arXiv subjects

Jan H. van Schuppen

Publications and source records attributed to Jan H. van Schuppen.

At least 19 recordsLinked to original sources

arXiv2026

Control and Stability of a Multilevel Power System for a Future Distribution Network

The growing integration of renewable energy sources into distribution networks poses significant challenges to frequency and voltage stability due to their intermittent nature and low-inertia dynamics. This paper proposes a multilevel control framework for a future decarbonized power system, using energy storage systems as power buffers to mitigate frequency and voltage fluctuations. A nonlinear interconnected model is formulated to characterize the complex dynamics across multiple levels of the distribution network. To reduce operational complexity and communication overhead of these dynamics, a distributed linear quadratic regulator control strategy is developed for information exchange in a bottom-up approach, where each level implements local feedback control within a short time horizon. Stability conditions for both open-loop and closed-loop systems are established using Lyapunov-based analysis. In addition, explicit performance bounds are derived to quantify the optimal difference between the proposed distributed strategy and the centralized control method, demonstrating the effectiveness of the proposed framework.

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arXiv2024

Control of the Power Flows of a Stochastic Power System

How to determine the vector of power supplies of a stochastic power system for the next short horizon, such that the probability is less than a prespecified value that any phase-angle difference of a power line of the power network exits from a safe set? The power system is modelled such that the differential equation of each frequency is affected by a Brownian motion process. A safe set can be selected to be any subset of the interval $(-π/2, ~ +π/2)$, which is a sufficient condition for not losing synchronization. That the controlled system has an improved performance is shown by numerical results of three academic examples including a particular eight-node academic network, a twelve-node ring network, and a Manhattan-grid network.

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arXiv2024

Optimal Control of a Stochastic Power System -- Algorithms and Mathematical Analysis

The considered optimal control problem of a stochastic power system, is to select the set of power supply vectors which infimizes the probability that the phase-angle differences of any power flow of the network, endangers the transient stability of the power system by leaving a critical subset. The set of control laws is restricted to be a periodically recomputed set of fixed power supply vectors based on predictions of power demand for the next short horizon. Neither state feedback nor output feedback is used. The associated control objective function is Lipschitz continuous, nondifferentiable, and nonconvex. The results of the paper include that a minimum exists in the value range of the control objective function. Furthermore, it includes a two-step procedure to compute an approximate minimizer based on two key methods: (1) a projected generalized subgradient method for computing an initial vector, and (2) a steepest descent method for approximating a local minimizer. Finally, it includes two convergence theorems that an approximation sequence converges to a local minimum.

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arXiv2023

Synchronization of Power Systems under Stochastic Disturbances

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.

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arXiv2023

Synchronization of Coupled Phase Oscillators with Stochastic Disturbances and the Cycle Space of the Graph

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 matrix 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 are evaluated. It is proven that adding a new line or increasing the coupling strength of a line affect 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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arXiv2023

Increasing the Synchronization Stability in Complex Networks

We aim to increase the ability of a of coupled phase oscillators to maintain the 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 the attraction, to measure the 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 the synchronization stability. In this method, a new metric for the 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 the disturbances. Furthermore, by this new metric, one may identify those edges which may lead to desynchronization with a high risk. A case study shows that the mean first hitting time is dramatically increased after solving the corresponding optimization problems and the vulnerable edges are effectively identified. It is also found that optimizing the 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 the synchronization stability.

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arXiv2022

Characterization of the Gray-Wyner Rate Region for Multivariate Gaussian Sources: Optimality of Gaussian Auxiliary RV

Examined in this paper, is the Gray and Wyner achievable lossy rate region for a tuple of correlated multivariate Gaussian random variables (RVs) $X_1 : Ω\rightarrow {\mathbb R}^{p_1}$ and $X_2 : Ω\rightarrow {\mathbb R}^{p_2}$ with respect to square-error distortions at the two decoders. It is shown that among all joint distributions induced by a triple of RVs $(X_1,X_2, W)$, such that $W : Ω\rightarrow {\mathbb W} $ is the auxiliary RV taking continuous, countable, or finite values, the Gray and Wyner achievable rate region is characterized by jointly Gaussian RVs $(X_1,X_2, W)$ such that $W $ is an $n$-dimensional Gaussian RV. It then follows that the achievable rate region is parametrized by the three conditional covariances $Q_{X_1,X_2|W}, Q_{X_1|W}, Q_{X_2|W}$ of the jointly Gaussian RVs. Furthermore, if the RV $W$ makes $X_1$ and $X_2$ conditionally independent, then the corresponding subset of the achievable rate region, is simpler, and parametrized by only the two conditional covariances $Q_{X_1|W}, Q_{X_2|W}$. The paper also includes the characterization of the Pangloss plane of the Gray-Wyner rate region along with the characterizations of the corresponding rate distortion functions, their test-channel distributions, and structural properties of the realizations which induce these distributions.

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arXiv2021

Transient Performance of Power Systems with Distributed Power-Imbalance Allocation Control

We investigate the sensitivity of the transient performance of power systems controlled by Distributed Power-Imbalance Allocation Control (DPIAC) on the parameters of the control law. We model the disturbances at power loads as Gaussian white noises and measure the transient performance of the frequency deviation and control cost by the H_2 norm. For a power system with a communication network of the same topology as the power network, analysis shows that the transient performance of the frequency can be greatly improved by accelerating the convergence of the frequencies to the nominal value through a singe control gain coefficient. However, the control cost increases linearly as this control gain coefficient increases. Hence, in the feedback control law, DPIAC, there is a trade-off between the frequency deviation and control cost which is determined by this control gain coefficient. By increasing another control gain coefficient, the control cost can be decreased with an accelerated consensus of the marginal costs during the transient phase. Furthermore, the behavior of the state approaches that of a centralized control law when the consensus of the marginal costs is accelerated.

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arXiv2021

A New Approach to Lossy Network Compression of a Tuple of Correlated Multivariate Gaussian RVs

The classical Gray and Wyner source coding for a simple network for sources that generate a tuple of multivariate, correlated Gaussian random variables $(Y_1,Y_2)$ is re-examined using the geometric approach of Gaussian random variables, and the weak stochastic realization of correlated Gaussian random variables. New results are: (1) The formulation, methods and algorithms to parametrize all random variables $W : Ω\rightarrow {\mathbb R}^n $ which make the two components of the tuple $(Y_1,Y_2)$ conditionally independent, according to the weak stochastic realization of $(Y_1, Y_2)$. (2) The transformation of random variables $(Y_1,Y_2)$ via non-singular transformations $(S_1,S_2)$, into their canonical variable form. (3) A formula for Wyner's lossy common information for joint decoding with mean-square error distortions. (4) The methods are shown to be of fundamental importance to the parametrization of the lossy rate region of the Gray and Wyner source coding problem, and the calculation of the smallest common message rate $R_0$ on the Gray and Wyner source problem, when the sum rate $R_0+R_1+R_2$ is arbitrary close to the joint rate distortion function $R_{Y_1, Y_2}(Δ_1, Δ_2)$ of joint decoding. The methods and algorithms may be applicable to other problems of multi-user communication, such as, the multiple access channel, etc. The discussion is largely self-contained and proceeds from first principles; basic concepts of weak stochastic realization theory of multivariate correlated Gaussian random variables are reviewed, while certain results are developed to meet the requirement of results (1)-(4).

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arXiv2020

Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks

The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : Ω\rightarrow {\mathbb R}^{p_1}$ and $X_2 : Ω\rightarrow {\mathbb R}^{p_2}$, with respect to square-error distortion at the two decoders is re-examined using (1) Hotelling's geometric approach of Gaussian RVs-the canonical variable form, and (2) van Putten's and van Schuppen's parametrization of joint distributions ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : Ω\rightarrow {\mathbb R}^n $ which make $(X_1,X_2)$ conditionally independent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions ${\bf E}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq Δ_i \in [0,\infty], i=1,2$, by the covariance matrix of RV $W$. From this then follows Wyner's common information $C_W(X_1,X_2)$ (information definition) is achieved by $W$ with identity covariance matrix, while a formula for Wyner's lossy common information (operational definition) is derived, given by $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = \frac{1}{2} \sum_{j=1}^n \ln \left( \frac{1+d_j}{1-d_j} \right),$ for the distortion region $ 0\leq Δ_1 \leq \sum_{j=1}^n(1-d_j)$, $0\leq Δ_2 \leq \sum_{j=1}^n(1-d_j)$, and where $1 > d_1 \geq d_2 \geq \ldots \geq d_n>0$ in $(0,1)$ are {\em the canonical correlation coefficients} computed from the canonical variable form of the tuple $(X_1, X_2)$. The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.

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arXiv2019

A Uniform Approach to Maximal Permissiveness in Modular Control of Discrete-Event Systems

In this paper, a uniform approach to maximal permissiveness in modular control of discrete-event systems is proposed. It is based on three important concepts of modular closed-loops: monotonicity, distributivity, and exchangeability. Monotonicity of various closed-loops satisfying a given property considered in this paper holds whenever the underlying property is preserved under language unions. Distributivity holds if the inverse projections of local plants satisfy the given property with respect to each other. Among new results, sufficient conditions are proposed for distributed computation of supremal relatively observable sublanguages.

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arXiv2018

Power-Imbalance Allocation Control of Power Systems-Secondary Frequency Control

The traditional secondary frequency control of power systems restores nominal frequency by steering Area Control Errors (ACEs) to zero. Existing methods are a form of integral control with the characteristic that large control gain coefficients introduce an overshoot and small ones result in a slow convergence to a steady state. In order to deal with the large frequency deviation problem, which is the main concern of the power system integrated with a large number of renewable energy, a faster convergence is critical. In this paper, we propose a secondary frequency control method named Power-Imbalance Allocation Control (PIAC) to restore the nominal frequency with a minimized control cost,in which a coordinator estimates the power imbalance and dispatches the control inputs to the controllers after solving an economic power dispatch problem. The power imbalance estimation converges exponentially in PIAC, both overshoots and large frequency deviations are avoided. In addition, when PIAC is implemented in a multi-area controlled network, the controllers of an area are independent of the disturbance of the neighbor areas, which allows an asynchronous control in the multi-area network. A Lyapunov stability analysis shows that PIAC is locally asymptotically stable and simulation results illustrates that it effectively eliminates the drawback of the traditional integral control based methods.

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arXiv2018

Multi-Level Power-Imbalance Allocation Control for Secondary Frequency Control of Power Systems

A consensus-control-based multi-level control law named Multi-Level Power-Imbalance Allocation Control (MLPIAC) is presented for a large-scale power system partitioned into two or more areas. Centralized control is implemented in each area while distributed control is implemented at the coordination level of the areas. Besides restoring nominal frequency with a minimal control cost, MLPIAC can improve the transient performance of the system through an accelerated convergence of the control inputs without oscillations. At the coordination level of the control areas, because the number of the areas is smaller than that of nodes, MLPIAC is more effective to obtain the minimized control cost than the purely distributed control law. At the level of the control in each area, because the number of nodes is much smaller than the total number of nodes in the whole network, the overheads in the communications and the computations are reduced compared to the pure centralized control. The asymptotic stability of MLPIAC is proven using the Lyapunov method and the performance is evaluated through simulations.

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arXiv2018

An Observable Canonical Form for a Rational System on a Variety

An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the canonical form is defined by the condition that two equivalent systems have the same response map. A proof is provided that the defined form is well-defined canonical form. Special cases are discussed.

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arXiv2017

Controllability of Linear Positive Systems: An Alternative Formulation

An alternative formulation for the controllability problem of single input linear positive systems is presented. Driven by many industrial applications, this formulations focuses on the case where the region of interest is only a subset of positive orthant rather than the entire positive orthant. To this end, we discuss the geometry of controllable subsets and develop numerically verifiable conditions for polyhedrality of controllable subsets. Finally, we provide a method to check for controllability of a target set based on our approach.

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arXiv2016

Rational observers of rational systems

Rational observers are to be constructed for rational systems while polynomial observers are to be constructed for polynomial systems. An observer synthesis procedure is formulated. First an output-based rational realization is synthesized for the considered rational system. Then a perturbation technique creates an observer. Finite algebraic observability of the rational system impies the existence of a output-based rational realization. Several examples of rational observers are provided including a polynomial system of which the state-space dimension of the polynomial observer is strictly higher than that of the corresponding system.

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arXiv2015

An Algorithm for System Identification of a Discrete-Time Polynomial System without Inputs

A subalgebraic approximation algorithm is proposed to estimate from a set of time series the parameters of the observer representation of a discrete-time polynomial system without inputs which can generate an approximation of the observed time series. A major step of the algorithm is to construct a set of generators for the polynomial function from the past outputs to the future outputs. For this singular value decompositions and polynomial factorizations are used. An example is provided.

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arXiv2015

Combined Top-down and Bottom-up Approach to Multilevel Supervisory Control

Recently, we have proposed two complementary approaches, top-down and bottom-up, to multilevel supervisory control of discrete-event systems. In this paper, we compare and combine these approaches. The combined approach has strong features of both approaches, namely, a lower complexity of the top-down approach with the generality of the bottom-up approach. We show that, for prefix-closed languages, a posteriori supervisors computed in the bottom-up manner do not alter maximal permissiveness within the three-level coordination control architecture, that is, the supremal three-level conditionally-controllable and conditionally-normal language can always be computed in a distributed way using multilevel coordination. Moreover, a general polynomial-time procedure for non-prefix closed case is proposed based on coordinators for nonblockingness and a posteriori supervisors.

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