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Detection and tracking of divers have become an important factor in port protection against underwater intruders. A problem arises from divers with open breathing systems because detections of the air bubbles they produce can mislead the tracking filter and sometimes result in a lost track. In this paper a probabilistic model is developed which reflects the probability that a false measurement originates from the bubbles. The novel contribution of this paper is the integration of this model in the probabilistic data association filter (PDAF) to improve the track continuity. The bubble detections may also cause confusion in the track initiation. To prevent this problem, a clustering method is proposed based on morphological operators which allows tracks to be initialized based on two-point differencing of the cluster centroids from succeeding scans. This morphological clustering method is included in a cell averaging constant false alarm rate (CA-CFAR) detector in such a way that both the point detections and their corresponding clusters can be fed to the tracking filter. These techniques are implemented and applied to real data of two divers, one with an open breathing system and the other with a closed breathing system, operating simultaneously in a coastal area. The real data were recorded from an active 90 kHz narrowband multibeam imaging sonar.

Call-completion probability, call-dropping probability, and handoff rate are important performance measures of wireless networks. In this paper, we study the joint effect of channel fading and handover failure on these performance measures. For the case of Rayleigh and lognormal fading channels and for generalized distributions of the cell dwell time and the call-holding time, we derive simple closed-form expressions that closely approximate these performance metrics. The results are given in terms of the moment-generating function (MGF) of the distribution for the call-holding time and may be useful in the cross-layer design and the optimization of wireless networks. Index Terms--Call-completion probability, call-holding time, cell dwell time, handoff rate.

In this paper, we study the asymptotic performance of absolute-threshold- and normalized-threshold-based generalized selection combining (AT-GSC and NT-GSC) schemes over generalized fading channels for high average signal-to-noise ratios (ASNRs). By evaluating the asymptotic moment-generating function of the threshold-based GSC output SNRs, we derive the diversity and combining gains for AT- and NT-GSCs with a large class of modulation formats and versatile fading conditions, including different types of fading channels and nonidentical SNR statistics across diversity branches. Our analytical results reveal that the diversity gains of AT- and NT-GSC are equivalent to that of maximal-ratio combining, and the differences in the combining gains for different threshold values and modulation and diversity formats are derived by using the concept of modulation factors. Index Terms--Combining gain, diversity gain, error rate, outage probability, threshold-based generalized selection combining (T-GSC).

This paper discusses the impact of uncertain power injections in the grid on the load margin. Two common analyses of voltage stability are those aiming for the closest saddle node bifurcation and those assuming a prefixed direction of load and production increase. In the case of large renewable based generation units or a significant degree of dispersed generation, the loading margin has to be interpreted as a stochastic variable itself. This allows to interpret load margins at different levels of probability of voltage collapse with or without corrective actions undertaken. The probabilistic margin is assessed with a minimum number of samples by use of a stochastic response surface method implementation. The method is illustrated on the IEEE 24-bus and 118-bus system considering stochastic wind generation and dispersed generation. Index Terms--Dispersed generation, stochastic response surface, voltage stability.

The utilization of wind power generation is increasing throughout the world and it is therefore important that these facilities be integrated in the existing generating capacity planning and operating protocols and procedures. This paper presents an approach to evaluate the contribution that wind power can make to the load carrying capability of a power generating system in an operating scenario. The basic concepts of unit commitment risk analysis are extended to include the inherent variability associated with wind power by developing short-term probability distributions of the wind speed and wind power output using auto-regressive moving average (ARMA) time series models. The operating capacity contributions attributable to wind power are illustrated by application to a small test system and are expressed in terms of the increased load carrying capability due to the wind power generating facilities. Index Terms--Load carrying capability, operating capacity, risk assessment, wind power probability models.

Transfer capability is used to schedule future transactions between areas and arrange commercial activities in multi-area power systems. Probabilistic evaluation can provide more information about transfer capability than deterministic approaches, potentially leading to more efficient operation of the transmission network. Probabilistic methods can also model generating units with variable outputs such as wind farms (WFs). Fluctuating nature of WF creates different impact on transfer capability compared to conventional generators. In this paper, three methods are proposed to compute transfer capability in multi-area power systems containing WF. In all the methods, Monte Carlo simulation is used to assess probabilistic variation of transfer capability. The methods differ in the manner of handling output of WF and load. In the first approach, system load data and power output of WF are directly used in the Monte Carlo simulation. In the second approach clustered power output of WF and system load data are used. In the third one, after clustering of input data, Monte Carlo simulation is run separately for each of the clusters. Incorporating clustering into Monte Carlo simulation can accelerate convergence speed, reduce variance and sometimes provide additional useful information. IEEE-RTS is used to demonstrate the effectiveness of the proposed approaches. Index Terms--Clustering, Monte Carlo Simulation, probabilistic evaluation, total transfer capability, wind farm.

A new form of the probabilistically strongest neighbor filter (PSNF) algorithm taking into account the number of validated measurements is proposed. The probabilistic nature of the strongest neighbor (SN) measurement in a cluttered environment is shown to be varied with respect to the number of validated measurements. Incorporating the number of validated measurements into design of the PSNF produces a consistent and cost effective data association method. Simulation studies show that the new filter is less sensitive to the unknown spatial clutter density and is more reliable for practical target tracking in nonhomogeneous clutter than the existing PSNF. It has similar performances to the probabilistic data association filter amplitude information (PDAF-AI) with much less computational complexities.

Outage probabilities in shadowed fading channels are evaluated when macrodiversity and microdiversity techniques are implemented. The existence of correlation at the micro and macro levels is taken into account for the computation of outage probabilities. By modeling shadowing using a gamma probability density function, it is shown that an analytical expression for the outage can be expressed as the product of marginal outages summed with weighting factors that depend on the correlation coefficients at the macro level, assuming a maximal ratio combining at the micro level and selection combining (SC) diversity at the macro level. Results demonstrate the potential use of the approach in the analyses of wireless systems in shadowed fading channels. Index Terms--Gamma probability density function (pdf), macrodiversity and microdiversity, maximal ratio combining (MRC), outage probability, selection combining (SC), shadowed fading channels.

We analyze the effect of time-selective fading on the information outage probability and diversity order of Alamouti transmit diversity with joint maximum likelihood, symbolwise linear maximum likelihood, zero forcing, and decision feedback detectors. In contrast to the existing bit error rate (BER) and bit error outage (BEO) analyses, we evaluate the information outage probabilities and corresponding asymptotic diversity orders for each detector. From the theoretical and simulation results, we demonstrate that, when the channel is time selective, the asymptotic diversity orders for the four detectors approach two, zero, one, and one, respectively. Index Terms--Alamouti transmit diversity, asymptotic diversity order, information outage probability, mutual information, time-selective fading channel.

The aim of this paper is threefold: 1) to propose a simple accurate closed-form approximation to the probability density function of the sum of arbitrarily distributed Nakagami-Ta random variables; 2) to propose a simple accurate closed-form approximation to the level crossing rate for the sum of Nakagami-m random processes; and 3) to show some possible applications for the proposed formulations. With such an aim, we choose the [alpha]-[mu] distribution for which the parameters are estimated from the sum of the Nakagami-m envelopes. As shall be shown from sample representative examples, the proposed approximations are simple, versatile, and highly accurate. The approach used here can easily be extended to other applications such as bit error rate and channel capacity calculations among others. Index Terms--Approximation methods, moment-based estimators, Nakagami-m sums, [alpha]-[mu] distribution.

A method to solve the stationary state probability is presented for the first-order bang-bang phase-locked loop (BBPLL) with nonzero loop delay. This is based on a delayed Markov chain model and a state flow diagram for tracing the state history due to the loop delay. As a result, an eigenequation is obtained, and its closed form solutions are derived for some cases. After obtaining the state probability, statistical characteristics such as mean gain of the binary phase detector and timing error variance are calculated and demonstrated. Index Terms--Bang-bang phase-locked loop (BBPLL), delayed Markov chain, nonzero loop delay.

Two basic types of low voltage (LV) distribution exist in practice. The systems based on the European approach have long LV feeders whose sizing requires more careful design consideration than systems based on North American practice with shorter LV feeders. The most important criterion in conductor sizing is the correct estimation of the design loads. This paper describes the development and application of a probabilistic design method based on the Beta probability density function (pdf). The paper investigates an approximate approach for estimating load parameters in countries where a large load database is not available. This approach estimates the Beta parameters from the group after-diversity-maximum-demand (ADMD) for the customers. Index Terms--Load modeling, low-voltage (LV) distribution, probabilistic methods.

We derive a new fiber tracking algorithm for DT-MRI that parts with the locally 'greedy' paradigm intrinsic to conventional tracking algorithms. We demonstrate the ability to precisely reconstruct a diverse range of fiber trajectories in authentic and computer-generated DT-MRI data, for which well-known conventional tracking algorithms are shown to fail. Our approach is to pose fiber tracking as a problem in computing shortest paths in a weighted digraph. Voxels serve as vertices, and edges are included between neighboring voxels. We assign probabilities (weights) to edges using a Bayesian framework. Higher probabilities are assigned to edges that are aligned with fiber trajectories in their close proximity. We compute optimal paths of maximum probability using computationally scalable shortest path algorithms. The salient features of our approach are: global optimality--unlike conventional tracking algorithms, local errors do not accumulate and one 'wrong-turn' does not spell disaster; a target point is specified a priori; precise reconstruction is demonstrated for extremely low signal-to-noise ratio; impartiality to which of two endpoints is used as a seed; and, faster computation times than conventional all-paths tracking. We can use our new tracking algorithm in either a single-path tracking mode (deterministic tracking) or an all-paths tracking mode (probabilistic tracking). Index Terms--All-paths tracking, diffusion tensor imaging (DTI), diffusion-weighted imaging (DWI), dynamic programming, fiber tracking, fiber trajectory, maximum probability path, magnetic resonance imaging (MRI), optimal path, shortest path, single-path tracking, tractography, white matter.

This paper presents a probabilistic optimal power flow (POPF) algorithm taking account of the variation of load power. In the algorithm, system load is taken as a random vector, which allows us to consider the uncertainties and correlations of load. By introducing the nonlinear complementarity problem (NCP) function, the Karush-Kuhn-Tucker (KKT) conditions of POPF system are transformed equivalently into a set of nonsmooth nonlinear algebraic equations. Based on a first-order second-moment method (FOSMM), the POPF model which represents the probabilistic distributions of solution is determined. Using the subdifferential, the model which includes nonsmooth functions can be solved by an inexact Levenberg-Marquardt algorithm. The proposed algorithm is verified by three test systems. Results are compared with the two-point estimate method (2PEM) and Monte Carlo simulation (MCS). The proposed method requires less computational burden and shows good performance when no line current is at its limit. Index Terms--First-order second-moment method, inexact Levenberg-Marquardt algorithm, nonlinear complementarity problem, probabilistic optimal power flow, subdifferential, uncertainty and correlation.

This paper gives a detailed account of the current practices used in Brazil for probabilistic reliability assessment of the national power grid, from the adequacy point of view. These procedures are utilized by the Brazilian independent electric system operator and were recently made mandatory by the National Electric Energy Regulatory Agency (ANEEL). Index Terms--Adequacy, operation, planning, probabilistic method, reliability.

This paper investigates the probability of a radar being able to resolve two point-targets in an environment in which additive noise is the primary limitation on resolution. The statistical properties of the radar cross-ambiguity function are developed and used to arrive at an expression for the target resolution probability. The analysis is verified through simulations.

In this paper, we present a framework for evaluating the bit error probability (BEP) of [N.sub.d]-branch diversity combining with imperfect channel estimates in correlated fading environments. The estimator structure is based on the maximum likelihood estimate and arises naturally as the sample mean of [N.sub.p] pilot symbols. Our results show that the diversity order of the system under consideration matches that of an ideal system operating in the same correlated fading environment, regardless of the number of pilot symbols.

In this paper, we investigate the outage performance of the full-duplex decode-and-forward relay system in a Rician fading environment. We derive an analytical expression for the outage probability of the highest achievable information rate of the system, which can be evaluated by employing numerical techniques. We obtain an upper bound and a lower bound on the outage probability. These bounds are given in simple closed forms and can be easily evaluated. Furthermore, we study the impact of transmit signaling on the outage performance. Index Terms--Decode-and-forward (DF), information rate, outage probability, relay channels, Rician fading.

In this letter, we derive a closed-form solution of the exact pairwise error probability (PEP) for Rayleigh fading orthogonal frequency-division multiplexing (OFDM) channels with multiple transmit and receive antennas. We begin with the case of frequency-selective quasistatic fading OFDM channel model. The analysis is then generalized to the frequency-selective block fading OFDM channel model. This is a suitable model for cooperative space-time coded OFDM systems, where, due to the spatial separation between the cooperating users, the signal-to-noise ratio (SNR), the number of antennas, and the number of nonzero channel taps during each user's transmission (hence, in each block) may be different. We provide numerical examples to illustrate the usefulness of the derived exact PEP expressions. Index Terms--Fading channels, multiple-input-multiple-output (MIMO) systems, orthogonal frequency-division multiplexing (OFDM).

The relation between the striking distance and interception probability is experimentally investigated under standard lightning impulse voltages of both polarities. By using the rod-plane gap as a reference case, results are presented relating to the discharge interception probability of an earthed rod inserted in the gap. There is a critical separation distance between the stressed and earthed rods at which breakdown occurs either to the plane or to the earthed rod with the same probability. Besides the impulse voltage polarity, the length of the reference gap and the earthed rod height are considered as parameters influencing on interception probability. Results are discussed on the basis of the breakdown mechanism and electric field calculations. The striking distance is defined with the aid of an expression that takes into account, besides lightning peak current, the interception probability and the effects of lightning polarity and struck object height. The results introduce the design of lightning protection systems on the basis of lightning interception probability. Index Terms--Air gaps, electric breakdown, interception probability, lightning, striking distance.

We observe that an effect of 'disk resource sharing' of multi-copy movie traffic has great impact on the blocking performance of a video-on-demand system. This observation leads us to establish a conjecture on how to balance the movie traffic load among 'combination' groups of disks to maximize the level of disk resource sharing. For a given file replication instance, the conjecture predicts in general an effective lower bound on the blocking performance of the system. It motivates the design of a numerical index that measures quantitatively the goodness of disk resource sharing on allocation of multi-copy movie files. It also motivates the design of a greedy file allocation method that decides a good quality heuristic solution for each feasible file replication instance. We further develop analytical formulas to obtain approximate results for the bound fast and accurately. These techniques can be utilized by an optimization program to find near-optimal file assignment solutions for the system computationally efficiently. Index Terms--Blocking probability, combination load balancing, disk resource sharing, fixed-point approximation, video-on-demand.

The importance of query processing over uncertain data has recently arisen due to its wide usage in many real-world applications. In the context of uncertain databases, previous works have studied many query types such as nearest neighbor query, range query, top-k query, skyline query, and similarity join. In this paper, we focus on another important query, namely, probabilistic group nearest neighbor (PGNN) query, in the uncertain database, which also has many applications. Specifically, given a set, Q, of query points, a PGNN query retrieves data objects that minimize the aggregate distance (e.g., sum, min, and max) to query set Q. Due to the inherent uncertainty of data objects, previous techniques to answer group nearest neighbor (GNN) query cannot be directly applied to our PGNN problem. Motivated by this, we propose effective pruning methods, namely, spatial pruning and probabilistic pruning, to reduce the PGNN search space, which can be seamlessly integrated into our PGNN query procedure. Extensive experiments have demonstrated the efficiency and effectiveness of our proposed approach, in terms of the wall clock time and the speed-up ratio against linear scan. Index Terms--Probabilistic group nearest neighbor queries, uncertain database.

In solving the electrical power systems economic dispatch (ED) problem, the goal is to find the optimal allocation of output power among the various generators available to serve the system load. With the continuing search for alternatives to conventional energy sources, it is necessary to include wind energy conversion system (WECS) generators in the ED problem. This paper develops a model to include the WECS in the ED problem, and in addition to the classic economic dispatch factors, factors to account for both overestimation and underestimation of available wind power are included. With the stochastic wind speed characterization based on the Weibull probability density function, the optimization problem is numerically solved for a scenario involving two conventional and two wind-powered generators. Optimal solutions are presented for various values of the input parameters, and these solutions demonstrate that the allocation of system generation capacity may be influenced by multipliers related to the risk of overestimation and to the cost of underestimation of available wind power. Index Terms--Economic dispatch, penalty cost, reserve cost, Weibull probability density function, wind energy.

In this paper, we investigate the statistical behavior of the adaptation partial cancellation weight (PCW) in partial parallel interference cancellation (PPIC) schemes based on least mean square (LMS) algorithms in which the adaptation PCW of the last chip in a symbol period reveals the information of the correctness of the bit decision for direct-sequence code-division multiple-access (DS-CDMA) communication systems. We obtained an adaptive decision threshold using a maximum a posteriori probability (MAP) detection criterion for the PCWs to judge the correctness of the bit decision. A bit-inversion procedure is proposed to invert the incorrect bit decisions. After bit inversion, full PIC is performed to cancel multiple-access interference (MAI). Simulation results show that the proposed three-stage LMS-based adaptive full PIC (AFPIC) multiuser detector reaches the performance of a single user bound over flat fading channels. Moreover, the AFPIC detector effectively cancels MAI over a two-path frequency-selective fading channel. Index Terms--Direct-sequence code-division multiple-access (DS-CDMA), maximum a posteriori probability, multiuser detection, parallel interference cancellation.

A multiuser multiple-input-multiple-output (MIMO) system with orthogonal space-time block coding (OSTBC) is analyzed for the uplink of a wireless communication system in a Rayleigh fading environment. In the first part of this paper, a simple and exact closed-form expression for the outage probability of the signal-to-interference-and-noise ratio (SINR) is derived at the input of the base station (BS) receiver by making the following two assumptions: 1) All the users transmit their data by using the same OSTBC; and 2) the users are power controlled by the same BS so that the interfering users are requested to transmit with the same power. In the second part of this contribution, the outage probability of the signal-to-interference ratio (SIR) is calculated at the output of the BS receiver, which, in our case, is a spatial matched filter. To be able to analytically solve the latter problem, the presented analysis is restricted to the case of a single interferer and a 2 x 2 MIMO system with Alamouti coding. Monte Carlo simulations are carried out to verify the proposed analytical expressions for the outage probability. Index Terms--Matched filter, multiple-input-multiple-output (MIMO) transmitter, orthogonal space-time block coding (OSTBC), outage probability.

Previous work on statistical language modeling has shown that it is possible to train a feedforward neural network to approximate probabilities over sequences of words, resulting in significant error reduction when compared to standard baseline models based on n-grams. However, training the neural network model with the maximum-likelihood criterion requires computations proportional to the number of words in the vocabulary. In this paper, we introduce adaptive importance sampling as a way to accelerate training of the model. The idea is to use an adaptive n-gram model to track the conditional distributions produced by the neural network. We show that a very significant speedup can be obtained on standard problems. Index Terms--Energy-based models, fast training, importance sampling, language modeling, Monte Carlo methods, probabilistic neural networks.

Earlier work by Fowler, Davaney, and Hagedorn showed that the morphology of an anode streamer could be modeled as stochastic growth of a branching fractal tree in point-plane geometry. This investigation reproduces the results of that earlier study. Because one of the concerns about the earlier work is that the electric field dependence appeared to be unphysical, the model was modified to operate under assumptions that are consistent with those that have proven useful in earlier investigations. Specifically, linear electric field dependence was assumed and there is an assumed variability in the number density of available electrons. Computations using this assumption also produce the same range of morphologies that has been measured in experiments, in addition, some assessments of sensitivity to other possible variables are made. First, the sharp cutoff in the electric field strength is replaced with a presumably more realistic exponential dependence on energy. Under this assumption, it is also possible to simulate the experimentally observed behavior of anode streamers. It is shown that three possible refinements to the model have small, and likely negligible, effects. The first is using variable streamer step lengths in the calculation rather than the fixed step length used in the earlier work. The second is to assume growth at one point in the streamer makes growth in other parts somewhat less likely. The third is the assumption that the probability of a streamer making the next step in growth is influenced by the distance of the inter-electrode gap that has already been traversed. Index Terms--Dielectric liquids, electric breakdown, trees, modeling, fractals, probability.

During the past decade, several methane/air explosions in abandoned or sealed areas of underground coal mines have been attributed to lightning. Previously published work by the authors showed, through computer simulations, that currents from lightning could propagate down steel-cased boreholes and ignite explosive methane/air mixtures. The presented work expands on the model and describes a methodology based on IEEE Standard 1410-2004 to estimate the probability of an ignition. The methodology provides a means to better estimate the likelihood that an ignition could occur underground and, more importantly, allows the calculation of what-if scenarios to investigate the effectiveness of engineering controls to reduce the hazard. The computer software used for calculating fields and potentials is also verified by comparing computed results with an independently developed theoretical model of electromagnetic field propagation through a conductive medium. Index Terms--Coal mining, lightning, methane explosions.

A novel robust linear receiver with worst-case probability guarantee is proposed in this correspondence. Based on multivariate Chebyshev inequality, deterministic expressions of the probabilistic constraints are derived, and then, an iterative second-order cone programming is developed to obtain the robust solutions. Compared to the existing receivers, our proposed receiver improves performance by allowing a small outage probability and a more general uncertainty model. Simulations illustrate the improvement of our proposed approach. Index Terms--Convex optimization, probability guarantee, robust linear receiver, second-order cone programming (SOCP).

We investigate an ad hoc network where node locations are distributed according to a homogeneous Poisson process with intensity [lambda]. We assume that all the nodes are equipped with an identical wireless transceiver capable of operating satisfactorily up to a certain maximal link loss. Our link model depends on the length of the link and on random log-normal fading. Each node functions as a source and a destination of data packets and may also serve as a repeater to transport the packets over multihop routes, as determined by the network muter. We study two important properties of the network. The first is connectivity, viz., given the peak transmit power of a node, the probability that a node cannot communicate with a random destination node at distance D when at most t hops are allowed. We provide the exact analytical results for t = 1 and t = 2 and an iterative lower bound for t > 2. We calculate the average number of hops of the minimum-hop-count route between a source and a destination at a distance D apart. The second property relates to power consumption--an important parameter when the nodes are battery operated. We derive the cumulative distribution function of the total transmit energy required per data packet when the distance between the source and the destination node is D, and only one or at most two hops, are allowed. We graphically show the benefit of allowing two hops over just one. Index Terms--Multihop network, outage probability, routing, transmit energy.

In the 1995 paper entitled Radar Detection Probabilities and their Calculation, chi-square density functions, normalized with respect to the clutter variance, were used for the clutter density. While mathematically correct, it makes better physical sense to use clutter constant false-alarm rate (CFAR) density functions both there and in the 1999 paper, Generalized Radar Clutter Model, that are normalized with respect to the receiver noise, as is done with the target models. We update the results of these papers by redoing the relevant equations with these more appropriate clutter density functions.

A novel missile guidance law that is dependent on the conditional probability density function of the estimated states is presented. The guidance law is derived by analyzing an interception scenario in the framework of an linear quadratic Gaussian (LQG) terminal control problem with bounded acceleration command. The nonlinear saturation function is represented by the equivalent random input describing function. Since for the investigated problem the certainty equivalence property is not valid, the resulting controller depends on the measurement noise level and on the saturation limit. In comparison to the classical optimal guidance law (OGL), the maximal value of the effective navigation gain is achieved during the engagement instead of near the terminal time. Thus, the saturation limit is reached earlier so as to have enough time to reduce the guidance errors. Using Monte-Carlo simulations, the superiority of the new guidance law over the classical OGL is shown. This validates the new approach of designing an estimation statistics dependent guidance law by using a random input describing function to approximate the missile's acceleration saturation.

The point and interval estimation of the survivor function for a geometric distribution is discussed. The performance of the uniformly minimum variance unbiased estimator and maximum likelihood estimator becomes equivalent to the relative mean square error when devices are greater than or equal to 50. The interval estimator was found to perform better than other asymptotic estimators when failure probability exceeds a certain value.

This paper develops a geometric perspective that ties together a number of graphically based techniques from classical control theory. In particular, in the frequency domain, a connection between the Nyquist diagram and the Bode plots is unfolded via a sequence of three-dimensional representations. A parallel development in the 'gain-domain' begins with the Evans root locus plot and leads to a set of gain plots that portray eigenvalue behavior as an explicit function of forward gain. The gain plots extend the standard root locus plot by depicting explicitly the influence of gain (or any system parameter) on the closed-loop system eigenvalues. This is similar to the way the Bode plots embellish the information of the Nyquist diagram by exposing frequency explicitly. The gain plots enable direct determination of gain values for which the closed-loop system is stable or unstable. By exposing the correspondence of gain values to specific eigenvalues, the plots serve as a pole-placement tool for identifying closed-loop designs meeting performance specifications. Furthermore, the gain plots reveal by inspection information about the closed-loop root sensitivity. We have found the gain plots as well as the underlying geometric development in both the frequency and gain domains invaluable in undergraduate and graduate controls education.

This letter analyses time series of two distribution systems (DS) fault records to find their probability distribution. Observation indicate that the complementary cumulative distribution function (CCDF) of number of fault per day has a power law tail and that location of fault has an exponential tail. The data ad to the knowledge that is a fundamental basis for developing methods for power system maintenance and reliability. Index Terms--Distribution system, fault, probability distribution.