Institute for Systems Research
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Item New Results on Modal Participation Factors: Revealing a Previously Unknown Dichotomy(2007) Hashlamoun, Wael A.; Hassouneh, Munther A.; Abed, Eyad H.This paper presents a new fundamental approach to modal participation analysis of linear time-invariant systems, leading to new insights and new formulas for modal participation factors. Modal participation factors were introduced over a quarter century ago as a way of measuring the relative participation of modes in states, and of states in modes, for linear time-invariant systems. Participation factors have proved their usefulness in the field of electric power systems and in other applications. However, in the current understanding, it is routinely taken for granted that the measure of participation of modes in states is identical to that for participation of states in modes. Here, a new analysis using averaging over an uncertain set of system initial conditions yields the conclusion that these quantities(participation of modes in states and participation of states in modes) should not be viewed as interchangeable. In fact, it is proposed that a new definition and calculation} replace the existing ones for state in mode participation factors, while the previously existing participation factors definition and formula should be retained but viewed only as mode in state participation factors. Examples are used throughout the paper to illustrate the issues addressed and results obtained.Item On Participation Factors for Linear Systems(1999) Abed, Eyad H.; Lindsay, David; Hashlamoun, Wael A.; ISRParticipation factors are nondimensional scalars that measure the interaction betweenthe modes and the state variables of a linear system. Since their introduction byVerghese, P'erez-Arriaga and Schweppe, participationfactors have been used for analysis, order reduction and controller design in a variety of fields.In this paper, participation factors are revisited, resulting in new definitions. The aim ofthese definitions is to achieve a conceptual framework that doesn't hinge on anyparticular choice of initial condition. The initial condition is modeled as an uncertainquantity, which can be viewed either in a set-valued or a probabilistic setting.If the initial condition uncertainty obeys a symmetry condition, the new definitionsare found to reduce to the original definition of participation factors.
Item Stationary Bifurcation Control for Systems with Uncontrollable Linearization(1999) Taihyun Kim; Abed, Eyad H.; ISRStationary bifurcation control is studied under the assumption thatthe critical zero eigenvalue is uncontrollable for thelinearized system. The development facilitates explicit constructionof feedback control laws that render the bifurcation supercritical.Thus, the bifurcated equilibria in the controlled system are guaranteedstable.Bothpitchfork bifurcation and transcritical bifurcation are addressed.The results obtained forpitchfork bifurcations apply to general nonlinear models smoothin the state and the control. For transcritical bifurcations,the results require the system to be affine in the control.
Item Closed-Loop Monitoring Systems for Detecting Incipient Instability(1998) Kim, Taihyun; Abed, Eyad H.; ISRMonitoring systems are proposed for the detection of incipientinstability in uncertain nonlinear systems. The work employsgeneric features associated with the response to noise inputsof systems bordering on instability. These features, called "noisy precursors" in the work of Wiesenfeld, also yield information onthe type of bifurcation that would be associated with thepredicted instability. The closed-loop monitoring systems proposedin the paper have several advantages over simple open-loop monitoring.The advantages include the ability to influence the frequencies atwhich the noisy precursors are observed, and the ability tosimultaneously monitor and control the system.Item Local Nonlinear Control of Stall Inception in Axial Flow Compressors(1993) Adomaitis, Raymond A.; Abed, Eyad H.; ISRA combination of theoretical and computational nonlinear analysis techniques are used to study the scenario of bifurcations responsible for the initiation of rotating stall in an axial flow compressor model. It is found that viscosity tends to damp higher-frequency modes and so results in a sequence of bifurcations along the uniform-flow solution branch to stall cells of different mode number. Lower-mode stalled flow solutions are born in subcritical bifurcations, meaning that these equilibria will be unstable for small amplitudes. Secondary bifurcations, however, can render them stable, leading to hysteresis. Using throttle position as a control, we find that while the stall bifurcations are not linearly stabilizable, nonlinear state feedback of the first mode amplitude will reduce the hysteresis. This improves the nonlinear stability of the compression system near the stall margin.Item Bifurcation Analysis of Nonuniform Flow Patterns in Axial-Flow Gas Compressors(1992) Adomaitis, Raymond A.; Abed, Eyad H.; ISRWe study the transition from steady, spatially uniform-flow to nonuniform and time-dependent gas axial velocity profiles in an axial flow compression system. Local bifurcation analysis of the uniform-flow solution reveals a series of bifurcations to traveling waves of different mode number as a function of throttle opening. The number of bifurcating modes is found to depend on the gas viscosity parameter, an effect introduced in this work. Using the local approximations of the bifurcating solutions as starting points of our numerical analysis, we uncover a complicated scenario of secondary bifurcations ultimately resulting in parameter ranges where locally asymptotically stable stalled-flow solutions of different mode number coexist.Item Active Stabilization of Rotating Stall: A Bifurcation-Theoretic Approach(1992) Liaw, Der-Cherng; Abed, Eyad H.; ISRActive control of the onset of stall instabilities in axial flow compressors is pursued using bifurcation analysis of a dynamical model proposed by Moore and Greitzer (1986). This model consists of three ordinary differential equations with state variables being the mass flow rate, pressure rise, and the amplitude of the first harmonic mode of the asymmetric component of the flow. The model is found to exhibit a stationary (pitchfork) bifurcation at the inception of stall, resulting in hysteresis. Using throttle opening as a control, analysis of the linearized system at stall shows that the critical mode (zero eigenvalue) is unaffected by linear feedback. Hence, nonlinear tools must be used to achieve stabilization. A quadratic feedback control law using the measurement of asymmetric dynamics is proposed which stabilizes the bifurcation and eliminates the undesirable hysteretic behavior.Item Nonlinear Dynamics of Axial-Flow Compressors: A Parametric Study(1992) Adomaitis, Raymond A.; Liaw, Der-Cherng; Abed, Eyad H.; ISRAnalysis of the post-instability dynamical behavior of an axial- flow compression system model is carried out in a bifurcation- theoretic setting. Using global analysis techniques, we uncover the sequence of bifurcations in parameter space which allows us to rigorously determine whether the compressor stalls or surges when the throttle is slowly closed beyond the instability margin. Using these computational techniques, we also determine the conditions under which stalled and/or surging flow solutions coexist with the desired uniform-flow operating point and quantify the perturbations which destabilize this operating point.Item Bifurcations, Chaos and Crises in Power System Voltage Collapse(1992) Wang, Hua O.; Abed, Eyad H.; Hamdan, Anan M. A.; ISRBifurcations occurring in power system models exhibiting voltage collapse have been the subject of several recent studies. Although such models have been shown to admit a variety of bifurcation phenomena, the view that voltage collapse is triggered by possibly the simplest of these, namely by the (static) saddle node bifurcation of the nominal equilibrium, has been the dominant one. The authors have recently shown that voltage collapse can occur "prior" to the saddle node bifurcation. In the present paper, a new dynamical mechanism for voltage collapse is determined: the boundary crisis of a strange attractor or synonymously a blue sky bifurcation. This determination is reached for an example power system model akin to one studied in several recent papers. The identified mechanism for voltage collapse amounts to the disappearance of a strange attractor through collision with a coexisting saddle equilibrium point. This mechanism results in solution trajectories containing both an oscillatory component (as predicted by recent analytical work), and a sharp, steady drop in voltage (as observed in the field). More generally, blue sky bifurcations (not necessarily chaotic) are identified as important mechanisms deserving further consideration in the study of voltage collapse.Item Dynamic Bifurcations in a Power System Model Exhibiting Voltage Collapse(1992) Abed, Eyad H.; Alexander, James C.; Wang, H.; Hamdan, Anan M. A.; Lee, Hsien-Chiarn; ISRDynamic bifurcations, including Hopf and period-doubling bifurcations, are found to occur in a power system dynamic model recently employed in voltage collapse studies. The occurrence of dynamic bifurcations is ascertained in a region of state and parameter space linked with the onset of voltage collapse. The work focuses on a power system model studied by Dobson and Chiang (1989). The presence of the dynamic bifurcations, and the resulting implications for dynamic behavior, necessitate a re- examination of the role of saddle node bifurcations in the voltage collapse phenomenon. The bifurcation analysis is performed using the reactive power demand at a load bus as the bifurcation parameter. Due to numerical ill-conditioning, a reduced-order model is employed in some of the computations. It is determined that the power system model under consideration exhibits two Hopf bifurcations in the vicinity of the saddle node bifurcation. Between the Hopf bifurcations, i.e., in the "Hopf window," period-doubling bifurcations are found to occur. Simulations are given to illustrate the various types of dynamic behaviors associated with voltage collapse for the model. In particular, it is seen that an oscillatory transient may play a role in the collapse.