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In power systems, effective reliability analysis is an essen- tial factor in long term and .... bility perspective as compared with the calendar age. In Type I ..... [6] Z. Li and J. Guo, “Wisdom about age,” IEEE Power Energy Mag., vol. 4, no. 3, pp.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 1, FEBRUARY 2010

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Reliability Modeling and Simulation in Power Systems With Aging Characteristics Hagkwen Kim, Student Member, IEEE, and Chanan Singh, Fellow, IEEE

Abstract—In power system reliability evaluation, alternating renewal process is generally used to model the failure and repair cycle of a component. This means that after repair, the component is assumed to be restored to as good as new condition from the reliability perspective. However, in practice, there may exist an aging trend in some components as they grow old. This paper describes how aging characteristics of components may impact the calculation of commonly used reliability indices such as loss of load expectation (LOLE). In order to construct the system failure and repair history of components, sequential Monte Carlo simulation method using stochastic point process modeling is introduced. Three methods are described for this purpose and this methodology is applied to the Single Area IEEE Reliability Test System. The results are analyzed and compared. Index Terms—Aging model, alternating renewal process, homogeneous Poisson process, loss of load expectation, nonhomogeneous Poisson process, power law process, power system reliability, rate function, repairable systems, sequential Monte Carlo simulation, stochastic point process, trend analysis.

I. INTRODUCTION

F

OR reliability evaluation purposes, the continuously operating systems are generally classified into two categories: repairable and non-repairable. For repairable systems, if a component of the system fails, it is repaired and the system is put back into operation. However, a non-repairable system dies when it fails, and it needs to be replaced by a new one. The power systems fall in the category of repairable systems and this paper is focused on such systems. In power systems, effective reliability analysis is an essential factor in long term and operational planning [1], [2]. Accurate reliability analysis of power systems helps to predict future failure behavior and make appropriate maintenance plans [3]–[5]. A number of power system equipments, such as generators, transmission/distribution lines, or transformers, have been increasingly growing old around the world. For example, it is said that the United States electricity infrastructure is the “first world grid” because of its size or complexity, but it is also the “third world grid” because of the age or a maintenance level of its components [6]–[8]. In many electric utilities, maintenance or investment planning does not cover growing load demand and Manuscript received January 21, 2009; revised April 10, 2009. First published September 18, 2009; current version published January 20, 2010. Paper no. TPWRS-00024-2009. The authors are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2009.2030269

aging of existing components. Low reliability due to aging not only declines a competitive advantage, or valuation in the power utilities market, but also requires greater operation and repair costs. In light of the current situation, it is more important than ever to identify the aging of equipment quantitatively and incorporate this into the estimation of future reliability of the system [3]. There may be a trade off between reliability and cost suggesting system performance optimization based on cost-reliability analysis [9], [10]. II. BACKGROUND In power system reliability evaluation, failure and repair states of a component are modeled as an alternating renewal process, an uptime being followed by a repair time [11]. If only up times are considered, they are typically modeled by a renewal process in which the inter-arrival times are considered independently and identically distributed. Most of the models in power systems further assume that these uptimes are distributed exponentially, leading to a homogeneous Poisson process (HPP) [12]. Reference [11] provides a comprehensive treatment of models when the uptimes in a renewal process are non-exponentially distributed. This assumption of independent identical distribution of successive uptimes of components was first examined in power system reliability literature in [13]. This reference paper collected data on ten generators from the Brazilian power network and analyzed the presence of aging trend using the theory of stochastic point process [11]–[13]. Later papers which analyzed power distribution data are [14] and [15]. Using statistical hypothesis testing, it is possible to detect and analyze the presence of aging trend for a given significance level [14]–[16]. In statistics, trend indicates a pattern of the information observed over time. There are three types of trend: zero, positive, and negative [13], [17], [18]. If inter-arrival times have neither pattern of improvement nor deterioration, the process has zero trend and is called renewal process. However, if failure rate is increasing over time, the process has positive trend and indicates aging. When the failure rate is a decreasing function with time, the process has negative trend and indicates reliability growth. In general, electromechanical equipment of power systems has positive tendency as the age of component increases. Although [13] analyzed the trend in generating units, the mechanism of incorporating aging in power system reliability evaluation has not received considerable attention in the power system literature [19]. This paper examines the issues related to incorporating aging in reliability evaluation of repairable power systems in detail and proposes some methods based on Monte Carlo simulation.

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by a renewal process with the consecutive inter-failure times independently and identically distributed. When these inter-failure times are exponentially distributed, the process becomes HPP and the intensity function is constant. 2) The component may be only as good as it was immediately before failure. This is called minimal repair and can be represented by an NHPP. 3) The component may be in between 1 and 2 or perhaps even better than new. A general model for dealing with aging in repairable systems can be formulated by using the concept of virtual age [28]–[30]. The virtual age is supposed to represent the age from the reliability perspective as compared with the calendar age. In Type I repair is given Kijima Model, for example, the virtual age at by

Fig. 1. Sample path of stochastic point process, fN(t); t  0g.

In this paper, loss of load expectation (LOLE), loss of load duration (LOLD), loss of load frequency (LOLF), and expected energy not supplied (EENS) [20]–[23] are calculated for quantitative reliability analysis. Section III describes problem formulation, and Sections IV–VI present reliability modeling and simulation methodology. The techniques are illustrated by the application to the Single Area 24 Bus IEEE Reliability Test Systems (RTS) [24], [25]. MATLAB is the software tool for this implementation. Conclusion is provided by Section VII, and the last section provides references. III. PROBLEM FORMULATION For purposes of reliability modeling, inter-transition times of a repairable component can be modeled as a stochastic point process [26]. Fig. 1 shows a sample path of stochastic point , where N(t) is the number of events process, occurred during time t. Index x is inter-arrival time, and t is arrival time, i.e., time event occurred. This is illustrated by the following: (1) (2) (3) The expected value of N(t) can be represented by (4) is called the rate function or intensity The derivative of of the process and in reliability analysis represents function the failure or repair rate depending upon whether up times or is constant and repair times are being modeled. In HPP, equal to . The HPP is basically a renewal process with exponentially distributed inter-failure times. A nonhomogeneous Poisson process (NHPP) [13] is, however, more general and can handle trend, aging, or reliability growth, through the proper . specification of intensity function A few comments can be made about aging in repairable systems. When a component fails and is repaired, the state of the component can fall into one of the following three categories [27]. 1) The component may be as good as new after repair. This is called perfect repair and is what is commonly assumed in power system reliability modeling. This can be modeled

(5) where q is the repair adjustment factor, x is the inter-failure time, and t is the arrival time. For a renewal process, q is zero, that is, after every repair, the virtual age is set to zero, meaning the component is as good as new after repair or it does not age from one inter-failure interval to the next. It is important to note that in this case, the component may age from the beginning of the inter-failure time to the end, but repair is assumed to restore the component to as good as new state so that there is no aging over the long run. For an NHPP, q is assumed to be one, i.e., the virtual age is equal to the real age experienced by the component, meaning after the repair, the component is only as good as before the failure, i.e., the component is aging. Other repair strategies can be represented by different values of q to represent aging or reliability growth. It should be reemphasized that unless the inter-failure time distribution is exponential, the failure rate of the component changes during the inter-failure period [11], but in the ordinary renewal process, this is reset after each repair to the as good as new state. Methods for dealing with this situation are described in detail in [11]. When, however, the repair is minimal as in NHPP, the failure rate continues to change after repair as if the component is continuing to operate incessantly, the repair process serving to restore functionality and not reliability. As is shown later, modeling technology can handle imperfect repair with q other than 1 or 0. However, it is difficult to obtain data for modeling this situation. If q can be estimated, by expert opinion or data or a combination, then imperfect repair can be handled. In case such an estimate cannot be obtained, the results obtained by 0 and 1 can be interpreted as lower and upper bounds. It should be noted that since in the aging components, the failure rate is continuously varying (generally increasing) with time, this introduces a correlation of the failure rate with the load which is also changing. Such correlation is not causal but only coincidental as the load changes and the failure rate steadily increases with time. However, at least conceptually, the use of an average probability of aging components is likely to introduce error because of this correlation. It appears that in such cases, the use of sequential Monte Carlo simulation will be the most appropriate choice.

KIM AND SINGH: RELIABILITY MODELING AND SIMULATION IN POWER SYSTEMS WITH AGING CHARACTERISTICS

IV. RELIABILITY MODELING USING SEQUENTIAL MONTE CARLO

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Step 7) The statistics are updated as described by step 6 and the process moves to step 2. The simulation is continued until convergence criterion is satisfied. Let

A. Steps of Sequential Monte Carlo [31]–[35] A general algorithm, for any type of probability distribution of component states, can be described in the following steps. Let us assume that the th transition has just taken place at time and the time to next state transition of component i is given by . Thus, the vector of times to component state tranand the simulation proceeds in the folsitions is given by lowing steps. Step 1) The time to next system transition is naturally given by the minimum of the component transition times:

value of reliability index (for example, number of hours of load loss) obtained from simulation data for year ; number of years of simulated data available; standard deviation of the estimate I. Then, estimate of the expected value of the index I is given in (10) and standard deviation is given in (11):

(6)

(10)

If this corresponds to , that is the th component, and then next transition takes place by the change of state of this component. Step 2) The simulation time is now advanced:

(11) where (12)

(7) (13) where x is obtained from (6). Step 3) The residual time to component transition is calculated by the following: (8) where is residual time to transition of component i. Step 4) The residual time for component p causing system transition becomes zero and time to its next transition is determined by drawing a random number. Step 5) The time is set as shown in the following:

Note that , the standard deviation of the estimate, , and will approach zero as goes to infinity. varies as Simulation is iterated until coefficient of variation (COV) is less than specific value, . Usually is set as 0.05. As can be seen from this algorithm, in the application of this simulation algorithm, the times to the next transition of components need to be determined. In the next section, we discuss the determination of these times to next transition for aging and non-aging components. V. SAMPLING THE TIME TO NEXT TRANSITION A. Time to Transition for Non-Aging Model

(9) to , the status of equipment Step 6) In the interval stays fixed and the following steps are performed. a) The load for each node is updated to the current hour. b) If no node has loss of load, the simulation proceeds to the next hour; otherwise, state evaluation module is called. c) If after remedial action all loads are satisfied, then simulation proceeds to next hour. Otherwise, this is counted as loss of load hour for those nodes and the system. Also if in the previous hour there was no loss of load, then this is counted as one event of loss of load. . d) Steps a)–c) are performed until

Symbol x denotes inter-arrival time, Z is a random number from a uniform distribution (0, 1], and function F is cumulative distribution function. Then the value of x can be obtained from (14) so that (15) When the distribution of x is exponential, (15) reduces to a simpler form: (16) where represents failure rate if x is time to failure and it represents repair rate if x is time to repair. Here is something to be kept in mind—if Z is equal to zero, interval time x goes to infinity. Therefore, Z has an interval (0, 1].

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B. Time to Transition of Aging Model It should be noted that aging is an issue with the uptimes only. The repair times may have an exponential or non-exponential distribution but there is no concern about aging for the repair time. So for the purpose of discussion of sampling of times to failure, we will ignore the down times, assuming that the time to repair can be sampled using (15). In principle, extension to sampling of time to failure of aging model is straightforward if we know the probability distributions of the successive times to failure. Then basically (15) can be used to draw the sample from the distribution of that particular inter-failure interval. In practice, however, it is impractical to find the distributions of individual consecutive interfailure times. A model that has been widely studied for modeling the aging failures is the NHPP, particularly one with the power law process (PLP) [2], [12], [17], [36]. PLP is expressed by (17)–(20). Here (17) gives the intensity function, (18) gives the cumulative intensity function, and (19) gives probabilities of a given number of failures and is a generalization of the formula for HPP. By substituting (17) into (19), (20) is derived:

where parameter q is repair adjustment factor addressed in Section III. If q is zero and is one, (21) represents exponential distribution function. If q is one, then this can be used to represent minimal repair. Intermediate values of q can represent various degrees of repair effectiveness. Now using (14) (22) Since 1-Z has the same probability distribution as Z, (22) can rewritten as (23) This gives (24)

(25)

(17) (18)

By substituting q equal to one in (24) and (25) for

(26)

(19) for

(20) This model is derived from Weibull power law process [37], [38]. Sometimes it is also called Duane model [36]. Since the time to first failure for power law process has Weibull distribution, it is also called a Weibull process. It should not, however, be confused with the Weibull distribution as the Weibull distribution applies only to a single inter-failure time. Parameters and are positive. When is greater than one, failure rate is increasing over time. If is equal to one, it refers to HPP model. On the other hand, if is less than one, failure rate is a decreasing function of time, so that the process represents reliability improvement or reliability growth. One reason for the popularity of the NHPP with PLP is that methods for estimating its parameters are well documented [37]. Reference [27] extends the use of this model with other popular distributions like normal and log-normal. There are several techniques [39]–[41] that can be used to sample the NHPP which assume minimal repair. Modified forms of three methods are briefly described in this paper. These modifications result by inclusion of the repair adjustment factor based on the concept of the virtual age in (5). 1) Using Inter-Failure Time Cumulative Distribution Function (IIM-Interval by Interval Method): Let us assume that a failure just occurred at , and then using (18)–(20), the CDF for the inter-failure time is given by

(21)

(27)

Using (24) and (25), the failure times can be sampled by drawing random number Z for any repair adjustment factor. For minimal repair, (26) and (27) can be used. 2) Time Scale Transformation (TST): This method is based on the result that are the points in an NHPP with the if and only if are the cumulative rate function points in an HPP with rate one [41], where for nonzero (28) From (28) (29)

(30) for

(31)

for

(32)

It may be inefficient to employ this method for complicated intensity functions, since it requires numerical calculation of inverse function. However, in the case of PLP, the calculation is easily developed, illustrated by (29). Symbol is inter-arrival

KIM AND SINGH: RELIABILITY MODELING AND SIMULATION IN POWER SYSTEMS WITH AGING CHARACTERISTICS

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TABLE I GENERATING UNIT 6: MEAN UP TIMES VARIATIONS WITH

of this paper is to illustrate the methodology for effect of modeling of aging components. If the reliability indices are significantly affected in the HL I study [44], then arguably they will be also affected in the composite system study. A. Effect of

Fig. 2. Flowchart for the thinning algorithm.

time by HPP with rate one, and is inter-arrival time by NHPP. is calculated by (16), and then inter-arrival time for aging model, , is obtained by (31) and (32). 3) Method Based on Thinning for a NHPP (Thinning Algorithm-TA): Fig. 2 describes the procedure for this approach. Parameter q is the repair adjustment factor. Arrival time with is thinned out with probability . increases, becomes small, and then, thinAs ning out process occurs less. On the other hand, as is getting decreased, the thinning out process occurs more often and interval times are increasing. Contrary to methods a) and b), this method does not need numerical inverse integral calculation of rate function. Also, Log-linear rate function, or exponential polynomial rate function method [42], [43], can be used for specific intensity rate function. VI. SIMULATION METHODOLOGY AND APPLICATIONS Simulation methodology used in this paper is sequential Monte Carlo which is described in Section IV. All the three methods of sampling time to next failure of aging components are tested. It is assumed that the repair times are described by exponential distribution, although other distributions can also be used. The reason for assuming exponential distribution for repair times is that emphasis of this paper is on how times to failure are effected by aging. The Single Area IEEE RTS used for application of proposed methodology, is described in [24] and [25]. This system consists of 24 buses, 32 generators, and 38 transmission lines and transformers. MATLAB is used for computer programming. By using percent of annual, weekly, and daily peak load data [24], [25], hourly peak load data during one year are obtained. The load data ranges from 861.62 [MW] to 2850 [MW] with a mean of 1750.63 [MW]. The least unit of time is one hour. The transmission constraints are ignored in these studies as the objective

on Consecutive Mean Up Times

In PLP, parameter determines the shape of rate function. To show how affects mean up times, the following case is considered first and described in Table I. Based on generating unit reliability data in [24] and [25], Table III describes generator capacities, failure, and repair rates. Interval by interval method is used for this simulation. The moment simulation completes ten up times of the generator 6, it is terminated. Then, to get reasonable estimates of ten mean up times, simulation is repeated until converged. Convergence criterion described in Section IV is employed with COV set to 5%. This value is used for each reliability index in this paper. For ten mean up times [h], when is equal to one, interval times are exponentially distributed. For greater than 1, q is assumed to be 1, i.e., minimal repair. Up to ten consecutive mean up times of generator 6 are estimated from , all the values are almost the simulation. As expected, for same and equal to reciprocal of failure rate of generator 6. On the other hand, if is greater than one, mean up times are getting decreased as age of the component grows, showing positive aging trend. For the sake of performing a fair comparison, in this case as well as in subsequent studies in the paper, for both non-aging and aging situations, a given component is assumed to have the same reliability initially, i.e., the mean time to first failure (MTTFF) is the same, as illustrated in Table I. To achieve this, the scale parameter in (17)–(20) should be modified as shape parameter is varied such that the MTTFF stays the same. This can be done as follows. The reliability or survivor function, i.e., the probability of not failing by time t, can be obtained from (20) by setting k to zero. Thus (33) The MTTFF can be obtained by integrating the reliability function from 0 to [11] and gives

(34) where

and

is a gamma function.

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TABLE II DESCRIPTION OF CASES OF APPLICATION

TABLE IV RESULTS FOR CASE 1

TABLE III RELIABILITY DATA OF GENERATORS

TABLE V RESULTS FOR CASE 2

that will give an MTTFF equivalent to that with be obtained from the following equation:

can

TABLE VI LOLE VARIATIONS FOR DEFRERENT PARAMETER

(35) where when is one. From (34), the equivalent is given by the following: (36) B. Effect of

TABLE VII EENS VARIATIONS FOR DEFRERENT PARAMETER

on Reliability Indices

To observe the quantitative effect of aging characteristics on reliability, two cases are considered as illustrated by Table II. In case 1, all components are modeled as an exponential renewal process. In case 2, generators 23–26, and 30 are modeled as a PLP whereas the remaining generators are model as an HPP. The repair adjustment factor is assumed to be 1. 1) Case 1: All components are modeled using HPP. Reliability indices are calculated and compared. LOLP is expressed as percent, LOLE and LOLD are in hours, LOLF is in per hour, and EENS is in MWh. The three NHPP methods are also implemented as an alternative for non-aging model by setting equal to one. As shown in Table IV, the results have similar values. The differences are attributed to randomness of estimation. 2) Case 2: In this case, it is assumed that generators 23–26 and 30 from Table III have positive aging trend. Unit 23 is located at bus 15, unit 24 at bus 16, and unit 25, 26, and 30 at bus 23. Three simulation methods are implemented for generating interval times: interval-by-interval method (IIM), time scale transformation (TST), and thinning algorithm (TA) and the results are shown in Table V for . As expected, the results by the three methods have similar values. As parameter is increased, reliability indices tend to grow. In Table V, simulation time of the three sampling method of an NHPP is also

compared. IIM is based on probability distribution of inter-arrival times; th interval time is directly derived by using th in (24) and (25). This method shows the best performance in terms of computational time requirements. TST is based on inverse integrated rate function. The th interval time of an NHPP is taken from th and th interval time of an HPP with rate one in (31) and (32). On the other hand, TA does not use integrated rate function; instead, it is based on thinning out process and calculation of . Each interval time of an NHPP can be derived only after thinning out failure times of an HPP with the highest rate and for the aging PLP model, failure rate steadily increases. So the thinning out process occurs less frequently as time progresses. In other words, in the IF condition from Fig. 2, more “Yes” answers occur over time. So this method requires more time than the previous ones. In conclusion, it appears the most efficient simulation method is IIM, considering computer time and storage requirements. Tables VI and VII show the variation of reliability indices, LOLE and EENS, as varies from 1.0 to 1.8 in Case 2.

KIM AND SINGH: RELIABILITY MODELING AND SIMULATION IN POWER SYSTEMS WITH AGING CHARACTERISTICS

Fig. 3. Five-year study of LOLE with beta variation.

From results of Cases 1 and 2, it can be seen that if some of the components begin to have positive aging trend, load loss event will occur much more than before. The increase depends on the value of , i.e., the degree of aging. These results indicate that it is important that the effect of aging, if present, be included in reliability evaluation; otherwise, the computed reliability may be optimistic. It is evident that the indices are sensitive to the value of . The value of to be used in a planning study will depend on the age of the component at the beginning of the study year and needs to be estimated from the field data. C. Multi-Year Study System planning studies are typically done over a span of several years. A multi-year simulation was carried out and results are shown in Fig. 3. The variation of LOLE with different values of during five years is displayed. The value of q is assumed to be 1. The LOLE index shown is only during the corresponding , study year and is not a cumulative value. In the case of except for estimation variations, LOLE does not change over time and the value is almost the same as the results in Table IV. . As This is because failure rate of PLP is constant for is increased, LOLE becomes higher as years progress. The value of used in this and the next study is from 1 to 1.8. The higher range is used to emphasize the effect of although it is understood that this value is unlikely to be as high as 1.8 in practice. D. Effect of Variation of Repair Adjustment Factor A study was conducted to see the variation of the repair adjustment factor q. This factor was varied from 0 to 1 and all the three methods were tested. The results obtained by all the three methods were very close, so only the results by the IIM method are shown in Fig. 4. As can be seen, the effect of q is not linear. It first increases rapidly and then more gradually. Of course the effect of q also depends on the value of . For example for equal to 1, the value of q will not have any effect since the component is not aging and so the failure rate at the beginning and end of an interval is the same. As the value of increases, the effect of the choice of q will have more pronounced effect. The

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Fig. 4. LOLE for different aging adjustment factor q with fixed .

difference in reliability indices for different values of q can be quite significant. VII. CONCLUSION Many components of power systems around the world are increasingly getting older. It is emphasized that it is essential to consider the aging of such components in computing quantitative reliability indices in the planning studies. This paper describes how aging characteristics can quantitatively influence reliability calculations in power systems. For non-aging model, an HPP is applied and inverse of exponential distribution function is used. For aging, PLP model for an NHPP is introduced for implementation. This model is able to accommodate data with zero, positive, or negative aging trend by proper choice of parameter . The model can also accommodate the effect of repair adjustment factor. For possible correlation introduced by the increasing failure rate, sequential simulation is preferred. Three methods, IIM, TST, and TA, are explored for generating inter-arrival time sequence, based on power law intensity function. The results are analyzed and compared. Of the three methods, IIM appears to be the best choice in terms of model simplicity and simulation time requirements. The proposed techniques are applied to Single Area IEEE RTS. Reliability indices are calculated and compared. As aging parameter is increased, reliability indices have a tendency to grow. It is, therefore, important that aging of components, if present, be considered in reliability computations; otherwise, the computed indices may present a too optimistic picture. The value of depends on the age of the component at the start of the study period and needs to be estimated from field data. It is hoped that this paper will stimulate discussion on data analysis of generators to determine the proper value of . If sufficient data are analyzed, perhaps some recommended values under different conditions can be determined. REFERENCES [1] A. R. Abdelaziz, “Reliability evaluation in operational planning of power systems,” Elect. Power Compon. Syst., vol. 25, no. 4, pp. 419–428, Jan. 1997.

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Hagkwen Kim (S’08) was born in Dae-gu, South Korea, on September 17, 1981. He received the B.S. degree from the Kangneung National University, Kangneung, Korea, and is pursuing the M.Sc. degree in the Power Systems Engineering Division of the Electrical and Computer Engineering Department at Texas A&M University, College Station. His specific research interest is reliability analysis and evaluation in power system network.

Chanan Singh (S’71–M’72–SM’79–F’91) is currently Regents Professor and Irma Runyon Chair Professor in the Department of Electrical and Computer Engineering, Texas A&M University (TAMU), College Station. From 1995 to 1996, he served as the Director of Power Program at the National Science Foundation, and from 1997 to 2005, he served as the Head of the Electrical and Computer Engineering Department at TAMU. His research and consulting interests are in the application of probabilistic methods to power systems. He has authored/co-authored around 300 technical papers and two books and has contributed to several books. He has consulted with many major corporations and given short courses nationally and internationally. Dr. Singh is the recipient of the 1998 Outstanding Power Engineering Educator Award given by the IEEE Power Engineering Society. For his research contributions, he was awarded a D.Sc. degree by the University of Saskatchewan, Saskatoon, SK, Canada, in 1997. In 2008, he was recognized with the Merit Award by the PMAPS International Society.