1252

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

Power System Voltage Regulation via STATCOM Internal Nonlinear Control Keyou Wang, Member, IEEE, and Mariesa L. Crow, Fellow, IEEE

Abstract—A new internal STATCOM control based on feedback linearization is proposed. The feedback linearization controller is developed without any simplifying assumptions to the STATCOM model. The proposed control is validated on the IEEE 118-bus system with full-order generator and network models as opposed to a small test system. Furthermore, the proposed control is benchmarked against published results. Lastly controllability issues associated with a singularity in the feedback linearization control (FBLC) coordinate transformation is identified, and a solution is provided to avoid instability. Index Terms—Feedback linearization, nonlinear control, singularity induced bifurcation, STATCOM, voltage regulation.

I. INTRODUCTION

I

N modern bulk power transmission systems, flexible ac transmission system (FACTS) devices can potentially provide improved controllability and transfer capability of the power network and are receiving greater adoption by industry [1]. The static synchronous compensator (STATCOM), which consists of three-phase voltage source converters (VSC) connected in shunt to the system bus, has been shown to be effective in providing reactive support to the power system. STATCOM control consists of “internal,” “external,” and “gate” control as shown in Fig. 1 [2]. An internal controller generates a fundamental output voltage waveform with a desired magnitude and phase angle in synchronism with the AC system. This fundamental signal is then used to drive the gating signals through an appropriate pulse width modulation scheme. An external controller responds to system conditions and determines how much reactive current the STATCOM should generate or absorb to meet the requirements of the system. Since its inception, a variety of control approaches have been proposed for STATCOM dynamic control [3]–[21]. The majority of approaches involve traditional linear control techniques in which the nonlinear equations of the VSC average value model are linearized at a specific equilibrium [3]–[11]. A proportional-integral (PI) STATCOM controller structure was initially proposed in [3]. Since that time, numerous PI controllers have been reported to exhibit satisfactory performance when the parameters are fine tuned, whether in the

Manuscript received January 14, 2010; revised April 08, 2010, June 03, 2010, and July 20, 2010; accepted August 26, 2010. Date of publication September 27, 2010; date of current version July 22, 2011. This work was supported in part by a grant from the National Science Foundation under ECCS 0701643. Paper no. TPWRS-00026-2010. The authors are with the Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409-0810 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2010.2072937

Fig. 1. Three-level STATCOM control: External, internal, and gate.

applications of two-level converter [4]–[6] or the multilevel converter [7]–[11]. The drawback of such PI controllers is that their performance might degrade with a change of operating condition, especially a large disturbance such as sudden load change or nearby short-circuit fault. To compensate for a change of operating condition, a variety of controls have been reported that provide satisfactory performance over a wide range of operating conditions [2], [12]–[21]. One approach is to adaptively change the gains of the PI controller in response to changes in operating condition. A number of intelligent techniques have been proposed to adapt the PI controller gains of STATCOMs such as artificial immunity [12], neural networks [13], and particle swarm optimization [14]. Secondly, adaptive control and linear robust controls have been reported in the literature [15]–[16]. In [15], a gradient-based estimation of the load conductance is proposed to account for the load variations. The concept of calculating a complete set of the admissible feedback gains is proposed in [16]. Another control approach is to apply model-based nonlinear control which directly compensates for system nonlinearities without requiring a linear approximation. Feedback linearization control (FBLC) has been proposed for the STATCOM with promising results [2], [17]–[21]. The first consideration of FBLC for the STATCOM appeared in [17]. In this approach, the authors introduced the STATCOM terminal voltage magnitude as a dynamic state ( and therefore ) to in addition to the usual STATCOM states , , and

0885-8950/$26.00 © 2010 IEEE

WANG AND CROW: POWER SYSTEM VOLTAGE REGULATION VIA STATCOM INTERNAL NONLINEAR CONTROL

simplify the FBLC coordinate transformation. The resulting linearized single-machine-infinite bus system was then controlled using pole-placement. An alternate approach was proposed in [2] using direct feedback linearization which linearized the STATCOM by setting the control inputs to the nonlinear portion of the system and did not use the traditional FBLC coordinate transformation. Furthermore, the authors considered only the STATCOM currents and assumed that the dc-link voltage was constant. The linearized STATCOM was then controlled using approach. In [18], the authors neglect the STATCOM an converter losses. This leads to a nonstandard STATCOM model and a simplified FBLC coordinate transformation. Their control was validated with PSCAD on a single-machine-infinite bus system. In [19], the authors proposed general VSC FBLC controller and applied it on SSSC and UPFC. In the last works [20]–[21], the performance of FBLC has been experimentally validated on a single-STATCOM-single-bus system. Most of the STATCOM FBLC works have been reported on small scale system application and EMTP-type simulation, but without full consideration of the STATCOM model and a large scale system application. In this paper, we extend the previous work in STATCOM FBLC in several significant ways. First, we apply the FBLC to the standard STATCOM model [3] without making any simplifying assumptions. Secondly, we validate the proposed control on the IEEE 118-bus system with full-order generator and network models as opposed to a small test system. Thirdly, we compare the proposed control against benchmarked results [5], and lastly we identify controllability issues associated with a singularity in the FBLC coordinate transformation and propose a solution to avoid instability.

1253

Fig. 2. STATCOM equivalent circuit.

The STATCOM power balance equations at bus are [6]

(8)

(9) where the summation terms represent the power flow equations, is the th element of the admittance matrix and is the number of buses in the system. The first set of terms indicate the active and reactive powers injected by the STATCOM (respectively), whereas the summation of power terms on the right are the power flow equations of the power system. Note that these equations—(8) and (9)—represent the only coupling of the STATCOM states with the power system. To develop the feedback linearization transformation, the STATCOM equations must first be written as an “affine nonlinear system” such that [22] (10)

II. STATCOM MODEL The nonlinear STATCOM state equations for the equivalent circuit model shown in Fig. 2 in the reference frame are given by [3], [6] (1)

(11) where , are smooth vector functions that are continuous and differentiable, is a smooth scalar function, and is the number of control variables. In the STATCOM model, and the state equations can be written as

(2) (12) (3)

where

where and are the injected STATCOM currents, is the voltage across the DC capacitor, represents the conand are the coupling transformer resisverter losses, tance and inductance, respectively, and the STATCOM RMS bus . The inputs and are given by voltage is

(13)

(14)

(4) (5) where and tively, and

(15)

are the modulation ratio and phase shift, respecwhere (6) (7)

1254

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

and

Once the appropriate control input is determined, the inverse coordinate transform yields the desired original control inputs from (19):

(22) Note, however, that (22) depends on values of . These values determined from the of are the target values , , and system level control. Two possible control target objectives are summarized below. A. Reactive Current and DC-Link Voltage Control

III. STATCOM CONTROL The first step in developing an FBLC is through a coordinate transformation that converts the nonlinear system into a linear system without loss of state information. The coordinate transform is chosen to convert the states to such that

This control objective is the most common STATCOM usage in which reactive current injection and the DC-link capacitor voltage are simultaneously controlled. In this case, and are externally specified. The value for can be obtained from the equilibrium of (12) such that

(16)

(23) where can be found in Appendix B.

is the Lie derivative of with respect to where [22]–[24]. The function is chosen to satisfy and . For the STATCOM

. The derivation details

B. Active and Reactive Current Control

(17) can be found in Appendix A. The derivation of Differentiating (16) and incorporating (17) yields

In some cases, it may be desirable to control both the active and reactive power injection (with no dc-link capacitor voltage control). In this case, and are externally specified and must be calculated. In this case

(24)

(18) IV. SINGULARITY INDUCED BIFURCATION If the new control vector

is defined such that (19)

then the new linear state variables can be written

One potential drawback with the proposed control scheme is the possibility of the existence of a singularity induced bifurcation during the application of the control. Singularity induced bifurcations may arise in systems of differential-algebraic equations (DAEs) [25]–[27]. The power system may generally be modeled as

(20) (25) (26)

where

This system is controllable since rank ; therefore, linear control techniques such as pole placement, LQR, , etc. can be applied without loss of generality [23], such that

where contains the dynamic variables such as those associated with generators, STATCOMs, dynamic loads, etc. The vector contains the algebraic variables such as those associated bus is typically considvoltage magnitudes and angles, and ered to represent the power flow equations. A singular surface is defined by the simultaneous conditions that

(21)

(27)

where

is the state feedback matrix.

(28)

WANG AND CROW: POWER SYSTEM VOLTAGE REGULATION VIA STATCOM INTERNAL NONLINEAR CONTROL

1255

Fig. 3. Two-area system.

When an operating point crosses the singular surface, the system matrix is no longer well-defined and the trajectory curves may oscillate erratically near this point. This is often referred to as a “singularity induced bifurcation” or SIB. Applying the proposed feedback linearization to a power system with STATCOMs may inadvertently cause the existence of one or more SIB, even though STATCOM control does not usually contribute to system instability. To illustrate this effect, consider the Jacobian matrix of the coordinate transformation:

(29) When the STATCOM is coupled with the power system through the power balance algebraic (8) and (9), the STATCOM bus vary as a function of , even though the bus voltages and voltage magnitude may remain relatively constant. As vary, the nonsingularity of the system Jacobian cannot be guaranteed for all operating points. For example, considering , then the situation in which (30) (31) Furthermore, if the STATCOM losses are small , then (32)

becomes ill-conditioned and possibly singular. This may cause unexpected uncontrollability and chaotic behavior in the STATCOM dynamics. To illustrate this behavior, consider the two-area system shown in Fig. 3 [28]. In this system, a three-phase to ground fault is applied at bus 3 and is cleared after 0.1 s. The resulting generator angular frequencies are shown in Fig. 4(a). The bus voltage magnitude and angle at the STATCOM bus (bus 11) are shown in Fig. 4(b) and (c), respectively. Note that the STATCOM bus angle passes through at approximately 2.2 s and again at approximately 4.0 s. When the STATCOM bus angle is near , large chaotic oscillations are induced in the STATCOM bus voltage which further propagate into the generator frequency responses. This is a result of the near singularity of the system Jacobian near these points.

Fig. 4. Two-area system response to a fault. (a) Generator frequencies. (b) STATCOM bus voltage magnitude. (c) STATCOM bus voltage angle.

Fortunately, however, the proposed FBLC method can be modified to avoid this ill-conditioned behavior. Recall that the STATCOM dynamic states (and therefore the control inputs) are coupled with the power system only through the algebraic equations of the STATCOM power balance equations and the bus voltage magnitude and angle. Furthermore, the STATCOM injected active and reactive powers are independent of the reference frame. If the reference is taken selection of the

1256

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

the Jacobian singularity is avoided and the system rapidly attains a stable operating point. V. POWER SYSTEM VOLTAGE REGULATION The most common control objective for the STATCOM is power system voltage regulation by reactive power injection [1]. The block diagram of the voltage regulation control is shown in Fig. 6. The left dotted block is the “external (system) control” in the diagram of Fig. 1. In this approach, a target voltage reference for the STATCOM bus is typically specified at a value determined by the system operator. Proportional-integral (PI) control has been effectively used to convert the error in bus voltage magand nitude to a reactive current target value [3]. The gains are the proportional and integral parts of the voltage reguis the lation controller, respectively. The feedback gain . droop factor [29] which can be neglected by setting The inputs of the “internal control” are the reactive current target value (which is generated by the “external control”), and the dc-link voltage reference value, which is also typically specified to provide adequate performance of the VSC. The right dotted block is the “internal (converter) control” proposed in Section III. VI. RESULTS AND DISCUSSION

Fig. 5. Two-area system response to a fault—new control reference frame. (a) Generator frequencies. (b) STATCOM bus voltage magnitude. (c) STATCOM bus voltage angle.

taken with respect to the STATCOM bus voltage rather than the power system reference frame, then the control is based on a and the control Jacobian is no longer reference angle of potentially singular. The results of the same fault in the adjusted reference frame are shown in Fig. 5(a)–(c). Note that prior to 2.2 s (when the first ill-conditioning previously occurred), the system responses are nearly identical. In the new case, however,

The proposed control with shifted reference frame has been applied to the IEEE 118-bus system shown in Fig. 7 [30]. To provide an even comparison against a benchmarked system, the proposed control is compared to the control results presented in [5]. In this comparison, a STATCOM is placed on bus 86, which is then subjected to sudden changes in load. The STATCOM placement and PI controller gains are chosen according to the results in [5]. The STATCOM parameters, PI control parameters, and the gain matrix for the proposed control are given in Appendix C. The block diagram of PI controller is shown in Fig. 8. Note that these PI controller gains have been fine tuned using pole placement techniques at a single operating condition. During the simulation, each generator in the system is fully modeled with a two-axis synchronous machine, excitation/ AVR, and a turbine/speed governor. The loadflow data of the IEEE 118-bus system can be found online [30]. The generator model and parameters are given in Appendix D. The simulation results are based on a full nonlinear time-domain simulation for DAEs—(25) and (26)—which are simulated using Matlab. The integration approach is the implicit trapezoidal method and the nonlinear system solver is based on the Newton-Raphson technique [31]. Although the impact of the proposed STATCOM control was not specifically studied with respect to other controllers in the power system (such as excitation controls), it is doubtful that the proposed control will cause any undesired dynamic interactions. This is because the proposed control is an internal control and therefore has little impact beyond the fast dynamics of the STATCOM itself. It is possible that external controls may cause unanticipated interactions, but that is beyond the scope of this paper. Similarly, it is unlikely that the proposed control will cause interactions between STATCOMs; therefore, the internal controls for each STATCOM may be developed independently.

WANG AND CROW: POWER SYSTEM VOLTAGE REGULATION VIA STATCOM INTERNAL NONLINEAR CONTROL

1257

Fig. 6. STATCOM voltage regulation scheme.

Fig. 7. IEEE 118-bus system.

Fig. 8. STATCOM PI controller scheme.

A. Case I—Small Load Change Figs. 9–14 show the system dynamic response to a 0.175 pu step increase in active power load (0.9 power factor lagging) at bus 86 at time 0.01 s. The objective of the voltage regulation control is to maintain the STATCOM bus voltage at the ini-

tial voltage value. This is the same bus as that studied in [5]. In Fig. 9, the voltage response of bus 86 with the proposed control strategy is compared against the PI control and the no-control case. The dashed line indicates the no-control case and shows that the steady-state voltage magnitude drops in response to the load change. The thin line and bold line denote the PI control and the proposed control, respectively, and both exhibit good performance to track the voltage to the set point as compared to no-control case. The thin line shows that the PI controller can recover the voltage to the target reference rapidly. However, the bold line shows that proposed control exhibits near perfectly damped and instantaneous performance as compared to the PI control. The STATCOM state responses are shown in Figs. 10–12. The proposed control exhibits better performance than PI control in terms of response time and damping profile although the difference between the response of these two controllers is small as is expected for small changes about the operating point.

1258

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

Fig. 9. Case I: Bus voltage magnitude.

Fig. 12. Case I: current i .

Fig. 10. Case I: DC-link capacitor voltage.

Fig. 13. Case I: fundamental frequency control voltage angle .

Fig. 11. Case I: current i .

Fig. 14. Case I: fundamental frequency control voltage modulation gain k .

B. Case II—Large Load Change A significant advantage of feedback linearization control is that this control method is nonlinear and therefore relatively independent of operating conditions. Nonlinear systems to which

linear feedback is applied often suffer from degradation of performance when the operating point is far from the point about which the system was originally linearized. Fig. 15 illustrates

WANG AND CROW: POWER SYSTEM VOLTAGE REGULATION VIA STATCOM INTERNAL NONLINEAR CONTROL

1259

VII. CONCLUSIONS Both power systems and STATCOMs are inherently nonlinear; thus, nonlinear control will typically provide better controllability and performance. In this paper, a new nonlinear control for the STATCOM is proposed. At the internal level, the proposed nonlinear control is based on feedback linearization and is general enough that it can be extended to the control of any voltage source converter. The nonlinear simulation results show that the proposed control exhibits more effective performance regardless of operating condition including small load change, large load change, and three-phase short circuits, when compared with traditional PI controls. Additionally, it reference frame for the was shown that the choice of the nonlinear control can avoid any potential singularity induced bifurcations in the algebraic manifold. Fig. 15. Case II: Bus voltage magnitude.

APPENDIX A LIE DERIVATIVES Given

If

where is the relative degree of , and this problem

Fig. 16. Case III: Bus voltage magnitude.

the robustness of nonlinear control for a large excursion from the initial operating point. In this case, a large step increase 0.875 pu in load at bus 86 (five times as the small load case) is applied at time 0.01 s. As shown in Fig. 15, the PI control is still able to maintain the bus voltage magnitude and return it to the voltage reference set point, but it requires roughly double the settling time with a much larger overshoot than the small loading case. C. Case III—Short-Circuit Fault In this example, a short-circuit is applied to illustrate the effectiveness of the proposed FBL control under severe disturbances. A three-phase fault is applied on bus 87 at 0.01 s and is cleared at 0.11 s. The dynamic responses of the two controllers are shown in Fig. 16. Similar to Case II, the PI controller returns the bus voltage to the desired value. However, the PI controller results in a large overshoot (greater than 10%), whereas the proposed FBL controller provides a very smooth response, even in the presence of the large disturbance.

Furthermore

with respect to , then in

1260

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

where

TABLE I STATCOM PARAMETERS

is one solution of Riccati matrix equation as follows:

where and are the matrices in (20). Note that the maand must be chosen with care because an imtrices of and may lead to instability of the enproper choice of tire closed-loop system. The selection of and are based on the approach in [5]. The closed-loop eigenvalues of (20) for the are given STATCOM parameters and optimal gain matrix , where

TABLE II STATCOM PI CONTROL PARAMETERS (PU)

APPENDIX B CONTROL TARGET OBJECTIVES From equilibrium point of the (12), we obtain (33) (34)

APPENDIX D GENERATOR MODEL AND PARAMETERS Two-Axis Generator Model:

(35) Reactive Current and DC-Link Voltage Control: In this case, and are externally specified. Substituting (33) and (34) into (35) results in

where . There are two possible solutions to this quadratic equation, but only one is stable:

Active and Reactive Current Control: In this case, and are externally specified. Substituting (33) and (34) into (35) results in

APPENDIX C STATCOM AND CONTROL PARAMETERS The parameters of the STATCOM are given in Table I. The per unit approach used is the same as [3] on a 100-MW, 110-kV base. and are for bus voltage magThe control parameters nitude control, which are used by both the proposed PI controls, and as shown in Figs. 6 and 8. The control parameters are for control, which is only used by the PI control (see Table II). (21) is derived using LQR control, The feedback matrix which can be found in [24]

Assumption: , and IEEE Type I Exciter/AVR Model:

Turbine Model:

Speed Governor Model:

WANG AND CROW: POWER SYSTEM VOLTAGE REGULATION VIA STATCOM INTERNAL NONLINEAR CONTROL

TABLE III GENERATOR PARAMETERS

TABLE IV EXCITER/AVR PARAMETERS

TABLE V TURBINE/GOVERNOR PARAMETERS

TABLE VI GENERATOR INERTIA CONSTANTS

Power Balance Equations: Generator Buses:

Load Buses:

With the exception of the inertia constant , all of the generator parameters are the same for each machine; see Tables III–VI.

1261

REFERENCES [1] N. G. Hingorani and L. Gyugyi, Understanding FACTS, Concepts and Technology of Flexible AC Transmission Systems. Piscataway, NJ: IEEE Press, 1999. [2] F. Liu, S. Mei, Q. Lu, Y. Ni, F. F. Wu, and A. Yokoyama, “The nonlinear internal control of STATCOM: Theory and application,” Elect. Power Energy Syst., vol. 25, pp. 421–430, 2003. [3] C. Schauder and H. Mehta, “Vector analysis and control of advanced static VAR compensators,” Proc. Inst. Elect. Eng. Gen, Transm., Distrib., vol. 140, no. 4, pp. 299–306, Jul. 1993. [4] P. W. Lehn and M. R. Iravani, “Experimental evaluation of STATCOM closed-loop dynamics,” IEEE Trans. Power Del., vol. 13, no. 4, pp. 1378–1384, Oct. 1998. [5] P. Rao et al., “STATCOM control for power system voltage control applications,” IEEE Trans. Power Del., vol. 15, no. 4, pp. 1311–1317, Oct. 2000. [6] L. Dong et al., “A reconfigurable FACTS system for university laboratories,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 120–128, Feb. 2004. [7] Y. Cheng, C. Qian, and M. L. Crow, “A comparison of diode-clamped and cascaded multilevel converters for a STATCOM with energy storage,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1512–1521, Oct. 2006. [8] M. S. El-Moursi and A. M. Sharaf, “Novel controllers for the 48-Pulse VSC STATCOM and SSSC for voltage regulation and reactive power compensation,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1985–1997, Nov. 2005. [9] M. Saeedifard, H. Nikkhajoei, and R. Iravani, “A space vector modulated STATCOM based on a three-level neutral point clamped converter,” IEEE Trans. Power Del., vol. 22, no. 2, pp. 1029–1039, Apr. 2007. [10] N. Hatano and T. Ise, “Control scheme of cascaded H-bridge STATCOM using zero-sequence voltage and negative-sequence current,” IEEE Trans. Power Del., vol. 25, no. 2, pp. 543–550, Apr. 2010. [11] R. Sternberger and D. Jovcic, “Analytical modeling of a square-wavecontrolled cascaded multilevel STATCOM,” IEEE Trans. Power Del., vol. 24, no. 4, pp. 2261–2269, Oct. 2009. [12] H. F. Wang, H. Li, and H. Chen, “Application of cell immune response modelling to power system voltage control by STATCOM,” Proc. Inst. Elect. Eng. Gen, Transm., Distrib., vol. 149, no. 1, pp. 102–107, Jan. 2002. [13] S. Mohagheghi, Y. del Valle, G. K. Venayagamoorthy, and R. G. Harley, “A proportional-integrator type adaptive critic design-based neurocontroller for a static compensator in a multimachine power system,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 86–96, Feb. 2007. [14] C. H. Liu and Y. Y. Hsu, “Design of a self-tuning PI controller for a STATCOM using particle swarm optimization,” IEEE Trans. Ind. Electron., vol. 57, no. 2, pp. 702–715, Feb. 2010. [15] A. Jain, K. Joshi, A. Behal, and N. Mohan, “Voltage regulation with STATCOMs: Modeling, control and results,” IEEE Trans. Power Del., vol. 21, no. 2, pp. 726–735, Apr. 2006. [16] V. Spitsa, A. Alexandrovitz, and E. Zeheb, “Design of a robust state feedback controller for a STATCOM using a zero set concept,” IEEE Trans. Power Del., vol. 25, no. 1, pp. 456–467, Jan. 2010. [17] N. C. Sahoo, B. K. Panigrahi, P. K. Dash, and G. Panda, “Application of a multivariable feedback linearization scheme for STATCOM control,” Elect. Power Syst. Res., vol. 62, pp. 81–91, 2002. [18] D. Soto and R. Pena, “Nonlinear control strategies for cascaded multilevel STATCOMs,” IEEE Trans. Power Del., vol. 19, no. 4, pp. 1919–1927, Oct. 2004. [19] B. Lu and B. T. Ooi, “Nonlinear control of voltage-source converter systems,” IEEE Trans. Power Electron., vol. 22, no. 4, pp. 1186–1195, Jul. 2007. [20] Q. Song, W. Liu, and Z. Yuan, “Multilevel optimal modulation and dynamic control strategies for STATCOMs using cascaded multilevel inverters,” IEEE Trans. Power Del., vol. 22, no. 3, pp. 1937–1937, Jul. 2007. [21] E. Song, A. F. Lynch, and V. Dinavahi, “Experimental validation of nonlinear control for a voltage source converter,” IEEE Trans. Control Syst. Technol., vol. 17, no. 5, pp. 1135–1144, Sep. 2009. [22] A. Isidori, Nonlinear Control Systems. Berlin, Germany: Springer, 1985. [23] K. Zhou and J. Doyle, Robust and Optimal Control. Upper Saddle River, NJ: Prentice-Hall, 1995.

1262

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 3, AUGUST 2011

[24] Q. Lu and Y. Z. Sun, Nonlinear Control Systems and Power System Dynamics. Norwell, MA: Kluwer, 2001. [25] V. Vekatasubrumanian, H. Schattler, and J. Zaborszky, “Local bifurcations and feasibility regions in differential-algebraic systems,” IEEE Trans. Autom. Control, vol. 40, no. 12, pp. 1992–2013, Dec. 1995. [26] S. Ayasun, C. O. Nwankpa, and H. G. Kwatny, “Computation of singular and singularity induced bifurcation points of differential-algebraic power system model,” IEEE Trans. Circuits Syst. I, vol. 51, no. 8, pp. 1525–1538, Aug. 2004. [27] W. Marszalek and Z. Trzaska, “Singularity-induced bifurcations in electrical power systems,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 312–320, Feb. 2005. [28] P. Kundur, Power System Stability and Control. New York: McGrawHill, 1994. [29] X. Jiang, “Operating modes and their regulation of voltage-sourced converter-based FACTS controllers,” Ph.D. dissertation, Rensselaer Polytechnic Inst., Troy, NY, 2007. [30] [Online]. Available: http://www.ee.washington.edu/research/pstca/ pf118/pg_tca118bus.htm. [31] M. L. Crow, Computational Methods for Electric Power Systems, 2nd ed. Boca Raton, FL: CRC, 2009.

Keyou Wang (M’08) received the B.S. and M.S. degrees in electrical engineering from Shanghai Jiaotong University, Shanghai, China, in 2001 and 2004, respectively, and the Ph.D. degree from the Missouri University of Science and Technology (formerly University of Missouri-Rolla) in 2008. He is currently a Research Associate of electrical engineering at the Missouri University of Science and Technology. His research interests include FACTS control and power system transient stability simulation.

Mariesa L. Crow (S’83–M’90–SM’94–F’10) received the B.S.E. degree from the University of Michigan, Ann Arbor, and the Ph.D. degree from the University of Illinois, Urbana-Champaign. She is currently the Director of the Energy Research and Development Center and the F. Finley Distinguished Professor of Electrical Engineering at the Missouri University of Science and Technology, Rolla. Her research interests include computational methods for dynamic security assessment and the application of power electronics in bulk power systems.