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infrastructures
Article
Agent Based Model to Estimate Time to Restoration of
Storm-Induced Power Outages
Tara Walsh 1ID , Thomas Layton 2,3, David Wanik 1and Jonathan Mellor 1,*
1Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT 06269, USA;
tara.walsh@uconn.edu (T.W.); david.w.wanik@gmail.com (D.W.)
2Department of Emergency Response, Eversource Energy, Berlin, CT 06037, USA;
thomas.layton@eversource.com
3Department of Emergency Management, Jacksonville State University, Jacksonville, AL 36265, USA
*Correspondence: jonathan.mellor@uconn.edu
Received: 12 July 2018; Accepted: 27 August 2018; Published: 31 August 2018
Abstract:
Extreme weather can cause severe damage and widespread power outages across utility
service areas. The restoration process can be long and costly and emergency managers may have
limited computational resources to optimize the restoration process. This study takes an agent based
modeling (ABM) approach to optimize the utility storm recovery process in Connecticut. The ABM is
able to replicate past storm recoveries and can test future case scenarios. We found that parameters
such as the number of outages, repair time range and the number of utility crews working can
substantially impact the estimated time to restoration (ETR). Other parameters such as crew starting
locations and travel speeds had comparatively minor impacts on the ETR. The ABM can be used to
train new emergency managers as well as test strategies for storm restoration optimization.
Keywords: agent based model; emergency management; utility restoration
1. Introduction
Electric utility consumers rely on consistent access to electricity for daily activities. Extreme
weather can cause power outages lasting for long durations and cost US consumers $20 to $55 billion
a year [
1
]. In the United States, utilities are required to report events that cause power loss to at
least 50,000 customers to the North American Electric Reliability Corporation. In 2017 there were
147 total outage events and 77 of those were caused by extreme weather. These 77 weather-related
events affected about 19 million utility customers [
2
]. With occurrences of extreme weather increasing,
there is a potential for increases in extended power outages. For example, climate change is likely to
increase the intensity and frequency of hurricanes along the eastern seaboard and the frequency of
extreme rainfall events [
3
]. Climate change is highly likely to increase risks from heat stress, storms
and extreme precipitation, inland and coastal flooding, sea level rise and storm surge [3].
A system restoration solution must be feasible, provide as much service to customers as possible,
be implemented as quickly as possible and not cause further damage to the system [
4
]. Utility
companies tend to have their own approach to prioritizing the restoration of their customers but
currently there are few resources or analytical tools available to aid in the decision-making process.
Utilities rely on past experience from emergency managers in crew allocation decisions. For example,
utilities have limited crews available and therefore there is a limit on the number of outages they
can repair per day. When the number of outages is high enough that restoration will take many
days, utilities may turn to mutual assistance groups to decrease the time to restoration. The mutual
assistance program allows utilities to allocate unused crews to areas that were more severely affected
by a storm. However, some storms are large and widespread and mutual assistance crews must travel
Infrastructures 2018,3, 33; doi:10.3390/infrastructures3030033 www.mdpi.com/journal/infrastructures
Infrastructures 2018,3, 33 2 of 21
large distances to provide the necessary support, costing utilities a significant amount of money and
delays in restoration.
Several models have been developed to study storm restoration. The Institute of Electrical and
Electronics Engineers (IEEE) created a model to test the organizational system of utility crews by
considering the boundaries of service territories and districts and then considering crew assignments
within those districts [
5
]. The IEEE model was mostly used to determine the optimal territory
configuration and the crew assignments within those territories and was not used for recovery methods,
leaving utility companies to continue to base their strategies off past experiences rather than specific
models. Nateghi et al. (2011) developed several regression models to estimate the outage duration
for individual outages. This model includes parameters specific to the power system, along with
weather and geological parameters and were applied to outages in Hurricane Ivan. They determined
which variables contributed negatively or positively to the outage duration time and noted that the
number of available crews is a very important factor in determining the length of the outage but was
not incorporated in the model [
6
]. Another model developed by Wanik et al. (2018) incorporated
the number of crews working and customer variables (the peak customers affected). Data was used
from Storm Irene, a 2011 October Nor’easter and Hurricane Sandy to develop an outage repair rate
based off the known number of outages fixed and the number of crews working to develop an ETR
model [
7
]. Liu et al. [
8
] proposed an expert system approach. This approach was justified because
they argued that the restoration process involves logical reasoning. The expert system approach
determines an optimal order of outage repairs for general system restoration or to minimize power
losses. This approach is based on the utility system itself and not the social system of the crews. System
restoration is difficult to solve using mathematical programming because of its combinatorial nature.
Ingram [
9
] modified an existing optimization model used by Atlantic Electric to determine the best
location to stage crews. Ingram states that the model can also be used to justify restoration decisions
to state regulators. The use of a model can be a consistent tool in cases where past experience of
decision-making personnel is limited due to infrequent events.
An alternate approach can be to describe electric utility grids as complex systems. Electric
distribution systems have a large number of elements, which makes modeling these systems very
intricate [
4
]. More specifically, power outage repairs have many factors that need to be considered
when estimating system restoration. These include the number of outages, the location of outages,
storm length and repair times, which can all determine whether it is beneficial for a utility company to
call in mutual assistance. There is quite a bit of complexity when assessing storm repair times.
Other factors that need to be considered includes how the crews are dispatched, which was not
included in prior research articles. Crews can be dispatched from centralized Area Work Centers or
dispersed randomly throughout the state. There are a number of basic questions that should be asked
to optimize storm recovery; such as (a) will repairing outages from most to least customers affected
without regards to travel distance be more beneficial?; (b) would it be better for a crew to go to the
nearest outage regardless of how many customers are affected?; (c) should a crew seek outages with
the most customers affected within a given radius of the nearest outage?
Agent based modeling (ABM) is a modeling technique comprised of a set of agents that are
given defined rules and allowed to operate in a given environment [
10
]. They have been used to
study evacuation routes after tsunamis [
11
], model crowdsourcing systems [
12
], risk-based flood
incident management [
13
], coupled human and natural systems [
14
] and to develop an electric power
and communication synchronizing simulator [
15
]. ABMs can be used to model complex systems,
such as human-environment interactions. The model is allowed to run on its own and is studied for
emergent behavior that may not be expected prior to utilizing the model. ABMs provide a platform
to implement an environment with its features, to forecast and explore future scenarios, experiment
with possible alternative decisions, set different values for decision variables and analyze the effects
of these changes [
16
]. Agents change the environment around them by following the simple rules
they are assigned. Agents must interact with their environment, be independent, have social ability,
Infrastructures 2018,3, 33 3 of 21
be reactive and be proactive [
17
]. The goal of the project is to develop a working ABM to simulate
power outage restoration that could be used to determine the optimal repair strategy. Unlike previous
work, the ABM could be used to better estimate a time to complete restoration. The model could be
used as a decision-making mechanism or as a training tool for new emergency managers. The ABM
incorporates real decisions for users to make, as well as accurately simulating the crew’s response to
those decisions and is validated with five historic storms.
2. Methods
2.1. Model Setup
In this paper, the ABM contains five different agent classes: utility crews, roads, power outages,
area work centers and utility lines. The characteristics for each of these classes were drawn from
existing datasets. The road dataset for Connecticut was obtained from the University of Connecticut
Map and Geographic Information Center [
18
]. The points from the data file were uploaded into
NetLogo software [
10
] and connected via links to make connected roadways for the crews to follow.
The utility line dataset was obtained from Eversource and imported into the model similarly to the
road system using links. The area work centers (AWC) are centralized locations around the state of
Connecticut from where distribution equipment is stockpiled and crews are dispatched. These three
agent sets are consistent in all model runs. The power outages were integrated into the model in one
of two ways. For past storms, the power outage locations are known and are loaded into the model.
If the user is interested in a what-if scenario, the power outages can be randomized and the model
places them anywhere along the road system within the state of Connecticut.
To optimize model performance, the outages were geolocated to the nearest roadway. This allows
the utility crew agents to move along the road network to the outage. The utility crews were treated as
independent agents and have rules assigned to them. Each crew operated independently but they may
survey nearby crews in order to make decisions about where to go next. It is important to note that
the model does not take power system dynamics and switching into account as outages are treated as
individual events that can be repaired by a single crew.
When the model begins, the roads and power lines are loaded first, followed by the outages and
then the AWCs. All of these except the outages were the same for every model run. The number of
outages and locations can vary and were determined by the user prior to model setup. In the ABM one
“tick” is equal to the time interval set by the user. The range can be varied from 5 to 15 min, depending
on how granular the output should be. All runs for this study were completed with a 15-min interval.
The travel speed for all roads in Connecticut were set equal as determined by the user and could be
varied for each model run from 25 to 50 mph. For each time step, a crew moves the distance equal to
the travel speed times the time interval, unless the crew was on break or at their assigned outages.
In this application, the ABM uses a distributed approach because the agents are equipped with
self-organizing rules to reach the end goal of system restoration [
14
]. The agents in the ABM are
independent because they act without direct control of a human or other device. They are social
because they communicate the outage they chose and their location with other agents. They are
reactive to their environment because they repair damaged outages and ignore repaired outages.
Lastly, the agents are proactive because the overall goal is to repair the outages according to the
assigned rules.
The user has multiple options for the rules assigned to the crews. First, the crews can start at
AWCs or they can be randomly placed across the State of Connecticut. The number of available crews
can be set by the user, as well as any mutual assistance crews and the time until their arrival from
out of state. During storms with restoration times over 24 h, crews will be required to take breaks.
The ABM utilizes a percentage approach. During an eight hour shift a user defined percentage of crews
will be working. This allows the user to set an overall number of crews but change the percent working
during different eight hour shifts to simulate crews working and on their breaks. This approach allows
Infrastructures 2018,3, 33 4 of 21
the user to differentiate between day, evening and night hours. It also provides a way to allow some
crews to keep working while others have stopped, instead of all crews working and on break during
the same time period. Using past storm data, the total number of crews for a storm was calculated by
adding the number of working crews and crews on break. Then a percentage of the total crews on
break was calculated from this total. One storm may take several days and the average percentage for
each of these time periods was calculated. For simulated storms, the average break period of eight
hours from the validation storms was used to determine the overall percentage of crews working or
on break during each time period.
2.2. Model Run
Once the ABM has gone through the setup process, the model follows an ordered procedure
for each tick. First, all of the crews determine the next outage they will go to if they do not already
have an outage assigned to them. The options are either the (i) nearest outage, (ii) the outage with the
most customers affected, (iii) finding the nearest outage then setting a radius around it and within
that radius choosing the outage with the most customers affected, (iv) outage with the fastest repair
time, (v) finding the nearest outage then setting a radius around it and within that radius choosing
the outage with the fastest repair time, or (vi) outage with the fastest repair time and most customers
affected. The third option (which will be simplified as “nearest within radius”) simulates when crews
can travel a little further in order to have a greater impact on the number of customers still without
power. Once the crew determines its next outage, it changes the outage it found from “not taken” to
“taken.” In the case where there are more crews than outages, crews may call off another crew if the
crew without an assigned outage is closer to the unrepaired outage. After a crew determines the outage
it will travel to, Dijkstra’s routing algorithm is used to determine the shortest distance for the crew to
travel along the road network to their assigned outage. The algorithm factors in the number and length
of links to determine the optimal path. Dijkstra’s algorithm will determine the optimal path by finding
the combination of the least amount of links to travel and the overall shortest path [
19
]. Dijkstra’s
algorithm has been used to model travelers taking public transportation and driving a vehicle [
20
].
However, Dijkstra’s algorithm does not take power flow into consideration to prevent crews from
working too close. Over a series of ticks, the crew will travel at a user defined speed until it reaches
its outage. During the travel time, the outage will remain as “taken” and “unrepaired.” The crew
will stay at the outage and “work” for the user defined repair time assigned to the outage during the
setup. Once the crew finishes the repair, it will update the outage to “repaired” and check if the crew
is next to take a break. If so, the crew’s break time will start and they will remain on break at their
current location for the next eight hours or the equivalent of one shift. Once the break is over, the crew
will select a new outage as long as there are still “unrepaired” outages. If all of the outages are set as
“taken” but do not yet have a crew there working, a crew can call off another crew if they are closer.
This simulates the end of a storm with utility companies trying to finish the remaining outages as
quickly as possible. The model stops once no more crews are working and all of the outages have been
repaired. Figure 1illustrates the decisions made by the model.
As shown in Figure 1, mutual assistance crews may be arriving throughout the storm. At each
tick, the model does a check to see if any mutual assistance crews will be arriving. If so, the model will
sprout new crews. Just like the initial crews, the mutual assistance crews will either start at AWCs
or randomly across the state depending on the user chosen parameters. Once initiated, the mutual
assistance crews operate identical to the original crews. Mutual assistance crews keep track of their
travel time, their work time and their travel time back to where they started from. The model assumes
mutual assistance crews begin traveling as the model starts running. Therefore, mutual assistance
crews can keep track of how long they traveled, the amount of time they worked and include their
travel time back home. If the crew will be assisting a different utility after completing their work in
Connecticut, the travel time back to their home state will not be included. The total time the mutual
assistance crew was traveling and working for the utility is used to calculate the cost of their aid.
Infrastructures 2018,3, 33 5 of 21
The cost of each crew is added together and the total cost of mutual assistance is displayed to the user
at the completion of the model run. The hourly rate of the mutual assistance crews can be varied by
the user.
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Figure 1. Flowchart of model decision making.
Figure 1. Flowchart of model decision making.
Infrastructures 2018,3, 33 6 of 21
2.3. Model Validation
Model validation was completed utilizing past storm data and tested using different combinations
of parameters. These data included outage locations, number of customers affected and number of
crews working. Since the nature of storm damage is different for each storm, outage repair times were
varied uniformly between lower and upper limits to optimize the fit compared to past storm restoration
curves. Moreover, the crew starting locations (area work center or random) and search strategy (nearest,
most customers affected, most customers affected within a radius of the nearest outage, fastest repair
time, fastest repair time within a radius and fastest repair time with most customers affected) were
also varied to optimize the validation. Model fits were assessed using R
2
, mean absolute error (MAE)
and standard deviation.
3. Results and Discussion
The first step in model validation was to compare modeled versus the actual restoration curves
obtained from the utility company. With multiple input parameters for the model, the first task was
running the model for all combinations of search strategies. The number of crews working for each
storm was known from information obtained from the utility company, along with outage locations
and number of customers affected at each outage, as shown in Table 1. The data obtained for the
number of crews varied over time. Crews are moved to different areas throughout a storm, which
results in fluctuations of the total number of crews on duty. In the model, the percent of crews working
during a given day, evening or night shift corresponds to data from the actual storm. A summary of
the total number of crews and the percentage working during day, evening and night hours is shown
in Table 1. Travel speeds were set to 25 miles per hour and the repair time range was set as indicated
in Table 1with a uniform distribution. Storm repair curves from five different storms are shown
in Figure 2. Outage repair time ranges were optimized for each storm and are shown in
Tables 1and 2
.
While all repair curves showed similar large-scale behavior, there were significant differences between
combinations of search strategies and starting location. However, neither the starting location nor the
search strategy showed consistent trends (Figure 2).
Table 1.
Storms used for model validation. The repair time range column is validated from the model
and can be seen in Figure 4. The system recovery column is from historic storm data.
Time to System Recovery (h) 17 51 50 22 20 –
Repair Time Range (h) 1–7 1–13 1–9 1–13 1–11 1–10
% Crews Working Night 100 72 31 29 100 66
% Crews Working Evening 100 93 100 100 100 99
% Crews Working Day 100 64 100 88 73 85
Crews 201 365 212 392 190 272
Peak Customers 20,377 54,431 11,207 88,341 15,840 38,186
Outages 657 1399 495 2056 661 1220
Weather
Snow, Wind Snow, Wind
Wind Wind Wind –
Month April February February January February –
Storm 1 2 3 4 5 Average
Table 2includes the R
2
value, mean absolute error (MAE) and standard deviation for the top three
strategies of each storm shown in Figure 2. Table S1 includes all strategies of each storm. Storm 1 was
best fit with crews starting at area work centers and searching for the outage with the fastest repair
time and the most customers affected. Storms 2 and 3 were best fit with crews starting at area work
centers and searching for the outage with the fastest repair time. Both Storms 4 and 5 were best fit with
crews searching for the fastest repair time within a radius of the nearest outage but Storm 4 favored
crews starting at area work centers while Storm 5 favored random crew starting locations. Optimized
fits had R2values ranging from 0.91 to 0.99, indicating adequate fits for all scenarios.
Infrastructures 2018,3, 33 7 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 7 of 22
in the beginning meaning that they overestimated customers restored early in the storm. The early
storm underestimates seen in Storms 1, 3 and 5 could be due to the fact that the model does not
include priority locations such as hospitals. Utilities are aware of outages that impact the most
customers and will restore these points first. Moreover, at the end of storms, there is typically a long
tail representing outages that are difficult and time-consuming to repair and single service outages.
The residuals in Figure 3 have been normalized to the maximum number of customers affected per
storm. The residuals were biased because they were not randomly positive and negative. Storms 1, 3
and 5 had the lowest residuals. Storms 2 and 4 have larger residuals and as seen in Figure 3. In all
cases, the residual values decrease towards zero at the end of storm recoveries.
Figure 2. Variation of model parameters for validation storms. Crew start location and search strategy
are indicated. Travel speed set to 25 mph, the total number of crews and percent working during each
shift shown in Table 1., repair time range for each storm shown in Table 2. Number of outages and
total customers affected for each storm shown in Table 1. MC is most customers affected, NWR is
nearest with radius, FRT is fastest repair time, FRTR is fastest repair time within radius and FRTMC
is fastest repair time with most customers affected.
Figure 2.
Variation of model parameters for validation storms. Crew start location and search strategy
are indicated. Travel speed set to 25 mph, the total number of crews and percent working during each
shift shown in Table 1., repair time range for each storm shown in Table 2. Number of outages and total
customers affected for each storm shown in Table 1. MC is most customers affected, NWR is nearest
with radius, FRT is fastest repair time, FRTR is fastest repair time within radius and FRTMC is fastest
repair time with most customers affected.
Table 2.
R
2
, MAE and standard deviation of the top 3 combinations of modeled restoration curves for
each storm from Figure 2.
Storm Crew Start Search Strategy Repair Time
Range (h) R2MAE (Customers
Per Hour)
Standard
Deviation
Storm 1 AWC Fastest Repair Time and Maximum Customers 1 to 7
0.97
908.9 6347.44
Storm 1 AWC Fastest Repair Time 1 to 7
0.97
962.35 6544.29
Storm 1 Random Fastest Repair Time within Radius 1 to 7
0.96
1150.88 6900.30
Storm 2 AWC Fastest Repair Time 1 to 13
0.97
3704.87 17,397.89
Storm 2 Random Fastest Repair Time and Most Customers 1 to 13
0.93
4264.41 18,501.83
Storm 2 Random Fastest Repair Time within Radius 1 to 13
0.91
5321.12 18,313.75
Storm 3 AWC Fastest Repair Time 1 to 9
0.97
662.23 3941.78
Storm 3 AWC Fastest Repair Time within Radius 1 to 9
0.95
735.78 4040.33
Storm 3 AWC Most Outages 1 to 9
0.95
754.85 4308.54
Storm 4 AWC Fastest Repair Time within Radius 1 to 13
0.99
2233.73 27,746.07
Storm 4 AWC Fastest Repair Time 1 to 13
0.98
2633.03 25,023.54
Storm 4 Random Nearest 1 to 13
0.99
2970.47 27,930.11
Storm 5 Random Fastest Repair Time within Radius 1 to 11
0.98
545.18 5231.42
Storm 5 AWC Fastest Repair Time 1 to 11
0.98
775.01 4440.09
Storm 5 AWC Nearest within Radius 1 to 11
0.95
1068.25 5896.76
An analysis of the residuals (Figure 3) indicates that they tend to be positive in the beginning of
storms but this is not always the case. Storms 2 and 4 were larger in size and had negative residuals in
the beginning meaning that they overestimated customers restored early in the storm. The early storm
underestimates seen in Storms 1, 3 and 5 could be due to the fact that the model does not include
Infrastructures 2018,3, 33 8 of 21
priority locations such as hospitals. Utilities are aware of outages that impact the most customers and
will restore these points first. Moreover, at the end of storms, there is typically a long tail representing
outages that are difficult and time-consuming to repair and single service outages. The residuals in
Figure 3have been normalized to the maximum number of customers affected per storm. The residuals
were biased because they were not randomly positive and negative. Storms 1, 3 and 5 had the lowest
residuals. Storms 2 and 4 have larger residuals and as seen in Figure 3. In all cases, the residual values
decrease towards zero at the end of storm recoveries.
Infrastructures 2018, 3, x FOR PEER REVIEW 8 of 22
Table 2. R2, MAE and standard deviation of the top 3 combinations of modeled restoration curves for
each storm from Figure 2.
Storm
Crew Start
Search Strategy
Repair Time
Range (h)
R2
MAE (Customers
Per Hour)
Standard
Deviation
Storm 1
AWC
Fastest Repair Time and Maximum Customers
1 to 7
0.97
908.9
6347.44
Storm 1
AWC
Fastest Repair Time
1 to 7
0.97
962.35
6544.29
Storm 1
Random
Fastest Repair Time within Radius
1 to 7
0.96
1150.88
6900.30
Storm 2
AWC
Fastest Repair Time
1 to 13
0.97
3704.87
17,397.89
Storm 2
Random
Fastest Repair Time and Most Customers
1 to 13
0.93
4264.41
18,501.83
Storm 2
Random
Fastest Repair Time within Radius
1 to 13
0.91
5321.12
18,313.75
Storm 3
AWC
Fastest Repair Time
1 to 9
0.97
662.23
3941.78
Storm 3
AWC
Fastest Repair Time within Radius
1 to 9
0.95
735.78
4040.33
Storm 3
AWC
Most Outages
1 to 9
0.95
754.85
4308.54
Storm 4
AWC
Fastest Repair Time within Radius
1 to 13
0.99
2233.73
27,746.07
Storm 4
AWC
Fastest Repair Time
1 to 13
0.98
2633.03
25,023.54
Storm 4
Random
Nearest
1 to 13
0.99
2970.47
27,930.11
Storm 5
Random
Fastest Repair Time within Radius
1 to 11
0.98
545.18
5231.42
Storm 5
AWC
Fastest Repair Time
1 to 11
0.98
775.01
4440.09
Storm 5
AWC
Nearest within Radius
1 to 11
0.95
1068.25
5896.76
Figure 3. Plot of the residuals normalized to the maximum number of customers affected for each
storm in Figure 2 for each of the crew start and search strategy combinations. FRT is fastest repair
time, FWR is fastest repair time within radius and FMC is fastest repair time and most customers
affected.
Each storm is different in the damage it produces and therefore the length of time that repairs
take on average. It is difficult to know the average repair time range of a storm before the storm
occurs. As shown in Figure 2 and Table 2, each storm is fit with a different repair time range. These
ranges were determined using the data in Figure 4 and Table 3. Table 3 includes the top three R2
value, mean absolute error (MAE) and standard deviation for each storm shown in Figure 4. R2 values
were all >0.93. Table S2 includes all storms and all strategies. In these simulations, crews started at
random locations and searched for the nearest outage. In all cases, as the repair time range was
increased starting at one hour, the MAE decreases as the R2 value increases until the combination of
highest R2 and lowest MAE is reached. Storm 1 appeared to match well with a lower maximum repair
Figure 3.
Plot of the residuals normalized to the maximum number of customers affected for each
storm in Figure 2for each of the crew start and search strategy combinations. FRT is fastest repair time,
FWR is fastest repair time within radius and FMC is fastest repair time and most customers affected.
Each storm is different in the damage it produces and therefore the length of time that repairs
take on average. It is difficult to know the average repair time range of a storm before the storm occurs.
As shown in Figure 2and Table 2, each storm is fit with a different repair time range. These ranges
were determined using the data in Figure 4and Table 3. Table 3includes the top three R
2
value, mean
absolute error (MAE) and standard deviation for each storm shown in Figure 4. R
2
values were all
>0.93. Table S2 includes all storms and all strategies. In these simulations, crews started at random
locations and searched for the nearest outage. In all cases, as the repair time range was increased
starting at one hour, the MAE decreases as the R
2
value increases until the combination of highest R
2
and lowest MAE is reached. Storm 1 appeared to match well with a lower maximum repair time early
in the restoration but a longer repair time later on. Storms 2 and 4 best fit to a 13 h maximum repair
time and were both larger storms in this data set with 1399 and 2056 outages respectively. Storms 1
and 5 were similar in size (657 and 661 outages) but the best fitting repair time was 7 h for Storm 1
and 11 h for Storm 5. The repair time range that best fits each storm depends on more than storm size
alone. Storms 1, 3 and 5 had similar number of total crews working with 201, 212 and 190, respectively.
Infrastructures 2018,3, 33 9 of 21
Storm 2 had 365 and Storm 4 had 392. The larger storms had more crews working, yet still had the
longer repair time range.
Infrastructures 2018, 3, x FOR PEER REVIEW 9 of 22
time early in the restoration but a longer repair time later on. Storms 2 and 4 best fit to a 13 h
maximum repair time and were both larger storms in this data set with 1399 and 2056 outages
respectively. Storms 1 and 5 were similar in size (657 and 661 outages) but the best fitting repair time
was 7 h for Storm 1 and 11 h for Storm 5. The repair time range that best fits each storm depends on
more than storm size alone. Storms 1, 3 and 5 had similar number of total crews working with 201,
212 and 190, respectively. Storm 2 had 365 and Storm 4 had 392. The larger storms had more crews
working, yet still had the longer repair time range.
Figure 4. Varying maximum repair time for each validation storm. Crews start at random locations
and search for nearest outage. Travel speed set to 25 mph, the total number of crews and percent
working during each shift shown in Table 1. Maximum repair time varied from 1 to 15 h, as indicated.
Number of outages and total customers affected for each storm shown in Table 1. Repair time ranges
varied for each storm. The fit was not constant throughout the storm. Lower repair time ranges fit
better early in restoration and longer repair time ranges fit better later in storm restoration.
Table 3. R-squared, MAE and standard deviation of the top 3 combinations of modeled restoration
curves from Figure 4.
Storm
Min. Repair Time (h)
Max Repair Time (h)
R2
MAE (Customers Per Hour)
Standard Deviation
Storm 1
1
7
0.96
411.92
4579.69
Storm 1
1
5
0.93
416.32
3983.13
Storm 1
1
6
0.95
459.68
4421.56
Storm 2
1
11
0.94
4240.07
17,295.84
Storm 2
1
12
0.94
3837.29
18,433.79
Storm 2
1
13
0.94
3625.22
18,630.13
Storm 3
1
8
0.98
405.6
3728.86
Storm 3
1
9
0.98
388.3
3702.99
Storm 3
1
10
0.94
575.03
3951.83
Storm 4
1
12
0.99
2363.04
27,024.72
Storm 4
1
13
0.99
2225.84
27,353.72
Storm 4
1
14
0.99
2553.41
27,823.18
Storm 5
1
9
0.96
937.36
5699.48
Figure 4.
Varying maximum repair time for each validation storm. Crews start at random locations and
search for nearest outage. Travel speed set to 25 mph, the total number of crews and percent working
during each shift shown in Table 1. Maximum repair time varied from 1 to 15 h, as indicated. Number
of outages and total customers affected for each storm shown in Table 1. Repair time ranges varied for
each storm. The fit was not constant throughout the storm. Lower repair time ranges fit better early in
restoration and longer repair time ranges fit better later in storm restoration.
Table 3.
R-squared, MAE and standard deviation of the top 3 combinations of modeled restoration
curves from Figure 4.
Storm Min. Repair Time (h) Max Repair Time (h) R2MAE (Customers Per Hour) Standard Deviation
Storm 1 1 7
0.96
411.92 4579.69
Storm 1 1 5
0.93
416.32 3983.13
Storm 1 1 6
0.95
459.68 4421.56
Storm 2 1 11
0.94
4240.07 17,295.84
Storm 2 1 12
0.94
3837.29 18,433.79
Storm 2 1 13
0.94
3625.22 18,630.13
Storm 3 1 8
0.98
405.6 3728.86
Storm 3 1 9
0.98
388.3 3702.99
Storm 3 1 10
0.94
575.03 3951.83
Storm 4 1 12
0.99
2363.04 27,024.72
Storm 4 1 13
0.99
2225.84 27,353.72
Storm 4 1 14
0.99
2553.41 27,823.18
Storm 5 1 9
0.96
937.36 5699.48
Storm 5 1 10
0.97
913.37 5601.22
Storm 5 1 11
0.98
745.3 5543.55
Infrastructures 2018,3, 33 10 of 21
3.1. Model Sensitivity
With a reasonably validated model, we next tested the sensitivity of the model to outage locations.
The model was run first using the known outage locations of historic storms. It was run again using
the same number of outages but randomly placed across the State of Connecticut. For consistency,
the average repair time range of 1 to 10 h will be used for all subsequent What-If scenarios. As seen in
Figure 5, the two outage location options produced nearly identical results. This means that knowledge
of actual outage locations is not necessary to reproduce accurate ETR curves. The model showed little
sensitivity to travel speed, so the time lost to travel is negligible. Therefore, the location would have
little impact on the ETR because any change due to travel distance is minor. For all storm simulations,
outages will be randomly located within the state.
Infrastructures 2018, 3, x FOR PEER REVIEW 10 of 22
Storm 5
1
10
0.97
913.37
5601.22
Storm 5
1
11
0.98
745.3
5543.55
3.1. Model Sensitivity
With a reasonably validated model, we next tested the sensitivity of the model to outage
locations. The model was run first using the known outage locations of historic storms. It was run
again using the same number of outages but randomly placed across the State of Connecticut. For
consistency, the average repair time range of 1 to 10 h will be used for all subsequent What-If
scenarios. As seen in Figure 5, the two outage location options produced nearly identical results. This
means that knowledge of actual outage locations is not necessary to reproduce accurate ETR curves.
The model showed little sensitivity to travel speed, so the time lost to travel is negligible. Therefore,
the location would have little impact on the ETR because any change due to travel distance is minor.
For all storm simulations, outages will be randomly located within the state.
Figure 5. Comparison of actual and random outage locations for Storm 5. Crews start at area work
centers and search for nearest outage. Travel speed set to 25 mph, 190 crews with 73% working during
day shift, 100% working during evening shift and 100% working during night shift, 1 to 8 h repair
time range. Storm 5 has 661 outages and 15,840 customers affected. The outages are located in the
actual locations and then in random locations within Connecticut. The ETR curves were insensitive
to outage locations.
Tests were conducted to determine the sensitivity of the model to parameter changes. A small
storm in these tests is characterized as having 1000 outages, large storms have 5000 outages and
extreme storms have 15,000 outages. The number of customers affected per outage was determined
by using an average from the five validation storms.
The model was tested for sensitivity to the outage repair time distribution. Previously shown
results used a uniform outage repair time from 1 h until the chosen maximum repair time. However,
it was unclear if different repair time distributions would lead to different overall restoration times.
Storm characteristics can influence the nature of the damage caused and therefore the repair time
distributions. Tested distributions included uniform, normal, exponential, gamma and Poisson and
Figure 5.
Comparison of actual and random outage locations for Storm 5. Crews start at area work
centers and search for nearest outage. Travel speed set to 25 mph, 190 crews with 73% working during
day shift, 100% working during evening shift and 100% working during night shift, 1 to 8 h repair
time range. Storm 5 has 661 outages and 15,840 customers affected. The outages are located in the
actual locations and then in random locations within Connecticut. The ETR curves were insensitive to
outage locations.
Tests were conducted to determine the sensitivity of the model to parameter changes. A small
storm in these tests is characterized as having 1000 outages, large storms have 5000 outages and
extreme storms have 15,000 outages. The number of customers affected per outage was determined by
using an average from the five validation storms.
The model was tested for sensitivity to the outage repair time distribution. Previously shown
results used a uniform outage repair time from 1 h until the chosen maximum repair time. However,
it was unclear if different repair time distributions would lead to different overall restoration times.
Storm characteristics can influence the nature of the damage caused and therefore the repair time
distributions. Tested distributions included uniform, normal, exponential, gamma and Poisson and
are detailed in Table 4. Figure 6shows that the distribution of outage repair times has minor impacts
on the final restoration time. As seen in the small storm, a gamma and Poisson distribution tends
to produce a step-like restoration curve. Gamma distributions usually fit best with data with large
standard deviation but the relatively low mean of the chosen repair times limits a large range. In the
case a negative repair time was chosen, the model picks a new repair time. When all outages have
Infrastructures 2018,3, 33 11 of 21
the same repair time a step-like curve is produced because the crews arrive to their outage at similar
times and are all working for the same duration. These steps become less defined over the course of
the restoration process.
Infrastructures 2018, 3, x FOR PEER REVIEW 11 of 22
are detailed in Table 4. Figure 6 shows that the distribution of outage repair times has minor impacts
on the final restoration time. As seen in the small storm, a gamma and Poisson distribution tends to
produce a step-like restoration curve. Gamma distributions usually fit best with data with large
standard deviation but the relatively low mean of the chosen repair times limits a large range. In the
case a negative repair time was chosen, the model picks a new repair time. When all outages have
the same repair time a step-like curve is produced because the crews arrive to their outage at similar
times and are all working for the same duration. These steps become less defined over the course of
the restoration process.
Figure 6. Outage repair time distributions for small, large and extreme storms. Crews start at area
work centers and find the nearest outage. Travel speed set to 25 mph, 272 crews with 85% working
during day shift, 99% working during evening shift and 66% working during night shift and nearest
outage search strategy. ETR curve was insensitive outage repair time distribution.
Table 4. Distribution characteristics from Figure 6.
Distribution
Mean (h)
Standard Deviation (h)
Uniform
10
NA
Normal
10
5
Gamma
10
5
Exponential
10
NA
Poisson
10
NA
The next test compared the initial starting location of work crews for each size storm. At the
beginning of the storm, crews could start at either area work centers or random locations. For this
test crews would search for the nearest outage. Travel speed was set to 25 miles per hour, the repair
time range was 1 to 10 h and 272 crews were working (the average number from the validated
storms). Figure 7 shows that there was little difference between the two starting location options. All
future storms will be run with crews starting at area work centers, which is more realistic.
Figure 6.
Outage repair time distributions for small, large and extreme storms. Crews start at area
work centers and find the nearest outage. Travel speed set to 25 mph, 272 crews with 85% working
during day shift, 99% working during evening shift and 66% working during night shift and nearest
outage search strategy. ETR curve was insensitive outage repair time distribution.
Table 4. Distribution characteristics from Figure 6.
Distribution Mean (h) Standard Deviation (h)
Uniform 10 NA
Normal 10 5
Gamma 10 5
Exponential 10 NA
Poisson 10 NA
The next test compared the initial starting location of work crews for each size storm. At the
beginning of the storm, crews could start at either area work centers or random locations. For this
test crews would search for the nearest outage. Travel speed was set to 25 miles per hour, the repair
time range was 1 to 10 h and 272 crews were working (the average number from the validated storms).
Figure 7shows that there was little difference between the two starting location options. All future
storms will be run with crews starting at area work centers, which is more realistic.
As seen towards the end of the storm in Figure 7, some of the model runs can result in a long tail
until complete restoration. This occurs when the last few outages have long repair times and when
there are many single service outages. These long tails also can occur as difficult or hard to reach
repairs are often left until the end. To highlight the major scenario differences, all ETR curves will be
cropped when the number of customers remaining without power was no more than 20 for the small
Infrastructures 2018,3, 33 12 of 21
storm and 100 for the large and extreme storms. Figures S1–S3 for small and extreme storms can be
found in the Supplementary Materials.
Infrastructures 2018, 3, x FOR PEER REVIEW 12 of 22
Figure 7. Start location for simulated large storms with 5000 outages. Crew start location varied as
indicated. Travel speed set to 25 mph, 272 crews with 85% working during day shift, 99% working
during evening shift and 66% working during night shift, 1 to 10 h repair time range and nearest
outage search strategy. ETR curve was insensitive to crew start location.
As seen towards the end of the storm in Figure 7, some of the model runs can result in a long tail
until complete restoration. This occurs when the last few outages have long repair times and when
there are many single service outages. These long tails also can occur as difficult or hard to reach
repairs are often left until the end. To highlight the major scenario differences, all ETR curves will be
cropped when the number of customers remaining without power was no more than 20 for the small
storm and 100 for the large and extreme storms. Figures S1–S3 for small and extreme storms can be
found in the Supplementary Materials.
Next, the crew search strategy was varied between nearest outage, the outage with the most
customers affected within a radius of the nearest outage (nearest within radius), the outage with the
most customers affected, the outage with the fastest repair time, the outage with the fastest repair
time within a radius of the nearest outage and the outage with the fastest repair time and most
customers affected. All of the parameters were kept the same as previously described and crews
started at area work centers. The radius was set to four miles for the nearest within radius and fastest
within radius search options. Figure 8 shows some sensitivity to search strategy, especially in the
large and extreme storms. For the small storm, there was little difference between the nearest outage,
nearest with radius and fastest repair time. The most customers affected option performed better in
the beginning but a longer tail at the end lengthened the final ETR. The large storm shows a bigger
difference between the nearest outage and nearest with radius options. However, the nearest within
radius performed similar to the most outages option in the beginning but ended faster than most
outages. The nearest and nearest within radius search strategies had similar ETRs for the large storm
but nearest within radius always had less customers still without power than nearest. The biggest
differences between search options came in the extreme storm situation. The nearest within radius
option performed best throughout the run. In the beginning of the simulation most outages
performed between nearest within radius option and nearest outage, until about the 400-h point.
After the 400-h mark, the most outages option reduced the number of customers affected most slowly.
The nearest within radius option reduced the number of customers affected the fastest and had an
ETR closest to the nearest search strategy. In both the large and extreme storms, the search strategies
using repair times have similar impacts on the ETR curves. Both fastest repair time and fastest with
most customers have the longest final ETR. The fastest within radius performs the best in the
beginning of the storm but then has a final ETR similar to the nearest and nearest with radius
strategies. The final ETR for each search strategy and storm size can be seen in Table 5.
Figure 7.
Start location for simulated large storms with 5000 outages. Crew start location varied as
indicated. Travel speed set to 25 mph, 272 crews with 85% working during day shift, 99% working
during evening shift and 66% working during night shift, 1 to 10 h repair time range and nearest outage
search strategy. ETR curve was insensitive to crew start location.
Next, the crew search strategy was varied between nearest outage, the outage with the most
customers affected within a radius of the nearest outage (nearest within radius), the outage with the
most customers affected, the outage with the fastest repair time, the outage with the fastest repair time
within a radius of the nearest outage and the outage with the fastest repair time and most customers
affected. All of the parameters were kept the same as previously described and crews started at area
work centers. The radius was set to four miles for the nearest within radius and fastest within radius
search options. Figure 8shows some sensitivity to search strategy, especially in the large and extreme
storms. For the small storm, there was little difference between the nearest outage, nearest with radius
and fastest repair time. The most customers affected option performed better in the beginning but
a longer tail at the end lengthened the final ETR. The large storm shows a bigger difference between
the nearest outage and nearest with radius options. However, the nearest within radius performed
similar to the most outages option in the beginning but ended faster than most outages. The nearest
and nearest within radius search strategies had similar ETRs for the large storm but nearest within
radius always had less customers still without power than nearest. The biggest differences between
search options came in the extreme storm situation. The nearest within radius option performed best
throughout the run. In the beginning of the simulation most outages performed between nearest
within radius option and nearest outage, until about the 400-h point. After the 400-h mark, the most
outages option reduced the number of customers affected most slowly. The nearest within radius
option reduced the number of customers affected the fastest and had an ETR closest to the nearest
search strategy. In both the large and extreme storms, the search strategies using repair times have
similar impacts on the ETR curves. Both fastest repair time and fastest with most customers have the
longest final ETR. The fastest within radius performs the best in the beginning of the storm but then
has a final ETR similar to the nearest and nearest with radius strategies. The final ETR for each search
strategy and storm size can be seen in Table 5.
Infrastructures 2018,3, 33 13 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 13 of 22
Figure 8. Search strategy for simulated storms. Crew start location set to area work center and search
strategy as indicated. Travel speed set to 25 mph, 272 crews with 85% working during day shift, 99%
working during evening shift and 66% working during night shift, 1 to 10 h repair time range. 1000
outages for small storm, 5000 outages for large storm, 15,000 outages for extreme storm. Nearest
within 4-mile radius led to faster restoration times.
Table 5. Time to system restoration (in days) based on search strategy from Figure 8.
Search Strategy
Small Storm
Large Storm
Extreme Storm
Nearest
1.46
6.67
21.46
Nearest with 4-mile radius
1.46
6.33
21.29
Most outages
1.46
8.33
29.54
Fastest repair time
1.55
13.41
39.05
Fastest repair time with 4-mile radius
1.48
8.17
23.36
Fastest repair time and most customers affected
1.86
14.47
30.11
As previously stated, a four-mile radius was used for the nearest within-radius search option.
Next, this radius was varied from one mile to five miles. Figure 9 shows that the impact of the change
in radius depends on the storm size and Table 6 shows the time to restoration in days for each storm
size and radius. The small storm was not sensitive to the radius, which confirms from Figure 8 that
there was little performance difference between the nearest outage option and nearest with radius
option. However, the large and extreme storms were both sensitive to search radius. In both cases, a
larger radius reduced the customers without power the quickest. However, the five-mile radius did
have the longest tail in both the large and extreme storm.
Figure 8.
Search strategy for simulated storms. Crew start location set to area work center and search
strategy as indicated. Travel speed set to 25 mph, 272 crews with 85% working during day shift,
99% working during evening shift and 66% working during night shift, 1 to 10 h repair time range.
1000 outages for small storm, 5000 outages for large storm, 15,000 outages for extreme storm. Nearest
within 4-mile radius led to faster restoration times.
Table 5. Time to system restoration (in days) based on search strategy from Figure 8.
Search Strategy Small Storm Large Storm Extreme Storm
Nearest 1.46 6.67 21.46
Nearest with 4-mile radius 1.46 6.33 21.29
Most outages 1.46 8.33 29.54
Fastest repair time 1.55 13.41 39.05
Fastest repair time with 4-mile radius 1.48 8.17 23.36
Fastest repair time and most customers affected 1.86 14.47 30.11
As previously stated, a four-mile radius was used for the nearest within-radius search option.
Next, this radius was varied from one mile to five miles. Figure 9shows that the impact of the change
in radius depends on the storm size and Table 6shows the time to restoration in days for each storm
size and radius. The small storm was not sensitive to the radius, which confirms from Figure 8that
there was little performance difference between the nearest outage option and nearest with radius
option. However, the large and extreme storms were both sensitive to search radius. In both cases,
a larger radius reduced the customers without power the quickest. However, the five-mile radius did
have the longest tail in both the large and extreme storm.
Infrastructures 2018,3, 33 14 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 14 of 22
Figure 9. Change of radius for nearest-with-radius search strategy. Crew start location set to area
work center and search strategy set to most customers affected within a radius of the nearest outage.
Search radius varied as indicated. Travel speed set to 25 mph, 272 crews with 85% working during
day shift, 99% working during evening shift and 66% working during night shift, 1 to 10 h repair time
range. 1000 outages for small storm, 5000 outages for large storm, 15,000 outages for extreme storm.
Next the model was tested for sensitivity to travel speed. The crew travel speed has little impact
on the ETR, as shown in Figure 10 for the large storms with 5000 outages and with the nearest outage
and nearest within radius search strategy. However, there are slight differences in the 25 mph, 50
mph and 75 mph speeds for most outages search strategy. Repeating for the small and extreme storms
yielded similar result. The small storm was not run for the nearest with radius strategy because
previous tests showed it did not vary from the nearest strategy. Figures S1–S3 for the small and
extreme storm can be seen in the Supplementary Materials. The biggest difference for all three storms
was between the 25 mph and 50 mph speeds for the most outages search strategy. Increasing from 50
mph to 75 mph further reduced the ETR but not as much as 25 to 50 mph. For all three storms, the
different travel speeds did not significantly alter the final restoration time.
Figure 9.
Change of radius for nearest-with-radius search strategy. Crew start location set to area
work center and search strategy set to most customers affected within a radius of the nearest outage.
Search radius varied as indicated. Travel speed set to 25 mph, 272 crews with 85% working during
day shift, 99% working during evening shift and 66% working during night shift, 1 to 10 h repair time
range. 1000 outages for small storm, 5000 outages for large storm, 15,000 outages for extreme storm.
Table 6. Time to system restoration (in days) based on search radius from Figure 9.
Search Radius Small Storm Large Storm Extreme Storm
1 mile 1.48 9.39 22.96
2 miles 1.48 6.65 21.11
3 miles 1.52 10.05 23.39
4 miles 1.48 9.35 23.44
5 miles 1.52 6.26 20.84
Next the model was tested for sensitivity to travel speed. The crew travel speed has little impact
on the ETR, as shown in Figure 10 for the large storms with 5000 outages and with the nearest outage
and nearest within radius search strategy. However, there are slight differences in the 25 mph, 50 mph
and 75 mph speeds for most outages search strategy. Repeating for the small and extreme storms
yielded similar result. The small storm was not run for the nearest with radius strategy because
previous tests showed it did not vary from the nearest strategy. Figures S1–S3 for the small and
extreme storm can be seen in the Supplementary Materials. The biggest difference for all three storms
was between the 25 mph and 50 mph speeds for the most outages search strategy. Increasing from
50 mph to 75 mph further reduced the ETR but not as much as 25 to 50 mph. For all three storms,
the different travel speeds did not significantly alter the final restoration time.
Infrastructures 2018,3, 33 15 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 15 of 22
Figure 10. Travel speeds for simulated large storms with 5000 outages. Crew start location set to area
work center and search strategy as indicated in plot title. Travel speed set as indicated in legend. 272
crews with 85% working during day shift, 99% working during evening shift and 66% working during
night shift, 1 to 10 h repair time range. ETR curve was relatively insensitive to travel speed but some
differences are seen in the most customers affected search strategy.
The next test varied the number of crews, as shown in Figure 11. For each storm size, initially
increasing the number of crews creates a decrease in the ETR. However, there is a threshold where
bringing in more crews will have less of an impact on the ETR. For the small storm, this occurred
around 250 crews and for the extreme storm it was somewhere between 400 to 450 crews. Table 7
shows the time to restoration of each storm size based on number of crews from Figure 11.
Figure 10.
Travel speeds for simulated large storms with 5000 outages. Crew start location set to area
work center and search strategy as indicated in plot title. Travel speed set as indicated in legend. 272
crews with 85% working during day shift, 99% working during evening shift and 66% working during
night shift, 1 to 10 h repair time range. ETR curve was relatively insensitive to travel speed but some
differences are seen in the most customers affected search strategy.
The next test varied the number of crews, as shown in Figure 11. For each storm size, initially
increasing the number of crews creates a decrease in the ETR. However, there is a threshold where
bringing in more crews will have less of an impact on the ETR. For the small storm, this occurred
around 250 crews and for the extreme storm it was somewhere between 400 to 450 crews. Table 7
shows the time to restoration of each storm size based on number of crews from Figure 11.
Table 7. Time to system restoration (in days) based on crew size from Figure 11.
Number of Crews Small Storm Large Storm Extreme Storm
50 7.01 39.31 120.20
100 4.76 18.96 59.74
150 3.28 12.47 39.18
200 1.94 10.66 29.01
250 1.43 9.25 22.97
300 1.44 7.81 19.03
350 1.01 7.13 16.58
400 0.97 6.10 14.34
450 0.92 5.80 14.00
500 0.83 5.38 11.78
Infrastructures 2018,3, 33 16 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 16 of 22
Figure 11. Changing number of crews for large and extreme storm. Crew start location set to area
work center and search strategy set to nearest outage. Travel speed set to 25 mph. 85% crews working
during day shift, 99% working during evening shift and 66% working during night shift, 1 to 10 h
repair time range. 1000 outages for small storm, 5000 outages for large storm, 15,000 outages for
extreme storm. Increasing the number of crews decreases ETR until a threshold, which varies by storm
size.
Table 6. Time to system restoration (in days) based on search radius from Figure 9.
Search Radius
Small Storm
Large Storm
Extreme Storm
1 mile
1.48
9.39
22.96
2 miles
1.48
6.65
21.11
3 miles
1.52
10.05
23.39
4 miles
1.48
9.35
23.44
5 miles
1.52
6.26
20.84
Table 7. Time to system restoration (in days) based on crew size from Figure 11.
Number of Crews
Small Storm
Large Storm
Extreme Storm
50
7.01
39.31
120.20
100
4.76
18.96
59.74
150
3.28
12.47
39.18
200
1.94
10.66
29.01
250
1.43
9.25
22.97
300
1.44
7.81
19.03
350
1.01
7.13
16.58
400
0.97
6.10
14.34
450
0.92
5.80
14.00
500
0.83
5.38
11.78
Figure 11.
Changing number of crews for large and extreme storm. Crew start location set to area work
center and search strategy set to nearest outage. Travel speed set to 25 mph. 85% crews working during
day shift, 99% working during evening shift and 66% working during night shift, 1 to 10 h repair time
range. 1000 outages for small storm, 5000 outages for large storm, 15,000 outages for extreme storm.
Increasing the number of crews decreases ETR until a threshold, which varies by storm size.
As previously mentioned, for storms with a lot of predicted outages, utility companies will call
in mutual assistance crews to aid in the recovery process. This test looked at the impact on the ETR
of bringing in mutual assistance crews at different times throughout the storm. First, only the time
to arrival of 150 mutual crews added to 200 initial crews was varied as shown in Figure 12 for the
extreme storm only. The ETR increases with increasing time for arrival. Next, the number of crews
added to 200 initial crews with a two-day arrival was varied as shown in Figure 13. The ETR decreases
with increasing number of added crews. Lastly, both the number of mutual assistance crews and the
time to arrival was varied as shown in Figure 14. Also, as expected, the more crews and faster time to
arrival decreased the ETR while less crews and longer time to arrival increased the ETR. There was
also a point where calling in more crews that would take longer to get there made less of an impact in
the ETR than calling in less crews that could arrive sooner. Towards the end of a storm there are less
outages to repair. If a large number of mutual assistance crews arrive later in the recovery process,
there may be more crews than outages or the cost of the added crews may not justify calling them in.
Infrastructures 2018,3, 33 17 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 17 of 22
As previously mentioned, for storms with a lot of predicted outages, utility companies will call
in mutual assistance crews to aid in the recovery process. This test looked at the impact on the ETR
of bringing in mutual assistance crews at different times throughout the storm. First, only the time to
arrival of 150 mutual crews added to 200 initial crews was varied as shown in Figure 12 for the
extreme storm only. The ETR increases with increasing time for arrival. Next, the number of crews
added to 200 initial crews with a two-day arrival was varied as shown in Figure 13. The ETR decreases
with increasing number of added crews. Lastly, both the number of mutual assistance crews and the
time to arrival was varied as shown in Figure 14. Also, as expected, the more crews and faster time
to arrival decreased the ETR while less crews and longer time to arrival increased the ETR. There was
also a point where calling in more crews that would take longer to get there made less of an impact
in the ETR than calling in less crews that could arrive sooner. Towards the end of a storm there are
less outages to repair. If a large number of mutual assistance crews arrive later in the recovery
process, there may be more crews than outages or the cost of the added crews may not justify calling
them in.
Figure 12. Changing time to arrival of 150 mutual assistance crews added to 200 initial crews for
extreme storm. Crew start location set to area work center and search strategy set to nearest outage.
Travel speed set to 25 mph. 85% crews working during day shift, 99% working during evening shift
and 66% working during night shift, 1 to 10-h repair time range. The horizontal red line shows the
median ETR of 0 added crews. The ETR increases with increasing time to arrival of the mutual
assistance crews.
Figure 12.
Changing time to arrival of 150 mutual assistance crews added to 200 initial crews for
extreme storm. Crew start location set to area work center and search strategy set to nearest outage.
Travel speed set to 25 mph. 85% crews working during day shift, 99% working during evening shift and
66% working during night shift, 1 to 10-h repair time range. The horizontal red line shows the median
ETR of 0 added crews. The ETR increases with increasing time to arrival of the mutual assistance crews.
Infrastructures 2018, 3, x FOR PEER REVIEW 18 of 22
Figure 13. Changing the number of mutual assistance crews added to 200 initial crews for extreme
storm with a two-day arrival time. Crew start location set to area work center and search strategy set
to nearest outage. Travel speed set to 25 mph. 85% crews working during day shift, 99% working
during evening shift and 66% working during night shift, 1 to 10 h repair time range. The ETR
decreases with increasing number of mutual assistance crews.
Figure 14. Changing number of mutual assistance crews and time to arrival for 15,000 outages. Crew
start location set to area work center and search strategy set to nearest outage. Travel speed set to 25
mph. 85% crews working during day shift, 99% working during evening shift and 66% working
during night shift, 1 to 10 h repair time range.
Figure 13.
Changing the number of mutual assistance crews added to 200 initial crews for extreme
storm with a two-day arrival time. Crew start location set to area work center and search strategy set to
nearest outage. Travel speed set to 25 mph. 85% crews working during day shift, 99% working during
evening shift and 66% working during night shift, 1 to 10 h repair time range. The ETR decreases with
increasing number of mutual assistance crews.
Infrastructures 2018,3, 33 18 of 21
Infrastructures 2018, 3, x FOR PEER REVIEW 18 of 22
Figure 13. Changing the number of mutual assistance crews added to 200 initial crews for extreme
storm with a two-day arrival time. Crew start location set to area work center and search strategy set
to nearest outage. Travel speed set to 25 mph. 85% crews working during day shift, 99% working
during evening shift and 66% working during night shift, 1 to 10 h repair time range. The ETR
decreases with increasing number of mutual assistance crews.
Figure 14. Changing number of mutual assistance crews and time to arrival for 15,000 outages. Crew
start location set to area work center and search strategy set to nearest outage. Travel speed set to 25
mph. 85% crews working during day shift, 99% working during evening shift and 66% working
during night shift, 1 to 10 h repair time range.
Figure 14.
Changing number of mutual assistance crews and time to arrival for 15,000 outages.
Crew start location set to area work center and search strategy set to nearest outage. Travel speed set to
25 mph. 85% crews working during day shift, 99% working during evening shift and 66% working
during night shift, 1 to 10 h repair time range.
3.2. Model Limitations
The goal of an ABM was to develop the simplest, yet accurate model possible. In order to
accomplish this, two simplifications were made. First, Dijkstra’s algorithm does not account for power
flow considerations on the utility lines. The model does not prevent multiple crews from working
on the same line. Secondly, the model does not account for different work rates of regular utility
crews versus mutual assistance crews or a change in work rate as the restoration process continues.
Typically, mutual assistance crews will have slower repair times because they are not familiar with the
system or they take longer to navigate to the outage. The model does not account for this and uses
the same repair time range for regular utility crews and mutual assistance crews. As mentioned in
Figure 3, the beginning of the storm typically fits better to lower repair time ranges but the end of the
storm fits better to longer repair time ranges, indicating a change in repair rates throughout a storm.
In the beginning of the storm there are more resources available and towards the end of the storm the
resources are less readily available.
There are several parameters where small changes in the value can result in large differences in
the ETR. As shown in Figure 4, the repair time range assigned to the outages has the biggest impact on
the ETR but it is also the most variable input parameter in the model. However, Figure 6shows the
model is insensitive to whether normal, gamma, exponential, Poisson or uniform distributions were
used to assign outage repair times. Although storms can be divided into categories such as snow, ice,
wind, rain and so forth, the repair time range of the outages can vary from storm to storm. Different
failure types can take different times to repair, as well as different number of crews available. The ABM
only assigns one crew to each outage and does not differentiate between outage types but increasing
the repair time range can account for losing multiple crews to one outage. The model also does not
differentiate between different crew types. In a utility company, crews are equipped for specific types
of repairs. Some outages will require two or more crews to each work on their specific task.
Infrastructures 2018,3, 33 19 of 21
The number of crews working and both the number and arrival time of mutual assistance crews
can vary throughout a storm. The model simplified these changes for the number of crews by having
a percentage of the total crews “resting.” The crews did not leave the model but did not contribute to
the restoration process during that time. During storm recovery, mutual assistance crews can arrive
at different times and in different groups. To allow the user to easily input any mutual aid crews,
all crews enter the model at the same time.
4. Conclusions
We developed an ABM using the NetLogo platform [
19
] to demonstrate that ABMs can be
an important approach to power outage restoration after storms and can be beneficial to utility
companies. The ABM shows that different outage search strategies result in different ETR curves;
the travel speed of crews has a minor impact on the ETR; increasing the number of crews will decrease
the ETR but only to a threshold; and the impact of mutual assistance crews depends on both the
number of crews and their time to arrival.
This decision support tool has the following advantages compared to current methods:
•
It is a quantitative tool based on empirical data that can be used by emergency managers to test
a variety of restoration strategies.
•
The model could be utilized prior to a storm based on outage predictions [
21
–
26
] or in real-time
as outages are discovered.
•
It is a socio-technical model that integrates human decisions constrained by the physical infrastructure.
•
This is a decision support tool for utility managers to supplement current restoration time
estimates. Utility managers can test decisions prior to or during a storm to make necessary
adjustments to the restoration process including the decision to hire foreign crews.
•
Providing a range of values for input variables can give a probabilistic range of outcomes for final
ETRs. These probabilistic forecasts can be useful for utility companies to provide customers with
a range of estimated restoration times.
•
The model is easily transferable to other states or regions and would only require the road
network dataset.
The developed ABM incorporates parameters not previously included in regression models, such
as the number of crews working and user defined rules to simulate crew behavior. The crews in the
model respond to the decisions made by the user, instead of using a statistical approach based on past
data. The model can be used to determine the appropriate number of crews in order to reach a desired
ETR and where and when foreign crews may be needed to achieve those goals. The model could be
used to help estimate restoration times for policymakers and customers.
An important disadvantage of the ABM as it is currently structured is that it is computationally
intensive. There are many input variables and running the model over a range of all of these variables
can take a long time, especially for larger storms. The current ABM does not incorporate power flow
considerations, like the expert systems approach developed by Liu et al. [
8
] does. The expert systems
approach determined the optimal repair order based on minimizing losses and does not incorporate
the social interactions of crews.
In the future, this novel technique could be incorporated with outage predictions before storms
hit [
23
–
26
] to give emergency managers a powerful tool to decrease restoration times in Connecticut
and elsewhere. Cost of restoration and mutual assistance crews can be easily added to the model.
This added feature would allow utility managers to see the impact their decision would have on
the cost to the utility company. The ABM could be used to explore the economic and restoration
time benefits of resilience measures, such as tree trimming. Lastly, the ABM could be developed as
a training tool for new emergency managers.
Overall, this model is an important first step in a new approach to power restoration that could
benefit both utility companies and utility customers.
Infrastructures 2018,3, 33 20 of 21
Supplementary Materials:
The following are available online at http://www.mdpi.com/2412-3811/3/3/33/s1,
Table S1: R
2
, MAE and standard deviation of modeled restoration curves from Figure 2, Table S2. R
2
, MAE and
standard deviation of modeled restoration curves from Figure 4. Figure S1: Start location of crews for small and
extreme storms, Figure S2: Crew travel speeds for small storms, Figure S3: Crew travel speeds for extreme storms,
Agent Based Model for Storm Recovery Code.
Author Contributions:
D.W. and J.M. developed the research idea; T.W. and J.M. were the primary program
developers; T.W. and J.M. conducted the data analysis; T.L. and D.W. provided data and industry expertise.
Funding:
This research was funded by Department of Education grant number P200A150311 and Eversource
Energy Center.
Acknowledgments:
The authors gratefully acknowledge the support provided by Emmanouil Anagnostou and
the UConn Eversource Energy Center.
Conflicts of Interest: The authors declare no conflict of interest.
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