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This article investigates the ability of graph neural networks (GNNs) to identify risky conditions in a power grid over the subsequent few hours, without explicit, high-resolution information regarding future generator on/off status (grid topology) or power dispatch decisions. The GNNs are trained using supervised learning, to predict the power grid's aggregated bus-level (either zonal or system-level) or individual branch-level state under different power supply and demand conditions. The variability of the stochastic grid variables (wind/solar generation and load demand), and their statistical correlations, are rigorously considered while generating the inputs for the training data. The outputs in the training data, obtained by solving numerous mixed-integer linear programming (MILP) optimal power flow problems, correspond to system-level, zonal and transmission line-level quantities of interest (QoIs). The QoIs predicted by the GNNs are used to conduct hours-ahead, sampling-based reliability and risk assessment w.r.t. zonal and system-level (load shedding) as well as branch-level (overloading) failure events. The proposed methodology is demonstrated for three synthetic grids with sizes ranging from 118 to 2848 buses. Our results demonstrate that GNNs are capable of providing fast and accurate prediction of QoIs and can be good proxies for computationally expensive MILP algorithms. The excellent accuracy of GNN-based reliability and risk assessment suggests that GNN models can substantially improve situational awareness by quickly providing rigorous reliability and risk estimates., Comment: Manuscript submitted to Applied Energy

We investigate the utility of graph neural networks (GNNs) as proxies of power grid operational decision-making algorithms (optimal power flow (OPF) and security-constrained unit commitment (SCUC)) to enable rigorous quantification of the operational risk. To conduct principled risk analysis, numerous Monte Carlo (MC) samples are drawn from the (foretasted) probability distributions of spatio-temporally correlated stochastic grid variables. The corresponding OPF and SCUC solutions, which are needed to quantify the risk, are generated using traditional OPF and SCUC solvers to generate data for training GNN model(s). The GNN model performance is evaluated in terms of the accuracy of predicting quantities of interests (QoIs) derived from the decision variables in OPF and SCUC. Specifically, we focus on thermal power generation and load shedding at system and individual zone level. We also perform reliability and risk quantification based on GNN predictions and compare with that obtained from OPF/SCUC solutions. Our results demonstrate that GNNs are capable of providing fast and accurate prediction of QoIs and thus can be good surrogate models for OPF and SCUC. The excellent accuracy of GNN-based reliability and risk assessment further suggests that GNN surrogate has the potential to be applied in real-time and hours-ahead risk quantification., Comment: Manuscript submitted to IEEE PES GM 2024

A DC OPF surrogate modeling framework is developed for Monte Carlo (MC) sampling-based risk quantification in power grid operation. MC simulation necessitates solving a large number of DC OPF problems corresponding to the samples of stochastic grid variables (power demand and renewable generation), which is computationally prohibitive. Computationally inexpensive surrogates of OPF provide an attractive alternative for expedited MC simulation. Graph neural network (GNN) surrogates of DC OPF, which are especially suitable to graph-structured data, are employed in this work. Previously developed DC OPF surrogate models have focused on accurate operational decision-making and not on risk quantification. Here, risk quantification-specific aspects of DC OPF surrogate evaluation is the main focus. To this end, the proposed GNN surrogates are evaluated using realistic joint probability distributions, quantification of their risk estimation accuracy, and investigation of their generalizability. Four synthetic grids (Case118, Case300, Case1354pegase, and Case2848rte) are used for surrogate model performance evaluation. It is shown that the GNN surrogates are sufficiently accurate for predicting the (bus-level, branch-level and system-level) grid state and enable fast as well as accurate operational risk quantification for power grids. The article thus develops tools for fast reliability and risk quantification in real-world power grids using GNN-based surrogates., Comment: Manuscript submitted to IEEE Transactions on Power Systems

In a grid with a significant share of renewable generation, operators will need additional tools to evaluate the operational risk due to the increased volatility in load and generation. The computational requirements of the forward uncertainty propagation problem, which must solve numerous security-constrained economic dispatch (SCED) optimizations, is a major barrier for such real-time risk assessment. This paper proposes a Just-In-Time Risk Assessment Learning Framework (JITRALF) as an alternative. JITRALF trains risk surrogates, one for each hour in the day, using Machine Learning (ML) to predict the quantities needed to estimate risk, without explicitly solving the SCED problem. This significantly reduces the computational burden of the forward uncertainty propagation and allows for fast, real-time risk estimation. The paper also proposes a novel, asymmetric loss function and shows that models trained using the asymmetric loss perform better than those using symmetric loss functions. JITRALF is evaluated on the French transmission system for assessing the risk of insufficient operating reserves, the risk of load shedding, and the expected operating cost.

As an emerging technology in the era of Industry 4.0, digital twin is gaining unprecedented attention because of its promise to further optimize process design, quality control, health monitoring, decision and policy making, and more, by comprehensively modeling the physical world as a group of interconnected digital models. In a two-part series of papers, we examine the fundamental role of different modeling techniques, twinning enabling technologies, and uncertainty quantification and optimization methods commonly used in digital twins. This second paper presents a literature review of key enabling technologies of digital twins, with an emphasis on uncertainty quantification, optimization methods, open source datasets and tools, major findings, challenges, and future directions. Discussions focus on current methods of uncertainty quantification and optimization and how they are applied in different dimensions of a digital twin. Additionally, this paper presents a case study where a battery digital twin is constructed and tested to illustrate some of the modeling and twinning methods reviewed in this two-part review. Code and preprocessed data for generating all the results and figures presented in the case study are available on GitHub.

As an emerging technology in the era of Industry 4.0, digital twin is gaining unprecedented attention because of its promise to further optimize process design, quality control, health monitoring, decision and policy making, and more, by comprehensively modeling the physical world as a group of interconnected digital models. In a two-part series of papers, we examine the fundamental role of different modeling techniques, twinning enabling technologies, and uncertainty quantification and optimization methods commonly used in digital twins. This first paper presents a thorough literature review of digital twin trends across many disciplines currently pursuing this area of research. Then, digital twin modeling and twinning enabling technologies are further analyzed by classifying them into two main categories: physical-to-virtual, and virtual-to-physical, based on the direction in which data flows. Finally, this paper provides perspectives on the trajectory of digital twin technology over the next decade, and introduces a few emerging areas of research which will likely be of great use in future digital twin research. In part two of this review, the role of uncertainty quantification and optimization are discussed, a battery digital twin is demonstrated, and more perspectives on the future of digital twin are shared.

We develop a systematic approach for surrogate model construction in reduced input parameter spaces. A sparse set of model evaluations in the original input space is used to approximate derivative based global sensitivity measures (DGSMs) for individual uncertain inputs of the model. An iterative screening procedure is developed that exploits DGSM estimates in order to identify the unimportant inputs. The screening procedure forms an integral part of an overall framework for adaptive construction of a surrogate in the reduced space. The framework is tested for computational efficiency through an initial implementation in simple test cases such as the classic Borehole function, and a semilinear elliptic PDE with a random source term. The framework is then deployed for a realistic application from chemical kinetics, where we study the ignition delay in an H2/O2 reaction mechanism with 19 uncertain rate constants. It is observed that significant computational gains can be attained by constructing accurate low-dimensional surrogates using the proposed framework.

Bulk thermal conductivity estimates based on predictions from non-equilibrium molecular dynamics (NEMD) using the so-called direct method are known to be severely under-predicted since finite simulation length-scales are unable to mimic bulk transport. Moreover, subjecting the system to a temperature gradient by means of thermostatting tends to impact phonon transport adversely. Additionally, NEMD predictions are tightly coupled with the choice of the inter-atomic potential and the underlying values associated with its parameters. In the case of silicon (Si), nominal estimates of the Stillinger-Weber (SW) potential parameters are largely based on a constrained regression approach aimed at agreement with experimental data while ensuring structural stability. However, this approach has its short-comings and it may not be ideal to use the same set of parameters to study a wide variety of Si-based systems subjected to different thermodynamic conditions. In this study, NEMD simulations are performed on a Si bar to investigate the impact of bar-length, and the applied thermal gradient on the discrepancy between predictions and the available measurement for bulk thermal conductivity at 300 K by constructing statistical response surfaces at different temperatures. The approach helps quantify the discrepancy, observed to be largely dependent on the system-size, with minimal computational effort. A computationally efficient approach based on derivative-based sensitivity measures to construct a reduced-order polynomial chaos surrogate for NEMD predictions is also presented. The surrogate is used to perform parametric sensitivity analysis, forward propagation of the uncertainty, and calibration of the important SW potential parameters in a Bayesian setting. It is found that only two (out of seven) parameters contribute significantly to the uncertainty in bulk thermal conductivity estimates for Si.

This paper proposes a new deterministic sampling strategy for constructing polynomial chaos approximations for expensive physics simulation models. The proposed approach, effectively subsampled quadratures involves sparsely subsampling an existing tensor grid using QR column pivoting. For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem. For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem. While we provide bounds on the condition number of such tall submatrices, it is difficult to ascertain how column pruning effects solution accuracy as this is problem specific. We conclude with numerical experiments on an analytical function and a model piston problem that show the efficacy of our approach compared with randomized subsampling. We also show an example where this method fails., Comment: 17 pages

Measure the similarity of the nodes in the complex networks have interested many researchers to explore it. In this paper, a new method which is based on the degree centrality and the Relative-entropy is proposed to measure the similarity of the nodes in the complex networks. The results in this paper show that, the nodes which have a common structure property always have a high similarity to others nodes. The nodes which have a high influential to others always have a small value of similarity to other nodes and the marginal nodes also have a low similar to other nodes. The results in this paper show that the proposed method is useful and reasonable to measure the similarity of the nodes in the complex networks., Comment: 6 figures

The local structure entropy is a new method which is proposed to identify the influential nodes in the complex networks. In this paper a new form of the local structure entropy of the complex networks is proposed based on the Tsallis entropy. The value of the entropic index $q$ will influence the property of the local structure entropy. When the value of $q$ is equal to 0, the nonextensive local structure entropy is degenerated to a new form of the degree centrality. When the value of $q$ is equal to 1, the nonextensive local structure entropy is degenerated to the existing form of the local structure entropy. We also have find a nonextensive threshold value in the nonextensive local structure entropy. When the value of $q$ is bigger than the nonextensive threshold value, change the value of $q$ will has no influence on the property of the local structure entropy, and different complex networks have different nonextensive threshold value. The results in this paper show that the new nonextensive local structure entropy is a generalised of the local structure entropy. It is more reasonable and useful than the existing one., Comment: 5 figures

How complex of the complex networks has attracted many researchers to explore it. The entropy is an useful method to describe the degree of the $complex$ of the complex networks. In this paper, a new method which is based on the Tsallis entropy is proposed to describe the $complex$ of the complex networks. The results in this paper show that the complex of the complex networks not only decided by the structure property of the complex networks, but also influenced by the relationship between each nodes. In other word, which kinds of nodes are chosen as the main part of the complex networks will influence the value of the entropy of the complex networks. The value of q in the Tsallis entropy of the complex networks is used to decided which kinds of nodes will be chosen as the main part in the complex networks. The proposed Tsallis entropy of the complex networks is a generalised method to describe the property of the complex networks., Comment: 12 pages

The structure entropy is one of the most important parameters to describe the structure property of the complex networks. Most of the existing struc- ture entropies are based on the degree or the betweenness centrality. In order to describe the structure property of the complex networks more reasonably, a new structure entropy of the complex networks based on the Tsallis nonextensive statistical mechanics is proposed in this paper. The influence of the degree and the betweenness centrality on the structure property is combined in the proposed structure entropy. Compared with the existing structure entropy, the proposed structure entropy is more reasonable to describe the structure property of the complex networks in some situations., Comment: 21 pages,10 figures

The fractal and self-similarity properties are revealed in many complex networks. In order to show the influence of different part in the complex networks to the information dimension, we have proposed a new information dimension based on Tsallis entropy namely Tsallis information dimension. The Tsallis information dimension can show the fractal property from different perspective by set different value of q., Comment: 14 pages, 4 figures

With an increasing emphasis on network security, much more attention has been attracted to the vulnerability of complex networks. The multi-scale evaluation of vulnerability is widely used since it makes use of combined powers of the links' betweenness and has an effective evaluation to vulnerability. However, how to determine the coefficient in existing multi-scale evaluation model to measure the vulnerability of different networks is still an open issue. In this paper, an improved model based on the fractal dimension of complex networks is proposed to obtain a more reasonable evaluation of vulnerability with more physical significance. Not only the structure and basic physical properties of networks is characterized, but also the covering ability of networks, which is related to the vulnerability of the network, is taken into consideration in our proposed method. The numerical examples and real applications are used to illustrate the efficiency of our proposed method.

As an equilibrium refinement of the Nash equilibrium, evolutionarily stable strategy (ESS) is a key concept in evolutionary game theory and has attracted growing interest. An ESS can be either a pure strategy or a mixed strategy. Even though the randomness is allowed in mixed strategy, the selection probability of pure strategy in a mixed strategy may fluctuate due to the impact of many factors. The fluctuation can lead to more uncertainty. In this paper, such uncertainty involved in mixed strategy has been further taken into consideration: a belief strategy is proposed in terms of Dempster-Shafer evidence theory. Furthermore, based on the proposed belief strategy, a belief-based ESS has been developed. The belief strategy and belief-based ESS can reduce to the mixed strategy and mixed ESS, which provide more realistic and powerful tools to describe interactions among agents., Comment: 26 pages, 3 figures

Efficient modeling on uncertain information plays an important role in estimating the risk of contaminant intrusion in water distribution networks. Dempster-Shafer evidence theory is one of the most commonly used methods. However, the Dempster-Shafer evidence theory has some hypotheses including the exclusive property of the elements in the frame of discernment, which may not be consistent with the real world. In this paper, based on a more effective representation of uncertainty, called D numbers, a new method that allows the elements in the frame of discernment to be non-exclusive is proposed. To demonstrate the efficiency of the proposed method, we apply it to the water distribution networks to estimate the risk of contaminant intrusion., Comment: 20 pages, 3 figures