IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004
657
A Probabilistic Approach for Secondary Arc Risk Assessment Jose H. Vivas Nava, Member, IEEE, Richard A. Rivas, Member, IEEE, and Alberto J. Urdaneta, Senior Member, IEEE
Abstract—A probabilistic methodology for secondary arc extinction time calculations is presented. The proposed approach permits the calculation of the probability of secondary arc extinction as a function of time and represents an extension of the well-known deterministic model proposed in [1], where the extinction time is obtained from EMTP-type simulations and groups of secondary arc reignition voltage curves. Instead of using only one pessimistic characteristic, a probability distribution is evaluated to randomly select the reignition voltage curve to use for each calculation. As opposed to the deterministic methods, the proposed probabilistic approach allows the assessment of the risk undertaken by a power system under different single-pole reclosing time schemes and distinct weather conditions alike. Index Terms—Electromagnetic Transients Program, reclosing time, risk assessment, secondary arc, single-pole switching.
NOMENCLATURE ,
Positive and zero sequence capacitances per unit length, respectively. Delayed unit step function, which has value , and a value equal to equal to zero when one when . Magnitude of fault primary current. Nominal steady-state value of secondary arc current (in A rms). Primary arc dependent part of the arc reignition voltage. Secondary arc component, dependent on the arc reignition voltage. Arc length time variation characteristic. Line length. Insulator string length. Probability function of secondary arc extinction. Time variable, measured after the initiation of the primary arc. Duration of primary fault. Time of any arc extinction (either intermediate or final), measured from the time of initiation of secondary arc current. Line-to-neutral nominal voltage. Secondary arc reignition voltage per unit length (in volts per meter). System frequency (in rad/s).
Manuscript received September 28, 2000. This work was supported in part by “Consejo Nacional para el Desarrollo de las Investigaciones Científicas y Tecnológicas (FONACIT),” Caracas, Venezuela. The authors are with the Departamento de Conversión y Transporte de Energía, Universidad Simón Bolívar, Caracas 89000, Venezuela (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRD.2003.820426
I. INTRODUCTION
S
IGNIFICANT research has been conducted to assess and improve the performance of single-pole switching (SPS) of overhead transmission lines [2]. However, after a short circuit on one phase of a transmission line, when only the circuit breakers of the faulty phase open, the transition from a high current (primary arc) to a low current (secondary arc) can jeopardize the security of the power system. This low current, maintained by the mutual electromagnetic coupling between the healthy phases and the faulty phase, requires consequently SPS to be delayed until the final selfextinction of the secondary arc. The secondary arc extinction time depends on factors such as arc length time variation speed and recovery voltage of the insulation, in this case, air. At the same time, those factors are affected by weather conditions such as temperature, humidity, rain, pressure, and wind speed. Traditionally, a secondary arc reignition voltage characteristic obtained from the most adverse weather conditions (e.g., extremely slow or almost zero wind speed), has been utilized to simplify the problem and predict a pessimistic extinction time. This technique was proposed by Johns and Al-Rawi [1], [3] and uses the field test results reported by Haubrich, et al. [4]. A random behavior, such as the one associated with the secondary arc extinction time, can be evaluated mathematically through a statistical approach. Therefore, given a universe of secondary arc reignition voltage curves, a universe of secondary arc extinction times can be generated by randomly changing the arc reignition voltage curve associated with each different weather condition. Subsequently, the results of such simulations can be applied to assess, from a probabilistic point of view, the risk undertaken by a power system under the influence of different single-pole reclosing time schemes. In order to develop a probabilistic method for secondary arc calculations, the following steps are followed: First, a new set of secondary arc reignition voltage characteristics is developed. Then, a statistical approach to evaluate secondary arc extinction times is formulated. Finally, results obtained are compared with those of field tests reported in the technical literature. II. DEVELOPMENT OF ARC REIGNITION VOLTAGE CHARACTERISTICS In this work, the ideas developed in [1] and [3] are applied. The creation of a new set of secondary arc reignition voltage characteristics suggests that information relating secondary arc extinction times to different weather conditions should be used. Haubrich [4] reported the results of many field and laboratory
0885-8977/04$20.00 © 2004 IEEE
658
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004
tests, in which secondary arc currents are associated with their respective extinction times. Johns [1], [3] summarized those results to develop the reignition voltage characteristic of a secondary arc model. A. Data Fig. 1 depicts the time limits of the field test results reported by Haubrich [4]. As shown in the figure, there is a wide interval of possible extinction times between the lower and upper limits. If weather conditions are adverse, the wind speed will tend to be low, and extinction times close to the upper limit will occur. On the contrary, if weather conditions are favorable, the wind speed will tend to be high, and extinction times close to the lower limit will arise. B. Methodology
Fig. 1.
Secondary arc extinction time limits from field tests.
Fig. 2.
System employed to determine Ks(jIsj).
The studies carried out by Johns [1], [3] describe the equations of both secondary arc current and voltage, as well as the expression of the secondary arc reignition voltage, as a function of time. If the voltage imposed by the power system is higher than the reignition voltage, when the secondary arc crosses through zero, the arc is held in a conduction state; otherwise, it is held in a partial or final extinction time condition. The methodology employed by Johns, which is used here, assumes that the secondary arc reignition voltage can be calculated through (1) (1) Johns [1] proposed the use of the function equal to , from experimental data, and the determinaby simulation. The objective was tion of the function such that the predicted extinction to find discrete values of times matched the experimental times close to the upper limit for pesshown in Fig. 1. With those values, a function simistic extinction times was estimated. To simulate the secondary arc phenomenon, a time arc length variation characteristic was defined as well. By using experimental data associated with wind speeds lower than 1 m/s, Johns recommended the use of the linear approximation given by (2) if if
ms ms
(2)
Since more data relating the wind speed to the arc length variation is not available, (2) was also used here to develop the new set of secondary arc reignition voltage characteristics. The above-mentioned restriction yields results in which unrealistic wave shapes of secondary arc current and voltage might arise. However, this represents no problem in as much as our work focuses on evaluating extinction times statistically within the range established by field test results, rather than on representing precisely the instantaneous values of arc current and voltage that may be present in the phenomenon. If required, more realistic waveforms can be obtained replacing (2) with the arc length variation characteristic proposed in [11]. Fig. 1 does not specify the type of system, fault location, nor faulty phase. Therefore, Johns’ deterministic methodology [1] is also based on the assumption that the field test results
match the average secondary arc extinction times obtained from the simulation of a hypothetical and unloaded 500-kV overhead transmission line between two short-circuit equivalents. Such a system is depicted in Fig. 2, and its conductor data can be obfor each , the line length tained from Table I. To generate must be modified discretely, as suggested in [1]. To determine the initial value of line length associated with a particular secondary arc current, the expression for uncompensated transmission lines proposed by Kimbark [5] is utilized (3)
C. Determination of Curve Parameters Obtaining for any time-current pair in Fig. 1 requires the use of an iterative procedure after estimating the initial value of the line length through (3). [1] First, steady-state secondary arc currents per phase at three locations of the line (e.g., sending-end, receiving-end, and half-way down the line), are calculated and averaged not only taking into account its capacitive coupling, but also considering its magnetic coupling. If the average current does not match the
VIVAS et al.: A PROBABILISTIC APPROACH FOR SECONDARY ARC RISK ASSESSMENT
TABLE I SYSTEM PARAMETERS EMPLOYED TO DETERMINE Ks(jIsj).
Fig. 3.
659
TABLE II EXTINCTION TIME (S) AS A FUNCTION OF SECONDARY ARC CURRENT (A RMS) FOR THE NINE CURVES OF FIG. 3
Curves of secondary arc extinction time versus arc current.
expected current, the line length is modified until both expected current and average current matchup. is Further on, as recommended in [1], an initial value of assumed for the chosen values of time, current, and line length. Then, nine different single-phase faults, including the corresponding single-pole switching sequences as well as Johns’ secondary arc model, are simulated. Three faults (phases a, b, and c) are applied to the sending-end, three to the receiving-end, and three halfway down the line. The nine secondary arc extinction times obtained are averaged and the result is compared with the experimental time extracted from Fig. 1. If the average does not is modified and the nine simmatch the experimental time, ulations are repeated again for the same time-current pair. Othis considered adequate, and a new time-current pair erwise, is chosen. D. Calculation of Secondary Arc Reignition Voltage Characteristics To determine the new set of secondary arc reignition voltage characteristics, both line parameters and arc currents were calculated using the well-known Electromagnetic Transients Program (EMTP) [6]. To simulate Johns’ secondary arc model, [1] the EMTP-ATP version was utilized [7]. Fig. 3 depicts the curves of secondary arc extinction time versus arc current employed. As shown in Fig. 3, seven uniformly spaced curves were generated in addition to the two curves originally depicted in Fig. 1 to widen the range of possible values when performing the sta, tistical study. To achieve an adequate representation of between 5 and 40 A discrete steps of 5 A were utilized for rms. The extinction time values as a function of secondary arc current for the group of curves depicted in Fig. 3 are indicated obtained in Table II. The coefficients of the function
Fig. 4. General flowchart of the proposed methodology.
through simulation and the use of least squares techniques are shown in the Appendix. III. FORMULATION OF PROBABILISTIC ASSESSMENT METHODOLOGY Customarily, a pessimistic reignition characteristic is used to assess both the recovery of the electric insulation and the secondary arc extinction time associated with single-pole switching operations. However, if a finite universe of secondary arc reignition voltage curves is utilized instead of only one pessimistic characteristic, and a specific probability distribution is assigned to the set of curves, then the proposed methodology permits the calculation of the probability function of the secondary arc ex. The flowchart of the proposed approach is detinction picted in Fig. 4. A. Construction of Extinction Time Probability Function Given any value of secondary arc current, the first step to construct the probability characteristic of secondary arc extinction as a function of time is the assignment of the type of probability distribution to be used by the set of reignition voltage curves. The choice of a specific probability distribution (e.g., uniform, Gaussian, Weibull, etc.) does not affect the application
660
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004
of the methodology, and is simply used to determine the desired frequency of occurrence of the secondary arc extinction time universe to be obtained. Without loss of generality, applications of both the uniform as well as the normal (Gaussian) probability distributions are analyzed in this paper to illustrate the methodology, but cases with other probability distributions can be studied as well since the proposed approach allows the use of other probability models. This may be a subject for future research. A uniform probability distribution assigns equal probability of occurrence to the different secondary arc reignition voltage characteristics, which may be particularly useful when information on the local weather conditions is not available. Also, a normal or a Weibull probability distribution assigns skewness to the frequency distribution, which can be advantageous when data on local weather conditions (e.g., wind speed) fluctuation is available. Each different weather condition can be associated with a reignition voltage characteristic and, hence, with the set of secondary arc extinction times obtained from the simulation. Since weather conditions such as wind speed, temperature, humidity, and rain vary randomly, one can think of assigning a probability distribution that gives more weight to those curves associated with the local weather conditions. For instance, any geographical region can be given three wind speed values: maximum, minimum, and average. Similarly, each value can be associated with minimum, maximum, and average secondary arc extinction time curves, respectively. As a result, the intermediate curves between the upper and lower limits depicted in Fig. 3 will represent feasible secondary arc extinction times associated with possible wind speed fluctuation. For example, if a normal distribution is utilized, the second step to construct the probability characteristic is the selection of the mean value ( ) and the standard deviation ( ). By assuming that the curve 5 (intermediate extinction time) is associated with average wind speed, and curves 1 (minimum extinction time) and 9 (maximum extinction time) are associated with high and low wind speeds, respectively, the parameters and of the , , distribution can be obtained as follows: ; therefore . The application of a Gaussian distribution is limited to symmetrical cases (with reference to wind speed values) without loss of generality. In a more general case, the asymmetrical Weibull distribution applies. IV. COMPARISON WITH FIELD TEST RESULTS To evaluate the methodology, three cases with available field test results were studied. The first of them is a 500-kV transposed line of the Tennessee Valley Authority (TVA) [8], the second is a 765-kV untransposed line of the American Electric Power Company (AEP) [9], and the third consists of a 750-kV, 50-Hz transposed transmission line [10]. The statistical results obtained are shown in Figs. 5–7, respectively. A. Test Case 1: 500-kV System Thirteen field tests performed with a TVA 500-kV line [8] yielded secondary arc currents between 6.5 and 22.5 A rms, and
Fig. 5. Probability of secondary arc extinction as a function of time. Test case 1: 500-kV system. jIsj = 22:5 A rms.
Fig. 6. Probability of secondary arc extinction as a function of time. Test case 2: 765-kV system. jIsj = 24 A rms.
Fig. 7. Probability of secondary arc extinction as a function of time. Test case 3: 750-kV system. jIsj = 26 A rms.
reclosing times between 2.25 and 24.75 cycles (37.5 and 412.5 ms), respectively. Fig. 5 shows the results obtained from the statistical assessment when using uniform and normal frequency distributions, as well as a 22.5-A secondary arc current value obtained from steady-state simulation of the system. As shown in Fig. 5, the extinction time of a 22.5-A secondary arc varies between 37.4 and 422 ms, and the probability of extinction for arcs no longer than 386 ms is equal to 90 and 100% for uniform and normal distributions, respectively. A successful reclosing at 386 ms (23.15 cycles) for a 22.5-A secondary arc current was reported in [8]. This result agrees with the high
VIVAS et al.: A PROBABILISTIC APPROACH FOR SECONDARY ARC RISK ASSESSMENT
probabilities of extinction statistically obtained for times shorter than 386 ms. It should be noted, however, that the statistical methodology evaluates extinction time probabilities while [8] reports reclosing times. Therefore, one can suggest that the probability of success associated with a 386-ms reclosing time should be lower than that obtained from statistical simulation, given that the reclosing time must always be greater than the secondary arc extinction time to achieve a successful reclosing.
661
Future research should combine the probability of arc extinction with the probability of arc reignition obtained from experimental data. APPENDIX This appendix contains the parameters of the nine curves for . is obtained in , and is given in A rms. Curve 1 (min)
B. Test Case 2: 765-kV System Six field tests performed with an AEP 765-kV line [10] yielded secondary arc currents between 8 and 24-A rms, and secondary arc extinction times between 11 and 340 ms, respectively. Fig. 6 shows the results obtained from the statistical assessment when using uniform and normal frequency distributions, as well as a 24 A secondary arc current value obtained from steady-state simulation of the system. As shown in Fig. 6, the extinction time of a 24-A secondary arc varies between 22.6 and 581 ms, and the probability of extinction of arcs no longer than 340 ms is equal to 70 and 90% for uniform and normal distributions, respectively. C. Test Case 3: 750-kV System Six field tests performed with a 750-kV, 50-Hz line [10] yielded secondary arc currents between 3 and 26 A rms, and secondary arc extinction times between 90 and 300 ms, respectively. Fig. 7 shows the results obtained from the statistical assessment when using uniform and normal frequency distributions, as well as a 26-A secondary arc current value obtained from steady-state simulation of the system. As shown in Fig. 7, the extinction time of a 26-A secondary arc varies between 15 and 220 ms, and the probability of extinction of arcs no longer than 200 ms is equal to 90 and 100% for uniform and normal distributions, respectively.
V. CONCLUSIONS A statistical methodology for risk assessment of single-pole switching schemes based on the use of a family of secondary arc reignition voltage characteristics and a probability distribution curve assigned to it have been presented. The proposed reignition voltage curves represent the behavior of the electric insulation under different weather conditions and the assigned probability distribution allows the estimation of the risk associated with distinct extinction time values. By using EMTP-type simulations and field test results, the proposed technique performs a statistical evaluation of the behavior of the residual current, thus allowing the calculation of the probability of secondary arc extinction as a function of time and the quantification of the risk associated with different reclosing time relay settings. As compared to deterministic techniques based on pessimistic results, the proposed methodology is a more reasonable option to estimate the entire range of possible extinction times present during a single-pole switching operation.
where
Curve 2
where
Curve 3
where
Curve 4
where
Curve 5
where
Curve 6
where
Curve 7
where
662
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004
Curve 8
[8] L. Edwards, J. W. Chadwick Jr, H. A. Riesch, and L. E. Smith, “Single-pole switching on TVA’s Paradise-Davidson 500 kV line: Design, concepts and staged fault test results,” IEEE Trans. Power App. Syst., vol. PAS-90, pp. 2436–2450, Nov./Dec. 1971. [9] A. J. Fakheri, T. C. Shuter, J. M. Schneider, and C. H. Shih, “Singlephase switching tests on the AEP 765 kV system extinction time for large secondary arc currents,” IEEE Trans. Power App. Syst., vol. PAS-102, pp. 2775–2783, Aug. 1983. [10] B. Shperling, H. Scherer, J. Chadwick, N. Belyakov, V. Rashkes, and K. Khoetsian, “Single-phase switching test on 765 and 750 kV transmission lines,” IEEE Trans. Power App. Syst., vol. PAS-104, pp. 1537–1548, June 1985. [11] R. Rivas et al., “A fuzzy model for secondary arc extinction-time calculations,” Proc. 31st IEEE North Amer. Power Symp., pp. 1–6, Oct. 1999.
where
Curve 9 (max)
where
ACKNOWLEDGMENT The authors would like to thank L. Linares, F. Moreira, and M. Lukic of the University of British Columbia, Vancouver, BC, Canada, for their helpful discussions and suggestions. REFERENCES [1] A. T. Johns and A. M. Al-Rawi, “Developments in the simulation of long-distance single-pole switched EHV systems,” Proc. Inst. Elect. Eng., vol. 131, no. 2, pp. 67–77, Mar. 1984. [2] E. W. Kimbark, “Bibliography on single-pole switching,” IEEE Trans. Power App. Syst., vol. PAS-94, pp. 1072–1076, May/June 1975. [3] A. T. Johns and A. M. Al-Rawi, “Digital simulation of EHV systems under secondary arcing conditions associated with single-pole autoreclosure,” Proc. Inst. Elect. Eng., vol. 129, no. 2, pp. 49–58, Mar. 1982. [4] H. Haubrich, G. Hosemann, and R. Thomas, “Single-Phase Autoreclosing in EHV Systems,” CIGRE, Paris, France, paper 31–09, 1974. [5] E. Kimbark, “Charts of three quantities associated with single-pole switching,” IEEE Trans. Power App. Syst., vol. PAS-94, pp. 388–395, Mar./Apr. 1975. [6] H. W. Dommel, “Digital computer solution of electromagnetic transients in single and multi-phase networks,” IEEE Trans. Power App. Syst., vol. PAS-88, pp. 388–399, Apr. 1969. [7] S. Goldberg, W. F. Horton, and D. Tziouvaras, “A computer model of the secondary arc in single-phase operation of transmission lines,” IEEE Trans. Power Delivery, vol. 4, pp. 586–595, Jan. 1989.
Jose H. Vivas Nava (M’00) was born in Caracas, Venezuela, in 1970. He received the electrical engineer and M.Sc. degrees in electrical engineering from Universidad Simón Bolívar in 1995 and 1999, respectively. Currently, he is an Assistant Professor with the Department of Energy Conversion and Delivery at Universidad Simón Bolívar, Caracas. His areas of interest include power system analysis, FACTS, electrical transients, and overvoltage phenomena.
Richard A. Rivas (M’01) was born in Caracas, Venezuela, in 1965. He received the electrical engineer and M.Sc. degrees from Universidad Simón Bolívar, Caracas, Venezuela, in 1988 and 1993, respectively, and the Ph.D. degree in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2001. Currently, he is an Associate Professor with the Department of Energy Conversion and Delivery at Universidad Simón Bolívar, where he has been since 1991. His areas of interest include electromagnetic transients, underground transmission, and power systems protection.
Alberto J. Urdaneta (SM’90) received the M.Sc. degree in electrical engineering and applied physics and the Ph.D. degree in systems engineering from Case Western Reserve University, Cleveland, OH, in 1983 and 1986, respectively. Currently, he is a Professor of electrical engineering with the Department of Energy Conversion and Delivery at Universidad Simón Bolívar, Caracas, Venezuela. He is also Former Dean of Professional Studies. His research interests include the areas of power system analysis and optimization. Dr. Urdaneta is former Chairman of the IEEE Venezuelan Section.
