arXiv subjects

Shriram Srinivasan

Publications and source records attributed to Shriram Srinivasan.

14 recordsLinked to original sources

arXiv2026

Conformal Margins for Electrical Transmission Capacity Under Fixed Balancing Policies

This article presents a data-driven approach to minimally adjust forecast-based steady-state generation dispatch in electrical transmission networks. The goal is to manage congestion caused by uncertainty in demand and renewable generation under a fixed automatic generation control (AGC) policy operating without saturation. Under this policy, we first show that the post-AGC residual injection distribution is invariant to the scheduled generation dispatch. This result separates line loading into a dispatch-dependent baseline and an uncertainty-induced residual component. Using historical forecast errors, we then apply split conformal prediction to construct transmission-line margins with distribution-free, finite-sample linewise or joint coverage guarantees. We also calibrate the aggregate pre-AGC mismatch to determine the generation headroom required for the prescribed AGC response. We incorporate these calibrated quantities into a convex secure re-dispatch problem that minimally adjusts the nominal schedule while reserving sufficient transmission capacity and AGC headroom. The resulting corrected dispatch provides finite-sample probabilistic guarantees on per-line security and unsaturated AGC operation. Case studies on the RTS-GMLC system demonstrate substantial reductions in out-of-sample line-limit violations with modest operating-cost increases.

math.OC↗

arXiv2026

The Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations via Graph Partitioning

The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.

physics.comp-ph↗

arXiv2026

Steady Flow of Natural Gas in Pipeline Networks via Solution of a Nonlinear Differential-Algebraic System of Equations

In the consideration of steady-state flow of gas in pipeline networks, the exclusion of gravity and nonlinear inertial effects (convective acceleration) leads to a fortuitous simplification in the governing equations to yield a system of nonlinear algebraic equations. Consequently, there are no studies that quantify the effect of gravity and inertial effects on the flow of gas in pipeline networks or delineate regimes of flow conditions wherein the effects are significant or negligible. In addressing this need, we consider the steady-state flow equations in pipeline networks without neglecting the gravitational and inertial terms and in place of a system of algebraic equations (one for each pipe), this approach results in a nonlinear system of first-order ordinary differential equations (ODEs) which are coupled through algebraic equations that appear in the form of boundary conditions on the pressure and balance of mass flows at either end. One of our main contributions in this article is to demonstrate how the Newton-Raphson algorithm can still be used to solve the coupled nonlinear differential-algebraic system by utilizing the appropriate forward sensitivity ODEs to evaluate the Jacobian terms arising in the iterative scheme. We also propose a variable transformation that alleviates the poor scaling of the ODE, and we introduce a two-point collocation scheme as a coarse approximation of the system from which to find initial guesses for the Newton iterations. Simulation studies were conducted for a single pipe as well as a large-scale pipeline network with real data. From these studies, we concluded that while the effect of gravity is important, the inertial effect was negligible in all cases. The proposed methodology is applicable to a wide class of pipeline and thermal-fluid networks beyond natural gas, including liquid pipelines and hydrogen transport.

math.GM↗

arXiv2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology and in general there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. This article proposes and describes such a procedure, an hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

math.NA↗

arXiv2024

Optimization of District Heating Network Parameters in Steady-State Operation

We examine the modeling, simulation, and optimization of district heating systems, which are widely used for thermal transport using steam or hot water as a carrier. We propose a generalizable framework to specify network models and scenario parameters, and develop an optimization method for evaluating system states including pressures, fluid flow rates, and temperatures throughout the network. The network modeling includes pipes, thermal plants, pumps, and passive or controllable loads as system components. We propose basic models for thermodynamic fluid transport and enforce the balance of physical quantities in steady-state flow over co-located outgoing and return networks. We formulate an optimization problem with steam and hot water as the outgoing and return carriers, as in legacy 20th century systems. The physical laws and engineering limitations are specified for each component type, and the thermal network flow optimization (TNFO) problem is formulated and solved for a realistic test network under several scenarios.

math.OC↗

arXiv2024

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision-support tools for system design and operation. While the nonlinear system is known to have a unique solution (when one exists), the absence of a definite result on the existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of a solution. In this letter, we show that for any fluid whose equation of state is a scaled monomial, a unique solution exists for such nonlinear systems if the term solution is interpreted in terms of potentials and flows rather than pressures and flows. However, for gases following the CNGA equation of state, while the question of existence remains open, we construct an alternative system that always has a unique solution and show that the solution to this system is a good approximant of the true solution. The existence result for flow of natural gas in networks also applies to other fluid flow networks such as water distribution networks or networks that transport carbon dioxide in carbon capture and sequestration. Most importantly, our result enables correct diagnosis of algorithmic failure, problem stiffness, and non-convergence in computational algorithms.

math.NA↗

arXiv2023

Relaxations of the Steady Optimal Gas Flow Problem for a Non-Ideal Gas

Natural gas ranks second in consumption among primary energy sources in the United States. The majority of production sites are in remote locations, hence natural gas needs to be transported through a pipeline network equipped with a variety of physical components such as compressors, valves, etc. Thus, from the point of view of both economics and reliability, it is desirable to achieve optimal transportation of natural gas using these pipeline networks. The physics that governs the flow of natural gas through various components in a pipeline network is governed by nonlinear and non-convex equality and inequality constraints and the most general steady-flow operations problem takes the form of a Mixed Integer Nonlinear Program. In this paper, we consider one example of steady-flow operations -- the Optimal Gas Flow (OGF) problem for a natural gas pipeline network that minimizes the production cost subject to the physics of steady-flow of natural gas. The ability to quickly determine global optimal solution and a lower bound to the objective value of the OGF for different demand profiles plays a key role in efficient day-to-day operations. One strategy to accomplish this relies on tight relaxations to the nonlinear constraints of the OGF. Currently, many nonlinear constraints that arise due to modeling the non-ideal equation of state either do not have relaxations or have relaxations that scale poorly for realistic network sizes. In this work, we combine recent advancements in the development of polyhedral relaxations for univariate functions to obtain tight relaxations that can be solved within a few seconds on a standard laptop. We demonstrate the quality of these relaxations through extensive numerical experiments on very large scale test networks available in the literature and find that the proposed relaxation is able to prove optimality in 92% of the instances.

math.OC↗

arXiv2022

Modeling and Optimization of Steady Flow of Natural Gas and Hydrogen Mixtures in Pipeline Networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

math.OC↗

arXiv2022

Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

We formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

eess.SY↗

arXiv2021

Demand Response Analogues for Residential Loads in Natural Gas Networks

Demand response for electrical power networks is a mature field that has yielded numerous efficiency and resilience benefits like managing the peaks and valleys of electricity usage, reduction in peak electricity usage, to name a few. However, only recently has the study of the counterpart to demand response for electric power in natural gas started to receive similar levels of attention. Natural gas systems are increasingly operating at or near capacity, which challenges these systems to meet all the needs for gas, especially during severe winter weather. However, unlike demand response programs in electrical networks, demand response in gas networks cannot shift peak usage. Here, we develop analogues to demand response that can help improve the resilience of natural gas systems by reducing peak consumption and thereby limiting potential disruptions such events can cause. This paper develops a mathematical formulation to support a residential-level demand response analogue for natural gas based on current and anticipated smart thermostat technologies. The mathematical formulation takes the form of an optimal control problem (OCP) that leverages physics constraints to model temperature changes, balance equitable service, and optimize the gas consumption for collections of houses. On test problems, the formulation demonstrates significant benefits, including the ability to cut peak demand by 15% while still ensuring equitable service to customers.

math.OC↗

arXiv2021

Operation of Natural Gas Pipeline Networks With Storage Under Transient Flow Conditions

We formulate a nonlinear optimal control problem for intra-day operation of a natural gas pipeline network that includes storage reservoirs. The dynamics of compressible gas flow through pipes, compressors, reservoirs, and wells are considered. In particular, a reservoir is modeled as a rigid, hollow container that stores gas under isothermal conditions and uniform density, and a well is modeled as a vertical pipe. For each pipe, flow dynamics are described by a coupled partial differential equation (PDE) system in density and mass flux variables, with momentum dissipation modeled using the Darcy-Wiesbach friction approximation. Compressors are modeled as scaling up the pressure of gas between inlet and outlet. The governing equations for all network components are spatially discretized and assembled into a nonlinear differential-algebraic equation (DAE) system, which synthesizes above-ground pipeline and subsurface reservoir dynamics into a single reduced-order model. We seek to maximize an objective function that quantifies economic profit and network efficiency subject to the flow equations and inequalities that represent operating limitations. The problem is solved using a primal-dual interior point solver and the solutions are validated in computational experiments and simulations on several pipeline test networks to demonstrate the effectiveness of the proposed methodology.

math.OC↗

arXiv2020

What is the gradient of a scalar function of a symmetric matrix ?

Perusal of research articles that deal with the topic of matrix calculus reveal two different approaches to calculation of the gradient of a real-valued function of a symmetric matrix leading to two different results. In the mechanics and physics communities, the gradient is calculated using the definition of a \frechet derivative, irrespective of whether the argument is symmetric or not. However, members of the statistics, economics, and electrical engineering communities use another notion of the gradient that explicitly takes into account the symmetry of the matrix, and this "symmetric gradient" $G_s$ is reported to be related to the gradient $G$ computed from the \frechet derivative with respect to a general matrix as $G_s = G + G^T - G \circ I$, where $\circ$ denotes the elementwise Hadamard product of the two matrices. We demonstrate that this relation is incorrect, and reconcile both these viewpoints by proving that $G_s = \mathrm{sym}(G)$.

math.NA↗

arXiv2018

Landmark Placement for Localization in a GPS-denied Environment

Path planning algorithms for unmanned aerial or ground vehicles, in many surveillance applications, rely on Global Positioning System (GPS) information for localization. However, disruption of GPS signals, by intention or otherwise, can render these plans and algorithms ineffective. This article provides a way of addressing this issue by utilizing stationary landmarks to aid localization in such GPS-disrupted or GPS-denied environment. In particular, given the vehicle's path, we formulate a landmark-placement problem and present algorithms to place the minimum number of landmarks while satisfying the localization, sensing, and collision-avoidance constraints. The performance of such a placement is also evaluated via extensive simulations on ground robots.

math.OC↗

arXiv2018

Fast Ewald summation for Green's functions of Stokes flow in a half-space

Recently, Gimbutas et al derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these half-space Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real-space (short-range interactions) and one in Fourier-space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier-space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison to the free-space evaluation, greater computational savings is also achieved when compared to their respective direct sums.

math.NA↗