514 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 TABLE I POWER–FLOW RESULTS OF THE IEEE 14 NODES SYSTEM WITH AN SVC is bigger than those required by the NR, each iteration is roughly five times faster than one iteration of Newton’s method. Table I summarizes the power-flow results obtained with both methods. IV. CONCLUSION A comprehensive SVCmodel suitable for a direct implementation in the fast decoupled load flow has been presented in this paper. In contrast to SVC models reported in open literature, the proposed model does not make use of the generator concept employed for the SVC repre- sentation. Instead, it uses the variable shunt susceptance concept, repre- senting actual SVC operationmore realistically. The SVC state variable is adjusted automatically to satisfy specified voltagemagnitude control. An important characteristic of the proposed model and algorithmic im- plementation is that the decoupling of the linearized load-flow equa- tions with constant load-flow matrices is maintained. The results ob- tained demonstrate the numerical efficiency of this approach. REFERENCES [1] G. Reed, J. Paserba, and P. Salavantis, “The FACTS of resolving transmission gridlock,” IEEE Power Energy, vol. 1, no. 5, pp. 41–46, Sep.–Oct. 2003. [2] I. A. Enrimez, Editor, “Static Var Compensators,”, Working Group 38-01, Task Force No. 2 on SVC, CIGRE, 1986. [3] H. Ambriz-Perez, E. Acha, and C. R. Fuerte-Esquivel, “Advanced SVC models for Newton–Rapshon load flow and newton optimal power flow studies,” IEEE Trans. Power Syst., vol. 15, no. 1, pp. 129–136, Feb. 2000. [4] A.Monticelli, A. Garcia, and O. R. Saavedra, “Fast decoupled load flow: Hypotesis, derivations and testing,” IEEE Trans. Power Syst., vol. 5, no. 4, pp. 1425–1431, Nov. 1990. [5] B. Stott and O. Alsac, “Fast decoupled load flow,” IEEE Trans. Power App. Syst., vol. PAS-93, no. 2, pp. 859–869, May 1974. [6] L. L. Freris and A. M. Sasson, “Investigation of the load-flow problem,” Proc. Inst. Elect. Eng., vol. 115, no. 10, pp. 1459–1470, Oct. 1968. [7] C. Taylor, G. Scott, A. Hammad, W. Wong, D. Osborn, A. J. P. Ramos, B. Johnson, D. McNabb, S. Arabi, D. Martin, H. L. Thanawale, J. Luini, R. Gonzalez, and C. Concordia, “Static VAR compensator models for power flow and dynamic performance simulation,” IEEE Trans. Power Syst., vol. 9, no. 1, pp. 229–240, Feb. 1995. Complementarity Model for Generator Buses in OPF-Based Maximum Loading Problems Codruta Roman, Student Member, IEEE, and William Rosehart, Member, IEEE Abstract—A mathematical program with complementary constraints is used to better model the relationship between the current or base operating point and the maximum loading point in a power system when solving the maximum loading problem. Index Terms—Complementarity programming, stability. I. INTRODUCTION The voltage stability limits of power networks are strongly influ- enced by operational limits of the system’s components associated with reactive power, particularly synchronous generator excitation limits [1], [2]. In [3], the influence of generator reactive limits on voltage stability is analyzed using singular value decomposition of the power flow Jacobian. In [4], the reactive power margin with respect to voltage collapse is computed in steps based on an optimization formulation. At each step, generators that reached their reactive output limit are switched from a PV bus to a PQ bus. In [1], the reactive limits of the generators are modeled by two inequalities: one for the limited field voltage and one for the automatic voltage regulator (AVR) equation, along with the condition that at least one of them has to be satisfied. In this paper, the simple “box” constrained generator model is en- hanced by a complementarity condition between the reactive power limits and the terminal voltage of generator buses. The complemen- tarity model is then used in an optimization problem to maximize the loading margin of the system. II. COMPLEMENTARITY MODEL FOR GENERATOR BUSES In optimization-based voltage stability assessment for a given op- erating point, the common practice is to consider a “box” constrained generator, i.e., the active and reactive output of the generator are be- tween some constant minimum and maximum values, with a fixed ter- minal voltage. The minimum and the maximum values for the reactive power generation reflect the limits imposed on the field current [3]. When one of reactive limits is reached, the reactive power output be- comes a fixed variable, and the terminal voltage becomes a free vari- able. This change of behavior can be modeled as a mixed complemen- tarity problem, i.e., find Q 2 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 515 power in the range [Qmin; Qmax]. In the underexcited limited mode, Q�Q min = 0. From the complementarity condition of (1a), the term V 1 can then take a positive value. In this case, Qmax � Q > 0, and V 2 = 0. In the overexcited limited mode, Qmax �Q = 0 in (1b). The complementarity condition is fulfilled for any positive value of V 2 . In (1a),Q�Qmin > 0 and V 1 is zero. The model (1) can be written com- pactly as Q min �Q � Q max ? V = V 1 � V 2 V gen =V gen + V; 8 V 2 516 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 the figure, incorporating complementarity allows for better modeling of generator behavior when reactive power limits are reached. V. CONCLUSIONS The papers presents a simple complementarity model that enables the change of behavior of the active and reactive power “box”-bounded generator in the maximum loading optimization problem. If the change of behavior of the generator buses is not considered, a breaking point solution can be encountered, even before any voltage level throughout the network has decreased under the minimum operating limit. The breaking point is a stable equilibrium point, and the load of the system still can be increased. Although the breaking points are important for the power system operation, the optimization problem should be able to find maximum loadability points that correspond to an operational limit, a limit-induced bifurcation, or a saddle-node bifurcation. REFERENCES [1] C. D. Vournas, M. Karystianos, and N. G. Maratos, “Bifurcation points and loadability limits as solutions of constrained optimization problems,” in Proc. IEEE Power Eng. Soc. Summer Meeting, vol. 3, Jul. 16–20, 2000, pp. 1883–1888. [2] I. Dobson and L. Lu, “Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encoun- tered,” IEEE Trans. Circuits Syst. I, vol. 39, no. 9, pp. 762–766, Sep. 1992. [3] P.-A. Löf, G. Andersson, and D. J. Hill, “Voltage dependent reactive power limits for voltage stability studies,” IEEE Trans. Power Syst., vol. 10, no. 1, pp. 220–228, Feb. 1995. [4] T. Van Cutsem, “A method to compute reactive power margins with re- spect to voltage collapse,” IEEE Trans. Power Syst., vol. 6, no. 1, pp. 145–156, Feb. 1991. [5] W. Rosehart, C. Cazinares, and V. Quintana, “Multiobjective optimal power flows to evaluate voltage security costs in power networks,” IEEE Trans. Power Syst., vol. 18, no. 2, pp. 578–587, May 2003. [6] M. C. Ferris, R. Fourer, and D. M. Gay, “Expressing complementarity problems in an algebraic modeling language and communicating them to solvers,” SIAM J. Optimizat., vol. 9, no. 4, pp. 991–1009, 1999. [7] H. Y. Benson, D. F. Shanno, and R. J. Vanderbei, “Interior-Point Algo- rithms for Nonconvex Nonlinear Programming: Complementarity Con- straints,” Tech. Rep. ORFE-02–02, Univ. Princeton, Princeton, NJ, 2002. A Novel Approach to Implement Generic Load Restoration in Continuation-Based Quasi-Steady-State Analysis Qin Wang, Member, IEEE, and Venkataramana Ajjarapu, Senior Member, IEEE Abstract—This paper presents a new approach to deal with load restora- tion in the Continuation-basedQuasi-Steady-State (CQSS) analysis. By ap- propriately choosing the continuation parameter, the differential equations representing the load restoration dynamics are eliminated inQuasi-Steady- State (QSS) simulation. This method will improve the speed of simulation and will take into account short-term load characteristics. Numerical re- sults are illustrated on a small test system. Index Terms—Continuationmethod, load restoration, quasi-steady-state analysis, voltage stability. I. INTRODUCTION In Quasi-Steady-State (QSS) simulation proposed in [1], loads are generally treated as the smooth differential equations, where the power consumed by the load at any time depends upon the instantaneous value of a load state variable. However, this method encounters numerical difficulties around the singularity-induced bifurcation (SIB) point [2], where the short- term load characteristic is tangential to the systemPV curve. In order to overcome this problem and readily identify the SIB point during the tracing, the Continuation-based Quasi-Steady-State (CQSS) simulation is proposed to consider load restoration. It also eliminates the need to include the differential equation in the simulation and simplify the whole tracing procedure. In addition, the equilibrium of fast dynamics is calculated only once to speed up the calculation. Moreover, the SIB point will be easily found by the change of the con- tinuation parameter. II. GENERAL MODEL FOR LOAD RESTORATION IN QSS The system model for the CQSS analysis can be formulated as [1] 0 = f(x; y; z D ; z C ; �) (1) 0 = g(x; y; z D ; z C ; �) (2) z D (k + ) =h D (x(k � ); y(k � ); z D (k � ); z C (k � ); �(k)) (3) z : C =h C (x; y; z D ; z C ; �) (4) where x is a short-term variable relating to the generator and its control devices. y includes bus voltage magnitudes and angles. The long-term dynamics of discrete-type controllers, such as OLTCs, can be repre- sented as z D . k is the time step used in the simulation. Equation (4) comes from generic load self-restoration [5] and other controllers. � is the load exponent that we choose as the continuation parameter, which will be explained later. Manuscript received April 26, 2003. Paper no. PESL-00076-2002. Q. Wang is with the EMC Corporation, Hopkinton, MA 01748 USA (e-mail:
[email protected]). V. Ajjarapu is with the Department of Electrical Engineering, Iowa State Uni- versity, Ames, IA 50011 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2004.841151 0885-8950/$20.00 © 2005 IEEE