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This paper introduces a novel approach to addressing uncertainty and associated risks in power system management, focusing on the discrepancies between forecasted and actual values of load demand and renewable power generation. By employing Economic Dispatch (ED) with both day-ahead forecasts and actual values, we derive two distinct system costs, revealing the financial risks stemming from uncertainty. We present a numerical algorithm inspired by the Integrated Gradients (IG) method to attribute the contribution of stochastic components to the difference in system costs. This method, originally developed for machine learning, facilitates the understanding of individual input features' impact on the model's output prediction. By assigning numeric values to represent the influence of variability on operational costs, our method provides actionable insights for grid management. As an application, we propose a risk-averse unit commitment framework, leveraging our cost attribution algorithm to adjust the capacity of renewable generators, thus mitigating system risk. Simulation results on the RTS-GMLC grid demonstrate the efficacy of our approach in improving grid reliability and reducing operational costs., Comment: 17 pages, 9 figures, 4 tables

In this paper, we investigate the transition between topological phases in a Su-Schrieffer-Heeger (SSH) model composed of springs and masses in which the intracellular Aubry-Andr\'e disorder modulates the spring constants. We analytically compute the eigenvectors and eigenvalues of the dynamical matrix for both periodic and fixed boundary conditions, and compare them with the dispersion spectrum of the original tight-binding SSH model. We observe the presence of a topological Anderson insulating (TAI) phase within a specific range of quasi-periodic modulation strength and calculate the phase transition boundary analytically. We examine the localization properties of normal modes using their inverse participation ratio (IPR) of eigenstates of the dynamical matrix, and the corresponding fractal dimension associated with quasiperiodic modulation. We also examine the stability of the TAI phase across a range of modulation strengths and comments on the presence of mobility edge that separate localized modes from non-localized ones. We demonstrate the fact that special analytical techniques are needed to compute an exact expression for mobility edges in such scenarios., Comment: 16 pages, 16 figures. This article is a simple and normal extension of the earlier article arXiv:2407.17883 by the same author

We examine the transition from trivial to non-trivial phases in a Su-Schrieffer-Heeger model subjected to disorder in a quasi-periodic environment. We analytically determine the phase boundary, and characterize the localization of normal modes using their inverse participation ratio. We compute energy-dependent mobility edges and provide evidence for the emergence of a topological Anderson insulator within specific parameter ranges. Whereas the phase transition boundary is affected by the quasi-periodic modulation, the topologically insulating Anderson phase is stable with respect to the chiral disorder in a quasi-periodic setup. Additionally, our results also uncover a re-entrant topological phase transition from non-trivial to trivial phases for certain values of quasi-periodic modulation with fixed chiral disorder., Comment: 20 pages, 15 figures

We study paycheck optimization, which examines how to allocate income in order to achieve several competing financial goals. For paycheck optimization, a quantitative methodology is missing, due to a lack of a suitable problem formulation. To deal with this issue, we formulate the problem as a utility maximization problem. The proposed formulation is able to (i) unify different financial goals; (ii) incorporate user preferences regarding the goals; (iii) handle stochastic interest rates. The proposed formulation also facilitates an end-to-end reinforcement learning solution, which is implemented on a variety of problem settings.

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. Finally, direct numerical simulations of the viscoelastic sub-diffusive channel flows (where the stress-constitutive law is obtained via coarse-graining the polymer relaxation spectrum at finer scale, Chauhan et. al., Phys. Fluids, DOI: 10.1063/5.0174598 (2023)), underscore the advantage of using these invariants in effectively quantifying the macrostructures., Comment: 19 pages, 2 figures

We propose a novel family of asymptotically stable, implicit-explicit, adaptive, time integration method (denoted with the $\theta$-method) for the solution of the fractional advection-diffusion-reaction (FADR) equations. This family of time integration method generalized the computationally explicit $L_1$-method adopted by Brunner (J. Comput. Phys. {\bf 229} 6613-6622 (2010)) as well as the fully implicit method proposed by Jannelli (Comm. Nonlin. Sci. Num. Sim., {\bf 105}, 106073 (2022)). The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of favorable dispersion for a limited range of Peclet number. The numerical inversion of the coefficient matrix is avoided by exploiting the sparse structure of the matrix in the iterative solver for the Poisson equation. The accuracy and the efficacy of the method is benchmarked using (a) the two-dimensional (2D) fractional diffusion equation, originally proposed by Brunner, and (b) the incompressible, subdiffusive dynamics of a planar viscoelastic channel flow of the Rouse chain melts (FADR equation with fractional time-derivative of order $\alpha=\nicefrac{1}{2}$) and the Zimm chain solution ($\alpha=\nicefrac{2}{3}$). Numerical simulations of the viscoelastic channel flow effectively capture the non-homogeneous regions of high viscosity at low fluid inertia (or the so-called `spatiotemporal macrostructures'), experimentally observed in the flow-instability transition of subdiffusive flows., Comment: 29 pages, 9 figures

The temporal and spatiotemporal linear stability analyses of viscoelastic, subdiffusive, plane Poiseuille and Couette flows obeying the Fractional Upper Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds number ($Re$) and Weissenberg number ($We$), is reported to identify the regions of topological transition of the advancing flow interface. In particular, we demonstrate how the exponent in the subdiffusive power{\color{black}-}law scaling ($t^\alpha$, with $0 {\color{black} < } \alpha \le 1$) of the mean square displacement of the tracer particle, in the microscale [Mason and Weitz, Phys. Rev. Lett. {\bf 74}, 1250-1253 (1995)] is related to the fractional order of the derivative, $\alpha$, of the corresponding non-linear stress constitutive equation in the continuum. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, $\alpha=\nicefrac{1}{2}$, and in Zimm chain solutions, $\alpha=\nicefrac{2}{3}$. The temporal stability analysis indicates that with decreasing order of the fractional derivative: (a) the most unstable mode decreases, (b) the peak of the most unstable mode shifts to lower values of $Re$, and (c) the peak of the most unstable mode, for the Rouse model precipitates towards the limit $Re \rightarrow 0$. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities and evanescent modes. The spatiotemporal phase diagram indicates an abnormal region of temporal stability at high fluid inertia, revealing the presence of a non-homogeneous environment with hindered flow, thus highlighting the potential of the model to effectively capture certain experimentally observed, flow-instability transition in subdiffusive flows., Comment: 28 pages, 8 figures. arXiv admin note: text overlap with arXiv:2109.08922

We study the formation of an optimal interbank network in a model where banks control both their supply of liquidity, through cash reserves, and their exposures to other banks' risky projects. The value of each bank's project may suddenly decline depending on their cash reserves and both the occurence and magnitude of liquidity shocks. In two distinct settings, we solve the system-wide optimal control problem and obtain explicit formulas for the unique optimal allocations of capital. In the first decentralized setting, banks seek only to maximize their own expected utility. Second, a central planner aims to maximize the sum of all banks' expected utilities. Both of the resulting networks exhibit a `core-periphery' structure in equilibrium. However, in the decentralized setting, banks elect to hold less cash reserves compared to the centralized setting, leading to greater susceptibility to liquidity shortages. We characterize the behavior of the planner's optimal allocation as the size of the system grows. Surprisingly, the relative welfare gap is of constant order. Finally, we derive co-investment requirements that allow the decentralized system to achieve the planner's optimal level of risk. In doing so, we find that banks in the network's core are subjected to the highest co-investment requirements -- ensuring that they are sufficiently incentivized to hold significant cash reserves. Our analysis may inform regulators' requirements on banks' liquidity reserves, as have been debated in the wake of the 2023 regional banking crisis in the US.