Roman Kuiava

Practical stability of switched systems without a common equilibria and governed by a time-dependent switching signal

In this paper, the problem of practical stability of some classes of continuous-time switched systems is studied. The main results of this paper include some sufficient conditions concerning practical stability of continuous-time switched nonlinear systems without a common equilibria for all subsystems. In this class of switched systems, the equilibrium point varies discontinuously according to a time-dependent switching signal. So, stability with respect to a set, rather than a particular point, is discussed. Using this preliminary result, we present sufficient conditions in the form of linear matrix inequalities (LMIs) for practical stability of a particular class of switched systems without common equilibria: the switched affine systems. An illustrative example in the power system stability area is presented to show the validity of the results.

Practical stability of continuous-time switched systems without a common equilibria and governed by a time-dependent switching signal

In this paper, the problem of practical stability of some classes of continuous-time switched systems is studied. The main results of this paper include some sufficient conditions concerning practical stability of continuous-time switched nonlinear systems without a common equilibria for all subsystems. In this class of switched systems, the equilibrium point varies discontinuously according to a time-dependent switching signal. So, stability with respect to a set, rather than a particular point, is discussed. Using this preliminary result, we present sufficient conditions in the form of linear matrix inequalities (LMIs) for practical stability of a particular class of switched systems without common equilibria: the switched affine systems. An illustrative example is presented to show the validity of the results.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Optimal operation of distribution networks with synchronous generators via transient stability constrained optimal power flow

A common approach to deal with transient stability constraints in optimization problems is the combination of a numerical discretization method with interior point methods (IPMs) for solving large-scale nonlinear programming (NLP). The numerical discretization is adopted to convert the differential equations that describe the electromechanical transients in power systems in a set of algebraic equations to be included into a conventional optimal power flow (OPF) problem. The resulting transient stability constrained optimal power flow (TSC-OPF) problem, however, suffers with the curse of dimensionality, high computational time and memory consumption to solve it, even for small systems. To relieve this computational burden, this paper proposes an algorithm that aims at solving a TSC-OPF problem via primal-dual IPM with only a few time steps of numerical discretization in post-fault period, which is enough to ensure the rotor angle first swing stability. Once the TSC-OPF problem is solved, a numerical discretization method is applied to calculate the system trajectory from the first rotor angle peak in post-fault period. An important contribution of the proposed algorithm is to provide a proper accuracy on the computation of constant admittance loads. Numerical results are obtained for a 9-bus distribution network with the purpose of providing a proper sizing of DG units based on synchronous generators and an optimal operation to the network.

Practical stability assessement of distributed synchronous generators under load variations

This paper proposes a method to assess the practical stability of distribution systems with synchronous generators subject to changes in the system equilibrium conditions due to load variations. The concept of practical stability deals with two known state-space regions Ω1 (which contains all the initial conditions reflecting the perturbations at which the system is subject during its operation) and Ω2 (which represents the operating security region of the power distribution system) satisfying Ω1 ⊂ Ω2. The practical stability problem and the focus of this paper is to determine the minimum amount of time in which the system is required to stay in a particular loading condition, such that the system trajectories from an initial condition in Ω1 will be confined in the security region of operation Ω2 for the whole time interval of interest. This study was carried out using a mathematical model of the distribution system with synchronous generators in the form of a switched affine system. Sufficient conditions for the power distribution system with synchronous generators described as a switched affine system to be practically stable with respect to Ω1 and Ω2 are given in the form of matrix inequalities constraints. Numerical results were obtained for a model of a cogeneration plant of 10 MW added to a distribution network constituted by a feeder and six buses.