A Probability Model for Grid Faults Using Incomplete Information

Transcription

1 A robability Model for Grid Faults Using Incomplete Information Lina Al-Kanj, Belgacem Bouzaiene-Ayari and Warren B. owell Operations Research and Financial Engineering Department, rinceton University, NJ, USA {lalkanj, belgacem, Abstract Utilities face the challenge of responding to power outages due to storms and ice damage, but most power grids are not equipped with sensors to pinpoint the precise location of fault causing the outage. Instead, utilities have to depend primarily on phone calls (trouble calls) from customers who have lost power. This paper presents an information model of the grid in the presence of outages; the developed model is used to estimate the probability of power line faults causing the outages. However, the computational complexity of the problem grows exponentially with the number of power lines in the grid. Thus, several methods are proposed for handling the combinatorial growth of events and the behavior is demonstrated using the data of a real power grid. erformance results show that power line fault detection can be achieved with high accuracy even with a very low percentage of customers calling to report an outage. Index Terms ower outage, Grid fault, Trouble call, robability of fault, Distribution system fault, ower line fault, robability model I. INTRODUCTION Climate change is producing more powerful storms, increasing the frequency and severity of outages in the power grid. According to the Edison Electric Institute [1], on average 55% of power outages in the U.S. are due to weather and it can reach up to 80% in some years [2]. Wind (primarily) and ice frequently bring trees and branches down on power lines creating sporadic outages that quickly spread through the grid due to the limited number of protective devices that are triggered due to a short circuit. It is estimated that 90% of customer outage-minutes are due to faults affecting the local distribution systems [2]. Whenever a power line is damaged, the closest upstream protective device disconnects to avoid overloading of the downstream network components which can form a major part of the distribution circuit; this results in power outage to all customers served through that protective device. Fast and accurate power fault detection has been a concern for utilities since the deployment of power systems which has driven the research community to develop various algorithms to detect the location of faults as summarized in [3], [4]. Different utilities use different methodologies for management and power restoration during an outage; while most utilities install measurement units on most of the transmission systems, distribution systems are not equally automated due to the cost of the equipments. Some utilities have advanced automated systems that include SCADA such as Carolina ower & Light utility [5], but the majority of the utilities do not have large scale installed sensors. Instead, utilities depend primarily on phone calls which are referred to as trouble calls from customers who have lost their power. Complicating the situation is that as few as one percent of customers call when they lose their power, creating tremendous uncertainty in the knowledge of the state of the grid. For the transmission system, most approaches rely on collected measurements; Ferreira et al. [6] propose the measurement of the electric current using magnetic field sensing coils instead of transformers to avoid erroneous measurements during fault transients. Fault location using wavelets multiresolution analysis to process the measurements is presented in [7], [8]. Some papers use the extracted measurements along with artificial intelligence techniques to detect power line faults such as fuzzy inference systems [7], [9] and neural networks [7]. ower line fault detection incorporating the effect of power swings is investigated in [10]. Distribution systems have different configurations; the most studied configuration in the literature is the radial distribution system where there is a single path for power flow from the substation to the consumers. This enables utilities and researchers to propose escalation algorithms for fault identification based on customer calls and grid topology [11], [12]. Escalation algorithms gather the set of calls and then for each customer call, up tracing is done till the first common location for all calls is located; this location would be identified as the faulted one. Escalation algorithms suit single fault scenarios but cannot capture the case of multiple fault scenarios. An improved escalation algorithm for a heat storm is presented in [13] where the escalation from the locations of calls depends on the type of the upstream devices. Artificial intelligence techniques that make use of customer calls to identify the fault locations are also investigated to provide better performance than the simple heuristics described above. For example a neural network is presented in [14] but the main limitation is in determining the sample training set. An approach based on fuzzy set theory and tabu search is proposed in [15] but the limitation is in the computational complexity that becomes intractable for large networks. Knowledge-based outage identification that make use of SCADA and automated meter reading to provide the electric utility center (EUC) with knowledge about the status of the distribution system on top of customer calls is proposed in [16]. There are also works that just rely on measurements to identify the locations of faults [17] [19]. But, most of the utilities do not have automated distribution systems and thus, primarily rely on the grid topology, the phone calls of the customers and the experience of the utility personnel to estimate the locations of outages. Moreover, since only a few customers will call, identifying the locations of the faults across the power system is still a major dilemma for the EUC which needs to restore the network as fast as possible. The main contribution of this paper is developing a probability model that estimates the likelihood that damage has

2 2 occurred on a power line based only on customer phone calls (trouble calls) and the configuration of the grid (which contains prior information in the form of prior probability of power line fault). However, the problem proves to be combinatorial and it grows exponentially with the number of power lines. Thus, several computational complexity reduction techniques are proposed by making use of the properties of the probability model. The performance of the probability model is tested using a simulator that simulates the power grid of the state of New Jersey by using real data provided by SE&G which is the main utility in the state. robabilistic modeling is a mainstay of artificial intelligence that provides essential information for analyzing data. The developed probability model, in this paper, provides basic and necessary information that is needed to route utility crews efficiently across the power grid in order to restore the power grid as fast as possible. In this case, the locations with higher probability of fault will be certainly given a higher priority to be visited first. It can be also used in developing a detector that detects power line faults within a certain confidence interval. robabilistic models have gained increased research attention in power systems for different objectives; for example, probabilistic models for consumer loads in distribution systems are developed in [20] [22] which can be later used for the dimensioning and optimization of distribution systems. Other probabilistic models are developed to estimate the required number of spare substation transformers [23] or wind farm generation for reliability studies [24]. The paper is organized as follows. Section II gives a mathematical model of the flow of information for a grid as some event such as a storm evolves producing faults and loss of power. Section III derives the posterior probability of power line faults based on phone calls from customers and the structure of the grid which provides prior information. The developed probability model is intractable; Section IV presents several strategies for reducing the computational complexity. erformance results and analysis are presented in Section V. Finally, conclusions are drawn in Section VI. II. SYSTEM MODEL ower systems are typically divided into stations/substations that supply the power, the transmission system that carries the power from the generating centers to the load centers, and the distribution system that feeds the power to the residential and industrial consumers/customers. Distribution systems would include medium-voltage power lines, substations and polemounted transformers that represent the final major component of the power system that serves several customers. This paper addresses a power distribution system consisting of a substation, protective devices, power lines, transformers and customers as shown in Figure 1. The objective is studying the power line fault probability of the core distribution system, i.e., the power lines connecting the substation, transformers and protective devices and not the power lines connecting individual customers to the transformers since a power line fault there due to a tree fall, for example, can be immediately reported by the customers. This means that the state of the network is known for the power lines connecting the customers to the transformers. Thus, we define the set of power lines N 0 C T Substation rotective Device Transformer ower Line House that called House Building N 1 C Fig. 1. ower system. T N 2 N 3 N 4 N 5 N 6 C T N 7 C T N 8 N 9 I = {i, i = 1,..., I} connecting two consecutive nodes where a node can be either a substation, transformer or protective device. The set of nodes is N = {N i, i = 1,..., I} where node N i is the immediate downstream node of power line i, i.e., power line i provides node N i with power. Let n i be the number of customers attached to node N i ; if node N i is a transformer, then n i > 0, otherwise n i = 0 because no customers are attached to the power generator or protective devices. Also, let n T be the total number of customers in the power system, i.e., n T = i I n i. Consider a radial distribution system where there is a single power flow path from the substation to any customer. Assume that a storm has occurred causing some of the power lines to fault which shuts down the protective device upstream to the damaged power line leaving all downstream customers of the protective device in outage. Based on this, some of the customers will call to notify the EUC of the outage as shown in Figure 1. So, given the location of the calls of the customers in outage, the protective device(s) that disconnected can be identified and then all downstream power lines are suspected of being damaged. Thus, the challenge to the EUCs is in identifying which of the power lines has faulted so that the problem can be fixed as fast as possible. For example, if a call is received from a customer on node N 8, this means that either protective device N 7 or N 1 has shut down. More specifically, N 7 will shut down, if a fault occurs on a power line connecting any two nodes in the set {N 7, N 8, N 9, N 10 } leaving all customers on nodes N 8 through N 12 in outage. Whereas, N 1 will shut down if a fault occurs on a power line connecting any two nodes in the set {N 1, N 2, N 3, N 4, N 7 } leaving all customers in the power system in outage. So, it is clear that an outage at node N i can only be caused by a fault of a subset of power lines. Using the configuration of the grid, we define D i to be the first upstream protective device for power line i and T i to be the set of nodes that become in outage if D i is triggered, i.e., T i = {N j, N j in outage if D i shuts down}; in other words, T i contains all downstream nodes of protective device D i. We also define a set Q i that contains the power lines which, after triggering the associ- N 12 N 11 N 10 C T

3 3 TABLE I NOTATIONS FOR OWER LINE FAULT I set of power lines in the power system, I = {i, i = 1,..., I} where power line i provides node N i with power N set of nodes of the power system N = {N i, i I} where node N i is the immediate downstream node of power line i n i number of customers attached to each node; if node N i is a transformer, then n i > 0, otherwise n i = 0 n T total number of customers in the power system, i.e., n T = i I n i n c i number of customers calling from node i to report an outage Ω countable sample space formed of a set of scenarios where each scenario ω corresponds to a set of received phone calls and a set of power line faults H i random variable representing the number of received phone calls from node N i H random vector representing the possible realizations of received calls from the nodes of the power system, i.e., H = {H i, i I} H(ω) vector representing a realization of H according to scenario ω, i.e., H(ω) = {H i (ω) = n c i, i I} H set of all possible call realizations, i.e., H = {H(ω), ω Ω} J(H) set of nodes from which calls were received, i.e., J(H) = {N i, H i > 0 for i I} L i random variable indicating whether power line i faults, i.e., L i = 1 if power line i faults and L i = 0 otherwise L random vector representing the possible realizations of power line faults, i.e., L = {L i, i I} L(ω) sample realization of L indicating the power line faults according to scenario ω, i.e., L(ω) = {L i (ω), i I} where L i (ω) = 1 if power line i faults and L i (ω) = 0 otherwise set of scenarios corresponding to the power line faults such Ω L (h) that the set of calls in all these scenarios is the realization h, i.e., Ω L (h) = {ω, H(ω) = h} L set of vectors representing all power lines fault combinations, J(L) i.e., L = {L(ω), ω Ω L (h)} set containing the subset of power lines of L that faulted, i.e., J(L) = {i, L i = 1 for i I} i ρ i probability of customers calling from node N i to report outage i.e., Q i = {j, N i becomes in outage if power line j faults} Q set of power lines that when fault cause an outage to node N i, D i T i T J(L) Z(H) R i I(H) first upstream protective device of power line i set of nodes that become in outage if D i is triggered, i.e., T i = {N j, N j in outage if D i shuts down} set of nodes that become in outage if protective devices i J(L) D i are triggered, i.e., T J(L) = i J(L) T i set containing the combinations of power lines that when fault trigger all customers in H to call, i.e., Z(H) = {L, J(L) Q i 1, N i J(H)} set of power lines downstream of power line i but with a different first upstream protective device, i.e., R i = {j, power line j is downstream of power line i & D i D j } subset of power lines for which the set of nodes between their upstream protective devices and the substation do not have a call, i.e., I(H) = I N i J(H) R i I(H) c complement set of I(H); subset of power lines for which the set of nodes between their upstream protective devices and the substation do have a call I(H) c = N i J(H) R i set of protective devices in the power system, i.e., = {N i, N i is a protective device} segment containing the power lines that trigger protective e i device N i, i.e., e i = {j, D j = N i for N i } set of segments in the power systems, i.e., E = E {e i, e i is the downstream segment of N i } Notations with generic vector X (X can be L) prior probability of fault occurring on the entities (power lines p(x) or segments) of X p(h) probability of the set of received calls being H probability of the set of received calls being H given the p(h X) faulted entities of X posteriori probability of fault of X given that the set of p(x H) received calls is H set containing a subset of vectors of X where the i th entry {X } i=xi of each vector is equal to the value of x i ated protective device, would leave node N i in outage, i.e., Q i = {j, N i becomes in outage if power line j faults}; in other words, Q i contains the power lines that trigger D i and all upstream power lines of D i. Let N be the set of protective devices in the power system, i.e., = {N i, N i is a protective device}. Now, we can define the set of segments E = {e i, e i is the downstream segment of N i } where each segment contains the power lines that trigger the same protective device, i.e., e i = {j, D j = N i for N i }. For example, the first segment is formed of the power lines that trigger node N 1 to shut down, i.e., e 1 = {2, 3, 4, 7}. Table I contains the main notations used in the calculation of the power line fault probability throughout the paper. III. ROBABILITY MODEL Define the probability space (Ω, F, ) formed of the countable sample space Ω, the set of events F and the function that assigns events to probabilities; Ω is formed of a set of scenarios where each scenario ω indicates a specific set of calls and a set of power lines that could have faulted together. Since Ω is countable then define F as the power set of Ω, i.e., F = 2 Ω which is the biggest σ-algebra using Ω. Thus, F can be omitted and one can just write (Ω, ) to define the probability space. Let p(ω) be the probability of scenario ω, where ω Ω p(ω) = 1. Then, we have (A) = ω A p(ω) for all A Ω. In what follows, more discussions are held on the calculation of p(ω). After a storm occurs, the EUC has two sources of information that it can rely on to estimate the potential fault of power lines and these are: 1) the phone calls of the customers and 2) the prior probability of power line fault which mainly depends on the grid structure and the storm pattern. First, let H = {H i, i = 1,..., I} be a random vector representing the possible realizations of received calls from the nodes of the power system where H i is a random variable representing the number of received phone calls from node N i. Let H(ω) be the vector representing a realization of H according to scenario ω. Thus, H(ω) = {H i (ω) = n c i, i = 1,..., I} where n c i is the number of received calls from node N i. Let H = {H(ω), ω Ω} be the set of all possible call realizations in the power system. If there are n T customers in the power system, then there is a maximum of H = 2 n T call realizations. Also, define the set J(H) = {N i, H i > 0 for i I} which represents the set of nodes from which calls were received. Over time, the EUC can build a database that stores the probability, ρ i, of customers calling from node N i to report an outage; then 1 ρ i is the probability of a customer not calling when experiencing an outage. Historical data provides a goods estimate for ρ i since people tend to behave in a consistent manner when experiencing an outage. In the worst case, a uniform calling probability from all nodes can be assumed and then other factors provide a good indicator of the location of fault such as the set of received of calls. Since p(h i > 0) + p(h i = 0) = 1, N i N and p(h) = N p(h i J(H) i > 0) N i N J(H) p(h i = 0), then (H) = 1 if H contains all phone calls realizations, i.e., if H = 2 n T. The EUC receives a specific set, say h, of phone calls from the customers. Then, based on this set of received phone calls,

4 4 we have to figure out the power line fault probability for all power lines in the power system. Let L = {L i, i = 1,..., I} be a random vector representing the possible realizations of power lines that could have faulted after the storm where L i is a random variable indicating whether power line i has faulted; we assume L i = 1 if power line i faults and L i = 0 otherwise. Let L(ω) = {L i (ω), i = 1,..., I} be a specific realization of the random vector L, i.e., L(ω) is a vector indicating the power lines that fault according to scenario ω. Then, Ω L (h) = {ω, H(ω) = h} represents the set of scenarios corresponding to the power line faults such that the set of calls in all these scenarios is the realization h. Thus, the sample space Ω can be expressed as Ω = h H ΩL (h). Second, the EUC can estimate a prior fault probability, p(l i ), of power line i, i I taking several factors into account such as 1) the storm path in case a storm has happened, 2) grid structure and 3) historical data. These sources of data combined provide an estimate of the prior fault probability as follows. First regarding the storm pattern, using meteorological tools, its path across the circuits and strength can be forecasted as it has already blown out which enables the estimation of good representative prior probabilities. Then, the EUC can update the priors based on its knowledge of the impacted area such as if there are a lot of surrounding trees that could potentially fall. Finally, the EUC can further update its estimation based on historical data, such as its knowledge of locations that frequently fault or old equipments in the grid that have a higher tendency to fault. In the worst case, the EUC can assume a uniform probability of fault for all power lines, and then the set of received of calls identify the ones that have most likely faulted. Given these two sources of information, we can calculate the posterior probability, p(l i H), of power line i faulting given that the set of received phone calls is H which is the information that we are looking for power line fault identification. In general, several power lines in the power system can fault together, thus we define the set J(L) = {i, L i = 1 for i I} that contains the subset of power lines of L that have faulted after the storm. Then, the EUC can also estimate the prior probability, p(l = L(ω)), of power line faults of realization L(ω) where the faults occur on the power lines having coefficient L i (ω) = 1. Note that it is assumed that the power lines fault independently from each other, thus p(l) = i J(L) p(l i = 1) i I J(L) p(l i = 0). In reality, all possible combinations of power lines in I can fault together. Thus, for I power lines, there are Ω L (h) = 2 I possible combinations of power lines faulting together. Now, define L = {L(ω), ω Ω L (h)} as the set of vectors representing the power line fault combinations of I. Since p(l i = 1) + p(l i = 0) = 1, i I and p(l) = i J(L) p(l i = 1) i I J(L) p(l i = 0), then (L) = p(ω L (h)) = 1 if L contains all power line fault combinations of I, i.e., when Ω L (h) = 2 I. For a specific set h of received phone calls, the power line fault probability should be figured out for all power lines in Ω L (h) only which is a subset of the sample space Ω. However, Ω L (h) becomes computationally intractable for large I. Thus, the EUC can start initially with a sampled set Ω L (h) that indicates the possible combinations of power lines faulting together. Then, based on the storm pattern and the previous observations, the EUC can update Ω L (h) based on a certain chosen policy. If Ω L (h) is sampled, then a preprocessing step should be applied to have (L) = (Ω L (h)) = 1 as will be explained in Section IV-C. The realization of received calls, h, can be used to define the likelihood, p(h = h L = L(ω)), which is the probability of the set of received calls being h given that J(L(ω)) is the set of power lines that faulted. In general, for any realization H, we can define the probability of the set of received phone calls H as: p(h) = L L p(h L)p(L), (1) With this information being collected at the EUC database then, for any realization of L and H, the posteriori probability, p(l H), of the power lines of L faulting given the phone calls of the customers, is obtained using Bayes theorem as follows: p(l H) = p(h L)p(L). (2) p(h) The power lines that can fault together can be in different locations in the power system which requires more than one utility truck to go and investigate the damaged locations at the same time to fix the faulted power lines rapidly. The information that the EUC is looking for is the probability of fault of each power line given the phone calls of the customers, i.e., p(l i H), which can be expressed as: L {L} i=li p(h, L) p(l i H) =, (3) p(h) L {L} i=li p(h L)p(L) = L L p(h L)p(L), (4) where {L} i=li is the set containing a subset of vectors of L where the i th entry of each vector is equal to L i. For example, {L} i=1 is the set containing all the combinations of power lines that can fault with power line i. In order to calculate p(l i H), we need to first construct L and p(l i ) and then compute p(h) and p(h L), L L. From (1), p(h) can be calculated knowing L, p(l) and p(h L). As explained earlier, L, p(l i ) and consequently p(l) are assumed to be known after the storm has occurred. So, p(h L) is the entity that should be computed in order to calculate p(h), according to (1), and consequently p(l i H), according to (4). In what follows, we show how to compute p(h L) given H and L. A. Calculation of the likelihood p(h L) This section proposes two probability models of phone calls and then it shows that both models yield the same posterior probability of power line fault. The set of received calls, h, is used to find the probability of the event H = h. In probability Model 1, the number of calls from the same transformer is taken into account. Thus, if H i = n c i > 0 known customers did call out of n i, then the probability of calling from node N i is ρ nc i i (1 ρ i ) ni nc i. In probability Model 2, it is assumed that one or more calls from the same transformer yield the same

5 5 information. Thus, the probability of calling, in this case, is equal to the probability of receiving at least one call from node N i, i.e., the probability of calling is 1 (1 ρ i ) ni for H i = n c i > 0. For both models, the probability of not receiving any calls from node N i, i.e., when H i = n c i = 0 is (1 ρ i) ni. Thus, the probability of calling from node N i given L can be expressed as: 0 for n c i > 0 & given L, N i should not be in outage but N i is in outage, (1 { ρ i ) ni for n c i = 0 & given L, N i is in outage, p(h i = n c i L) = ρ nc i i (1 ρ i ) ni nc i Model 1 for n c 1 (1 ρ i ) ni i Model 2 > 0 & given L, N i is in outage, 1 for n c i = 0 & given L, N i is not in outage. (5) The first condition in (5) indicates the following. If there are calls from node N i, it means that it is in outage. However, given L, if the power lines of L that faulted do not leave node N i in outage, this means that there is a contradiction. Thus, given L, the event H i = n c i cannot happen, i.e., the probability p(h i = n i c L) = 0. For example, if a call is received from a customer on node N 6 only (in Figure 1), then, if we are conditioning on L with J(L) = {10} only, the likelihood p(h 6 = 1 L) = 0 because a fault at power line 10 will shut down its first upstream protective device which is N 7, and thus, it is not possible that node N 6 would have been affected. So, the probability of the event H 6 = 1 given that only power line 10 has faulted is 0. The probability of calls from all nodes in the power system, represented by the vector H given L can be expressed as: p(h L) = i p(h i L), (6) since customers call independently from each other to report that they are experiencing outages. For any realizations of H and L, the likelihood, p(h L), can be expressed as: { N p(h L) = i T J(L) p(h i L), if L Z(H), (7) 0, otherwise, where T J(L) is the set of nodes that become in outage if the protective devices, i J(L) D i, of the power lines that fault are triggered, i.e., T J(L) = i J(L) T i. And, Z(H) is the set containing the combinations of power lines that when fault trigger all customers in H to call which can be expressed as: Z(H) = {L, J(L) Q i 1, N i J(H)}. (8) The basic probability model and the statement of Bayes theorem for the posterior probability would apply to any distribution system configuration. The probability model takes advantage of the characteristics of the radial configuration only for computing the likelihood of the calls given the faults, i.e., p(h L) given in (7). For non-radial distribution systems, there is redundancy in the infrastructure where a customer can have different alternative paths for power flow in case one of the paths is in outage. The model can be extended to non-radial networks, but the assumptions should be specified first to incorporate their functionality in the calculation of the posterior distribution. 1) Computational Complexity of the Likelihood p(h L): The computational complexity of p(h L), depends on the complexity of 1) checking whether L Z(H) and if this condition applies, then 2) compute T J(L). The computational ( complexity of checking whether L Z(H) is ) O N J(L) Q i J(H) i according to (8). The computational complexity of T J(L) is equal to the computational complexity of ( ) i J(L) T i which is O i J(L) T i in the worst ( case. Thus, the computational complexity of p(h L) is O J(L) N Q i J(H) i + ) i J(L) T i if L Z(H) and ( O J(L) ) N Q i J(H) i otherwise. 2) Effect of the Number of Calls on The ower Line fault robability: The next theorem shows that Model 1 and Model 2 yield the same posterior probability of power line fault. Theorem 1: The posterior probability of power line i faulting given H under Model 1, p M1 (L i H) is equal to that under Model 2, p M2 (L i H), i.e., p M1 (L i H) = p M2 (L i H). (9) The proof is derived in Appendix I. Consequently, p(l i H) depends only on the transformers from which calls were received but is independent of the number of received calls from those transformers. In other words, one call will provide the same information as 10 calls from the same transformer which indicate that there is a power line fault affecting the customers attached to the transformer. Thus, using either of the calling models, in the third condition of (5), will give the same power line fault probability. 3) Examples on Multiple ower Line faults: Using probability Model 1, the likelihood p(h L) can be expressed as: { Ni TJ(L) p(h L) = ρnc i i (1 ρ i ) ni nc i, if L Z(H), (10) 0, otherwise, where Z(H) follows the general formula provided in (8). Now, if calls are received from nodes in the set J(H) = {N 6, N 8, N 12 } as shown in Figure 1, then Z(H) = {L, J(L) Q i 1, N i {N 6, N 8, N 12 }}. Assume that two power lines i and j have faulted then p(h L) = p(h L i = 1, L j = 1) with, e.g., p(h L 5 = 1, L 9 = 1) = ρ 3 (1 ρ) n5+n6+n8+n9+n11+n12 3, p(h L 5 = 1, L 11 = 1) = 0 and p(h L 3 = 1, L 11 = 1) = ρ 3 (1 ρ) n T 3.

6 6 TABLE II STATISTICS OF 319 OWER CIRCUITS mean minimum maximum Number of ower Lines Number of rotective Devices B. Computational Complexity of the osterior robability of ower Line fault p(l i H) So far, the computation and complexity of the likelihood, p(h L), have been discussed which is needed for the calculation of p(h), according to (1), and consequently p(l i H), according to (4). For each realization of H and L, the likelihood, p(h L), can be calculated according to (7) with a computational complexity discussed in Section III-A.1. Thus, the overall computational complexity of p(h) is O J(L) Q i + T i. (11) L L N i J(H) i J(L) Also, the worst case computational complexity of p(l i H) is mainly due to the computational complexity of p(h) and thus, it is equal to (11). So, the developed probability model is intractable for a power grid with a large number of power lines. IV. SAMLE SACE AND COMUTATIONAL COMLEXITY The computational complexity of power line faults of a given circuit is mainly affected by the size of the sample space Ω L (H). As discussed in Section III, any combination of power lines in I can fault together which results in Ω L (H) = 2 I possible combinations. Considering all possible combinations returns the exact solution but this is computationally intractable for large I. For example, if there is only 25 power lines in the power system, then there are over 33 million combinations to be computed, and actually networks are much larger. Table II shows the statistics over 319 circuits for SE&G; it is shown that the average number of power lines is 724. Thus, the run time for obtaining exact power line fault probabilities would make the simulator unusable for practical purposes. This creates a need to decrease the number of scenarios while still choosing the set of scenarios that are the most representative of the power line faults in the power system. In this paper, we propose three approaches for decreasing the number of scenarios to be computed as will be discussed. The proposed approaches complement each other and should be applied in the presented hierarchy. A. Informative ower Line Set for the Calculation of the ower Line fault robability p(l i H) This section shows that given the set of calls H, we do not have to consider all power line fault combinations in the power system as the probability of fault of some power lines is directly equal to the prior. Definition 1: An Informative set of power lines is the set for which the posterior probability of fault is not equal to the prior, i.e., p(l i H) p(l i ); the informative set provides new information about the power line fault probability. Let I(H) I be the informative subset of power lines. More specifically I(H) represents the subset of power lines for which the set of nodes between their upstream protective devices and the substation do not have a call. Let I(H) c be the complement of the set I(H), i.e., I(H) c = N R i J(H) i where R i represents the set of power lines downstream of power line i but not protected by the same protective device, i.e., R i = {j, power line j is downstream of power line i & D i D j }. Thus, I(H) c represents the subset of power lines for which the set of nodes between their upstream protective devices and the substation do have a call. For example, in Figure 1, given that there are calls from the set of nodes J(H) = {N 6, N 8, N 12 }, then I(H) = {1,..., 10} and I(H) c = {11, 12}. The received set of calls H does not provide any new information on the probability of fault of the power lines of I(H) c as stated in the following theorem. Theorem 2: i I(H) c, the posterior probability of fault of power line i is equal to the prior, i.e., p(l i H) = p(l i ), i I(H) c. (12) The proof is derived in Appendix II. According to Theorem 2, it is only required to do computations for p(l i H), i I(H). In this case, p(l i H) is computed according to (4); however, L should represent the power line combinations of I(H) only and thus the set of vectors L that needs to be computed has a maximum number of 2 I(H) power line combinations instead of 2 I. In summary, for each power line i I(H) c, j I(H) that is upstream to power line i and that must have faulted; this naturally might trigger a call at the set of nodes covered by I(H) c whether any of the power lines of I(H) c faulted or not. Thus, we have proven that the posterior probability of fault of the power lines in I(H) c are equal to the prior which means that the phone calls do not provide any new information about the power lines in I(H) c as long as there is an upstream power line that has definitely faulted. B. Aggregating the ower Lines of the Same Segment This section shows that the computational complexity can be decreased further using another property of the power system. According to the functionality of a protective device, all power lines that trigger the same protective device to shut down (i.e., the power lines belonging to the same segment) cause the circuit to respond in the same way in the event of a fault along one of them. Thus, according to the derived likelihood in (10), any set of power lines triggering the same protective device will be assigned the same likelihood. Note that, it is not required to do the computations for all the protective devices of the power system denoted by the set. The needed computations are only for the set of protective devices that cover the power lines in I(H) since it is already known that p(l i H) = p(l i ), i I(H) c. Thus, the set of protective devices that should be considered is (H) = {N i, N i is a protective device for a power line in I(H)}. However, the problem is written in a generic way over the set. Recall that the set of segments is E = {e i, e i is the downstream segment of N i }. Let

7 7 S = {S i, e i E} be a random vector representing the possible realizations of segments that could have faulted after the storm where S i is a random variable indicating whether segment e i has faulted; we assume S i = 1 if segment e i faults and S i = 0 otherwise. The prior probability of fault, p(s i = 1), of segment S i is equal to the probability that at least one power line j S i faults, i.e., p(s i = 1) = 1 j e i p(l j = 0). Now, it is straightforward to see that p(h S)={ e i T J(S) k e i ρ nc k k (1 ρ k) n k n c k, if S Z (H), 0, otherwise, (13) where J(S) = {e i, S i = 1}, T J(S) = e T i J(S) e i, Z (H) = {S, J(S) Q ei 1, N j J(H) with j e i } and Q ei = {e j, nodes of e i become in outage if segment e j faults}. Obviously, (10) and (13) are equivalent j e i, i.e., p(h S i ) = p(h L j ), j e i. This result favors the investigation of the probability of fault on the segment level instead of the power line level from a computational complexity point of view while still obtaining exact power line fault probabilities. Table II shows that the average number of protective devices is only 41 when the average number of power lines is 724. Thus, the number of protective devices/segments is on average 18 times smaller than the number of power lines. Accordingly, calculating the probability of fault of the segments instead of the power lines requires a much smaller number of scenarios. In this case, the size of the sample space, Ω S (H), of the segments that fault together given H is Ω S (H) = 2 which is much smaller compared to the size of the power line sample space Ω L (H) = 2 I. Now, p(h) can be calculated as follows: p(h) = S S p(h S)p(S), (14) where S = {S(ω), ω Ω S (H)}. We will prove that the probability of calls, p(h), is the same using either (1) or (14) in Theorem 3. The computational complexity of p(h) becomes: O J(S) Q ej + T ei. (15) S S N i J(H),i e j e i J(S) The probability of fault of segment e i given the phone calls of the customers, i.e., p(s i H), is calculated using the same steps for the calculation of p(l i H), but over sample space Ω S (H) instead of Ω L (H) which can be expressed as: S {S} i=si p(h S)p(S) p(s i H) =, (16) p(h) where {S} i=si is the set containing a subset of the vectors of S where the i th entry of each vector is equal to S i. Also, the worst case computational complexity of p(s i H) is mainly due to the computational complexity of p(h) and thus, it is equal to (15). To calculate the power line fault probability from the segment fault probability, the following theorem is proposed. Theorem 3: j e i, the posterior probability of fault of power line j is given by p(l j H) = p(s i H) p(l j )/p(s i ), j e i. (17) The proof is derived in Appendix III. Intuitively, (17) shows that for a given segment e i, the posterior fault probability of all power lines j e i, changes with respect to the prior in a constant proportion. This can be justified by the fact that a fault across any of the power lines of segment e i will trigger the same protective device and consequently the whole system to respond in the same way. As indicated earlier, the required computations are only for the set of segments of I(H) since p(l i H) = p(l i ), i I(H) c which decreases the sample space of the set of segments from 2 to 2 (H). C. Monte Carlo Simulations The computational complexity using the segment sample space is much smaller compared to using the power line sample space, but still can be highly computationally complex according to the statistics of Table II where the average number of protective devices is 41 and can reach up to 307. Using the set of received calls H decreases the number of segments to be computed but still the number of segments covered by the protective devices in (H) can be high. If the number of protective devices in (H) is large, e.g., higher than 30, then one of the methods to reduce the sample space Ω S (H) is to sample it according to a certain policy based on the storm pattern and the previous observations. But, if the EUC does not have enough information to do this sampling, then the only remaining tool to use is Monte Carlo simulations. In Monte Carlo simulations, a sample space, Ω M (H), is generated of a predefined size to reduce the computational complexity of the power line fault problem. Let S M = {S(ω), ω Ω M (H)} where S(ω) represents one of the realizations of segments faulting together. Assume that S(ω) = {S i (ω), S i (ω) B(p(S i = 1))}, i.e., the random variable S i (ω) of each segment in S(ω) follows a Bernoulli distribution, B(p(S i = 1)), with a probability of success equal to the prior probability of fault, p(s i = 1), of segment S i. Thus, the higher the probability of fault of segment S i, the more likely S i (ω) = 1 in the generated realizations. First, M samples are generated according to the described model with a uniform probability of 1/M. Then, if sample S is repeated m times, its prior probability of fault will be p(s) = m/m. V. ERFORMANCE RESULTS To assess the performance of the proposed approaches, the simulated power grid is constructed using real data provided by SE&G which describes the structure of circuits in their electrical distribution network. The data corresponds to the northeastern portion of SE&G s power grid in New Jersey and it is formed of 319 circuits. The statistics on the average number of protective devices and power lines per circuit have been already presented in Table II. The data identifies the type and location of each component in the circuits such as substations, protective devices, power lines and transformers. Note that there are several types of protective devices in power grids installed based on their hierarchy in the circuit such as reclosers, switches, circuit

8 8 breakers, capacitor banks, fuses, etc. But all these protective devices shut down when a downstream power line or node faults to protect the electric components of the grid. The simulator is also programmed to generate storms that pass across the grid generating power line faults causing total or partial circuit power outages. The obtained outages trigger some of the affected customers to call to report the outage. The simulator can generate four types of storms with different intensities: weak, moderate, severe, and hurricane. The intensity of the storm determines its radius where weak storms have the smallest impact radius and hurricanes have the largest. As the storm passes across the grid, each power line i along its way is associated a prior probability of power line fault, at time step t, according to its distance d ti from the center of the storm as follows: ( p t (L i ) = θ 1 d ) ti, (18) d max where d max is the maximum radius of the storm and θ is the maximum fault probability of a power line. According to this model, the closer the power line is to the center of the storm, the higher its prior probability of fault will be. The simulated storm s path is split into a set of discrete time steps t = 1,..., T. At each time step t, power line i has a different probability of fault p t (L i ) depending on its distance from the center of the storm. For each value of p t (L i ), we generate a Bernoulli random variable with a probability of success equal to p t (L i ) to determine whether power line i has faulted or not; it is obvious that the higher the prior probability of fault p t (L i ) is, the more likely a power line fault will be generated. For each power line i, if at one of the time steps it faults according to the generated Bernoulli random variable, it is considered as a faulted power line. Now, it can be easily seen that the value of θ affects the prior probabilities of fault and consequently the number of generated faults in the circuit. Over all time steps, we assume that the prior probability of fault of power line i is equal to the probability of faulting at least once at one of the time steps, i.e., p(l i ) = 1 t (1 p t (L i )). (19) When a power line faults, the simulator finds the parts of the circuit that become in outage due to this fault. For each node N i, if there are attached customers experiencing a power outage, then a Bernoulli random variable is generated for each customer with a probability of success equal to the probability of calling ρ i to determine whether the customer will call or not; according to this model, the higher the probability of calling ρ i, the higher the number of customers calling will be. After collecting the priors for power line faults, the obtained faults and the customers that called, the simulator executes the proposed power line fault probability model. Figure 2 shows the obtained performance for a part of an impacted circuit. The thickness of a power line represents the value of the obtained posterior probability of power line fault. Results show two important observations; the first observation is that the proposed model detects the power lines that have faulted which are marked by a yellow circle. The second observation Fig. 2. erformance of the proposed probability model for power line fault detection for a part of an impacted circuit. The thickness of a power line represents the value of the obtained posterior probability of fault. The power line that faulted are marked by circles and the customers that called are marked by rectangles. is that the power lines that have been identified as faulted but did not actually fault are the neighboring power lines to the ones that have faulted; this is due to the fact that close power lines have a correlated prior probability of fault and that if have faulted would trigger the same customers to call. But, the good news is that the neighboring power lines to the actual one that faulted lie within a close distance which enables the utility crew to look it up efficiently given the information provided by the posterior probabilities. To evaluate the performance of the proposed probability model, we measure the gain provided by the posterior probability of power line fault compared to the prior. First, the following terminologies are defined: faulted power lines: power lines that have faulted. OFF power lines: power lines through which no power is passing whether they have faulted or not (because an upstream protective device has shut down). ON power lines: power lines through which power is still passing. Note that the EUC does not know which power lines are ON or OFF; it is expecting that the probability model will inform it about the likelihood of power line faults. However, using simulations, the efficacy of the proposed probability model for ON and OFF power lines can be evaluated. Figures 3 and 4 present the corresponding prior and posterior fault probabilities for OFF and ON power lines, respectively considering one storm simulation. The results are sorted in ascending order of the prior probabilities assuming that the customer s calling probability to report outage is ρ i = 0.1, i I. With probability of fault θ = and a moderate storm strength, 31 power lines have faulted which is a typical number compared to what happens in reality after a storm. We notice that the obtained prior and posterior probabilities are relatively low since the value of θ is which already generated 31 power line faults in the power system. Increasing the value of θ will increase

9 osterior rior osterior rior ower line fault probability ower line fault probability ower line ower line Fig. 3. rior and posterior probabilities of fault for OFF power lines sorted in ascending order of the prior probabilities. Fig. 4. rior and posterior probabilities of fault for ON power lines sorted in ascending order of the prior probabilities. the prior probabilities of power line fault and consequently the posteriors; however, this will also increase the number of generated power line faults. Note that the developed simulator is generic and accepts any value of θ. But, this paper presents the value of θ that generates a logical number of faults in reality for moderate storms since what matters in the end is the relative behavior of the posterior probability with respect to the prior where the power lines that have most likely faulted are the ones with highest obtained posterior probabilities in the power system. Figure 3 shows that the posterior probability of the OFF power lines is higher than the prior. The peaks refer to either the power line that have actually faulted or their immediate neighbors; any power line that is OFF should trigger the impacted customers to call and thus the posterior probabilities for the OFF power lines are, in general, higher than the priors. The average prior and posterior fault probabilities of OFF power lines are and , respectively. Whereas, on the contrary, the posterior probabilities for the ON power lines is lower than or equal to the prior for more than 98% of the samples as shown in Figure 4; the average prior and posterior fault probabilities of ON power lines are and , respectively. For very few instances, the posterior probability of the ON power lines is greater than the prior mainly when they are very close to OFF power lines. When a protective device shuts down due to a downstream power line fault, it preserves power flow in the upstream power lines. But, since there can be calls from the downstream customers of the ON power line, its posterior probability of fault can increase. This behavior is also affected by the number of customers along the segment of the ON power lines. For example, if there is a high number of customers along that segment and none of them is definitely calling, then the probability model detects easily that there is no fault across it since p(h L i ) is proportional to (1 ρ) n S j, i Sj where n Sj is the number of customers along segment S j. Thus the higher n Sj is, the lower p(h L i ) will be and consequently p(l i H) will be lower. Whereas, if n Sj is small and there are downstream calls, then p(l i H) can be higher than the prior even if it is ON. However, still when this happens, the value of the obtained posterior probability, which is in Figure 4, is much smaller compared to the obtained posteriors of the power lines that have actually faulted, which is equal to 0.2 in Figure 3; this shows the ability of the proposed probability model to highly detect faulted, OFF and ON power lines. Figure 5 shows the prior and posterior power line fault probabilities versus the calling probability averaged over generated storms ranging from weak to hurricane. The chosen value of θ is which generated on average power line faults per storm with a minimum of 1 and a maximum 364 which highly reflects actual experience. The measurements are conducted for the power lines that fault, remain ON or become OFF but do not fault. First, the posterior probability of the power lines that faulted is the highest and increases significantly with respect to the prior which implies that the faulted power lines have been identified. Second, the posterior probability of the ON power lines is the lowest and also decreases significantly with respect to the prior, highly indicating that those power lines have not been affected by the storm. Thus, whenever the EUC experiences a dramatic increase or decrease of the posterior probability with respect to the prior, it is informed that the corresponding power lines have either faulted or not. Third, the posterior probability for the OFF power lines that did not fault increases as logically justified previously since any power line that is OFF should trigger the affected customers to call. However, we notice that, on average, the posterior probability of the OFF power lines that did not fault does not exceed those that have actually faulted and thus it is less likely that the EUC will confuse the one that have actually faulted from the ones that did not; this has been captured in the probability model by the set Z(H) which contains the combinations of power line faults that trigger the affected customers to call otherwise the OFF power line, say i, is identified as have not faulted by setting p(l H) = 0 with L i = 1 if the chosen set of faulted power lines L does not trigger all affected customers to call. Consequently, the developed probability model can identify

10 10 ower line fault probability osterior: Faulted osterior: ON osterior: OFF NF rior: Faulted rior: ON rior: OFF NF Calling probability (ρ) Reduction in distance error (%) Expected value: θ = Standard deviation: θ = Expected value: θ = Standard deviation: θ = Calling probability (ρ) Fig. 5. rior and posterior power line fault probabilities for faulted, ON and OFF not faulted (NF) power lines versus the customer s calling probability to report power outage. Gain in power line fault identification (G(H)) Calling probability (ρ) Fig. 6. Relative gain in the posterior probabilities of the power lines that have faulted with respect to the ones that did not fault (G(H) = i F p(l i H) i F c p(l i H)) as a function of the calling probability. whether a power line is OFF due to a fault of a neighboring power line that triggers a different set of calls from the power line that is OFF and has actually faulted. Figure 5 also shows the posterior probability of power line fault as a function of the calling probability where a different behavior is noticed for the ON and OFF power lines. Recall that the customers that call are the ones experiencing outage, i.e., the customers that are typically fed by the power lines that are currently OFF whether they have faulted or not. Whereas, the customers fed by the power lines that are still ON will never call as the calling probability increases. Thus, the posterior probability of the ON power lines decreases as the calling probability increases since more calls are providing more information about the location of the OFF power lines. Whereas, for the OFF power lines whether faulted or not, the same behavior is obtained as a function of the calling probability where both increases or decrease jointly. The joint increase or decrease depends on the simulation parameters such as the values of the prior probabilities and the number of customers calling to report outage due to the OFF power lines. But what matters is the behavior of the relative gain of Fig. 7. Reduction of the expected value and standard deviation of the posterior distance error with respect to the prior as a function of the calling probability. Distance error represents the expected distance between projected faults and the nearest location of an actual fault. the posterior probability of the power lines that have actually faulted with respect to the ones that did not fault since the power lines with the highest posteriors are the more likely to fault. Let F be the set of power lines that have faulted and F c be its complement, i.e., F c contains the ON and OFF power lines that did not fault. We define the relative gain as G(H) = i F p(l i H) i F p(l i H). c Figure 6 shows the relative gain in the probabilities of the power lines that have actually faulted with respect to the ones that did not fault as a function of the calling probability. It is obvious that as the calling probability increases, the gain increases as expected. This result emphasizes the proposition that more calls provide more information about the power lines that have faulted in the power system. Figure 7 shows the reduction in distance error of the posterior with respect to the prior as a function of the calling probability. Distance error represents the expected distance between projected faults and the nearest location of an actual fault. Let d kj be the distance between power lines k and j and let d f j be the distance of power line j from the closest fault, i.e., d f j = min k F d kj. The prior and posterior expected distance errors are E [ d ] f = p(l j) j i I p(li)df j and E [ d f H ] = p(l j H) j i I p(li H)df j, respectively and the reduction gain is ( [ ] E d f E [ d f H ]) /E [ d ] f. Figure 7 shows the reduction gain of the expected value and standard deviation of the distance error for θ = and θ = 0.004, respectively. The reduction gain depends on the simulation parameters such as the number of generated faults and the area of the impacted grid. Within the same impacted area, for θ = and θ = 0.004, the expected number of generated faults is and 10.78, respectively whereas the prior expected distance error is meters and meters, respectively and the prior standard deviation of distance error is meters and meters, respectively. The smaller the number of faults within the same area, the larger the distance, d f j, of a nonfaulted power line to a faulted one will be. The positive reduction gains in terms of expected value and standard deviation of the distance error indicate that the

11 11 posterior probability provides a better indication about the location of fault with respect to the prior. For example, when 10% of the customers call, the reduction in the expected value and standard deviation of the distance error are 43.8% and 34.9% for θ = 0.015, respectively and 60.2% and 39.6% for θ = 0.004, respectively. It is also revealed that as the calling probability increases, the reduction gains increase too as expected. Finally, for a smaller θ, the distance of the projected faulted power lines with respect to the faulted is larger. But, since the probability model is able to identify the locations of faults by allocating higher probabilities only to the locations that have actually faulted or their immediate neighbors as explained above, higher reduction gains are obtained for a smaller θ since the prior distance error is larger. VI. CONCLUSION We have developed a probability model that estimates the probability of power line faults. erformance results have been conducted using data for a real power system. It is shown that even with a small percent of customers calling to report an outage, significant gains in terms of power line fault identification can be achieved where the obtained posterior probabilities of the faulted power lines were the highest; this provides a good indication about the power lines that have most likely faulted. Thereby, the developed probability model is providing fundamental and basic information that is necessary for several future research directions such as utility truck routing for fast power restoration and power line fault detector development with guaranteed confidence intervals. ACKNOWLEDGEMENT This research was funded in part by NSF contract ECCS and the SA Initiative for Energy Systems Research. REFERENCES [1] K. L. 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Jabr, Statistical representation of distribution system loads using gaussian mixture model, IEEE Transactions on ower Systems, vol. 25, pp , February [22] J. D. Hobby, A. Shoshitaishvili, and G. H. Tucci, Analysis and methodology to segregate residential electricity consumption in different taxonomies, IEEE Transactions on Smart Grids, vol. 3, pp , March [23] A. M. L. da Silva, J. G. de Carvalho-Costa, and A. A. Chowdhury, robabilistic methodologies for determining the optimal number of substation spare transformers, IEEE Transactions on ower Systems, vol. 25, pp , February [24] A.. Leite, C. L. T. Borges, and D. M. Falcao, robabilistic wind farms generation model for reliability studies applied to brazilian sites, IEEE Transactions on ower Systems, vol. 21, pp , November AENDIX I ROOF OF THEOREM 1 In (7), by satisfying the condition of L Z(H), then the first condition of (5) is no more applicable to any L Z(H). Also by looping over the nodes in T J(L), we are only studying the probability of the calls from nodes in outage in the power system given L. That is, the fourth condition of (5) is not applicable anymore to any node N i T J(L). Thus, the nodes of T J(L) can be divided into two sets; one represents the set from which calls were received, i.e., J(H) and the probability of calling of those nodes satisfies the third condition of (5) which is referred to as p c (H i L). And, the other set represents the nodes from which no calls were received, i.e., T J(L) J(H) and the probability of calling of those nodes satisfies the second condition of (5) which is referred to as p n (H i L). Thus, the first condition of (7) can be written as p(h L) = N i J(H) pc (H i L) N i T J(L) J(H) pn (H i L), L Z(H). Now, Theorem 1 can be proved as follows. roof: It is shown that the likelihood p(h L) = N i J(H) pc (H i L) N i T J(L) J(H) pn (H i L) if L Z(H)

12 12 and p(h L) = 0 otherwise. Let p c(mk) (H i L) be the probability of calling under Model k. From the definition of Z(H), any realization L = L(ω), ω Ω L (H) should trigger all the nodes of H to call or in other words J(H) T J(L), L L. Since J(H) is a constant set for all L L, then the term C Mk = N i J(H) pc(mk) (H i L) is also constant and is common for all L L under Model k. Now, the likelihood p(h L) under Model k can be written as p Mk (H L) = { C Mk N i T J(L) J(H) (1 ρ i) ni, if L Z(H), 0, otherwise, (20) where C M1 = N i J(H) ρnc i i (1 ρ i ) ni nc i under Model 1 and C M2 = N (1 (1 ρ i J(H) i) ni ) under Model 2. Using Model k, the posterior probability, p(l i H), of fault of power line i given H, is calculated according to (4) and it becomes p Mk L {L} i=li p Mk (H L)p(L) (L i H) = = = L L pmk (H L)p(L) L {Z(H)} i=li C Mk N i T J(L) J(H) (1 ρ i) ni p(l) L Z(H) CMk N i T J(L) J(H) (1 ρ i) ni p(l) (21) N i T J(L) J(H) (1 ρ i) ni p(l) L Z(H) N i T J(L) J(H) (1 ρ (22) i) ni p(l) L {Z(H)} i=li where the equality in (21) holds since p Mk (H L) = 0 if L / Z(H) as per (7) and the equality in (22) holds since C Mk can be canceled out from the numerator and denominator. Thereby, we have proved that p Mk (L i H) is independent of Model k which means p M1 (L i H) = p M2 (L i H) which concludes the proof of the Theorem 1. AENDIX II ROOF OF THEOREM 2 An important property of the set R i is that if power line i faults then all nodes connected to the power lines of R i become in outage. In this case, T j T i j R i, thus the probability that the set of received calls is H given that power lines i and j have faulted is p(h L i L j ) = p(h L i ), j R i according to (7). For example, if calls are received from nodes N 8 and N 12 only, then there is a definite information that a fault has happened on one of the power lines of Q 8 so that a customer on N 8 would call. But, it is not known definitively if a fault has happened on the power lines of R 8 = {11, 12} or not since a fault on Q 8 will trigger calls from all downstream devices. In this case, p(h L 8 L 12 ) = p(h L 8 ), p(h L 8 ) = ρ 2 (1 ρ) n8+n9+n11+n12 2 whereas p(h L 12 ) = 0. In order to prove the claim that p(l i H) = p(l i ), i I(H) c, we first need to show an important property about the power lines in the set I(H) c. The next proposition states that each power line i I(H) c can either fault or not while still triggering the set of calls of H and power line i has no effect on the likelihood p(h L) for any L Z(H). roposition 1: i I(H) c, if L = {L m, L i = 0 and L m = x m, m I\i} Z(H) then L = {L m, L i = 1 and L m = x m, m I\i} Z(H) for the same set of values of x m, m I\i. In addition, p(h L ) = p(h L). roof: Recall that I(H) c = N j J(H) R j, thus for each i I(H) c, N j J(H) such that i R j, i.e., there is a power line in Q j that is upstream to power line i and that must have faulted so that a call is received from node N j. If i R j, this directly implies that Q j Q i, i.e., the upstream power lines of power line i are a subset of the upstream power lines of power line j. Recall, L Z(H) holds only if J(L) Q j 1, N j J(H). Thus, even if calls are received from nodes N i and N j, but if Q j Q i, then J(L) Q j 1, L. In this case, whether power line i faulted or not, node N i is in outage since an upstream power line in Q j must have faulted. Thus, i I(H) c, if L = {L m, L i = 0 and L m = x m m I\i} Z(H) then L = {L m, L i = 1 and L m = x m m I\i} Z(H). Also, p(h L) = N m T J(L) p(h m L) if L Z(H) where T J(L) = m J(L) T m. Recall that i I(H) c, there exist an upstream node N j J(H) such that i R j, and we have T i T j. Consequently, i I(H) c and L Z(H), T J(L) = m {J(L)} T m = m {J(L)\i} T m whether power line i faulted, i J(L), or not, i / J(L). Now, for each i I(H) c, if L = {L m, L i = 0 and L m = x m m I\i} Z(H), then L = {L m, L i = 1 and L m = x m m I\i} Z(H) and p(h L ) = p(h L) = N m T {J(L)\i} p(h m L). Now, we want to show that this property does not hold i I(H). Assume that we receive a call from node N i and there is no call from any upstream node. Then, there is a power line in Q i that must have faulted. For i I(H), consider the vector L = {L m, L i = 1 and L m = 0 m I\i}; L indicates that power line i faulted which can trigger a call from node N i, i.e., L Z(H). However, the vector L = {L m, L i = 0 and L m = 0 m I\i} / Z(H) which means that p(h L) = 0 and consequently p(h L) p(h L ). Thus, it is shown that the properties that hold i I(H) c, do not hold i I(H). Now, using the properties of the power lines of I(H) c stated in roposition 1, we can prove that the received set of calls H does not provide any new information on the probability of fault of the power lines of I(H) c as stated in Theorem 2. roof: According to (4), we have:

13 13 p(li = 1 H) = L {L}i=1 L L = p(h L)p(L) p(h L)p(L) = L {Z(H)}i=1 L {Z(H)}i=1 L Z(H) p(h L)p(L) p(h L)p(L), (23) p(h L)p(L), (24) p(h L)p(L) Q p(li =1) L {Z(H)}i=1 p(h L) Lj L\Li p(lj ) Q Q =, (25) p(li =1) L {Z(H)}i=1 p(h L) Lj L\Li p(lj ) + p(li =0) L {Z(H)}i=0 p(h L) Lj L\Li p(lj ) Q p(li = 1) L {Z(H)}i=1 p(h L) Lj L\Li p(lj ) Q =, (26) L {Z(H)}i=1 p(h L) Lj L\Li p(lj ) [p(li = 1) + p(li = 0)] L {Z(H)}i=1 p(h L)p(L) + L {Z(H)}i=0 = p(li = 1), (27) where the equality in (23) holds since p(h L) = 0 if L / Whereas, according to (3), the posterior probability of fault of Z(H) as pers(7). The equality in (24) holds since Z(H) = power line j ei is c {Z(H)}i=0 {Z(H)}. The equality in p(h L)p(L) p(h L) X i=1, i I(H) Q (26) holds since p(h L) Lj L\Li p(lj ) = p(lj H) = L {L}j=Lj = p(l), L {Z(H)} i=1 Q p(h) p(h) L {L}j=Lj L {Z(H)}i=0 p(h L) Lj L\Li p(lj ) because according to roposition 1, for each L = {Lm, Li = 0 and Lm = xm m (29) I\i} Z(H) then L0 = {Lm, Li = 1 and Lm = xm m p(h Si ) p(lj ), (30) = I\i} Z(H) for the same set of values of xm, m I\i, p(h) in addition p(h L0 ) = p(h L). Thus, p(li H) = p(li ), i 0 where I(H)c which concludes the proof of Theorem 2. L {L}j=Lj p(l) = p(lj ) L0 {L}0j p(l ) where 0 {L}j contains all power line combinations of segment ei \{j}, A ENDIX III thus, L0 {L}0 p(l0 ) = 1. Consequently, (29) is equal to (30). j ROOF OF T HEOREM 3 From (28) and (30), the following holds In order to prove Theorem 3, we first show that the p(lj ) probability of calls, p(h), is the same when calculated using, j ei, (31) p(lj H) = p(si H) p(si ) either the power line space as per (1) or the segment space as per (14). The proved property in Section IV-B should be also which concludes the proof of the Theorem 3. used where it states that j ei, p(h Si ) = p(h Lj ). These two properties allow to prove Theorem 3 as follows: roof: Consider a single segment, denoted as ei, in the power system. If there are calls along segment ei, then p(h Si = 0) = p(h L = 0) = 0 otherwise p(h Si = Lina Al-Kanj received the h.d. degree in Electri0) = p(h L = 0) = p(h Si = 1) = p(h L), L L, as cal and Computer Engineering from the American shown in Section IV-B, where L contains all combinations of University of Beirut in She was a ostdocpower line faults on segment ei and 0 represents the vector toral Research Fellow at the Electrical Engineering Department at rinceton University in Since of all zeros, i.e., none of the power lines of the segment February 2014, she is a ostdoctoral Research Asdid fault. According to (14), p(h) = p(h Si = 1)p(Si = sociate at the Operations Research and Financial 1) Engineering Department at rinceton University. Her + p(h Si = 0)p(Si = 0) and according to (1), p(h) = research interests include integer and stochastic opl L p(h L)p(L) = p(h L) L L {0} p(l) + p(h L = timization with applications to cooperative wireless 0)p(L = 0). The probability p(h) is the same using either (1) networks, radio network planning and smart grids. or (14) since it is shown that: 1) p(h Si = 0) = p(h L = 0) and p(h Si = 1) = p(h L), L L {0} and 2) p(si = 0) = p(l = 0) and p(si = 1) = L L {0} p(l) which both hold by definition. More explicitly, the probability p(si = 1) Belgacem Bouzaiene-Ayari is a senior research is equal to the probability of having at least one power line staff in the Department of Operations Research and fault on segment ei ; thus, it is equal to the probability of Financial Engineering at rinceton University. He assuming 1, 2,..., ei faults on the segment or in other words joined the CASTLE Labs team in 1997 where he worked on a number of large scale and complex by considering all possible power line fault combinations on resource management problems. He developed and segment ei. implemented a number of optimization models that According to (16), the posterior probability of fault of are currently used by our sponsors to answer fundamental business questions at the strategic, tactical segment ei is p(h Si ) p(si H) = p(si ). p(h) and real-time levels. (28)

14 14 Warren B. owell is a professor in the Department of Operations Research and Financial Engineering at rinceton University, where he has taught since His research specializes in computational stochastic optimization, with applications in energy, transportation, health and finance. He has authored/coauthored over 200 publications and two books. He founded and directs CASTLE Labs ( specializing in fundamental contributions to computational stochastic optimization with a wide range of applications.