Entropy Minimization & Locating. Faults Across the Electrical Network. Using Customer No Light Calls

Transcription

1 Entropy Minimization & Locating Faults Across the Electrical Network Using Customer No Light Calls Kevin Cen Advisor: Professor Warren 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 2014

2 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. Kevin Cen 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. Kevin Cen

3 Abstract This thesis develops a probability model that uses information from no light calls to estimate the probabilities of faults across the electrical distribution network. We first consider a single fault per circuit scenario, before developing the logic for the multiple fault scenario. Due to the problem of high dimensionality in the multiple fault logic, we cannot calculate the probabilities based on our assumptions exactly. Instead, we use Monte Carlo methods to estimate the edge fault probabilities. The Monte Carlo method for estimating probabilities is then implemented in the robust grid simulator, which allows us to run storm simulations and observe the results. Shannon entropy is introduced as a metric to measure our uncertainty about the edge fault probabilities over a circuit. In the last part of the thesis, we observe how changing major parameters, such as the no light call probability and setting of our prior, affects circuit entropies. We use two simple heuristic policies, pure exploitation and Boltzmann exploration to send lookup trucks to measure edges with the objective of minimizing circuit entropy. iii

4 Acknowledgements First, I would like to thank my advisor Professor Warren Powell. Despite juggling classes, conferences and research projects all at once, he always made time to be available for his advisees, and provided guidance at the end, when it was most needed. Thank you to Dr. Belgacem Bouzaiene-Ayari for his time and hard work on the robust grid simulator, which this thesis uses heavily. I hope that the robust grid project moves forward successfully and will look forward to keeping tabs on its progress. My sincere gratitude to the entire Operations Research and Financial Engineering Department. The faculty and students in this department are hard working and exceptionally intelligent. I am proud to be a member of the Princeton ORFE community. Good to all my friends at Princeton, who have made my undergraduate experience a blast. Four years have gone by in the blink of an eye, and I will definitely look back on my time here with a big smile. And finally, thank you to my beloved family: to my sister Sarah, who is also an undergraduate here, I hope you enjoy the rest of your time at this great institution - and to my parents who have supported me my entire life, I could never have gotten here without you. iv

5 v To my family and friends.

6 Contents Abstract Acknowledgements List of Tables List of Figures iii iv viii ix 1 Introduction An Overview of Electrical Grids The Electric Transmission Network The Electric Distribution Network Protective devices Electric Utilities & PSEG Causes of outages Grid Improvements & Smart Grids Overview of Data and Simulator Hurricane Forecasting The Grid Data The Robust Grid Simulator The Probability Model The Single Fault Scenario Fault Occurance vi

7 3.1.2 Using Bayes Theorem to Calculate Edge Fault Probabilities Without Conditioning on a Storm Example Single Fault Scenario Calculation Circuit reduction with protective devices Additional Single Fault Scenario Examples with Circuit Reduction The Multiple Fault Scenario Monte Carlo Implementation Estimating Fault Probabilities with Monte Carlo methods Overview of Monte Carlo methods Applying MC Methods to the Multiple Fault Scenario Monte Carlo implementation results Results with accurate priors Results with uniform prior Circuit Entropy Minimization Definition of Entropy Entropy results from storm simulations Entropy minimization policies Pure exploitation Boltzmann exploration Entropy minimization policy results Discussion and Conclusion Discussion Further research A Selected code 69 vii

8 List of Tables 3.1 Ex1. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex2. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex3. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex4. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex5. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex6. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Ex7. Fault probabilities with diverse node populations & uniform failure distribution & 0.2 call probability Ex8. Fault probabilities with diverse node populations & non-uniform failure distribution & 0.01 call probability Pure exploitation (PE) vs. Boltzmann Exploration (BE) with ρ = 5 and uniform priors. Averaged over 10 storm simulations viii

9 List of Figures 1.1 New York City blackout on October 29, Source: Taylor, Structure of the Electric Grid. Source: Quarkology U.S. Electric Utilities. Source: APPA, PSEG s Electric Territory. Source: Power2Switch R&D Expenditure in Different Industries. Source: DOE Example Estimates of the Cost of Power Outages. Source: DOE Cumulative Distribution of Five-Year Atlantic Basin Cyclone Track Forecast Errors. Source: NOAA, Sandy Forecasts Four Days Before Landfall. Source: Blake, Robust grid simulator visualization of PSEG distribution circuits Severe storm simulation Sample Radial Circuit Single no light call received from N 12. Fault probabilities associated with each edge shown in red, assuming uniform distribution of edge failures and two customers served by each transformer Reduced sample circuit from Figure 3.2, showing only positioning of protective devices Larger Sample Reduced Circuit A portion of circuit KUL ix

10 4.2 Portion of circuit CED8021 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior Southern portion of circuit CED Circuits CLF8023 & KUL8022 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior Circuit COR8044 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior Circuit COR8044 after severe storm simulation with 200,000 fault scenario samples per circuit and uniform prior Circuit entropy values with θ = and accurate priors. Approximately 10 simulations per storm type and call probability value Circuit entropy values with θ = and accurate priors. Approximately 10 simulations per storm type and call probability value Circuit entropy values with θ = 0.01 and accurate priors. Approximately 10 simulations per storm type and call probability value Circuit entropies with uniform priors and θ = Approximately 10 simulations per storm type and call probability value Percentage change in circuit entropy with accurate priors, θ = Percentage change in circuit entropy with uniform priors, θ = Tuning the ρ parameter for Boltzmann exploration. Uniform priors with θ = Tuning the ρ parameter for Boltzmann exploration. Accurate priors with θ = Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy x

11 5.10 Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy xi

12 Chapter 1 Introduction Hurricane Sandy, often referred to as Superstorm Sandy, came ashore near Atlantic City, New Jersey at 8 p.m. on October 29, By landfall, Sandy was in fact no longer considered a hurricane, but a post-tropical cyclone. However, Sandy s path up from the southeast increased the height of its storm surge since a cyclone s highest storm surge and strongest winds are to the front and right of its circulation. In addition, Sandy s landfall coincided with the full moon and high tide - in New York this resulted in a nearly 14 foot storm surge, which set a record for New York Harbor (Drye, 2012). Hurricane Sandy is estimated to have caused over $50 billion in damages, which makes it the second costliest hurricane on record in United States history - behind only Hurricane Katrina in 2005, which caused an estimated $108 billion in damages (Drye, 2012). Utility companies in New Jersey and New York, such as Public Service Electric & Gas (PSEG), were especially hard hit by Hurricane Sandy. Out of 2.2 million total PSEG electric customers, an unprecedented 1.7 million, or over 75%, lost electric power - across the eastern U.S. an estimated 8 million customers lost power (PSEG, 2012). The problems for PSEG and other utilities were further compounded by a nor easter storm that followed shortly after, from November 7 to November 10, 1

13 Figure 1.1: New York City blackout on October 29, Source: Taylor, This nor easter brought significant early season snow and cold temperatures to the East Coast, which was still recovering from the aftermath of Hurricane Sandy. This combination of wintery conditions and widespread power outages raised safety concerns, making the speedy restoration of electricity to customers of high importance. In the two weeks following Hurricane Sandy, PSEG made 2.1 million electric service restorations, a record for any utility in the United States over such a time span (PSEG, 2012). Ninety-six substations, or 39% of all PSEG substations, were affected, a third of transmission lines interrupted, and over 1,000 transformers were damaged. Almost 4,500 foreign utility crew members and 1,500 foreign crew trucks were brought in to assist with PSEG s restoration efforts (PSEG, 2012). Clearly the process of restoring electric power to customers after Hurricane Sandy was a largescale project. Although damages were eventually repaired, Hurricane Sandy and the events that followed highlight the need to improve the reliability of PSEG s electric network. Currently, one major problem is that much of the state of PSEG s grid is unknown, meaning the locations of power outages are often not automatically detectable. The 2

14 most important source of information in such cases happens to be customers that call PSEG s customer service center to report a loss of power - referred to as no light calls by utility companies. In this thesis, we seek to improve the efficiency and speed of the outage restoration process by using information gained from no light calls to predict the location of faults that cause power outages with greater accuracy. Chapter 2 will provide an overview of the grid data and storm simulator, as well as a brief overview of storm forecasting. Chapter 3 will detail the probability model, beginning with a single-fault scenario and moving on to the more realistic multifault scenario. Chapter 4 will introduce Monte Carlo methods as a tool to combat the problem of high dimensionality in the multiple fault scenario. Chapter 5 will introduce the concept of entropy as a measurement of the uncertainty in our belief of the edge fault probabilities across a circuit. We will then use two simple policies to decide where to send lookup trucks to gather fault information in an effort to minimize circuit entropy. Finally, in Chapter 6 we will discuss our results and future research areas. 1.1 An Overview of Electrical Grids Electrical grids exist to deliver electricity from suppliers to customers. This delivery requires a complex network that we will briefly describe in this section. In general, an electric grid can be split into two main sections: the electric transmission network and the electric distribution network. Power generating stations produce electrical power that is carried via high-voltage transmission lines to the distribution networks that serve individual customers. After providing an overview of the transmission and distribution systems, we describe the main types of protective devices. In this thesis, we will end up focusing mainly on the distribution network, and the layout of protective devices in the distribution network, since this is the part of the electrical 3

15 grid from which no light calls are received. Figure 1.2: Structure of the Electric Grid. Source: Quarkology The Electric Transmission Network The electric transmission network transfers large amounts of electricity from power generating plants to the electrical substations, located closer to population centers, that serve customer demands. The U.S. transmission network is made up of more than 200,000 miles of high voltage transmission lines - which typically means lines with a voltage of 230 kilovolts (kv) or greater (EEI). Power generators produce electricity with a voltage around 25 kv. Since these power generating plants are located far from the distribution networks they serve, the generated electricity must be transported in bulk over long distances. Electricity transmission can be done more efficiently at higher voltages. Thus, generated electricity is first sent to transmission substations, which step up the voltage from around 25kV to 230 kv or above. This change in voltage is performed by electrical devices called transformers. A transformer transfers energy from one circuit to another and can be used as a voltage converter - it can change the voltage at its input to a higher or lower voltage at its output. Once the electric power has been stepped up by transformers at transmission substations, it is delivered to the transmission lines. Overhead transmission lines are 4

16 usually used as underground power transmission carries significantly higher costs and limitations. Another important limitation is that, in general, electrical energy cannot be stored and must be generated as needed to satisfy demand. Therefore, the transmission network must work to ensure that electric power generation matches demand closely - a delicate balancing act that requires a complex control system. If supply and demand are not balanced well, power generation plants and the transmission network can shut down, leading to large regional blackouts such as the Northeast blackout of 2003, which affected over 50 million people (Minkel, 2008). For this reason, electric transmission networks are interconnected into regional and national networks, which provide several redundant alternate routes for power to flow should weather or equipment failures disrupt the network. The subtransmission network shown in Figure 1.2 is a part of the transmission network that runs at lower voltages to serve larger industrial customers. There is no set cutoff between the transmission and subtransmission networks, or the subtransmission and distribution networks, and the voltage ranges often have overlap The Electric Distribution Network The electric distribution network is the part of the electric grid that directly serves residential, commercial and smaller industrial customers, the final stage of electric power delivery. The distribution network transports electricity at lower voltages, typically between 2 kv to 34.5 kv. Transmission voltages are higher than these levels, so additional transformers located at distribution substations must step down the voltage. Since distribution substations are positioned much closer to the end customers than the generating plants, the decreased efficiency of transporting at lower voltages is no longer such a limiting consideration. These distribution substations feed the rest of the distribution system through primary distribution feeders that typically emanate radially from the substation (McCalley, 2005). 5

17 The majority of primary distribution feeders are three-phase and four-wire. The length of the feeders is based on the load density at the location - for areas where customer load density is high, the primary feeders will end close to the customers, meaning the secondary feeders will be shorter, while in areas where load density is lower, the secondary feeders will generally be longer. This difference is a big reason why a distinction is often made between distribution networks in rural areas and urban areas. The distribution substation and primary feeders form the primary distribution network. Branching off from the primary network are secondary feeders, often referred to as laters, or taps, which form the secondary distribution network. Distribution transformers between the primary and secondary distribution networks further step down the voltages from primary feeders to voltage levels for residential service - in the U.S. this is 120V or 240V. The most common distribution system is a radial circuit as it is the simplest, and cheapest to implement. There are also several other configurations for distribution networks such as primary auto loop systems, primary or secondary selective systems, spot network systems, etc. However, at any moment in time, given information on the network configuration, we know the direction that current is flowing across the circuit Protective devices Located throughout the distribution network are various types of equipment such are fuses, reclosers, circuit breakers and switches, These components operate to prevent damage to the electric distribution system, and are referred to as protective devices (Hosseinzadeh, 2008). At any given time, each protective device is set to either opened or closed, meaning its status can be described with a binary variable. The objective of protective devices is to maintain the stability of the distribution system by isolating 6

18 parts of the network that have faulted, thereby leaving the rest of the network still functional. Circuit breakers function by detecting a fault condition and automatically opening to interrupt the flow of current. Reclosers are circuit breakers that are equipped with a mechanism that allows them to automatically close after the breaker has opened due to a fault. Fuses are also capable of sensing and disconnecting faults. Each fuse has a set load limit - if there is a surge of current due to a fault that surpasses this limit, the fuse automatically deteriorates, disconnecting the portion of the circuit downstream of it. Therefore, unlike circuit breakers and reclosers, fuses must be manually replaced once they are triggered. However, fuses are a cheaper alternative, making them more common on secondary distribution networks, while circuit breakers and reclosers are often placed closer to distribution substations. Switches are different in that they are not automatically triggered in the event of a fault. Switches can be opened and closed to divert current and reconfigure the network. Switches are safe to open under a normal load current, while circuit breakers, fuses and reclosers are safe to open under a fault current. Protective devices do not always function as desired, and when they fail to do so, fault currents spread to other parts of the network. These fault currents quickly ramp up in magnitude and the excess current can tear apart equipment like transformers, power lines and buses. A short current also heats up circuit parts with poor conductivity, which is a common cause of fires. However, in this thesis, we assume that protective devices operate as intended. 1.2 Electric Utilities & PSEG There are more than 3,000 electric utilities spread across the United States, serving over 137 million customers (APPA, 2013). These utilities sell $400 billion worth of 7

19 electricity each year, which is sent over 2.7 million miles of power lines (Chris Martin, 2013). Figure 1.3: U.S. Electric Utilities. Source: APPA, There are four major electric utility companies in New Jersey that satisfy the electricity demand of the state s nearly nine million residents. Figure 1.4 shows the territory of the four electric utility companies. PSEG s territory corresponds to a narrow corridor from central to northeastern NJ. However, despite what appears to be a limited area, this corridor coincides with many of the most densely populated areas in the country - after all NJ is the most densely populated state. In fact, the approximately 2.2 million customers that PSEG serves makes PSEG the largest electric distribution and transmission utility company in NJ, and the ninth largest in the entire country. Presently, PSEG, along with many other electric utilities, cannot monitor the status of their distribution system at the individual customer level, meaning much of the 8

20 Figure 1.4: PSEG s Electric Territory. Source: Power2Switch. state of the system is unknown. This creates a problem when trying to locate where outages have occurred across the network. Instead, PSEG relies heavily on customer no light calls. PSEG highly encourages its customers to report outages and any additional information that may help in the power restoration process. Anecdotally, we believe approximately one percent of PSEG customers that experience an outage will make a no light call. This percentage seems surprisingly low, although it likely varies across regions and may increase during high stress situations. For instance, over the two-week period following Hurricane Sandy, more than 2.1 million calls were made to PSEG s customer service center, suggesting a call percentage higher than one for such scenarios. PSEG s Outage Management System (OMS) groups together no light calls until 9

21 enough calls are received from a part of the network to identify a possible outage. However, this current OMS system uses relatively simple rules, and depends largely on the experience and human judgement of its operators. In fact, in the immediate aftermath of Hurricane Sandy, damage to the network was so widespread that the OMS was not utilized. Instead, lookup and repair trucks were dispatched from distribution substations and instructed to drive along circuits, reporting or fixing any damage they came across. Lookup trucks collect information about the location of outages and assess damage - they are so named because they often do so by looking up at the overhead power lines they drive past and inspect - but do not perform repairs themselves. The repairs are done by repair trucks that carry with them the necessary materials - repair trucks are often dispatched after information has been collected by lookup trucks. However, lookup and repair trucks take time, are expensive to dispatch and repair trucks cannot carry parts to fix every possible type of fault. Thus, PSEG wishes to send lookup trucks to locations where the expected value of information gained is greatest, and repair trucks to locations likely to have faults - and if a fault is indeed near the location, PSEG wants the repair truck to have the necessary parts to perform the repair. 1.3 Causes of outages A wide range of circumstances can cause faults that result in power outages. According to the Edison Electric Institute, approximately 70% of power outages in the U.S. are due to weather. Lightening seeks to reach the ground, meaning utility poles, wires, transformers and other electrical equipment are easy targets - lightening also often hits trees, which can cause them to fall across power lines. Strong winds from storms can knock down trees across power lines, or cause power lines to swing and 10

22 come together, creating a short circuit. Snow and ice can build up on wires, causing them to break - or weigh down trees enough to topple them onto power lines. Heavy rain can cause flooding which damages electrical equipment. Natural disasters such as Hurricane Sandy are a major cause for these weather events. Animals that come into contact with power lines account for a surprisingly high number of outages, approximately 11% (Supply). When birds, squirrels and other animals climb on equipment such as fuses or transformers, they can cause a short circuit - animal guards are sometimes installed on equipment to prevent this from happening. The remaining percentage of outages are mostly man made - caused by vehicle or construction accidents, maintenance or human error. 1.4 Grid Improvements & Smart Grids Figure 1.5: R&D Expenditure in Different Industries. Source: DOE. For the past thirty years, growth in peak demand for electricity has exceeded 11

23 transmission growth by nearly 25% each year (DOE). However, as seen in Figure 1.5, research and development in the electrical utility industry as a percentage of revenue is among the lowest of any industry. Power outages and quality issues are estimated to cost American businesses more than $100 billion each year. Reliability is one area in need of significant improvements. In the past forty years, there have been five blackouts on a massive scale, with three of the five having occurred in the past decade (DOE). Without improvements, the blackouts and brownouts will continue to occur more often, and inflict more economic damage, due to lack of automated measurements and analytics, lack of information about the state of the grid, etc. Figure 1.6: Example Estimates of the Cost of Power Outages. Source: DOE. In other words, the electrical grid is struggling to keep up with increased consumer demand and the advancement of technology - it is in great need of improvements. The Department of Energy and the U.S. Government are pushing for a transition towards a smarter grid. This transition involves the implementation of many different technologies. One example is grid visualization technology that will provide more situational awareness, using real-time sensor data, weather information, and grid modeling (DOE). The hope is that such visualization technology will eventually allow utility companies to examine the state of the grid even at the street level, providing more 12

24 information about outages, power quality and system operation. In fact, the robust grid simulator introduced in the next chapter is a step towards this visualization. A transition towards a smarter grid requires investments not typically made by electrical utilities. However, a smarter grid can generate cost reductions due to improved efficiency, reliability and energy savings. Whether smart or not, additional infrastructure must inevitably be built to handle increased demand and improve grid function - The Brattle Group estimates that about $1.5 trillion will need to be spent between 2010 and For instance, PSEG s Energy Strong program calls for $3.9 billion in investments over the next 10 years to improve infrastructure against failures and severe weather. This includes $200 million to create additional redundancies in the system, and $454 million for integration of smart grid technologies (PSEG, 2013). It is clear that there is increasingly wide acknowledgement of the fact that electrical networks across the country need more upgrades and investment. However, implementing improvements is a costly process, with many benefits only coming into full effect over the longer term. PSEG is a publicly traded utility company, which must do all it can to lower costs - meaning the goal of a fully smart grid able to give operators information about the state of the grid at the individual customer level is likely many years from being reached. These practical limitations are a major motivation for this thesis. The probability model that this thesis outlines relies on information from no light calls - information that is already available at no additional cost. It will seek to improve upon PSEG s existing OMS without the need to implement smart grid technologies - an improvement that will be useful over the near future, and possibly even after the grid becomes much smarter. 13

25 Chapter 2 Overview of Data and Simulator This chapter begins with a brief overview of hurricane forecasting. We then describe the data provided by PSEG on the layout of their electrical grid. This data forms the foundation for the robust grid simulator being constructed. We review the parts of the simulator relevant to this thesis - the parts that involve generation of storms and the resulting faults, outages and no light calls. 2.1 Hurricane Forecasting The National Oceanic and Atmospheric Administration (NOAA) is the federal agency charged with forecasting and warning against hurricanes, and other extreme weather events. The NOAA aggregates information from several different types of hurricane forecasting models to form their prediction. Currently, the best hurricane forecasting models are global models that solve mathematical equations describing the behavior of the atmosphere around the globe - called dynamical models (Masters, 2011). The European Center for Medium-Range Weather Forecasting (ECMWF) model is one such global dynamical model that has produced very accurate hurricane forecasts since Forecasts rely on data gathered about the storm. Multiple satellites take mea- 14

26 surements that are combined with compute-based climate models to provide information on the storm, such as sea surface temperatures, rain, wind speed, and wave height (NOAA). Buoys in the water can provide further measurements of wind speed, pressure, and water and air temperature. Finally, perhaps most important is data collected by U.S. Air Force and NOAA hurricane hunter aircraft, which fly through the eye of the storm, and NOAA s G-IV jet which flies around the storm (NOAA). In general, about thirty-six to forty-eight hours before predicted landfall, the National Hurricane Center goes into overdrive in communicating to the public their forecasts. The NOAA evaluates storm forecasts by comparing them with a cyclone s best track database - which consists of estimates of the cyclone s center location, maximum wind, and other parameters taken every six hours, in addition to any other post-storm data (NOAA). We can see from Figure 2.1 that, for example, two thirds of 24 hour forecasts have track errors of 52 nautical miles or less. Figure 2.1: Cumulative Distribution of Five-Year Atlantic Basin Cyclone Track Forecast Errors. Source: NOAA, The track forecasts for Hurricane Sandy are widely considered a success. Almost 15

27 four days prior to Hurricane Sandy s landfall in New Jersey, the consensus forecast from the National Hurricane Center showed the storm making a sharp left hook into the Atlantic Coast. This sudden turn from east to west was something that no recorded storm had ever done before - demonstrating how hurricane forecasting models have improved in recent years. However, although track models have become much more accurate, they still often have errors of many miles, which is very significant when considering that the length of PSEG s grid is only about 100 miles. Moreover, forecasting models still have a tough time predicting storm intensity and what is truly going on inside the storm. Figure 2.2: Sandy Forecasts Four Days Before Landfall. Source: Blake, The reason we cover hurricane forecasting and measurement is because of the robust grid simulator being constructed - which we cover in the rest of this chapter. If we know a storm forecast beforehand, the hope is to eventually be able to simulate the approximate impact of the storm on PSEG s electrical grid. This section demonstrates that despite forecasting improvements, we are still a far way off from being able to accurately anticipate a hurricane s effects. 16

28 2.2 The Grid Data The data provided by PSEG describes the structure and arrangement of circuits in their electrical distribution network. The data is initially split by component type - recloser, switch, overhead and underground line segments, circuit breaker, capacitor bank, fuse, etc. The geographic coordinates of each component are given. For components that correspond to protective devices, the default status, open or closed, is also given. Currently only data corresponding to 320 circuits in the northeastern portion of PSEG s electric territory is provided. However, for the purposes of this thesis, the provided data is sufficient. The probability model to be outlined in Chapter 3 can be automatically applied to circuits in the simulator constructed from new data. We also have a file containing a partial history of outages on PSEG s network, which has not yet been processed - the records are also unfortunately quite vague as faults are generally associated with the substation upstream of the fault, rather than the actual location of the fault. We do not have any information regarding customer no light calls. If such information could be obtained, it would help estimate the probability of customers making a no light call in the event of an outage, and whether that probability changes across different regions in the network, or for different types of outage events. 17

29 Figure 2.3: Robust grid simulator visualization of PSEG distribution circuits 18

30 2.3 The Robust Grid Simulator The simulator being constructed for the robust grid project treats the distribution network as a collection of point nodes and connections between these nodes. Each node has an associated type that corresponds to the type of device located at that node. Nodes are connected to each other by overhead or underground power lines to form the network - we call the connections between nodes edges. Nodes are placed using the provided geographic coordinates and connected with line segments that have matching coordinates on one end or the other. Once the circuits in the network have been constructed, the robust grid simulator has the ability to generate storms and run them across the grid. These storm simulations are currently quite far from being realistic. However, the hope is that over the months, or years, after the completion of this thesis, the storm simulation will become increasingly accurate. For this purpose, we do have students from other relevant disciplines working on the robust grid project. The simulator can generate four types of storms of differing intensities - weak, moderate, severe, and Sandy. The difference between the four is the radius of the storm, with weak storm simulations having the smallest impact radius and Sandy storm simulations having the largest. Once a storm is generated, we generate faults across the circuits, and if a fault has taken out customers, which customers without power will make an no light call. Faults are generated as the storm makes its way across the grid. A simulated storm s path is split into a set number of discrete steps. At each step, any point of a circuit within a certain radius of the center of the storm is eligible to fail - the probability of a point failing increases with its proximity to the center of the storm. We coin flip to see if a point has failed for each step of the storm in which the point falls within the failure radius. If a point does fail, then we immediately calculate which part of the circuit will go down as a result, and thus which distribution transformers 19

31 Figure 2.4: Severe storm simulation serving customers lose power. For each customer behind a transformer that has gone down, we coin flip again to see if the customer makes a no light outage call. Note that with this method of generating faults, we cannot limit the number of faults per circuit, as we had initially attempted to do. We can however change the fault probability parameter to lower or raise the likelihood of faults occurring. The failure generation radiuses are 100m, 200m, 300m and 400m for weak, moderate, severe and Sandy storm simulations respectively. After the storm has finished making its way across the electric grid, we store the results of the storm - where faults have occurred, and which customers have called. In addition, for each point we store the probability of that point failing under the same simulated storm. For example, a point may have a 0.1 probability of faulting 20

32 under a stored simulated storm if that same storm path and intensity were run again - whether the point actually faulted does not affect this probability. This probability can be obtained by calling a function within the simulator, and we later use these probabilities as a setting for our prior belief of edge failures. The end goal is to use the robust grid simulator as a platform for testing algorithms, policies and rules. This will be done by running repeated iterations of storms on the grid data, and recording the results. Although the simulator does not yet sufficiently approximate the effects of a real storm on the grid, we can still use it in this thesis to test edge fault probability calculations and observe how these calculations change as we vary the parameters of the simulator. 21

33 Chapter 3 The Probability Model In this chapter we will outline the probability model that we will use to predict the location of faults across the distribution network. We begin by assuming a maximum of one fault occurrence per circuit. The single fault calculations will reveal a way to simplify the structure of each circuit. Finally, we move on to the logic for a multiple fault scenario, where we remove the cap on the number of faults per circuit. 3.1 The Single Fault Scenario The large majority of PSEG s distribution circuits are radial and therefore we develop the probability logic assuming a radial circuit model. A radial circuit has no closed loops and a single power supply from which power is supplied downstream to the rest of the circuit. This means that from each node, there is a unique path by which we can trace upstream to the root node. Figure 3.1 shows a small sample circuit. Nodes are labeled N i, i = 0, 1,..., M, meaning there are M + 1 total nodes in the circuit with N 0 corresponding to the root node from which power is supplied. Other nodes correspond to protective devices, distribution transformers, or dummy nodes used to preserve the structure of the circuit. Edges are labelled E j, j = 1, 2,..., M, with E j corresponding to the edge 22

34 between N j and the node directly previous. A quick note that this notation will be changed slightly when we move on to the multi-fault scenario. Figure 3.1: Sample Radial Circuit Fault Occurance When a fault occurs on an edge, we assume the network responds by opening the first protective device reached in an upstream trace from the faulted edge. When a protective device opens in response to a fault, power is shut off to all nodes downstream of the opened protective device. This means that customers at the distribution transformer nodes downstream will lose power. When a customer loses power, that customer immediately becomes a candidate for making a no light call. We assume that a customer will not make a false no light call - if a customer has not lost power, they will call to report an outage with probability zero. 23

35 For example, in the sample circuit in Figure 3.1, if a fault occurs along E 5, E 6 or E 7, the protective device at N 4 will open in response. This will cut off power to N 5, N 6 and N 7 - all customers served by the transformers at these nodes will lose power and become candidates for making a no light call Using Bayes Theorem to Calculate Edge Fault Probabilities We initially limit the number of possible faults per circuit to one. This is because the single fault scenario is theoretically and computationally simpler, allowing for the introduction of the logic we will use. The goal of the probability model is to estimate the probability that a fault has occurred along a certain edge in a circuit, given a record of no light calls and a known storm path and intensity. In this section, we condition on the storm because we assume that the approximate path and intensity of a storm that has just hit the grid can be measured and recorded - which is generally true in real world scenarios. This allows for a better estimate for the prior belief of edge failure probabilities. Nevertheless, conditioning on the storm can be easily removed as shown in the following section. 1, if E i has faulted Let F i = 0, otherwise Let S be a random variable describing the storm. Let H be a random variable describing the record of no light calls, or the call history for short. Let C m be a random variable equal to the number of no light calls received from N m in the circuit. Let D m be a random variable equal to the number of customers who did not make a no light call from N m. We can now define a realization of H as describing the realizations of C m and D m for each N m in the circuit. At any point in time after a storm, we know H. We also know the approximate 24

36 path and intensity of the storm described by S, from data collected as the storm passes by. Therefore, we wish to calculate P[F i = 1 H = h, S = s], i. Note that because F i is an indicator variable, P[F i = 1 H = h, S = s] = E[F i H = h, S = s]. We can then use Bayes Theorem to expand this term. P[F i = 1 H = h, S = s] = P[H = h, S = s F i = 1]P[F i = 1] P[H = h, S = s] = P[H = h F i = 1, S = s]p[f i = 1 S = s]p[s = s] P[H = h S = s]p[s = s] = P[H = h F i = 1, S = s]p[f i = 1 S = s] P[H = h S = s] = P[H = h F i = 1, S = s]p[f i = 1 S = s] j P[H = h F j = 1, S = s]p[f j = 1 S = s] In most cases, the denominator of the term obtained using Bayes Theorem is the trickier calculation. This is true here, where calculating the denominator involves calculating the numerator. Let p m be the probability that a customer served by transformer node N m makes a no light call if N m loses power. For instance, if N m serves two customers and loses power, the probability both customers will make no light calls = p 2 m. As mentioned previously, we believe p m be approximately 0.01 on average. In this thesis, we assume that p m is uniform for all transformer nodes m - using this notation allows for this assumption to be removed in the future. By conditioning on a single fault scenario F i = 1 and a storm S = s, the probability of observing H = h can be computed. This is done by first figuring out which parts of the circuit will fail under the fault scenario. Then, we know which customers have lost power, and can calculate the probability of observing the given call history H, using our assumption about the value of p m. 25

37 Define Φ i (N m, h) = 0, if under F i = 1, N m does not lose power and c h m > 0 p ch m m (1 p m ) dh m, if under Fi = 1, N m loses power 1, otherwise Now that we have defined this function, we use it to calculate the probability of observing a call history given a fault scenario and storm. Note that the storm being conditioned on does not effect the following term, and can be essentially ignored in the calculation. M P(H = h F i = 1, S = s) = Φ i (N m, h) We must calculate this term for each possible fault scenario. Since we are looking at the single fault scenario, if there are M 1 edges in the circuit, then there are M 1 fault scenarios. The other term that must be calculated for each fault scenario is P(F i = 1 S = s). This is our prior belief of the edge fault probabilities under storm S = s, which we set before doing the calculations. We have now outlined how to obtain values for P(H = h F i = 1, S = s) and P(F i = 1 S = s), for all i. Finally, we calculate P[F i = 1 H = h, S = s] for all i from these terms using the equations above. m= Without Conditioning on a Storm What if we do not wish to condition on a specific storm S? The probability calculations above can still be done - all that changes is that we now set the prior belief of the probabilities of each fault scenario assuming we do not have information about the storm. This means we need a prior for P[F i = 1] instead of P[F i = 1 S = s]. P(H = h F i = 1, S = s) = P(H = h F i = 1) because given a fault scenario, 26

38 the probability of observing a history is not dependent on the storm - it is based only on the arrangement of and number of customers who have called and who have not called, and the probability of each customer making a no light call. Therefore, conditioning on a storm only changes what value we assign to the probability that an edge fails, the probability of a certain single fault scenario occurring. If we condition on a storm path, this may give us a better estimate of the probability of a fault scenario - for example, we might assign higher probabilities of faulting to edges that had a closer proximity to areas hit hardest by the storm. However, if we do not know enough about the storm to confidently do so, we can simply set a value for P[F i = 1] that we believe may work best, and perform the calculations with those values. Then, we use the following equation, where we take out the storm condition from each term. P[F i = 1 H = h] = P[H = h, F i = 1]P[F i = 1] P[H = h] = P[H = h F i = 1]P[F i = 1] j P[H = h F j = 1]P[F j = 1] In this thesis, we will generally condition on a storm S, since after a storm we at least know approximately where it has hit the circuit. However, as shown in this section, we can quickly remove this condition, as it only changes the setting of the prior belief of a fault scenario Example Single Fault Scenario Calculation We use the same sample circuit in Figure 3.1 to illustrate an example calculation. Figure 3.2 depicts a scenario where a no light call is received from N 12. Suppose that each transformer node serves two customers, and set P(F i = 1 S = s) = 1 M 1, i, 27

39 where M = 13, the total number of nodes in the circuit - a uniform setting for our prior. Let p m = 0.01, m. We loop over each edge E i and and calculate E[F i H = h, S = s]. For E 2 this calculation is done as follows: P[F 2 = 1 H = h, S = s] = P[H = h F 2 = 1, S = s]p[f 2 = 1 S = s] P[H = h S = s] = P[H = h F 2 = 1]P[F 2 = 1 S = s] 12 j=1 P[H = h F j = 1]P[F j = 1 S = s] = P[F 2 = 1 S = s] 12 m=0 Φ 2(N m, h) 12 j=1 P[F j = 1 S = s] 12 m=0 Φ j(n m, h) = p(1 p) p(1 p) p(1 p) , where p = 0.01 For the single fault scenario, the sum of the edge failure probabilities must equal one. In other words, i P[F i = 1 H = h, S = s] = 1. The probabilities for each edge are displayed in Figure 3.2. We see that certain edges have a probability of zero of having failed. This is because we are limiting the number of faults to one. The probability of any of E 5, E 6 or E 7 failing is zero because if any of these edges failed, then the protective device at N 4 would trigger. This would mean that only N 5, N 6 and N 7 would lose power. Since there is a single no light call from N 12, this fault scenario is not possible under the assumption that customers do not make false no light calls - an assumption that seems reasonable in a real-world situation. In this way, many edges under the single fault scenario can be quickly written off as candidates for failure. 28

40 Figure 3.2: Single no light call received from N 12. Fault probabilities associated with each edge shown in red, assuming uniform distribution of edge failures and two customers served by each transformer. 3.2 Circuit reduction with protective devices From the probabilities displayed in Figure 3.2 we notice that the probabilities for edges between protective devices are the same. In fact, we can reduce the structure of each circuit by only considering edges between protective devices, rather than edges between all nodes. This means all relevant information needed to calculate edge fault probabilities can be described by the positions of the protective devices in the circuit, and the number of calls and non-calls downstream of each protective device but upstream of the next protective devices reached by tracing downstream. Figure 3.3 shows the same sample circuit from Figure 3.1, but with the described structure reduction. Using this reduced circuit structure requires a slight modification of the previous notation. E i now corresponds to the portion of the circuit downstream 29

41 of the protective device at N i, but upstream of any downstream protective devices. C i now describes the total number of no light calls received from all nodes along E i, while D i describes the total number of customers that did not call from all nodes along E i. Figure 3.3: Reduced sample circuit from Figure 3.2, showing only positioning of protective devices F i = 1 means that at least one fault has occurred somewhere along E i. For instance, if several trees have fallen down across E i, we treat that as a single fault along F i = 1. This means we are calculating the probability that one or more faults has occurred somewhere along E i - the probability E i has failed - not the specific number of faults across E i. This reduction of circuit structure makes sense given how we believe protective devices function. If multiple edges cause the circuit to respond in the same way in the event of a fault along one of them, they will be assigned the same probabilities, and we can therefore aggregate them into a single edge. 30

42 3.3 Additional Single Fault Scenario Examples with Circuit Reduction We perform some additional example calculations on a more complex circuit, pictured in Figure 3.4. In the first two examples we change the number of people that make outage calls from E 7. Notice that the edge fault probabilities do not change as a result of the increased calls from E 7. In examples 3 and 4, the probability of E 7 becomes zero because a call is received from somewhere along E 6. Only E 1 and E 4 are now possible candidates. Changing the number of calls from E 6 and E 7 in Ex. 3 & Ex. 4 does not change the fault probabilities, and in fact, the proportion of the fault probability of E 1 to E 4 does not change in any of the first four examples. Figure 3.4: Larger Sample Reduced Circuit What happens if we change the number of customers associated with each edge? Interestingly, if we only change the populations of the edges downstream of E 4, shown in Table 3.5, the probabilities do not change. This makes sense since in the denominator term of Bayes equation P(H), changing the number of customers downstream of E 4 results in changes that can be factored out in both the numerator and denominator, and cancelled out. However, if we change the populations of all the edges, shown 31

43 Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / / Table 3.1: Ex1. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / / Table 3.2: Ex2. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /7 0 Table 3.3: Ex3. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability 32

44 in Table 3.6, there is a change in the probabilities - changing the number of customers upstream of E 4 will result in changes in both the numerator and denominator that cannot be factored out. This shows that in our probability logic, we must take into account not only if customers along an edge called or not, but also the number of customers that did so. Notice however, that despite doubling the number of customers for each edge upstream of E 4 in Ex.6, the probabilities changed only very slightly. What happens if we raise the probability of making a no light call? We have used a no light call probability of 0.01 because this is what PSEG believes the call probability is on average. However, this call probability undoubtedly changes across regions, and for different outage situations. Instead, Table 3.7 shows how the probabilities are affected if we set the call probability to 0.2. Suddenly, the probability of the fault having occurred on E 4 rises significantly to almost 1. This result makes sense because if customers are much more likely to make a call when experiencing an outage, then the scenario of E 1 faulting and no customers upstream from E 4 calling becomes highly unlikely. This suggests that the higher the call probability, the more certainty we have about where the fault on the circuit has occurred. In fact, this increases the importance of this thesis problem - PSEG believes the call probability is very low, there is more uncertainty for us to deal with. Another very important parameter is our setting of the priors for each fault scenario. In the examples 1-7, we used a uniform setting for each edge. In Table 3.8, we increase our prior for E 1 compared to E 4. As expected, this increases the calculated probability for the fault scenario where E 1 faults. The ratio of the probability assigned to E 1 in comparison to E 4 changes proportionally to the ratio of the prior probability assigned to E 1 in comparison to E 4. We realize that the probability calculation only depends on the relative proportions of the assigned prior fault probabilities assigned 33

45 Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /7 0 Table 3.4: Ex4. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /7 0 Table 3.5: Ex5. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /7 0 Table 3.6: Ex6. Fault probabilities with diverse node populations & uniform failure distribution & 0.01 call probability 34

46 Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /7 0 Table 3.7: Ex7. Fault probabilities with diverse node populations & uniform failure distribution & 0.2 call probability Edge Calls Non-Calls P(F i = 1) P(F i = 1 H = h) / / / / / / /50 0 Table 3.8: Ex8. Fault probabilities with diverse node populations & non-uniform failure distribution & 0.01 call probability to each edge. For example, if we assigned a weight of 1 for each edge fault scenario in Ex.1-7 instead of a uniform probability of 1 7, P(F i = 1 H = h) would not change. 3.4 The Multiple Fault Scenario So far, we have limited the number of faults per circuit to one. Unfortunately, this restriction is unrealistic when a storm rolls across the grid. During sunny days, when the electrical grid is not placed under severe stress due to weather, it is indeed highly unlikely that multiple failures will occur on the same circuit simultaneously. However, during a storm, the possibility of multiple faults per circuit increases greatly for circuits that are put under stress. Since dealing with post-storm situations is the major focus of the robust grid project, in this section we develop the probability 35

47 logic when we remove the limit on the number of faults per circuit. We still seek to calculate E[F i H = h, S = s], i. Let F be a vector of zeros and ones of length M, where M is the number of edges in the circuit. Each entry of the vector F describes the functional status of an edge in the network - for example, if the entry of F corresponding to E 1 is one, then the vector describes a fault scenario in which F 1 = 1. Let F c i correspond be a vector of zeros and ones of length M 1 describing the fault status of all edges in the network besides E i. Finally, let Ω correspond to the set containing all possible settings of the vector F, and let Ω c i correspond to the set containing all possible settings of the vector F c i. If F has M entries, and each entry can be set to zero or one, then Ω will contain ( ) ( ) ( ) M M M different settings of the vector F. Ω c 1 2 M i, with M 1 ( ) ( ) ( ) M 1 M 1 M 1 entries, will contain different settings of Fi c. 1 2 M 1 Instead of using combination notation, we can say there are 2 M possible settings of a vector of length M where each entry can be one or zero - the combination notation is simply more intuitive when thinking about a x-fault scenario. Note that due to the circuit reduction introduced in section 3.2, M corresponds to the number of protective devices in the circuit, plus the root node - not the total number of nodes in the circuit. We calculate E[F i H = h, S = s] using Bayes Theorem, as before. Define Φ i,f c i (N m, h) = 0, if under F i = 1 and F c i, N m does not lose power and c h m > 0 p ch m m (1 p m ) dh m, if under Fi = 1 and F c i, N m loses power 1, otherwise 36

48 Define Φ F (N m, h) = 0, if under F, N m does not lose power and c h m > 0 p ch m m (1 p m ) dh m, if under F, Nm loses power 1, otherwise P[F i = 1 H = h, S = s] = = F c i Ωc i = F c i Ωc i P[F i = 1, F c i H = h, S = s] P(H = h F i = 1, Fi c, S = s)p(f i = 1, Fi c S = s) F Ω P(H = h F, S = s)p(f S = s) F c i Ωc i [P(F i = 1, Fi c S = s) M m=0 Φ i,fi c(n m, h)] F Ω [P(F S = s) M m=0 Φ F (N m, h)] The two functions Φ F (N m, h) and Φ i,f c i (N m, h) are essentially the same, but we define them separately to keep the notation clean. In addition, note that we may again easily remove the restriction of conditioning on the storm. We still have to somehow set a prior belief for P(F ), F. It is likely that the setting of each entry of the vector F for each circuit in the grid is correlated with the settings of the other entires. In this thesis, we assume the entries are independent, when conditioning on a storm or not - using correlated beliefs may be a topic for future research. 37

49 P[F i = 1 H = h] == = F c i Ωc i F c i Ωc i P(H = h F i = 1, Fi c )P(F i = 1, F c F Ω P(H = h F )P(F ) i ) [P(F i = 1, Fi c ) M m=0 Φ i,fi c(n m, h)] F Ω [P(F ) M m=0 Φ F (N m, h)] Note that the logic we have developed in this chapter applies only applies to circuits from which a call is received, and thus for which we have a call history H. This makes sense because without a call history, we would simply set the edge fault probabilities equal to our priors, and estimating the priors is a different problem altogether. This also means that in the robust grid simulator, we only loop over circuits from which at least one call has been made - since we assume that false no light calls are not made, we know such circuits have at least one fault. 38

50 Chapter 4 Monte Carlo Implementation In this chapter we use Monte Carlo methods to calculate estimates of the edge fault probabilities for the multiple fault scenario. We use Monte Carlo sampling because of the high dimensionality of the vector F, which makes it computationally infeasible to calculate a probability for every possible setting of F. We then implement the outlined sampling technique in the robust grid simulator code and examine the resulting outputs. 4.1 Estimating Fault Probabilities with Monte Carlo methods The motivation for using Monte Carlo sampling for this problem is because of the large number of possible fault scenarios, and therefore the large number of settings of the vector F. Recall that F is a vector of zeros and ones where each entry describes the fault status of an edge in the circuit. Figure 4.1 shows the major portion of circuit KUL8014. Fuse nodes are highlighted in yellow, and reclosers in blue - both are protective devices. If we count the number of protective devices in KUL8014, we surpass 40. This means the vector F describing 39

51 Figure 4.1: A portion of circuit KUL8014 the circuit will contain at least forty entries, each of which can be set to 0 or 1. ( ) 40 Therefore, there are at least = 658, 008 possible five fault scenarios. If we wish 5 to consider twenty fault scenarios, then there are over 137 billion possible settings of F. This is a very large number of scenarios to loop over, and we must also consider that some large circuits may have dozens more protective devices. Although theoretically, looping over all these possible fault scenarios is just a iterative calculation, within the context of the robust grid simulator, the run time for obtaining exact edge fault probabilities would make the simulator unusable for practical purposes. Because in the multiple fault scenario we have to deal with a high dimensional variable F, we turn to Monte Carlo sampling, which is commonly used when the distribution of a variable is known, but high dimensionality raises problems. 40

52 4.1.1 Overview of Monte Carlo methods First, we provide a brief overview of Monte Carlo methods. Monte Carlo methods are used to solve two types of problems: 1) generating samples of a random variable or vector x from a given probability distribution P (x) and 2) estimating the expectations of of functions under the distribution P (x), such as E[φ(x)] = φ(x)p (x)dx (Mackay). Once we have solved the first problem, sampling, the solution for the second problem can be calculated from the samples. With random samples, {x (r) R r=1, the estimator for E[φ(x)] is Ê[φ(x)] = 1 φ(x (r) ) R An interesting property of Monte Carlo simulation is that the accuracy of the Monte Carlo estimate is independent of the dimensionality of the space being sampled. In other words, the variance of r Ê[φ(x)] goes as σ2 R, where σ2 is the variance of φ (Mackay). Rather it is the number of samples generated that is important. Generating samples from a distribution can often be quite difficult. However, in this thesis, we are generating a vector of binary random variables, where we assume each entry is independent of other entries. Thus, to generate samples we coin flip for each entry. For example, suppose we wish to generate samples of a vector Y with two entries, each of which, independent of the other entry, takes on a value of 0 with probability 0.5 and a value of 1 with probability 0.5. For the first entry, we use a random number generator to generate a number uniformly between 0 and 1. If this number is less than 0.5, we set the first entry to 0, or 1 otherwise. The same step is repeated for the second entry. Say that we get y 1 = [0, 1]. This is one sample of the vector Y. We store this sample, rinse and repeat until we have the desired number of 41

53 samples. Then, the expected value of each entry of Y can be computed by averaging the value of the entry over all the samples Applying MC Methods to the Multiple Fault Scenario In order to use Monte Carlo methods for estimating fault probabilities in the multiple fault scenario, we must generate sample fault scenarios. We generate these samples based on our settings of prior beliefs for the probabilities of each fault scenario. If we wish to condition on a storm S, then we generate the samples based on how we set P[F S = s]. If we do not wish to condition on a storm S, then we generate samples from our settings of P[F ]. With the current method of fault generation by the robust grid simulator, each entry of F is independent when conditioning on a storm, which simplifies the process of generating samples. In other words, we can take from the simulator a probability for each entry of F that that entry is 0 or 1, where each entry is conditionally independent given a storm S. If we do not wish to condition on a storm, it is reasonable, perhaps even more realistic, to set a prior where the entries of F are correlated. However, in this thesis, when we generate without conditioning on a storm, we also assume each entry is independent. Let us say we wish to generate X sample fault scenarios for each circuit to use in estimating fault probabilities. With our prior for the distribution of F, to generate a single sample we coin flip for each entry by generating a random number uniformly between 0 and 1. If this randomly generated number is less than the probability of failure of the protective device in the circuit corresponding to the entry of F, then we set the entry to 1. This process is repeated X times for each circuit. Therefore, if we have Z circuits, we generate a total of X Z samples for each storm simulation. If we generate a sample F where the settings of all the entries is zero, we throw out that sample. This is because we are doing the calculations only for circuits from which 42

54 a call has been made, which means we know with certainty at least one fault has occurred. Next, we take the samples generated for each circuit and say that these samples describe all possible fault scenarios for that circuit. Each sample is weighted equally by 1. A fault scenario may occur more than once in the X samples. If the same fault X scenario F occurs λ times in the X samples, then the estimate for ˆP[F ] or ˆP[F S = s] is λ. Note that we use a hat in the notation to denote that we are using Monte Carlo X methods to get an estimate of the probability values. We loop over all the possible fault scenarios in the samples and sum up ˆP[H = h F ] ˆP[F S = s] or ˆP[H = h F ] ˆP[F ] to get ˆP[H S = s] or ˆP[H]. For each sampled fault scenario F, we can calculate ˆP[H = h F ] like before, as M m=0 Φ F (N m, h) For edge i, we loop over each sample F where F i = 1. For each such sample F, we compute M m=0 Φ F (N m, h) and again weight the result by 1. We sum these X values to get an estimate of ˆP[H = h F i = 1]. Finally, using these estimates, we calculate estimates of the edge fault probabilities - take out S = s from each term if not conditioning on a storm. ˆP[F i = 1 H = h, S = s] = ˆP[H F i = 1]ˆP[F i = 1 S = s] ˆP[H = h S = s] Thus, by using Monte Carlo simulation, we are able to get an estimate for the fault probability of each edge with a reasonable number of computations. 4.2 Monte Carlo implementation results The Monte Carlo sampling method for estimating Ê[F i H = h, S = s] is implemented in the StormAnalyzer class in the robust grid simulator. We run several storms across the grid, and using the robust grid simulator s visualization examine the results to 43

55 see the probability calculations and how they relate to the circuit configurations. In general, we can generate about 50,000 samples per circuit for our estimates, and still have each storm run last less than or not much more than 60 seconds Results with accurate priors We initially use accurate priors. By accurate priors, we mean setting the distribution of fault scenarios used for Monte Carlo sampling equal to the actual simulator s fault generating distribution. The simulator generates faults in the following way. The fault probability parameter, θ affects the simulator s generation probabilities of failure, at each time step, for each point in following equation, where d is the distance from the point to the center of the storm path and maxdist is the radius of the storm. P oint F ailure P robability = θ (1.0 d maxdist ) This means that a point right in the center of a storm at time step k will have a probability of θ of failing, while a point that is maxdist or more meters away from the center at time step k will have a probability of zero of failing - the decline in probabilities as we move away from the center is linear. We use 50,000 sampled fault scenarios for each impacted circuit to estimate the probabilities. Figure 4.2 shows a portion of circuit CED8021 after running a severe storm simulation. In the simulator graphics, we weight the edges by their estimated probability of failure - a thicker edge has a higher estimated probability of having faulted, and vice versa. The fault on the left occurred on an edge, say E a, assigned a probability of 1 and the fault at the top occurred on an edge assigned a probability of 0.6. At first, a probability of 1 being calculated for E a may seem like a mistake - it means that for all generated samples that can have caused the observed no light 44

56 calls and where each sample has at least one fault, E a faulted. Figure 4.2: Portion of circuit CED8021 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior. However, if we move down to see more of the circuit in Figure 4.3, we notice that E a continues uninterrupted by protective devices for a long distance. E a is a very long edge which will have many points that can be hit by a simulated storm and subsequently fault. This means that under the current fault generation algorithm, the probability of no faults occurring across is E a is much lower compared to the rest of the edges in the circuit due to its length and the number of points that are located along it. E a is a three phase line, meaning it is considered a primary feeder. There are multiple secondary taps that Figure 4.3: Southern portion of circuit CED8021 branch off from E a, with fuses placed between the connections. These secondary taps mostly have fault probabilities significantly lower than 45

57 one. Edges similar to E a seem quite common in PSEG s distribution circuits. Most circuits have a three phase primary feeder that serves many branching secondary taps. These primary feeders usually run for relatively long distances without interruption by protective devices. Figure 4.4 shows two other circuits impacted in a severe storm simulation. Both have a long primary feeder line that is assigned a probability of 1 of having faulted. This suggests that such a circuit configuration makes it harder to predict faults on other edges. This is because the primary feeder serves many of the secondary taps and thus many call histories are possible under a scenario where the primary feeder faults. Therefore, it is harder to distinguish, in a multiple fault scenario, whether a call history is caused just by a primary feeder fault, or if there is another fault elsewhere. Figure 4.4: Circuits CLF8023 & KUL8022 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior. 46

58 Placing more protective devices along the primary feeder, to split it up into more edges, may be helpful in locating faults. Figure 4.5 shows a portion of circuit COR8044. Notice that in this circuit configuration, there are several reclosers, pictured by the blue nodes, on the primary three-phase feeder. The portion with the thickest red weight is assigned a probability of approximately 1, but this probability cuts off when the line hits the reclosers. The edges downstream of the reclosers are assigned probabilities close to zero. This circuit configuration demonstrates how protective devices, especially on the primary feeder which is most vulnerable to faulting in our simulations, can narrow down which parts we believe to have faulted. Figure 4.5: Circuit COR8044 after severe storm simulation with 50,000 fault scenario samples per circuit and accurate prior. The above figures show how the probabilities tend to be significantly higher, for three-phase primary feeders than for other secondary taps, under our current simulation model. However, this is not necessarily wrong, as they also show the majority 47

59 of faults are in fact on these edges Results with uniform prior Figure 4.6: Circuit COR8044 after severe storm simulation with 200,000 fault scenario samples per circuit and uniform prior. What happens if we modify our prior beliefs for the distribution of fault scenarios? This means that we sample from a different distribution which we believe is correct, but which may be quite far from the actual distribution of fault scenarios. We test the effect of a bad prior by setting a uniform probability of failure for each edge as our prior and running storm simulations. We lack the data to come up with a reasonable level for the uniform probability setting. Thus, we use an arbitrary uniform setting for the sampling distribution given that we do not have data to reasonably estimate this distribution. For each circuit, we take the simulator s probability of generating a failure for each edge under the simulated storm, and sum up these probabilities 48

60 over all edges in the circuit. We divide this sum by the number of edges, and set the prior belief for the probability of failure of each edge equal to this number - then we generate samples from this prior. This way, we are guaranteed to have a fault probability setting that is between 0 and 1, uniform for all edges in a circuit, as well as partially based on the storm being simulated and not completely arbitrary. As expected, we see in Figure 4.6 that the failure probabilities are closer together. Anecdotally, there are no edges anymore that are assigned a probability of one, or nearly one. Instead, the highest probabilities appear to be between 0.5 and 0.6. On the other hand, the probabilities assigned to other edges seem to have risen, with almost no edges having a probability of less than Before, with accurate priors, there were often edges with probabilities of nearly zero. Therefore, from a few storm simulations we see that a uniform prior changes the probabilities greatly in comparison to using accurate priors. However, the results in this chapter are largely anecdotal. In the next chapter, we will further examine and test these results by running repeated simulations, as well as introducing entropy as a measurement statistic. 49

61 Chapter 5 Circuit Entropy Minimization This chapter introduces the concept of entropy and entropy minimization. We use entropy as a measurement of the amount of certainty we have in the edge fault probabilities over a circuit. We calculate a circuit entropy value by summing the entropy values of each edge over the entire circuit. We modify different parameters to see how the resulting entropy values change. Finally, we test pure exploitation and Boltzmann exploration policies for sending lookup trucks to traverse the grid with the objective of minimizing circuit entropy. 5.1 Definition of Entropy Assume we have a belief about a discrete random variable µ. Then the entropy H of µ is defined as H(µ) = i P(µ i) log(µ i ). The entropy we use is actually referred to as Shannon entropy. The idea behind Shannon entropy is to measure the uncertainty in a random variable. For instance, consider flipping a coin. The entropy of the coin toss is highest when the coin is fair - meaning it will come up heads with probability 0.5 and tails with probability 0.5. In this case, H(µ) = [0.5 log(0.5) log(0.5] = On the other hand, entropy of the coin toss is lowest when the result of the coin toss is known 50

62 with complete certainty - when both sides are heads or tails. In this case, H(µ) = [1 log(1)+0 log(0)] = 0. Note that for entropy, we set log(0) = 0. Generally, log with base 2 is used, however for our purposes the natural logarithm works (Warren B. Powell, 2012). Using the Monte Carlo sampling method outlined in Chapter 4, we can come up with numerical estimates for the probabilities of fault for each edge of an impacted circuit. For instance, if we believe an edge E i has a 0.1 probability of having faulted, then the entropy of the edge is H(E i ) = [0.1 log(0.1) log(0.9)] = Entropy results from storm simulations What happens to the entropy values for an impacted circuit when we modify the probability of making a no light call? Or if we modify the density of faults caused by the simulated storms? In this section, we modify several of the major parameters in the robust grid simulator to see the effect on the entropy values over each impacted circuit. This is done by running multiple storm iterations, of differing intensities across PSEG s electrical grid. Then, in each impacted circuit, we compute the entropy value for each edge based on the calculated fault probabilities. The entropy value for an impacted circuit is obtained by summing over the entropy values for all the edges in the circuit. Note that summing the entropies of each edge together is reasonable since we assume that each edge s probability of failure is independent of other edges in the circuit. Finally, we record the average summed entropy value for the impacted circuits for each storm iteration. Circuit j entropy = i H(E j i ) We begin by modifying the no light call probability for customers who have lost 51

63 power. As mentioned before, we have been told this probability is around However, other electric utilities seem to have expressed surprise at such a low call probability. Is this because PSEG s New Jersey customers are busier and do not have the time to make the call - or perhaps the low percentage is due to laziness? Whatever the reason, and whether or not PSEG s estimate is accurate, we wish to see what happens when we change the call probability parameter. Figure 5.1: Circuit entropy values with θ = and accurate priors. Approximately 10 simulations per storm type and call probability value. Figure 5.1 shows the entropy values summed over all edges in a circuit, per circuit, with a setting of the fault probability parameter θ, and using accurate priors. We see that as the call probability increases, the average summed entropy per circuit slopes downwards. This suggests that as a higher percentage of people call in, we know with greater certainty which edges have faulted. This downward trend seems to be more pronounced the larger the storm being simulated - entropy values for simulations of severe and Sandy storms decline noticeably as the call probability is increased from 0.01 to 0.3, while the entropy values for weak storm simulations 52

64 Figure 5.2: Circuit entropy values with θ = and accurate priors. Approximately 10 simulations per storm type and call probability value. remain relatively constant. We also see that the magnitude of the entropy values gets progressively larger with the storm size. This makes sense since a larger storm is more likely to generate more faults across an impacted circuit. This means there may be multiple edge faults that take down the same portions of the circuit, making it harder to distinguish which edge(s) is responsible for the no light calls. Figures 5.2 and 5.3 show circuit entropies with different settings of the fault probability parameter, θ. With a modified θ, we still observe the same general trends, with the entropy values declining as the call probability is increased, and larger storms having greater entropy. When θ is increased we see that the magnitude of the circuit entropies follow suit, and vice versa, for each setting of call probability. This is not surprising since a storm that generates more faults will likely have more uncertainty about which edges have failed. As in chapter 4, we ask what happens when our setting of the prior used for sampling changes. We change the prior to be uniform for edges across the same circuit. This is done using the same uniform setting as before where we sum up the 53

65 Figure 5.3: Circuit entropy values with θ = 0.01 and accurate priors. Approximately 10 simulations per storm type and call probability value. probabilities across the circuit generated by the simulator, and divide by the number of edges. With uniform priors, it is no surprise that the circuit entropy values are larger. In Figure 5.4 we see that the entropy values seem to decline at a steadier pace compared as the call probability increases, in comparison to when we use accurate priors. The magnitude of the declines also appears greater, especially for weak and moderate storm simulations. This suggests that when we use a uniform prior, which is a poor prior in the context of our simulations, circuit entropy is much higher, but additional calls are even more helpful in reducing circuit entropy, particularly for storms that generate fewer faults. Before we take into account information from no light calls to modify edge fault probabilities, we have a prior belief for the fault probabilities. Using just this prior, we can calculate a circuit entropy value. We wish to measure how this prior circuit entropy value compares to circuit entropy values after using our probability logic on the call history. Figures 5.5 and 5.6 show the percentage changes that result. When 54

66 Figure 5.4: Circuit entropies with uniform priors and θ = Approximately 10 simulations per storm type and call probability value. using accurate priors, we see that for non-weak storms, the percentage change in circuit entropy is negative, which is good since we wish to minimize entropy. Weak storms appear as the anomaly, and seem to require a significant increase of the call probability from 0.01, to above 0.05 before a negative entropy change is observed - perhaps due to the narrow damage path of weak storms and the low number of faults. Interestingly, for each storm intensity, the decline in entropy is quite steep, and appears to be approximately linear. When we use a uniform setting for the fault probability prior, surprisingly we see that circuit entropies actually increase for low call probabilities. For weak storms, we again observe that the call probability must be increased further above 0.05 before a negative entropy change occurs, and that the entropy values are highly volatile. Thus, we focus mainly on the moderate to Sandy storm levels. For non-weak storms, a clear negative change in circuit entropy is only observed once the call probability increases past The downward trend takes a concave parabolic shape. This suggests that with a poor prior such as our uniform setting, as the call probability rises, it has 55

67 Figure 5.5: Percentage change in circuit entropy with accurate priors, θ = an even stronger effect on decreasing uncertainty than with an accurate prior. The percentage changes with a uniform prior are also greater in magnitude than with an accurate prior. 5.3 Entropy minimization policies In this section, we use two relatively simple measurement policies, pure exploitation and Boltzmann exploration, as rules for sending lookup trucks with the goal of minimizing circuit entropy. In the real world, lookup trucks are dispatched to gather information about faults - they drive along the part of the circuit they are sent to, looking for faults, and report their findings back to their utility company. For our purposes, we assume that lookup trucks are available to be dispatched instantly and that they will accurately report whether or not an edge they have examined has faulted. We do not consider the problem of optimal pathing for the lookup trucks. Both pure exploitation and Boltzmann exploration are heuristic policies that are popular for 56

68 Figure 5.6: Percentage change in circuit entropy with uniform priors, θ = discrete selection problems, where the set of measurement decisions is discrete and not too large - this works in the context of our problem since the number of edges per circuit is not huge. The decision we make at each time step for each circuit is which edge to measure. Since we wish to minimize circuit entropy, we choose min π H n (E j i ) j i as our objective function, where H n (E j i ) is the entropy of edge i in impacted circuit j after the nth lookup truck measurement. At each time step n, we have a binary decision to make for each edge E i - whether or not we should send a lookup truck to E i to determine if it has faulted. If we do decide to measure an edge E j i, we always observe the true setting of E j i - meaning there is no variance in our measurements and after we measure an edge, its entropy value is immediately set to zero. 57

69 5.3.1 Pure exploitation A pure exploitation policy picks the best decision based on our current set of estimates (Warren B. Powell, 2012). At any point in time, we have a set of estimates for the entropy values of each edge in a circuit. We can exploit our knowledge by sending a lookup truck to measure the edge with the highest estimated entropy, eliminating the uncertainty for that edge and reducing its entropy to zero. Therefore, our decision function for circuit j is described by E j,n i = arg max E j i Ej H n (E j i ) Boltzmann exploration Boltzmann exploration is a policy that mixes pure exploration and pure exploitation (Warren B. Powell, 2012). For each circuit j, we measure edge i with probability p n E j i = e ρ Hn (E j i ) e ρ E j Hn (E j i ) i Ej However, in this equation, the exponent ρ H n (E j i ) can become very large, making it hard to calculate e ρ Hn (E j i ). Therefore, we instead use the following method for each circuit j µ j,n = max E j i Ej H n (E j i ) 58

70 p n E j i = e ρ (Hn (E j i ) µj,n ) e ρ E j (Hn (E j i ) µj,n ) i Ej ρ is a tunable parameter that we need to find a reasonable setting for. If ρ = 0 then we measure each edge with equal probability, which is pure exploration. As ρ increases and approaches infinity, we sample the edge with the highest entropy with a probability near one, which is the same as pure exploitation. Figures 5.7 and 5.8 show the average circuit entropy after one lookup truck is sent (n = 1) using the Boltzmann exploration policy to make a decision, with different settings of ρ. We notice that for a low setting of ρ, such as between one and ten, average circuit entropy after the first measurement is much higher than for values of ρ greater than 50. For both a uniform and accurate setting of our prior for generating samples, it appears that a setting of around 50 for ρ is quite good. We will therefore set ρ equal to 50 for subsequent simulations Entropy minimization policy results We now wish to examine the performance of these two policies in dispatching lookup trucks to minimize circuit entropy. In order to evaluate the performance of a policy, we calculate, at each time step n, the percentage change in average estimated circuit entropy from time step 0 to n. We limit n from 0 to 3, although this limit can be easily lifted. Note that at time step n, there have been n lookup truck measurements made - for example, at n = 1, we have sampled one edge and discovered its status. At each time step, after calculating the entropy of each edge in a circuit, we use one of the above two policies to dispatch a lookup truck to measure an edge. We keep track of the edges which we have measured and their settings, sampling fault scenarios of the other non-measured edges for the probability calculations. Running simulations to 59

71 Figure 5.7: Tuning the ρ parameter for Boltzmann exploration. Uniform priors with θ = see the average percentage change in circuit entropy from time 0 onwards will provide numerical data on which policy reduces circuit entropy after 1, 2 or 3 lookup truck measurements at the fastest rate. Keep in mind that we still assume independence of edge fault probabilities. Figures 5.9 through 5.11 show the comparison between the pure exploitation and Boltzmann exploration policies when tested with the robust grid simulator. Looking at the percentage changes in entropy as a result of the lookup truck dispatch decisions by the two policies, it is hard to pick a clear favorite - pure exploitation and Boltzmann exploration seem to perform quite similarly. Our choice of the tuning parameter for Boltzmann exploration may explain this - setting ρ equal to 50 is a high value and thus our Boltzmann exploration policy may actually be quite similar to pure exploration. These figures also show how efficiently the policies are in reducing circuit entropy. In each of the graphs, it is clear that the first lookup truck measurement has the most profound affect on minimizing circuit entropy. For each setting of the call probability 60

72 Figure 5.8: Tuning the ρ parameter for Boltzmann exploration. Accurate priors with θ = parameter p and for each policy and type of storm, the first lookup measurement results in a percentage decrease in circuit entropy greater than the next two measurements combined. This result is not completely unexpected since after we take a measurement, we resample the possible fault scenarios with the measured edge set deterministically. Therefore, we are likely to see decreasing returns with each additional measurement. Since we set ρ to be so high for the above simulations, let us see how Boltzmann performs when we have a much lower setting. In this way we can somewhat test how a pure exploration policy would do. In fact, as mentioned before, with a tuning parameter of 0, Boltzmann exploration becomes pure exploration. However, we do not expect a policy close to pure exploration to do well because edges in the distribution network vary widely, as we saw from some of the figures in Chapter 4. 61

73 Figure 5.9: Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy. n PE BE PE-BE Table 5.1: Pure exploitation (PE) vs. Boltzmann Exploration (BE) with ρ = 5 and uniform priors. Averaged over 10 storm simulations. Boltzmann exploration with ρ = 5 does not perform as well as pure exploitation, which is not a surprise given that low values of ρ performed poorly while tuning the parameter. On the other hand, we again observe the importance of the first measurement - after the first lookup truck has sampled an edge, the average difference in entropy is about -0.25, but after two more samples, the difference between the two policies has not increased. 62

74 Figure 5.10: Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy. The numerical results in this section suggest that pure exploitation - where we send a lookup truck to measure the edge in each circuit that we estimate has the highest entropy at each time step - is a decent policy, especially for the first measurement. Overall, using circuit entropy as a metric has allowed us to gain an idea of how our belief of the uncertainty in a circuit s edge fault probabilities changes as we change the major parameter inputs in our model. 63

75 Figure 5.11: Average % change in circuit entropy from time 0 to n, with uniform priors & no light call probability p = simulations per storm level and policy 64