holekamp,mark final thesis - princeton...
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Keeping the Lights On:
An Analysis of the Dynamic Allocation Problem of
Assigning Utility Repair Trucks to Outages
Mark Holekamp
Advisor: Warren B. Powell
Submitted in partial fulfillment
of the requirements for the degree of
Bachelor of Science in Engineering
Department of Operations Research and Financial Engineering
Princeton University
June 2013
I hereby declare that I am the sole author of this thesis.
I authorize Princeton University to lend this thesis to other institutions or
individuals for the purpose of scholarly research.
/s/ Mark Holekamp
Mark Holekamp
I further authorize Princeton University to reproduce this thesis by photocopying
or by other means, in total or in part, at the request of other institutions or
individuals for the purpose of scholarly research.
/s/ Mark Holekamp
Mark Holekamp
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Acknowledgements
I would first like to thank my thesis advisor Warren Powell for introducing me to
this project and for his guidance throughout. I appreciated the opportunity to
work on a project that could be applied to real world issues with the chance of
helping improve New Jersey and the Princeton community.
I would also like to thank Belgacem Bouzaiene-‐‑Ayari for his extensive help with
the programming in this thesis. Your patience with me as I struggled to
familiarize myself with Java was appreciated, and your hard work in translating
PSE&G’s data into a functional simulated grid made this thesis possible.
I’d like to thank my friends here at Princeton for making the past four years such
an unforgettable journey. My experiences in club lacrosse, club squash, and the
ever-‐‑welcoming Cloister community will stay with me forever. I’d especially like
to thank Bee Keeler for making my time here so special.
Lastly I would like to thank my family for their unwavering support and love. I
am very thankful to my parents for giving me the opportunity to attend such an
amazing University.
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Abstract
For electrical providers like PSE&G, the main goal of the company is to
provide consistent and reliable services to their customers. However, these
companies are often challenged by an overwhelming number of outages
following a storm, and restoring power to their customers as quickly and
efficiently as possible with a limited amount of repair resources is their highest
priority. This thesis aims to analyze the process of assigning utility repair crews
to outages by modeling the PSE&G electrical grid and simulating problems that
may arise following a storm. It will then test and compare various dynamic
allocation policies for assigning repair crews to outages to determine the most
effective policy for resolving potential storm-‐‑based issues in the future.
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Table of Contents
Acknowledgements ......................................................................................................... 1
Abstract .............................................................................................................................. 2
Table of Contents .............................................................................................................. 3
Chapter 1: Introduction .................................................................................................. 4 1.1 Background of PSE&G ...................................................................................................... 4 1.2 Standards of PSE&G’s Electrical Service ........................................................................ 7 1.3 Critical Customers ........................................................................................................... 10 1.4 Impact of Storms on Electrical Grid .............................................................................. 12 1.5 Assignment of Repair Crews ......................................................................................... 15 1.6 Thesis Overview .............................................................................................................. 17
Chapter 2: Dynamic Resource Allocation Model .................................................... 18 2.1 Introduction to Dynamic Resource Allocation ............................................................ 18 2.2 Cost Function and Policy Function Approximations ................................................. 20 2.3 Mathematical Model ....................................................................................................... 23 2.31 State Variable ............................................................................................................. 23 2.32 Decision Variable ...................................................................................................... 26 2.33 Exogenous Information ........................................................................................... 29 2.34 Transition Function .................................................................................................. 30 2.35 Objective Function .................................................................................................... 32
Chapter 3: Policies and Simulator .............................................................................. 34 3.1 Overview of Assignment Policies ................................................................................. 34 3.2 The Simulator ................................................................................................................... 39 3.21 Initial Truck Placement ............................................................................................ 40 3.22 Task Generation ........................................................................................................ 42 3.23 Fixed Parameters ...................................................................................................... 46
Chapter 4: Simulation Data and Analysis ................................................................ 49 4.1 Initial Policy Search ......................................................................................................... 49 4.12 Tunable Parameter Test Results ............................................................................. 50 4.2 Policy Testing and Comparison .................................................................................... 56 4.21 10 Truck Scenarios Results ...................................................................................... 56 4.22 20 Truck Scenarios Results ...................................................................................... 59 4.23 25 Truck Scenarios Results ...................................................................................... 62 4.24 Policy Conclusions ................................................................................................... 65
Chapter 5: Conclusion .................................................................................................. 69
Appendix ......................................................................................................................... 74
References ....................................................................................................................... 77
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Chapter 1: Introduction
This chapter is designed to provide an extensive background on the
problem of assigning repair crews in order to resolve any problems that occur as
a result of a storm. It will delve into not only the electrical company whose grid
is being modeled (PSE&G) but also how storms can affect an electrical grid and
why repair crews can be very important to the level of service provided to
customers. After familiarizing the reader with the background issues of this
problem, this chapter will then summarize the motivation and goals of this
thesis.
1.1 Background of PSE&G
PSE&G is one of the largest and oldest publicly owned gas and electric
companies in the United States. Initially formed in 1903 and given its current
name in 1948, PSE&G services almost three quarters of New Jersey’s population
including 2.2 million electric customers. This electrical provider covers a large
corridor of the state including New Jersey’s six largest cities (PSE&G 2014). For a
visual of PSE&G’s area of electrical services, see the following Figure 1.1 that
depicts the company’s territory in yellow:
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* Note: the map is from PSE&G’s Outage Center, and the blue triangles are
outages that were impacting the grid at the time the screenshot was taken
Figure 1-‐‑1: PSE&G Coverage Map
PSE&G’s electrical grid is part of an East Coast power infrastructure that
ranks among the oldest in the nation (Lavelle 2012). With over 22,225 miles of
overhead electricity wire and around 8,000 miles of underground wire, PSE&G
has almost three fourths of its electrical grid above ground (Marques 2014). This
(PSEG Outage Center)
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fact combined with the age of the infrastructure makes PSE&G’s electrical grid
vulnerable to power outages given New Jersey’s generally wooded landscape.
In fact, a significant proportion of PSE&G’s electrical customer base
experiences power outages at least once a year. Of its 2.2 million customers,
PSE&G has had an average of 1,568,992 customers experience blackouts annually
over the past ten years with a low of 1,339,468 in 2004 and a high of 1,840,608 in
2010 (Marques). While these numbers are not unusual for the electrical utility
industry, the high proportion of customers experiencing outages each year
makes electrical repair services an important area for PSE&G. The following table
provides a breakdown of PSE&G’s customer base by county:
County Customers Served
Camden 161,298
Somerset 107,769
Union 201,364
Bergen 339,387
Burlington 174,138
Essex 347,035
Gloucester 34,497
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Hudson 307,623
Hunterdon 56
Mercer 148,358
Middlesex 250,544
Monmouth 1,663
Morris 99
Passaic 173,150
Total 2,246,981
Table 1-‐‑1: Customer Numbers by County
1.2 Standards of PSE&G’s Electrical Service
PSE&G measures the reliability of its services using a set of indices
defined in the Institute of Electrical and Electronic Engineers (IEEE) Standard
1366. This standard, most recently revised in 2012, is a guide designed to identify
factors that affect service reliability and aid in consistent measurement and
reporting of service reliability in the utilities industry. Provided in the IEEE
Xplore digital library, Standard 1366 defines four statistical indices that are
currently used by PSE&G to measure its reliability.
(PSEG Outage Center)
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Basic variables and terms:
CN = Number of distinct customers that experience interruptions in an
area
IMi = Number of momentary interruptions
Momentary Interruption: A loss of electrical services less than five
minutes in duration
Ni = Number of customers affected by sustained interruption event i
Nmi = Number of customers affected by momentary interruption event i
NT = Total number of customers served in an area
ri = Restoration time for interruption event i
Sustained Interruption: A loss of electrical services for longer than five
minutes
𝑆𝐴𝐼𝐷𝐼 = Σ Customer Minutes of Interruption𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑆𝑒𝑟𝑣𝑒𝑑 =
Σ 𝑟!𝑁! 𝑁!
SAIDI, also called the System Average Interruption Duration Index, effectively
measures the total duration of sustained electrical service interruption for the
average customer in an area during a defined period of time. The unit of this
measurement is typically minutes or hours of interruption.
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𝐶𝐴𝐼𝐷𝐼 = Σ 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑀𝑖𝑛𝑢𝑡𝑒𝑠 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛
𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝐼𝑛𝑡𝑒𝑟𝑟𝑢𝑝𝑡𝑒𝑑 = Σ 𝑟!𝑁!Σ 𝑁!
CAIDI, known as the Customer Average Interruption Duration Index, calculates
the average time required to restore electrical services to the average customer
affected by a sustained interruption in an area. This index is also normally
measured in minutes or hours of interruption.
𝐶𝐴𝐼𝐹𝐼 = Σ 𝑇𝑜𝑡𝑎𝑙 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑆𝑢𝑠𝑡𝑎𝑖𝑛𝑒𝑑 𝐼𝑛𝑡𝑒𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛𝑠
𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐷𝑖𝑠𝑡𝑖𝑛𝑐𝑡 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝐼𝑛𝑡𝑒𝑟𝑟𝑢𝑝𝑡𝑒𝑑 = Σ 𝑁!𝐶𝑁
CAIFI, the Customer Average Interruption Frequency Index, measures the
average frequency of sustained interruptions for the customers in an area that are
experiencing such interruptions. The unit of this index is interruptions per
interrupted customer.
𝑀𝐴𝐼𝐹𝐼 = Σ 𝑇𝑜𝑡𝑎𝑙 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑀𝑜𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝐼𝑛𝑡𝑒𝑟𝑟𝑢𝑝𝑡𝑖𝑜𝑛𝑠
𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑆𝑒𝑟𝑣𝑒𝑑 = Σ 𝐼𝑀! ∗ 𝑁!"
𝑁!
MAIFI, also called the Momentary Average Interruption Frequency Index,
calculates the average frequency of momentary interruptions in an area. It is
typically measured in interruptions per customer.
PSE&G consistently measures the statistical factors that make up these four
indices in order to track the reliability of its services from year to year. These
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indices allow the company to calculate the duration of sustained outages for the
average customer (SAIDI), the average time required to restore electricity to
affected customers (CAIDI), the frequency of sustained interruptions for
customers experiencing such outages (CAIFI), and the frequency of momentary
interruptions for the average customer (MAIFI). Since these indices measure the
extent to which customers are affected by momentary and sustained electrical
service interruptions, it is PSE&G’s goal to minimize these indices in order to
provide the most reliable service to its customers.
1.3 Critical Customers
Critical customers such as hospitals, utilities, police stations, etc. are
especially important to PSE&G because it is vital that they remain running and
operational for the services they provide to society. As electricity is the basis for
being operational, consistent electrical service is paramount for these customers.
This section will discuss how PSE&G values critical customers and their
heterogeneous attributes and electrical requirements.
Providing consistent electrical service to certain customers is especially
important for PSE&G. These critical customers can be significantly impacted by
prolonged power outages; when outages do occur, restoring power to them as
quickly as possible is of utmost importance because the length of the blackout
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may be very impactful not only to the critical customers themselves but also to
the communities they serve.
Hospitals are one example of a critical electrical customer because they are
constantly in need of electricity. Power is needed around the clock to provide
lighting for surgeries and to run medical equipment such as diagnostic machines
or life support. Without electricity, a hospital would be unable to perform its
critical life-‐‑saving functions, and as a result the hospital and its patients would
suffer from prolonged power outages. Although most hospitals do in fact have
backup generators that come online automatically in the case of a loss of power,
these are typically diesel generators with a limited fuel supply. Most hospitals
would be able to operate for ten or twelve hours on electricity from these
generators, but in cases much longer than this the hospital will run out of fuel
and be helpless without power. Thus, it is essential that PSE&G utility repair
crews respond to and resolve grid problems affecting hospitals as quickly as
possible.
Utility buildings and certain residential customers are also entities that
can be especially sensitive to lengthy blackouts. Water plants depend on
electrical services to consistently provide its products to its area. Although these
buildings often have backup generators as well, again these generators are
usually diesel-‐‑fuelled and thus limited in their time of operation. Elderly,
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disabled, or sick residential customers are another example of critical customers
for electrical providers. These customers are frequently immobile and susceptible
to the negative effects of blackouts, especially during periods of extreme
temperature during the peak of winter or summer. Without power for a
prolonged period of time during such weather, their homes can become
dangerously hot or cold without electricity to run their heating or air
conditioning. These residents would be unable to leave their homes and could
face significant health risks if power is not restored in a timely manner.
Because of critical customers, power outages are not always homogeneous
in their time dependency. Problems affecting certain areas of PSE&G’s electrical
grid need to be resolved especially quickly depending of the composition of the
customers in these locales. Critical customers make quick and efficient handling
of power outages by utility repair crews even more important for PSE&G.
1.4 Impact of Storms on Electrical Grid
Storms are a large and continual source of concern for electrical providers
like PSE&G. Not only are they capable of causing direct physical damage to
electrical grids by themselves but also the localized high winds often generated
by storms can bring down trees. Downed trees are the main problem for many
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electrical companies as a large majority of transmission wires are strung above
ground in close proximity to
trees.
A single tree falling on a
segment of wires as a result of a
storm can cause a power outage
for a significant number of
customers depending on where
in the line the breakage occurs.
This event is both impossible to predict and fairly common even during a typical
storm. Because of this, storms are the biggest challenge in providing quality,
consistent electrical services to customers because they are the most frequent
sources of blackouts.
Hurricane Sandy in 2012 was a perfect example of the damage storms can
wreak on electrical grids. Sandy, the second most costly storm in the history of
the United States with damages estimated to be upwards of $50 billion, hit New
Jersey particularly hard. Total business losses in New Jersey amounted to around
$8.3 billion, a significant chunk of the state’s GDP (Huffington Post 2013). This
staggering amount of physical damage caused by Hurricane Sandy was not the
only cost of the storm, however. Loss of electrical power, a critical part of
An example of storm damage (Powell 2014)
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modern life, crippled not only businesses but many New Jersey communities as
well. Businesses without generators, especially smaller ones, were without
power for a substantial period of time due to Sandy and were unable to operate
without electricity. This resulted in further economic losses for New Jersey in
addition to the physical damages since these businesses were essentially closed
as long as they did not have power.
PSE&G itself was one of the main victims of this costly storm, as
Hurricane Sandy rolled through much of its coverage area including Newark,
one of the most populous areas in which PSE&G provides electrical services. 1.7
million PSE&G customers, many in Newark, were without power shortly
following the storm due to the widespread and extensive damage to the grid
(Swetha 2012). PSE&G was able to restore power to large sections of Newark
within 24 hours, including the critical area of Newark Airport (Caroom 2012).
However, over 750,000 PSE&G customers were still without power three days
after the storm hit, many of these outages due to fallen trees and downed lines
(Gopinath 2012). Some customers experienced blackouts lasting almost two
weeks, an enormous amount of time to be completely devoid of power.
Although Hurricane Sandy was not a typical storm in terms of its power
and scope, it demonstrated on a large scale the difficulties that power companies
like PSE&G face immediately following a storm. Repairing the damage done to
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the electrical grid is time consuming, and the time it takes to bring the grid back
to 100% functionality is very noticeable to the customers who are experiencing
blackouts. Because of this, PSE&G is under extreme pressure to quickly restore
power to every single one of its customers following a storm. Thus, minimizing
the amount of time it takes to restore power to its customers after a storm is of
utmost importance to PSE&G.
1.5 Assignment of Repair Crews
Clark Gellings, a fellow at the Electric Power Research Institute, points out
that “it is virtually impossible to protect the system from a storm like Sandy . . .
can we do a better job at putting it all back together?” (qtd. in Lavelle 2012).
Storms will always be an issue for electrical providers like PSE&G since the
systemic damage they cause is simply unavoidable. Although there are some
strategies for minimizing potential storm damage such as burying wires
underground, issues will simply always arise in the electrical grid following
storms. Thus, addressing the various problems that cause blackouts in the grid
through the assignment of utility repair crews is an important aspect of electrical
companies like PSE&G in providing quality service to their customers.
In most cases of outages, PSE&G uses its own workforce of utility repair
crews to resolve the problem. Additionally, the company has a number of “first
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call” contracts with local contractors throughout New Jersey. These contracts
essentially give PSE&G first priority in the event that the services of the local
contractors are needed. First call contracts are generally used when company
utility crews are insufficient and PSE&G needs additional repair resources in a
certain area.
In more extreme cases when PSE&G is completely overwhelmed by grid
problems, the company can make use of its assistance group. PSE&G is currently
a member of the North Atlantic Mutual Assistance Group (NAMAG), a
collection of 21 utility companies. NAMAG is a Regional Mutual Assistance
Group (RMAG) that follows guidelines dictated by the Edison Electric Institutes
(EII)—members of the group agree to assist one another in terms of utility repair
resources. NAMAG’s membership includes companies from Baltimore at its
southern most reaches to Nova Scotia at its northern point and companies as far
west as Ohio. In the event of a national event like Hurricane Sandy, the EII will
supervise and coordinate utility crew distribution to the NAMAG members as
needed. While NAMAG offers a large source of outside utility repair resources,
PSE&G typically appeals to its mutual assistance group peers for support only
after the rare major storm (Marques 2014).
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1.6 Thesis Overview
PSE&G faces the constant challenge of responding to outages and
resolving them as quickly as possible. Its primarily aboveground, old
infrastructure and wooded terrain means that issues arise frequently in the grid,
especially after storms. A large proportion of PSE&G’s customers experience
power loss each year, and restoring power through efficient use of its utility
repair crews is of utmost importance to providing as reliable of service as
possible. Critical customers make it even more vital to address issues in certain
areas of the grid.
Since replacing its grid infrastructure with newer technology or burying
large parts of its overhead wires would require an enormous investment of
capital, optimizing its allocation of repair crews to problems as they arise would
be the easiest and most cost effective method of improving the reliability of
PSE&G’s service. This is the motivation for this thesis, as it aims to find the most
efficient policy of crew allocation by modeling PSE&G’s electrical grid and
simulating post-‐‑storm issues. In the following chapters, the development of both
the grid model and the simulator will be laid out as well as the results of the
simulations and the policy analysis.
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Chapter 2: Dynamic Resource Allocation Model
This chapter will first give an introduction to dynamic resource allocation
problems. It will then give a summary of two types of policies that can be used to
solve resource allocation problems, followed by a detailed layout of the
mathematical model behind this thesis.
2.1 Introduction to Dynamic Resource Allocation
Dynamic resource allocation is a fundamental problem that involves the
allocation of limited resources over a period of time. These resources can be
physical—trucks, raw materials, and specialized machines— or human. The
management of these resources includes dynamic information processes that can
significantly complicate the problem (Powell and Van Roy 2004). For example, a
flower delivery service must consider a myriad of information processes like
outstanding customer orders, weather, road construction, and more when
allocating their drivers and vehicles to effectively fulfill their business
commitments. Although making optimal allocation decisions can be extremely
complex, the basis of dynamic resource allocation problems can be represented
by a fairly simple visual:
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In theory, dynamic resource allocation problems can be regarded as
Markov decision processes and solved through dynamic programming
algorithms; however, solving dynamic resource allocation problems on a
practical scale in such a way is often infeasible (Powell and Van Roy 2004). In
their paper titled “Approximate Dynamic Programming for High-‐‑Dimensional
Resource Allocation Problems”, Powell and Van Roy point out that there are
three “curses of dimensionality” that typically plague realistic resource allocation
problems: the number of state variables, the number of decision variables, and
the number of random variables. As the number of these three entities grows in a
dynamic resource allocation problem, the computation time and memory
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required by classic programming algorithms grow exponentially. Thus, resource
problems on a practical scale—that is, problems that involve many variables—
cannot be feasibly solved by typical dynamic programming algorithms.
2.2 Cost Function and Policy Function Approximations
Instead of using dynamic programming algorithms, resource allocation
problems can be approached as a sequential stochastic optimization problem. In
the paper by Powell et al. “Approximate Dynamic Programming in
Transportation and Logistics: A Unified Framework”, two classes of policies are
presented that can be used to solve sequential stochastic optimization problems.
These two classes of policies, cost function approximations and policy function
approximations, have distinct characteristics but share the same ability to
address resource allocation problems that would otherwise be infeasible with the
aforementioned dynamic programming algorithms. Cost function
approximations and policy function approximations avoid the exponentially
increasing time and data requirements that would otherwise hamper large scale
problems through their simple, defined decision making processes. Although in-‐‑
depth discussion of the specific policies used in this thesis will occur later in
Chapter 3, the following will give a summary of the basic structure and
characteristics of these two classes of policies.
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Cost function approximations, also called myopic policies, are a class of
policies that optimize costs/rewards at the current time step and do not consider
the future or future decisions. These policies also commonly include tunable
parameters that can be changed to improve the performance of the policy
(Powell 2010). The basic form of cost function approximations is:
𝑋!"#(𝑆!) = 𝑎𝑟𝑔𝑚𝑖𝑛!!∈!!𝐶(𝑆! , 𝑥!|𝜃)
Where XCFA is the cost function approximation policy, St is the state variable at
time t, xt is a decision variable at time t, θ is the tunable parameter, and C() is a
cost function (Powell et al. 2012). This type of policy essentially chooses the
decision xt that minimizes the cost function given the current state and the
tunable parameter. If C() is a reward function, then the policy would choose a
decision xt that maximizes the function given the state and tunable parameter.
Cost function approximations are a fairly simple type of policy, but they are
often used in resource allocation problems because they can handle high-‐‑
dimensional problems. Since they ignore the future and the future effects of a
decision in the current time period, these policies can be solved quickly with a
basic linear program (Powell 2010). A simple example of a cost function
approximation policy would be a policy that assigns salespeople to customers in
order to maximize a revenue function given a current state and tunable
parameter.
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Policy function approximations are a class of policies that choose a
decision given a state, without any form of internal optimization or consideration
of the future (Powell 2010). These types of policies are generally used when there
is a desired policy structure. The basic form of policy function approximations
can vary depending on the problem. One example from this class of policies is an
inventory reordering policy that is discussed in Powell et al.’s “Approximate
Dynamic Programming in Transportation and Logistics” paper:
𝑋!"# 𝑅! = 0 𝑖𝑓 𝑅! ≥ 𝑞𝑄 − 𝑅! 𝑖𝑓 𝑅! < 𝑞
Where XPFA is the policy function approximation, Rt is the inventory of some
good at time t, q is some predetermined lower limit of inventory, and Q is
another preselected quantity of inventory. This policy essentially does nothing if
the inventory is above the lower limit and orders an amount Q minus the current
inventory Rt if the inventory falls below the lower limit. This policy function
approximation has an obvious structure to it and does not have any sort of
optimization like in the cost function approximations. Rather, it only depends on
the state, which in the example above is the inventory of some good at time t.
However, policy function approximations are similar to cost function
approximations in that they require little computational time or data, making
them applicable for complicated problems like resource allocation that would
otherwise be overwhelming.
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2.3 Mathematical Model
The aim of this thesis is to determine the optimal policy for assigning
electric repair crews to problems following a storm by modeling PSE&G’s
electrical grid and comparing assignment policies through simulation. However,
before this problem can be solved, it must first be carefully modeled
mathematically. Sequential stochastic decision problems can be broken down
into five components: state variable, decision variable, exogenous information,
transition functions, and objective function. The following sections will lay out
these five components of this thesis’ mathematical model.
2.31 State Variable
Although there are a number of definitions for the state variable in the
academic community, in this case it will be defined as the minimally
dimensioned function of history that is necessary and sufficient to compute the
decision function, the transition function, and the contribution function (Powell 1
Oct. 2013). In other words, the state variable is the information you need to make
a decision, calculate its impact, and move on to the next time step and nothing
more. The information required to do these steps includes not only static
information but also dynamic information that changes as time progresses.
However, only dynamic information is typically included in the state variable
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while static information is stated separately to keep the model and the state
variable succinct. For this thesis the state variable will be referred to as St, and it
is defined in this model as follows:
𝑆! = 𝛤!,𝛶!
Where
Γt = the list of available repair trucks at time t
Υt = the list of unassigned or uncompleted tasks (outages) at time t
Each of these elements of the state variable is a dynamic set of information whose
individual elements have multiple attributes. The truck and task elements will be
detailed below.
The repair truck element Γt includes all trucks that are available for
assignment in the current time step t. In other words, all trucks included in this
list are not working on a task at time t. A given truck in this list, Tri, has a set of
attributes that provide important information about the truck at time t.
Γ! = (Tr!,Tr!,… ,Tr!,… )
Where
Tri = (t, Lt,i, IDi)
t = the current time
Lt,i = the current location of the truck
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IDi = the unique identification tag for the particular truck
The current time t and the current location Lt,i are dynamic attributes while IDi is
static and does not change, but it is necessary to keep the identification
information because the trucks need to be able to be recognized as individuals.
The task element Υt includes all tasks that are awaiting service at the
current time step t and have not yet been assigned a truck. A given task in this
list, Tsi, has various attributes:
Υ! = (Ts!,Ts!,… ,Ts!,… )
Where
Ts! = (t, L!,!, ID,T!"!#$,!,D!,P!,N!)
t = the current time
Lt,i = the current location of the task
IDi = the identification tag for the particular task
Tevent,i = time at which the task was learned about
Di = the duration of the task
Pi = priority of the task
Ni = the number of customers affected by the task
26
Of the above attributes of the task, only the current time is dynamic information.
All of the other attributes are static and do not change as time progresses, but
they are very important in the decision making process.
2.32 Decision Variable
At the simplest level, the decision being made in this thesis’ mathematical
model during a given time step is to what unassigned task is an available truck
being allocated. Since there can be multiple repair resources available at the same
time, in some cases multiple decisions are made in a single time step. While the
decision itself is fairly simple, the decision variable xt is in fact a multifaceted
variable with several elements. Each of these elements is an important piece of
information:
x! = (T!"#$",T!"#, L!"#$#%, L!"#$, ID!"#$%, ID!"#$,C)
Where
Tstart = the start time of the decision (essentially the current time)
Tend = the end time of the decision, i.e. when the task will be completed
Lorigin = the current location of the truck being assigned
Ldest = the current location of the task being resolved
IDtruck = the identification tag of the truck being assigned
IDtask = the identification tag of the task being resolved
27
C = the contribution of the decision
All of the above elements are in fact static information as none of them change as
time progresses from the time step that this decision is made, which is time t in
this case. Both Tend and C are slightly more complicated than the other elements,
as they involved some calculation using other bits of information. The
calculations that define these elements are detailed below:
C = (Ntask, Dtotal)
Where
Ntask = the number of customers affected by the task being resolved by the
decision
Dtotal = Tend – Tevent, task
Tevent, task = the time at which the task being resolved was learned about
The contribution of a decision effectively holds two pieces of information—the
number of customers being affected by the task being resolved by this decision
and the total time it took to resolve the problem since it was first learned about.
These two elements are important because they are part of the objective function,
discussed later in section 2.35. The calculation of Tend, the end time of the
decision, is a little more complicated:
28
Tend = t + Ttravel + Di
Where
Di = duration of the task being resolved, in this case task i
Ttravel = d/s
d = distance between Lorigin and Ldest
s = speed at which a truck travels
The speed at which the truck travels is a fixed parameter in the simulation, and it
will be discussed further in the next chapter. The distance between Lorigin and
Ldest, essentially the distance between the truck being assigned and its task at the
time the decision is being made, is calculated using the haversine formula to
determine the great-‐‑circle distance between two points. In other words, this
distance is the shortest distance over the surface of the earth between the two
points (Veness 2014). It is calculated using the following equation:
d = R ∗ 2 ∗ atan2( b, 1− b)
Where
b = sind!"#2 ∗ sin
d!"#2 + sin
d!"#2 ∗ sin
d!"#2 ∗ cos lat!!"#$#% ∗ cos lat!!"#$
R = Earth’s radius in miles = 3961 miles (Rosenberg 2012)
d!"# = lat!!"!! − lat!!"#$#%
d!"# = lon!!"#$ − lon!!"#$#%
29
lat!!"#$ = latitude of the location L!"#$
lat!!"#$#% = latitude of the location L!"#$#%
lon!!"#$ = longitude of the location L!"#$
lon!!"#$ = longitude of the location L!"#$#%
2.33 Exogenous Information
Exogenous information in this model is any information that arrives at a
time period t. It is helpful to model the new information coming into the system
as a single variable, which in this case will be Wt. This variable represents a
realization of the information that arrives at time period t (Powell 1 Oct. 2013).
This variable is not random, but it contains information that is very important to
making a decision during the time step. The exogenous information in a given
time step t is detailed below:
W! = (ΔΓ!,ΔΥ!)
Where
ΔΓt = the trucks that come into the available truck list at time step t
ΔΥt = the tasks that come into the unassigned task list at time step t
The lists of available trucks and unassigned tasks, parts of the state variable, are
constantly changing as we progress through time. While the trucks and tasks
30
removed from the list as a result of decisions are not considered exogenous
information in this model (this will be covered more in the next section), the
trucks and task that are added to the list are included in Wt. Trucks reenter the
available truck list as they finish working on a task, and previously unknown
tasks are added to the list of uncompleted tasks as they are learned about. The
exogenous information coming in during time steps is extremely important
because they impact the state variable and determine which trucks and tasks can
be considered. The exact way in which exogenous information is taken into
account is discussed in the following section on transition functions.
2.34 Transition Function
The transition function is essentially a function that captures the evolution
of the system over time (Powell 1 Oct. 2013). The form of the transition function
at its most overarching level in this thesis is:
S!!! = S!(S!, x!,W!!!)
This function embodies the transition that goes on in the model when it shifts
between time step t and time step t+1. In this thesis, the difference in time
between two time steps is actually five minutes, but for the sake of simplicity this
change will be referenced as going from time step t to t+1. The function above,
31
however, is just the general form of the transition function. The transition
function actually includes additional equations that illustrate more specifically
how the state variable evolves as time changes. Suppose that decision xt,1, xt,2, … ,
xt,n are made during time step t. Also, exogenous information Wt+1=(ΔΓt, ΔΥt)
arrives at the start of time step t+1. The way in which the lists of available trucks
and uncompleted tasks in the state variable change is:
Γ!!! = Γ! − Tr!!,!
!
!!!
+ ΔΓ!!!
Υ!!! = Υ! − Ts!!,!
!
!!!
+ ΔΥ!!!
Where
Tr!!,! = the truck being assigned by decision x!,!
Ts!!,! = the task being resolved by decision x!,!
Again, ΔΓt and ΔΥt are the trucks and tasks that are becoming available as a
result of the exogenous information Wt+1. As the functions above demonstrate,
the truck and task list elements of the state variable are updated as decisions are
made in the current time step t and as exogenous information comes in as time
moves to t+1. Any trucks and tasks that are included in the decisions are
removed from the list, and any that come into availability as the exogenous
information rolls in are added to the list. Although the transitions functions seem
32
rather straightforward and obvious, these functions are key because they allow
the model to change and update as time progresses from step to step.
2.35 Objective Function
The objective function is essentially the same as the goal of this thesis: to
find the best policy of allocating repair crews to outages. The mathematical
definition of the objective function is:
min!E!{ C S!,X! S! }!
Where
C(St, Xµμ(St)) = a cost function
Xµμ(St) = the decision function based on policy π
The entire expression is a minimum with µμ as an argument because the objective
is to find the policy µμ that minimizes the expected cost over all the time periods
being examined (represented by Σt). In this thesis, the cost function is based on
the indices used by PSE&G to measure its reliability that are defined in section
1.2. Specifically, SAIDI and CAIDI will be considered by the cost function
because they track the total duration of outage time for the average customer and
the average time required to restore electrical services to the average customer
affected by outages respectively. These two statistics are directly influenced by
33
how a policy allocates repair crews, and will give a good indication of how
effective the policy is. CAIFI and MAIFI, on the other hand, are not considered in
this thesis because they measure the frequency at which customers experience
outages, which is not influenced by how well repair trucks are managed. Thus,
since SAIDI and CAIDI depend on the number of customers affected by outages
and the cumulative duration of customer outages, the cost function C takes into
account the contribution element of each decision made since it holds the
information needed to calculate these indices (see section 2.32 for contribution
element definition).
34
Chapter 3: Policies and Simulator
Chapter 3 will revolve around the simulation of the storm itself, both how
the simulation will work and its parameters. It will be designed to provide the
reader with an understanding of the simulation before the data and results of the
simulations are presented.
3.1 Overview of Assignment Policies
Two classes of policies were discussed back in section 2.2: cost function
approximations and policy function approximations. Cost function
approximations, also called myopic policies, make decisions based on
straightforward optimization of cost/rewards in the current time period. Policy
function approximations, on the other hand, revolve around clearly structured
decisions based on the state of the system at the current time step without any
optimization.
The four assignment policies being tested and compared in this thesis are
all hybrid policies. That is, each of them does not fall under a single
categorization of either cost function approximation or policy function
approximation. Rather, they all include optimizations in a given time step but are
based on a desired structure at their root. The four resource allocation policies of
35
this thesis, which will be called FIFO, Distance Exploitation, Weighted Priority
and Weighted Priority Plus Distance, are discussed in detail below.
The first policy that will be examined, FIFO, stands for First In First Out.
A common system for servicing retail customers, this assignment policy is based
on the desired structure that the customer who has been waiting the longest for
service should get served first. In terms of this thesis’ resource allocation
problem, that means that the outage that was learned about first will have first
priority for repair truck assignment. The practical application of this desired
assignment structure depends on a simple optimization in the current time step:
ID!"#$ = max!" t− T!"#$%,! for all i ∈ Υt
To translate the above expression into words, the ID of the task that is chosen for
a decision is the ID of the task in the list of unassigned tasks that has been
waiting the longest for service. This length of waiting is calculated by subtracting
the time at which the task was learned about from the current time. FIFO ensures
that customers who have been experiencing outages the longest get service first,
a policy which is generally considered to be “fair” to the customers.
Distance Exploitation also includes both a desired baseline structure and
internal optimization. The basic thinking behind this policy is that repair trucks
should be assigned to tasks that are the closest to them in order to get repair
work started as quickly as possible after the truck is available. Distance
36
Exploitation seeks to minimize the time wasted while a repair truck travels to its
outage destination. This assignment policy is referred to as “exploitation”
because it makes a decision based simply on what appears to be best at the
current time step. It identifies the task that is closest to an available truck and
picks this as the best task to address right now. Distance Exploitation is carried
out in practice by actually generating all possible decisions based on Γt and Υt
and performing the following optimization:
x! = min! d! for all i ∈ X!
Where
Xt = the list of all possible decisions
di = distance between L!"#$#%! and L!"#$!
L!"#$#%!= the current location of the truck that would be assigned in
decision i
L!"#$!= the current location of the task that would be resolved in decision i
The distance di above would be calculated using the haversine formula (see
section 2.32 for details). The decision picked by Distance Exploitation is the one
in the list of possible decisions that has the smallest distance di between the truck
and the task that would be involved with the decision.
Weighted Priority is the third hybrid assignment policy that is considered
in this thesis. The desired structure behind this policy is that relatively short
37
tasks that affect a large number of customers should be addressed first in an
attempt to minimize the cumulative duration of customer outages. Weighted
Priority also has a term that includes a tunable parameter that can be adjusted to
influence how much the waiting time of customers is considered. The
optimization involved with this assignment policy is:
ID!"#$ = max!"{N!50.5−
D!135+ 𝜃 ∗ t− T!"#$%,! } for all i ∈ Υt
The θ in the formula above is the tunable parameter, and adjusting it up or down
will impact the magnitude of influence of the waiting time of customers. This
optimization essentially picks the task in the list of unassigned tasks that has the
largest priority value calculated by the expression within the brackets. The fact
that the duration fraction is subtracted from the number of customers fraction
favors tasks that have a large number of customers affected Ni and a small
duration of repair Di. Also, since the expression involving the tunable parameter
is added, this policy favors tasks that have been waiting a long time. The
denominators of the fractions, 50 and 135 for number of customers and duration
respectively, were chosen because they are the expected mean of the two
randomly generated attributes. The number of customers affected by the task is a
random integer between 1 and 100, so the expected mean number of customers is
50.5. Likewise, the duration of a task is a random integer between 30 and 240, so
the expected mean duration is 135. By dividing the two attributes by their
38
expected mean, this policy essentially normalizes the two attributes so that they
have approximately equal impact on the priority value. Although the normalized
influences of the two attributes are not exactly the same due to differences in
their ranges of possible values, they are very close to being equal and will be
used in this way for the sake of simplicity.
Weighted Priority Plus Distance (WPPD) is very similar to Weighted
Priority, but it includes travel distance as an additional element in its attempt to
minimize the cumulative duration of customer outages. As its name implies, this
policy is simply an extension of the Weighted Priority policy both in structure
and optimization to include distance. The internal optimization run each time
step by WPPD is very similar to Weighted Priority except for a distance element
added as a fraction:
ID!"#$ = max!"{N!50.5−
D!135−
d!6 + 𝜃 ∗ t− T!"#$%,! } for all i ∈ Υt
The θ element is again a tunable parameter designed to adjust the influence of
the customer waiting time on the decision making process. The distance fraction
is divided by 6 because the mean distance that trucks must travel in the
simulated grid is around this value. By dividing by this number, the influence of
the distance attribute is approximately normalized with the number of customers
and duration elements. Note that the fact that the distance fraction is subtracted
favors tasks that have a short travel distance.
39
3.2 The Simulator
While the mathematical model defined in Chapter 2 is the fundamental
basis of looking at PSE&G’s resource allocation problem, the actual simulation of
this scenario occurs through a programmed representation of the company’s
electrical grid. Provided with a great deal of data by PSE&G, Belgacem
Bouzaiene-‐‑Ayari of Princeton’s CASTLE Lab did a majority of the work in
organizing, refining, and translating this information into a digital model of a
chunk of PSE&G’s electrical grid. This model is used as a basis for some
programming written in this thesis to actually run a simulation of outages
following a storm and the allocation of repair trucks to these problems. The
following subsections will go into detail about the programming done in this
thesis to place trucks, generate outage tasks, and set up a post-‐‑storm scenario
that can be used to test resource assignment policies. Figure 3.1 below is a
snapshot of the digital grid created by Belgacem, which includes most of
Newark, NJ. For a zoomed-‐‑in look at a section of the grid and its structure see
Figure 1 of the Appendix.
40
Figure 3.1: Snapshot of Digital Grid (Bouzaiene-‐‑Ayari)
3.21 Initial Truck Placement
PSE&G’s work force of repair trucks for its entire electrical grid is
primarily centered in nine locations, four headquarters and five sub-‐‑
headquarters (Marques 2014). These locations serve both as storage facilities
when trucks are idle and basing points for when they are sent out to address
problems with the grid. As the simulated electrical grid in this thesis
encompasses a single area centered around Newark, it will assume that the
41
repair crews in this area have one headquarter location in the center of the grid (a
single location node was selected to be the headquarters).
At the start of a simulation, all truck resources are located at this
headquarters location and are available. This initial setup simulates a scenario in
which a storm has just passed through the area and trucks have not yet been
dispatched out into the area. The total number of repair trucks available in this
area in a given simulation is a parameter set before the start depending on what
kind of scenario is desired. So, before a simulation even starts, a list of N trucks is
generated by a program that was written for this thesis, where N is the desired
total number of trucks, and a given truck in this list, Tri, has the starting
following attributes (see section 2.31 for details on truck attributes):
Tri = (t, Lt,i, IDi)
t = the starting time of the simulator
Lt,i = the headquarter location
IDi = “TR” + i
Essentially, a given truck in the generated list is located at the headquarter
location at the beginning of the simulation and has an identification tag in String
format that is simply the String “TR” plus a number i concatenated to the end of
the String. This list of available trucks is actually written into a file outside of the
simulator, which then sends the truck information into the simulator to create the
42
available truck resources part of the initial state variable, Γ. All of the truck data
is sent to the start time of the simulator. See Figure 2 of the Appendix for a flow
chart that demonstrates this interaction.
3.22 Task Generation
The outage tasks to be solved in this resource allocation problem are
generated in the same program as the truck resources. Like the trucks in the
previous section, generated tasks are written to an outside file, which then sends
the information to the simulator itself. Tasks are sent into the simulator at their
event time, meaning tasks are inserted into the list of unassigned tasks Υ at the
time that they are learned about by the company. Tasks with event times before
the start time of the simulator are sent to the start time. See Figure 2 of the
Appendix for a diagram of this process. The total number of tasks to be resolved
is a parameter set before the simulator begins, and the attributes of each of these
tasks are generated through a series of random processes. As a reminder, an
outage task is made up of seven attributes (see section 2.31 for details):
Ts! = (t, L!, ID!,T!"!#$,!,D!,P!,N!)
Six of these attributes—t, Lt,i, IDi, Tevent,i, Di, and Ni—are generated by this
program ahead of time while the priority Pi is calculated in each time step by the
assignment policies. The current time t is fairly simple, as it is just set to be the
43
start time of the simulation, and it will change as time progresses. The other five
attributes are static, meanwhile, and the values generated by the program remain
fixed throughout the simulation. The way in which each of these attributes is
generated will be discussed one by one below.
The location of the task, Lt, is generated by randomly generating an
integer between 0 and Q, where Q is the number of possible locations in the
modeled grid. Each of these possible locations has an integer location tag
between 0 and Q, so the randomly generated integer for Lt,i is in fact a location
tag for a point in the grid. Thus, the task is assigned to the location whose
location tag matches the randomly generated number. The grid has thousands of
possible locations, so the probability that two tasks in a simulation are in the
same location is very small since the total number of tasks is small in
comparison.
The identification tag of each task is generated in a similar way to the
truck resources identification. The identification tag IDi assigned to a given task
Tsi is simply a String comprised of “TS” plus a number i concatenated at the end.
So, if we are generating J tasks, each of the identification tags for tasks Ts0, Ts1, …
, TsJ is “TS0”, “TS1”, … , “TSJ” respectively.
The time at which we learned about a task, Tevent,i is another randomly
generated attribute. If the start time in minutes of the simulation is Tstart, then the
44
event time for a task is determine by generating a random integer F between 1
and 500 and setting:
Tevent,i = Tstart + 200 – F
This process essentially assigns a task a random event time between 300 minutes
before the simulation start time and 200 minutes after the start time. Thus,
approximately three-‐‑fifths of the outage tasks are already recognized at the
beginning of the simulation, while the other two-‐‑fifths will become known in the
first 200 minutes of the simulation. This means that some tasks will be
dynamically added to the list of unassigned tasks Υ of the state variable as the
simulation progresses. This setup attempts to simulate a scenario in which a
large number of outages are known shortly following a storm, and more are
learned about while time progresses and the company actively allocates its repair
resources.
The duration of a task in minutes, Di, is generated through another
random process. A random integer F is generated between 0 and 210 and the
duration is simply assigned as:
Di = 30 + F
Thus, the duration of a task can be anywhere between 30 and 240 minutes.
Although outage events in reality can take much longer than four hours to repair
and the average repair time is around four hours, since the simulated model is
45
only a chunk of the actual grid the possible range of task durations is scaled
down to keep this attribute from being the dominating factor in cumulative
customer outage time. In the real world, outages can be far away from repair
resources, so travel time can be rather long. Since the scope of the simulated grid
is limited (the maximum distance to travel is only around 15 miles), the duration
of outages must be scaled down to account for the fact that truck travel distance
is limited to smaller values than in reality.
Lastly, the number of customers affected by an outage task Ni is generated
by this program. This number of customers is generated through the
straightforward process of randomly generating an integer between 1 and 100
and setting Ni equal to this value. This attribute is generated in this way in order
to keep outage tasks fairly simple and independent of one another. In reality, the
number of customers affected by an outage depends mainly on where in a
branch of the electrical grid the problem occurs. Customers on a certain branch of
the grid can actually be affected by multiple problems at the same time if they
occur close to one another on the same line. This scenario would make modeling
and simulating much more difficult though more realistic, and it is avoided
through the way that Ni is generated in this thesis. For more discussion on this
issue see Chapter 5.
46
3.23 Fixed Parameters
There are also a number of fixed parameters in the simulator that are
significant. One of these that was briefly mentioned in section 2.32, truck travel
speed, is similar to task duration in that it is set to offset the limited size of the
simulated grid. The speed at which the trucks travel between locations is fixed at
25 miles per hour. Although the average speed of a repair truck in reality would
likely be higher due to faster speeds available on highways and large roads, the
limited size of the grid means that travel distances are less than what they could
be in the real world. However, following a storm there may also be
transportation issues like downed trees on roads or accidents, so 25 miles per
hour may not be all that low of an estimate for the average speed over a trip. Like
shorter duration times for tasks, the fixed travel speed for trucks is designed to
make travel distance slightly more influential in terms of time spent addressing
outages.
Two other fixed parameters of note are the simulation start and end times.
Although the start time is trivial, the end time is important because is dictates the
amount of time over which the resource allocation problem takes place. In a
limited time setting, poorly performing assignment policies may not be able to
complete all the outage tasks depending on the number of trucks and tasks and
the distribution of tasks. However, in order to provide a full comparison of the
47
policies, the start time and end time (essentially the simulation run length) are set
so that in a given scenario all four tasks are able to resolve all outages. This way,
the full amount of time needed to allocate trucks to every outage is provided,
and there is no uncertainty as to how much longer a policy would have needed
to finish.
Lastly, there is another fixed parameter that plays an important role on the
data generation side of the simulator—the random generation seed. In Java, the
programming language in which this simulator is written, the random
generation of numbers can be given a “seed” which serves as a basis for
generating random numbers. These generated numbers aren’t in fact purely
random, but the random generation function in Java is pretty close. One
characteristic of the random generation seed that is particularly important is that
if a certain seed is used to generate a set of random numbers, using the same
seed again to generate the same number of random numbers will result in the
same output. Thus, the random generation seed will be kept the same for a given
scenario so that all four policies will face the exact same randomly generated sets
of tasks. This will help eliminate noise in the resulting data that may have been
otherwise caused by differences in the generation of tasks for the same scenario.
However, a different seed will be used between scenarios so that there are totally
independent sets of tasks. In other words, while the same seed will be used when
48
running the simulation four times for the four different assignment policies in a
given scenario (say 10 repair trucks and 100 tasks), a different seed will be used
when comparing the four policies in a different scenario like 10 repair trucks and
120 tasks.
49
Chapter 4: Simulation Data and Analysis
4.1 Initial Parameter Tuning
As described in section 3.1, two of the assignment problems tested in this
thesis included a tunable parameter θ. This parameter essentially determines
how much the amount of time a task has been waiting for service impacts the
calculation of the priority of the task in the Weighted Priority and Weighted
Priority Plus Distance policies. θ is set prior to the simulation, so the value it is
given has a significant impact on the performance of the policy since it changes
the relative influence of different task attributes in its priority calculation. Since
the goal of this thesis is to determine the best assignment policy for PSE&G’s
resource allocation problem, the tunable parameter θ in each of the two policies
should first be set in such a way that the policies perform most efficiently when
they are tested and compared to the other policies.
In order to determine what values of θ are best for each of the policies, a
policy search must be implemented. A policy search is essentially the process of
simulating the performance of the policies with different values of θ to
determine which works best (Powell 3 Oct. 2013). Thus, the Weighted Priority
and WPPD policies were simulated with various tunable parameter values over
three scenarios to determine which works best, and the results are in the
following section.
50
4.12 Tunable Parameter Test Results
The first scenario in which we simulated the two policies with a range of
tunable parameter values was where there were 10 repair trucks and a total of
100 outage tasks. The performance of the policies in this scenario in terms of the
average value of the reliability index SAIDI was:
Figure 4.1: Average SAIDI Performance Over θ in 10 Truck Scenario The performance of the policies in this scenario of 10 trucks and 100 total tasks
was also measured in terms of the average value of CAIDI, and the results are
displayed in Figure 4.2 below:
127.8
128
128.2
128.4
128.6
128.8
129
0 0.1 0.2 0.4 0.8
Average SAIDI (minutes)
Tunable Parameter Value
Average SAIDI vs Tunable Parameter Value (10 trucks, 100 tasks)
Weighted Priority
Weighted Priority Plus Distance
51
Figure 4.2: Average CAIDI Performance Over θ in 10 Truck Scenario As demonstrated in the graphs above, the performance of both Weighted Priority
and WPPD decreased in this scenario as the value of θ increased. In other words,
making the customer waiting time more influential in the resource allocation
decision part of these two policies made them less efficient at resolving the tasks
quickly.
The second scenario in which the policy search was conducted involved
20 trucks with 100 tasks. This setup sought to determine if the impact of the
tunable parameter θ was different when the beginning ratio of available truck
resources to unassigned tasks was larger. The results in terms of SAIDI can be
viewed in the following Figure 4.3:
632
633
634
635
636
637
638
0 0.1 0.2 0.4 0.8
Average CAIDI (minutes)
Tunable Parameter Value
Average CAIDI vs Tunable Parameter Value (10 trucks, 100 tasks)
Weighted Priority
Weighted Priority Plus Distance
52
Figure 4.3: Average SAIDI Performance Over θ in 20 Truck Scenario And the impact on CAIDI was:
Figure 4.4: Average CAIDI Performance Over θ in 20 Truck Scenario Again, it was apparent that increasing the tunable parameter value negatively
affected both assignment policies. The total duration of sustained electrical
68.2%
68.25%
68.3%
68.35%
68.4%
68.45%
68.5%
68.55%
0% 0.1% 0.2% 0.4% 0.8%
Average'S
AIDI'(m
inutes)'
Tunable'Parameter'Value'
Average'SAIDI'vs'Tunable'Parameter'Value'(20'trucks,'100'tasks)'
Weighted%Priority%
Weighted%Priority%Plus%Distance%
338#338.2#338.4#338.6#338.8#339#
339.2#339.4#339.6#339.8#
0# 0.1# 0.2# 0.4# 0.8#
Average'C
AIDI'(m
inutes)''
Tunable'Parameter'Value'
Average'CAIDI'vs'Tunable'Parameter'Value'(20'trucks,'100'tasks)'
Weighted#Priority#
Weighted#Priority#Plus#Distance#
53
service interruption for the average customer in an area (SAIDI) increased as the
tunable parameter values increased, and the average time required to restore
electrical services to the average customer affected by a sustained interruption
(CAIDI) also increased as θ went from 0.0 to 0.8.
The last scenario in which the impact of the tunable parameter in the
Weighted Priority and WPPD policies was tested involved a much larger number
of tasks. In this situation, there were only 25 repair trucks and 400 total tasks to
resolve. This scenario not only examined the impact of θ when there was a low
ratio of repair resources to tasks but also when there was a much larger number
of tasks and hence longer time frame involved. The findings of this policy search
test were:
Figure 4.5: Average SAIDI Performance Over θ in 25 Truck Scenario
193.5&194&
194.5&195&
195.5&196&
196.5&197&
197.5&198&
0& 0.1& 0.2& 0.4& 0.8&
Average'S
AIDI'(m
inutes)'
Tunable'Parameter'Value'
Average'SAIDI'vs'Tunable'Parameter'Value'(25'trucks,'400'tasks)'
Weighted&Priority&
Weighted&Priority&Plus&Distance&
54
And the CAIDI results can be seen in Figure 4.6 below:
Figure 4.6: Average CAIDI Performance Over θ in 25 Truck Scenario Once again, as the value of θ increased, the performance of the policies in terms
of both SAIDI and CAIDI decreased.
The aggregation of results from the three scenarios clearly indicated that
the tunable parameter θ had a negative affect on both policies. Giving more and
more priority to customers who have been waiting for a longer time made the
reliability of electrical services decrease as the policies became less efficient. The
tunable parameter also appeared to have had a greater negative impact on the
WPPD policy than on Weighted Policy, as indicated by the larger positive slope
in the graphs in each interval.
One aspect of the results to note was the scale of the vertical axes in each
of the result graphs. Gradually changing the value of θ from 0.0 to 0.8 in each of
960$
965$
970$
975$
980$
985$
0$ 0.1$ 0.2$ 0.4$ 0.8$
Average'C
AIDI'(m
inutes)''
Tunable'Parameter'Value'
Average'CAIDI'vs'Tunable'Parameter'Value'(25'trucks,'400'tasks)'
Weighted$Priority$
Weighted$Priority$Plus$Distance$
55
the policies did not result in an enormous change in performance, though there
was a negative change. Though this may appear to indicate that θ does not affect
the performance of the policy significantly, the design of the policy search was
important to recognize in this issue. The three scenarios were run 1000
independent times for each of the policies, and the results were the average
values of the reliability indices over these 1000 trials. Since the values of SAIDI
and CAIDI were averages over a large number of independent trials, any
changes due to policy shifts were relatively small in relation to comparing the
potential impact of θ on the policies in one or two trials. The design of the
simulator and the randomness of the task attributes may also dampen the
impacts of θ on the policy performances, though this issue will be primarily
discussed later in this chapter.
Nevertheless, this policy search has concluded that the optimal value of θ
for Weighted Priority and WPPD in terms of overall performance is 0.0. In other
words, disregarding customer waiting time when calculating the task priorities
in each of the policies results in the most efficient resource allocation. Since the
most efficient versions of Weighted Priority and WPPD should be used when
comparing them to FIFO and Distance Exploitation, the tunable parameter θ will
be set to 0.0 in each of the policies when they are tested and compared.
56
4.2 Policy Testing and Comparison
In the following sections, the four allocation policies being considered by
this thesis will be tested and compared in three batches of simulation scenarios.
Like in the policy search in the previous section, the three sets of scenarios will
involve 10 trucks, 20 trucks, and 25 trucks respectively. However, each of the
scenario sets will test five different total numbers of tasks in order to provide
analysis over a wider range of situations. The policies will be tested and
compared in each of the five situations through 1000 independent simulations.
The final section will then sum the results of the various scenarios and make a
conclusion as to the best assignment policy for this dynamic resource allocation
problem. For a snapshot of the data generated by the simulations and a brief
description, see Figure 3 in the Appendix.
4.21 10 Truck Scenarios Results
For the first set of scenarios, the four policies were implemented in
multiple situations involving 10 trucks and a range of total tasks. There were five
situations total, with 80, 90, 100, 110, and 120 total tasks respectively. The
performance results of the policies in these scenarios in terms of SAIDI are
depicted in Figure 4.7 below:
57
Figure 4.7: Average SAIDI Performance in 10 Truck Scenarios
The results in terms of CAIDI are available in the following Figure 4.8:
Figure 4.8: Average CAIDI Performance in 10 Truck Scenarios
As demonstrated in the above graphs, Weighted Policy and WPPD were
both significantly better than FIFO and Distance Exploitation in terms of both
70#
90#
110#
130#
150#
170#
190#
210#
230#
250#
80# 90# 100# 110# 120#
Average'SA
IDI'(minutes)'
Number'of'Tasks'
Average'SAIDI'vs'Number'of'Tasks'(10'trucks)'
FIFO#
Distance#Exploita:on#
Weighted#Priority#
Weighted#Priority#Plus#Distance#
450$
550$
650$
750$
850$
950$
1050$
80$ 90$ 100$ 110$ 120$
Average'CA
IDI'(minutes)'
Number'of'Tasks'
Average'CAIDI'vs'Number'of'Tasks'(10'trucks)'
FIFO$
Distance$Exploita;on$
Weighted$Priority$
Weighted$Priority$Plus$Distance$
58
SAIDI and CAIDI over the range of task totals, with FIFO performing the most
poorly. The two weighted policies are virtually indistinguishable in the graphs,
however, and a closer look was needed to determine if one or the other was
statistically better. To do this, confidence intervals were constructed for the
average SAIDI and CAIDI values in each scenario. For this analysis, 95%
confidence intervals were used of the form:
CI!"% = µμ ± z ∗𝜎N
Where
µμ = the measured mean of the values
z = the upper critical value for a standard normal distribution
σ = the measured standard deviation of the values
N = the number of trials conducted
In this case, µμ was simply the observed average SAIDI and CAIDI values in each
situation, z = 1.96 for 95% confidence level, σ was the observed standard
deviation of the SAIDI and CAIDI values in each situation, and N = 1000 since
we conducted 1000 simulations for each scenario. See Table 4.1 below for a table
summary of the 95% confidence intervals of the average SAIDI and CAIDI levels
in each policy:
59
Table 4.1: Confidence Intervals for 10 Truck Scenarios
It was statistically conclusive that Weighted Priority and WPPD were
better than Distance Exploitation and FIFO in each situation because their
confidence intervals for both SAIDI and CAIDI were lower than the others’ with
no overlap. Also, Distance Exploitation was distinctly better than FIFO because it
had lower confidence intervals than FIFO in all five instances with no overlap.
However, since the SAIDI and CAIDI confidence intervals for Weighted Priority
and WPPD overlapped in all five cases, neither one was conclusively better than
the other statistically. Thus, even though the average values of SAIDI and CAIDI
were in fact slightly lower for WPPD than for Weighted Priority, the simulations
did not provide definitive evidence that it was a better assignment policy.
4.22 20 Truck Scenarios Results
The next scenarios in which the assignment policies were tested involved
20 available repair trucks. The range of total number of tasks were the same as
SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDIFIFO
Lower+Bound 103.44 641.46 129.41 715.66 159.25 791.58 193.05 872.84 230.63 952.91Upper+Bound 104.68 646.93 130.83 721.44 160.95 797.52 194.94 878.89 232.67 959.23
Distance,ExploitationLower+Bound 92.72 574.96 116.28 643.03 142.34 707.53 171.93 777.30 205.09 847.30Upper+Bound 93.84 579.99 117.57 648.39 143.86 713.14 173.72 783.29 207.04 853.46
Weighted,PrioirityLower+Bound 83.05 515.02 103.30 571.19 126.91 630.73 152.96 691.47 182.87 755.41Upper+Bound 84.12 520.19 104.54 576.65 128.36 636.31 154.64 697.36 184.75 761.62
Weighted,Priority,Plus,DistanceLower+Bound 82.82 513.55 102.94 569.20 126.36 628.02 152.29 688.44 181.87 751.29Upper+Bound 83.88 518.70 104.18 574.63 127.80 633.56 153.97 694.33 183.74 757.47
10,trucks,,110,tasks 10,trucks,,120,tasks10,trucks,,80,tasks 10,trucks,,90,tasks 10,trucks,,100,tasks
60
the previous test, with 80, 90, 100, 110, and 120 total tasks in the five simulated
situations. The number of trucks was doubled while keeping the total number of
tasks the same in order to compare the performance of the policies in situations
that were not quite as overwhelming. In this batch of simulations, the ratio of
trucks to tasks ranged from 1:4 with 80 tasks to 1:6 with 120 tasks. In the previous
section, the ratio ranged from 1:8 to 1:12. The performances of the four policies in
terms of SAIDI and CAIDI are displayed below in Figures 4.9 and 4.10:
Figure 4.9: Average SAIDI Performance in 20 Truck Scenarios
40#
50#
60#
70#
80#
90#
100#
110#
120#
130#
80# 90# 100# 110# 120#
Average'SA
IDI'(minutes)'
Number'of'Tasks'
Average'SAIDI'vs'Number'of'Tasks'(20'trucks)'
FIFO#
Distance#Exploita<on#
Weighted#Priority#
Weighted#Priority#Plus#Distance#
61
Figure 4.10 Average CAIDI Performance in 20 Truck Scenarios
As in the previous section, Weighted Priority and WPPD are both
significantly more efficient than FIFO and Distance Exploitation in terms of
SAIDI and CAIDI. Again, the two weighted policies have such similar
performance results that they are indistinguishable in the graphs. 95%
confidence intervals were constructed for these scenarios using the same formula
as before in order to compare these two policies in more detail. The results are
available in Table 4.2 below:
250$
300$
350$
400$
450$
500$
80$ 90$ 100$ 110$ 120$
Average'CA
IDI'(minutes)'
Number'of'Tasks'
Average'CAIDI'vs'Number'of'Tasks'(20'trucks)'
FIFO$
Distance$Exploita:on$
Weighted$Priority$
Weighted$Priority$Plus$Distance$
62
Table 4.2: Confidence Intervals for 20 Truck Scenarios
As demonstrated in the table, it was statistically apparent that Weighted
Priority and WPPD were superior to FIFO and Distance Exploitation, with FIFO
being the least efficient assignment policy. However, once again it was not
conclusive as to whether or not Weighted Priority was better than WPPD
because their confidence intervals overlapped for SAIDI and CAIDI in all five
situations. Like the previous set of tests, the Weighted Priority policy’s average
values of SAIDI and CAIDI were slightly lower than those of WPPD, but these
simulations unfortunately did not provide conclusive evidence that one was
better than the other.
4.23 25 Truck Scenarios Results
The last set of scenarios that were used to test the assignment policies in
this thesis involved 25 available repair trucks and a much higher range of total
numbers of tasks. The five sets of task totals were 360, 380, 400, 420, and 440.
SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDIFIFO
Lower+Bound 53.64 331.98 67.48 371.03 82.15 407.89 98.90 445.58 117.90 485.11Upper+Bound 54.31 335.04 68.24 374.15 83.03 411.13 99.89 448.83 119.07 488.49
Distance,ExploitationLower+Bound 49.42 305.84 61.80 339.78 75.19 373.33 90.03 405.58 106.88 439.74Upper+Bound 50.05 308.71 62.50 342.72 76.01 376.50 90.95 408.72 107.96 442.95
Weighted,PrioirityLower+Bound 45.53 281.73 56.36 309.82 67.97 337.45 80.88 364.35 96.22 395.80Upper+Bound 46.12 284.57 57.05 312.79 68.76 340.55 81.77 367.43 97.26 399.09
Weighted,Priority,Plus,DistanceLower+Bound 45.47 281.37 56.27 309.33 67.83 336.74 80.66 363.35 95.95 394.73Upper+Bound 46.06 284.19 56.95 312.29 68.62 339.84 81.54 366.41 97.00 398.00
20,trucks,,110,tasks 20,trucks,,120,tasks20,trucks,,80,tasks 20,trucks,,90,tasks 20,trucks,,100,tasks
63
These significantly higher numbers were chosen in hopes that a larger number of
assignment decisions would provide more opportunity for allocation
improvement. Having each policy make more decisions in each simulation could
potentially highlight differences between policies, especially between Weighted
Policy and WPPD. Also, this range of task totals was chosen to see if the policies’
performances compared differently in much more overwhelming circumstances.
In these tests, the ratio of trucks to tasks ranged from 1:14.4 with 360 tasks to
1:17.6 with 440 tasks. This expanded the range of examined ratios upwards from
what was previously tested in the previous two batches of scenarios. The
performance of the four assignment policies in the five different scenarios are
displayed in Figures 4.11 and 4.12 below:
Figure 4.11 Average SAIDI Performance in 25 Truck Scenarios
40#
240#
440#
640#
840#
1040#
1240#
1440#
360# 380# 400# 420# 440#
Average'SA
IDI'(minutes)'
Number'of'Tasks'
Average'SAIDI'vs'Number'of'Tasks'(25'trucks)'
FIFO#
Distance#Exploita9on#
Weighted#Priority#
Weighted#Priority#Plus#Distance#
64
Figure 4.12 Average SAIDI Performance in 25 Truck Scenarios
The previously observed trends in performance between policies were
continued in this set of scenarios. FIFO was clearly the worst, followed by
Distance Exploitation, while Weighted Priority and WPPD were right along side
each other. In the above graphs, however, both the green and purple lines
representing Weighted Priority and WPPD respectively can be partially seen,
indicating that there may have been a slightly larger margin of difference
between the two policies than in the past two sets of scenarios. 95% confidence
intervals were again constructed in order to determine if there was a statistically
superior assignment policy. The results can be seen in the table below:
250$
450$
650$
850$
1050$
1250$
1450$
360$ 380$ 400$ 420$ 440$
Average'CA
IDI'(minutes)'
Number'of'Tasks'
Average'CAIDI'vs'Number'of'Tasks'(25'trucks)'
FIFO$
Distance$Exploita:on$
Weighted$Priority$
Weighted$Priority$Plus$Distance$
65
Table 4.3: Confidence Intervals for 25 Truck Scenarios
As expected the confidence intervals confirmed that FIFO was statistically
the least efficient assignment policy, followed by Distance Exploitation.
However, the above table also concluded that WPPD was in fact superior to
Weighted Priority as indicated by the lack of any overlap in their confidence
intervals in all five scenarios.
4.24 Policy Conclusions
Based on the results from the previous sections, it is evident that both
Weighted Priority and WPPD are significantly better than FIFO and Distance
Exploitation, with WPPD consistently having slightly lower average values of
SAIDI and CAIDI than Weighted Priority. In the first set of scenarios involving
10 repair trucks, WPPD was on average 20.62% better than FIFO, 11.19% better
than Distance Exploitation, and 0.41% better than Weighted Priority over the five
different situations. Although the confidence intervals for these scenarios
SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDI SAIDI CAIDIFIFO
Lower+Bound 821.50 1130.83 911.00 1189.16 1012.26 1253.15 1113.44 1315.77 1220.48 1375.26Upper+Bound 825.93 1135.25 915.76 1193.76 1017.40 1257.82 1119.09 1320.52 1226.42 1380.02
Distance,ExploitationLower+Bound 729.62 1004.33 810.29 1057.66 898.41 1112.17 985.88 1165.01 1081.51 1218.63Upper+Bound 733.74 1008.48 814.57 1061.92 903.05 1116.52 991.02 1169.38 1086.83 1223.04
Weighted,PrioirityLower+Bound 641.74 883.32 712.63 930.16 789.32 977.06 867.15 1024.64 951.38 1071.94Upper+Bound 645.58 887.35 716.70 934.31 793.76 981.45 871.92 1028.96 956.49 1076.41
Weighted,Priority,Plus,DistanceLower+Bound 634.90 873.90 704.65 919.75 780.33 965.93 856.54 1012.10 939.10 1058.09Upper+Bound 638.73 877.92 708.67 923.83 784.70 970.24 861.24 1016.35 944.18 1062.55
25,trucks,,420,tasks 25,trucks,,440,tasks25,trucks,,360,tasks 25,trucks,,380,tasks 25,trucks,,400,tasks
66
conclusively indicated that WPPD was superior to FIFO and Distance
Exploitation, it was not conclusive as to whether or not it was better than
Weighted Priority statistically. In the second set of scenarios involving 20 repair
trucks, WPPD was on average 17.23% better than FIFO, 9.44% better than
Distance Exploitation, and 0.21% better than Weighted Priority over the five
different task totals. Again, WPPD was definitively more efficient than FIFO and
Distance Exploitation, but there was not conclusive evidence regarding Weighted
Priority. In the third batch of scenarios involving 25 trucks, however, WPPS was
statistically proven to be the best assignment policy. It was better than FIFO by
22.86%, Distance Exploitation by 13.07%, and Weighted Priority by 1.17% on
average over the five scenarios.
The fact that WPPD was decisively better than Weighted Priority in the
scenarios involving a larger number of tasks indicates that this assignment policy
is in fact superior, but many allocation decisions are needed to provide statistical
evidence of the difference. Thus, this thesis has found that of the four policies
tested, WPPD is the best assignment policy for PSE&G’s dynamic resource
allocation problem. However, the exact margins by which this policy was more
efficient than the others should not necessarily be extrapolated to the real world.
Various elements of the thesis’ mathematical model and simulator significantly
influence the magnitude of the policies’ performances. The way in which tasks
67
and their attributes were generated, various fixed parameters, and the structure
of the simulated grid all influence the resulting values of SAIDI and CAIDI, and
they should not be considered expected results of applying the assignment
policies to real world situations.
Another element that might be relevant to consider regarding the best
policy for PSE&G is the impact on certain customers. FIFO was demonstrated to
be the least efficient of the policies for PSE&G’s allocation problem, but it is
commonly used in retail because customers see it as fair. If PSE&G implemented
the WPPD policy with θ = 0.0 (no influence by wait time on decision process), the
company risks a negative response from certain customers. Under this policy,
customers who live in sparsely populated rural areas, though among the first to
experience a power outage in some cases, may be among the last to receive repair
services due to the priority system. Although the system reduces the total
number of outage hours experienced by customers and thus the overall
reliability of PSE&G’s electrical services, it may alienate these select customers,
potentially angering them enough to seek the services of another electrical
provider if possible. Thus, PSE&G may want to consider balancing policy
performance with customer backlash if they do not want to risk losing a small
number of customers. If so, the company may want to consider implementing
68
WPPD with a higher value of θ to sacrifice performance for customer
satisfaction.
69
Chapter 5: Conclusion
This thesis has succeeded in providing a baseline comparison of four
potential assignment policies for PSE&G’s dynamic resource allocation problem
involving assigning utility repair trucks to power outages. Through the
construction of a mathematical model and simulator, it has tested the policies
over a range of scenarios and found that the Weighted Priority Plus Distance
policy is the superior assignment policy. However, the margin by which the
policy was more efficient should not be extended into the real world due to some
limitations of this thesis. The model and simulator greatly simplify PSE&G’s
dynamic allocation problem, providing a good environment for testing and
comparison but making the margin of results not necessarily realistic. Thus, there
are several areas of improvement and future research for this problem.
One significant way in which this thesis’ work could be extended and
improved would be to build out the digital grid to encompass PSE&G’s entire
grid. Although this would require an enormous amount of work considering the
great deal of time Belgacem Bouzaiene-‐‑Ayari had to commit to modeling just a
chunk of the grid, doing so would enable certain aspects of the simulator to
become more realistic. Since truck travel distance would be more accurate, travel
speed and task duration could be set to more closely mirror the real world. Task
duration especially could be more realistic by generating the duration based on a
70
distribution made from historical data on outages PSE&G has resolved in the
past, if such data is available. Additionally, the actual locations of PSE&G’s
regional headquarters and sub-‐‑headquarters for their repair resources could be
built into the simulator, and initial placement of trucks would be at these places.
Critical customers are also an area in which the assignment policy
comparison could be improved. Restoring power quickly to customers like
hospitals and police stations are of utmost importance to PSE&G following a
storm, and this thesis does not currently consider their influence. If the simulated
grid was built out, locations of critical customers could be compiled through
extensive research and included in the grid. Changing the policies to consider
these customers during the initial stages of a simulation could impact their
performance relative to one another.
One other way in which the simulator could be expanded would be to
consider the actual structure of the grid itself. Specifically, making the simulator
recognize that there can be multiples problems on the same stretch of grid
affecting the same customers would make this problem much more realistic and
complex. In such as situation, multiple problems would have to all be resolved in
order to restore power to the affected customers, complicating the process of
calculating how long restoring power would take and how to assign trucks to the
tasks. Such a change would make this resource allocation problem similar in
71
some ways to the machine-‐‑scheduling problem discussed by Simon French in his
book Sequencing and Scheduling: An Introduction to the Mathematics of the Job-‐‑Shop.
In the scheduling problem, a limited number of machines are available to do
work on various stages of a list of jobs. Each job has a varying number of stages
left before it is totally completed, and the assignment policies considered by
French take into account this attribute. The assignment policies considered in
PSE&G’s utility repair truck problem could be extended to consider the number
of tasks requiring completion before a group of customers regain power if the
grid structure is considered.
An area of future research for this problem would be on the location and
type of outages following a storm. This thesis simplifies the generation of tasks
through random generation of attributes and by considering outages as a single
type of task. In reality, outages can be caused by an array of issues ranging from
broken support poles or fallen trees on the lines to blown transformers, and these
various issues require different equipment and services in order to be resolved.
The work being done by Kevin Cen for his thesis on damage assessment could be
applied to this area. He is working on generating probability distributions for the
location and type of outages following a storm based on historical data. His
findings could potentially be applied to this resource allocation problem to make
the generation of tasks more realistic. Additionally, the location of outages
72
following a storm is not truly random, as elements like the path of the storm, its
strength, local topography, and density of trees in a certain area all likely affect
where outages occur. Research being done by Belgacem Bouzaiene-‐‑Ayari on
storm damage generation could certainly be applied to make task generation
better. Belgacem is working on simulating storms of various strengths and their
potential damage to an electrical grid based on its path. Applying this research to
PSE&G’s dynamic resource allocation could make the current random generation
of task locations much more realistic.
Lastly, the work done in this thesis could be extended to include another
set of assignment policies—look ahead policies. This thesis intentionally chose to
compare four simple cost function and policy function approximation policies
because they are computationally quick and could be relatively easily
implemented by an electrical provider like PSE&G. However, look ahead
policies, policies that consider the future when making decisions, could
potentially be much better than the four policies that were tested in this thesis.
Look ahead policies are much more mathematically complicated and
computationally difficult because they often involve solving a linear program
over multiple periods (Powell 3 Oct. 2013). Although simulating and comparing
look-‐‑ahead policies would be much more difficult than cost and policy
73
approximations, future research in this area would be worth pursuing based on
the potential for higher allocation performance.
74
Appendix
Figure 1: Snapshot of PSE&G’s Modeled Grid
(Bouzaiene-‐‑Ayari)
75
Figure 2: Truck and Task Generation Process
Trucks and Tasks Generated By Program
Data Sent to
Outside File Data
Stored in File
Simulation Start Time
All Truck Data and Tasks Occurring Before Start Time Sent to Simulator at Start
Time
Tasks Occurring After Start Time Sent to Simulator at Their Designated Time
76
Figure 3: Snapshot of Simulation Data
Note: # of Tasks Assigned, Total Resolve Time, Total # of Customers, and Total
Duration of Tasks are results of the simulations, while SAIDI and CAIDI are
calculated afterwards. Tasks Assigned, and Duration of Tasks primarily serve to
check that all tasks are resolved and that the same tasks are addressed in each
iteration between scenarios. 1000 total iterations are run for each scenario.
Iteration #*of*Tasks*Assigned Total*Resolve*Time Total*#*of*Customers Total*Duration*of*Tasks*ServedSAIDI CAIDI1 80 2828776 3961 11276 113.15 714.162 80 2175780 3674 10136 87.031 592.213 80 2859552 4235 10595 114.38 675.224 80 2720391 4443 9965 108.82 612.295 80 3048192 4357 11362 121.93 699.616 80 2668416 3958 10994 106.74 674.187 80 2726547 4218 11158 109.06 646.418 80 2650536 4162 10358 106.02 636.849 80 2370940 3817 10062 94.838 621.1510 80 2643058 4240 10828 105.72 623.3611 80 2796201 4030 11305 111.85 693.8512 80 2380117 3964 10357 95.205 600.4313 80 2518563 4113 11154 100.74 612.3414 80 3014063 4471 11625 120.56 674.1415 80 2602575 3942 10894 104.1 660.2216 80 2478313 3854 10020 99.133 643.0517 80 2438539 3751 10772 97.542 650.118 80 2697612 4424 10877 107.9 609.7719 80 2722168 3927 10782 108.89 693.1920 80 2406925 4497 9573 96.277 535.2321 80 2929085 4385 10645 117.16 667.9822 80 3015398 4278 10904 120.62 704.8623 80 2261044 3900 10235 90.442 579.7524 80 2948159 4280 11427 117.93 688.8225 80 2129672 3839 10908 85.187 554.75
77
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