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Here, the authors correct the proof in the reference when explaining that the produced plateaued functions have no non-zero linear structures. Moreover, a new class of plateaued functions with the best algebraic degree is given.

For improved discrimination of a ballistic warhead from a decoy, a simple 3D feature vector (3DFV) is proposed that is composed of the fundamental frequency, the bandwidth, and the sinusoidal moment of the time–frequency image. Compared to two existing methods, the 3DFV was more efficient and gave results that were less sensitive to noise.

A new characterization of balanced rotation symmetric (*n*, *m*)-functions is presented. Based on the characterization, the nonexistence of balanced rotation symmetric (*p* ^{ r }, *m*)-functions is determined, where *p* is an odd prime and *m* ≥ 2. And there exist balanced rotation symmetric (2^{ r }, *m*)-functions for 2 ≤ *m* ≤ 2^{ r } − *r*. With the help of these results, we also prove that there exist rotation symmetric resilient (2^{ r }, *m*)-functions for 2 ≤ *m* ≤ 2^{ r } − *r* − 1.

In recent years, two new types of irreducible pentanomials, i.e. Type C.1 and Type C.2 pentanomials, and their associated generalised polynomials bases (GPBs) have been proposed to yield efficient bit-parallel multiplier architectures. The GPB squarer for Type C.1 pentanomial is also investigated previously. But no GPB squarer for Type C.2 pentanomial is given as these pentanomials are far more complicated. In this Letter, the authors give explicit GPB squarer formulae for all Type C.2 pentanomials by re-classifying these pentanomials into certain sub-groups, which is based on the parities of pentanomial parameters. As the main contribution of this Letter, the authors show that the GPB squarers for most Type C.2 pentanomials match the fastest results.

Using the linear representation of symmetric group in the structure vector of finite games as its representation space, the inside structures of several kinds of symmetric games are investigated. First of all, the symmetry, described as the action of symmetric group on payoff functions, is converted to the product of permutation matrices with structure vectors of payoff functions. Second, in the light of the linear representation of the symmetric group in structure vectors, the algebraic conditions for the ordinary, weighted, renaming and name-irrelevant symmetries are obtained as the invariance under the corresponding linear representations. The semi-tensor product of matrices is a fundamental tool in this approach.

Toeplitz matrix–vector product (TMVP) decomposition is an important approach for designing and implementing subquadratic multiplier. In this Letter, a symmetric matrix (SM), which is the sum of a symmetric TM and Hankel matrix, is proposed. Applying the symmetry property, 2-way, 3-way and *n*-way splitting methods of SMVP is presented. On the basis of 2-way splitting method, the recursive formula of SMVP is presented. Using the two cases *n* = 4 and 8, the SMVP decomposition approach has less space complexity than 2-way TMVP, TMVP block recombination and symmetric TMVP for even-type Gaussian normal basis multiplication.

The right and left con-Sylvester matrices for two polynomials are defined, and then the criteria for right and left coprimeness of two polynomials in the framework of conjugate product are given in terms of the determinant of the right and left con-Sylvester matrices, respectively. The results in this study can be viewed as a generalisation of the well-known Sylvester resultant criterion for coprimeness of two polynomials in the framework of ordinary product.

This study is concerned with the finite-horizon bounded synchronisation and state estimation for the discrete-time complex networks with missing measurements based on the local performance analysis. First, a new local description of the bounded synchronisation performance index is proposed, which considers only the synchronisation errors among neighbours. In addition, a more general sector-bounded condition is presented, where the parameter matrices are different for different node. Next, by establishing the vector dissipativity-like for the complex network dynamics, the synchronisation criterion is derived in term of the locally coupled conditions for each node. These conditions implemented in a cooperative manner can judge whether the complex network reaches synchronisation. Similarly, the existence conditions for the estimator on each node are obtained, and then the estimator parameters are designed via the recursive linear matrix inequalities. Notably, these conditions on each node by cooperation among neighbours can achieve the desirable performance index. The distinctive features of the authors' algorithms are low complexity, scalability, and distributed execution. At last, two numerical examples are utilised to verify the effectiveness and applicability of the proposed algorithms.

The positive fraction vector fitting (PFVF) is a special method to guarantee the passivity of rational models such as frequency-dependent network equivalents. It involves constraints that enforce each fraction of the rational model to be passive, which are much stricter than the original passive requirements. PFVF lacks theoretical foundation but works well in practise. This study explains the rationality of PFVF by revealing important features of rational models that the complex-pole fractions corresponding to dominant resonance peaks can be adjusted passive through a minor change. The numerical case corroborates the theoretical analysis.

Sequential order one negative exponential (SOONE) function is used to measure the sparsity of a two-dimensional (2D) signal. A 2D gradient projection (GP) method is developed to solve the SOONE function and thus the 2D-GP-SOONE algorithm is proposed. The algorithm can solve the sparse recovery of 2D signals directly. Theoretical analysis and simulation results show that the 2D-GP-SOONE algorithm has a better performance compared with the 2D smoothed L0 algorithm. Simulation results also show that the proposed algorithm has a better performance and requires less computation time than 2D iterative adaptive approach.

This appendix contains a short overview of the concept of vectors and linear transformations (dyadics) of vectors, and how these are represented in terms of their components. Transformations between different rotated coordinate systems are also reviewed as well as the corresponding transformation of the components of a dyadic. The appendix ends with a short overview of quaternions and their use to represent rotations.

In this chapter, we consider the representation of vector functions (often referred to as “vector fields”) with low-order (constant and linear) polynomial basis functions on simple cells, such as triangles or quadrilateral cells in two dimensions or tetrahedrons and bricks in three dimensions. As will soon be apparent, there are multiple ways of defining vector basis functions, and therefore the approach requires some consideration. The proper representation of a function depends on what will be done with it-do we need to compute the curl of the function, for instance? If so, the representation might be different than if we need to compute the divergence of the function. We use the term curl conforming to denote the space of vector functions that maintain first-order tangential-vector continuity throughout the domain and can be differentiated via the curl operation, without producing unbounded or generalized functions (Dirac delta functions) in the process. The term divergence conforming is used to denote the complementary space of vector functions that maintain first-order normal-vector continuity throughout the domain and can therefore be differentiated via the divergence operation. (First order or C_{0} continuity is continuity of the function itself, but not necessarily continuity of its first derivatives.) The simple low-order polynomial vector basis functions in widespread use are either curl conforming or divergence conforming; seldom we will use functions that maintain complete continuity and belong to both the curl-conforming and divergence-conforming spaces, although it is possible to define such functions.

This paper describes an efficient implementation of a form of linear semi-infinite programming (LSIP). We look at maximizing (minimizing) a linear function over a set of constraints formed by positive trigonometric polynomials. Previous studies about LSIP are formulated using semi-definite programming (SDP), this is typically done by using the Kalman Yakubovich Popov (KYP) lemma or using a trace operation involving a Grammian matrix, which can be computationally expensive. The proposed algorithm is based on simplex method that directly solves the LSIP without any parameterization. Numerical results show that the proposed LISP algorithm is significantly more efficient than existing SDP solvers using KYP lemma and Grammian matrix, in both execution time and memory.

Given a graph G = (V, E) with a set W ⊆ V of vertices, we enumerate colorings to W such that for every two enumerated colorings c and c' the corresponding colored graphs (G, c) and (G, c') are not isomorphic. This problem has an important application in the study of isomers of chemical graphs such as generation of benzen isomers from a tree-like chemical graph structure. The number of such colorings can be computed efficiently based on Polya's theorem. However, enumerating each from the set of these colorings without using a large space is a challenging problem in general. In this paper, we propose a method for enumerating these colorings when the automorphisms of G are determined by two axial symmetries, and show that our algorithm can be implemented to run in polynomial delay and polynomial space.

Word search is a classical puzzle to search for all given words on a given assignment of letters to a rectangular grid (matrix). This problem is clearly in P. The inverse of this problem is more difficult, which asks to assign letters in a given alphabet to a matrix of given size so that every word in a given wordset can be found horizontally, vertically, or diagonally. This problem is in NP; it admits a trivial polynomial-size certificate. We prove its NP-hardness. It turns out to be so even under the following restrictions: 1) the alphabet size is 2 (binary) and 2) all the words to be found are of length at most 2. These results are optimal in the sense that decreasing these bounds 2 to 1 makes the problem be trivially in P.

Polynomial parahermitian matrices can accurately and elegantly capture the space-time covariance in broadband array problems. To factorise such matrices, a number of polynomial EVD (PEVD) algorithms have been suggested. At every step, these algorithms move various amounts of off-diagonal energy onto the diagonal, to eventually reach an approximate diagonalisation. In practical experiments, we have found that the relative performance of these algorithms depends quite significantly on the type of parahermitian matrix that is to be factorised. This paper aims to explore this performance space, and to provide some insight into the characteristics of PEVD algorithms.

This study deals with the design of a robust fault estimation and fault-tolerant control for vehicle lateral dynamics subject to external disturbance and unknown sensor faults. Firstly, a descriptor state and fault observer is designed to achieve the system state and sensor fault estimates simultaneously. Secondly, based on the information of on-line fault estimates, a robust fault-tolerant controller based on static output-feedback controller (SOFC) design approach is developed. To provide linear matrix inequalities of less conservatism, the results are conducted in the non-quadratic framework dealing with unmeasurable premise variables case. Simulation results show the effectiveness of the proposed control approach when the vehicle road adhesion conditions change and the sideslip angle is unavailable for measurement.

This study considers the pinning synchronisation in a network of coupled Lur’e dynamical systems under directed topology. By using tools from M-matrix theory, S-procedure and Lyapunov functional method, some simple pinning criteria in terms of linear matrix inequalities, whose dimensions are just determined by the size of a single Lur’e node, are derived for Lur’e networks with fixed and designed inner coupling matrices, respectively. A selective pinning scheme is proposed for a directed Lur’e network such that the network can be globally asymptotically pinned to a homogeneous state. Simulation results are provided to illustrate the effectiveness of the theoretical analysis.

We consider the computation of r-th roots in finite field-s. For the computation of square roots, there are two typical probabilistic methods: the Tonelli-Shanks method and the Cipolla-Lehmer method. The former method can be extended to the case of r-th roots, which is called the Adleman-Manders-Miller(AMM) method. The latter method had been generalized to the case of r-th roots with r prime. In this paper, we extend the Cipolla-Lehmer to the case of r-th root with r prime power and give the expected running time of our algorithm.

An approach for eigenvalue assignment in linear descriptor systems via output feedback is proposed. Sufficient conditions in order that a given set of eigenvalues is assignable are established. Parametric form of the desired output feedback gain matrix is also given. The approach assigns the full number of generalised eigenvalues, guarantees the closed-loop regularity and overcomes the defects of some previous works.

This study presents new sufficient conditions for Hurwitz and Schur stability of interval matrices. Tight bounds for the spectrum of interval matrices are estimated using computationally simple optimisation problems. The conservativeness is reduced further by application of ordinary similarity transformation. A necessary and sufficient vertex based criterion for the stability of a subclass of interval systems in continuous and discrete-time cases is also proposed. This enables the spectra for this class of interval systems to be determined exactly. A selection of various examples adopted from existing literature is used to demonstrate the utility of the proposed criteria.

Presented is a square root algorithm in _{q} which generalises Atkins's square root algorithm [see reference 6] for *q* ≡ 5 (mod 8) and Müller's algorithm [see reference 7] for *q* ≡ 9 (mod 16). The presented algorithm precomputes a primitive 2^{s}-th root of unity ξ where ** s** is the largest positive integer satisfying 2

_{s}|

*q*− 1, and is applicable for the cases when

*s*is small. The proposed algorithm requires one exponentiation for square root computation and is favourably compared with the algorithms of Atkin, Müuller and Kong

*et al.*

In the textile industry, the classfication of woven fabric is usually manual. To improve work efficiency, this paper purposes a novel approach to extract image features for woven fabric's recognition automatically. Firstly, the local binary pattern method and the gray level co-occurrence matrix are adopted to compute the fabric image features. Then, the principal component analysis is utilized to reduce the high dimensional feature data. Lastly, a support vector machine is used as a classifier to recognize the woven fabric type. The experiments show that these methods can automatically and accurately classify the plain weave fabrics, twill weave fabrics and satin weave fabrics.

This paper is concerned with the inverse eigenvalue problems for symmetric Toeplitz matrices. A kind of inverse problem for constructing a real symmetric Toeplitz matrix from the given k eigenpairs is proposed. By using the special structure of symmetric Toeplitz matrices, the Kronecker product and the vec operator of matrices, the problem is transformed into the system of linear equations. Some necessary and sufficient conditions for the solvability of the problem are given. The general solutions of the problem are presented.

This study presents a mathematical method to calculate the generating function of the specialized mechanisms in type synthesis with design constraints. First, the link permutation group is derived from a candidate kinematic chain according to the combinatorial theory. Next, according to the types of design constraints, joint, path, and adjacent (or non-adjacent) pair groups are derived from the link group respectively. Then, the generating function of these groups can be calculated mathematically based on four well defined operators and basic groups. The generating function is a polynomial to express the results of specialized mechanisms subjected to a specific design constraint. The benefit of the method is that the results of the type synthesis can be calculated directly by mathematical manner without any inspection of isomorphism.

The problem of norm invariance property of two-spin half systems is discussed in this paper. Based on the basis of tensor space, the Liouville-von Neumann equation of two-spin half systems is transformed into the coordinate differential equation by using matrices of adjoint operators of two-spin half systems. Then, norm invariance property of trajectory of the transformed coordinate differential system is established.

The study paper deals with sensor fault detection problem for a class of linear uncertain systems with bounded disturbances and non-zero constant reference inputs. The sensor faults are modelled as multi-mode, named as fault-free mode and faulty modes. A steady-state-based approach is proposed to detect the sensor faults. The steady-state-based approach can be used to detect lock-in-place sensor faults with arbitrary small magnitudes, which has not been well investigated in the literature. A convergent iterative algorithm based on linear matrix inequalities is given to obtain the solutions. A numerical example is given to illustrate the effectiveness of the proposed methods.

This part reviews complex-variable algebra.

Transient conditions such as switching operations and faults cause the generation of travelling waves on power transmission lines. Spatial current and voltage distributions on the transmission line are computed in the time domain using the state-space technique. The state-space representations of the transmission lines for short- and open-circuit faults and for various types of terminations are given. The results obtained by the method are compared with the solutions obtained in the frequency domain.

Although the full radix-4 CORDIC algorithm is efficient compared to the standard radix-2 version, the scale-factor overhead causes its improvement to be limited. In this work, an algorithm and its associated architecture have been proposed for parallel compensation of the scale factor for the radix-4 CORDIC algorithm in the rotation mode. The proposed method, which makes no prior assumptions about the elementary angles of rotation, reduces the latency from *n* to (*n*/2)+3, where *n* is the precision length in bits, at the cost of a reasonable increase in hardware complexity. The architecture presented relates to the redundant signed-digit number system. The architecture has been modelled in VHDL and simulated to establish its functional validity.

The speed of simulation of power system dynamics has been one of the topics most concerned with on-line security assessment. This paper proposes a new method for forming the constant Jacobian matrix for enhancing the speed of system simulation. By using the constant Jacobian matrix approach in system simulations, both for post-fault and fault-on duration, simulation efficiency is greatly enhanced. Techniques for speeding up the constant Jacobian matrix approach are discussed. Theoretical analysis is presented to demonstrate how the constant Jacobian matrix approach can be extended to a power system with generator controls. Simulation results with the constant Jacobian matrix approach in the 10-generator New England test system and the North China power system are compared with the results obtained by the use of commercial software BPA.

Various ray-tracing methods have been developed for wireless propagation predication. Most of them are hybrid 2D and 3D models (see Athanasiadou, G.E. and Nix, A.R., IEEE Trans. Veh. Technol., vol.49, no.4, p.1152-68, 2000; Liang, G. and Bertoni, H.L., IEEE Trans. Antennas Propagat., vol.46, no.6, p.853-63, 1998). They assume the walls are vertical, roofs and ceilings are horizontal and the ground is flat. These assumptions are not always true. This paper presents a new 3D ray-tracing method based on 3D geometry and vector calculations. Propagation path concepts of triangular reflection pyramid ray-tubes and diffraction hollow cones have been developed. This method applies to different terrains and both indoor and outdoor environments. Virtual reality (VR) is used to visualise the environments and line-of-sight (LOS) and non-LOS signal paths and allows us to verify the methods we used.

Obtaining the steady-state operation of a power electronic device by means of brute force computer simulation is not feasible in many practical cases. Fast steady-state algorithms that formulate the steady-state problem as a boundary problem and solve it using Newton's method have been proposed to overcome this difficulty. These algorithms are known as shooting algorithms. An extension of the shooting algorithm for piecewise linear circuits is provided. The complete Jacobian matrix that takes into account the switching instants variation is analytically derived for a state variable formulation of the steady-state problem of a piecewise linear circuit. A computer program PWiseSS based on this algorithm is used to solve a previously proposed test circuit of difficult convergence as well as to solve a realistic six-pulse converter of interest to the power electronics engineer.

A direct method for frequency stability assessment of power systems is proposed in this paper, which is based on the latest Jacobian matrix of the Newton-Raphson load flow calculation. The method can directly calculate the final system frequency after the last switching without step-by-step integral, and the system frequency stability can be determined. The proposed method has been applied to the New England 68-bus system, and the results show that the method is of good accuracy and the fine prospect of online application.

In many applications neural networks use temporal information, i.e. any kind of information related to a time series. Temporal information can be either represented within a network using a dynamic network paradigm or embodied in the input features of a network. The paper presents two methods for an explicit use of temporal information in input features. A least-squares approximation of signals with orthogonal polynomials is used to infer information about trends in a signal (average, increase, curvature, etc.). Input information about the length of a time series up to a certain point in time may act as a decreasing threshold making the network more and more sensible to changes in other input features. The advantages of the two methods are demonstrated by means of a real-world application example, tool wear monitoring in turning.

Often in process control, it is necessary to use extra measurements, or so-called secondary outputs for monitoring processes. Traditionally, the selection of these outputs is based on criteria which need to be calculated scheme-by-scheme resulting in a combinatorial problem. In this work, it is shown that these secondary outputs can be selected using an efficient measure, the output effectiveness (OE) when the outputs are scaled in a special way. This method does not need any scheme-by-scheme calculation so that the combinatorial problem is avoided. Since the OE measure is dependent on output scaling, its sensitivity to this scaling is also discussed. A scaling sensitivity matrix is derived for the designer to quickly determine the effect of different choices of scaling factors on the OE measure. This information can then be used for efficient output selection. This is demonstrated on a distillation example.

In this paper, a new fault isolation method is proposed. This method is based on the inverse sensitivity analysis. The idea is to identify input command and parameter perturbations using the information provided by the output trajectory deviation and the sensitivity matrices. The estimated perturbation is then used for fault isolation and possibly system reconfiguration. The proposed fault isolation method has been applied to a simulated aircraft control system and the results confirm the effectiveness of this sensitivity analysis approach.

Local model (LM) networks are applied to the identification of the global nonlinear dynamics of a turbogenerator excitation loop. A hybrid algorithm is used to optimise the learning process. Extensive use of plant `a priori' information is used to form an initial estimate for the nonlinear interpolation regions. The resulting model was found to describe the behaviour of the nonlinear generator-exciter system over its entire operating range.

The problem of quality control of radiological systems is discussed in relation to the management and quantitative tools normally applied to it. An alternative approach is to consider the problem in terms of a control loop in which the sets of quality control measurements are compared with their previous values in the context of a numerical model of the system to produce quantitative corrections. A preliminary study of the method applied to mammographic quality control is discussed. By modelling the system with an appropriate set of simultaneous linear equations relating changes in the controlled parameters to the quality control measurements obtained, it is possible to identify quantitative changes in the controlled parameters from measured quality control data using the inverse of the model in matrix form. This method essentially encapsulates knowledge about the system in the form of the linear model. This is distinct from current work attempting to apply artificial intelligence knowledge-based techniques to the problem. (4 pages)

Sparse-matrix solution is a dominant part of execution time in simulating VLSI circuits by a detailed simulation program such as SPICE. The paper develops a parallel-block partitionable sparse-matrix-solution algorithm which exploits sparsity at the matrix block level as well as within a nonzero block. An efficient mapping scheme to assign different matrix blocks to processors is developed which maximises concurrency and minimises communication between processors. Associated reordering and efficient sparse storage schemes are also developed. Implementation of this parallel algorithm is carried out on a transputer processor array which plugs into a PC bus. The sparse matrix solver is tested on matrices generated from a transistor-level expansion of ISCAS-85 benchmark logic circuits. Good acceleration is obtained for all benchmark matrices up to the number of transputers available.

This paper outlines the method of obtaining the frequency response functions of a rotor system with strong gyroscopic effect and simulating it using MATLAB. The rotating effect causes unsymmetry in the system matrices, resulting in complexity in decoupling the mathematical models of the system for the purpose of modal analysis. A different method is therefore required, which can handle general system matrices rather than symmetrical matrices, which is normal for passive structures. In this paper the mathematical model of an overhung rotor system with two degrees of freedom is expressed. This model is then used to extract the right and left eigenvalues/vectors and subsequently the frequency response functions are extracted and simulated. MATLAB is used to carry out such simulation, since it has good capability for eigen analysis and also good graphical facility. The reflection of splitting of critical speeds of a rotating rotor on the frequency response functions are shown.

This paper presents a new parallel-in-space algorithm for power system transient stability simulations. The nonlinear differential equations are discretized by applying the trapezoidal rule and solved together with the nonlinear algebraic equations for each time step. A network partitioning scheme, which is based on the subdivision of the factorization path tree of the network matrix, is proposed to exploit the parallelism-in-space of the transient stability problem. The parallel version of the very dishonest Newton (VDHN) method, in which the parallel algorithm for solving large sparse network matrix equations is incorporated, is developed and tested on a distributed memory message passing multicomputer. Test results on a sample power system are presented to show the performance of the proposed algorithm.

Geometrical symmetry often occurs in computational electromagnetics. However, it is generally not taken into account when the excitations do not share this symmetry. A rationale is required, and group theory gives the only valuable tool for this purpose. It provides a symmetric decomposition of any problem that allows its study on a reduced part of the initial geometry. This way of treating field problems generally leads to substantial computational savings. The symmetry concept is applied in the paper to 3-D linear eddy-current analysis, where the numerical scheme used is a mixed FEM–BEM method. Detailed examples are presented to demonstrate the efficiency of the use of symmetry.

This paper presents the method used to derive the oscillation condition by using symbolic calculus. The program is based on the full nonlinear Barkhausen criterion method. The behaviour of an oscillator is described by a complex polynomial called the characteristic polynomial. This polynomial enables us to calculate the steady state features of the oscillation as well as the differential equation for transient analysis in the time domain. The literal determination of this characteristic polynomial involves lengthy algebraic calculations and cannot be done by hand as the electronic oscillator circuit involves too many components. We recently developed a formal calculus program allowing to automatically obtain all necessary equations for oscillation analysis. We propose new methods to calculate them in an optimal form.

This paper reports the use of a general technique to combine several different methods to solve complex systems of algebraic equations in the context of load flow calculations of electrical power networks. Such a combinations of methods, referred to as 'team algorithms', seem specially well suited to be used with distributed memory computer systems, in an asynchronous environment. Experimental results solving example problems in a commercially available parallel computer system show that a 'synergetic effect' with considerable speedup can be obtained using these 'team algorithms'.

The paper is devoted to the development of a neural network architecture which implements the Newton-Raphson algorithm for solving the set of nonlinear equations of power-system load-flow analysis. The principal context is that of online network analysis in energy management systems with particular reference to the optimal power-flow function. The author shows that the complete Newton-Raphson load-flow formulation maps into an array of two-layer neural networks. The development starts from a formulation for solving as a minimisation problem the linearised equation system to which the Newton-Raphson sequence leads at each iteration. For that purpose, an objective function in quadratic form is derived. A neural network structure is given which implements the steepest descent method for minimising this objective function. It is shown that the weighting coefficients of neural networks are formed from element values in the Jacobian matrix of Newton-Raphson load-flow analysis. When the Jacobian matrix is nonsingular, the quadratic objective function derived has a unique and global minimum. A principal feature of the extensive parallel processing capability of the architecture developed is that the computing time of load-flow analysis is independent of the number of nodes in the power network for which analysis is carried out. For a sample section of a power network, and by software simulation, the architecture which the paper seeks to report gives solutions which are identical with those from a standard sequential processor load-flow program.

The authors present the application of the technique of least absolute value (LAV) dynamic filtering to optimal tracking of power system harmonics. The proposed technique uses digitised samples of the voltage and current waveform at a power system bus, where a harmonics standard is contemplated. The proposed technique can easily handle time-varying harmonic parameters. Two models are developed and tested. In the first, the measurements matrix varies with time, with an identity transition matrix. In the second model, the state transition matrix is a function of the sampling rate and the number of harmonics chosen. The algorithm is tested using a simulated and actual recorded data set. A sample set of results is reported.

Voltage collapse is associated with a stress condition of power systems. Control actions must provide the desired results, otherwise a system may operate in an unknown condition. It has been shown that this unknown condition is associated with two regions of operation and the boundary between them. The boundary between the two regions is related to a singular load-flow Jacobian. To identify the critical bus, a reduction of the load-flow Jacobian in relation to each load bus is developed. To reduce the computational burden associated with large power networks, a network partitioning is proposed which is based on voltage variation at each load bus in relation to load variation at the other load busses. Comparison between the proposed method and network partitioning using Sanchis' method is presented. The weak area of a power system is identified in the network partitioning; a weak area is defined as the area that contains the critical bus of the power network. To calculate the margins for each load bus of the weak area, the relation between load variation at each load bus and voltage magnitude and phase angle variations at the critical bus is normalised, one by one. The busses strongly connected to the critical bus have smaller load variation in relation to the busses weakly connected to the critical bus. The proposed method has been tested using the IEEE 24-bus and 300-bus systems.

This paper develops an economic dispatch algorithm for the determination of the global or near global optimum dispatch solution. The algorithm is based on the simulated annealing technique. In the algorithm, the load balance constraint and the operating limit constraints of the generators are fully accounted for. In the development of the algorithm, transmission losses are first discounted and they are subsequently incorporated in the algorithm through the use of the B-matrix loss formula. The algorithm is demonstrated by its application to a test system. The results determined by the new algorithm are compared to those found by dynamic programming with a zoom feature.