arXiv subjects

Hung D. Nguyen

Publications and source records attributed to Hung D. Nguyen.

At least 37 records · Page 2Linked to original sources

arXiv2022

The generalized Langevin equation in harmonic potentials: Anomalous diffusion and equipartition of energy

We consider the generalized Langevin equation (GLE) in a harmonic potential with power law decay memory. We study the anomalous diffusion of the particle's displacement and velocity. By comparison with the free particle situation in which the velocity was previously shown to be either diffusive or subdiffusive, we find that, when trapped in a harmonic potential, the particle's displacement may either be diffusive or superdiffusive. Under slightly stronger assumptions on the memory kernel, namely, for kernels related to the broad class of completely monotonic functions, we show that both the free particle and the harmonically bounded GLE satisfy the equipartition of energy condition. This generalizes previously known results for the GLE under particular kernel instances such as the generalized Rouse kernel or (exactly) a power law function.

math.PR↗

arXiv2022

Gaussian Process-based Approach for Bilevel Optimization in the Power System -- A Critical Load Restoration Case

Bilevel optimization problems can be used to represent the collaborative interaction between a power system and grid-connected entities, called the followers, such as data centers. Most existing approaches assume that such followers' response behaviors are made available to the power system in the operation decision-making, which may be untenable in reality. This work presents a novel idea of solving bilevel optimization problems without assuming power systems' omniscience. The followers' responses will be represented by a function of the power system's decisions using Gaussian Process Regression. Then the two layers in the bilevel problem can be solved separately by the power system and its followers. This not only avoids the omniscience assumption, but also significantly increases the computational efficiency without compromising accuracy, especially for the problems with a complex lower layer. Moreover, a bilevel critical load restoration model is developed to test the proposed technique. Compared to the conventional methods, the proposed restoration model considers the load-side operation and the varying load marginal value, and can accurately estimate load-side loss and achieve better restoration solutions. Two case studies validate the advantages of the proposed approaches from different perspectives.

eess.SY↗

arXiv2021

On the Hölder regularity of a linear stochastic partial-integro-differential equation with memory

In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in~\cite{mckinley2018anomalous}, we provide sufficient conditions on the differential operator and the noise to obtain the existence as well as Hölder regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is Hölder $(1/2-\ep)$ and $(1/4-\ep)$ respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the noise. Here, we show that the particular form of colored-in-time memory that arises in viscoelastic diffusion applications, satisfying what is called the Fluctuation--Dissipation relationship, yields sample paths that are Hölder $(1/2-\ep)$ and $(1/2-\ep)$ in space and time.

math.PR↗

arXiv2021

A Convexification Approach for Small-Signal Stability Constrained Optimal Power Flow

In this paper, a novel convexification approach for Small-Signal Stability Constraint Optimal Power Flow (SSSC-OPF) has been presented that does not rely on eigenvalue analysis. The proposed methodology is based on the sufficient condition for the small-signal stability, developed as a Bilinear Matrix Inequality (BMI), and uses network structure-preserving Differential Algebraic Equation (DAE) modeling of the power system. The proposed formulation is based on Semi-definite Programming (SDP) and objective penalization that has been proposed for feasible solution recovery, making the method computationally efficient for large-scale systems. A vector-norm based objective penalty function has also been proposed for feasible solution recovery while working over large and dense BMIs with matrix variables. An effectiveness study carried out on WECC 9-bus, New England 39-bus, and IEEE 118-bus test systems show that the proposed method is capable of achieving a stable equilibrium point without inflicting a high stability-induced additional cost.

math.OC↗

arXiv2021

Asymptotic analysis of the mean squared displacement under fractional memory kernels

The generalized Langevin equation (GLE) is a universal model for particle velocity in a viscoelastic medium. In this paper, we consider the GLE family with fractional memory kernels. We show that, in the critical regime where the memory kernel decays like $1/t$ for large $t$, the mean squared displacement (MSD) of particle motion grows linearly in time up to a slowly varying (logarithm) term. Moreover, we establish the well-posedness of the GLE in this regime. This solves an open question from [Mckinley 2018 Anomalous] and completes the answer to the conjecture put forward in [Morgado 2002 Relation] on the relationship between memory kernel decay and anomalously diffusive behavior. Under slightly stronger assumptions on the memory kernel, we construct an Abelian-Tauberian framework that leads to robust bounds on the deviation of the MSD around its asymptotic trend. This bridges the gap between the GLE memory kernel and the spectral density of anomalously diffusive particle motion characterized in [Didier 2017 Asymptotic].

math.PR↗

arXiv2021

The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit.

math.PR↗

arXiv2020

A Gaussian Process-based Price-Amount Curve Construction for Demand Response Provided by Internet Data Centers

For a Demand Response (DR) program with internet data centers (IDC), the Price-Amount curve that estimates how the potential DR amount depends on the DR price determined by power systems is crucial. Constructing this curve is challenging mainly due to the uncertainty in IDCs' operation. A novel Gaussian Process Regression-based estimation method is thus proposed. The variance of resulting curve reflecting the IDC operational uncertainty is also calculated.

eess.SY↗

arXiv2020

A Framework for Health-informed RUL-constrained Optimal Power Flow with Li-ion Batteries

Battery energy storage systems are widely adopted in grid-connected applications to mitigate the impact of intermittent renewable generations and enhance power system resiliency. Degradation of the battery during its service time is one of the major concerns in the deployment that strongly affects the long-term lifetime. Apart from environmental factors, this intrinsic property of a battery depends on the daily operating conditions. Thus, optimally engaging the daily operation of the battery based on its current status in order to meet the required remaining useful life becomes a practical and demanding need. To address this issue, this paper proposes a health-informed RUL-constrained optimal power flow framework to characterize the corresponding optimal feasible operation space. The targeted service lifespan is achieved if the battery's working condition is confined within this feasible domain. Equivalent box constraints are then constructed for better computational efficiency in solving the optimization problem. In this framework, a Monte Carlo-based data-driven approach and a health indicator (HI) representing the battery's current states are introduced. The performance of the proposed method is illustrated with the IEEE 39-bus system.

eess.SY↗

arXiv2020

Health-Focused Optimal Power Flow

In this paper, we propose a novel Health-Focused Optimal Power Flow (HF-OPF) to take into account the equipment health in operational and physical constraints. The health condition index is estimated based on the possible fault characteristics for generators and batteries. The paper addresses the need for understanding the relationship between health condition index and the operational constraints in OPF problems. Such a relationship is established through Finite Element Method (FEM)-based simulations. The simulations for generator and battery faults indicate the presence of an inflection point after which effect of fault becomes severe. A microgrid system is used to illustrate the performance of the proposed HF-OPF. The results show that health condition inflicts high cost of generation and can lead to infeasibility even with less critical faults.

math.OC↗

arXiv2020

A Two-Layer Framework with Battery Temperature Optimal Control and Network Optimal Power Flow

Battery energy storage is an essential component of a microgrid. The working temperature of the battery is an important factor as a high-temperature condition generally increases losses, reduces useful life, and can even lead to fire hazards. Hence, it is indispensable to regulate the temperature profile of the battery modules/packs properly in the battery energy storage during the operation. In view of this, a two-layer optimal control and operation scheme is proposed for a microgrid with energy storage. In the first layer, an optimal control model is formed to derive the optimal control policy that minimizes the control efforts, consisting of the fan speed and battery current magnitude, in order to achieve a temperature distribution reference over the battery modules. In the second layer, the system operator of the microgrid performs an optimal power flow to search for the optimal temperature distribution reference used in the first stage and the corresponding operating current of the battery that minimize the operation cost of the entire microgrid system. This two-layer scheme offers a great computational benefit that allows for large-scale integration of batteries. A case study is performed on the proposed two-layer model to illustrate its performance.

eess.SY↗

arXiv2020

Gaussian Process Learning-based Probabilistic Optimal Power Flow

In this letter, we present a novel Gaussian Process Learning-based Probabilistic Optimal Power Flow (GP-POPF) for solving POPF under renewable and load uncertainties of arbitrary distribution. The proposed method relies on a non-parametric Bayesian inference-based uncertainty propagation approach, called Gaussian Process (GP). We also suggest a new type of sensitivity called Subspace-wise Sensitivity, using observations on the interpretability of GP-POPF hyperparameters. The simulation results on 14-bus and 30-bus systems show that the proposed method provides reasonably accurate solutions when compared with Monte-Carlo Simulations (MCS) solutions at different levels of uncertain renewable penetration as well as load uncertainties, while requiring much less number of samples and elapsed time.

eess.SY↗

arXiv2020

Non-parametric Probabilistic Load Flow using Gaussian Process Learning

In this work, we propose a non-parametric probabilistic load flow (NP-PLF) technique based on the Gaussian Process (GP) learning to understand the power system behavior under uncertainty for better operational decisions. The technique can provide "semi-explicit" power flow solutions by implementing the learning and testing steps which map control variables to inputs. The proposed NP-PLF leverages upon GP upper confidence bound (GP-UCB) sampling algorithm. The salient features of this NP-PLF method are: i) applicable for power flow problem having power injection uncertainty with an unknown class of distribution; ii) providing probabilistic learning bound (PLB) which further provides control over the error and convergence; iii) capable of handling intermittent distributed generation as well as load uncertainties, and iv) applicable to both balanced and unbalanced power flow with different type and size of power systems. The simulation results performed on the IEEE 30-bus and IEEE 118-bus system show that the proposed method can learn the voltage function over the power injection subspace using a small number of training samples. Further, the testing with different input uncertainty distributions indicates that complete statistical information can be obtained for the probabilistic load flow problem with average percentage relative error of order $10^{-3}$\% on 50000 test points.

eess.SY↗

arXiv2019

Exploration of the Applicability of Probabilistic Inference for Learning Control in Underactuated Autonomous Underwater Vehicles

Underwater vehicles are employed in the exploration of dynamic environments where tuning of a specific controller for each task would be time-consuming and unreliable as the controller depends on calculated mathematical coefficients in idealised conditions. For such a case, learning task from experience can be a useful alternative. This paper explores the capability of probabilistic inference learning to control autonomous underwater vehicles that can be used for different tasks without re-programming the controller. Probabilistic inference learning uses a Gaussian process model of the real vehicle to learn the correct policy with a small number of real field experiments. The use of probabilistic reinforced learning looks for a simple implementation of controllers without the burden of coefficients calculation, controller tuning or system identification. A series of computational simulations were employed to test the applicability of model-based reinforced learning in underwater vehicles. Three simulation scenarios were evaluated: waypoint tracking, depth control and 3D path tracking control. The 3D path tracking is done by coupling together a line-of-sight law with probabilistic inference for learning control. As a comparison study LOS-PILCO algorithm can perform better than a robust LOS-PID. The results shows that probabilistic model based reinforced learning is a possible solution to motion control of underactuated AUVs as can generate capable policies with minimum quantity of episodes.

eess.SY↗

arXiv2019

Probabilistic Robust Small-Signal Stability Framework using Gaussian Process Learning

While most power system small-signal stability assessments rely on the reduced Jacobian, which depends non-linearly on the states, uncertain operating points introduce nontrivial hurdles in certifying the system's stability. In this paper, a novel probabilistic robust small-signal stability (PRS) framework is developed for a power system based on Gaussian process (GP) learning. The proposed PRS assessment provides a robust stability certificate for a state subspace, such as that specified by the error bounds of the state estimation, with a given probability. With such a PRS certificate, all inner points of the concerned subspace will be stable with at least the corresponding confidence level. To this end, the behavior of the critical eigenvalue of the reduced Jacobian with state points in a state subspace is learned using GP. The proposed PRS certificate along with the Subspace-based Search and Confidence-based Search mechanisms constitute a holistic framework catering to all scenarios. The proposed framework is a powerful approach to assess the stability under uncertainty because it does not require input uncertainty distributions and other state-specific input-to-output approximations. Further, the critical eigenvalue behavior in a state subspace is analyzed using an upper bound of the eigenvalue variations and their inferences are discussed in detail. The results on three-machine nine-bus WSCC system show that the proposed certificate can find the robust stable state subspace with a given probability.

eess.SY↗

arXiv2019

Region of Attraction for Power Systems using Gaussian Process and Converse Lyapunov Function -- Part I: Theoretical Framework and Off-line Study

This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available, and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need of an analytic Lyapunov function and learning the unknown Lyapunov function with the Gaussian Process (GP) approach. In addition, a Gaussian Process Upper Confidence Bound (GP-UCB) based sampling algorithm is designed to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Within the constructed ROA, it is guaranteed in probability that the system state will converge to the stable equilibrium with a confidence level. Numerical simulations are also conducted to validate the assessment approach for the ROA of the single machine infinite bus system and the New England $39$-bus system. Numerical results demonstrate that our approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

eess.SY↗

arXiv2019

Steady-state Voltage Profile and Long-term Voltage Stability of Electrified Road with Wireless Dynamic Charging

Wireless dynamic charging technologies are becoming a promising alternative solution to plug-in ones as they allow on-the-move charging for electric vehicles. From a power network point of view, this type of charging makes electric vehicles a new type of loads--the moving loads. Such moving loads are different from the traditional loads as they may change their locations constantly in the grids. To study the effect of these moving loads on power distribution grids, this work focuses on the steady-state analysis of electrified roads equipped with wireless dynamic charging. In particular, the voltage profile and the long-term voltage stability of the electrified roads are considered. Unusual shapes of the voltage profile are observed such as the half-leaf veins for a one-way road and the harp-like shape for a two-way road. Voltage swings are also detected while the vehicles move in the two-way road configuration. As for the long-term voltage stability, continuation power flow is used to characterize the maximum length of a road as well as the maximum number of vehicles that the road can accommodate.

eess.SP↗

arXiv2019

Convex Restriction of Power Flow Feasibility Sets

The convex restriction of the power flow feasible sets identifies the convex subset of power injections where the solution for power flow is guaranteed to exist and satisfy the operational constraints. In contrast to convex relaxations, the convex restriction provides a sufficient condition for power flow feasibility and is particularly useful for problems involving uncertainty in the power generation and demand. In this paper, we present a general framework of constructing convex restriction of an algebraic set defined by equality and inequality constraints and apply the framework to power flow feasibility problem. The procedure results in convex quadratic constraints that provide a sufficiently large region for practical operation of the grid.

math.OC↗

arXiv2018

A Framework for Robust Steady-State Voltage Stability of Distribution Systems

Power injection uncertainties in distribution power grids, which are mostly induced by aggressive introduction of intermittent renewable sources, may drive the system away from normal operating regimes and potentially lead to the loss of long-term voltage stability (LTVS). Naturally, there is an ever increasing need for a tool for assessing the LTVS of a distribution system. This paper presents a fast and reliable tool for constructing \emph{inner approximations} of LTVS regions in multidimensional injection space such that every point in our constructed region is guaranteed to be solvable. Numerical simulations demonstrate that our approach outperforms all existing inner approximation methods in most cases. Furthermore, the constructed regions are shown to cover substantial fractions of the true voltage stability region. The paper will later discuss a number of important applications of the proposed technique, including fast screening for viable injection changes, constructing an effective solvability index and rigorously certified loadability limits.

math.OC↗