Risk-Based Composite Power System Vulnerability Evaluation to Cascading Failures Using Importance Sampling

1 Risk-Based Composite Power System Vulnerability Evaluation to Cascading Failures Using Importance Sampling Quan Chen, Student Member, IEEE and Lamine Mili, Senior Member, IEEE Abstract Large-scale blackouts typically result from cascading failure in power systems operation. Their mitigation in power system planning calls for the development of methods and algorithms that assess the risk of cascading failures due to relay overtripping, short-circuits induced by overgrown vegetation, voltage sags, line and transformer overloading, transient instabilities, voltage collapse, to cite a few. This paper describes such a method based on composite power system reliability evaluation via sequential Monte Carlo simulation. One of the impediments of the study of these phenomena is the prohibitively large computational burden involved by the simulations. To overcome this difficulty, importance sampling technique utilizing the Weibull distribution is applied. It is shown that the method significantly reduce the number of samples that need to be investigated while maintaining the accuracy at a given level. To illustrate the developed approach, a case study is conducted and analyzed on the IEEE reliability test system. Index Terms Composite power systems; cascading failures; risk assessment; Monte Carlo methods; importance sampling. I. ITRODUCTIO OWER Systems are operating under the risk of major P disturbances that may induce large-scale blackouts, which are costly to society. One typical example is the 2003 U.S.-Canada blackout, whose total estimated costs incurred by the United States economy range between 4 to 10 billion U.S. dollars. In Canada, the gross domestic product suffered in August 2003 a decrease of 0.7% while the work hours losses amounted to 18.9 million. In Ontario, the manufacturing shipments decreased by 2.3 billion Canadian dollars [1]. This blackout has prompted the development of new approaches and methodologies aimed at assessing and managing the risks of cascading failures in power systems. The need to secure a continuous supply of electric energy for a modern nation has raised major concerns about power system reliability. These concerns have promoted system reliability analysis as one major research endeavor in power systems. Seminal work in this area has been carried out by Billinton et al. [3], [6], [14]. However, the methods and algorithms proposed so far do not model cascading failures. The latter include cascading overloads, failures of protection This work was sponsored by SF under grant EFRI 0835879. Quan Chen and Lamine Mili are with the Bradley Department of Electrical and Computer Engineering, orthern Virginia Center, Virginia Tech, Falls Church, VA 22043, USA (e-mails: quanchen@vt.edu; lmili@vt.edu ). 1 devices, transient instability, forced or unforced initiating failures, shortage of reactive power, voltage instabilities and voltage collapse, computer failures at the control center, unavailability of Energy Management System (EMS) functions for security monitoring and analysis such as power system state estimation and contingency analysis, lack of situational awareness, communications mistakes, and operational errors. In general, large-scale blackouts stem from the occurrence of a combination of several of these failures. While in the literature, a few papers (i.e. [2], [4], [8]) have investigated each of these failures independently from each other, they do not provide a general framework for analyzing them as a combined phenomenon. This paper meets this need by proposing a method that extends composite reliability analysis to account for cascading failures in transmission networks. As indicated in [4], protective relays play an important role in 73.5% of major disturbances. Consequently, protection system should be taken into account when analyzing the risk of cascading failures. In composite power systems, the major protective devices involve over-and under-voltage relays on generator, under-voltage relays on large-capacity motors, and impedance relays on transmission lines. Thorp et al. [2], [5] apply an empirical hidden failure probabilistic model to the zone 3 relays of all the lines incident to a given line while Mili et al. [7], [8] apply short-circuit analyses to identify all the relays exposed to hidden failures. The former model has not been proved while the later method is cumbersome. According to our study of the 2003 U.S.-Canada blackout [1], tree contact is one of the critical factors that push the system operating point toward the edge of collapse. The combination of heavy-loaded transmission lines and overgrown trees may increase the risk of tree contact failure. Although some researchers have developed models of sagtension of overhead transmission lines through conductor thermal strains and physical characteristics, ambient temperature, creep time and so on [9], [10], [11], there is still a need of a simple but realistic model of this phenomenon for cascading failure simulations. Since an exhaustive computation of detailed modeling for all possible combinations of failures is infeasible, compromises are needed in modeling and analyzing cascading failures. For example, there is an extensive literature on

cascading failure that takes a high level approach and neglects the power loading of a power system [12], [13]. However, power flow pattern changes after each outage, which has an effect on the likelihood of subsequent outages. Some attempt to reduce the sampling size by applying importance sampling technique has been made on transmission line failures to calculate the probability of system collapse [2], but it is difficult to extend this approach to a composite reliability analysis. Billinton at al. [3], [6] have made some progress by applying two types of variance reduction techniques, namely control variates and antithetic variates, in power system reliability evaluation to decrease the computing time burden of the simulations. The speedup provided by these techniques can reach a factor of 3.33 or more for the 6-bus Roy Billinton Test System (RBTS). But their efficiencies are very much model dependent [6]; for example, it drops to a factor of 2 for 24-bus IEEE Reliability Test System (RTS). This has prompted us to investigate further these techniques when simulating cascading failures. The paper describes a risk-based method for composite power system reliability evaluation to cascading failures via sequential Monte Carlo simulations. umerous scenarios with a feasible sampling size are considered so that the likelihood of cascading failure is reasonably estimated when achieving power system expansion. To decrease the computational burden, importance sampling technique utilizing the Weibull distribution is applied. To illustrate and evaluate the developed approach, a case study is conducted and analyzed on the IEEE reliability test system. The paper is organized as follows. Chapter 2 deals with sequential Mote Carlo modeling considering cascading failures. Chapter 3 is devoted to the implementation of the importance sampling method while Chapter 4 describes a case study on the IEEE reliability test system. II. SEQUETIAL MOTE CARLO MODELIG COSIDERIG CASCADIG FAILURES When modeling cascading failures in composite reliability analysis, sequential Monte Carlo simulations require the implementation of prohibitively large sampling sizes. This is mainly due to the need to model many types of failures and many different mechanisms by which failures propagate if all possible real system scenarios are accounted for. The simulations also involve a variety of modeling requirements at multiple timescales since electromechanical phenomena occur in a millisecond timeframe while voltage support devices and thermal heating effect react in tens of minutes. It is therefore necessary to find a trade-off between simulation accuracy and computational burden. The way to achieve this tradeoff is explained next. A. Sequential Monte Carlo Simulations Since system failures may cascade in numerous ways, Monte Carlo simulations are the methods of choice. These techniques estimate system responses by implementing and executing a 2 series of plausible scenarios. Generally speaking, there are three different simulation approaches in reliability evaluation. First, in state sampling approach, a uniformly distributed random variable is generated to decide whether the component state is in failure or not. That is, if the random variable is larger than the failure probability, the component is in the normal state; otherwise, it fails. The system state is the combination of all component states. Second, state duration sampling approach uses the component state duration distribution functions. In a two-state component representation, these are operating and repair state duration distribution functions, which are usually assumed to be exponentially distributed. Finally, in state transition sampling approach, there is a transition probability that represents the probability of the departure from the previous state to the present state in the time domain. This approach will not be implemented in our simulations as suggested by Billinton and Li [14]. The state sampling and state transition sampling approach are not considered here because of their inability to model cascading failures as a sequence of dependent events that propagate successively over a time span with varying transition probability. The latter characteristic stems from the changing likelihood of subsequent outages that hinge on system power flow patterns. It turns out that the state duration sampling approach does not have these weaknesses. Combined with annual chronological load curves, it provides a relatively realistic framework for simulating cascading events. It is this method that has been implemented and tested. It will be described next. B. Basic Reliability Model Sequences of dependent cascading failures involve the actions of various components and devices, which include over- and under-voltage relays on generators and large-capacity motors, zone 3 impedance relays on transmission lines, and vegetation under transmission lines. ow, the actions of these components hinge on system power flow and nodal voltage patterns. Consequently, their simulations require the execution of full AC power flow calculations. They also involve the execution of optimal power flow calculations, where each generators capacities instead of theirs exact output power are provided. ote that these simulations need the specification of the forced outage rates of the generators and the transmission lines. C. Effects of Relays on Cascading Failures Over- and under-voltage relays protect most generators while under-voltage relays protect large-capacity load motors ( 750 MVA) [15] and some particular equipments. While in general these relays operate as intended, their operations will reduce the angular and voltage stability margins of the system in the course of a sequence of cascading failures. Impedance protective relays are the major protective devices of high voltage transmission lines. Generally, they operate when the measured impedance falls within the relays 2

3 setting range. Unfortunately, as observed during the 2003 US- Canada blackout, they may unduly overtrip during cascading failures, due for example to voltage sags caused by line overloads. The latter make the measured impedance by a relay smaller than its setting, simulating a nearby fault on the system. ote that among impedance relays, zone 3 relays are the most sensitive to voltage dip due to its large setting range. D. Effects of Vegetation on Cascading Failures Some papers (see for example [3]) classify line outages due to tree contact faults as forced outages, which is a reasonable assumption under normal operating conditions. During abnormal conditions, heavily-loaded transmission lines have high conductor temperature due to the thermal characteristic of the conductors, particularly when the ambient temperature is high. This results in large line sags, which in turn increases the probability of tree contacts. This is one of the reasons why more cascading failures occur during summer time than during colder seasons. The detailed model of overhead conductor sag is complex, which includes modeling of conductor metal thermal strain, elastic strain, settling strain and creep strain, conductor arc length, half-span and weight, line current, ambient temperature, wind speed, wind direction and so on. To improve the efficiency of these calculations, we propose a simplified model that is effective and efficient in cascading failure simulation. A crude relationship between conductor temperature and current is proposed in [16]. It is based on the assumption that the conductor temperature change is proportional to the heat quantity produced by the conductor, which is proportional to the square of the current in the conductor as follows: θ F = θ 0 + (θ e θ 0 ) I F I e 2. (1) Here θ 0 is the ambient temperature around the conductor; θ e denotes the maximum limit of the conductor temperature in normal operation; I F is the maximum long-term current in the conductor; I e is the maximum limit of the current in the conductor, which is a modified value of the rated current of the conductor. It is given by from (1) and (2), the probability of a tree contact is given by the cumulative probability distribution function of the normal distribution. III. IMPORTACE SAMPLIG METHOD The exponential failure model for the chronological state change of generators has been widely used in reliability analysis. In this model, the random variable T drawn in the Monte Carlo simulation follows an exponential distribution with the probability density function (p.d.f.) given by f e (t) = λe λt, (3) where λ is the mean value of this distribution, which is also the generator mean-time-of-failure given by MTTF = 0 tf e (t)dt. (4) The estimation results, be they system risks or other reliability indices, are related to MTTF. Let us now choose a probability density function (pdf) that is similar to the scaled shape of g(x) = xf e (x), while λ = 0.001. In these simulations, we use the Weibull distribution expressed as f w (x) = β α x α β 1 e x α β. (5) This function has two parameters α and β, which are fixed to α = 2105 and β = 1.55, respectively. ote that the Weibull parameter value is obtained via an estimation method, for example via a least square method. The comparison to the scaled p.d.f. function is shown in Fig. 1. Applying the importance sampling method, we get MTTF = xf e (x) 0 f w (x) f w (x)dx, (6) I e = I e θ e θ 0 θ e θ 0e. (2) Here I e denotes the rated current of the conductor, θ 0e is the ambient temperature around the conductor. In (2), the rated current I e is set together with the temperature limit, θ e. The graph relating the measured sag and the conductor temperature shown in Fig. 5 and 6 in [9] and Fig. 6 in [10] exhibits an approximate linear relationship. It is therefore reasonable to assume in our simulations that the sag is proportional to the conductor temperature. Another assumption that we make is that the height of the vegetation under the transmission lines is assumed to be a normal random variable. This means that when using the sag value calculated 3 Fig. 1. Comparison of g(x) and scaled p.d.f of the Weibull distribution.

4 where f w (x) is the Weibull pdf given by (5). A least-square estimator of I is expressed as MTTF = 1 X if e (X i ) i=1. (7) f w (X i ) Here the random variables X i are drawn from the Weibull distribution. Since the variance of Xf e(x) is smaller than that of f w (X) T, the variance of the new estimator based on the importance sampling method is smaller than the conventional exponential random sampling method. This leads to a greater simulation precision under the same number of samples. ote that the same procedure is applied to the mean-time-to-repair, but with a different parameter μ. IV. A CASE STUD USIG THE IEEE RELIABILIT TEST SSTEM The developed method has been applied to the IEEE RTS, whose data are provided in [17], [18]. The initial condition for the simulations is a normal operating case of the system without any generator or transmission line outages. The average temperature of the city of Falls Church in Virginia [19] is taken as the ambient temperature. Since only hourly data are available for the load curves, the time unit of the simulations is the hour. Fig. 2 shows the procedure of our simulation. In these simulations, we consider voltage collapse has occurred if the AC power flow disconverge owing to the ill-conditioned or singular Jacobian matrix. In this case, all the loads are arbitrarily curtailed for 10 hours to recover the system. The outcomes of the Monte Carlo simulation include the sample mean μ, and the sample variance σ 2, of the expected energy not supplied (EES) from a sample of size, which allows us to calculate the coefficient of variation C v defined as C v = σ2. (8) μ The algorithm outlined in Fig. 2 is executed until the same 0.91% coefficient of variation for both conventional method and importance sampling method are achieved. All relays and tree contact probabilities are reset to their initial values t = t + 10 AC PF converge? Start Determine the failure or repair times for each generator according to the current state of them in hour t Combined with the under- or over- voltage relay states on the buses, determine the generation capacity for each generation bus and each load state in hour t Determine if the lines are tripped in hour t due to the forced outage rate and the tree contact probability Combined with the zone 3 impedance relay states on the lines, determine which lines are operating Any state change of either bus or line? Store energy not supplied in hour t AC OPF converge? Determine the state of each relay and tree contact probability of each line for hour t + 1 End of one year? Calculate energy not supplied for this year Fig. 2. Flowchart of the developed Monte Carlo procedure. Max simulation year reached? End t = t + 1 t = 0 Fig. 3 and 4 show the convergence of the EES. The conventional method requires 250 simulation years, while the importance sampling approach only requires 81 simulation years for the same convergence criterion. Therefore, the speedup ratio of this approach is Speedup = 250 81 = 3.09, (10) which is higher than the antithetic variates algorithm given in [6]. The final data is provided in Table 1. All these results show that the importance sampling algorithm significantly outperform the conventional method. Fig. 3. EES comparison between the direct and the importance sampling algorithm 4

5 Fig. 4. EES variance comparison between direct and importance sampling algorithm Table 1. EES comparison between direct and importance sampling algorithm. EES ( 10 5 MWh) Simulation years Conventional method Importance sampling method 1.2879 1.2846 0.26 250 81 67.6 V. COCLUSIOS % Difference A risk-based composite power system vulnerability evaluation has been developed. Unlike conventional methods, it models cascading failures in power transmission networks due to various mechanisms observed in actual blackouts, including relay overtripping, short-circuits due to overgrown trees, voltage sags, to cite a few. Since cascading failures involve a sequence of dependent outages, a sequential Monte Carlo simulation approach has been used. To reduce the computational burden while maintaining the accuracy of the results at a given level, the importance sampling approach has been applied. Simulation results performed on the IEEE RTS has revealed that the proposed approach is able to significantly reduce the number of samples that need to be executed. ACKOWLEDGEMET The authors gratefully acknowledge the support of SF under grant SF EFRI 0835879. REFERECES [1] U.S.-Canada Power System Outage Task Force, Final report on the August 14, 2003 blackout in the United States and Canada: causes and recommendations, April 2004. [2] J. S. Thorp, A. G. Phadke, S. H. Horowitz, S. Tamronglak, Anatomy of power system disturbances: importance sampling, Electrical Power & Energy Systems, vol. 20, no. 2, pp. 147-152, 1998. [3] A. Sankarakrishnan, R. Billinton, Sequential Monte Carlo simulation composite power system reliability analysis with time varying loads, IEEE Trans on Power Systems, vol. 10, o. 3, August 1995, pp 1540-1545. [4] S. Tamronglak, A. G. Phadke, S. H. Horowitz, J. S. Thorp, Anatomy of power system blackouts: Preventive relaying strategies. IEEE Transaction on Power Delivery, 1996, 11(2), 708-715. [5] H. Wang, J. S. Thorp, Optimal locations for protection system enhancement: A simulation of cascading outages, IEEE Trans. Power Delivery, vol. 16, no. 4, October 2001, pp. 528-533. [6] R. Billinton, A. Jonnavithula, Composite system adequacy assessment using sequential Monte Carlo simulation with variance reduction techniques, IEE Proc.-Gener. Transm. Distrib. Vol 144, no. 1, January 1997, pp 1-6. [7] L. Mili, Q. Qiu, A. G. Phadke, Risk assessment of catastrophic failures in electric power systems, International Journal of Critical Infrastructures, vol. 1, no. 1, pp. 38-63, 2004. [8] J. De La Ree,. Liu, L. Mili, A. G. Phadke, L. Dasilva, Catastrophic failures in power systems: Causes, analyses, and countermeasures, Proceedings IEEE, vol. 93, no. 5, May 2005, pp. 956-964. [9] J. S. Barrett, S. Dutta, O. igol, A new computer model of ACSR conductors, IEEE Trans. Power Apparatus and System, vol. PAS-102, no. 3, March 1983, pp. 614-621. [10] S. L. Chen, W. Z. Black, M. L. Fancher, High-temperature sag model for overhead conductors, IEEE Trans. Power Delivery, vol. 18, no. 1, January 2003, pp. 183-188. [11] J. F. Hall, A. K. Deb, Prediction of overhead transmission line ampacity by stochastic and deterministic models, IEEE. Trans. Power Delivery, vol. 3, no. 2, April 1988, pp. 789-800. [12] I. Dobson, K. R. Wiezbicki, B. A. Carreras, V. E. Lynch, D. E. ewman, An estimator of propagation of cascading failure, 39 th Hawaii International Conference on System Sciences, January 2006, Kauai, Hawaii. [13] K. R. Wiezbicki, I. Dobson, An approach to statistical estimation of cascading failure propagation in blackouts, CRIS, Third International Conference on Critical Infrastructures, Alexandria, Virginia, Sept 2006. [14] R. Billinton, W. Li, Reliability assessment of electric power system using Monte Carlo methods, Plenum Press, ew ork, 1994. [15] E. L. Davis, D. L. Funk, Protective relay maintenance and application guide, EPRI P-7216 research project 2814-89 final report, December 1993. [16] Xinying Xiong, ongli Zhu, Electrical sections of power plant (3 rd edition), China Electric Power Press, Beijing, 2004 [17] Reliability Test System Task Force, IEEE reliability test system, IEEE Trans. Power Apparatus and Systems, vol. PAS- 98, no. 6, ov./dec. 1979, pp. 2047-2054. [18] Reliability Test System Task Force, The IEEE reliability test system 1996, IEEE Trans. Power Systems, vol. 14, no. 3, August 1999, pp. 1010-1020. [19] http://www.weather.com/weather/climatology/usva0265?clim omonth=7&cm_ven=usatoday&promo=0&site=www.usatoda y.com&cm_ite=citypage&par=usatoday&cm_cat=www.usatod ay.com&cm_pla=wxpage. BIOGRAPHIES Quan Chen (S 10) received the B.S. degree in Electrical Engineering from Southeast University, anjing, China in 2006, and dual M.S. degrees in Electrical and Computer Engineering from Georgia Institute of Technology and Shanghai Jiaotong University, Shanghai, China in 2009. She is currently a PhD candidate at the Electrical Engineering Department of Virginia Tech. Her 5

6 research interests include power system planning and reliability, power system analysis and control. Lamine Mili (S 82, M 88, SM 93) received the B.S. degree from the Swiss Federal Institute of Technology, Lausanne, in 1976, and the Ph.D. degree from the University of Liege, Belgium, in 1987. He is presently a Professor of Electrical and Computer Engineering at Virginia Tech. His research interests include robust statistics, robust signal processing, and power systems analysis and control. 6