Ant algorithms for the optimal restoration of distribution feeders during cold load pickup

Abstract

The ant colony algorithm is a new technique for combinatorial optimization borrowed from swarm intelligence. This paper outlines an ant colony algorithm to compute the optimal order of restoring sections in a power distribution system. Restoration of distribution feeders after long interruptions creates cold load pickup conditions due to loss of diversity among the loads. The distribution system load may have to be restored step-by-step using sectionalizing switches under such conditions to prevent overheating of substation transformer. The restoration time is dependent on the order in which sections are restored. Results obtained using this method for two test cases are presented including a comparison with the simulated annealing algorithm.

Full text

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Ant Algorithms for the Optimal Restoration of Distribution
Feeders During Cold Load Pickup
Indira MohantyJugal KalitaSanjoy Das Anil PahwaErik Buehler
Department of Computer Science
University of Colorado
Colorado Springs, CO 80933
Department of Electrical & Computer Engineering
Kansas State University
Manhattan, KS 66506
Abstract: The ant colony algorithm is a new technique for
combinatorial optimization borrowed from swarm
intelligence. This paper outlines an ant colony algorithm to
compute the optimal order of restoring sections in a power
distribution system. Restoration of distribution feeders after
long interruptions creates cold load pickup conditions due to
loss of diversity among the loads. The distribution system
load may have to be restored step-by-step using
sectionalizing switches under such conditions to prevent
overheating of substation transformer. The restoration time
is dependent on the order in which sections are restored.
Results obtained using this method for two test cases are
presented including a comparison with the simulated
annealing algorithm.
Keywords: Distribution system restoration, cold load
pickup, ant colony optimization.
I. INTRODUCTION
COLD LOAD PICKUP
Generally speaking, loads can be classified into two
categories, cyclic and non-cyclic. Cyclic loads are those
that, during normal operation, turn on and off
intermittently. Typically, these cyclic loads are
thermostatically controlled and include air conditioners,
refrigerators, water heaters etc. Non-cyclic loads, such as
lighting are typically turned off and on manually.
In spite of the presence of cyclic loads, the overall power
delivered to a distribution system is approximately
constant at a given time. This is because the distribution
system generally incorporates such a large number of
individual cyclic loads that only a small fraction of them
are turned on at any time. In other words, under normal
operation, a diversity of loads persists. As a result of load
diversity, the overall load on the distribution system is
fairly evenly distributed, and significantly less than the
sum total of its rated load. The installed transformer rating
follows this general principle.
The cold load pickup has several stages (Ucak, Pahwa
1994). Although for a transient period, the inrush current
is upto 15 times the normal load, this condition lasts for
only about 15 seconds. Following this the enduring stage
with load equal to 2 to 5 times normal load and it may last
for several hours depending on rate of load decay. Loss of
diversity due to long interruptions is the cause of the
enduring stage and thus the enduring demand on persists
until the normal diversity among the load is regained.
Under certain conditions all of the system load during
cold load pickup cannot be restored simultaneously as
overloading of transformers due to enduring load demand
may result in excessive transformer heating.
STEP-BY-STEP RESTORATION
Sectionalizing switches can be used on distribution
system feeders to restore the load in steps in case total
load on the system cannot be restored in one step. The
short-term load level and its duration during step-by-step
restoration can be determined based on thermal
characteristics and loading limitations of the transformer.
To model the aggregate load during cold load pickup a
delayed exponential model as shown in figure 1 is used
(Ucak, Pahwa 1994). Where S(t) is load as a function of
time, Tout is the outage duration, SU is the undiversified
load, SD is the diversified load, α is the rate of load
decay, and Δ is the undiversified load duration.
Figure 1: A model of cold load pickup
It should be noted that because of the serial arrangement
of the sections in each feeder, the sectionalizing switch of
a section causes supply to be cut off not only from the
corresponding section, but from all subsequent sections as
well. Therefore, turning on a switch that appears later on
in a feeder has no effect unless all previous switches have

S D
S U
S(t)
t (time)
α α
Δ Δ
T
out

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already been closed. This imposes a certain restriction on
the order in which sections in a feeder can be turned on.
We will refer to this as the precedence constraint. For
instance, in the distribution system shown in figure 2(a),
since sections 1, 2, 3; 4, 5; 6, 7 and 8, 9, 10 are on four
different feeders, in the order mentioned, the precedence
constraints are 1st  2nd  3rd ; 4th  5th ; 6th  7th and
8th  9th  10th. In other words section 3 cannot be
restored unless sections 1 and 2 are restored.
The sequence in which sectionalizing switches are closed
is important since different sequences would result into
different restoration times, as undiversified loads of
sections are not the same. Restoration time is the time at
which the last sectionalizing switch in the sequence is
closed. Some of the restoration sequences are not valid
because of precedence constraints. For example, in the
10-section system, (1 4 6 2 8 7 9 10 3 5) is a valid
sequence whereas (4 3 6 8 1 7 5 9 2 10) is not so, since
section 3 appears before 1 and 2 in the sequence.
Figure 2(a). A distribution system with 10 sections.
Figure 2(b). A distribution system with 12 sections.
The restoration time of different sequences can be
calculated from published results by Ucak, Pahwa, 1994,
which is given by,
−
=
lU
S
)
)(
ln(
)(
][][
][][
][
][
∑
∈
l
∑
∈
l
Δ+
−
−
R
T
lD
R
e
lDiUMT
i
l
S
SSS
T
α
(1)
where,
loading. The section numbers are shown within square
brackets. The set R contains all the previously restored
sections. The undiversified loads and diversified loads for
two different distribution systems of Figure 2 are given in
Tables 1 and 2. We need to optimize the restoration
sequence of sections so that restoration time is minimized.
Some of the combinations would give the same
restoration time since more than one section can be
restored in the first step.
MT
S
is the maximum allowable transformer
2
THE ANT COLONY APPROACH
It is no surprise that ant colony algorithms based on
cooperative artificial ant-like agents (Dorigo, Maniezzo,
Colorni, 1996; Dorigo, Gambardella, 1997) were
originally proposed as a method for the traveling
salesperson problem (Garey, Johnson, 1977) where the
problem is also one of determining a minimum distance
path. But subsequently it has been shown to work well for
other difficult permutation problems too, such as
quadratic assignment and job shop scheduling (Dorigo, Di
Caro 1999; Dorigo, Di Caro, Garmbardella, 1999).
DESCRIPTION OF THE ALGORITHM
In each iteration of the algorithm, an ant is allowed to
execute a move, leaving behind a pheromone trail for
others to follow. Although several attempts at enabling
multiple ants trace out paths simultaneously have been
successfully implemented’ such as in a variation called
the ant colony systems approach (Dorigo, Gambardella,
1997), where multiple ants have been used to
simultaneously traverse different routes in the traveling
salesperson problem (see later), such a scheme can easily
be construed as only as an improvement of the basic
algorithm, where pheromone updating is carried out after
multiple ant runs rather than after each. Therefore, ACO
is not a population-based method. The extended ACO
approach proposed by Das et al. (Das, Mohanty,Yang,
2002) also considers multiple interacting ants to solve
Steiner trees. Yet that approach requires very close
coordination between these ants. Each ant is used
generate different branches of the Steiner tree, and
therefore the function of these interacting are non-
overlapping. Ultimately these ants merge together to form
a single ant in this scheme. The extended ACO too is
therefore not a population-based technique. In other
words, ACO is very unlike other algorithms based on
biological phenomena, such as genetic algorithms, which

Circuit breaker
Sectionalizing Switch
1 2 3
4 5
6 7
8 9 10
Transformer

12
Circuit breaker Sectionalizing Switch
1 2 3
4
7
5
8
10 11
Transformer

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necessarily have to maintain a significant population of
individuals within each iteration.
In each iteration of the algorithm, an ant produces a
candidate solution for the optimization problem. The
situation is analogous to nature, where each ant defines a
path from the ant nest to a food source. Each ant defines a
candidate solution piece-by-piece, probabilistically
selecting a component at a time, until an entire solution is
obtained. We shall hereafter refer to the process by which
an ant assembles a complete solution in a component-wise
manner, as the ant’s traversal along a solution. Each
solution, or a portion of a full solution, shall be referred to
variously, as a path taken by the ant. Each separate
component, or path, can be assigned a cost metric, and it
is the sum of all the individual components that define the
function to be minimized by the ACO.
Because each ant’s traversal yields a single solution of the
optimization problem, the ACO approach can be regarded
as a search through the solution space of the problem
instance. As we will see, the search converges gradually
onto regions where potentially good solutions can be
found.
While traversing a solution, an ant picks components in a
stochastic manner. However, to limit the ACO from
wasteful iterations that explore unfeasible regions of the
search space, the ant’s traversal is guided by the
pheromone trail (sometimes, referred to simply as trail).
The stronger the pheromone concentration along any path,
the more likely an ant is, to include that path in defining a
solution. Therefore, ACO is an exploitative algorithm. It
seeks solutions from information gathered previously, and
performs its search in the vicinity of good solutions.
Needless to say, an algorithm that would solely be guided
by pheromone concentration, with no probability of using
paths with low or zero accumulations would be a greedy
strategy. Since solution components are not picked in a
greedy manner, ACO is also an exploratory algorithm that
samples a wide range of solutions in the solution space.
This exploratory-exploitative approach is characteristic of
many heuristic based optimization approaches, including
evolutionary algorithms, tabu search and particle swarms.
In particular, it is also an attribute of the simulated
annealing algorithm that serves as the basis of comparing
the effectiveness of the proposed algorithm.
In contrast to the basic genetic algorithm or simulated
annealing which make no use of common sense
heuristics, ant colony algorithms incorporate this feature
well within their basic structure. Whenever an ant makes
a move, to add a new path to its trajectory, the
probabilistic selection of the new path is governed not
only by the pheromone level, but also by means of some
heuristic that provides a rough measure of the quality of
the paths to add. During the initial stages of the algorithm,
this factor, called the desirability, plays a very important
role. It restricts the exploration to regions in the search
space that are most likely to contain better solutions.
During the course of defining a solution, each ant
maintains a data structure, called the tabu list. This list is
merely the set of all paths that have already been
traversed by it. The tabu list is necessary to prevent the
ant from including the same path again, producing
redundancies, and often, invalid solutions. In most
implementations, the solution produced in each iteration
can be extracted directly from this set.
The process of pheromone deposition is critical to the
success of the algorithm. As opposed to real ant systems
where pheromones are deposited on-line, e.g.
continuously while the ant is moving, within ACO, the
trail intensities is updated at the end, only when a
complete solution is defined. Therefore, it is updated only
at the end of each iteration. Hence, trail intensity
increment is an off-line process in ACO. Furthermore, the
trail increment is inversely related to the quality of the
solution generated. The reason for this deviation is simply
because component paths that constitute a good solution
need to be reinforced more than others, and the solution
quality can only be determined at the end of the iteration
only when it is fully defined.
Closely linked to, and playing an inverse role to
pheromone deposition is pheromone evaporation.
Pheromones are subject to dissipation even in real ants. In
ACO, pheromones are evaporated from the ant trails to
prevent them from increasing monotonically to
undesirable levels. Evaporation is controlled by a
parameter, the evaporation rate that controls how quickly
trails are updated. A very slow evaporation rate would
make the algorithm converge towards good
neighborhoods in the solution space more slowly. But on
the other hand, a very high rate of evaporation would
result in an undue amount of trail being deposited along
the path of any ant, thereby causing subsequent ants to
follow closely, if not exactly, the same path. This prevents
the algorithm from being exploratory enough, and results
in premature convergence. The right rate of evaporation is
crucial to the performance of the algorithm, and all paths
are subject to the same rate of evaporation regardless of
the quality. In order to associate the trail deposition with
the evaporation rate, the deposition is controlled by the
evaporation rate also, with a higher evaporation rate
amounting to more pheromone deposition. The next
section introduces two algorithms for computing optimal
restoration sequences of any distribution system. These
are, the Absolute Switch Position Ant Colony Algorithm
(ASP-ACO) and the Relative Switch Position Ant Colony
Algorithm (RSP-ACO).
THE ASP-ACO ALGORITHM
Restoring switches with larger undiversified loads let
these loads to assume stable diversified levels at an earlier
stage of the restoration. This usually results in quicker

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overall restoration. Accordingly, we define the
desirability
l η , of picking a switch l for restoration as,
∑
m
=
mU
lU
S
l
S
][
][
η
. (2)
The summation in the above equation is carried over all
sections. The algorithm requires several trials to compute
an optimal sequence. In each trial, an ant produces a valid
sequence of sections for restoration in an iterative
manner, adding a new section per iteration. Starting from
an empty sequence each ant adds new sections to it, once
per iteration, while maintaining the precedence
constraints. The probability that a section k will be
selected, as the pth section to restore is given by,
∑
l
=
lpl
k
η
(
pk
τ
(
pk
p
βα
βα
η
(
τ
(
))
))
. (3)
The summation in the denominator is carried out over all
sections l that are ready for restoration in iteration p. The
quantities α and β are parameters of the algorithm. The
quantity
pk
τ
, the pheromone concentration, simply
reflects the effectiveness of restoring section k at the pth
position. The pheromone level of each pair of sections is
updated at the end of each trial when the ant has
generated a valid sequence. If a section k appears in
position p in the generated sequence, the pheromone
concentration is updated as,
kpkpkp
τρτ
)
ρτΔ+−=
1 (
. (4)
Here the quantity
kp
τΔ
is the pheromone intensity
increment, which is given by,
γ
total
T
τ
kp
Q
=Δ
. (5)
The quantity
by summing the individual restoration times of all
sections in proper order. The index γ and the coefficient
Q are algorithm constants. The quantity ρ is the usual
evaporation rate. For all entries in the pheromone
concentration matrix that were not updated by equations 4
and 5, the evaporation is done in accordance with,
total
T
is the total restoration time computed
kpkp
τ
)
ρτ
1 ( −=
. (6)
THE ASP-ACO ALGORITHM
The algorithm starts by initializing its basic parameters;
initial pheromone concentration τo, pheromone
evaporation rate ρ, pheromone increment Q and the
constant exponents α, β. The desirability η is computed in
the usual manner shown in equation 2. In this variation of
the algorithm instead of starting with an empty sequence
of sections, an ant selects the first section randomly and
computes its restoration time using equation 1. Then new
sections are added to the sequence iteratively based on
their probability given by equation 3 and the total
restoration time is calculated. The main difference of
these two proposed algorithms lies in the configuration of
the pheromone concentration matrix. The update rule for
the pheromone concentration in this case is as follows,
ρτ−=
1 (
where, the pheromone increment Δτ mn is given by,
mnmnmn
τρτ
)
Δ+
, (7)
γ
total
τ
mn
T
Q
=Δ
. (8)
Here, each pair of sections in the generated sequence
gets a reinforcement given by equations 7 and 8 whereas
the remaining entries in the matrix subject to uniform
evaporation in accordance with,
τ=
The subscripts in the above equations represent
consecutive sections in the generated sequence.
mnmn
τ
)
ρ
1 ( −
. (9)
SIMULATED ANNEALING
Simulated annealing (S. Kirkpatrick, C. D. Gellatt, M. P.
Vecchi) is a stochastic approach for continuous as well as
discrete optimization. Simulated annealing is a relatively
simple algorithm, and designing such an algorithm for
obtaining the optimal restoration sequences is
straightforward. The only aspect of the algorithm that is
specific to the problem is the method by which a solution
is perturbed. In this case, any given sequence of switches
is perturbed by interchanging the positions of two
switches. The switches are picked randomly, while
ensuring that after the swapping operation, the validity
constraints as described earlier 3 are preserved. As an
instance, consider a sequence (1 2 10 4 5 7 8 6 3 11 9 12)
of the 12 section problem. This sequence conforms to the
aforementioned restrictions. Swapping the positions of
switches 8 and 11 yields the new sequence (1 2 10 4 5 7
11 6 3 8 9 12) where validity continues to be maintained.
But if switch 7 is picked for swapping with switch 11,
instead of switch 8, the sequence (1 2 10 4 5 11 8 6 3 7 9
12) is obtained. This is an invalid sequence as in every
valid sequence, the restoration of switch 8 can only follow
that of switch 7, since both 7 and 8 belong to the same
feeder, and 7 precedes 8. Any move that would create
invalid solutions is not done by the algorithm.
IMPLEMENTATION ISSUES
Our programs were implemented entirely in MATLAB.
We used the standard MATLAB interpreter to run these
programs. All our experiments were carried out in a
Pentium processor running Windows NT.

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3
RESULTS AND DISCUSSION
We fine-tuned all the parameters of the algorithm
empirically, to get sufficiently good results from our
limited runs. The data was obtained from the real-world
distribution system described earlier (Ucak, Pahwa, 1994;
Chavali, Pahwa, Das 2002). The diversified and
undiversified loads for the the 10 section and 12 section
problems are shown in table 1 and 2. The transformer
capacity
MT
S
was 45.0 MVA. The load decay and the
undiversified load duration of each section were equal to
0.5/hr.
Each algorithm was executed 50 times for each of the two
problem instances. The parameters for each algorithm
were adjusted to obtain the best possible performance. For
the ant colony algorithms, the exponents _ and _ were
kept at unity. For simulated annealing (SA), the initial
temperature was 100. It was cooled to 90% of its value
after every 100 iterations.
Figure 3 shows the convergence of the programs. In each
case, a variable containing the restoration time of the best
sequence obtained by the algorithm was maintained
separately. It was updated only if at the end of each
iteration, a sequence associated with a lower restoration
time than the stored value was obtained. The plot shows
this minimum restoration time (y-axis) as a function of
iterations (x-axis), averaged over 50 runs. Although the
ant colony algorithms converged very quickly, each
program was run for a total of 600 iterations to allow the
simulated annealing algorithm enough time to converge.
Figure 4 is a similar plots for the distribution system
containing 12 sections. It shows similar trends as
previously.
Both RSP-ACO and ASP-ACO are ant colony
algorithms. The difference between them is the manner in
which pheromone reinforcement is carried out. Although
both strategies of pheromone reinforcement have similar
code and algorithmic complexities, the choice of
pheromone reinforcement greatly affected the
convergence. RSP-ACO clearly converges faster than
ASP-ACO.
The present study was restricted to two relatively
small optimization problems. This is because in
practically all distribution system installations, the total
number of sections rarely exceeds 10 or 12.
Because of their recent nature, ant colony algorithms
have been applied thus far, only to simple optimization
problems. This research extends the applicability of this
class of stochastic optimization algorithms to practical
engineering problems. It was found that the method used
to reinforce pheromone levels is crucial to the algorithm’s
performance. Our results also suggest that designing an
appropriate algorithm for any particular problem is very
important. This observation should be of great value to
future research, particularly since the theoretical
convergence properties of these algorithms have not been
studied at length.
Table 1: The distribution system with 10 sections.
Section
Number
Diversified
Load
(MVA)
Undiversified
Load
(MVA)
13.09.0
2 3.05.5
32.56.0
41.55.0
53.0 5.0
66.09.5
72.5 6.0
83.56.5
9 4.511.0
10 3.58.0
Total33.071.5
Table 2: The distribution system with 12 sections.
Section
Number
Diversified
Load
(MVA)
Undiversified
Load
(MVA)
12.5 7.5
22.5 4.5
32.05.0
41.53.0
52.56.0
6 2.05.6
72.5 6.5
8 2.04.0
9 3.0 9.0
103.06.0
111.5 4.8
12 2.0 6.0
Total27.0 67.9