network Flow - If selected, all switches to use are closed to obtain a lower bound on the
losses. Closing all switches will typically create loops, requiring a network power flow solution.
Due to topology assumptions of the network power flow, it may not be possible to close all
switches and convergence may be difficult.
Allow Backtracking - If selected, the algorithm will keep track of alternate tie switches each time
it completes an open loop. If more load is added to the open loop, these alternate ties will be
re-evaluated. This option improves the algorithm’s performance with, usually, a negligible
increase in computational effort.
Check Device Coordination - If selected, the algorithm will check that overcurrent protective
devices remain coordinated at each step. Note that voltage, current, and radiality constraints are
always checked.
Percent Low Normal Voltage - Switching operations that allow any voltage to drop below this
value are disallowed, under normal operating conditions.
Percent Low Emergency Voltage - Switching operations that cause any voltage to drop below
this value are disallowed, under emergency operating conditions. The selection of normal or
emergency condition is made later, using the control center.
Percent Overload Allowed - Switching operations that cause any component current overload to
exceed this value are disallowed, under emergency operating conditions. The allowed current
overload for normal conditions is 0%.
A.3 Control Center - Loss Minimization
When the loss minimization algorithm runs, the control center shown in Figure 20 appears in a
floating window. The control center allows you to select normal or emergency ratings, specify
any failed components, operate switches, start the analysis, and view results. Some of the buttons
in Figure A.3 are not enabled until you run the algorithm.
Thomas E. McDermott
Appendix A - User Manual
48
Figure A.3 - Control dialog for loss minimization - before execution
Any switches that had Allow Device Movement turned off in the Schematic Control Center will
be highlighted with a gray ellipse on the schematic. Any switches manually operated through the
control center’s Lock Switches button will be highlighted with a magenta ellipse on the schematic.
None of these switches will be operated automatically by the DAOP algorithm; they will be fixed
in their existing positions when you select Execute. The highlighting is removed when you close
the Loss Minimization Control Center, but note that switches with the gray highlights are still
locked for other DEW algorithms.
Normal/Emergency - These radiobuttons determine whether normal or emergency component
current ratings are used as constraints, and also whether the normal or emergency low voltage
constraint is used. Current ratings are established from the parts tables and voltage ratings are
established from the setup panel.
Lock Switches - The chooser cursor appears to manually operate switches. Each time you click
on a switch, the state toggles through:
1.
locked in the original state
2.
toggled from the original state and locked
3.
automatically operable in the original state
Press Esc to exit this mode. Switches remain locked only for the loss minimization algorithm.
Restore - Return switch states and failed component states to their original condition, before
invoking the control center.
Spec Failed - The chooser cursor appears to select failed components. Each time you click on a
component, the state toggles between failed and good. Press Esc to exit this mode.
Configurations - Access the switching configurations dialog, which lets you save and restore
various switch configurations (but not failed component selections).
Execute - Run the algorithm to isolate any failed components and reconfigure the switches for
minimum loss. This enables the result buttons, as shown in Figure A.4.
Thomas E. McDermott
Appendix A - User Manual
49
Figure A.4 - Control dialog for loss minimization - after execution
System Result - Brings up a dialog showing the algorithm’s recommended switches to open and
close, with a command button to highlight each switch.
Cmp Result - The chooser cursor appears to select components for individual power flow results
and messages. Press Esc to exit this mode.
Report - Formats a report of system results to save to disk, or view in Notepad.
A.4 Switching Configurations - Loss Minimization
The open/close state of each switch in the system can be saved and recalled using the dialog
shown in Figure A.5. A switch configuration comprises a list of all switches in the system by
name, with their current states.
Figure A.5 - Dialog to manage switching configurations
Switch Configurations - This list shows all of the switch configurations you have saved by name.
Select one from this list to recall or delete it.
Save Current Configuration - A dialog appears for you to supply a unique name. All existing
switch states will be saved as a configuration under this name.
Thomas E. McDermott
Appendix A - User Manual
50
Delete Selected Configuration - Delete the selected configuration.
Implement Selected Configuration - Set all switches to the states specified in the selected
configuration.
A.5 Outputs - Loss Minimization
The loss minimization algorithm produces four kinds of output:
Switch Configuration - Operable switches will be opened or closed to isolate failed components
and minimize losses. The switch configuration may be viewed on the schematic, and saved using
the switching configurations dialog accessed through the control center.
System Results - A list of switches operated by the algorithm is displayed in a listbox, segregated
by those operated to isolate failed components and those operated to minimize losses. Figure A.6
shows the dialog used to display the switching operations. The Pan button centers and highlights
the selected switch on the schematic.
Figure A.6 - Switching operation results dialog for loss minimization
Component Results - Individual component results include those available from the final power
flow solution. There may also be messages on certain components reaching constraints.
Text Files - The formatted text output includes system losses and unserved loads for the initial
system, DAOP solution, and network system. Figure A.7 shows a sample text report. In each of
the three reported systems, any failed components have been isolated already by opening switches.
The network system’s numbers will be zero, unless the user requested a network power flow in
the setup panel.
The formatted text output also lists the switches to open or close, separately for isolating failed
components and for minimizing the subsequent losses. If the user specified failed componentes,
or manually locked any switches from the control center, those components are also listed.
Thomas E. McDermott
Appendix A - User Manual
51
The algorithm also writes a text file called lm_times.txt in DEWorkstation’s directory. This file is
written each time DAOP runs. It shows the time spent in full power flow calculations,
approximate loss evaluations, network power flow, and open-loop backtracking.
Figure A.7 - Sample report from loss minimization
Appendix B
Derivation of Approximate Loss Equation
This appendix presents a derivation of the approximate loss equation (3.2). The system losses
occur in overhead lines, cables, transformers, and other series-connected devices. Figure B.1
shows a three-phase line or transformer, with its series impedance separated into real and
imaginary parts. The phase currents, denoted I1, I2, and I3, create a series voltage drop in the
device. To calculate the real power loss, we only need the resistive part of the series voltage drop
in each phase, which is denoted V1, V2, and V3 in Figure B.1. The phase conductors are coupled,
so that both R and X are three-by-three matrices. The currents and voltages are both phasor
quantities, while R is real.
I1
I2
I3
V1
V2
[R]
j[X]
V3
Figure B.1 - A three-phase lossy device.
The real power loss is given by the complex power equation, (B.1). Even though this is a phasor
expression, note that the result should be a real number.
S = I 1 V &1 + I2 V &2 + I3 V&3
(B.1)
We can replace the voltages in (B.1) with an expansion of the resistance matrix multiplication:
S = I 1 (R 11 I&1 + R 12 I&2 + R 13 I&3 ) + I2 (R 21 I &1 + R 22 I &2 + R 23 I &3 ) + I 3 (R 31 I&1 + R 32 I&2 + R 33 I&3 )
(B.2)
The R matrix is symmetric, so that R12 = R21, R13 = R32, and R23 = R32. Rearranging the terms in
(B.2), we obtain:
S = R 11 I 1 I &1 + R 22 I 2 I &2 + R 33 I 3 I &3 + R 12 (I 1 I &2 + I2 I&1 ) + R 13 (I1 I &3 + I 3 I &1 ) + R 23 (I2 I&3 + I 3 I &2 )
(B.3)
The product of a phasor current and its conjugate is the magnitude squared, denoted I2, so:
S = R 11 I 21 + R 22 I 22 + R 33 I 23 + R 12 (I 1 I &2 + I2 I&1 ) + R 13 (I 1 I &3 + I3 I&1 ) + R 23 (I2 I &3 + I 3 I &2 )
52
(B.4)
Thomas E. McDermott
Appendix B - Derivation of Approximate Loss Equation
53
The first three terms in (B.4) are real, and account for the resistive loss in each conductor with no
coupling. Each of the three cross terms will also reduce to a real number, as shown for two
arbitrary phasors, A and B, in (B.5).
AB & + BA & = (A re + jA im )(B re − jB im ) + (B re + jB im )(A re − jA im )
= A re B re + A im B im + jA im B re − jA re B im + A re B re + A im B im + jA re B im − jA imB re
(B.5)
= 2(A re B re + A im B im)
By defining complex decompositions of the phase currents in (B.6)-(B.8), we can write the real
power loss in (B.9).
I 1 h A re + jA im
(B.6)
I 2 h B re + jB im
(B.7)
I 3 h C re + jC im
(B.8)
P = I 21 R 11 + I22 R 22 + I 23 R 33 + 2R 12 (A re B re + Aim Bim ) + 2R 13 (A re C re + Aim Cim) + 2R 23 (Bre C re + B imC im)
(B.9)
Equation (B.9) is quite similar to (3.2), except for the prime superscripts in (3.2) indicating a
modified current phase reference. To use (B.9), we can estimate the current in each phase
independently, in (B.10)-(B.15).
A re = kWflow1 /kV 1
(B.10)
A im = kVARflow1 /kV 1
(B.11)
B re = kWflow2 /kV 2
(B.12)
B im = kVARflow2 /kV 2
(B.13)
C re = kWflow3 /kV 3
(B.14)
C im = kVARflow3 /kV 3
(B.15)
The real and reactive power flows in (B.10)-(B.15) are known from the system load models,
which generally specify the load in real and reactive power. The total downstream real and
reactive load served by each line segment or transformer is summed to obtain the flows, and then
any downstream capacitor banks are subtracted from the reactive power flow. The
phase-to-ground voltage magnitudes in (B.10)-(B.15) are obtained from the most recent load
flow solution. Note these voltages are not the series voltage drops V1, V2, and V3 shown in
Figure B.1.
Thomas E. McDermott
Appendix B - Derivation of Approximate Loss Equation
54
The three phase currents estimated in (B.10)-(B.15) have phase angles referenced to their
individual phase-to-ground voltages. To use these estimated currents in (B.9), they must be
converted to the same phase reference. The three phase-to-ground voltages will have phase
differences of approximately 120 degrees. Unbalanced source voltages, loads, or line impedances
may cause variations of a few degrees in the voltage phase angles, but usually the difference is less
than one degree. (The currents will often be much more unbalanced than the voltages.) The
phase 1 voltage is arbitrarily chosen to be the phase reference at zero degrees, in (B.16). We then
assume phase 2 lags phase 1 by 120 degrees (B.17) and that phase 3 leads phase 1 by 120 degrees
(B.18).
Â
I 1 = (A re + jA im )
(B.16)
Â
3
2
) = − 12 B re +
3
2
B im − j 12 B im − j
3
2
B re
(B.17)
Â
3
2
) = − 12 C re −
3
2
C im − j 12 C im + j
3
2
C re
(B.18)
I 2 = (B re + jB im )(− 12 − j
I 3 = (C re + jC im )(− 12 + j
A new set of estimated currents, rotated to the same phase reference, is given in (B.19)-(B.24).
Â
A re = A re
Â
A im = A im
Â
B re = −0.5B re + 0.866B im
Â
B re = −0.5B im − 0.866B re
Â
C re = −0.5C re − 0.866C im
Â
C re = −0.5C im + 0.866C re
(B.19)
(B.20)
(B.21)
(B.22)
(B.23)
(B.24)
Replacing the currents in (B.9) with (B.19)-(B.24), we obtain (3.2).
The approximate loss formula is applied to components energized by a candidate switch, and also
to components in the feeder path from the candidate switch back to the source. These feeder path
components have a current increase due to the additional load. Another approximate loss formula
was developed for the loss increment in the feeder path components, but it proved more efficient
to apply (3.2) to the incremented feeder path currents, and then subtract the pre-switching losses
as determined from the existing load flow solution.
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Thomas E. McDermott
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Thomas E. McDermott
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Vita
Thomas E. McDermott was born in St. Louis, Missouri in 1958. After receiving the Rensselaer
Medal in high school, he attended Rensselaer Polytechnic Institute in Troy, New York. In 1980,
he received the Bachelor of Science Degree in Electric Power Engineering from Rensselaer. He
continued in Rensselaer’s graduate school as an IEEE Fortescue Fellow, receiving the Master of
Engineering Degree in Electric Power Engineering in 1981. While at RPI, he completed three
cooperative education assignments at American Electric Power in New York City and in St.
Joseph, Michigan. His first professional experience was at Westinghouse Electric Corporation’s
Advanced Systems Technology Division in Pittsburgh, Pennsylvania. At Westinghouse, he
conducted staged switching surge tests and transient simulation studies using the Electromagnetic
Transients Program. After Westinghouse sold the division to ABB, he worked at Power
Technologies, Inc., also in Pittsburgh. There, he developed lightning protection design software
for the Electric Power Research Institute, and performed a number of other consulting projects in
electric power systems. He has also worked at Ansoft Corporation in Pittsburgh, developing
commercial software for time-domain simulations of electromechanical devices based on finite
element models. Since April 1997, he has worked for Electrotek Concepts of Knoxville,
Tennesee, based at a home office in Pittsburgh. His projects at Electrotek focus on distribution
planning and electrical power quality. In January 1994, he was admitted to the Ph.D. program at
Virginia Tech, where he worked under the supervision of Dr. Robert Broadwater in electric
power distribution system planning, design, and simulation. Mr. McDermott’s areas of technical
interest include distribution system planning, time-domain simulation of electrical circuits, and
custom software development.
Publications
Cole, W. H. and T. E. McDermott, "A Computer Study and Model Demonstration of Safety
Ground Locations During High Voltage Line Maintenance," IEEE Transactions on Power
Apparatus & Systems, vol. PAS-103, no. 3, pp. 455-462, March 1984.
McDermott, T. E. and S. F. Mauser, "Digital Computer Simulation of Electromagnetic Transients
in the Power Industry," Society for Computer Simulation Conference, Norfolk, VA, April
1984.
Frick, T. H., J. R. Stewart, A. R. Hileman, C. R. Chowaniec and T. E. McDermott,
"Transmission Line Insulation Design at High Altitude," IEEE Transactions on Power
Apparatus & Systems, vol. PAS-103, no. 12, pp. 3672-3680, December 1984.
Abi-Samra, N. C., R. F. Smith, T. E. McDermott and M. B. Chidester, "Analysis of
Thyristor-Controlled Shunt SSR Countermeasures," IEEE Transactions on Power Apparatus
& Systems, vol. PAS-104, no. 3, pp. 584-597, March 1985.
Bhasavanich, D., T. E. McDermott, D. M. Milone, J. S. Barnick, L. S. Frost and C. F. Kropp,
"Digital Simulations and Field Measurements of Transients Associated with Large Capacitor
Bank Switching on Distribution Systems," IEEE Transactions on Power Apparatus &
Systems, vol. PAS-104, no. 8, pp. 2274-2282, August 1985.
65
Thomas E. McDermott
Vita
66
Broadwater, R. P., J. C. Thompson and T. E. McDermott, "Pointers and Linked Lists in Electric
Power Distribution Circuit Analysis," IEEE PICA Conference Proceedings, Baltimore, pp.
16-21, May 1991.
Elkins, D. L., A. C. Jain, T. E. McDermott and V. Rees, "Transients During 138-kV SF6
Breaker Switching of Low Inductive Currents," IEEE Transactions on Industry Applications,
vol. IAS-29, no. 4, pp. 721-726, July/August 1993.
McDermott, T. E., T. A. Short and J. G. Anderson, "Lightning Protection of Distribution Lines,"
IEEE Transactions on Power Delivery, vol. PWRD-9, no. 1, pp. 138-152, January 1994.
McDermott, T., P. Zhou, J. Gilmore and Z. Cendes, "Simulation Models Magnets that Move,"
Machine Design, pp. 79-85, December 14, 1995.
McDermott, T., S. Hess and Z. Cendes, "A New Approach to Using Finite Element Solutions in
System Simulation," International Symposium on Automotive Technology and Automation,
paper 96VR006, Florence, Italy, June 1996.
McDermott, T., P. Zhou, J. Gilmore and Z. Cendes, "Electromechanical System Simulation with
Models Generated from Finite Element Solutions," IEEE Transactions on Magnetics, vol.
MAG-33, no. 2, pp. 1682-1685, March 1997.
Zhou, P., T. McDermott, Z. Cendes and M. Rahman, "Steady State Analysis of Synchronous
Generators by a Coupled Field-Circuit Method," paper TA1-3, IEEE International Electric
Machines and Drives Conference Proceedings, May 18-21, 1997, Milwaukee.
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