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.

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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.@en

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.

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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.

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Concepts and results of probability theory are presented in this chapter which complement those of Chapter 2. Concepts covered in detail include the canonical variable decomposition of a tuple of Gaussian random variables, a set of stable probability distribution functions, conditional probability, conditional Gaussian random variables, the conditional independence relation, a measure transformation, the P-essential infimum, and metrics on the set of probability measures.

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The concept of a dissipative system is defined for a deterministic linear system and is satisfied if there exists a storage function and a supply rate such that the dissipation inequality holds. It is proven that a deterministic linear system is dissipative if and only if a related function is a positive-definite function. Any covariance function of a stochastic process is a positive-definite function. An equivalence condition is proven for a deterministic linear system to be dissipative in terms of an algebraic condition, a matrix has to satisfy a linear matrix inequality.

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Several sets of stochastic systems are defined in this chapter. The sets are selected based on the sets in which the outputs take values. Conditions are provided for the selection of the output-state conditional distribution function and for the selection of the conditional distribution function of the next-state on the current state. Useful are a Poisson–Gamma stochastic system, a Beta–Gamma stochastic system, an output-finite–state-polytopic stochastic system, a σ -algebraic system, and a multiple conditional independent relation.

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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.

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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.

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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.

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This book helps students, researchers, and practicing engineers to understand the theoretical framework of control and system theory for discrete-time stochastic systems so that they can then apply its principles to their own stochastic control systems and to the solution of control, filtering, and realization problems for such systems. Applications of the theory in the book include the control of ships, shock absorbers, traffic and communications networks, and power systems with fluctuating power flows.
The focus of the book is a stochastic control system defined for a spectrum of probability distributions including Bernoulli, finite, Poisson, beta, gamma, and Gaussian distributions. The concepts of observability and controllability of a stochastic control system are defined and characterized. Each output process considered is, with respect to conditions, represented by a stochastic system called a stochastic realization. The existence of a control law is related to stochastic controllability while the existence of a filter system is related to stochastic observability. Stochastic control with partial observations is based on the existence of a stochastic realization of the filtration of the observed process.​@en

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.

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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.

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A stochastic control problem is to determine a control law within a rather general set of control laws such that the closed-loop system meets prespecified control objectives. A stochastic control problem is motivated by control problem of engineering, economics, or other areas of the sciences. The concepts of an information pattern, a control law, and of a closed-loop stochastic system are defined. The main control objectives are stability, optimization of performance, and robustness in case of changes in the control system. Control synthesis at a theoretic level is described. Statistical decision problems are treated as elementary stochastic control problems in which there is no dynamics.

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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.

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The Lyapunov equation and the algebraic Riccati equation are treated in depth. The Lyapunov equation arises as the equation for the asymptotic covariance matrix of the state of a stationary Gaussian system. The algebraic Riccati equation arises in the Kalman filter, in stochastic control, and in stochastic realization of a Gaussian system. Results for both equations are provided on: the existence of a solution, uniqueness with respect to conditions, a description of the set of all solutions if applicable, particular properties of solutions, and on the computation of solutions.

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The filter problem is to derive an expression for the conditional distribution of the state of a stochastic system conditioned on the past outputs of the considered system and a recursion of the parameters of that conditional distribution. In this chapter the filter problem for a Gaussian stochastic system is solved. The Kalman filter is derived, including the time-varying, the time-invariant Kalman filter, and the conditional Kalman filter. The relation of the Kalman filter with stochastic realization is discussed. The Kalman filter is used in a very large number of research areas including signal processing, information theory, communication theory, weather prediction, hydrology, etc.

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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.

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Optimal stochastic control problems are considered for a time-invariant stochastic control system with partial observations on an infinite horizon. Such problems can be solved by a dynamic programming method for partial observations. Both the average cost and the discounted cost functions are considered. Treated as special cases are optimal control problems for a Gaussian stochastic control system and for a finite stochastic control system.

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