Order Subsystems Research Articles

LFC Using a Perturbational Approach for a Two-Area Power System With Detailed Modelling

A large power system may be viewed as an interconnection of several lower order subsystems, each of which is required to operate with as much autonomy as possible. Owing to the complexity of the present-day interconnected pouer systems, the application of multi-level control strategies for load frequency control situations is assuming importance. This paper reports the application of a two-level control scheme to a multiarea power system model, which includes the effects of the excitation system, the speed-governing mechanism and a single time-delay voltage regulator. Apart from the local controls, corrective controls are provided to minimise the deviation from the decoupled optimum, due to the presence of interconnections. The global controller used attempts to utilise certain possible beneficial aspects of the interconnections to give more desirable system performances.

CALCULATION OF PHASE DIAGRAMS OF IONIC SYSTEMS FROM FUNDAMENTAL SOLUTION THEORIES∗

Fundamental concepts and theories of molten salt solutions appear to be useful in a number of technologies (e.g., high-temperature batteries, fuels cells, aluminum production). A simple but exact cycle permits one to calculate the solubility product of, for example, oxides and sulfides in halide melts. The use of this cycle is well illustrated by calculations related to nuclear fuel processing and to advanced battery technology. For concentrated solutions, the conformal ionic solution theory which is a statistical mechanical perturbation theory is used to calculate the phase diagrams of terna ry systems solely based on information from the lower order (unary and binary) subsystems. Thus, one can numerically generate phase diagrams a priori, reliably extend phase diagrams into unmeasured regions and check the thermodynamic self consistency of published measurements.

Error minimization in multiple-connected subsystems

A set of fully interconnected n th order discrete-time subsystems is considered. The subsystems errors are minimized with respect to the coupling, feedback, and forward gain parameters, using nonlinear programming. It was demonstrated that the error functions for subsystems of the third order and higher, are independent of the coupling gains and part of the feedback gains. Increasing the number of subsystems tends to increase the error cost function.

Stability criteria for multivariable non-linear feedback systems consisting of low order subsystems

Abstract Popov-type stability criteria are given for two types of multivariable feedback systems. The systems are regarded as interconnections of several subsystems and the composite-system method is used to derive the criteria. The values of constants which are needed in the criteria are calculated and tabulated for first- and second-order subsystems. By using these values, we can immediately obtain a stability criterion for the system consisting of first- and/or second-order subsystems. Examples show that the criteria obtained are as sharp as or sharper than previously reported criteria.

Stability of Interconnected Systems Described by Stochastic Nonlinear Volterra Integral Equations

New results are established for the stochastic stability of interconnected systems (also called composite systems or large scale systems) described by nonlinear Volterra integral equations with uncertain weighting functions, uncertain nonlinearities and stochastic inputs. We utilize the viewpoint of analyzing complex systems in terms of lower order (and simpler) subsystems and in terms of the system interconnecting structure. Whenever appropriate, frequency domain techniques (including circle criteria and Popov- like conditions) are emphasized. The results are applied to systems described by stochastic nonlinear differential equations. In addition, a specific example is also presented.

Second‐Order Stochastic Input‐Output Stability and Instability of Interconnected Systems

AbstractNew results for the second‐order stochastic input‐output stability and instability of large classes of continuous time and discrete time interconnected stochastic systems are established. The analysis is accomplished in terms of lower order subsystems and in terms of system interconnecting structure. When appropriate, frequency domain interpretations are emphasized. The method of analysis advanced herein is applied to three specific examples.

On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.

Two-time-scale feedback design of a class of nonlinear systems

For a class of nonlinear singularly perturbed systems, feedback stabilizing and near-optimal controls are designed using two separate lower order subsystems. The two-time-scale properties greatly simplify the stability analysis and the nonlinear controller design. Two electric machine examples (a dc motor and a synchronous generator) illustrate the proposed design procedure.

Singular perturbation analysis of systems with sustained high frequency oscillations

Using singular perturbation techniques, a system with high frequency oscillations is decomposed into two lower order subsystems, one containing only the slowly varying part and the other containing only the fast oscillatory part. Eigenvalue and state approximations achieved by the subsystems are given. A mass-spring-damper example shows that a stiff spring can be regarded as a perturbation of a rigid rod and an interconnected power system example illustrates the occurrence of coherency and inter-machine oscillations.

Qualitative Analysis of Interconnected Systems Described on Banach Spaces: Well Posedness, Instability and Lagrange Stability

AbstractNew qualitative results for interconnected dynamical systems described on Banach spaces are established. Items of interest include well posedness, instability, complete instability, and Lagrange stability (i.e., boundedness and ultimate boundedness of solutions). In all cases the objective is to analyze complex systems in terms of lower order subsystems and in terms of the system interconnecting structure. The results are applied to a specific hybrid dynamical system.

On the controllability of singularly perturbed systems

Sufficient conditions for controllability of high-order systems of singularly perturbed type are formulated in terms of two lower order subsystems. At first, linear time-varying systems are considered and then the results are extended to a class of nonlinear systems.

On vector Lyapunov functions for stochastic dynamical systems

Vector Lyapunov functions are used in the stability analysis of large-scale stochastic systems described by Itô differential equations (with stochastic disturbances in the subsystems and in the interconnecting structure). Sufficient conditions for asymptotic stability and exponential stability with probability 1 and in probability are established, in all cases the objective is the same: to analyze large-scale systems in terms of their lower order (and simpler) subsystems and in terms of their interconnecting structure. Use of the method presented makes it often possible to circumvent difficulties usually encountered when the Lyapunov method is applied to high-dimensional systems and to systems with complicated interconnecting structure. In order to demonstrate the usefulness of the present approach, a specific example is considered.

Stability of stochastic composite systems

In a recent paper [15] results for the asymptotic stability and exponential stability (with probability one) of a class of continuous parameter stochastic composite systems, with disturbances confined to the sub-system structure, were established. In this short paper these results are extended to allow stability analysis of systems for which stochastic disturbances may not only enter into the subsystem structure but also into the interconnecting structure of composite systems. As in previous related results, the objective is to analyze composite systems in terms of their lower order subsystems and in terms of their interconnecting structure.

Computation of the viscous shock layer on blunted cones

A numerical method is described for solving the equations of the compressible viscous shock layer on smooth spherically blunted axisymmetric cones at zero angle of attack and flow of a perfect gas. Effective use is made of the scheme of separating the original system of equations into parabolic (second order) and inviscid (first order) subsystems, which are solved by intrinsic methods. The results of the computations are presented. The method is capable of natural generalization to the case of nonequilibrium physical and chemical processes and diffusion. In most published papers dealing with computation of the compressible shock layer, the authors examine either the vicinity of the stagnation point or a certain region of spherical blunting [1–5]. In all the papers except [4, 5], a number of simplified assumptions have been made regarding the flow picture. Very few papers [6–8] have calculated the viscous shock layer on the forward surface of blunted bodies. In [6, 7] an approximate examination was made only of hyperboloids and paraboloids of revolution, which have very favorable geometry. Reference [8] used a approximate Karman—Polhausen integral method for a very simple system of equations. The method proposed here is essentially an accurate numerical method for solution of the viscous shock layer equations.

Stability of Large Scale Power Systems

In this paper a new approach to the stability analysis of large scale power systems using Vector Lyapunov functions is introduced. By decomposing the composite power system into a number of low order subsystems, Lyapunov functions for these subsystems are constructed in the usual way. These scalar Lyapunov functions are then used as components of a vector Lyapunov function for the composite system. Using this function together with the stability properties of the individual subsystems and the nature of interactions between the subsystems criteria for the stability of the overall system are derived. Such an approach of piecewise analysis provides an attractive alternative to the methods employing stability criteria of multinonlinear systems to power system problems. It is also possible with this technique to investigate the stability of the system due to structural changes in the system. The procedure is illustrated for a 3 machine power system.

Stability analysis of stochastic composite systems

New results for asymptotic stability and exponential stability with probability one of several classes of continuous parameter and discrete parameter stochastic composite systems are established. In all cases the objective is to analyze composite systems in terms of their lower order subsystems and in terms of their interconnecting structure. The results are applied to three specific examples.

Stability and trajectory behavior of composite systems

New results concerned with the Lyapunov stability and with estimates of trajectory behavior and bounds of composite or interconnected systems, described by ordinary differential equations, are established. In all cases the objective is the same: to analyze composite systems in terms of their lower order subsystems and in terms of their interconnecting structure. In order to demonstrate the usefulness of the present method, four specific examples are considered.

The structure of certain triple systems

For each prime power q ≡ 7 ( mod 12 ) q \equiv 7 \pmod {12} , there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each q > 1000 q > 1000 . Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order p > 1000 p > 1000 were found to have no proper subsystems of order greater than 3. In the remaining case, q = 343 q = 343 , there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with 7 > q > 1000 7 > q > 1000 is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)

The Structure of Certain Triple Systems

For each prime power $q \equiv 7 \pmod {12}$, there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each $q < 1000$ . Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order $p < 1000$ were found to have no proper subsystems of order greater than 3. In the remaining case, $q = 343$, there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with $7 < q < 1000$ is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when q is a power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.)

Stability Analysis of Stochastic Large‐Scale Systems

AbstractSufficient conditions for the Lyapunov stability, with probability one, of several classes of (a) continuous‐parameter stochastic composite systems, and, (b) discrete‐parameter stochastic composite systems are established. In all cases the objective is the same: to analyze the stability of large‐scale composite systems in terms of their lower order subsystems and in terms of their interconnecting structure. In order to demonstrate the usefulness of the present approach, several specific examples are considered.