arXiv subjects

Anatoly Zlotnik

Publications and source records attributed to Anatoly Zlotnik.

At least 19 recordsLinked to original sources

arXiv2026

General Revenue Adequacy Conditions for Energy Transport Networks

Optimization is widely used to determine the physical and financial exchange of wholesale electricity in organized markets. Guarantees of solution optimality and feasibility rest largely on convexity, which is not in general a characteristic of the governing equations for power grid and gas pipeline networks. Policy decisions that base the scheduling and locational pricing of electricity transactions on optimization rely on the guarantee of revenue adequacy, which ensures that the market administrator will collect enough payments in congestion rents to settle financial transmission rights. Developing a similar mechanism for locational trade valuation of natural gas also requires assurance that pricing outcomes are revenue adequate, and also cover the costs of gas compressor operation. However, it has been shown that the AC power flow equations are in general non-convex and hence conditions for guaranteeing revenue adequacy in optimal power flow solutions are challenging to generalize. In this study, we develop a general formal mathematical setting for nonlinear physical network flows and examine the conditions for revenue adequacy. The result is verified for DC and AC power flow as well as steady-state gas flow in a pipeline network.

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arXiv2026

Robust Quantum Control for Bragg Pulse Design in Atom Interferometry

We formulate a robust optimal control algorithm to synthesize minimum energy pulses that can transfer a cold atom system into various momentum states. The algorithm uses adaptive linearization of the evolution operator and sequential quadratic programming to iterate the control towards a minimum energy pulse that achieves optimal target state fidelity. Robustness to parameter variation is achieved using Legendre polynomial approximation over the domain of variation. The method is applied to optimize the Bragg beamsplitting operation in ultra-cold atom interferometry. Even in the presence of 10-40% variability in the initial momentum dispersion of the atomic cloud and the intensity of the optical pulse, the algorithm reliably converges to a control protocol that robustly achieves unprecedented momentum levels with high fidelity for a single-frequency multi-photon Bragg diffraction scheme (e.g. $|\pm 40\hbar k\rangle$). We show the advantages of our method by comparison to stochastic optimization using sampled parameter values, provide detailed sensitivity analyses, and performance of the designed pulses is verified in laboratory experiments.

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arXiv2026

Simulating Arbitrage Optimization for Market Monitoring in Gas and Electricity Transmission Networks

We examine market outcomes in energy transport networks with a focus on gas-fired generators, which are producers in a wholesale electricity market and consumers in the natural gas market. Market administrators monitor bids to determine whether a participant wields market power to manipulate the price of energy, reserves, or financial transmission rights. If economic or physical withholding of generation from the market is detected, mitigation is imposed by replacing excessive bids with reference level bids to prevent artificial supply shortages. We review market monitoring processes in the power grid, and present scenarios in small interpretable test networks to show how gas-fired generators can bid in the gas market to alter outcomes in a power market. We develop a framework based on DC optimal power flow (OPF) and steady-state optimal gas flow (OGF) formulations to represent two interacting markets with structured exchange of price and quantity bids. We formulate optimization-based methods to identify market power in a power grid, as well as to identify market conditions that indicate market power being exerted by a generator using gas market bids.

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arXiv2025

Robust Quantum State Generation in Symmetric Spin Networks

In this work, we consider a parameterized Ising model with long-range symmetric pairwise interactions on a network of spin $\frac{1}{2}$ particles. The system is designed with symmetric dynamics, allowing for the reduction of the state space to a subspace defined by the set of Dicke states. We propose a method for designing robust electromagnetic amplitude pulses based on a moment quantization approach. The introduced parameter accounts for uncertainties in the electromagnetic field, resulting in a family of distinct Hamiltonians. By employing a discretized moment-based quantization technique, we design a control pulse capable of simultaneously steering an infinite collection of dynamical systems to compensate for parameter variations. This approach benefits from the duality between the infinite-dimensional parameterized system and its finite-dimensional trucnated moment dynamics. Simulation results demonstrate the efficacy of this method in achieving states of significant interest in quantum sensing, including the GHZ and W states.

quant-ph↗

arXiv2025

Linear System Analysis and Optimal Control of Natural Gas Dynamics in Pipeline Networks

We design nonlinear and adaptive linear model-predictive control (MPC) techniques to minimize operational costs of compressor-actuated dynamics in natural gas pipeline networks. We establish stability of the local linear system and derive rigorous bounds on error between the nonlinear and linear system solutions. These bounds are used to quantify conditions under which the linear MPC can substitute the nonlinear MPC without significant loss of predictive accuracy. Furthermore, we prove and numerically verify that the computational cost of the linear MPC is orders of magnitude lower than that of solving the baseline optimal control problem. Numerical simulations are performed on nontrivial networks to demonstrate that the proposed MPC can effectively adapt to varying load conditions while maintaining nearly 95% optimality.

math.OC↗

arXiv2025

Robust Quantum Gate Preparation in Open Environments

We develop an optimal control algorithm for robust quantum gate preparation in open environments with the state of the quantum system represented using the Lindblad master equation. The algorithm is based on adaptive linearization and iterative quadratic programming to progressively shape the control signal into an optimal form. Robustness is achieved with exponential rates of convergence by introducing uncertain parameters into the master equation and expanding the parameterized state over the basis of Legendre polynomials. We prove that the proposed control algorithm reduces to GRadient Ascent Pulse Engineering (GRAPE) when the robustness portion of the algorithm is bypassed and signal restrictions are relaxed. The control algorithm is applied to prepare Controlled NOT and SWAP gates with high precision. Using only second order Legendre polynomials, the examples showcase unprecedented robustness to 100% parameter uncertainty in the interaction strength between the qubits, while simultaneously compensating for 20% uncertainty in signal intensity. The results could enable new capabilities for robust implementation of quantum gates and circuits subject to harsh environments and hardware limitations.

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arXiv2024

Convergence of Iterative Quadratic Programming for Robust Fixed-Endpoint Transfer of Bilinear Systems

We present a computational method for open-loop minimum-norm control synthesis for fixed-endpoint transfer of bilinear ensemble systems that are indexed by two continuously varying parameters. We suppose that one ensemble parameter scales the homogeneous, linear part of the dynamics, and the second parameter scales the effect of the applied control inputs on the inhomogeneous, bilinear dynamics. This class of dynamical systems is motivated by robust quantum control pulse synthesis, where the ensemble parameters correspond to uncertainty in the free Hamiltonian and inhomogeneity in the control Hamiltonian, respectively. Our computational method is based on polynomial approximation of the ensemble state in parameter space and discretization of the evolution equations in the time domain using a product of matrix exponentials corresponding to zero-order hold controls over the time intervals. The dynamics are successively linearized about control and trajectory iterates to formulate a sequence of quadratic programs for computing perturbations to the control that successively improve the objective until the iteration converges. We use a two-stage computation to first ensure transfer to the desired terminal state, and then minimize the norm of the control function. The method is demonstrated for the canonical uniform transfer problem for the Bloch system that appears in nuclear magnetic resonance, as well as the matter-wave splitting problem for the Raman-Nath system that appears in ultra-cold atom interferometry.

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arXiv2024

Setpoint Tracking and Disturbance Attenuation for Gas Pipeline Flow Subject to Uncertainties using Backstepping

In this paper, we consider the problem of regulating the outlet pressure of gas flowing through a pipeline subject to uncertain and variable outlet flow. Gas flow through a pipe is modeled using the coupled isothermal Euler equations, with the Darcy-Weisbach friction model used to account for the loss of gas flow momentum. The outlet flow variation is generated by a periodic linear dynamic system, which we use as a model of load fluctuations caused by varying consumer demands. We first linearize the nonlinear equations around the equilibrium point and obtain a 2-by-2 coupled hyperbolic partial differential equation (PDE) system expressed in canonical form. Using an observer-based PDE backstepping controller, we demonstrate that the inlet pressure can be manipulated to regulate the outlet pressure to a setpoint, thus compensating for fluctuations in the outlet flow. Furthermore, we extend the observer-based controller to the case when the outlet flow variation is uncertain within a bounded set. In this case, the controller is also capable of regulating the outlet pressure to a neighborhood of the setpoint by manipulating the inlet pressure, even in the presence of uncertain fluctuations in the outlet flow. We provide numerical simulations to demonstrate the performance of the controller.

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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

Boundary Control for Suppressing Chaotic Response to Dynamic Hydrogen Blending in a Gas Pipeline

It is known that periodic forcing of nonlinear flows can result in a chaotic response under certain conditions. Such non-periodic and chaotic solutions have been observed in simulations of heterogeneous gas flow in a pipeline with periodic, time-varying boundary conditions. In this paper, we examine a proportional feedback law for boundary control of a parabolic partial differential equation system that represents the flow of two gases through a pipe. We demonstrate that periodic variation of the mass fraction of the lighter gas at the pipe inlet can result in the chaotic propagation of gas pressure waves, and show that appropriate flow control can suppress this response. We examine phase space solutions for the single pipe system subject to boundary control, and use numerical experiments to characterize conditions for the controller gain to suppress chaos.

math.DS↗

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

Simulation of Gas Mixture Dynamics in a Pipeline Network using Explicit Staggered-Grid Discretization

We develop an explicit second order staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge domain boundary values that are significantly more complex than in the case of a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

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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

Stochastic Active Discretizations for Accelerating Temporal Uncertainty Management of Gas Pipeline Loads

We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based on the stochastic finite volume (SFV) framework that circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. The initial boundary value problem (IBVP) on a set of network-connected one-dimensional domains that represent a pipeline is represented using active discretization of the physical and stochastic spaces, and we evaluate the propagation of uncertainty through network nodes by solving a junction Riemann problem. The adaptivity of our method in refining discretization based on error metrics enables computationally tractable evaluation of intertemporal uncertainty in order to support decisions about timing and quantity of pipeline operations to maximize delivery under transient and uncertain conditions. We illustrate our computational method using simulations for a representative network.

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arXiv2024

Optimal Ensemble Control of Matter-Wave Splitting in Bose-Einstein Condensates

We present a framework for designing optimal optical pulses for the matter-wave splitting of a Bose-Einstein Condensate (BEC) under the influence of experimental inhomogeneities, so that the sample is transferred from an initial rest position into a singular higher diffraction order. To represent the evolution of the population of atoms, the Schroedinger's equation is reinterpreted as a parameterized ensemble of dynamical units that are disparately impacted by the beam light-shift potential in a continuous manner. The derived infinite-dimensional coupled Raman-Nath equations are truncated to a finite system of diffraction levels, and we suppose that the parameter that defines the inhomogeneity in the control applied to the ensemble system is restricted to a compact interval. We first design baseline square pulse sequences for the excitation of BEC beam-splitter states following a previous study, subject to dynamic constraints for either a nominal system assuming no inhomogeneity or for several samples of the uncertain parameter. We then approximate the continuum state-space of the ensemble of dynamics using a spectral approach based on Legendre moments, which is truncated at a finite order. Control functions that steer the BEC system from an equivalent rest position to a desired final excitation are designed using a constrained optimal control approach developed for handling nonlinear dynamics. This representation results in a minimal dimension of the computational problem and is shown to be highly robust to inhomogeneity in comparison to the baseline approach. Our method accomplishes the BEC-splitting state transfer for each subsystem in the ensemble, and is promising for precise excitation in experimental settings where robustness to environmental and intrinsic noise is paramount.

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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

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom (DoFs) where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for non-linear hyperbolic PDEs based on Burgers' and Euler's equations.

math.NA↗

arXiv2023

Transitions from Monotonicity to Chaos in Gas Mixture Dynamics in Pipeline Networks

The blending of hydrogen generated using clean energy into natural gas pipeline networks is proposed in order to utilize existing energy systems for their planned lifetimes while reducing their reliance on fossil fuels. We formulate a system of partial differential equations (PDEs) that govern the flow dynamics of mixtures of gases in pipeline networks under the influence of time-varying compressor and regulator control actions. The formulation is derived for general gas networks that can inject or withdraw arbitrary time-varying mixtures of gases into or from the network at arbitrarily specified nodes. The PDE formulation is discretized in space to form a nonlinear control system that is used to prove that homogeneous mixtures are well-behaved and heterogeneous mixtures may be ill-behaved in the sense of monotone-ordering of solutions. We use numerical simulations to compute interfaces in the parameter region of sinusoidal boundary conditions that delimit monotonic, periodic, and chaotic system responses. The interfaces suggest that any solution in the monotonic response region is not chaotic and will eventually approach a periodic orbit. The results are demonstrated using examples for a single pipeline and a small test network.

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