Copyright ro IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998 A passivation approach to power systems stabilization R.Ortega Lab des Signaux et Systemes Supelec Plateau de Moulon 91192 Gif sur Yvette FRANCE rortegaOlss.supelec.fr A. Stankovic and P. Stefanov Northeastern University Electrical and Computer Eng . 303 Dana Research Center Boston, Mass., 02115 USA astankovOcdsp.neu.edu Abstract: In this paper we address the problem of supression of low frequency oscil- lations in power systems. These oscillations appear in strongly interconnected networks because of load and topology changes, and they may cause loss of synchronism and gen- erator tripping . We propose the utilisation of passivation techniques to design power system stabilizers for the synchronous generators. The generator to be controlled is de- scribed by a standard lagrange model, with three forcing terms : the mechanical torque coming from the turbine, the terminal voltage of the network and the field voltage, which is our control variable. In view of the significant differences between the mechanical and the electrical time scales, the first signal can be treated as a constant disturbance . The terminal voltage may be viewed as the output of an operator , -defined by the remaining part of the network- , which is in feedback interconnection with the generator . Our basic assumption is that the network is always absorbing energy from the generator, whence the interconnection subsystem (as viewed from the generator) is passsive. The control objective is then to close a loop around the field voltage so as to passivize the generator system. We characterize, in terms of a simple linear matrix inequality, a class of linear state-feedback controllers which achieve this objective. Copyright ro 1998 IFAC Keywords: Power systems, nolinear control, passivity-based control compensation, slid- ing mode control. 1. Introduction We consider in this paper the problem of designing decentralized controllers for synchronous generators to supress low frequency oscillations in power distribu- tion networks. These oscillations appear in strongly interconnected networks because of load and topology changes, and they may cause loss of synchronism and generator tripping. passive (resp. output strictly passive) if 30 0 (resp. > 0 ) and 3/3 E 'R such that This problem is usually studied in the power systems literature [4] adopting a sinusoidal quasi-steady state approximation that ignores the dynamics of the network circuit elements. Further, the interaction between the generator and the network is characterized via terminal voltage phasor and complex power pairs. As pointed out in [9], this sinusoidal characterization may prove in- adequate in some studies. For instance, in the present problem -which boils down to the stabilization of an equilibrium-, the proliferation of switching controls in modern power systems typically deviates these equilib- ria from the assumed sinusoidal form. In this paper we adopt a detailed (Lagrangian) model of the generator and propose the utilisation of passiva- tion techniques to design power system stabilizers. Our work may be viewed as an extension, to the nonlinear case, of the one reported in [10] . Notation. 'R- field of real numbers; I . I - Euclidean norm ; (.) T - transposition ; In - n X n identity matrix ; Omxn-an m x n matrix of zeros; .c 2 ,.c 2e - spaces of n- dimensional square integrable functions and its exten- sion; - space of n-dimensional essentially bounded functions; a causal system L : .c 2e 1-7 .c 2e is said to be 309 < ulEu loT uT (t)(Eu)(t)dt loT I(Eu)(t)1 2 dt + /3 , 'Vu E \:IT O. It is incrementally passive (resp. incrementally output strictly passive) if 30 0 (resp. > 0 ) and 3/3 E 'R such that 'VUI, U2 E C 2e , \:IT 0 we have < 'Ill - u21Eul - EU2 OIlUI - + /3. 2. Problem Formulation As shown in appendix A the dynamics of a synchronous generator (in the dq coordinate frame of the machine and per unit notation) are described by Dx + [J"(x) + R]x = Y (1) (2) where x = [Xl, · ··, X7]T E 'R7 is the state vector, U = [UI ,· ·· ,U4]T , Y = [Yl,·· ·, Y4]T E'R4 are "input" and "output" vectors, D = DT > 0, R = RT > 0, J"(x) = -J"T (x), and M E 'R 7X4 are defined in appendix A. The generator is interconnected to the "outside world" via the mechanical shaft torque U4 and the termi- nal volt ages 'Ill, u2 and currents YI, Y2 . Given the sig- nificant differences between the electrical and mechan- ical time scales it is reasonable to treat U4 as a con- stant disturbance. On the other hand, the generator is