Tools and analysis for improving the small-signal stability of a stochastic power system by optimal power dispatch in each short time horizon, such as five-minute intervals, are provided in this paper. An objective function which characterizes the maximal exit probability from the static stability region (−π/2,π/2) across the phase-angle differences of all power lines is formulated. This objective function is proven to be Lipschitz continuous, nondifferentiable, and nonconvex, with a finite minimum defined over the region of power supply vectors. The formulas of the generalized subgradient and directional derivative of the objective function are provided, and based on these formulas, a two-step algorithm is designed to approximate a minimizer accompanied by the convergence proof: (1) using a projected generalized subgradient method to compute an effective initial vector, and (2) applying the steepest descent method to approximate a local minimizer. The algorithms have been verified using a synthesized power network, demonstrating computational validity and effectiveness in minimizing the maximal exit probability of all power lines.

Serious fluctuations caused by disturbances may lead to instability of power systems. With the disturbance modeled by a Brownian motion process, the fluctuations are often described by the asymptotic variance at the invariant probability distribution of an associated Gaussian stochastic process. Here, we derive the explicit formula of the variance matrix for the system with a uniform damping-inertia ratio at all the nodes, which enables us to analyze the influences of the system parameters on the fluctuations and investigate the fluctuation propagation in the network. With application to systems with complete graphs and star graphs, it is found that the variance of the frequency at the disturbed node is significantly bigger than those at all the other nodes. It is also shown that adding new nodes may prevent the propagation of fluctuations from the disturbed node to all the others. Finally, it is proven theoretically that larger line capacities accelerate the propagation of the frequency fluctuation and larger inertia of synchronous machines help suppress the fluctuations of the phase differences, however, these acceleration and suppression are quite limited.

The synchronization stability of a complex network system of coupled phase oscillators is discussed. In case the network is affected by disturbances, a stochastic linearized system of the coupled phase oscillators may be used to determine the fluctuations of phase differences in the lines between the nodes and to identify the vulnerable lines that may lead to desynchronization. The main result is the derivation of the asymptotic variance matrices of the phase differences which characterizes the severity of the fluctuations. It is found that the cycle space of the graph of the system plays a role in this characterization. With theory of the cycle space, the effect of forming small cycles on the fluctuations is evaluated. It is proven that adding a new line or increasing the coupling strength of a line affects the fluctuations in the lines in any cycle including this line, while it does not affect the fluctuations in the other lines. In particular, if the phase differences at the synchronous state are not changed by these actions, then the affected fluctuations reduce.

The synchronization of power generators is an important condition for the proper functioning of a power system, in which the fluctuations in frequency and the phase angle differences between the generators are sufficiently small when subjected to stochastic disturbances. Serious fluctuations can prompt desynchronization, which may lead to widespread power outages. Here, we model the stochastic disturbance by a Brownian motion process in the linearized system of the non-linear power systems and characterize the fluctuations by the variances of the frequency and the phase angle differences in the invariant probability distribution. We propose a method to calculate the variances of the frequency and the phase angle differences. For the system with uniform disturbance-damping ratio, we derive explicit formulas for the variance matrices of the frequency and the phase angle differences. It is shown that the fluctuation of the frequency at a node depends on the disturbance-damping ratio and the inertia at this node only, and the fluctuations of the phase angle differences in the lines are independent of the inertia. In particular, the synchronization stability is related to the cycle space of the network. We reveal the influences of constructing new lines and increasing capacities of lines on the fluctuations in the phase angle differences in the existing lines. The results are illustrated for the transmission system of Shandong Province of China. For the system with non-uniform disturbance-damping ratio, we further obtain bounds of the variance matrices.

We aim to increase the ability of coupled phase oscillators to maintain synchronization when the system is affected by stochastic disturbances. We model the disturbances by Gaussian noise and use the mean first hitting time when the state hits the boundary of a secure domain, that is a subset of the basin of attraction, to measure synchronization stability. Based on the invariant probability distribution of a system of phase oscillators subject to Gaussian disturbances, we propose an optimization method to increase the mean first hitting time and, thus, increase synchronization stability. In this method, a new metric for synchronization stability is defined as the probability of the state being absent from the secure domain, which reflects the impact of all the system parameters and the strength of disturbances. Furthermore, by this new metric, one may identify those edges that may lead to desynchronization with a high risk. A case study shows that the mean first hitting time is dramatically increased after solving corresponding optimization problems, and vulnerable edges are effectively identified. It is also found that optimizing synchronization by maximizing the order parameter or the phase cohesiveness may dramatically increase the value of the metric and decrease the mean first hitting time, thus decrease synchronization stability.

Filter problems are formulated for stochastic systems which are not Gaussian systems. Both the estimation problem, the sequential estimation problem, and the filter problem are treated. A sufficient condition for the existence of a finite-dimensional filter system is formulated. The concept of a family of invariant conditional distributions is defined. It is described how to solve the filter problem by a measure transformation method. Cases treated include the Poisson–Gamma filter and the filter of an output-finite–state-finite stochastic system.

Optimal stochastic control problems with complete observations and on an infinite horizon are considered. Control theory for both the average cost and the discounted cost function is treated. The dynamic programming approach is formulated as a procedure to determine the value and the value function; from the value function, one can derive the optimal control law. Stochastic controllability is in general needed to prove that there exists a control law with a finite average cost in case of positive cost. Special cases treated in depth are: the case of a Gaussian stochastic control system and of a finite stochastic control system.

Specialized topics on the theory of stochastic processes are described which are used in the body of this book. Defined are a filtration and stochastic processes relative to a filtration. Elementary martingale theory is discussed. Stopping times and a stochastic process indexed by a stopping time are defined. The supermartingale convergence theorem is established. A brief introduction to ergodicity is provided.

Concepts and results of the geometric structure of the set of state-variance matrices of a time-invariant Gaussian system are provided in this chapter. With respect to a condition, the set is convex with a minimal and a maximal element. In case the noise variance matrix satisfies a nonsingularity condition, the matrix inequality is equivalent to an inequality of Riccati type. The singular boundary matrices of the set of state variances play a particular role. Finally, the classification of all elements of the set of state variances can be described in terms of an increment above the minimal element or below the maximal element, which increments satisfy a Lyapunov equation.

The concept of a stochastic control system is defined as a map from a tuple of the current state and the current input to the conditional probability distribution of the tuple of the next state and the current output. A Gaussian stochastic control system representation is defined which represents such a stochastic system. Stochastic controllability is defined as the property that there exists an input trajectory such that the probability distribution of a future state equals a prespecified particular probability distribution. A characterization of stochastic controllability of a Gaussian stochastic control system is provided.

In stochastic control with partial observations, the control law at any time can depend only on the past outputs and the past inputs of the stochastic control system. Neither is available to the control law in the current state nor the past states. Control theory for stochastic systems with partial observations is poorly developed and poorly understood. The approach to this control problem is to first determine a stochastic realization of the stochastic control system based on the information structure and second to apply a modified dynamic programming procedure. The approach is illustrated for a Gaussian stochastic control system and for an output-finite–state-finite stochastic control system. The tracking problem is treated.

This chapter concerns positive matrices which are matrices with elements of the positive real numbers. The motivations for the inclusion of the algebraic structure of positive matrices are the problems (1) of stability of the system of probability measures of the Markov process of a finite stochastic systems and (2) of minimal positive matrix factorization motivated by the stochastic realization problem of an output-finite stochastic system. Using the concepts of similarity, of equivalence, of a prime in the positive matrices, and of an extremal cone, one can investigate a minimal positive matrix factorization. This appendix is to be considered as a reference chapter.

Concepts and theorems of the system theory of deterministic linear systems are summarized. Controllability, observability, and a realization are formulated. Realization theory includes necessary and sufficient conditions for the existence of a realization, a characterization of the minimality of a realization, and a classification and description of all minimal realizations. Attention is restricted to linear control systems after a general introduction.

A stochastic system (without input) is a mathematical model of a dynamic phenomenon exhibiting uncertain signals. Such a system is mathematically characterized by the transition map from the current state to the joint probability distribution of the next state and the current output. Only Gaussian systems are treated in this chapter. Forward and backward Gaussian stochastic systems are defined and related. Stochastic observability and stochastic co-observability are defined and characterized.

Elementary concepts and results of the theory of stochastic processes are summarized in this chapter. Concepts presented include a stochastic process, equivalent processes, a Gaussian process, stationarity, time-reversibility, and a Markov process. It is shown how to go from the definition of a Gauss–Markov process to a representation of such a process in the form of a state equation of a stochastic system. Advanced topics of the theory of stochastic processes are provided in Chapter 20.

Stochastic realization problems are presented for a tuple of Gaussian random variables, for a tuple of σ -algebras, for a σ -algebra family, and for a finite stochastic system. The solution of the weak and of the strong stochastic realization of a tuple of Gaussian random variables is provided. The main theoretical contribution is the description of the strong stochastic realization of a tuple of σ -algebras. This is followed by stochastic realization of a family of σ -algebras. Finally the stochastic realization problem for finite-valued output processes is discussed.

The reader finds in this short appendix concepts and results of various topics of mathematics. These topics are used in the body of the book but are not part of control theory. Topics covered are: algebra of set theory; a canonical form; algebraic structures including monoids, groups, and rings; linear algebra and matrices; analysis; geometry; convex sets; affine sets; and optimization.

The weak stochastic realization problem is to determine all stochastic systems whose output equals a considered output process in terms of its finite-dimensional distributions. Such a system is then said to be a stochastic realization of the considered output process. The problem encompasses: (1) an equivalent condition for the existence of a realization, (2) characterizing when such a system is a minimal stochastic realization, and (3) classifying all minimal stochastic realizations. The concepts of stochastic realization theory are the basis of filter theory and of control theory. In this chapter the stochastic realization is presented for stationary Gaussian stochastic processes. Particular concepts discussed include: covariance realization, minimality of a stochastic realization, a classification map, and a canonical form for a stochastic system.

Stochastic control issues of a general character are presented. Problems of control theory are mentioned which require research interest the coming years. A general method for sufficient and necessary conditions for the existence of an optimal control law is discussed. The framework covers arbitrary cost functions, including additive and multiplicative functions. In addition, the approach of a measure transformation is formulated and illustrated for stochastic control of a partially observed stochastic control system.

The study of control of stochastic systems requires knowledge of probability and of stochastic processes. Probability is summarized in this chapter in a way which is sufficient for studying the control and system theory of the subsequent chapters. Additional concepts and results of probability theory are provided in an appendix labeled Chapter 20. The main concepts of probability needed for the reading of this book are probability distribution functions, probability measures, random variables, Gaussian random variables, conditional expectation, and conditional independence.