Singularity Induced Bifurcation Points Research Articles

Analysis and behavior control of a modified singular prey–predator model

In this paper, a harvested singular prey–predator model is proposed based on governing differential equations, in addition to an algebraic equation. After comparing other nonsingular models with the proposed singular model, its advantages are explained. The model is analyzed and its stability is studied. This model has singularity induced bifurcation (SIB) points causing various complexities in the behavior of the model. We suggest a state feedback controller, which eliminates one of the SIB points, to stabilize the singular model system around an interior equilibrium. Then, the local stability of the system is studied by the Lyapunov nonlinear method. Finally, the results are numerically simulated to verify our analytical approach.

Fold Points and Singularities in Hall MHD Differential–Algebraic Equations

We consider the singularity crossing phenomenon in differential-algebraic equations (DAEs) of Hall MHD systems in one spatial dimension. The Hall MHD DAEs have singularities with impasse points, pseudoequilibrium points, or singularity-induced bifurcation (SIB) points. The pseudoequilibrium and SIB points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Riemann sheets. Within the singular pseudoequilibrium points, there may exist only one analytic trajectory crossing the sonic curve (as in the case of SIB point), two analytic and two other trajectories of lower degree of smoothness in the case of pseudosaddle points, or two analytic and an uncountable number of trajectories of lower smoothness in the case of singular pseudonodes. In this paper, we show examples of singular points in Hall MHD systems described by DAEs and explain the singularity (also called the sonic or forbidden curve) crossing phenomenon by using the recent developments in the qualitative analysis of DAEs.

Singularity-induced bifurcations in lumped circuits

A systematic analysis of singular bifurcations in semistate or differential-algebraic models of electrical circuits is presented in this paper. The singularity-induced bifurcation (SIB) theorem describes the divergence of one eigenvalue through infinity when an operating point or equilibrium locus of a parameterized differential-algebraic model crosses a singular manifold. The present paper extends this result to cover situations in which several eigenvalues diverge; we prove a multiple SIB theorem which states that a minimal rank (resp. index) change makes it possible to compute the number of diverging eigenvalues in terms of an index (resp. rank) change in the matrix pencil characterizing the linearized problem. The scope of the work comprises quasi-linear ordinary differential equations, semiexplicit index-1 differential-algebraic equation (DAEs), and Hessenberg index-2 DAEs, describing different electrical configurations. The electrical features from which singularities and, specifically, singular bifurcations stem are extensively discussed. Examples displaying simple, double, and triple SIB points illustrate different ways in which the spectrum may diverge.

An efficient method to compute singularity induced bifurcations of decoupled parameter-dependent differential-algebraic power system model

In this paper, we present an efficient method to compute singular points and singularity induced bifurcation points of differential-algebraic equations (DAEs) for a multi-machine power system model. The algebraic part of the DAEs brings singularity issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the algebraic equations, but at which the vector field is not defined. In terms of power system dynamics, around singular points, the generator angles (the natural states variables) are not defined as a graph of the load bus variables (the algebraic variables). Thus, the causal requirement of the DAE model breaks down and it cannot predict system behavior. Singular points constitute important organizing elements of power system DAE models. This paper proposes an iterative method to compute singular points at any given parameter value. Generator angles are parameterized through a scalar parameter in the constraint manifold and identification of singular points is formulated as a bifurcation problem of the algebraic part of the DAEs. Singular points are determined using a second order Newton–Raphson method. Moreover, the decoupled structure of the DAE model is exploited to find new set of parameters and parameter increase pattern which will result in a set of equilibria including singularity induced bifurcations. The simulation results are presented for a 5-bus power system and singular and singularity induced bifurcation points are depicted together with equilibria to visualize static and dynamic stability limits.

Computation of Singular and Singularity Induced Bifurcation Points of Differential-Algebraic Power System Model

In this paper, we present an efficient algorithm to compute singular points and singularity-induced bifurcation points of differential-algebraic equations for a multimachine power-system model. Power systems are often modeled as a set of differential-algebraic equations (DAE) whose algebraic part brings singularity issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the algebraic equations, but at which the vector field is not defined. In terms of power-system dynamics, around singular points, the generator angles (the natural states variables) are not defined as a graph of the load bus variables (the algebraic variables). Thus, the causal requirement of the DAE model breaks down and it cannot predict system behavior. Singular points constitute important organizing elements of power-system DAE models. This paper proposes an iterative method to compute singular points at any given parameter value. With a lemma presented in this paper, we are also able to locate singularity induced bifurcation points upon identifying the singular points. The proposed method is implemented into voltage stability toolbox and simulations results are presented for a 5-bus and IEEE 118-bus systems.

Double SIB points in differential-algebraic systems

A singularity-induced bifurcation (SIB) describes the divergence of one eigenvalue through infinity when an equilibrium locus of a parameterized differential-algebraic equation (DAE) crosses a singular manifold. The present note extends the analysis of this behavior to cover double SIB points, for which two eigenvalues diverge. The key assumption supporting this phenomenon is that the Kronecker index jumps by two at the singularity. In this situation, double SIB points are shown to undergo generically a transition from a spiral to a saddle in the linearized problem, after restricting the analysis to the corresponding invariant subspace. Typical examples arise in the context of nonlinear RLC circuits. The setting for the study is that of semi-explicit DAEs in Hessenberg form with arbitrary index.

Computation of the singularity induced bifurcation points in DAEs via extended system reduction

Singularity-induced bifurcation (SIB) with one parameter is considered. This kind of bifurcation arise in parameter dependent differential–algebraic equations (DAEs) of the form x ̇ =f , 0= g. The extended system reduction is introduced as a convenient method to compute the SIB points. Non-degeneracy conditions on the functions f and g are derived. Then under verification of these conditions, SIB points are associated with the non-degenerate equilibrium points of the extended system. An iterative method (e.g., Newton–Raphson) then can be used to compute the non-degenerate equilibrium points of the extended system which including the SIB points of the original DAEs. An example is given to illustrate the idea of this paper.

Double singularity-induced bifurcation points and singular Hopf bifurcations

The singularity-induced bifurcation and singular Hopf bifurcation theorems and the degeneracies that arise when Newton's laws are coupled to Kirchhoff's laws are explored. Such models are used in the electrical engineering literature to describe electrical power systems and they can take the form of either an index-1 differential-algebraic equation (DAE) or a singularly perturbed ordinary differential equation (ODE). As a consequence of the debate in the engineering literature as to which class of system is the 'true' representation of power systems, a discussion is included of the consequences of the power engineer's 'load-flow singularity' for both ODE and DAE.