arXiv subjects

Kaarthik Sundar

Publications and source records attributed to Kaarthik Sundar.

At least 37 records · Page 2Linked to original sources

arXiv2024

Locational Marginal Pricing of Energy in Pipeline Transport of Natural Gas and Hydrogen with Carbon Offset Incentives

We propose an optimization formulation for locational pricing of energy transported through a pipeline network that carries mixtures of natural gas and hydrogen from distributed sources to consumers. The objective includes the economic value provided by the pipeline to consumers of energy and suppliers of natural gas and green hydrogen, as well as incentives to lower carbon emissions by consuming the latter instead of the former. The optimization is subject to the physics of gas flow and mixing in the pipeline network as well as engineering limits. In addition to formulating this mathematical program, we synthesize the Lagrangian and derive analytical expressions for the dual variables. We propose that the dual solution can be used to derive locational marginal prices of natural gas, hydrogen, and energy, as well as the decarbonization premium paid by consumers that receive hydrogen. We derive several properties of solutions obtained using the proposed market mechanism, and demonstrate them using case studies for standard 8-node and 40-node pipeline test networks. Finally, we show that optimization-based analysis of the type proposed here is critical for making sound decisions about economic policy and infrastructure expansion for blending green hydrogen into existing natural gas pipelines.

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arXiv2024

Deep Reinforcement Learning-Based Approach for a Single Vehicle Persistent Surveillance Problem with Fuel Constraints

This article presents a deep reinforcement learning-based approach to tackle a persistent surveillance mission requiring a single unmanned aerial vehicle initially stationed at a depot with fuel or time-of-flight constraints to repeatedly visit a set of targets with equal priority. Owing to the vehicle's fuel or time-of-flight constraints, the vehicle must be regularly refueled, or its battery must be recharged at the depot. The objective of the problem is to determine an optimal sequence of visits to the targets that minimizes the maximum time elapsed between successive visits to any target while ensuring that the vehicle never runs out of fuel or charge. We present a deep reinforcement learning algorithm to solve this problem and present the results of numerical experiments that corroborate the effectiveness of this approach in comparison with common-sense greedy heuristics.

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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.

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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.

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arXiv2024

Stochastic Finite Volume Method for Uncertainty Management in Gas Pipeline Network Flows

Natural gas consumption by users of pipeline networks is subject to increasing uncertainty that originates from the intermittent nature of electric power loads serviced by gas-fired generators. To enable computationally efficient optimization of gas network flows subject to uncertainty, we develop a finite volume representation of stochastic solutions of hyperbolic partial differential equation (PDE) systems on graph-connected domains with nodal coupling and boundary conditions. The representation is used to express the physical constraints in stochastic optimization problems for gas flow allocation subject to uncertain parameters. The method is based on the stochastic finite volume approach that was recently developed for uncertainty quantification in transient flows represented by hyperbolic PDEs on graphs. In this study, we develop optimization formulations for steady-state gas flow over actuated transport networks subject to probabilistic constraints. In addition to the distributions for the physical solutions, we examine the dual variables that are produced by way of the optimization, and interpret them as price distributions that quantify the financial volatility that arises through demand uncertainty modeled in an optimization-driven gas market mechanism. We demonstrate the computation and distributional analysis using a single-pipe example and a small test network.

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arXiv2024

Dynamic Optimization and Optimal Control of Hydrogen Blending Operations in Natural Gas Networks

We present a dynamic model for the optimal control problem (OCP) of hydrogen blending into natural gas pipeline networks subject to inequality constraints. The dynamic model is derived using the first principles partial differential equations (PDEs) for the transport of heterogeneous gas mixtures through long distance pipes. Hydrogen concentration is tracked together with the pressure and mass flow dynamics within the pipelines, as well as mixing and compatibility conditions at nodes, actuation by compressors, and injection of hydrogen or natural gas into the system or withdrawal of the mixture from the network. We implement a lumped parameter approximation to reduce the full PDE model to a differential algebraic equation (DAE) system that can be easily discretized and solved using nonlinear optimization or programming (NLP) solvers. We examine a temporal discretization that is advantageous for time-periodic boundary conditions, parameters, and inequality constraint bound values. The method is applied to solve case studies for a single pipe and a multi-pipe network with time-varying parameters in order to explore how mixing of heterogeneous gases affects pipeline transient optimization.

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arXiv2024

Exact and Heuristic Approaches for the Stochastic N-k Interdiction in Power Grids

The article introduces the stochastic N-k interdiction problem for power grid operations and planning that aims to identify a subset of k components (out of N components) that maximizes the expected damage, measured in terms of load shed. Uncertainty is modeled through a fixed set of outage scenarios, where each scenario represents a subset of components removed from the grid. We formulate the stochastic N-k interdiction problem as a bi-level optimization problem and propose two algorithmic solutions. The first approach reformulates the bi-level stochastic optimization problem to a single level, mixed-integer linear program (MILP) by dualizing the inner problem and solving the resulting problem directly using a MILP solver to global optimality. The second is a heuristic cutting-plane approach, which is exact under certain assumptions. We compare these approaches in terms of computation time and solution quality using the IEEE-Reliability Test System and present avenues for future research.

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arXiv2023

Fairly Extreme: Minimizing Outages Equitably

This paper focuses on the problem of minimizing the outages due to extreme events on the power grid equitably among all customers of the grid. The paper presents two ways of incorporating fairness into the existing formulations that seek to minimize the total outage in the power grid. The first method is motivated by existing literature on incorporating fairness in optimization problems and this is done by modifying the problem's objective function. The second method introduces a novel notion of fairness, termed $\varepsilon$-fairness, that can be incorporated into existing problem formulations through a single second-order cone constraint. Both these methods are very general and can be used to incorporate fairness in existing planning and operational optimization problems in the power grid and beyond. Extensive computational case studies that examine the effectiveness of both these methods to characterize fairness is presented followed by conclusions and ways forward.

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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.

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arXiv2023

Optimization of Hydrogen Blending in Natural Gas Networks for Carbon Emissions Reduction

We present an economic optimization problem for allocating the flow of natural gas and hydrogen blends through a large-scale transportation pipeline network. Physical flow of the gas mixture is modeled using a steady-state relation between pressure decrease and flow rate, which depends on mass concentration of the constituents as it varies by location in the network. The objective reflects the economic value provided by the system, accounting for delivered energy in withdrawn flows, the cost of natural gas and hydrogen injections, and avoided carbon emissions. The problem is solved subject to physical flow equations, nodal balance and mixing laws, and engineering inequality constraints. The desired energy delivery rate and minimum hydrogen concentration can be specified as upper and lower bound values, respectively, of inequality constraints, and we examine the sensitivity of the physical pressure and flow solution to these parameters for two test networks. The results confirm that increasing hydrogen concentration requires greater energy expended for compression to deliver the same energy content, and the formulation could be used for valuation of the resulting mitigation of carbon emissions.

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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.

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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.

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arXiv2022

Heuristics for Multi-Vehicle Routing Problem Considering Human-Robot Interactions

Unmanned ground vehicles (UGVs) are being used extensively in civilian and military applications for applications such as underground mining, nuclear plant operations, planetary exploration, intelligence, surveillance and reconnaissance (ISR) missions and manned-unmanned teaming. We consider a multi-objective, multiple-vehicle routing problem in which teams of manned ground vehicles (MGVs) and UGVs are deployed respectively in a leader-follower framework to execute missions with differing requirements for MGVs and UGVs while considering human-robot interactions (HRI). HRI studies highlight the costs of managing a team of follower UGVs by a leader MGV. This paper aims to compute feasible paths, replenishments, team compositions and number of MGV-UGV teams deployed such that the requirements for MGVs and UGVs for the missions are met and the path, replenishment, HRI and team deployment costs are at minimum. The problem is first modeled as a a mixed-integer linear program (MILP) that can be solved to optimality by off-the-shelf commercial solvers for small-sized instances. For larger instances, a variable neighborhood search algorithm is offered to compute near optimal solutions and address the challenges that arise when solving the combinatorial multi-objective routing optimization problem. Finally, computational experiments that corroborate the effectiveness of the proposed algorithms are presented.

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arXiv2021

A Branch-and-Price Algorithm for a Team Orienteering Problem with Fixed-Wing Drones

This paper formulates a team orienteering problem with multiple fixed-wing drones and develops a branch-and-price algorithm to solve the problem to optimality. Fixed-wing drones, unlike rotary drones, have kinematic constraints associated with them, thereby preventing them to make on-the-spot turns and restricting them to a minimum turn radius. This paper presents the implications of these constraints on the drone routing problem formulation and proposes a systematic technique to address them in the context of the team orienteering problem. Furthermore, a novel branch-and-price algorithm with branching techniques specific to the constraints imposed due to fixed-wing drones are proposed. Extensive computational experiments on benchmark instances corroborating the effectiveness of the algorithms are also presented.

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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.

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arXiv2021

Deployable, Data-Driven Unmanned Vehicle Navigation System in GPS-Denied, Feature-Deficient Environments

This paper presents a novel data-driven navigation system to navigate an Unmanned Vehicle (UV) in GPS-denied, feature-deficient environments such as tunnels, or mines. The method utilizes landmarks that vehicle can deploy and measure range from to enable localization as the vehicle traverses its pre-defined path through the tunnel. A key question that arises in such scenario is to estimate and reduce the number of landmarks that needs to be deployed for localization before the start of the mission, given some information about the environment. The main focus is to keep the maximum position uncertainty at a desired value. In this article, we develop a novel vehicle navigation system in GPS-denied, feature-deficient environment by combining techniques from estimation, machine learning, and mixed-integer convex optimization. This article develops a novel, systematic method to perform localization and navigate the UV through the environment with minimum number of landmarks while maintaining desired localization accuracy. We also present extensive simulation experiments on different scenarios that corroborate the effectiveness of the proposed navigation system.

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arXiv2021

Routing Problem for Unmanned Aerial Vehicle Patrolling Missions -- A Progressive Hedging Algorithm

The paper presents a two-stage stochastic program to model a routing problem involving an Unmanned Aerial Vehicle (UAV) in the context of patrolling missions. In particular, given a set of targets and a set of supplemental targets corresponding to each target, the first stage decisions involve finding the sequence in which the vehicle has to visit the set of targets. Upon reaching each target, the UAV collects information and if the operator of the UAV deems that the information collected is not of sufficient fidelity, then the UAV has to visit all the supplemental targets corresponding to that target to collect additional information before proceeding to visit the next target. The problem is solved using a progressive hedging algorithm and extensive computational results corroborating the effectiveness of the proposed model and the solution methodology is presented.

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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.

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