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 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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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

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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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.

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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.

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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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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.

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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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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.

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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 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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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.

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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.

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Several examples of engineering control problems are described for which control of stochastic systems has been developed. Examples treated include control of a mooring tanker, control of freeway traffic flow, and control of shock absorbers. A list of additional control problems is provided.

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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 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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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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