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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 1, FEBRUARY 2011

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Eliminating Subsynchronous Oscillations With an Induction Machine Damping Unit (IMDU) Sujit Purushothaman, Student Member, IEEE, and Francisco de León, Senior Member, IEEE

Abstract—The IEEE First and Second Benchmark Models for subsynchronous resonance (SSR) are used to analyze the damping properties of an induction machine damping unit (IMDU) coupled to the shaft of a turbo-generator set. This paper investigates the rating and location of the induction machine that, without the aid of any controllers, effectively damps subsynchronous resonance for all line series compensation levels. Eigenvalue analyses are performed on linearized models of the shaft system including the induction machine to find the optimum location. The best location of the IMDU, providing maximum damping, is next to the HP turbine at the end of the shaft. Time domain simulations are conducted to find the adequate rating of the induction machine. It is observed that a small size, high power (about 10% of the generator rating), low energy machine effectively damps SSR. The IMDU reduces peak torques in shaft sections during transients. In the paper, it is demonstrated that the addition of an IMDU at the end of the shaft would have prevented the SSR events of the Mohave Desert shafts. Index Terms—Damping oscillations, induction machine damping unit (IMDU), subsynchronous resonance (SSR).

I. INTRODUCTION

S

ERIES capacitive compensation has long been used as a means to increase the power transfer capability of a transmission line by reducing the inductive reactance of the line. This, however, may lead to subsynchronous resonance (SSR) [1]–[4]. The electrical equivalent of the inertia of the mechanically coupled masses may be seen as a capacitance. The rotational speed of the shaft is equivalent to the voltage across the capacitance [2], [5]. The coupling shafts are viewed as inductances, connecting these masses together. The resonant modes of this equivalent “LC” circuit are typically below synchronous frequency. The natural frequency of resonance for a series compensated line is given by [1]

(1) where equivalent system inductance, and series compensation of line. If the natural resonant frequency coincides with the 60-Hz complement of the coupled mass system, subsynchronous oscillations may be induced in the system [3]. Each mode has a parManuscript received March 18, 2010. First published May 10, 2010; current version published January 21, 2011. Paper no. TPWRS-00217-2010. The authors are with the Polytechnic Institute of NYU, Brooklyn, NY 11201 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.2010.2048438

ticular frequency of resonance which is indicated by the eigenvalues of the system. A strategy capable of curbing torsional oscillations at each resonant mode may be utilized as a method to mitigate SSR oscillations. Damping SSR oscillations has been a topic of great interest and research. Early strategies suggested dissipating the energy during resonance in resistor banks [6]. Countermeasures utilizing TCSC, NGH schemes [7]–[9], phase shifters [10], excitation controllers, and static VAR compensators [11] have been extensively researched through the years. Numerous modeling techniques and improvements on these schemes have also been given [12], [13]. The use of stored magnetic energy has been published in [14]–[16]. The concept of the induction machine damping unit (IMDU) as a countermeasure to SSR was devised in [17] and extended in [18]. These papers stated the possibility of damping SSR using an IMDU in conjunction with additional control for static VAR systems. A system similar to the IEEE First Benchmark Model (FBM) [19] was utilized in these papers, while assuming negligible mass for the IMDU. In this paper, we use an IMDU to eliminate SSR from both the true FBM and the IEEE Second Benchmark Model (SBM) [20]. An IMDU is a special high-power, low energy induction machine, with small rotor resistance and leakage reactance values, designed to operate close to synchronous speed. It is mechanically coupled to the turbo-generator (T-G) shaft and electrically connected to the generator bus. The main contributions of this paper include: putting forward the possibility of damping SSR with only an IMDU (no controllers needed) coupled to the shaft of the T-G; the identification of the location and rating of the damping unit; and the derivation of a set of specifications of an IMDU that yields a SSR-free generator set (rotor resistance, leakage reactance, and minimum shaft size to transmit effective damping). The IEEE First and Second Benchmark Models for subsynchronous studies were used to conduct eigenvalue analyses and time domain simulations. Time domain simulation studies were conducted with the IMDU in-between two masses and at the shaft ends. Simulations to study large transients were conducted. The coupling coefficient between masses was varied, and its effects on SSR mitigation and transient stability were observed. The best location providing maximum damping is next to the HP turbine at the end of the shaft as shown in Fig. 1. The capabilities of the damping unit were tested from 0%–100% line compensation levels, from zero to a maximum power transfer of 0.9 pu, and with zero mechanical damping. A study to find the power and energy dissipated by the machine was conducted to size the IMDU. It is found that an IMDU

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Fig. 1. Model of the T-G shaft including IMDU.

providing 10% of the generator power for 10 s would have prevented the SSR events at Mohave.

Fig. 2. Electrical equivalent used to model dynamics of IMDU and coupling shaft.

II. ANALYSIS The physical setup and mechanical coupling of the IMDU to the SBM shaft is depicted in Fig. 1. The damping unit is coupled to the HP turbine through a shaft. The spring constant of the coupling shaft was varied to determine its size. For simulation purposes, the inertia and mechanical damping of the IMDU are considered equivalent to the exciter which would give insight on how additional mass added on the shaft system would affect damping.

C. Adding Mass To study the effect of added mass on the T-G shaft, simulations were conducted using a modified IEEE SBM model without the IMDU and adding the equivalent mass of the damping unit on the shaft. A negligible change in the eigenvalues confirmed the results from time domain studies that the added mass of the IMDU produced no significant damping.

A. IMDU

D. Eigenvalue Analysis

The damping unit is an induction machine running at synchronous speed during steady state, consequently consuming or generating no power. In fact, it can even be electrically disconnected, drawing no magnetizing current. Deviations from synchronous speed due to subsynchronous resonance induce slip frequency currents in the rotor, and hence, damping torque is produced. It is observed (see Section IV-A) that the IMDU power reduces in a few seconds from the inception of SSR; hence, it is rated for very short duty cycle. The IMDU, therefore, is a high power and low energy induction machine. When shaft speed (at the IMDU location) is less than synchronous speed, the IMDU acts as a motor accelerating the shaft. When shaft speed exceeds synchronous speed, the machine becomes a generator decelerating the shaft towards synchronous speed. At synchronous speed (normal operating conditions), the IMDU remains dormant, producing no torque.

A linear model (electrical equivalent) of the SBM T-G system with the dynamics of the IMDU and its coupling shaft is illustrated in Fig. 2. The field exciter dynamics is not modeled and the excitation is held constant at 1 pu, as done in [20]. The time constant of the mechanical system is very large when compared to the electrical system, and hence, the speed governor dynamics are not included, keeping the input power to the turbines constant. Variables give the torque in the concerned shaft section, which are functions of the difference of the torsion angle at the ends of the shaft. The differential equations governing the model of Fig. 2 are as follows [3], [5]:

(3a)

B. Modeling the Damping Unit

(3b)

For the eigenvalue analysis, the torque-speed characteristics of an induction machine, with small rotor resistance, to be used as IMDU can be considered linear between synchronous speed and the critical slip (maximum torque) when operated at constant terminal voltage and frequency. Therefore, the torque of the damping unit, , can be modeled as being proportional to speed deviation, (deviation from synchronous speed) [21]. Load tests, for speeds close to synchronous, were performed on the selected induction machine running as motor and generator. Torque values were measured for every small increment in speed deviation. The slope of the torque-speed characteristic line was found as

(3c) (3d) (3e) (3f) (3g) (3h) (3i)

(2)

(3j)

PURUSHOTHAMAN AND DE LEÓN: ELIMINATING SUBSYNCHRONOUS OSCILLATIONS WITH AN INDUCTION MACHINE DAMPING UNIT (IMDU)

where represent the angular speed of individual masses, the and mechanical inertia of which is given by capacitances damping by resistances . Inductances represent the spring constant of the shaft section used to connect adjacent masses. is the constant input torque to the turbines and is the output torque of the generator. The acceleration equations (3a), (3c), (3e), (3g), and (3i) are obtained from the application of Kirchhoff’s current law at each node. Voltage at each node is equivalent to speed of each mass. Equations (3b), (3d), (3f), and (3h) are obtained by applying Kirchhoff’s voltage law between adjacent nodes. For example, (3a) defines the acceleration of the exciter and (3b) relates the torsion on shaft section with speed difference between exciter and generator. The power (3j) defines the relation between gen, voltages erator torque , equivalent system reactance at generator and infinite bus and , and the power angle between the two buses . Variable in (3i) linearizes the operation of the IMDU around synchronous speed. Increasing yields a steeper torque speed characteristic around synchronous speed, which indicates a reduction of the rotor resistance . The reduction in rotor resistance means an increase in the power rating of the machine. Hence, may be utilized as a representation of the power of the IMDU machine (more details are given in Section IV). in (3h) represents the spring constant of the coupling shaft and is an indication of the shaft size.

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TABLE I MOST UNSTABLE EIGENVALUES FOR FBM

TABLE II MOST UNSTABLE EIGENVALUES FOR SBM

III. NUMERICAL RESULTS To cover the maximum of power transfer capability curve of the generator, calculations were done for initial power transfer varying from 0 to 0.9 pu. The eigenvalues were found for a wide range of line compensation varying from nil to total compensation. A. Location of IMDU The location of IMDU is of great importance to this study. The damping unit may be placed in various locations on the T-G shaft, between masses or at either end. When located between two masses, and , which were coupled with spring conand with the same stant , the IMDU is coupled to spring constant . When the IMDU is at the end of the shaft, the spring constant has been reduced to the minimum necessary to damp oscillations by reducing shaft diameter (67% of maximum in the SBM; the value is computed in Section IV-B). The initial power transfer was held constant at 0.9 pu. The eigenvalues were computed for the damping unit located in various positions on the T-G shaft. The real parts of the eigenvalues increase (become less negative or more positive) as the compensation increases, but leaving the imaginary part constant. The variations of the eigenvalues are given in Table I for the FBM and in Table II for the SBM. Note that when the IMDU is not present, there are eigenvalues with positive real parts and only when the IMDU is coupled to the HP turbine all eigenvalues have negative real parts. The IMDU coupled to the HP turbine at the end of the T-G shaft yields best damping results as it stabilizes all unstable modes. The addition of a mass at this location on the T-G shaft adds a stable super synchronous torsional mode. Coupling the

Fig. 3. Eigenvalues for second and third torsion modes for the SBM and damped system with compensation.

IMDU at the end of the shaft not only allows for a reduced size shaft but also opens the possibility of retrofitting existing installations. B. Capacitive Compensation Level A study was done to compare the contribution to the damping by the IMDU with the IEEE SBM as a function of the compensation level. Fig. 3 shows the variation of the eigenvalues for the second and third torsion modes for the SBM with and without the damping unit. One can see from Fig. 3 that the IMDU is very effective eliminating risks of SSR since the real parts of the eigenvalues are substantially negative over the entire range of compensation. The eigenvalues for the other modes are stable and show negligible change with compensation.

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TABLE III MECHANICAL PARAMETERS AND RATING OF THE IMDU

C. Time Domain Simulations The Alternative Transient Program (ATP) was used to conduct time domain simulation of the system with the damping unit. The IMDU speed-torque characteristic is modeled in full, and it has not been linearized for time domain simulations. The IEEE FBM and SBM used for the simulation was as per [19] and [20]. No controllers were modeled. The excitation voltage and turbine torque were held constant. The line compensation was kept constant at 70% in FBM and 55% for the SBM. Successive iterations were carried out with damping units of increasing MW ratings until SSR was damped for power transfers varying from 0 to 0.9 pu. The data for the IMDU with minimum MW rating that suppresses SSR are given in Table III. It must be noted that the minimum MW rating is one tenth of the rating of their respective generators. Smaller damping units produced insufficient damping hence unable to mitigate SSR oscillations for the entire study range. The Indsync program (part of ATP) was used for the calculation of the electrical and mechanical parameters of the IMDU [22], [23]. A small disturbance on the infinite bus (“BUS1” in the IEEE benchmark model) was used to trigger SSR oscillations while keeping the initial power transfer at 0.5 pu. The graphs of generator torque versus time in the un-damped SBM system and with the damping unit are given in Figs. 5(a) and (b), respectively. Fig. 4(b) shows the damped torque oscillations in the FBM system with the damping unit. The figures show how effectively the damping unit mitigates SSR oscillations. Note that the initial response (for about 2 s) with and without IMDU is similar. Therefore, the IMDU can remain electrically disconnected in steady state and be switched on when a torsional interaction problem is detected.

Fig. 4. (a) Generator torque oscillations for un-damped FBM system. (b) Generator torque oscillations for FBM system with IMDU.

D. Effect on Transient Stability The effect of the damping unit on the critical clearing time was studied to get an insight on the transient stability effect of the IMDU. A sustained short circuit was created on the infinite bus. The compensated line was disconnected when the fault was cleared. The critical clearing time showed no change with and without the damping unit. E. Initial Power Angle The IEEE SBM was used to study the effects of the power transmitted on the lines prior to the initiation of SSR. Fig. 6

Fig. 5. (a) Generator torque oscillations for un-damped SBM system. (b) Generator torque oscillations for SBM system with IMDU.

shows the variation of the second and third eigenvalues versus the steady state (pre-SSR) power angle with and without the IMDU. One can observe that the system (with no IMDU) is stable for low power transfer, but it becomes increasingly unstable as the power transfer increases. The inclusion of the

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Fig. 8. Peak G-LP torque versus fault clearing time. Fig. 6. Eigenvalues for second and third torsion modes for the SBM with and without IMDU (55% line compensation). TABLE IV EIGENVALUES OF SYSTEM WITH ZERO MECHANICAL DAMPING

enough to solve the torque amplification problem. Fig. 8 compares the peak torque in Gen-LP shaft section (this section undergoes the maximum stresses) in the IEEE Second Benchmark Model and system with IMDU. Therefore, although the IMDU does not solve the torque amplification problem, it also causes no ill-effects. G. Mechanical Damping

Fig. 7. Variation of eigenvalues with compensation and power transferred for the SBM system with IMDU.

IMDU leads to stable eigenvalues for entire range of power transfers. Sections III-B and III-E show the dependency of system damping with line compensation and power transfer, respectively. Fig. 7 gives a consolidated graph showing the variation of system damping for all resonant modes with both of these parameters. This is done for three particular power transfers: 0, 0.5, and 0.9 pu. The inclusion of the IMDU shifts the eigenvalues for all the resonant modes into the stable region for all line compensation and power transfers, consequently completely eliminating SSR. F. Torque Amplification Study Time domain simulations were conducted to study the torque amplification SSR for the SBM. With conditions as stated in [20], peak torques in all the shaft sections were observed to be reduced by up to 14.8%. This is a small improvement, not large

To verify the effectiveness of the IMDU coupled to the HP turbine end of the IEEE SBM T-G shaft, the mechanical damping was set to zero which makes all the torsional modes unstable. Table IV gives the eigenvalues with zero mechanical damping. In the second benchmark model, all the torsional modes become unstable. With the IMDU coupled to the HP turbine, it was observed that all modes except the one at 51 Hz become stable. There is a negligible shift in this eigenvalue with or without IMDU. Hence, we can infer that this mode is more dependent on the mechanical damping of the masses than the presence of the IMDU. However, the IMDU provides practical damping to the other modes. The FBM does not incorporate mechanical damping. However, the IMDU provides sufficient damping to all modes. This subsection illustrates how the inclusion of the IMDU in the IEEE FBM and SBM with zero mechanical damping provides sufficient damping to all modes (except one for the SBM). This fact establishes the SSR mitigation capability of the IMDU. IV. RATING AND SIZING THE IMDU AND SHAFT FOR THE IEEE SECOND BENCHMARK MODEL A. IMDU The IMDU torque opposes oscillating torques to mitigate SSR. A natural inference could be to connect a high power damping unit to produce more damping in the system. Fig. 9 shows the torque-speed characteristics of an induction machine as a function of rotor resistance and leakage reactance. It can be seen that the slip at maximum torque is proportional

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Fig. 10. Variation of system damping with k of IMDU.

Fig. 9. Effect of rotor resistance and leakage reactance on torque speed characteristics of induction machine close to synchronous speed. Fig. 11. Decaying IMDU power with fault time.

to the rotor resistance , while the maximum torque itself is independent of . The maximum torque is principally affected by the leakage reactance , while the slip at maximum torque is directly proportional to the ratio. For the purposes of and should be as small damping SSR oscillations, both yields a larger slope in (2) while as possible. Reducing gives a larger maximum torque. Fig. 9 shows the reducing variation of the torque-speed characteristics when the rotor resistance and leakage reactance are divided by two and by four. It can be seen that reducing the rotor resistance produces greater slope around the synchronous speed. Reducing the leakage reactance increases the maximum torque extending the operating region over a wider range. These two features are desirable characteristics for an IMDU. Fig. 10 plots the variation of eigenvalues for resonant modes pertaining to 24 and 32 Hz with as in (3i) for the damping unit. It is observed that damping increases by increasing , which is equivalent to reducing . The energy to be dissipated by the IMDU could be of prime importance for the sizing of the machine. Time domain simulation results were used to plot the power developed by the IMDU versus time after initialization of SSR oscillations. The fault clearing time affects the peak torque in the different shaft sections and therefore the torque requirements from the IMDU. Fig. 11 shows the profile of decaying IMDU power for the two extreme clearing times (2 ms and 100 ms). Two milliseconds is used to reproduce the case illustrated in [20] and 100 ms is the critical clearing time. A sustained fault longer than 100 ms results in loss of synchronism. It can be seen from Fig. 11 that the peak torque developed by the IMDU in both cases is the same, but the effective power

during the mitigation period varies with the fault time. One can also see that maximum power is required only when SSR is initialized and that power requirements drop quickly as the oscillations are damped. The maximum energy dissipated by the IMDU for the mitigation of SSR for 100-ms fault is calculated as the integral of the power and was found to be 71.34 MJ. The equivalent power over this period is 7.97 MW. The sizing of the machine can be done by finding the appropriate size of conductors which depends on the amount of heat generated due to the current flowing in the machine. The temperature rise in the current carrying conductors was calculated using the adiabatic heating method [24], which is known to yield conservative results. The current was found from time domain simulations using the power of Fig. 11 and keeping the voltage at the generator bus constant. For the given duty cycle of the IMDU, a relatively copper conductor produces rise above amsmall . Fig. 12 shows the adiabatic temperbient temperature of copper conductor for the IMDU current. ature rise in an B. Sizing of Coupling Shaft The spring constant required by the coupling can be used as a parameter to size the shaft. The effect on the system damping due to variation of coupling shaft spring constant from 0 to maximum is given in Fig. 13, with all other system parameters kept constant as in Section III-C. The maximum refers to the Gen-LP turbine shaft in the benchmark model.

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From the analysis of this section, we conclude that heating will not determine the size of the IMDU, but resistance and reactance requirements will be the design constraining parameters. The design details for the IMDU are not within the scope of this paper and are the subject of our future research. V. CONCLUSIONS

Fig. 12. Temperature rise in 8 mm copper wire for IMDU duty cycle.

Fig. 13. Variation of system damping with spring constant of coupling shaft.

The system damping is observed to be almost constant for (20% of spring constants greater than maximum spring constant on SBM T-G shaft). The spring conof a uniform circular shaft with diameter , material stant rigidity modulus , and length is given by [5]

(4) It infers that a coupling shaft with minimum spring constant and same length as the Gen-LP shaft would require only 67% of the diameter to mitigate SSR oscillations. Any further increase in the shaft size will not produce any significant increase in system damping. It must be noted that any reduction in length of the coupling shaft would allow a further reduction of its diameter.

Eigenvalue studies and time domain simulations conducted on the IEEE First and Second Benchmark Models show the damping benefits on SSR of an IMDU coupled with the shaft of a T-G. The major findings of the study are as follows. 1) The inclusion of the IMDU on the T-G shaft can eliminate torsional interaction type of SSR oscillations without the aid of any other controller. 2) The IMDU can suppress SSR over the entire range of power transfer conditions and line compensation levels. 3) The IMDU does not eliminate torque amplification; however, it reduces the torsional stresses on the shaft sections during large transients by up to 14.8%. 4) The IMDU is a small-size high power and low energy induction machine. 5) The HP turbine at the end of the shaft offers the best location for the IMDU for both the FBM and the SBM. 6) In the case of the SBM system, the diameter of the coupling shaft can be 33% smaller than the maximum diameter of the T-G shaft. 7) The IMDU does not affect the initial dynamic response of the T-G; therefore, it can be switched on only after a disturbance exciting torsional interaction is detected. 8) The IMDU does not affect the response of the generator to transient stability. 9) An IMDU with an instantaneous rating of one tenth of that of the continuous rating of the generator provides sufficient damping to all modes in the SBM and FBM systems. The damping capabilities of the IMDU were tested on the IEEE FBM which is based on real values taken from the Navajo Project SSR case. The positive results obtained, i.e., an IMDU coupled at the end of the shaft would have prevented the occurrence of SSR, validate the effectiveness of the IMDU on a real system. SSR oscillations can be eliminated without modifying the electrical or mechanical parameters of the generator and turbine. Therefore, it is potentially possible to retrofit existing installations.

C. Observations The system damping is greatly dependent on the power rating of the induction machine. The value of k relates to the instantaneous power rating. ATP simulations show that the peak torque of the IMDU lasts for a very short time. The energy required to be dissipated by the machine also reflects the reduced size of the machine. The unit must have high instantaneous power capacity (small and ) but it can be a low energy machine due to its small duty cycle. The dependency of system damping on the spring constant of the shaft section connecting the damping unit to the HP turbine is negligible as long as it is greater than or one fifth of that of the Gen-LP shaft.

ACKNOWLEDGMENT The authors would like to thank the Can/Am EMTP User Group for providing documentation on the original work done in subsynchronous resonance in ATP. REFERENCES [1] IEEE Subsynchronous Resonance Working Group, “Terms, definitions and symbols for subsynchronous oscillations,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 6, pp. 1326–1334, Jun. 1985. [2] B. L. Agarwal, P. M. Anderson, and J. E. Van Ness, Subsynchronous Resonance in Power Systems. Piscataway, NJ: IEEE Press, 1990. [3] K. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems. Boston, MA: Kluwer, 1999.

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[4] J. W. Ballance and S. Goldberg, “Subsynchronous resonance in series compensated transmission lines,” IEEE Trans. Power App. Syst., vol. PAS-92, no. 5, pp. 1649–1658, Sep. 1973. [5] P. Kundur, Power System Stability and Control. New York: McGrawHill, 2007. [6] O. Wasynczuk, “Damping shaft torsional oscillations using a dynamically controlled resistor bank,” IEEE Trans. Power App. Syst., vol. PAS-100, no. 7, pp. 3340–3349, Jul. 1981. [7] E. Gustafson, A. Aberg, and K. J. Astrom, “Subsynchronous resonance. A controller for active damping,” in Proc. 4th IEEE Conf. Control Applications, Sep. 1995, pp. 389–394. [8] N. Kakimoto and A. Phongphanphanee, “Subsynchronous resonance damping control of thyristor-controlled series capacitor,” IEEE Power Eng. Rev., vol. 22, no. 9, p. 63, Sep. 2002. [9] H. Sugimoto, M. Goto, W. Kai, Y. Yokomizu, and T. Matsumura, “Comparative studies of subsynchronous resonance damping schemes,” in Proc. Int. Conf. Power System Technology, Oct. 2002, vol. 3, pp. 1472–1476. [10] M. R. Iravani and R. M. Mathur, “Damping subsynchronous oscillations in power systems using a static phase-shifter,” IEEE Trans. Power Syst., vol. 1, no. 2, pp. 76–82, May 1986. [11] L. Wang and Y. Y. Hsu, “Damping of subsynchronous resonance using excitation controllers and static VAr compensations: A comparative study,” IEEE Trans. Energy Convers., vol. 3, no. 1, pp. 6–13, Mar. 1988. [12] B. K. Perkins and M. R. Iravani, “Dynamic modeling of a TCSC with application to SSR analysis,” IEEE Trans. Power Syst., vol. 12, no. 4, pp. 1619–1625, Nov. 1997. [13] X. Zhao and C. Chen, “Damping subsynchronous resonance using an improved NGH SSR damping scheme,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Jul. 1999, vol. 2, pp. 780–785. [14] W. Li, L. Shin-Muh, and H. Ching-Lien, “Damping subsynchronous resonance using superconducting magnetic energy storage unit,” IEEE Trans. Energy Convers., vol. 9, no. 4, pp. 770–777, Dec. 1994. [15] A. H. M. A. Rahim, A. M. Mohammad, and M. R. Khan, “Control of subsynchronous resonant modes in a series compensated system through superconducting magnetic energy storage units,” IEEE Trans. Energy Convers., vol. 11, no. 1, pp. 175–180, Mar. 1996. [16] O. Wasynczuk, “Damping subsynchronous resonance using energy storage,” IEEE Trans. Power App. Syst., vol. PAS-101, no. 4, pp. 905–914, Apr. 1982. [17] S. K. Gupta, A. K. Gupta, and N. Kumar, “Damping subsynchronous resonance in power systems,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 149, no. 6, pp. 679–688, Nov. 2002. [18] K. Narendra, “Damping SSR in a series compensated power system,” in Proc. IEEE Power India Conf., 2006, p. 7.

[19] IEEE Subsynchronous Resonance Task Force, “First benchmark model for computer simulation of subsynchronous resonance,” IEEE Trans. Power App. Syst., vol. PAS-96, no. 5, pp. 1565–1572, Sep. 1977. [20] IEEE Subsynchronous Resonance Working Group, “Second benchmark model for computer simulation of subsynchronous resonance,” IEEE Trans. Power App. Syst., vol. PAS-104, no. 5, pp. 1057–1066, May 1985. [21] G. Slemon, Electric Machines and Drives. Reading, MA: Addison Wesley, 1992, pp. 185–201. [22] H. W. Dommel, Electromagnetic Transients Program (EMTP) Theory Book, 1987, pp. 311–338. [Online]. Available: http://www.emtp.org. [23] A. Rifaldi and R. B. Lastra, ATP Rulebook, ch. IX. [24] Calculation of Thermally Permissible Short-Circuit Currents, Taking Into Account Non-Adiabatic Heating Effects, IEC std. 949, 1988. Sujit Purushothaman (S’09) received the B.E. degree in electrical engineering from Mumbai University (Sardar Patel College of Engineering), Mumbai, India, in 2005 and the M.S. degree from the Polytechnic Institute of NYU, Brooklyn, NY, where he is currently pursuing the Ph.D. degree. His work experience includes testing and development of medium voltage switchgear for Siemens India. His research interest includes power system transients, machine design and modeling, and interconnection of distributed generators to the power grid.

Francisco de León (S’86–M’92–SM’02) received the B.Sc. and the M.Sc. (Hons.) degrees in electrical engineering from the National Polytechnic Institute, Mexico City, Mexico, in 1983 and 1986, respectively, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 1992. He has held several academic positions in Mexico and has worked for the Canadian electric industry. Currently, he is an Associate Professor at the Polytechnic Institute of NYU, Brooklyn, NY. His research interests include the analysis of power definitions under nonsinusoidal conditions, the transient and steady-state analyses of power systems, the thermal rating of cables and transformers, and the calculation of electromagnetic fields applied to machine design and modeling.