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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.

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.

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.

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.

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.

We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th order error bound in the enlarged energy norm. This scheme is three-point in each spatial direction, and it exploits additional sought functions which approximate 2nd order non-mixed spatial derivatives of the solution to the equation. At the first time level, a similar two-level in time scheme is applied, with no derivatives of the data. No iterations are required to implement the scheme. We also present results of various 3D numerical experiments that demonstrate a very high accuracy of the scheme for smooth data, its advantages in the error behavior over the classical explicit 2nd order scheme for nonsmooth data as well and an example of the wave in a layered medium initiated by the Ricker-type wavelet source function., Comment: 19 pages, 6 figures, 8 tables

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.

A posteriori reduced-order models, e.g. proper orthogonal decomposition, are essential to affordably tackle realistic parametric problems. They rely on a trustful training set, that is a family of full-order solutions (snapshots) representative of all possible outcomes of the parametric problem. Having such a rich collection of snapshots is not, in many cases, computationally viable. A strategy for data augmentation, designed for parametric laminar incompressible flows, is proposed to enrich poorly populated training sets. The goal is to include in the new, artificial snapshots emerging features, not present in the original basis, that do enhance the quality of the reduced-order solution. The methodologies devised are based on exploiting basic physical principles, such as mass and momentum conservation, to devise physically-relevant, artificial snapshots at a fraction of the cost of additional full-order solutions. Interestingly, the numerical results show that the ideas exploiting only mass conservation (i.e., incompressibility) are not producing significant added value with respect to the standard linear combinations of snapshots. Conversely, accounting for the linearized momentum balance via the Oseen equation does improve the quality of the resulting approximation and therefore is an effective data augmentation strategy in the framework of viscous incompressible laminar flows., Comment: 27 pages, 8 figures, 2 tables

Yahoo's native advertising (also known as Gemini native) serves billions of ad impressions daily, reaching a yearly run-rate of many hundred of millions USD. Driving the Gemini native models that are used to predict both click probability (pCTR) and conversion probability (pCONV) is OFFSET - a feature enhanced collaborative-filtering (CF) based event prediction algorithm. \offset is a one-pass algorithm that updates its model for every new batch of logged data using a stochastic gradient descent (SGD) based approach. Since OFFSET represents its users by their features (i.e., user-less model) due to sparsity issues, rule based hard frequency capping (HFC) is used to control the number of times a certain user views a certain ad. Moreover, related statistics reveal that user ad fatigue results in a dramatic drop in click through rate (CTR). Therefore, to improve click prediction accuracy, we propose a soft frequency capping (SFC) approach, where the frequency feature is incorporated into the OFFSET model as a user-ad feature and its weight vector is learned via logistic regression as part of OFFSET training. Online evaluation of the soft frequency capping algorithm via bucket testing showed a significant 7.3% revenue lift. Since then, the frequency feature enhanced model has been pushed to production serving all traffic, and is generating a hefty revenue lift for Yahoo Gemini native. We also report related statistics that reveal, among other things, that while users' gender does not affect ad fatigue, the latter seems to increase with users' age., Comment: In Proc. CIKM'2019. arXiv admin note: text overlap with arXiv:2111.07866 by other authors

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.

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.

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.

Few-layer graphene deposited on semiconductor nanorods separated by undoped spacers has been studied in perspective for the fabrication of stable nanoresonators. We show that an applied bias between the graphene layer and the nanorod substrate affects the graphene electrode in two ways: 1) by a change of the carrier concentration in graphene and 2) by inducing strain, as demonstrated by the Raman spectroscopy. The capacitance of the investigated structures scales with the area of graphene in contact with the nanorods. Due to the reduced contact surface, the efficiency of graphene gating is one order of magnitude lower than for a comparable structure without nanorods. The shift of graphene Raman modes observed under bias clearly shows the presence of electrostatically-induced strain and only a weak modification of carrier concentration, both independent of number of graphene layers. A higher impact of bias on strain was observed for samples with a larger contact area between the graphene and the nanorods which shows perspective for the construction of sensors and nanoresonator devices.

We deal with the reduced four-equation model for the dynamics of heterogeneous compressible binary mixtures with the stiffened gas equations of state. We study its further reduced form, with the excluded volume concentrations, and with a quadratic equation for the common pressure of the components; this form can be called a quasi-homogeneous form. We prove new properties of the equation, derive simple formulas for the squared speed of sound and present an alternative proof for a formula that relates it to the squared Wood speed of sound; also, a short derivation of the pressure balance equation is given. For the first time, we introduce regularizations of the heterogeneous model (in the quasi-homogeneous form). Previously, regularizations of such type were developed only for the homogeneous mixtures of perfect polytropic gases, and it was unclear how to cover the case considered here. In the 1D case, based on these regularizations, we construct new explicit two-level in time and symmetric three-point in space finite-difference schemes without limiters, and provide numerical results for various flows with shock waves., Comment: 19 pages, 7 figures, 3 tables

We examine nonlinear and adaptive linear control systems that model compressor-actuated dynamics of natural gas flow in pipeline networks. A model-predictive controller (MPC) is developed for feedback control of compressor actions in which the internal optimization over the local time horizon is constrained by the dynamics of either the nonlinear system or the adaptive linear system. Stability of the local linear system is established and a rigorous bound on the error between the solutions of the nonlinear and linear systems is derived and used to devise situations when the linear MPC may be used instead of the nonlinear MPC without a significant difference between their respective predictions. We use several test networks to compare the performances of various controllers that involve nonlinear and adaptive linear models as well as moving-horizon and single-interval optimization. Our results demonstrate that the proposed moving-horizon MPC is well-equipped to adapt in local time to changes in system parameters and has the ability to reduce total computational costs by orders of magnitude relative to conventional transient optimization methods.

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.

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.

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., Comment: 12 pages

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., Comment: In Proceedings of the 2023 American Control Conference

We formulate a control system model for the distributed flow of mixtures of highly heterogeneous gases through large-scale pipeline networks with time-varying injections of constituents, withdrawals, and control actions of compressors. This study is motivated by the proposed blending of clean hydrogen into natural gas pipelines as an interim means to reducing end use carbon emissions while utilizing existing infrastructure for its planned lifetime. We reformulate the partial differential equations for gas dynamics on pipelines and balance conditions at junctions using lumped elements to a sparse nonlinear differential algebraic equation system. Our key advance is modeling the mixing of constituents in time throughout the network, which requires doubling the state space needed for a single gas and increases numerical ill-conditioning. The reduced model is shown to be a consistent approximation of the original system, which we use as the dynamic constraints in a model-predictive optimal control problem for minimizing the energy expended by applying time-varying compressor operating profiles to guarantee time-varying delivery profiles subject to system pressure limits. The optimal control problem is implemented after time discretization using a nonlinear program, with validation of the results done using a transient simulation. We demonstrate the methodology for a small test network, and discuss scalability and potential applications., Comment: In Proceedings of the 2023 American Control Conference

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach, and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. We demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

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., Comment: 12 pages