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Investigation of SSR Characteristics of Hybrid Series Compensated Power System with SSSC

Investigation of SSR Characteristics of Hybrid Series Compensated Power System with SSSC Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 621818, 8 pages doi:10.1155/2011/621818 Research Article Investigation of SSR Characteristics of Hybrid Series Compensated Power System with SSSC 1 1 2 R. Thirumalaivasan, M. Janaki, and Nagesh Prabhu School of Electrical Engineering, VIT University, Vellore 632014, India Canara Engineering College, Benjanapadavu, Bantwal, Mangalore 574219, India Correspondence should be addressed to R. Thirumalaivasan, thirumalai.r@vit.ac.in Received 2 November 2010; Accepted 13 March 2011 Academic Editor: Henry S. H. Chung Copyright © 2011 R. Thirumalaivasan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The advent of series FACTS controllers, thyristor controlled series capacitor (TCSC) and static synchronous Series Compensator (SSSC) has made it possible not only for the fast control of power flow in a transmission line, but also for the mitigation of subsynchronous resonance (SSR) in the presence of fixed series capacitors. SSSC is an emerging controller and this paper presents SSR characteristics of a series compensated system with SSSC. The study system is adapted from IEEE first benchmark model (FBM). The active series compensation is provided by a three-level twenty four-pulse SSSC. The modeling and control details of a three level voltage source converter-(VSC)-based SSSC are discussed. The SSR characteristics of the combined system with constant reactive voltage control mode in SSSC has been investigated. It is shown that the constant reactive voltage control of SSSC has the effect of reducing the electrical resonance frequency, which detunes the SSR. The analysis of SSR with SSSC is carried out based on frequency domain method, eigenvalue analysis and transient simulation. While the eigenvalue and damping torque analysis are based on linearizing the D-Q model of SSSC, the transient simulation considers both D-Q and detailed three phase nonlinear system model using switching functions. 1. Introduction a power system by dynamically controlling the power-angle curve of the system [1, 2]. With SSSC, working in capacitive Power transfer capability of long transmission line is limited mode, net reactance is reduced, and, during the first by the transient stability limit. The first swing stability limit swing, sufficient decelerating area is introduced to count- of a single machine infinite bus system can be determined erbalance the accelerating area. However, in the subsequent through well-known equal-area criterion [1, 2]. During swings, the SSSC provides better damping than that of the faulted period, the electrical output power of the machine STATCOM when supplementary modulation controllers are decreases drastically while the input mechanical power incorporated [3]. remains more or less constant. Thus, the machine acquires The series capacitor compensation for long-distance excess energy and is used to accelerate the machine. The power transmission line helps in enhancing power transfer excess energy during faulted period can be represented by and is an economical solution to improve the system stability an area called accelerating area. To maintain stability, the compared to addition of new lines. A series capacitor machine must return the excess energy once the fault is compensated line exhibits a resonant minimum in its cleared. The excess energy returning capability of the impedance at a frequency f = f = X /X ,where X is er 0 C L C machine in postfault period is represented by another area the capacitive compensating reactance, X is inductive line called decelerating area. Thus, the stability of the system can reactance, and f is the synchronous frequency of the power be improved by enlarging the decelerating area, and it system. The resonant frequency f of the compensated er requires raising the power-angle curve of the system. Flexible line depends on the level of compensation of the line AC transmission systems (FACTS) devices are found to be inductance but is always subsynchronous since, in practice, very effective in improving both stability and damping of the compensation ratio is less than unity. It is the coupling of 2 Advances in Power Electronics this subsynchronous electrical transmission line resonance to V i the mechanical resonances of a multistage turbine-generator that gives rise to the phenomenon of SSR. The problem of self-excited torsional frequency oscillations (due to torsional interactions) was experienced at Mohave power station in USA. in December 1970 and October 1971 [3, 4]. The hybrid compensation consisting of suitable combi- nation of passive elements and active FACTS controller such as TCSC or SSSC can be used to mitigate SSR [5]. SSSC is a new generation series FACTS controller based on VSC VSC and has several advantages over TCSC based on thyristor controllers. An ideal SSSC is essentially a pure sinusoidal AC voltage source at the system fundamental frequency. The dc voltage is always injected in quadrature with the line current, dc thereby emulating an inductive or a capacitive reactance in series with the transmission line [3]. SSSC output impedance at other frequencies is ideally zero. Thus, SSSC does not resonate with the inductive line impedance to initiate sub- synchronous resonance oscillations. However, in hybrid series compensation, fixed capacitor element contributes for series resonance. Figure 1: Schematic representation of SSSC. The SSSC has only one degree of freedom (i.e, reactive voltage control, unless there is an energy source connected on the DC side of VSC which allow for real power exchange) which is used to control active power flow in the line [3]. a multipulse or a multilevel configuration. The elimination The VSC based on three-level converter topology greatly of voltage harmonics requires multi-pulse configuration of reduces the harmonic distortion on the AC side [3, 6, 7]. VSC. The multi-pulse converters are generally of TYPE-2 In this paper fixed series capacitor and active compensation where only the phase angle of converter output voltage can provided by three-level twenty four-pulse VSC-based SSSC be controlled and modulation index of the converter remains are considered. The constant reactive voltage control of SSSC fixed [3]. is considered. The major objective is to investigate SSR char- When the DC voltage is constant, the magnitude of ac acteristics of the series compensated system with SSSC using output voltage of the converter can be changed by Pulse both linear analysis and nonlinear transient simulation. It Width Modulation (PWM) with two-level topology which is shown that the constant reactive voltage control of SSSC demands higher switching frequency and leads to increased has the effect of reducing the electrical resonance frequency, losses. In three level converter topology, both the magni- which detunes SSR. tude and phase angle of converter output voltage can be The study is carried out based on frequency domain controlled. This converter is classified as TYPE-1 converter method, eigenvalue analysis and transient simulation [8]. The modelling of the system neglecting VSC is detailed [9], where DC bus voltage is maintained constant and the magnitude of converter output voltage is controlled by (including network transients) and can be expressed in DQ varying dead angle β with fundamental switching frequency variables or (three) phase variables. The modeling of VSC modulation [3, 10]. The harmonics are dependent on the is based on (1) DQ variables (neglecting harmonics in the capacitance and the operating point of the SSSC. The detailed output voltages of the converters) and (2) phase variables and three-phase model of SSSC is developed by modelling the the use of switching functions. The damping torque analysis, converter operation by switching functions. The switching eigenvalue analysis, and the controller design is based on function for phase “a”is shown in Figure 2. the DQ model while the transient simulation considers both models of VSC. The results based on linear analysis The switching functions of phases b and c are similar are validated using transient simulation based on nonlinear but phase shifted successively by 120 in terms of the system model. fundamental frequency. Assuming that the dc capacitor The paper is organized as follows. Section 2 describes the voltages V = V = V /2, the converter terminal voltages dc1 dc2 dc modelling of SSSC whereas the different methods of analysis with respecttothe midpointof dc side“N”can be obtained of SSR are discussed in Section 3. Section 4 describes a case as study and investigates the SSR characteristics with SSSC. The major conclusions of the paper are given in Section 5. ⎡ ⎤ ⎡ ⎤ V P (t) aN 2. Modelling of SSSC with Three-Level VSC ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dc ( ) ⎢ V ⎥ = ⎢ P t ⎥ ,(1) bN ⎣ ⎦ ⎣ ⎦ Figure 1 shows the schematic representation of SSSC. In the P (t) V c cN power circuit of an SSSC, the converter is usually either Advances in Power Electronics 3 1.5 where 2π 1 ( ) S t = S t + 1x 1x ω 12 (6) π 1 P (t) a S (t) = S t + , x = a, b, c. 1x x ω 24 0.5 The switching functions for second twelve-pulse converter are given by 0 2β S (t) = S (t) + S (t) − S (t) , 2a 2a 2a 2c −0.5 S (t) = S (t) + S (t) − S (t) , (7) 2b 2b 2b 2a −1 ( ) ( ) √ ( ) ( ) Line current i (t) S t = S t + S t − S t , 2c a 2c 2c 2b where −1.5 0.42 0.425 0.43 0.435 0.44 0.445 0.45 2π 1 Time (s) S (t) = S t + 2x 2x ω 12 Figure 2: Switching function for a three level converter. (8) π 1 S (t) = S t − , x = a, b, c. 2x x ω 24 and the converter output voltages with respect to the neutral The switching functions for a twenty four-pulse con- of transformer can be expressed as verter are given by ⎡ ⎤ ⎡ ⎤ 24 12 12 S (t) = S (t) + S (t), x = a, b,and c. (9) V S (t) a x 1x 2x an ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V S (t) V ⎢ ⎥ = ⎢ ⎥ ,(2) b dc bn If the switching functions are approximated by their fun- ⎣ ⎦ ⎣ ⎦ i damental components (neglecting harmonics) for a 24-pulse V S (t) cn three level converter, we get where V = V cos β sin ω t + φ + γ , (10) dc o an P (t) P (t) + P (t) + P (t) π a a b c S (t) = − . (3) 2 6 i ◦ and V , V are phase shifted successively by 120 . bn cn The line current is given by i = 2/3I sin(ω + φ)and a a o S (t) is the switching function for phase “a” of a 6-pulse 3- i , i are phase shifted successively by 120 .Notethat γ is b c level VSC. Similarly for phase “b”, S (t), and for phase “c ”, the angle by which the fundamental component of converter S (t) can be derived. The peak value of the fundamental and output voltage leads the line current. It should be noted that harmonics in the phase voltage V are found by applying an γ is nearly equal to ±π/2 depending on whether SSSC injects Fourier analysis on the phase voltage and can be expressed as inductive or capacitive voltage. Neglecting converter losses, we can get the expression for dc capacitor current as V = V cos hβ , (4) dc an(h) hπ ⎡ ⎤ ⎢ ⎥ where, h = 1, 5, 7, 11, 13, and β is the dead angle (period) ⎢ ⎥ 24 24 24 i S (t) S (t) S (t) i =− ⎢ ⎥ . (11) dc b a b c during which the converter pole output voltage is zero. We ⎣ ⎦ can eliminate the 5th and 7th harmonics by using a twelve- i pulse VSC, which combines the output of two six-pulse con- A particular harmonic reaches zero when 2β = 180 /h.At verters using transformers. β = 3.75 , the three level 24-pulse converter behaves optimum The switching functions for first twelve-pulse converter nearly like a two-level 48-pulse converter as 23th and 25th are given by harmonics are negligibly small. S (t) = S (t) + S (t) − S (t) , 1a 1a 1a 1c 2.1. Modelling of SSSC in D-Q Variables. When switching functions are approximated by their fundamental frequency S (t) = S (t) + S (t) − S (t) , (5) 1b 1a components, neglecting harmonics, SSSC can be modelled 1b 1b by transforming the three-phase voltages and currents to D- 12   Q variables using Kron’s transformation [2]. The SSSC can S (t) = S (t) + S (t) − S (t) , 1c 1c 1c 1b be represented functionally as shown in Figure 3. P (t) a 4 Advances in Power Electronics I∠φ + dc ref K Σ p Σ γ P(se)(ord) γ and β se R X st st 1 calculator V ∠φ + γ 1+ sT md V se R(se)(ord) dc Figure 3: Equivalent circuit of SSSC as viewed from AC side. Figure 4: Type-1 controller for SSSC. In Figure 3, R and X are the resistance and reactance of st st the interfacing transformer of VSC. The magnitude control + ΔS of converter output voltage V is achieved by modulating the ΔT m Mechanical system conduction period affected by dead angle of converter while G(s) dc voltage is maintained constant. The converter output voltage can be represented in D-Q ΔT frame of reference as Electrical system 2 2 i i i KH (s) V = V + V , D Q i (12) V = k V sin φ + γ , m dc Figure 5: Interaction between mechanical and electrical system. V = k V cos φ + γ , m dc performance. In three level 24-pulse converter, dc voltage ref- where k = kρ cos β ; k = 4 6/π for a 24-pulse converter. ρ m se erence may be adjusted by a slow controller to get optimum is the transformation ratio of the interfacing transformer. harmonic performance at β = 3.75 in steady state. From a control point of view, it is convenient to define the se The structure of type-1 controller for SSSC is given in active voltage (V )and reactive (V ) voltage injected P(se) R(se) i i Figure 4.In Figure 4, γ and β are calculated as se by SSSC in terms of variables in D-Q frame (V and V )as D Q follows R(se)(ord) -1 i i γ = tan , V = V cos φ − V sin φ, R(se) D Q P(se)(ord) (13) i i ⎡ ⎤ (15) V = V sin φ + V cos φ. 2 2 P(se) D Q V + V P(se)(ord) R(se)(ord) −1 ⎣ ⎦ β = cos se k V Here, positive V implies that SSSC injects inductive volt- m dc R(se) age and positive V implies that it draws real power to P(se) meet losses. 3. Analysis of SSR The dc side capacitor is described by the dynamical equa- tion as The two aspects of SSR are [4](i) steady-state SSR (induction generator effect (IGE) and torsional interaction (TI)) (ii) g ω dV ω dc c b b =− V − i , (14) dc dc shaft torque amplification due to transients. The analysis of dt bc bc steady-state SSR can be done by linearized models at the where i =−[k sin(φ + γ)I + k cos(φ + γ)I ], I and I operating point and include damping torque analysis and dc m D m Q D Q are the D-Q components of the line current. eigenvalue analysis. The analysis of shaft torque amplification due to transients requires transient simulation of the nonlin- ear model of the system. For the analysis of SSR, it is adequate 2.2. Type-1 Controller. In this type of controller, both mag- to model the transmission line by lumped resistance and nitude (modulation index k ) and phase angle of converter inductance where the line transients are also considered. output voltage (γ) are controlled. The capacitor voltage is The generator stator transients are also considered by using maintained at a constant voltage by controlling the active detailed (2.2) model of the generator. component of the injected voltage V . The real voltage P(se) The analysis of SSR is carried out based on damping reference V is obtained as the output of DC voltage P(se)(ord) torque analysis, eigenvalue analysis, and transient simula- controller. The reactive voltage reference V may R(se)(ord) tion. be kept constant or obtained from a power scheduling controller. However, for the SSR analysis, constant reactive voltage control is considered. 3.1. Damping Torque Analysis. Damping torque analysis is a It should be noted that harmonic content of the SSSC- frequency domain method which can be used to screen the injected voltage would vary depending upon the operating system conditions that give rise to potential SSR problems point since magnitude control will also govern the switching. involving torsional interactions. It also enables the planners The capacitor voltage reference can be varied (depending on to decide on a suitable countermeasure for the mitigation of reactive voltage reference) so as to give optimum harmonic the detrimental effects of SSR. Damping torque method gives Advances in Power Electronics 5 V ∠θ g g E ∠0 R X X X L L C SYS Generator R X T T SSSC VSC ω + b HP IP LPA LPB EXC c GEN (a) Electrical System (b) Six mass mechanical system Figure 6: Modified IEEE first benchmark model with SSSC. torsional filter, and the transmission line with SSSC. The SSSC equations (12)–(14) along with the equations repre- senting electromechanical system [2, 4] (in D-Q variables), are linearized at the operating point, and eigenvalues of −10 system matrix are computed. The stability of the system is determined by the location of the eigenvalues of system −20 matrix. The system is stable if the eigenvalues have negative real parts. −30 With SSSC 3.3. Transient Simulation. The eigenvalue analysis uses equa- −40 tions in D-Q variables neglecting the harmonics. To validate the results obtained from damping torque and eigenvalue −50 analysis, the transient simulation should be carried out using Without SSSC detailed nonlinear three-phase model of SSSC which con- −60 siders the switching in the thyristors/three-phase converters. 0 50 100 150 200 250 300 350 The actual converter switching of the SSSC based on three Frequency (rad/s) level 24-pulse converter is modelled by generating switching functions. Figure 7: Damping torque with admittance function in D-Q axes. 4. A Case Study a quick check to determine the torsional mode stability. The The system considered is a modified IEEE FBM [12]. The system is assumed to be stable if the net damping (including complete electromechanical system is represented schemat- electrical and mechanical) at any of the torsional mode ically in Figure 6, which consists of a generator, turbine, and frequency is positive. series compensated long transmission line with SSSC inject- The interaction between the electrical and mechanical ing a reactive voltage in series with the line. The electrical system can be represented by the block diagram shown in system data is taken from [8]. Figure 5.(ΔS ) is the p.u. deviation in generator rotor speed, The modelling aspects of the electromechanical system and (ΔT ) is the p.u. change in electric torque [11]. comprising the generator, and the mass-spring mechanical The transfer function relating (ΔT )to(ΔS )is KH (s). e m system, the excitation system, power system stabilizer (PSS) At any given oscillation frequency of the generator rotor, the with torsional filter, and the transmission line containing component of electrical torque (ΔT ) in phase with the rotor the conventional series capacitor are discussed in [4]. The speed (ΔS ) is termed as damping torque. The damping analysis is carried out on the IEEE FBM based on the torque coefficient (T (ω)) is defined as follows: de following initial operating condition and assumptions. ΔT jω T (ω) = = H jω . (16) de (1) The generator delivers 0.9 p.u. power to the transmis- K=1 ΔS jω sion system. In obtaining (16), it is necessary to express the impedance (2) The input mechanical power to the turbine is as- function [Z ] of SSSC in D-Q frame [8]. sumed constant. (3) The total series compensation level is set at 0.6 p.u. 3.2. Eigenvalue Analysis. In this analysis, the detailed gen- erator model (2.2) [2] is considered. The electromechanical (4) For transient simulation, a step decrease of 10% me- system consists of the multimass mechanical system, the gen- chanical input torque applied at 0.5 sec and removed erator, the excitation system, power system stabilizer (PSS), at 1 sec is considered in all case studies. de 6 Advances in Power Electronics 85 2 1.5 0.5 70 −0.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Time (s) (a) (b) Figure 8: System response without SSSC. Table 1: Eigen values of the combined system. 4.1. Damping Torque Analysis. The damping torque due to electrical network is evaluated in the range of frequency of Eigenvalue 10–360 rad/sec for the following cases using (14). Torsional mode Without SSSC With SSSC (1) Without SSSC, −1.7366 ± j 0 −1.2987 ± j 8.1094 8.9279 (2) With SSSC. 1 −0.2143 ± j 99.4580 −0.2132 ± j 99.135 In case (1), the series compensation of 60% is completely 20.6658 ± j 127.000 −0.0695 ± j 127.050 met by fixed capacitor and in case (2), hybrid compensation 3 −0.6459 ± j 160.420 −0.6438 ± j 160.210 is used wherein 45% of compensation is met by fixed capac- 4 −0.3646 ± j 202.820 −0.3694 ± j 202.800 itor and the remaining 15% by SSSC. The variation of damping torque with frequency for both cases is shown in 5 −1.8504 ± j 298.170 −1.8504 ± j 298.170 Figure 7. Network mode −1.9029 ± j 126.950 −1.4918 ± j 149.980 It is to be noted that, in case (1), the damping torque Network mode −2.9906 ± j 626.790 −2.4803 ± j 582.980 is maximum negative at a frequency of around 127 rad/sec which matches with the natural frequency of torsional mode-2 and adverse torsional interactions are expected. In SSSC has been carried out using MATLAB-SIMULINK [13]. case (2), maximum undamping occurs at a frequency of The system response for simulation without SSSC is shown about 150 rad/sec. Since this network frequency mode is not in Figure 8. The simulation results of combined system with coinciding with any of the torsional modes, the system is detailed three phase model of SSSC is shown in Figure 9.It stable. is to be noted that the system is stable with the inclusion of SSSC for the constant reactive voltage control. 4.2. Eigenvalue Analysis. In this analysis, generator model (2.2) is considered. The SSSC equations along with the equa- 4.4. Discussion. The representation of impedance function tions representing electromechanical system considering of SSSC in single-phase basis (Z (jω)) from that of D-Q s(1ph) mechanical damping are linearized at the operating point. axes [Z ][8] is approximate and is given below The eigenvalues of system matrix are computed and are given in Table 1. It is to be noted that inclusion of SSSC leads to a stable system and reduces the potential risk of SSR problem. Z = Z j (ω − ω ) + Z j (ω − ω ) s(1ph) sDD 0 sQQ 0 +j Z j (ω − ω ) − Z j (ω − ω ) . 4.3. Transient Simulation. The eigenvalue analysis uses equa- sDQ 0 sQD 0 tions in D-Q variables where the switching functions are (17) approximated by their fundamental components (converter switchings are neglected). To validate the results obtained The resistance R and reactance X of SSSC on single se se from damping torque and eigenvalue analysis, the transient phase basis as a function of frequency ω are computed for er simulation should be carried out using detailed model of X = 0.15 with constant reactive voltage control. It is found sssc SSSC which considers the switching of three-phase converter. that the resistance is negligible while the reactance X is se Hence, the three level 24-pulse converter is modelled by practically constant with frequency. generating switching functions. The transient simulation of The effect of inclusion of SSSC on the resonance the combined system with detailed three-phase model of frequency is shown in Figure 10 for cases 1 and 2. δ (deg) LPA-LPB torque (p.u) Advances in Power Electronics 7 85 0.8 0.75 0.7 0.65 0.6 70 0.55 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Time (s) (a) (b) Figure 9: System response with detailed three phase model of SSSC. 3.5 however, it offers a reactance which remains practically con- stant with frequency. 2.5 5. Conclusion In this paper, we have presented the analysis and simulation of a hybrid series compensated system with SSSC. The 1.5 −(X + X ) C se modelling details of 24-pulse three level VSC-based SSSC is presented. The application of D-Q model is validated by the −X transient simulation of the three-phase model of SSSC. 0.5 There is no appreciable difference in the resonance frequency of the electrical network as the total series com- se pensation (in a hybrid compensation scheme) is increased by 216 227 −0.5 increasing the series reactive voltage injected, instead of the 50 100 150 200 250 300 350 series capacitor. This reduces the risk of SSR as the fixed ω (rad/s) er capacitor can be chosen such that the electrical resonance frequency does not coincide with the complement of the Figure 10: Graphical representation of resonance conditions with torsional modal frequency (which is practically independent and without SSSC. of the electrical network). It is observed that the injected reactive voltage can be adjusted to detune the SSR. The case studies indicated that the SSSC is not strictly SSR neutral When the fixed capacitor provides 45% compensation, however, it offers a reactance which remains practically the resonance occurs at ω = 216 rad/sec where X = er C constant with frequency. X . When the additional compensation of 15% is provided by SSSC, the effective capacitive reactance (X + X )is C se obtained by adding the constant reactance offered by SSSC References to that offered by fixed capacitor. The variation of effective capacitive reactance (X + X ) with frequency is also [1] N.G.Hingorani and L. Gyugyi, Understanding FACTS, IEEE C se shown in Figure 10.Now,the resonance occurs ata higher Press, New York, NY, USA, 2000. [2] K.R.Padiyar, Power System Dynamics—Stability and Control, frequency of ω = 227 rad/sec where (X + X ) = X and er C se L B.S.Publications, Hyderabad, India, 2nd edition, 2002. this is consistent with the subsynchronous network mode [3] K. R. Padiyar, FACTS Controllers in Power Transmission and frequency (ω -ω = 377-227 = 150) of about 150 rad/sec 0 er Distribution, New Age International (P) Limited, New Delhi, as obtained with damping torque analysis with SSSC. India, 2007. The effect of providing additional series compensation by [4] K.R.Padiyar, Analysis of Subsynchronous Resonance in Power SSSC to supplement the existing fixed capacitor is to increase Systems, Kluwer Academic Publishers, Boston, Mass, USA, the electrical resonance frequency of the network. However, this increase in frequency is not significant as compared to [5] K. R. Padiyar and N. Prabhu, “Analysis of SSR with three-level that obtained with the equivalent fixed capacitor offering twelve-pulse VSC-based interline power-flow controller,” IEEE additional compensation (case 1) ω = 250 rad/sec in this er Transactions on Power Delivery, vol. 22, no. 3, pp. 1688–1695, case. This indicates that the SSSC is not strictly SSR neutral 2007. Reactance δ (deg) LPA-LPB torque (p.u) 8 Advances in Power Electronics [6] R. W. Menzis and Y. Zhuang, “Advanced static compensa- tionusing a multilevel GTO thyristor inverter,” IEEE Transac- tions on PowerDelivery, vol. 10, no. 2, 1995. [7] J. B. Ekanayake and N. Jenkins, “Mathematical models of a three-level advanced static VAr compensator,” IEE Proceedings, vol. 144, no. 2, pp. 201–206. [8] K. R. Padiyar and N. Prabhu, “Analysis of subsynchronous resonance with three level twelve-pulse VSC based SSSC,” in Proceedings of the IEEE Confernce on Covergent Technologies for the Asia-Pacific Region (TENCON ’03), pp. 76–80, October [9] C. Schauder and H. Mehta, “Vector analysis and control of advanced static VAR compensators,” IEE Proceedings C,vol. 140, no. 4, pp. 299–306, 1993. [10] K. K. Sen and E. J. Stacey, “UPFC—Unified Power Flow Con- troller: theory, modeling, and applications,” IEEE Transactions on Power Delivery, vol. 13, no. 4, pp. 1453–1460, 1998. [11] N. Prabhu and K. R. Padiyar, “Investigation of subsyn- chronous resonance with VSC-based HVDC transmission systems,” IEEE Transactions on Power Delivery, vol. 24, no. 1, pp. 433–440, 2009. [12] IEEE Subsynchronous Resonance Task Force, “First bench mark model for computer simulation of Subsynchronous res- onance,” IEEE Transactions on Power Apparatus and Systems, vol. 96, no. 5, pp. 1565–1572, 1977. [13] The Math works Inc, “Using MATLAB-SIMULINK,” 1999. 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Investigation of SSR Characteristics of Hybrid Series Compensated Power System with SSSC

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Copyright © 2011 R. Thirumalaivasan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract

Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 621818, 8 pages doi:10.1155/2011/621818 Research Article Investigation of SSR Characteristics of Hybrid Series Compensated Power System with SSSC 1 1 2 R. Thirumalaivasan, M. Janaki, and Nagesh Prabhu School of Electrical Engineering, VIT University, Vellore 632014, India Canara Engineering College, Benjanapadavu, Bantwal, Mangalore 574219, India Correspondence should be addressed to R. Thirumalaivasan, thirumalai.r@vit.ac.in Received 2 November 2010; Accepted 13 March 2011 Academic Editor: Henry S. H. Chung Copyright © 2011 R. Thirumalaivasan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The advent of series FACTS controllers, thyristor controlled series capacitor (TCSC) and static synchronous Series Compensator (SSSC) has made it possible not only for the fast control of power flow in a transmission line, but also for the mitigation of subsynchronous resonance (SSR) in the presence of fixed series capacitors. SSSC is an emerging controller and this paper presents SSR characteristics of a series compensated system with SSSC. The study system is adapted from IEEE first benchmark model (FBM). The active series compensation is provided by a three-level twenty four-pulse SSSC. The modeling and control details of a three level voltage source converter-(VSC)-based SSSC are discussed. The SSR characteristics of the combined system with constant reactive voltage control mode in SSSC has been investigated. It is shown that the constant reactive voltage control of SSSC has the effect of reducing the electrical resonance frequency, which detunes the SSR. The analysis of SSR with SSSC is carried out based on frequency domain method, eigenvalue analysis and transient simulation. While the eigenvalue and damping torque analysis are based on linearizing the D-Q model of SSSC, the transient simulation considers both D-Q and detailed three phase nonlinear system model using switching functions. 1. Introduction a power system by dynamically controlling the power-angle curve of the system [1, 2]. With SSSC, working in capacitive Power transfer capability of long transmission line is limited mode, net reactance is reduced, and, during the first by the transient stability limit. The first swing stability limit swing, sufficient decelerating area is introduced to count- of a single machine infinite bus system can be determined erbalance the accelerating area. However, in the subsequent through well-known equal-area criterion [1, 2]. During swings, the SSSC provides better damping than that of the faulted period, the electrical output power of the machine STATCOM when supplementary modulation controllers are decreases drastically while the input mechanical power incorporated [3]. remains more or less constant. Thus, the machine acquires The series capacitor compensation for long-distance excess energy and is used to accelerate the machine. The power transmission line helps in enhancing power transfer excess energy during faulted period can be represented by and is an economical solution to improve the system stability an area called accelerating area. To maintain stability, the compared to addition of new lines. A series capacitor machine must return the excess energy once the fault is compensated line exhibits a resonant minimum in its cleared. The excess energy returning capability of the impedance at a frequency f = f = X /X ,where X is er 0 C L C machine in postfault period is represented by another area the capacitive compensating reactance, X is inductive line called decelerating area. Thus, the stability of the system can reactance, and f is the synchronous frequency of the power be improved by enlarging the decelerating area, and it system. The resonant frequency f of the compensated er requires raising the power-angle curve of the system. Flexible line depends on the level of compensation of the line AC transmission systems (FACTS) devices are found to be inductance but is always subsynchronous since, in practice, very effective in improving both stability and damping of the compensation ratio is less than unity. It is the coupling of 2 Advances in Power Electronics this subsynchronous electrical transmission line resonance to V i the mechanical resonances of a multistage turbine-generator that gives rise to the phenomenon of SSR. The problem of self-excited torsional frequency oscillations (due to torsional interactions) was experienced at Mohave power station in USA. in December 1970 and October 1971 [3, 4]. The hybrid compensation consisting of suitable combi- nation of passive elements and active FACTS controller such as TCSC or SSSC can be used to mitigate SSR [5]. SSSC is a new generation series FACTS controller based on VSC VSC and has several advantages over TCSC based on thyristor controllers. An ideal SSSC is essentially a pure sinusoidal AC voltage source at the system fundamental frequency. The dc voltage is always injected in quadrature with the line current, dc thereby emulating an inductive or a capacitive reactance in series with the transmission line [3]. SSSC output impedance at other frequencies is ideally zero. Thus, SSSC does not resonate with the inductive line impedance to initiate sub- synchronous resonance oscillations. However, in hybrid series compensation, fixed capacitor element contributes for series resonance. Figure 1: Schematic representation of SSSC. The SSSC has only one degree of freedom (i.e, reactive voltage control, unless there is an energy source connected on the DC side of VSC which allow for real power exchange) which is used to control active power flow in the line [3]. a multipulse or a multilevel configuration. The elimination The VSC based on three-level converter topology greatly of voltage harmonics requires multi-pulse configuration of reduces the harmonic distortion on the AC side [3, 6, 7]. VSC. The multi-pulse converters are generally of TYPE-2 In this paper fixed series capacitor and active compensation where only the phase angle of converter output voltage can provided by three-level twenty four-pulse VSC-based SSSC be controlled and modulation index of the converter remains are considered. The constant reactive voltage control of SSSC fixed [3]. is considered. The major objective is to investigate SSR char- When the DC voltage is constant, the magnitude of ac acteristics of the series compensated system with SSSC using output voltage of the converter can be changed by Pulse both linear analysis and nonlinear transient simulation. It Width Modulation (PWM) with two-level topology which is shown that the constant reactive voltage control of SSSC demands higher switching frequency and leads to increased has the effect of reducing the electrical resonance frequency, losses. In three level converter topology, both the magni- which detunes SSR. tude and phase angle of converter output voltage can be The study is carried out based on frequency domain controlled. This converter is classified as TYPE-1 converter method, eigenvalue analysis and transient simulation [8]. The modelling of the system neglecting VSC is detailed [9], where DC bus voltage is maintained constant and the magnitude of converter output voltage is controlled by (including network transients) and can be expressed in DQ varying dead angle β with fundamental switching frequency variables or (three) phase variables. The modeling of VSC modulation [3, 10]. The harmonics are dependent on the is based on (1) DQ variables (neglecting harmonics in the capacitance and the operating point of the SSSC. The detailed output voltages of the converters) and (2) phase variables and three-phase model of SSSC is developed by modelling the the use of switching functions. The damping torque analysis, converter operation by switching functions. The switching eigenvalue analysis, and the controller design is based on function for phase “a”is shown in Figure 2. the DQ model while the transient simulation considers both models of VSC. The results based on linear analysis The switching functions of phases b and c are similar are validated using transient simulation based on nonlinear but phase shifted successively by 120 in terms of the system model. fundamental frequency. Assuming that the dc capacitor The paper is organized as follows. Section 2 describes the voltages V = V = V /2, the converter terminal voltages dc1 dc2 dc modelling of SSSC whereas the different methods of analysis with respecttothe midpointof dc side“N”can be obtained of SSR are discussed in Section 3. Section 4 describes a case as study and investigates the SSR characteristics with SSSC. The major conclusions of the paper are given in Section 5. ⎡ ⎤ ⎡ ⎤ V P (t) aN 2. Modelling of SSSC with Three-Level VSC ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ dc ( ) ⎢ V ⎥ = ⎢ P t ⎥ ,(1) bN ⎣ ⎦ ⎣ ⎦ Figure 1 shows the schematic representation of SSSC. In the P (t) V c cN power circuit of an SSSC, the converter is usually either Advances in Power Electronics 3 1.5 where 2π 1 ( ) S t = S t + 1x 1x ω 12 (6) π 1 P (t) a S (t) = S t + , x = a, b, c. 1x x ω 24 0.5 The switching functions for second twelve-pulse converter are given by 0 2β S (t) = S (t) + S (t) − S (t) , 2a 2a 2a 2c −0.5 S (t) = S (t) + S (t) − S (t) , (7) 2b 2b 2b 2a −1 ( ) ( ) √ ( ) ( ) Line current i (t) S t = S t + S t − S t , 2c a 2c 2c 2b where −1.5 0.42 0.425 0.43 0.435 0.44 0.445 0.45 2π 1 Time (s) S (t) = S t + 2x 2x ω 12 Figure 2: Switching function for a three level converter. (8) π 1 S (t) = S t − , x = a, b, c. 2x x ω 24 and the converter output voltages with respect to the neutral The switching functions for a twenty four-pulse con- of transformer can be expressed as verter are given by ⎡ ⎤ ⎡ ⎤ 24 12 12 S (t) = S (t) + S (t), x = a, b,and c. (9) V S (t) a x 1x 2x an ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V S (t) V ⎢ ⎥ = ⎢ ⎥ ,(2) b dc bn If the switching functions are approximated by their fun- ⎣ ⎦ ⎣ ⎦ i damental components (neglecting harmonics) for a 24-pulse V S (t) cn three level converter, we get where V = V cos β sin ω t + φ + γ , (10) dc o an P (t) P (t) + P (t) + P (t) π a a b c S (t) = − . (3) 2 6 i ◦ and V , V are phase shifted successively by 120 . bn cn The line current is given by i = 2/3I sin(ω + φ)and a a o S (t) is the switching function for phase “a” of a 6-pulse 3- i , i are phase shifted successively by 120 .Notethat γ is b c level VSC. Similarly for phase “b”, S (t), and for phase “c ”, the angle by which the fundamental component of converter S (t) can be derived. The peak value of the fundamental and output voltage leads the line current. It should be noted that harmonics in the phase voltage V are found by applying an γ is nearly equal to ±π/2 depending on whether SSSC injects Fourier analysis on the phase voltage and can be expressed as inductive or capacitive voltage. Neglecting converter losses, we can get the expression for dc capacitor current as V = V cos hβ , (4) dc an(h) hπ ⎡ ⎤ ⎢ ⎥ where, h = 1, 5, 7, 11, 13, and β is the dead angle (period) ⎢ ⎥ 24 24 24 i S (t) S (t) S (t) i =− ⎢ ⎥ . (11) dc b a b c during which the converter pole output voltage is zero. We ⎣ ⎦ can eliminate the 5th and 7th harmonics by using a twelve- i pulse VSC, which combines the output of two six-pulse con- A particular harmonic reaches zero when 2β = 180 /h.At verters using transformers. β = 3.75 , the three level 24-pulse converter behaves optimum The switching functions for first twelve-pulse converter nearly like a two-level 48-pulse converter as 23th and 25th are given by harmonics are negligibly small. S (t) = S (t) + S (t) − S (t) , 1a 1a 1a 1c 2.1. Modelling of SSSC in D-Q Variables. When switching functions are approximated by their fundamental frequency S (t) = S (t) + S (t) − S (t) , (5) 1b 1a components, neglecting harmonics, SSSC can be modelled 1b 1b by transforming the three-phase voltages and currents to D- 12   Q variables using Kron’s transformation [2]. The SSSC can S (t) = S (t) + S (t) − S (t) , 1c 1c 1c 1b be represented functionally as shown in Figure 3. P (t) a 4 Advances in Power Electronics I∠φ + dc ref K Σ p Σ γ P(se)(ord) γ and β se R X st st 1 calculator V ∠φ + γ 1+ sT md V se R(se)(ord) dc Figure 3: Equivalent circuit of SSSC as viewed from AC side. Figure 4: Type-1 controller for SSSC. In Figure 3, R and X are the resistance and reactance of st st the interfacing transformer of VSC. The magnitude control + ΔS of converter output voltage V is achieved by modulating the ΔT m Mechanical system conduction period affected by dead angle of converter while G(s) dc voltage is maintained constant. The converter output voltage can be represented in D-Q ΔT frame of reference as Electrical system 2 2 i i i KH (s) V = V + V , D Q i (12) V = k V sin φ + γ , m dc Figure 5: Interaction between mechanical and electrical system. V = k V cos φ + γ , m dc performance. In three level 24-pulse converter, dc voltage ref- where k = kρ cos β ; k = 4 6/π for a 24-pulse converter. ρ m se erence may be adjusted by a slow controller to get optimum is the transformation ratio of the interfacing transformer. harmonic performance at β = 3.75 in steady state. From a control point of view, it is convenient to define the se The structure of type-1 controller for SSSC is given in active voltage (V )and reactive (V ) voltage injected P(se) R(se) i i Figure 4.In Figure 4, γ and β are calculated as se by SSSC in terms of variables in D-Q frame (V and V )as D Q follows R(se)(ord) -1 i i γ = tan , V = V cos φ − V sin φ, R(se) D Q P(se)(ord) (13) i i ⎡ ⎤ (15) V = V sin φ + V cos φ. 2 2 P(se) D Q V + V P(se)(ord) R(se)(ord) −1 ⎣ ⎦ β = cos se k V Here, positive V implies that SSSC injects inductive volt- m dc R(se) age and positive V implies that it draws real power to P(se) meet losses. 3. Analysis of SSR The dc side capacitor is described by the dynamical equa- tion as The two aspects of SSR are [4](i) steady-state SSR (induction generator effect (IGE) and torsional interaction (TI)) (ii) g ω dV ω dc c b b =− V − i , (14) dc dc shaft torque amplification due to transients. The analysis of dt bc bc steady-state SSR can be done by linearized models at the where i =−[k sin(φ + γ)I + k cos(φ + γ)I ], I and I operating point and include damping torque analysis and dc m D m Q D Q are the D-Q components of the line current. eigenvalue analysis. The analysis of shaft torque amplification due to transients requires transient simulation of the nonlin- ear model of the system. For the analysis of SSR, it is adequate 2.2. Type-1 Controller. In this type of controller, both mag- to model the transmission line by lumped resistance and nitude (modulation index k ) and phase angle of converter inductance where the line transients are also considered. output voltage (γ) are controlled. The capacitor voltage is The generator stator transients are also considered by using maintained at a constant voltage by controlling the active detailed (2.2) model of the generator. component of the injected voltage V . The real voltage P(se) The analysis of SSR is carried out based on damping reference V is obtained as the output of DC voltage P(se)(ord) torque analysis, eigenvalue analysis, and transient simula- controller. The reactive voltage reference V may R(se)(ord) tion. be kept constant or obtained from a power scheduling controller. However, for the SSR analysis, constant reactive voltage control is considered. 3.1. Damping Torque Analysis. Damping torque analysis is a It should be noted that harmonic content of the SSSC- frequency domain method which can be used to screen the injected voltage would vary depending upon the operating system conditions that give rise to potential SSR problems point since magnitude control will also govern the switching. involving torsional interactions. It also enables the planners The capacitor voltage reference can be varied (depending on to decide on a suitable countermeasure for the mitigation of reactive voltage reference) so as to give optimum harmonic the detrimental effects of SSR. Damping torque method gives Advances in Power Electronics 5 V ∠θ g g E ∠0 R X X X L L C SYS Generator R X T T SSSC VSC ω + b HP IP LPA LPB EXC c GEN (a) Electrical System (b) Six mass mechanical system Figure 6: Modified IEEE first benchmark model with SSSC. torsional filter, and the transmission line with SSSC. The SSSC equations (12)–(14) along with the equations repre- senting electromechanical system [2, 4] (in D-Q variables), are linearized at the operating point, and eigenvalues of −10 system matrix are computed. The stability of the system is determined by the location of the eigenvalues of system −20 matrix. The system is stable if the eigenvalues have negative real parts. −30 With SSSC 3.3. Transient Simulation. The eigenvalue analysis uses equa- −40 tions in D-Q variables neglecting the harmonics. To validate the results obtained from damping torque and eigenvalue −50 analysis, the transient simulation should be carried out using Without SSSC detailed nonlinear three-phase model of SSSC which con- −60 siders the switching in the thyristors/three-phase converters. 0 50 100 150 200 250 300 350 The actual converter switching of the SSSC based on three Frequency (rad/s) level 24-pulse converter is modelled by generating switching functions. Figure 7: Damping torque with admittance function in D-Q axes. 4. A Case Study a quick check to determine the torsional mode stability. The The system considered is a modified IEEE FBM [12]. The system is assumed to be stable if the net damping (including complete electromechanical system is represented schemat- electrical and mechanical) at any of the torsional mode ically in Figure 6, which consists of a generator, turbine, and frequency is positive. series compensated long transmission line with SSSC inject- The interaction between the electrical and mechanical ing a reactive voltage in series with the line. The electrical system can be represented by the block diagram shown in system data is taken from [8]. Figure 5.(ΔS ) is the p.u. deviation in generator rotor speed, The modelling aspects of the electromechanical system and (ΔT ) is the p.u. change in electric torque [11]. comprising the generator, and the mass-spring mechanical The transfer function relating (ΔT )to(ΔS )is KH (s). e m system, the excitation system, power system stabilizer (PSS) At any given oscillation frequency of the generator rotor, the with torsional filter, and the transmission line containing component of electrical torque (ΔT ) in phase with the rotor the conventional series capacitor are discussed in [4]. The speed (ΔS ) is termed as damping torque. The damping analysis is carried out on the IEEE FBM based on the torque coefficient (T (ω)) is defined as follows: de following initial operating condition and assumptions. ΔT jω T (ω) = = H jω . (16) de (1) The generator delivers 0.9 p.u. power to the transmis- K=1 ΔS jω sion system. In obtaining (16), it is necessary to express the impedance (2) The input mechanical power to the turbine is as- function [Z ] of SSSC in D-Q frame [8]. sumed constant. (3) The total series compensation level is set at 0.6 p.u. 3.2. Eigenvalue Analysis. In this analysis, the detailed gen- erator model (2.2) [2] is considered. The electromechanical (4) For transient simulation, a step decrease of 10% me- system consists of the multimass mechanical system, the gen- chanical input torque applied at 0.5 sec and removed erator, the excitation system, power system stabilizer (PSS), at 1 sec is considered in all case studies. de 6 Advances in Power Electronics 85 2 1.5 0.5 70 −0.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Time (s) (a) (b) Figure 8: System response without SSSC. Table 1: Eigen values of the combined system. 4.1. Damping Torque Analysis. The damping torque due to electrical network is evaluated in the range of frequency of Eigenvalue 10–360 rad/sec for the following cases using (14). Torsional mode Without SSSC With SSSC (1) Without SSSC, −1.7366 ± j 0 −1.2987 ± j 8.1094 8.9279 (2) With SSSC. 1 −0.2143 ± j 99.4580 −0.2132 ± j 99.135 In case (1), the series compensation of 60% is completely 20.6658 ± j 127.000 −0.0695 ± j 127.050 met by fixed capacitor and in case (2), hybrid compensation 3 −0.6459 ± j 160.420 −0.6438 ± j 160.210 is used wherein 45% of compensation is met by fixed capac- 4 −0.3646 ± j 202.820 −0.3694 ± j 202.800 itor and the remaining 15% by SSSC. The variation of damping torque with frequency for both cases is shown in 5 −1.8504 ± j 298.170 −1.8504 ± j 298.170 Figure 7. Network mode −1.9029 ± j 126.950 −1.4918 ± j 149.980 It is to be noted that, in case (1), the damping torque Network mode −2.9906 ± j 626.790 −2.4803 ± j 582.980 is maximum negative at a frequency of around 127 rad/sec which matches with the natural frequency of torsional mode-2 and adverse torsional interactions are expected. In SSSC has been carried out using MATLAB-SIMULINK [13]. case (2), maximum undamping occurs at a frequency of The system response for simulation without SSSC is shown about 150 rad/sec. Since this network frequency mode is not in Figure 8. The simulation results of combined system with coinciding with any of the torsional modes, the system is detailed three phase model of SSSC is shown in Figure 9.It stable. is to be noted that the system is stable with the inclusion of SSSC for the constant reactive voltage control. 4.2. Eigenvalue Analysis. In this analysis, generator model (2.2) is considered. The SSSC equations along with the equa- 4.4. Discussion. The representation of impedance function tions representing electromechanical system considering of SSSC in single-phase basis (Z (jω)) from that of D-Q s(1ph) mechanical damping are linearized at the operating point. axes [Z ][8] is approximate and is given below The eigenvalues of system matrix are computed and are given in Table 1. It is to be noted that inclusion of SSSC leads to a stable system and reduces the potential risk of SSR problem. Z = Z j (ω − ω ) + Z j (ω − ω ) s(1ph) sDD 0 sQQ 0 +j Z j (ω − ω ) − Z j (ω − ω ) . 4.3. Transient Simulation. The eigenvalue analysis uses equa- sDQ 0 sQD 0 tions in D-Q variables where the switching functions are (17) approximated by their fundamental components (converter switchings are neglected). To validate the results obtained The resistance R and reactance X of SSSC on single se se from damping torque and eigenvalue analysis, the transient phase basis as a function of frequency ω are computed for er simulation should be carried out using detailed model of X = 0.15 with constant reactive voltage control. It is found sssc SSSC which considers the switching of three-phase converter. that the resistance is negligible while the reactance X is se Hence, the three level 24-pulse converter is modelled by practically constant with frequency. generating switching functions. The transient simulation of The effect of inclusion of SSSC on the resonance the combined system with detailed three-phase model of frequency is shown in Figure 10 for cases 1 and 2. δ (deg) LPA-LPB torque (p.u) Advances in Power Electronics 7 85 0.8 0.75 0.7 0.65 0.6 70 0.55 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Time (s) (a) (b) Figure 9: System response with detailed three phase model of SSSC. 3.5 however, it offers a reactance which remains practically con- stant with frequency. 2.5 5. Conclusion In this paper, we have presented the analysis and simulation of a hybrid series compensated system with SSSC. The 1.5 −(X + X ) C se modelling details of 24-pulse three level VSC-based SSSC is presented. The application of D-Q model is validated by the −X transient simulation of the three-phase model of SSSC. 0.5 There is no appreciable difference in the resonance frequency of the electrical network as the total series com- se pensation (in a hybrid compensation scheme) is increased by 216 227 −0.5 increasing the series reactive voltage injected, instead of the 50 100 150 200 250 300 350 series capacitor. This reduces the risk of SSR as the fixed ω (rad/s) er capacitor can be chosen such that the electrical resonance frequency does not coincide with the complement of the Figure 10: Graphical representation of resonance conditions with torsional modal frequency (which is practically independent and without SSSC. of the electrical network). It is observed that the injected reactive voltage can be adjusted to detune the SSR. The case studies indicated that the SSSC is not strictly SSR neutral When the fixed capacitor provides 45% compensation, however, it offers a reactance which remains practically the resonance occurs at ω = 216 rad/sec where X = er C constant with frequency. X . When the additional compensation of 15% is provided by SSSC, the effective capacitive reactance (X + X )is C se obtained by adding the constant reactance offered by SSSC References to that offered by fixed capacitor. The variation of effective capacitive reactance (X + X ) with frequency is also [1] N.G.Hingorani and L. Gyugyi, Understanding FACTS, IEEE C se shown in Figure 10.Now,the resonance occurs ata higher Press, New York, NY, USA, 2000. [2] K.R.Padiyar, Power System Dynamics—Stability and Control, frequency of ω = 227 rad/sec where (X + X ) = X and er C se L B.S.Publications, Hyderabad, India, 2nd edition, 2002. this is consistent with the subsynchronous network mode [3] K. R. Padiyar, FACTS Controllers in Power Transmission and frequency (ω -ω = 377-227 = 150) of about 150 rad/sec 0 er Distribution, New Age International (P) Limited, New Delhi, as obtained with damping torque analysis with SSSC. India, 2007. The effect of providing additional series compensation by [4] K.R.Padiyar, Analysis of Subsynchronous Resonance in Power SSSC to supplement the existing fixed capacitor is to increase Systems, Kluwer Academic Publishers, Boston, Mass, USA, the electrical resonance frequency of the network. However, this increase in frequency is not significant as compared to [5] K. R. Padiyar and N. Prabhu, “Analysis of SSR with three-level that obtained with the equivalent fixed capacitor offering twelve-pulse VSC-based interline power-flow controller,” IEEE additional compensation (case 1) ω = 250 rad/sec in this er Transactions on Power Delivery, vol. 22, no. 3, pp. 1688–1695, case. This indicates that the SSSC is not strictly SSR neutral 2007. Reactance δ (deg) LPA-LPB torque (p.u) 8 Advances in Power Electronics [6] R. W. Menzis and Y. Zhuang, “Advanced static compensa- tionusing a multilevel GTO thyristor inverter,” IEEE Transac- tions on PowerDelivery, vol. 10, no. 2, 1995. [7] J. B. Ekanayake and N. Jenkins, “Mathematical models of a three-level advanced static VAr compensator,” IEE Proceedings, vol. 144, no. 2, pp. 201–206. [8] K. R. Padiyar and N. Prabhu, “Analysis of subsynchronous resonance with three level twelve-pulse VSC based SSSC,” in Proceedings of the IEEE Confernce on Covergent Technologies for the Asia-Pacific Region (TENCON ’03), pp. 76–80, October [9] C. Schauder and H. Mehta, “Vector analysis and control of advanced static VAR compensators,” IEE Proceedings C,vol. 140, no. 4, pp. 299–306, 1993. [10] K. K. Sen and E. J. Stacey, “UPFC—Unified Power Flow Con- troller: theory, modeling, and applications,” IEEE Transactions on Power Delivery, vol. 13, no. 4, pp. 1453–1460, 1998. [11] N. Prabhu and K. R. Padiyar, “Investigation of subsyn- chronous resonance with VSC-based HVDC transmission systems,” IEEE Transactions on Power Delivery, vol. 24, no. 1, pp. 433–440, 2009. [12] IEEE Subsynchronous Resonance Task Force, “First bench mark model for computer simulation of Subsynchronous res- onance,” IEEE Transactions on Power Apparatus and Systems, vol. 96, no. 5, pp. 1565–1572, 1977. [13] The Math works Inc, “Using MATLAB-SIMULINK,” 1999. 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