arXiv subjects

Kaihua Xi

Publications and source records attributed to Kaihua Xi.

11 recordsLinked to original sources

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

A Critical Escape Probability Formulation for Enhancing the Transient Stability of Power Systems with System Parameter Design

For the enhancement of the transient stability of power systems, the key is to define a quantitative optimization formulation with system parameters as decision variables. In this paper, we model the disturbances by Gaussian noise and define a metric named Critical Escape Probability (CREP) based on the invariant probability measure of a linearised stochastic processes. CREP characterizes the probability of the state escaping from a critical set. CREP involves all the system parameters and reflects the size of the basin of attraction of the nonlinear systems. An optimization framework that minimizes CREP with the system parameters as decision variablesis is presented. Simulations show that the mean first hitting time when the state hits the boundary of the critical set, that is often used to describe the stability of nonlinear systems, is dramatically increased by minimizing CREP. This indicates that the transient stability of the system is effectively enhanced. It also shown that suppressing the state fluctuations only is insufficient for enhancing the transient stability. In addition, the famous Braess' paradox which also exists in power systems is revisited. Surprisingly, it turned out that the paradoxes identified by the traditional metric may not exist according to CREP. This new metric opens a new avenue for the transient stability analysis of future power systems integrated with large amounts of renewable energy.

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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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arXiv2023

Explicit formulas for the Variance of the State of a Linearized Power System driven by Gaussian stochastic disturbances

We look into the fluctuations caused by disturbances in power systems. In the linearized system of the power systems, the disturbance is modeled by a Brownian motion process, and the fluctuations are described by the covariance matrix of the associated stochastic process at the invariant probability distribution. We derive explicit formulas for the covariance matrix for the system with a uniform damping-inertia ratio. The variance of the frequency at the node with the disturbance is significantly bigger than the sum of those at all the other nodes, indicating the disturbance effects the node most, according to research on the variances in complete graphs and star graphs. Additionally, it is shown that adding new nodes typically does not aid in reducing the variations at the disturbance's source node. Finally, it is shown by the explicit formulas that the line capacity affect the variation of the frequency and the inertia affects the variance of the phase differences.

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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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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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arXiv2016

Synchronization of cyclic power grids: equilibria and stability of the synchronous state

Synchronization is essential for proper functioning of the power grid. We investigate the synchronous state and its stability for a network with a cyclic topology and with the evolution of the states satisfying the swing equations. We calculate the number of stable equilibria and investigate both the linear and nonlinear stability of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. The nonlinear stability, which we calculated by comparing the potential energy of the type-1 saddles with that of the stable synchronous state, depends on the network size ($N$) in a more complicated fashion. In particular we find that when the generators and consumers are evenly distributed in an alternating way, the energy barrier, preventing loss of synchronization approaches a constant value. For a heterogeneous distribution of generators and consumers, the energy barrier will decrease with $N$. The more heterogeneous the distribution is, the stronger the energy barrier depends on $N$. Finally, we found that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of $N\to\infty$. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.

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