Predictive Storm Damage Modeling and Optimizing Crew
Response to Improve Storm Response Operations
by
Sean David Whipple
B.S.E., University of Michigan, Ann Arbor, 2008
Submitted to the MIT Sloan School of Management and the Engineering Systems Division in
partial fulfillment of the requirements for the degrees of
MS
ETkf
Master of Business Administration
MASSACHUSETTS A'81tTT
OF TECHNOLOGY
and
Master of Science in Systems Engineering
JUN 13 2014
in conjunction with the Leaders for Global Operations Program at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
L IRRARIES
June 2014
@ Sean David Whipple, MMXIV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper
and electronic copies of this thesis document in whole or in part in any medium now known or
hereafter created.
Author ..........
..................
Signature redacted .......
MIT Sloan School of Management and the Engineerfng Systems Division
May 9, 2014
Certified by.........
Signature--------redacted
-- -- --- --- ---I ............................
-V
Certified by.......................
James Kirtley, Thesis Supervisor
Professor of Electrical Engineering
Department of Electrical Engineering and Computer Science
Signature redacted
Georgia Perakis, Thesis Supervisor
William F. Pounds Professor of Management Science
MIT Sloan School of Management
Approved by .
Signature redacted.,
. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Richard Larson
Mitsui Professor of Engineering Systems
Chair, Engineering Systems Division Education Committee
Approved by...........Signature
redacted .........................
Maura Herson
Director, MBA Program
MIT Sloan School of Management
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2
Predictive Storm Damage Modeling and Optimizing Crew Response to
Improve Storm Response Operations
by
Sean David Whipple
Submitted to the MIT Sloan School of Management and the Engineering Systems Division
on May 9, 2014, in partial fulfillment of the
requirements for the degrees of
Master of Business Administration
and
Master of Science in Systems Engineering
Abstract
Utility infrastructures are constantly damaged by naturally occurring weather. Such damage
results in customer service interruption and repairs are necessary to return the system to normal operation. In most cases these events are few and far between but major storm events (i.e.
Hurricane Sandy) cause damage on a significantly higher scale. Large numbers of customers
have service interrupted and repair costs are in the millions of dollars. The ability to predict
damage before the event and optimize response can significantly cut costs.
The first task was to develop a model to predict outages on the network. Using weather data
from the past six storms as well as outage data from the events, asset information (framing, pole
age, etc.), and environmental information were used to understand the interactions that lead to
outages (forested areas are more likely to have outages than underground assets for example).
Utilizing data mining and machine learning techniques we developed a model that gathers the
data and applies a classification tree model to predict outages caused by weather.
Next we developed an optimization model to allocate repair crews across Atlantic Electric
staging locations in response to the predicted damage to ensure the earliest possible restoration
time. Regulators impose constraints such as cost and return to service time on utility firms
and these constraints will largely drive the distribution of repair crews. While the model starts
with predicted results, the use of robust optimization will allow Atlantic Electric to optimize
their response despite the uncertainty of why outages have occurred, which will lead to more
effective response planning and execution across a variety of weather-related outages.
Using these models Atlantic Electric will have data driven capability to not only predict how
much damage an incoming storm will produce, but also aid in planning how to allocate their
repair crews. These tools will ensure Atlantic Electric can properly plan for storm events and
as more storms occur the tools will increase their efficacy.
Thesis Supervisor: James Kirtley
Title: Professor of Electrical Engineering
Department of Electrical Engineering and Computer Science
Thesis Supervisor: Georgia Perakis
Title: William F. Pounds Professor of Management Science
MIT Sloan School of Management
3
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4
Acknowledgments
First and foremost I would like to thank my advisers Prof. Georgia Perakis and Prof. James
Kirtley for their invaluable insight, guidance, and support on this project and on my MIT experience as a whole.
In addition to Georgia Perakis, the work done on this paper was also done with great collaboration with Prof. Vivek Farias (Sloan), Matthieu Monsch (PhD MIT ORC), and Anna
Papush (PhD candidate MIT ORC) and want to thank them for their support as well as the
Atlantic Electric team for making my internship a challenging and rewarding experience.
I would also like to thank my fellow LGO and Sloan classmates for making my graduate experience at MIT and Sloan one I will always treasure.
Finally I would like to thank my friends, family, and girlfriend, Lisa Kurajian, for their encouragement, love, and support during my time at MIT.
5
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6
Contents
1
Introduction to Storm Response at Atlantic Electric
1.1
Current Storm Response Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2
Literature Review
1.3
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1
Outage Prediction Literature Review . . . . . . . . . . . . . . . . . . . . . 15
1.2.2
Optimization Literature Review
. . . . . . . . . . . . . . . . . . . . . . .
17
Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Predicting Outages
2.1
19
D ata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1
Electrical Asset Information . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2
Land Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.3
Historical Outages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4
Historical Weather Information . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2
Segment Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3
Weather Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4
3
13
2.3.1
Weather Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2
Weather Forecasts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1
Maximum Likelihood Estimator
. . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2
Classification Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6
Implementation of Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
31
Repair Crew Optimization
. . . . . . . . . . . . . . . . 31
3.1
Introduction to Crew Allocation at Atlantic Electric
3.2
Solving the Master Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7
Relaxing the Problem
3.4
Comparison of Master and Relaxed Formulations . . . . . . . . . . . . . . . . . .
3.5
3.6
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3
3.4.1
Test Evenly Distributed Workload Assumption . . . . . . . . . . . . . . . 35
3.4.2
Comparison of Crew Assignments Between Master and Relaxed Formulation 36
Optimization Under Uncertainty
. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
. . . . . . . . . . . . . . . . 38
3.5.1
Robust Optimization Using Box Constraints
3.5.2
Robust Optimization Using Bertsimas-Sim Uncertainty Sets . . . . . . . . 39
3.5.3
Determining Robustness of Solution . . . . . . . . . . . . . . . . . . . . . 41
3.5.4
Comparison of Optimization Results . . . . . . . . . . . . . . . . . . . . . 44
C onclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
47
Conclusions and Future Work
4.1
4.2
4.3
35
Improvements to Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.1
Bootstrapping
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.2
Parallelization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.3
Principal Component Analysis on Weather Information
. . . . . . . . . . 48
. . . . . . . . . . . . . . . . . . . . . . . . 48
Improvements to Optimization Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1
Data Inaccuracies
4.2.2
In Storm Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.3
Customer Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
C onclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A In Sample and Out of Sample Prediction Results
51
B Complete Maximum Likelihood Formulation
55
B.1 Without Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
B.2 W ith Censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
59
C Complete Variable Notation
C.A Outage Prediction Variable Notation . . . . . . . . . . . . . . . . . . . . . . . . . 59
. . . . . . . . . . . . . . . . . . . . . . . . 59
C.2 Crew Optimization Variable Notation
61
Bibliography
8
List of Figures
2-1
Asset Description . . . . . . . . . . . . . . . .
20
2-2
Land Cover Distribution . . . . . . . . . . . .
21
2-3
Weather Station Locations . . . . . . . . . . .
22
2-4
2010 Storm Results . . . . . . . . . . . . . . .
28
2-5
Web Application User Interface . . . . . . . .
29
2-6
Web Application Output Window . . . . . . .
29
3-1
Job Allocation
. . . . . . . . . . . . . . . . .
35
3-2
Crew Workload Comparison . . . . . . . . . .
36
3-3
Comparison of Results in Master and Relaxed Formulations
37
3-4
Historical Non-Storm Outage Histogram . . .
42
3-5
r Simulation Histogram . . . . . . . . . . . .
43
3-6
Comparison of Optimization Results.....
44
9
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10
List of Tables
2.1
Hisotrical Outages
2.2
Sensitivity and Specificity Results of Outage Prediction
27
3.1
Master Formulation Variable Notation . . . . . . . . . .
32
3.2
Relaxed Formulation Variable Notation
34
A.1
October 2011 Winter Storm Results
. . . . . . . . . . 52
A.2
December 2010 Winter Storm Results
. . . . . . . . . . 52
A.3
Hurricane Sandy . . . . . . . . . . . .
. . . . . . . . . . 53
A.4
Hurricane Irene . . . . . . . . . . . . .
. . . . . . . . . . 53
A.5
February 2010 Snow Storm
. . . . . .
. . . . . . . . . . 54
A.6
December 2008 Snow Storm . . . . . .
. . . . . . . . . . 54
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
21
C.1 Outage Prediction Variable Notation
59
C.2 Crew Optimization Variable Notation
59
11
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12
Chapter 1
Introduction to Storm Response at Atlantic
Electric
Severe weather often causes significant damage to Atlantic Electric 1 assets, prompting Atlantic
Electric to respond by dispatching crews to repair any damage and return the system to normal
operation. Large storms such as hurricanes are particularly problematic because they not only
require significant costly repairs and interrupt service to hundreds of thousands of customers,
but the Department of Public Utilities (DPU) can fine Atlantic Electric and other utility firms
if they determine that a storm response plan was not properly created and implemented.
Because Atlantic Electric often does not have enough crews to repair the damage in an acceptable amount of time, they need to bring in outside contract crews to meet the need. This is
an industry standard, most firms are staffed for normally operation and maintenance. Staffing
for very large storms would add unnecessary cost throughout normal operating periods. Since
large storms affect large areas, it results in many utilities competing for the same limited supply
of repair crews. As a result Atlantic Electric and other utilities try to to obtain these crews
several days in advance of a major storm to allow travel and planning time so that the utility
is able to respond to the storm as efficiently as possible.
During storm events, Atlantic Electric stations its repair crews at sites called platforms. These
platforms are the home base for these crews throughout the duration of the storm (or until they
re-allocate crews). Once stationed, crews repair damage to the Atlantic Electric system when
outages occur in locations that are associated with their platform. Storm response is not only
a large driver of public opinion of Atlantic Electric, it also has a significant impact on Atlantic
'The name of the actual utility has been replaced with Atlantic Electric
13
Electric's costs. Large storms incur total repairs that number in the millions of dollars, much
of which may not be reimbursed by the DPU.
To motivate the problem we can examine the response of Atlantic Electric to Hurricane Irene
and a snowstorm in October 2011. After reviewing Atlantic Electric's response to these events,
Attorney General Martha Coakley recommended that the DPU seek $16 million in penalties,
the largest ever sought by the DPU [22].
This was after Atlantic Electric had already spent
$47 million in repairs for Hurricane Irene alone. Large storms have a significant direct financial
impact on a utility's bottom line. Additionally because utilities are essentially legal monopolies, government regulators limit the amount of earnings a utility can obtain based upon their
customer satisfaction level (customers tend to not be happy when they are without power for
significant periods of time).
1.1
Current Storm Response Process
Currently Atlantic Electric's process for determining the number of crews they need and implementing an effective approach for stationing them is very human. Regional managers estimate
damage based on their previous experience and request a specific amount of external crews.
Each manager has their own methods and experience driving that decision point, there is no
central decision making criteria. Atlantic Electric aggregates the information and tries to meet
this requested budget. Once the storm has hit Atlantic Electric has three meetings per day
where regional managers update progress on repairs and report their surplus/deficit of crews.
Again these surplus/deficit numbers are driven by intuition of what managers believe they need
to efficiently return the system to normal operation. Atlantic Electric managers again do their
best to meet all the needs using their experience.
Atlantic Electric's process is inherently human and can be quite flawed. While the managers
are all skilled and experienced, the lack of data driven approaches makes their decisions hard
to defend to regulators such as the DPU. Customers and the public in general are skeptical of
analyses that are solely driven by human intuition. It also creates a serious knowledge transfer
problem. When managers are unavailable to work or leave the company all of their expertise
goes with them. Understanding the effect of weather on Atlantic Electric assets is not a simple
one and takes years of experience to develop. A data driven model that is well maintained can
be operated by someone with significantly less training, their only requirements would be to
feed the model new data as it becomes available and interpret results. Such a system would
14
yield consistent, continually improving results as storm data is collected over time.
1.2
Literature Review
1.2.1
Outage Prediction Literature Review
Society's increasing dependence on technology, media and communication has created a growing reliance on the electrical supply industry. Stemming from this dependence, weather-based
power outages have recently become a very significant concern for both distributors and consumers. In the past decade there has been a good deal of research done with respect to this
particular problem, from multiple angles. Most of the literature related to this field can largely
be categorized into three distinct lines of work.
The first of these is research related to climate change and weather-based forecasting.
Cli-
mate variation over the past several decades has sparked a great deal of academic interest both
in terms of data collection and modeling. Synoptic weather typing is a scheme for the classification of weather conditions into distinct types. It has frequently come into use as a valuable
tool for work in climate impact applications such as pollution, disaster planning, agriculture
and human health. In [10] and [17], we see an automated and manual approach to this kind
of weather analysis. The work in [10] predicts occurrences of freezing rain by using automated
synoptic typing on differentiations in air mass. By studying hourly meteorological readings,
they identify weather types correlated with freezing rain and apply stepwise logistic regression
to predict its likelihood. Similarly, [17] also employs airborne particle concentrations and daily
weather data to build a manual synoptic typing that categorizes storm types in advance. This
branch of research also encompasses the effects of climate change on a socio-economic level.
Through time series modeling, [13] predicts daily variability in ski resort attendance based on a
combination of surrounding urban and mountain weather. The authors of [24] and [3] consider
the potentially harmful impacts of climate variability on temperature-related mortality and air
pollution-related health effects by analyzing correlations with weather parameters.
A second branch of the literature considers electrical system reliability with respect to weather
and the environment. Foundational work in this direction is done in [5], [6] and [7]. These papers
propose single and two-state weather models, then expand these to a three state weather model
that captures normal, adverse and extreme weather circumstances. Through the resulting calculations of reliability indices, they demonstrate the need for weather to be considered in practical
15
system assessments. Sensitivity studies show that disregarding weather effects produces overly
optimistic system appraisals, and that inclement weather conditions must be divided into a
minimum of two types. Prior to this work, multiple investigative studies considered specific
types of weather events and their impacts on system reliability. These works, such as [16], [1],
and [8], ultimately aim to improve reliability through system redesign. The work in [16] analyzes drought conditions and their resulting effects on tree faults; by using the Palmer Drought
Index, they present the influence of drought on tree-caused power outages. The latter two works
consider lightning storms and ice storms, respectively. By modeling system response and storm
characteristics, [1] presents a Monte Carlo simulation that evaluates system reliability and helps
identify weaker areas for system redesign. Using a similar approach, [8] models weather, vulnerability and restoration times in order to estimate system component reliability during severe ice
storms. More recently, there have been general weather reliability studies following Billinton's
works. In order to present a cost-benefit analysis for overhead-to-underground line conversions,
the work in [28] estimates damage rates based on hurricane wind speeds and simulates the
resulting restoration process. The paper by Caswell et al. [9] considers correlations between
reliability indices and various weather parameters to account for system variability.
The third line of work entails the prediction of weather-related electrical power outages. Some
of the earlier considerations of this problem are demonstrated in [23], [11] and [12]. The approach in [23] utilizes artificial neural networks (ANNs) in order to predict the number of power
interruptions based on inputted weather parameters. This approach combines time series and
regression to develop a learning algorithm. The follow-up works [11] and [12] consider the effects of normal daily weather conditions on distribution system interruptions; by using Poisson
regression models they determine the significant weather parameters that contribute most to
daily outages. Later work such as [27] and [29] show the incorporation of other statistical techniques. In [27], contingency probability estimators are computed through the use of maximum
likelihood (ML), to predict a transmission failure rate, and multiple linear regression on transformed weather data. Using both a Poisson regression model and a Bayesian network model,
[29] proposes a method for predicting the number of annual overhead distribution line failures
caused by weather.
The seminal series of papers by Liu et al., including [19] and [20], address a statistical approach to predicting spatial distribution of power outages and restoration times resulting from
hurricane and ice storm damage. They employ a generalized linear mixed regression model
16
(GLMM), however instead of using quantitative characteristics of each storm, they created
indicator variables that map each outage to its respective storm. Furthermore, their model
predicts damage on an outage level, meaning that it indicates whether a given device will open.
This lacks granularity in that an outage may be caused by 5 trees falling across the lines or
only by one broken pole. Their spatial prediction is executed on a 3 km x 3 km grid cell in
a given area serviced by a utility company. Building on this approach, [14] uses generalized
linear models (GLM) as well as generalized additive models (GAM), in addition to measurable
storm data that replaced indicator variables. In order to avoid variable collinearity, the data
was transformed using principal component analysis (PCA), which insures that the data is not
correlated. This work also predicts on a grid level, now 3.66 km x 2.44 km, in order to estimate
numbers of outages, customers without power and damaged poles and transformers. Although
this approach increases the prediction granularity, it still makes several assumptions on conditions that cause outages, such as the wind speed necessary to down a pole or uproot a tree.
In the more recent work by Hongfei et al. [15], a Bayesian hierarchical statistical modeling
approach is used to predict the number of outages and capture uncertainty in the outage data.
Although the prediction is not categorized by type of outage, the model also geographically
displays the uncertainty of the damage forecasts.
1.2.2
Optimization Literature Review
Optimizing operations in industry is not a novel concept and has been utilized across many
fields and industries. Balwani showed that a stochastic programming optimization can reduce
overtime repairs on Atlantic Electric's gas assets [2]. However the model presented in [2] contained data certainty, a fundamental constraint we will be facing in storm response is most of
the data will not be certain. [4] utilizes mixed integer programming to solve classic optimization
problems under uncertainty (ex. the traveling salesman problem). Other efforts had been made
to create optimization formulations with data uncertainty but were often accepting sub-optimal
results. Bertsimas and Sim create a more attractive trade off of data uncertainty and solution
optimality using probabilistic bounds of uncertainty in the data.
1.3
Thesis Outline and Contributions
This thesis will first present the methodology used to generate predicted outages in Chapter 2.
First we will discuss the data available to us and outline the methodology used to transform
that data into outage predictions. While models currently exist that attempt to solve similar
problems our solution provides analysis at a new level of granularity that currently does not
17
exist. The model presented will give Atlantic Electric a unique understanding of the incoming
damage from a storm at the device level (as opposed to the town level or other more aggregate
levels). Because our methods are purely data driven they are well suited to continuously improve as additional data becomes available.
In Chapter 3 we rely heavily on Bertsimas-Sim uncertainty sets to show that once Atlantic
Electric begins planning crew allocations (based upon predicted outages or actual damage retrieved from the field), restoration time can be significantly reduced without complete knowledge of the damage profile from any given storm. Reducing restoration time will allow Atlantic
Electric to release external contract crews earlier and utilize less manpower overall resulting
in significantly lower storm response cost. Additionally, restoration time reduction is critical
for Atlantic Electric's public appearance and regulatory filings to show that Atlantic Electric is
sufficiently planning for, and responding to, major weather storms.
18
Chapter 2
Predicting Outages
2.1
Data
In order to construct the model we built a database utilizing both Atlantic Electric asset information, outage information from past storm events, and historical weather information relevant
to those storms for the state of Massachusetts. Because we want to predict to outages during future weather events for improving storm response planning, it is critical to understand
the asset vulnerabilities on the Atlantic Electric network utilizing both their asset features,
environmental information, and relevant weather information.
2.1.1
Electrical Asset Information
In order to predict outages on the network we first need to have an understanding of the
electrical asset information. This analysis is focused on the distribution network of Atlantic
Electric, that is everything down stream of a substation. Substations and everything upstream
of them in the power generation and distribution system were not used in this analysis (Figure
2-1 illustrates this scope).
19
Circuit
Power station
Substations
Low voltage power lines
Houses
Segments
Figure 2-1: High level representation of the distribution network. This analysis focused solely
on everything downstream of substations. Everything upstream (substation, high voltage transmission lines, and power generation) was not considered.
The electrical distribution network is composed of segments that form a distribution tree downstream from the substation. These segments are any electrical equipment (poles, underground
lines, etc.) that connect devices (fuses, breakers, transformers, etc.). In total, Atlantic Electric
has approximately 60,000 devices and 280,000 segments (some, not all, of the segments have
devices located on them) on 300 circuits across Massachusetts serving approximately 1.2 million customers. In order to fully understand the asset information we obtained the physical
attributes of the 280,000 segments across Massachusetts. These segments are represented by
35 physical properties (insulation, density, age, framing, length, wiring, etc.). Utilizing this we
can create a define a new term, known as an asset. An asset is a device grouped with all the
segments that are downstream of that device until the next device.
2.1.2
Land Cover
Environmental information such as land cover and altitude was added to the segment description
to aid in capturing key interactions that lead to outages. For example, a bare framed pole with
thinner insulation is more likely to go out in a forested area due to falling trees compared to
flat open ground. Using data from MassGIS, Massachusetts has 33 different land cover types
(forested, low density residential, commercial, etc.).
Figure 2-2 illustrates the distribution of
Atlantic Electric segments and devices on those land cover types..
2.1.3
Historical Outages
In order to train the model we used the outage history across several storms as responses to the
weather and asset interactions. Table 2.1 contains summary statistics of historical outages that
20
40000
10000
||.'I.,
Figure 2-2: This histogram depicts the variety of land cover types where segments lie. The
green bars represent the total number of segments corresponding to each category, while the
blue bars represent the total number of segments corresponding to each category that are a part
of an asset which has been damaged in one of the six storms considered in this work.
Atlantic Electric experienced. It is important to note that in this table that the term "outage"
refers to the failure of a device, not a segment. Because one or more damaged segments can
lead to an outage, an outage in itself is not a good indicator of damage at or near the device
that went out.
Table 2.1: Hisotrical Outages
Days
Outages
Customers Out
2008-12-12
10
1784
185931
Wind Storm February 2010
2010-02-24
6
615
151350
Winter Storm December 2010
2010-12-26
4
444
106346
Tropical Storm Irene 2011
2011-10-29
11
2746
291672
Hurricane Sandy 2012
2012-10-29
7
1466
180416
Storm Name
First Outage
Winter Storm December 2008
It is important to note here that even the worst storm has approximately 4.5% of the the total
devices going out. This presents a particularly difficult prediction problem as we have so few
positive samples to work with.
21
................
......................
_ _ _ ..................
........................................
..............
............................
.....
2.1.4
Historical Weather Information
When examining historical weather there were two sources available. The first source was a
set of weather logs provided by a third party vendor, which includes time series data at several
hundred locations across MA and contains information on weather features such as temperature, wind (speed, gust, and direction), precipitation rates, etc. Each station contained one
data point per hour throughout each storm.
The second source was a collection of weather forecasts aggregated from several freely available weather providers that regularly report to Atlantic Electric. While this data was free and
Atlantic Electric already had the mechanisms in place to obtain it, it was considerably lacking
in granularity compared to the weather logs mentioned previously in this section. This data
set has fewer locations, fewer features (i.e. does not contain precipitation rates or wind gust
information) and only contains one data point per day per storm. This reduction in granularity
significantly impacts our efficacy in outage prediction. Figure 2-3 illustrates the locations where
weather information is available.
4%
%
4S N
4P
2%
~
f
Figure 2-3: Locations of available weather stations across MA. Blue stations represent locations
of weather provided by a private vendor. Orange locations are publicly available weather stations
that readily provide forecasts.
2.2
Segment Clustering
The first step in organizing the data is to group the different segments into types. We do this
using a clustering algorithm: k-means (also referred to as Lloyd's algorithm, [21]). This algo-
22
rithm aims to partition the segments into k clusters in which each segment is associated with
the center of its corresponding cluster. All segments associated with any given cluster then are
considered equivalent and have the same properties. The number of centers (k) was initially
set to 20, and eventually brought down to 10 to simplify the model and increase generality.
A higher number of centers leads to better representation of the segments but also more parameters to optimize in the final model. In order for the clustering algorithm to work, some
of the raw features were normalized by the length of the segment: the total number of poles
(respectively total number of customers) was converted to pole density (respectively customer
density). Without this preprocessing step, the clustering algorithm gives very poor results as
there is a large variance in segment lengths (from 1 meter to 86 miles).
Each segment type is then further categorized by land cover (the 33 land cover categories
were brought down to 10 by grouping similar ones together and removing land cover types that
covered negligible areas of land): forested areas, highly residential areas, open rural areas, etc.
This brings the total number of segment types to 100 (this is generated from all combinations
of our 10 types from k-means clustering and the 10 land cover types).
Now we can represent all Atlantic Electric assets with the following
ai = di :
{l, 12, ... , 11001i
Where ai represents asset i, di is the associated device on that asset, and the set {li,1,
.. , IOO}i is
the length of each type of segment directly downstream of di (downstream meaning all segments
between di and the immediate downstream devices of di in the distribution network).
2.3
Weather Profiles
When exploring how to use the weather information, two different approaches had to be taken
depending on whether the weather logs or forecasts were being used as described in Section 2.1.4.
While the weather logs are clearly more ideal for outage prediction, the logistics of working with
that vendor to provide tailored forecasts has not yet been completed. Therefore the weather
forecasts are necessary for prediction until tailored forecasts from the vendor are available for
use in predictions in advance of major storm events. We will present approaches for both data
sources.
23
2.3.1
Weather Logs
To make the weather information usable each weather feature was binned into dummy variables
where each bin contained the number of logs that fell into that particular range. Each feature
(wind speed, temperature, etc.) was set to have 10 bins evenly distributed across the range of
lowest to highest value for that particular feature. Increasing the number of bins could potentially increase accuracy of the model but in doing so we lose degrees of freedom and take away
from the generality of the model.
For each device all weather stations that were within a 5 mile radius were used to create
the weather profile for that device. Unfortunately not all of Atlantic Electric's assets fall within
a 5 mile radius of a weather station and therefore had to be excluded from use in the model.
However this only accounted for approximately 5% of Atlantic Electric's assets in this case.
2.3.2
Weather Forecasts
Because the forecasts lack the granularity to bin the information as we did with weather logs,
we simply applied the worst case scenario value of each forecast feature from every station
within a 20 mile radius (the radius had to be extended due to the forecasts being available at
fewer locations than the logs). This results in a significant decrease in prediction capability and
therefore the model captures much less variance when compared to the model using weather logs.
The loss in prediction quality is not surprising given the amount of data loss. However when
examining the data we can actually see it would be difficult for any model to yield quality
predictions using only the forecasts. We looked at the expected value of all the weather features
at each asset, given the outage response at that asset. This can be written mathematically as
follows
"EWfja 1 = ],
E [wf,|Oa = 0]
Where wfa is the weather feature value for feature
1Vf,a
f
(2.1)
at asset a. Because the above ratio was
approximately 1 for most of the weather features, there is little to no signal to generate quality
predictions.
24
2.4
Model Formulation
2.4.1
Maximum Likelihood Estimator
Much of the work from this section was done in collaboration with Matthieu Monsch (PhD
Massachusetts Institute of Technology Operations Research Center) and can be found in his
dissertation [25]. The complete model formulation from [25] can be found in Appendix B, but
we will summarize it here.
To begin we will make some assumptions about damages caused by weather events.
Assumption 1. Damaging events occur independently on Atlantic Electric's network as a Poisson process
It is safe to assume that damaging events that occur are unlikely to influence damaging events
at other locations (i.e. if a fuse fails in Worcester, that failure is unlikely to effect the failure
rates of other devices across MA). It then follows that damaging events can occur as Poisson
process dependent on weather and asset features.
Assumption 2. Damaging rates are uniform across each segment
This assumption is safe as after all the segments have been clustered into types it follows that
any one segment would be subject to the same weather vulnerabilities as segments of the same
type.
Combining Assumptions 1 and 2 we can state that damaging events occur with the following
AS't = lsge*gwt
where
I is the length of segment s, g,, is the vector of vulnerabilities for that particular
segment, and wt is a vector of the weather features surrounding that segment. We make no
assumption on particular weather features and allow for segments to be more vulnerable to
damage under certain weather conditions compared to others [25]. We cannot compute damage
at the segment level because of data granularity, but we can aggregate damage at the asset level
with the following:
Aa,t
is~* wt
=
s~a
la,cge*wt
=
C
-lawt
25
Or more generally
Aa,t
y*Xa,t
The maximum likelihood estimator (,y*) can be found efficiently and due to the non-linear nature
of this estimator, it provides a better prediction than those of linear estimators found in some
of the existing literature (e.g. logistic regression) [25].
2.4.2
Classification Trees
While the earlier assertion that a maximum likelihood estimator is better than typical linear
models is true, one can note that classification trees also present a viable option as they hold
true to those assumptions in general. Classification trees only require that the data be linear
within regions of the predictor variable space, not the entire variable space.
Recursive partitioning in classification trees can effectively identify regions of the predictive
variable space that are prone to outages while not being influenced by other areas of the space
that often do not lead to outages (i.e. underground segments almost never experience outages
due to weather) [26].
We used the scikit-learni package (implementation to be discussed in
detail later) in Python which uses random tree generation to lower generalization error and
produce more diverse classification trees [18].
2.5
Results
In order to test the model we did both in and out of sample testing. In sample testing used all
storms as training data and out of sample testing held back the storm that was being tested
and trained on the other five storms. This manner of testing prevented against over fitting but
also simulated the environment in which it would be used at Atlantic Electric; incoming storm
information would never be part of the training set. The in sample results however demonstrate
the model's ability to learn new outage responses as the data becomes available.
'http://scikit-learn.org/
26
Table 2.2 details results using sensitivity and specificity numbers where
sensitivity =
number of true positives
number of true positives + number of false negatives
specificity =
number of true negatives
number of true negatives + number of false positives
Table 2.2: Columns listed with "IS" denote in sample results and columns denoted with "OS"
denote out of sample results
Sensitivity (OS)
Specificity (OS)
Sensitivity (IS)
Specificity (IS)
Winter Storm Dec '08
.702
.989
.733
.999
Wind Storm Feb '10
.967
.991
.755
.999
Winter Storm Dec '10
.713
.991
.819
.999
Tropical Storm Irene
.978
.976
.767
.999
Hurricane Sandy
.669
.998
.724
.999
Storm Name
It is also useful to visualize the results to aid users in internalizing results and interpreting
them. Figure 2-4 is one such example where we can see how well we predict when compared to
an actual storm. We aggregated outages around Atlantic Electric's crew platform locations for
demonstration purposes. This visual was produced using Google Maps 2 for this document, later
we will show screen captures of the production system built to allow Atlantic Electric users to
interface with the model.
A complete summary of the model prediction results can be found in Appendix A. For simplicity and space all results are given in a table format as opposed to the visualization method
seen in Figure 2-4.
2
http://map.google.com
27
Figure 2-4: Out of sample results for Winter Storm 2010. Outages are grouped to the nearest
Atlantic Electric staging platform by location. Circle diameter varies linearly with the number of
outages associated with that platform. Red circles identify the number of outages that actually
happened in the storm. Blue circles show out of sample results from the prediction model.
2.6
Implementation of Prediction Model
The model previously described was built into a Python tool3 which utilized several packages
(scipy4 , numpy5 , scikit-learn 6 , pandas7 ) all interfacing with a MySQL database. The entire
project is deployed using Flask8 as a web application that can be accessed with a browser.
These Python packages provide a variety of numerical tools that enable us to do the analysis
described in this chapter. Because of the size of the computation necessary we hosted this
project on Amazon Web Services9 which has ample computing capacity to handle the demands
of the model. This allows Atlantic Electric to only pay for the computing power they need and
have multiple users easily access the tool from multiple
locations.
Shttp://python.org
4
http://scipy.org/
5
http://numpy.org
6
http://scikit-learn.org/
7
http://pandas.org/
8
http://flask.pocoo.org/
9
http: //aws.amazon.com/
28
..................
I I
-
Outage prdkton
VfW*W6*M
Gmm
Pawrahn
4 ftow*
* wm
Figure 2-5: This user interface allows users to select training storms, determine which platforms
to be opened, input data, and run the model.
O0~g pmdioton
t.,._-
0
P6001MIN
BANOW"
00""
NOWq.
4tim
Figure 2-6: After the model has predicted outages given the user input, the web application
visualizes the data by grouping outages to their closest opened platform allowing users to
visually internalize the results of the model. Outages are only grouped to platforms that have
been enabled, or opened, by the user.
2.7
Conclusion
The methods presented in this chapter outline a machine learning approach that will allow
Atlantic Electric to quantify the amount expected damage from an impending weather event
without any human bias. Even with the limited data available, we were able to predict outages with sufficient accuracy and that accuracy will continue to improve as Atlantic Electric
experiences more storm events. Because the model can clearly articulate the expected outages
29
I'll.. "I
from weather, Atlantic Electric is well suited to better contract necessary resources from outside
firms, deploy storm response resources effectively both before the storm occurs and during the
storm, and justify their actions to the public and government regulators. The next chapter will
detail the methods developed that will aid Atlantic Electric in making deployment decisions
based on the outage prediction results we are making here.
30
Chapter 3
Repair Crew Optimization
3.1
Introduction to Crew Allocation at Atlantic Electric
During storm events Atlantic Electric stations its repair crews at sites called platforms. These
platforms, or staging areas, are the home base for these crews throughout the duration of the
storm (or until they re-allocate crews). These locations are generally Atlantic Electric facilities
but can also include municipal locations such as schools or parks where Atlantic Electric can
temporarily stage crews during a storm. Once stationed crews repair damage to the Atlantic
Electric system when outages occur in locations that are associated with their platform. Storm
response is not only a large driver of public opinion of Atlantic Electric, it also has a significant
impact on Atlantic Electric's costs. Large storms cause damage that can require millions of
dollars in repairs. Atlantic Electric can petition for reimbursement from the DPU for these
repairs. However, if the DPU determines that Atlantic Electric did not sufficiently plan for and
respond to the storm, they can refuse reimbursement or even fine Atlantic Electric.
Currently Atlantic Electric uses a very manual and human process to station their repair crews
in a storm response scenario. Atlantic Electric managers use their experience and intuition
to make educated guesses to determine the number of crews they need to effectively repair
the system within an acceptable amount of time. Atlantic Electric managers have tremendous
experience and insight when making these decisions. However human decisions will always be
inherently sub-optimal and it is often difficult to transfer that knowledge and expertise to other
Atlantic Electric employees who do not have the same years of experience. Since storms do
not happen often it generally takes employees years to see enough storms in order to effectively
manage the storm response process.
31
The previous model produced predicted outages at the platform level based upon weather
forecast input. The next step is to utilize those predictions to aid in Atlantic Electric's storm
response planning. An understanding of where they anticipate damage will allow Atlantic Electric to station crews so that they are best suited to repair damage in the fastest time possible.
This will ultimately return service to customers quicker, reduce costs for Atlantic Electric, and
aid Atlantic Electric in justifying their actions to the DPU in their regulatory filings.
3.2
Solving the Master Problem
To solve the problem of storm response planning we created a formulation that ultimately determines where to station crews and what jobs they will be responsible for completing. While
this is not a scheduler it ultimately gives Atlantic Electric the ability to better understand
the anticipated work of each repair crew as a result of expected outages from the outage prediction model. The complete formulation and explanation of variables is given below (and all
formulation notation can also be found in Appendix C).
Table 3.1: Master Formulation Variable Notation
Notation
Decisions
j
at platform k
Xijk
Crew i assigned to job
Xik
Crew i assigned to platform k
Z
Data
Description
Yjk
Mk
Objective value equal to the worst case repair time
k
Time required to do job j from platform
Crew capacity for platform k
It should be noted that we are currently examining Yjk values that take on a known, nominal
value. In reality these repair times are unknown, as repairs can be caused by any combination
of problems with varying repair times. These unknowns will have serious effects on both the
solution and the optimization formulation. The uncertainty of -Yjk and its effects on the optimization problem will be discussed later in this chapter.
The model can be represented mathematically as the following:
32
Objective: Minimize Z subject to the following constraints
Vi
(3.1)
Vk
(3.2)
Vj
(3.3)
Vi
(3.4)
X jk < Xik
Vij,k
(3.5)
Xijk, Xik c {0, 1}
Vi, j, k
(3.6)
'-jkXijk < Z
5 ik<
5Xijk
5
k
Xi 5 1
k
All notation for this formulation can be found in Table 3.1 (and Appendix C). Platform capacities are ensured by 3.2 and 3.3 ensures that all jobs are completed. Constraints 3.4 and
3.5 ensure that crews are assigned to only one station and they can only repair jobs that are
assigned to that station. Constraint 3.1 stipulates that no repair crew's total repair time can
exceed that of the worst repair time given by the objective value Z.
This solution does obtain optimal solutions however at very long solve times. Given the nature
of variables and constraints, serious storm events will have a significant increase in the number
of decision variables and constraints. Consider a case of 300 crews, 600 outages, and 6 platforms
to station those crews. The resulting formulation has approximately 2 million decision variables
and constraints. Solve time for this particular case was on the order of days (relaxing Xijk to
a continuous variable on the range [0, 1] still produced integral solutions but the solve time was
still insufficiently long). It is possible that the software was exploring an extraneous number of
solutions given the potential symmetry of those solutions. However even achieving gap values
that were 3% of the optimal solution took hours. Given the importance of timing in Atlantic
Electric's storm response planning an alternate simplified formulation was explored.
3.3
Relaxing the Problem
Deeper analysis of the previous formulation showed that while we were obtaining optimal solutions, crews were essentially splitting the work evenly at each platform. In order to achieve
the best state wide completion time, the optimization would drive solutions such that all crews
were completing their work at approximately the same time. By making the assumption that
33
crews evenly split the work at each platform we can drastically simplify the problem. We no
longer need to assign crews to jobs and platforms. Now our decisions are reduced to assigning
jobs to platforms and the the number of crews to each platform.
Table 3.2: Relaxed Formulation Variable Notation
Notation
Xjk
Decisions
Job j assigned to platform k
Ck
Platform k fractional workload
C*
Number of crews assigned to platform k
C
Data
Description
Objective value indicating worst platform fractional workload
j
from platform k
Yjk
Time required to do job
Mk
Crew capacity for platform k
C*
Total number of crews available
The new mathematical model can be represented as the following:
Objective: Minimize C subject to the following constraints
Ck<
(3.1)
C
k
SJkXJk < Ck
kXj
>1
Vk
(3.2)
Vj
(3.3)
k
We interpret the number of crews at each platform as the following:
Ck
Ck = C C*
(3.4)
Vk
Ck* < Mk
(3.5)
The constraints are analogous to those in the master formulation. The major difference being
that now we ensure a statewide completion time with a combination of constraints (3.1) and
(3.2). Constraints for ensuring that all jobs are completed and all platform capacities are met
are similarly modified from the original formulation.
34
- ..
..
..........
-...- - - -...........
-1--.1..........
11111 "I'll
................
...
1 11....................
..
.................
-I--I I-- i .
..--..
....
....
..
0
Figure 3-1: This demonstrates the method used by the optimization to assign each job to a
platform. The about of time to complete a job is represented by the dashed red line, which is
unique to each platform.
This new formulation is a significant decrease in problem complexity. In our master solution
we examined one potential scenario of 300 crews, 600 jobs, and 6 platforms which contained
approximately 2 million decision variables and constraints. The relaxed formulation reduces the
problem to approximately 7 thousand decision variables and constraints and solve time is now
on the order of seconds.
3.4
Comparison of Master and Relaxed Formulations
While the improvement in computation time for the relaxed solution will allow Atlantic Electric
to adequately utilize the model within the time constraints imposed by storm operations, it is
important to verify that the new formulation produces adequately optimal results given the
assumptions made. In order to validate the relaxed model we will first look at the accuracy of
the assumption that the model allocates work among crews evenly in the master solution. Next
we will compare workload of crews in the master formulation and the relaxed formulation at a
platform level'.
3.4.1
Test Evenly Distributed Workload Assumption
In order to test the validity of our assumption we ran the master formulation on several notional
data set scenarios. The data sets were created by randomly sampling outages from previous
storms and only opening two platforms for stationing. These are representative of actual storm
scenarios that Atlantic Electric faces but are on a smaller scale to manage solve time. The
'It should be noted that here we are still using nominal -Yjk values as input data into our models.
35
histograms below show the crew workloads for each platform for all three scenarios.
1.0
Crew Workload I by Platform
,
64
2-
0.8-
0.6-
*i0.4 -f
E
U
1
0
'
0.2J
0.0[
'
'
360.0 360.5 361.0 361.5 362.0 362.5 363.0
4.0
335
6
3.5-
5
4-
. .
3.0
2.5E 2.0-
340
345
350
355
360
365
3
1.5 -
2
1.0
0.5
352 353 354 355 3 56 35 7 358 359 360 361
3.0
2.5
2.0.E 1.51.0
354 355 356
9
8
76-
I
357 358 359 360 361
5124Liii
335 340
345 3 50
355 360
365 370
PI atform 1
325 330 335 340 345 350 355 360 365 370
Platform 2
Figure 3-2: Crew workloads for three randomly sampled sets
Ultimately the crew workload differ by only minutes (compared to hours of total work). The
difference is due to the fact that we don't allow crews to go "help" another crew that is still
finishing a job if they have finished early. Overall these differences suggest that our assumption
of equal distribution of workload is a reasonable one.
3.4.2
Comparison of Crew Assignments Between Master and Relaxed Formulation
Now we will examine the same three scenarios and compare the optimization results from the
original and relaxed formulations. Ultimately we want to ensure that both optimization results
are producing similar numbers of crews at the platform level. The crew allocations for the three
test scenarios used earlier are given in Figure 3-3.
36
40
32
4033
Simulation 1
20
10
0
40
6
7
Pkonn I
Pktbrn 2
29
Smulaon 2
31
30
20
10
0
30
Pufon I
Phdi*rm 2
28
S mulation 3
27
20
10
0
10
11
PWtOrM I
Pkfflwrm 2
Figure 3-3: Comparison of Results in Master and Relaxed Formulations
Our relaxed formulation never deviates from the master solution by more than a few crews.
Given the level of granularity of data and our current accuracy for outage prediction, we believe
that this is sufficiently close to the optimum solution. Individual crew workloads may differ
by values more than 10% on occasion but the ultimate operational question we are trying to
answer is quite close to the master counterpart.
3.5
Optimization Under Uncertainty
As stated earlier we have been dealing with known, nominal values of 'Yjk in our optimization
formulations. However this will never be the case given that:
" Our weather forecast information, W, always contains inaccuracy and is an input we are
constrained by
" Predictions currently only assign P(OutagejW) (or a classification of an outage)
" Any given outage can be caused by a combination of issues (i.e. an outage can be caused
by a broken pole and three down trees or it can be caused by icing on a transformer
causing open breakers and downed wires) which lead to variable repair times
It is impossible to escape these uncertainties so we must program them into our optimization
to ensure that our solutions are always valid despite the variation in potential workload values.
Recall from our relaxed formulation that the only constraint using our unknown workload (-Yjk)
37
values was the following:
-YjkXjk < Ck Vk
We can amend this constraint to include all possible values of 'Yjk and ensure that any solution
given by our optimization will be a valid one with the following:
>
YkXjk < Ck,Vk, -Y,k EUV
Here
U is the set of all values that -y can potentially take on. Given this new constraint all
solutions from the model are guaranteed to be valid. However in this current form we no
longer have a mixed integer program (MIP). Understanding the nature of
U is important to
re-modeling the formulation back to a MIP [4].
3.5.1
Robust Optimization Using Box Constraints
The simplest method is to assume all
where
-yjk E [ljk, Ujk].
-Yjk
values reside in a "box" (a one dimensional range
Now our uncertainty set U takes the following form:
Uk
-Y
IVi,1j k <- -Yjk <- Ujk}IVk
Recall our original constraint which contained uncertainty:
S kXjk
CkYJk E Ujk,Vk
The above constraint can now be re-written as:
UjkXjk < CkVk
While the implementation of this solution is easy note that we now assume that all jobs take
on the worst case scenario value. Given that we assumed all predictions of outages happen
independently, we can assume that their respective repair times are also independent random
variables. Assuming that all of the restoration times will take on their upper bound value
simultaneously is highly unlikely and yields significantly lower and conservative objective value
[4).
38
3.5.2
Robust Optimization Using Bertsimas-Sim Uncertainty Sets
Instead of assuming all restoration times assume a worst case value we can now consider a case
where the uncertainty set
U is comprised of values where a fraction assume a worst case scenario
and the remainder are forced to their nominal values. Consider the following representation of
the uncertainty set:
'YYj
Uk
{-YIVi, Yk C [-Yjk
In the above specification of UA,
±7jk]
k
c:'YF
F}k
-jl
-j
-)7k represents the nominal value of Yjk and 73k represents the
F then bounds the total deviation from the
half width on the interval of which ^Yjk can reside.
nominal -Yjk values in the uncertainty set. Ultimately F is a parameter that specifies the number
of values that assume the extreme values ('yr,- 73k, 7Yjk + 73k) [4]. Selection of the F parameter
will be discussed later in this chapter.
To make the constraint robust we can dictate that:
'Yjk =Jk + -YjkUjk
where all 'Yjk are the nominal workload values,
73k is the potential deviation from the nominal
workload value, and ujk indicates the direction of that deviation (positive or negative). The
total deviation from nominal for all Yjk must be bounded and therefore all Ujk must reside in
the following uncertainty set:
Uk,u
=
ulV , Ujk
E [-
Taking this representation of our uncertainty set
11;
lUjk|I < IF
Uk ,u the original constraint can now be repre-
sented as the following:
ZkXjk
i
-
max ZUjk-YkXjk < Ck
U A.
39
The max problem on the right hand side is a linear optimization of the following problem:
Z(ujk -
maximize
uj-k)-jkXjk
j
subject to
U+ + u- < F,
Vk
u,jk' uk
jk <1,
Vj, k
k
0
The above optimization has a bounded finite region, thus attaining a finite optimum value. By
strong duality, the following optimization is also feasible and will obtain the same optimum
value [4].
minimize
FRk +
(
±'---j
subject to
Rk + r+ > yJkX~k,
Vj, k
Rk + rT
Vj, k
-> jkXjk,
Vj, k
Rk, r ,rj
Using the above transformations, our original robust constraint:
E kXjk + max
UIEUhtu
i
5UjkyjkXjk 5 Ck
Is equivalent to the following set of constraints:
ZkXjk +r+ +A
Rk +
K
Rk +
Vi--k. > --
R, rA,r
+FRk
YjkX~k,
Ck
Vk
VJ
kXjk,
Vj
>0
Vj, k
Combining this with our original relaxed formulation we now have a robust solution that does
not unnecessarily limit the objective value of the solution. Note however that the above solution does include the possibility that some constraints will be violated (i.e. it is possible for
the solution to be infeasible) [4]. The probability of this occurrence is dictated by F. Larger F
values will produce more robust formulations but hinder the resulting objective.
40
Our formulation can now be written completely as the following:
Objective: Minimize C subject to the following constraints
[_ _kXjk -+-+,+ r4-k,] + r R&
k <Ck
l+
'-j
±T.k
ZK~k
Rk -rAik
_> YkXjk,
Rk + rik- > -JkX3k,
Rk, r+ r
jk'
>0
(3.1)
VJ
(3.2)
Vj
(3.3)
Vj, k
(3.4)
Vj
(3.5)
Jk >-
Xk > 1
(
Vk
k
(3.6)
, Ck < C
k
We interpret the number of crews at each platform as the following:
C*
Ck* = Ck
C
(3.7)
Vk
CE < A4k
(3.8)
Similar to the earlier formulations we discussed, equation 3.8 ensures that platform capacity Mk.
(the maximum number of crews that can be station at platform k) is not exceeded, equation
3.5 ensures all jobs are completed and equation 3.6 ensures the earliest state wide completion
time. Ultimately the model is optimizing Ck, the number of crews that are assigned to each
platform (refer to Table 3.2 for a complete description of model variables).
In the next section we will discuss the trade offs between different choices of F by examining the historical repair information from Atlantic Electric and noting how that informs our
decisions when choosing our half width values (-Y)
3.5.3
and F.
Determining Robustness of Solution
In order to choose the parameters of our robust optimization sufficiently we must first obtain
an understanding of previous damaging events.
Because our prediction model only predicts
outages and not damage (which is a limitation of the data available and not the model),
examining historical information will be critical.
Atlantic Electric's information on damage is rather is limited but they have a much richer
data set on previous outages.
The data set contains data on what device opened, when it
41
opened, how long until it was restored, and how many customers were affected by this event. In
order to build a proxy for repair time we decided to use the difference between the time a device
initially went out (open time) and it when it was re-energized (closed time) and exclude outages
during storm events. During a storm a particular outage may be repaired but the circuit might
not be energized for a number of reasons
" A circuit will not be re-energized if other crews are repairing damage that a given device
feeds to
" Only qualified Atlantic Electric employees can re-energize a circuit, a contract crew may
make the repairs but the system will not turn on until a Atlantic Electric employee reenergizes the circuit thus affecting the turn on time
" Storms have "emergency" mode where crews are only repairing damage that causes a
risk to the public. Other non-threatening outages therefore have a longer downtime not
because repairs take longer but simply because crews are not authorized to work
Because of these operational differences a restoration time is not a good proxy for repair time
during a storm. Excluding storm outages from the historical data set we see the following
restoration profile.
Historical Non-Storm Outages
7000
-I
ijk
=
166.67
6000F
jk = 544.09
1 50000
.0
04000 -
E
z 30002000
1000
400
600
800
1000
1200
Repair Time (mins)
Figure 3-4: Historical Non-Storm Outage Histogram
While the data clearly does not show a normal (or even centered) distribution choosing a nominal ^yjk that equals the average of this distribution and a half width value that covers a large
42
portion of the histogram is sufficient. The above histogram then yields that -y3k = 166.67 and
a half width value -Yjk = 544.09 (2 times the standard deviation of the data). It should be
noted that the above values were all non-storm values restoration times so an average value
that is higher than the median is still reasonable. Similarly for our half width value, ysk, we
have selected to encompass all of the distribution (with a select few major outliers excluded).
These values are a bit conservative but they are still a valid representation of the data and the
implementation of the robust solution is significantly easier.
We can generate a P histogram by simulating random draws with replacement from the historical repair time histogram and calculating the cumulative relative error from the average repair
time as given below
z
p
Yik
7Yk
-.
-Yj k
Running 50 simulations with a scenario that contains 600 outages, using random draws from
our historical data with replacement produces the following result
Imputed
r Simulation
Histogram
3.5
90th percentile
3.0C
r=
15.81
2.5
E
2.02
z
1.5
1.0.
0.5
0.01
12
13
14
15
Imputed
Figure 3-5:
r Value
16
17
18
F Simulation Histogram
F parameter such that it encompasses a significant portion of this histogram
will ensure that our solution will remain valid with very high probability. A 90th percentile F
Selecting a final
= 15.81 and is sufficiently robust to meet the operational situation that Atlantic Electric faces.
While even large deviations (both in number of outages and simulations) yield relatively similar
F, values it is recommended that the simulation be re-run with each storm.
43
While the nominal 7jk value is given by 166.67 in our scenario, we still apply a weighted
distance to that value when applying it to each of the platforms. Even though the repair time
will be the same the added time helps to account for variables such as travel. This weighting
can be tuned by the user who can make estimates based upon weather conditions, logistical
constraints, and other factors that will increase the time to until a device is repaired that are
independent of the repairs themselves.
3.5.4
Comparison of Optimization Results
We can compare our model's results with actual Atlantic Electric operations using data from
Hurricane Irene. Figure 3-6 shows the number of crews stationed at each platform for three
scenarios: actual deployment, base model results, and increased robustness.
The increased
robustness scenario is a scenario in which I is higher than the previously calculated value to
further ensure feasibility of the solution.
70
60
50
Number of
Crews
--
----
- Grid
Model
U Increased r
UNational
40
URobust
30
-
20
10
0
/////Q
, #k#Q
Z
04
Figure 3-6: Comparison of Optimization Results
We can see that Atlantic Electric staged the vast majority of their crews at Platforms H and
Q. This practice
is common as those platforms are the largest and Atlantic Electric often waits
until damage assessment processes have been completed before stationing crews. Because our
outage prediction model can give us an advanced look of where damage will likely occur, we
can use our model to station our crews optimally. In this case Atlantic Electric would have
44
benefited from a more even distribution of their crews across their staging locations.
3.6
Conclusion
The work presented in this chapter provides Atlantic Electric with clear guidance for stationing
their repair crews despite the fact that there are significant unknowns in this process. The
model presented accounts for data error and ensures that a sufficiently optimal result is reached
to ensure an appropriate state wide completion time.
Some factors were not considered in the model that may affect Atlantic Electric's decisions.
For example, Atlantic Electric may want to stage more crews in a specific location not because
it has more outages, but because it has more customers or more important customers (schools,
hospitals, etc). Given that these factors are not present in the model it is unlikely that Atlantic
Electric will follow the model results precisely, however it will still be able to provide concrete
intuition as to how they should generally station their crews to best serve their customers.
45
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46
Chapter 4
Conclusions and Future Work
4.1
Improvements to Prediction Model
While the work presented here will aid Atlantic Electric in significantly improving their storm
response operations, there are several improvements to these models that can still add significant
value to Atlantic Electric.
4.1.1
Bootstrapping
As mentioned in Section 2.1.3 we have very few positives samples to train the model. It would
be possible to aid the prediction model by bootstrapping the positive samples to match the
number of negative samples. Bootstrapping is the process of sampling with replacement from
the existing positive sample set to create a new data set equal in size to the number of negative
samples. This aids the model in capturing the variance of the processes we are trying to model
and evenly weights the positive and negative samples.
An even weighting of positive and negative samples would help ensure that the model is not over
fitting to non-outages. It is possible that bootstrapping could lead to lower tree diversity [18]
and reduce our model specificity. However given the imbalance between positive and negative
samples and the current level of accuracy in specificity, it is likely that a beneficial trade-off
exists by using bootstrapping.
4.1.2
Parallelization
Currently the model is completely serial and computation time grows directly with the number
of storms being applied. As Atlantic Electric continues to gather more data from future storms
the model will take longer to train. Decision trees in themselves cannot be parallelized but other
47
models such as random forests can be easily parallelized. Because of this inherent parallelization
Atlantic Electric can utilize larger computing machines on AWS with multiple cores to train
the prediction model faster. It is possible that a random forest might not train as well as the
original model but with sufficient data and enough trials it is likely that a random forest model
will yield sufficiently optimal results.
4.1.3
Principal Component Analysis on Weather Information
Reducing the number of features in the model improves generality and prevents over fitting.
While we removed features that would obviously not impact outages (i.e. indoor temperature
in the weather logs) and features that had extremely low levels of variance, it is possible that
feature reduction could be implemented further using principal component analysis (PCA)
on the available weather features and/or asset features. It is possible that using PCA could
identify features, or combinations of features, that have little or no effect on producing outages.
Removing such features would not only improve model generality but also create the opportunity
to improve the model further. Because more degrees of freedom would be available, new feature
combinations can be experimented with to improve the model's efficacy.
4.2
4.2.1
Improvements to Optimization Model
Data Inaccuracies
Atlantic Electric currently lacks a data set that can capture the operational reality of repairs
during a storm situation. Even their restoration data is laced with variables that are difficult to
quantify because currently the only well documented aspect of a storm is what devices opened
and when they were closed. Because we do not know exactly when crews began work on a
particular damage point and when they finished (re-energizing a circuit does not necessarily
occur when the damage is repaired), we cannot extrapolate precise repair times. A richer data
set that details the repair times of outages is critical to building an optimization formulation
that can produce an optimal solution without the requirement of remaining robust.
4.2.2
In Storm Planning
Once a storm has passed, Atlantic Electric crews scout assets and do damage assessment of
the storm. Currently these processes are all done on paper and all damaging events are not
documented. Digitizing this information would allow the optimization model to be re-run after
the storm has hit using actual damage data to continue to help in crew deployment and re-
48
deployment. Atlantic Electric is currently in development of a new damage assessment system
that will use tablets and cell phones for damage assessment data entry. With the information
in a digital format it can be easily utilized in the optimization model and re-run after the storm
has passed. Because Atlantic Electric continually repositions crews throughout the storm the
optimization model can be continually re-run with updated damage information as the storm
response progresses to aid in operational decisions.
4.2.3
Customer Constraints
Atlantic Electric often places crews to ensure that they are repairing outages that reach larger
numbers of customers first, or their more important customers (schools, hospitals, etc). These
constraints are not currently added into the model largely because of the amount of time it
would take to quantify these constraints effectively.
Adding constraints such as these will
ensure that the model is more closely aligned with Atlantic Electric's incentives with regards
to their customers.
4.3
Conclusions
Traditionally if a firm provided a poor service or product, customers would simply move to another provider. While this is not the case in utilities, their revenues are directly tied to customer
service quality as determined by their government regulators. Additionally weather causes significant damage to their assets that also require costly repairs. The damage from weather and
the subsequent power outages are unavoidable, but utilities such as Atlantic Electric must rely
on proper planning and operational execution after the event to ensure that assets are repaired
in a quick, cost-effective manner.
The models outlined in this paper provide a data driven approach to improve Atlantic Electric's
ability to analyze incoming weather patterns and properly create a response plan. As regulator
demands are increasing, response plans that are built on data driven models present a strong
case to regulators to justify the operational response of utilities. These models also allow for
a more seamless transfer of knowledge, which is crucial given that utilities currently require
subject matter experts to make educated guesses for storm response planning. The knowledge
transfer of these subject matter experts is a long process that requires new employees to obtain
years of experience. The models presented here still require knowledge from critical employees
but do not require years of training and experience to run. Furthermore, as higher quality data
becomes available, the model efficacy will continue to improve. As Atlantic Electric continues to
49
encounter damaging weather events, the model will continue to develop new understanding on
weather and asset feature interactions that lead to outages, which will aid in Atlantic Electric's
ability to repair any resulting damage and return service to its customers.
50
Appendix A
In Sample and Out of Sample Prediction
Results
The following tables show the model's predicted outage results and observed outages for each
storm. There are four sets of model results for each storm created by combinations of using
logs and forecasts and in sample and out of sample testing. In sample tests train on all storms
including the storm being tested. Out of sample tests train on all storms except for the storm
being tested. This is a better indicator of how well the model will do when in operation as a
storm's precise weather information will not be available to assist in training the model.
51
1
14
1
Forecasted In Sample
Forecasted Out Of Sample
Actual
78
1
20
Forecasted Out Of Sample
Actual
20
70
Forecasted In Sample
Logs In Sample
70
Logs In Sample
Logs Out of Sample
37
Logs Out of Sample
23
2
1
2
41
31
42
36
79
43
25
47
47
157
46
67
30
1
8
27
17
25
24
124
48
10
55
56
40
235
127
235
203
94
11
14
12
13
68
156
50
155
151
51
419
38
366
109
22
3
41
2
2
80
31
27
30
32
61
23
61
22
41
24
124
124
123
310
0
2
218
2
0
11
104
181
99
92
119
0 4.
2
22
2
2
24
00
2
197
2
1
69
8
3
12
12
12
2
18
2
2
52
1
85
1
0
23
2
6
27
5
4
39
178
31
178
169
63
4.C-Y
166
1
0
56
Table A.2: December 2010 Winter Storm Results
425
115
37
104
102
128
00
289
85
75
87
55
77
C)
0:Z
Table A.1: October 2011 Winter Storm Results
3
33
3
0
1
121
26
113
11
3
7
106
5
3
17
202
91
196
133
91
0
.
110
11
34
11
12
86
4.
385
38
381
380
55
29
42
2
36
2
2
15
60
45
Q
84
7
27
9
9
87
298
160
294
260
378
2214
375
314
749
2551
973
2447
1851
1358
81
22
23
59
25
Logs In Sample
Forecasted In Sample
Forecasted Out Of Sample
Actual
57
89
58
15
67
51
Actual
Logs Out of Sample
53
41
Forecasted Out Of Sample
/
58
45
Forecasted In Sample
227
96
223
199
255
55
13
41
Logs In Sample
14
37
Logs Out of Sample
/
0
49
82
51
41
230
128
43
126
94
94
80
52
80
69
35
122
27
92
96
86
80
17
100
75
6
245
77
47
150
73
145
35
59
125
5
244
387
15
320
247
33
33
32
18
29
64
50
32
52
63
61
52
38
54
68
78
40
78
77
35
/
81
36
45
60
43
32
81
79
142
196
0
39
44
40
41
59
60
39
59
59
160
Table A.4: Hurricane Irene
81
79
84
84
75
Table A.3: Hurricane Sandy
14
52
70
23
35
11
11
72
32
22
43
23
18
32
32
10
20
57
5
15
8
88
58
58
58
53
120
33
5
0
2
63
89
80
44
20
24
18
3
5
60
88
86
63
91
109
110
109
76
89
49
79
109
106
66
107
91
75
0
84
88
83
80
156
332
35
21
32
17
20
172
0
141
165
141
137
77
1225
1598
87
95
1528
1289
83
89
2839
118
1323
1109
1287
991
931
102
22
96
Actual
75
Actual
Forecasted Out Of Sample
92
Forecasted Out Of Sample
91
72
Forecasted In Sample
Forecasted In Sample
72
Logs In Sample
Logs In Sample
103
Logs Out of Sample
189
26
230
57
26
111
26
11
35
46
38
44
164
2
60
2
2
45
7
115
7
7
73
32
59
25
3
19
12
77
12
12
43
k
4
45
5
4
18
~
23
120
23
20
32
22
98
24
23
44
16
82
16
16
48
12
53
9
1
40
5
27
5
5
32
20
33
31
19
15
15
1
0
14
21
15
13
3
12
3
3
259
32
317
220
1
8
4
1
91
22
94
67
19
32
16
8
Table A.6: December 2008 Snow Storm
102
102
105
83
121
0
Table A.5: February 2010 Snow Storm
32
8
26
8
15
61
9
8
50
81
30
100
82
62
140
58
55
93
0
3
19
0
42
29
20
0
20
32
0-
0
158
22
175
59
0;
10
136
10
5
26
0
42
197
44
43
110
212
44
223
210
0.
0
2
12
3
1
3
6
3
1
32
0
42
296
17
311
262
9
209
10
10
1595
563
1738
1157
494
1825
479
376
977
Appendix B
Complete Maximum Likelihood Formulation
This appendix is a complete description of the model formulation using maximum likelihood
techniques to solve the outage prediction problem as developed in [25].
Failure events happen independently across the network and at a rate proportional to their
length, and linearly dependent in the the surrounding weather features wt:
Ast = lsg*,Wt
Where
i is the length of the segment s and g,, is the vector of vulnerabilities for segment type
cS (c. being the type segment s). This makes no assumption of weather features and allows for
certain segment types to be vulnerable to damage under different weather conditions.
Because of data granularity Monsch aggregated damaging events at the asset level. Because of
the independence assumption we can represent this damage at each asset a with the following:
1,g*Cc Wt
Aa,t =
sEa
Z la,cgc wt
C
=
where
l*gwt
la is the vector of lengths of each segment type in asset a, and g is the matrix consisting of
the vectors ge assembled column-wise. The previous equation is linear in terms of the coefficients
of the matrix g. By rearranging the terms, we can therefore rewrite in a more common way as
55
a product of vectors:
AaytXa,t
where Xa,t is the vector representation of the matrix latw* which contains features for an asset
under specific weather conditions. We will use i to represent the (a, t) pairs from now on. By
definition of a Poisson process, the total number Y of events on each asset is a Poisson random
variable with parameter:
Yi ~ P(Y*xi)
From here we also get the probability distribution of a failure happening on a given asset. Let
Zi be the indicator variable corresponding to at least one event having happened on an asset:
Zi = 1 if and only if at least one event occurred on one of its segments, i.e. Zi = min(Yi, 1).
Therefore Zi is distributed according to the following exponential distribution:
Zi ~ E(1 - ey*xi)
Now when estimating y there are two cases to consider. First the case where Y is observable
(i.e. we have access to the underlying event data), then the case where only Zi. The next
sections show that the maximum likelihood estimator for
y can be efficiently estimated in both
cases.
B.1
Without Censoring
In this case, we are able to recover the exact count of events that occurred on an asset. The
corresponding likelihood function is therefor given by:
L(7) = P(Vi, Y = yjIy)
As is commonly done in literature, we will focus on the log-likelihood loss function (where K is
a constant that does not affect the optimal choice parameters):
L(-y)
-InL(y) + K
=[-*xi - yiln(y*xi)]
We then have the following result:
Theorem 1. The following maximum likelihood problem is convex:
56
max L()
-y
Proof. Convexity is immediate when observing that the Hessian matrix of the likelihood function
can be written as follows:
y Xkil
H=
(k,l)
= (Dx) * (Dx)
.
D is a diagonal matrix with entries Di,i =
B.2
H is therefore a semi definite matrix.
With Censoring
Because of the available data Monch focused on the following likelihood function instead:
L(y) = P(Vi, Zi = zi y)
And the following log-likelihood function:
L(y) =-InL(y) + K
>3*Xi -
=
zi=o
1: ln(1 - e--<* i)
zi=1
A similar result is found in this setting:
Theorem 2. The following maximum likelihood problem is convex:
maxL(-y)
Proof. Once again, the Hessian matrix is a positive semi-definite as can be seen in the following
expression:
H=
~Zi
(>z
i,kii,l
(ey*xi
_ 1)2 (k,1)
x~,x~,l
= (Dx) * (Dx)
where D is a diagonal matrix with entries Di,i =
57
.
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58
Appendix C
Complete Variable Notation
C.1
Outage Prediction Variable Notation
Table C.1: Outage Prediction Variable Notation
Variable
Description
Variable
Description
k
The number of segment clusters selected
to describe Atlantic Electric assets
di
Device i
1m
Length of segment of type m (k total)
ai
Asset indexed by i, linked with device
Wf.
Value for weather feature
Oa
Outage at asset a
A,,t
Probability of damage at segment s on
storm t
Is
Length of segment s
Vector of vulnerability values for each
type of segment
Wt
Weather for storm
Aa,t
C.2
f
at asset a
t
Probability of outage at asset a on storm t
Crew Optimization Variable Notation
Table C.2: Crew Optimization Variable Notation
Variable
Description
Variable
Description
Xijk
Crew i assigned to job j at platform k
Xik
Crew i assigned to platform k
10A
Time required to do job
AIAk
Crew capacity for platform k
Z
Worst repair time in master formulation
Xk
Job
CA
Fractional workload at platform k
C*
Number of crews assigned to platform k
Worst fractional workload simplified
C*
Total number of crews available
j
from platform k
formulations
59
j
assigned to platform k
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60
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