arXiv subjects

Takayuki Ishizaki

Publications and source records attributed to Takayuki Ishizaki.

At least 19 recordsLinked to original sources

arXiv2026

Voltage Stability Kernel: A Cofactor Theory of Voltage Stability in Lossy Power Systems

This paper introduces the voltage stability kernel (VSK), a cofactor-based bus-wise representation of voltage stability in lossy power systems. The VSK is defined as the vector of principal cofactors of the voltage stability Laplacian (VSL), a reduced Jacobian that retains voltage source internal angles while eliminating the other variables. We show that the VSK constitutes the left kernel of the VSL, which is typically nonsymmetric in lossy power systems. We also define the voltage stability margin (VSM) as the sum of all VSK components and show that it is equal to the product of all eigenvalues of the VSL except the trivial zero eigenvalue due to phase-shift symmetry. Thus, the VSK provides a bus-wise decomposition of the VSM. Furthermore, the VSK offers an algebraic interpretation of CPF calculations with a fixed slack bus. The singularity of the Jacobian in CPF calculations obtained by deleting the slack-bus row and column is characterized by the vanishing of the VSK component selected by the slack bus. In contrast, the static bifurcation is characterized by the vanishing of the VSM. Since these two conditions are generally different, our theory explains why a CPF nose point does not necessarily correspond to a static bifurcation in lossy cases.

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arXiv2026

Hierarchical Distributed Architecture for the Least Allan Variance Atomic Timing

In this paper, we propose a hierarchical distributed timing architecture based on an ensemble of miniature atomic clocks. The goal is to ensure synchronized and accurate timing in a normal operating mode where Global Navigation Satellite System (GNSS) signals are available, as well as in an emergency operating mode during GNSS failures. At the lower level, the miniature atomic clocks employ a distributed control strategy that uses only local information to ensure synchronization in both modes. The resulting synchronized time or generated time scale has the best frequency stability, as measured by the Allan variance, over the short control period. In the upper layer, a supervisor controls the long-term behavior of the generated time scale. In the normal operating mode, the supervisor periodically anchors the generated time scale to the standard time based on GNSS signals, while in the emergency operating mode, it applies optimal floating control to reduce the divergence rate of the generated time scale, which is not observable from the measurable time difference between the miniature atomic clocks. This floating control aims to explicitly control the generated time scale to have the least Allan variance over the long control period. Finally, numerical examples are provided to show that the proposed architecture achieves an Allan variance on the order of $10^{-23}$ over averaging times ranging from one second to several days. This demonstrates its effectiveness and feasibility for high-precision, GNSS-resilient atomic timing.

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arXiv2026

Explicit Ensemble Mean Synchronization for Time Scale Generation with Mixed Atomic Clock Ensembles

In this paper, we consider a mixed ensemble containing a mixture of cesium-type and hydrogen maser-type atomic clocks. For the mixed ensemble, the conventional Kalman filtering algorithm has certain limitations due to divergence of the error covariance matrix. To overcome these limitations, we obtain a Kalman filtering algorithm based on observable canonical decomposition that does not have any diverging terms. We use the estimates from the transformed Kalman filter to propose a time scale generation algorithm called explicit ensemble mean synchronization algorithm for the mixed ensemble. In this algorithm, we synchronize the time deviation of each clock from the ideal clock behavior to the unobservable ensemble mean of the phases where the weighting can be decided by the user. By regulating the free-running dynamics associated with the unobservable state, through choosing an appropriate weight vector, the frequency stability of the generated time scale or the synchronized time shared by the clocks is optimized over shorter (resp. longer) intervals, as measured by Hadamard variance. An illustrative example is given to demonstrate the efficiency of our algorithm.

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arXiv2026

Optimal GNSS Time Tracking for Long-term Stable Time Realisation in Synchronised Atomic Clocks

In this manuscript, we propose a novel optimal Global Navigation Satellite System (GNSS) time tracking algorithm to collectively steer an ensemble consisting of synchronising miniature atomic clocks towards standard GNSS time. The synchronising miniature atomic clocks generate a common synchronised time which has good short term performance but its accuracy and precision, which is measured by Allan variance, deteriorates in the long run. So, a supervisor designs and periodically broadcasts the proposed GNSS time tracking control to the ensemble miniature atomic clocks that steer the average of ensemble towards the average of GNSS receivers, which are receivers of GNSS time. The tracking control is constructed using a Kalman filter estimation process that estimates the difference in average of GNSS receivers and average of ensemble clocks by using relative clock readings between GNSS receivers and their adjacent ensemble clock. Under the influence of the periodically received tracking control, the stabilised ensemble clocks have better long term accuracy and precision over long averaging periods. Since the tracking control is designed to solely influence the average of the ensemble, the tracking process does not interfere with the synchronisation process and vice versa. The feedback matrix associated with the tracking control is obtained from an optimisation problem that minimises steady-state Allan variance. Numerical results are provided to show the efficacy of the proposed algorithm for enhancing long term performance.

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arXiv2026

Explicit Ensemble Mean Clock Synchronization for Optimal Atomic Time Scale Generation

This paper presents a novel theoretical framework, called explicit ensemble mean (EEM) synchronization. This framework unifies time scale generation, clock synchronization, and oscillator frequency regulation within the systems and control theory paradigm. By exploiting the observable canonical decomposition of a standard atomic ensemble clock model, the system is decomposed into two complementary components: the observable part, which represents the synchronization error, and the unobservable part, which captures the synchronization destination. Within this structure, we mathematically prove that standard Kalman filtering, which is widely used in current time scale generation, not only performs observable state estimation, but also significant unobservable state estimation, and it can be interpreted as a special case of the proposed framework that optimizes long-term frequency stability in terms of the Allan variance. Furthermore, applying state feedback control based on Kalman filtering to each component achieves optimal time scale generation, clock synchronization, and oscillator frequency regulation in a unified manner. The proposed framework provides a foundation for developing explainable timing systems.

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arXiv2025

Robust Continuous-Time Generation Scheduling under Power Demand Uncertainty: An Affine Decision Rule Approach

Most existing generation scheduling models for power systems under demand uncertainty rely on energy-based formulations with a finite number of time periods, which may fail to ensure that power supply and demand are balanced continuously over time. To address this issue, we propose a robust generation scheduling model in a continuous-time framework, employing a decision rule approach. First, for a given set of demand trajectories, we formulate a general robust generation scheduling problem to determine a decision rule that maps these demand trajectories and time points to the power outputs of generators. Subsequently, we derive a surrogate of it as our model by carefully designing a class of decision rules that are affine in the current demand, with coefficients invariant over time and constant terms that are continuous piecewise affine functions of time. As a result, our model can be recast as a finite-dimensional linear program to determine the coefficients and the function values of the constant terms at each breakpoint, solvable via the cutting-plane method. Our model is non-anticipative unlike most existing continuous-time models, which use Bernstein polynomials, making it more practical. We also provide illustrative numerical examples.

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arXiv2025

Small-Signal Stability Condition of Inverter-Integrated Power Systems: Closed-Form Expression by Stationary Power Flow Variables

This paper shows that a necessary and sufficient condition for the small-signal stability of an inverter-integrated power system can be expressed in terms of semidefinite matrix inequalities determined only by the synchronous reactance of the components, the susceptance matrix of the transmission network, and the stationary values of the power flow distribution. To derive the stability condition, we consider a class of grid-forming inverters corresponding to a singular perturbation of the synchronous generator. The resulting matrix inequality condition, which has twice as many dimensions as the number of buses and is independent of the dynamics of the connected components, is expressed in terms of each component compensating in a decentralized manner for the loss of frequency synchronization caused by the reactive power consumption in the transmission network. A simple numerical example using a 3-bus power system model shows that a grid-forming inverter load improves power system synchronization, while a grid-following inverter load disrupts it.

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arXiv2024

Equilibrium-Independent Passivity of Power Systems: A Link Between Classical and Two-Axis Synchronous Generator Models

We study the equilibrium-independent (EI) passivity of a nonlinear power system composed of two-axis generator models. The model of our interest consists of a feedback inter-connection of linear and nonlinear subsystems, called mechanical and electromagnetic subsystems. We mathematically prove the following three facts by analyzing the nonlinear electromagnetic subsystem. First, a lossless transmission network is necessary for the EI passivity of the electromagnetic subsystem. Second, the convexity of a strain energy function characterizes the largest set of equilibria over which the electromagnetic subsystem is EI passive. Finally, we prove that the strain energy function for the network of the two-axis generator models is convex if and only if its flux linkage dynamics is stable, and the strain energy function for the network of the classical generator models derived by singular perturbation approximation of the flux linkage dynamics is convex. Numerical simulation of the IEEE 9-bus power system model demonstrates the practical implications of the various mathematical results. In particular, we validate that the convex domain of the strain energy function over which the electromagnetic subsystem is EI passive is almost identical to the set of all stable equilibria. This result is also generalized to lossy power systems based on our finding that the convexity of the strain energy function is equivalent to the positive semidefiniteness of a synchronizing torque coefficient matrix.

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arXiv2023

Structured Kalman Filter for Time Scale Generation in Atomic Clock Ensembles

In this article, we present a structured Kalman filter associated with the transformation matrix for observable Kalman canonical decomposition from conventional Kalman filter (CKF) in order to generate a more accurate time scale. The conventional Kalman filter is a special case of the proposed structured Kalman filter which yields the same predicted unobservable or observable states when some conditions are satisfied. We consider an optimization problem respective to the transformation matrix where the objective function is associated with not only the expected value of prediction error but also its variance. We reveal that such an objective function is a convex function and show some conditions under which CKF is nothing but the optimal algorithm if ideal computation is possible without computation error. A numerical example is presented to show the robustness of the proposed method in terms of the initial error covariance

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arXiv2023

Higher-Order Allan Variance for Atomic Clocks of Arbitrary Order: Mathematical Foundation

In this paper, we perform a time-domain analysis of the higher-order Allan variance for atomic clock models of arbitrary order. Adopting a standard atomic clock model where the time series of the clock reading deviation is expressed as a Wiener or integrated Wiener process, we define the higher-order Allan variance as the mean squared higher-order difference of the clock reading deviation. The main results of this paper are threefold. First, we prove that the higher-order difference operation of the clock reading deviation, which can be interpreted as a linear aggregation with binomial coefficients, is not only sufficient but also necessary for a resulting aggregated time series to be an independent and identically distributed Gaussian process. Second, we derive a complete analytical expression of the higher-order Allan variance, which consists of both time-dependent and time-independent terms. Third, we prove that the higher-order Allan variance is time-independent if and only if the order of difference operation is greater than or equal to the order of the atomic clock model.

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arXiv2023

Relations Between Generalized JST Algorithm and Kalman Filtering Algorithm for Time Scale Generation

In this paper, we present a generalized Japan Standard Time algorithm (JST-algo) for higher-order atomic clock ensembles and mathematically clarify the relations of the (generalized) JST-algo and the conventional Kalman filtering algorithm (CKF-algo) in the averaged atomic time and the clock residuals for time scale generation. In particular, we reveal the fact that the averaged atomic time of the generalized JST-algo does not depend on the observation noise even though the measurement signal is not filtered in the algorithm. Furthermore, the prediction error of CKF- algo is rigorously shown by using the prediction error regarding an observable state space. It is mathematically shown that when the covariance matrices of system noises are identical for all atomic clocks, considering equal averaging weights for the clocks is a necessary and sufficient condition to ensure equivalence between the generalized JST-algo and CKF-algo in averaged atomic time. In such homogeneous systems, a necessary and sufficient condition for observation noises is presented to determine which algorithm can generate the clock residuals with smaller variances. A couple of numerical examples comparing the generalized JST-algo and CKF-algo are provided to illustrate the efficacy of the results.

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arXiv2021

Disconnection-aware Attack Detection and Isolation with Separation-based Detector Reconfiguration

This study addresses incident handling during an adverse event for dynamical networked control systems. Incident handling can be divided into five steps: detection, analysis, containment, eradication, and recovery. For networked control systems, the containment step can be conducted through physical disconnection of an attacked subsystem. In accordance with the disconnection, the equipped attack detection unit should be reconfigured to maintain its detection capability. In particular, separating the detection subunit associated with the disconnected subsystem is considered as a specific reconfiguration scheme in this study. This paper poses the problem of disconnection-aware attack detection and isolation with the separation-based detector reconfiguration. The objective is to find an attack detection unit that preserves its detection and isolation capability even under any possible disconnection and separation. The difficulty arises from network topology variation caused by disconnection that can possibly lead to stability loss of the distributed observer inside the attack detection unit. A solution is proposed based on an existing controller design technique referred to as retrofit control. Furthermore, an application to low-voltage power distribution networks with distributed generation is exhibited. Numerical examples evidence the practical use of the proposed method through a benchmark distribution network.

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arXiv2021

Parameterization of All Output-Rectifying Retrofit Controllers

This study investigates a parameterization of all output-rectifying retrofit controllers for distributed design of a structured controller. It has been discovered that all retrofit controllers can be characterized as a constrained Youla parameterization, which is difficult to solve analytically. For synthesis, a tractable and insightful class of retrofit controllers, referred to as output-rectifying retrofit controllers, has been introduced. An unconstrained parameterization of all output-rectifying retrofit controllers can be derived under a technical assumption on measurability of particular signals. The aim of this note is to reveal the structure of all output-rectifying retrofit controllers in the general output-feedback case. It is found out that the existing developments can be generalized based on the notions of state projection and an inverse system. The result leads to the conclusion that output-rectifying retrofit controllers can readily be designed even in the general case.

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arXiv2020

Distributed Design of Glocal Controllers via Hierarchical Model Decomposition

This paper proposes a distributed design method of controllers having a glocal (global/local) information structure for large-scale network systems. Distributed design, independent design of all subcontrollers that constitute a structured controller, facilitates scalable controller synthesis. While existing distributed design methods confine attention to the decentralized or distributed information structures, this study addresses distributed design of glocal-structured controllers. Glocal control exploits the nature that network system's behavior can typically be represented as a superposition of spatially local fluctuations and global interarea oscillations by incorporating a global coordinating subcontroller with local decentralized subcontrollers. The key idea to distributed design of glocal controllers is to represent the original network system as a hierarchical cascaded system composed of reduced-order models representing the global and local dynamics, referred to as hierarchical model decomposition. Distributed design is achieved by independently designing and implementing subcontrollers for the reduced-order models while preserving the cascade structure. This paper provides a condition for existence of the hierarchical model decomposition, a specific representation of the hierarchical system, a clustering method appropriate for the proposed approach, and a robust extension. Numerical examples of a power grid evidence the practical relevance of the proposed method.

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arXiv2020

Modularity-in-Design of Dynamical Network Systems: Retrofit Control Approach

In this paper, we develop a modular design method of decentralized controllers for linear dynamical network systems, where multiple subcontroller designers aim at individually regulating their local control performance with accessibility only to their respective subsystem models. First, we derive a constrained version of the Youla parameterization that characterizes all retrofit controllers for a single subcontroller, defined as an add-on type subcontroller that manages a subsystem. The resultant feedback system is kept robustly stable for any variation in the neighboring subsystems, other than the subsystem of interest, provided that the original system is stable prior to implementing the retrofit control. Subsequently, we find out a unique internal structure of the retrofit controllers, assuming that the interaction input signal from the neighboring subsystems is measurable. Furthermore, we show that the simultaneous implementation of multiple retrofit controllers, designed by individual subcontroller designers, can improve the upper bound of the overall control performance. Finally, the practical significance of the method is demonstrated via an illustrative example of frequency regulation using the IEEE 68-bus power system model.

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arXiv2020

Parameterization of Retrofit Controllers

This study investigates a parameterization of all retrofit controllers. Retrofit control can accomplish modular design of control systems, i.e., independent design of subcontrollers only with its corresponding subsystem model in a dynamical network system. In the retrofit control framework, the network system to be controlled is regarded as an interconnected system composed of the subsystem of interest and an environment. Existing studies have revealed that all retrofit controllers can be characterized as a constrained Youla parameterization under the technical assumption that the subsystem of interest is stable. It has also been discovered that all of the retrofit controllers that belong to a particular tractable class have a distinctive internal structure composed of a rectifier and an internal controller under the additional technical assumption that the interconnection signal from the environment to the subsystem of interest is measurable or that the internal state of the subsystem of interest is measurable. Due to the internal structure, the retrofit controller design problem under the assumptions can be reduced to a standard controller design problem to which existing controller synthesis techniques can be applied. The aim of this paper is to extend the above results without the technical assumptions. It is found that the existing developments can naturally be generalized through the Youla parameterization for unstable systems and an explicit description of the inverse system with the geometric control theory. The result leads to the conclusion that retrofit controllers can readily be designed even in the general case.

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arXiv2020

Disconnection-aware Attack Detection in Networked Control Systems

This study deals with security issues in dynamical networked control systems. The goal is to establish a unified framework of the attack detection stage, which includes the four processes of monitoring the system state, making a decision based on the monitored signal, disconnecting the corrupted subsystem, and operating the remaining system during restoration. This paper, in particular, considers a disconnection-aware attack detector design problem. Traditionally, observer-based attack detectors are designed based on the system model with a fixed network topology and cannot cope with a change of the topology caused by disconnection. The disconnection-aware design problem is mathematically formulated and a solution is proposed in this paper. A numerical example demonstrates the effectiveness of the proposed detector through an inverter-based voltage control system in a benchmark model.

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arXiv2019

Day-Ahead Energy Market as Adjustable Robust Optimization: Spatio-Temporal Pricing of Dispatchable Generators, Storage Batteries, and Uncertain Renewable Resources

We present modeling and analysis of day-ahead spatio-temporal energy markets in which each competitive aggregator aims at making the highest profit by managing a complex mixture of different energy resources, such as conventional generators, storage batteries, and uncertain renewable resources. First, we develop an energy market model in terms of an adjustable robust convex program. This market modeling is novel in the sense that the prosumption cost function of each aggregator, which evaluates the cost to realize an amount of spatio-temporal energy prosumption, is a multi-variable function resulting from a "parameterized" max-min program, in which the variable of the prosumption cost function is involved as a continuous parameter and the variable of dispatchable resources is involved as an adjustable variable for energy balance. This formulation enables to reasonably evaluate a reward for intertemporal dispatchability enhancement and a penalty for renewable energy uncertainty in a unified way. In addition, it enables to enforce a market regulation in which every aggregator is responsible for absorbing his renewable energy uncertainty by managing his own dispatchable energy resources. Second, in view of social economy as well as personal economy, we conduct a numerical analysis on the premise of several photovoltaic penetration levels. In this numerical analysis, we demonstrate that renewable generators do not always have priority of energy supply higher than conventional generators due to their uncertainty and limited dispatchability, meaning that the merit order of conventional and renewable generators can reverse. Furthermore, we analyze long-term evolution of competitive energy markets demonstrating that there can be found a social equilibrium of battery penetration levels, at which maximum personal profit with respect to battery system enhancement is attained.

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