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Geothermal district heating development has been gaining momentum in Europe with numerous deep geothermal installations and projects currently under development. With the increasing density of geothermal wells, questions related to the optimal and sustainable reservoir exploitation become more and more important. A quantitative understanding of the complex thermo-hydraulic interaction between tightly deployed geothermal wells in heterogeneous temperature and permeability fields is key for a maximum sustainable use of geothermal resources. Motivated by the geological settings of the Upper Jurassic aquifer in the Greater Munich region, we develop a computational model based on finite element analysis and gradient-free optimization to simulate groundwater flow and heat transport in hot sedimentary aquifers, and numerically investigate the optimal positioning and spacing of multi-well systems. Based on our numerical simulations, net energy production from deep geothermal reservoirs in sedimentary basins by smart geothermal multi-well arrangements provides significant amounts of energy to meet heat demand in highly urbanized regions. Our results show that taking into account heterogeneous permeability structures and a variable reservoir temperature may drastically affect the results in the optimal configuration. We demonstrate that the proposed numerical framework is able to efficiently handle generic geometrical and geological configurations, and can be thus flexibly used in the context of multi-variable optimization problems. Hence, this numerical framework can be used to assess the extractable geothermal energy from heterogeneous deep geothermal reservoirs by the optimized deployment of smart multi-well systems., Leibniz-Gemeinschaft https://doi.org/10.13039/501100001664

The correct choice of interface conditions and effective parameters for coupled macroscale free-flow and porous-medium models is crucial for a complete mathematical description of the problem under consideration and for accurate numerical simulation of applications. We consider single-fluid-phase systems described by the Stokes–Darcy model. Different sets of coupling conditions for this model are available. However, the choice of these conditions and effective model parameters is often arbitrary. We use large-scale lattice Boltzmann simulations to validate coupling conditions by comparison of the macroscale simulations against pore-scale resolved models. We analyse three settings (lid-driven cavity over a porous bed, infiltration problem and general filtration problem) with different geometrical configurations (channelised and staggered distributions of solid grains) and different sets of interface conditions. Effective parameters for the macroscale models (permeability tensor, boundary layer constants) are computed numerically for each geometrical configuration. Numerical simulation results demonstrate the sensitivity of the coupled Stokes–Darcy problem to the location of the sharp fluid–porous interface, the effective model parameters and the interface conditions., Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, Projekt DEAL

We construct an explicit transitive free action of a Banach space of H\"older functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths., Comment: Final version to appear in Proceedings of the London Mathematical Society. 34 pages

The interaction between automated and human-driven vehicles in mixed (human/automated) platoons is far from understood. To study this interaction, the notion of head-to-tail string stability was proposed in the literature. Head-to-tail string stability is an extension of the standard string stability concept where, instead of asking every vehicle to achieve string stability, a lack of string stability is allowed due to human drivers, provided it can be suitably compensated by automated vehicles sparsely inserted in the platoon. This work introduces a theoretical framework for the problem of head-to-tail string stability of mixed platoons: it discusses a suitable vehicle-following human driver model to study mixed platoons, and it gives a reduced-order design strategy for head-to-tail string stability only depending on three gains. The work further discusses the safety limitations of the head-to-tail string stability notion, and it shows that safety improvements can be attained by an appropriate reduced-order design strategy only depending on two additional gains. To validate the effectiveness of the design, linear and nonlinear simulations show that the string stability/safety trade-offs of the proposed reduced-order design are comparable with those resulting from full-order designs.

In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.

International audience

Automation of Managed Pressure Drilling (MPD) allows for fast and accurate pressure control in drilling operations. The achievable performance in automated MPD with model-based controllers is determined by the controller and, indirectly, also by the hydraulics model used for controller synthesis. On the one hand, such a hydraulics model should accurately capture essential flow dynamics of the system such as, e.g., wave propagation effects, for which typically complex models are needed. On the other hand, a suitable model should be simple enough to facilitate high-performance controller design as well as to support fast simulation studies supporting well scenario analysis. This paper shows that low-order models in terms of delay differential equations can effectively meet these requirements. Moreover, we propose a data-based model reduction technique to construct these low-order delay models. Next, based on this reduced-complexity model, a novel controller is designed to regulate the downhole pressure. Simulation results confirm that this controller outperforms existing pressure controllers in realistic drilling scenarios related to the mitigation of liquid kicks and mud losses encountered when drilling into high- or low-pressure zones.

We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size.

This paper proposes a data-driven model reduction approach on the basis of noisy data with a known noise model. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a large-dimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lower-dimensional QMI by applying the data reduction concept. Additionally, we present alternative procedures to compute a priori and a posteriori upper bounds with respect to the true system generating the data. Finally, the proposed techniques are illustrated by means of application to an example of a system of a cart with a double-pendulum.

We study identifiability for linear, continuous time-invariant systems. We state sufficient conditions on an input trajectory u and a finite number of its derivatives, in order to be able to deduce all differential equations describing the data-generating system from any corresponding input-output sequence (u, y) on a finite interval.

In this two-part paper we develop a unifying framework for the analysis of the feasibility of the power flow equations for DC power grids with constant-power loads. In Part I of this paper we present a detailed introduction to the problem of power flow feasibility of such power grids, and the associated problem of selecting a desirable operating point which satisfies the power flow equations. We introduce and identify all long-term voltage semi-stable operating points, and show that there exists a one-to-one correspondence between such operating points and the constant power demands for which the power flow equations are feasible. Such operating points can be found by solving an initial value problem, and a parametrization of these operating points is obtained. In addition, we give a full characterization of the set of all feasible power demands, and give a novel proof for the convexity of this set.

Effective design of autopilots for fixed-wing unmanned aerial vehicles (UAVs) is still a great challenge, due to unmodeled effects and uncertainties that these vehicles exhibit during flight. Unmodeled effects and uncertainties comprise longitudinal/lateral cross-couplings, as well as poor knowledge of equilibrium points (trimming points) of the UAV dynamics. The main contribution of this article is a new adaptive autopilot design, based on uncertain Euler-Lagrange dynamics of the UAV and where the control can explicitly take into account under-actuation in the dynamics, reduced structural knowledge of cross-couplings and trimming points. This system uncertainty is handled via appropriately designed adaptive laws: stability of the controlled UAV is analyzed. Hardware-in-the-loop tests, comparisons with an Ardupilot autopilot and with a robustified autopilot validate the effectiveness of the control design, even in the presence of strong saturation of the UAV actuators.

In this paper, we study positive invariance and attractivity properties for nonlinear control systems which are monotone with respect to proper cones. Monotonicity simplifies such analysis for specific sets defined by the proper cones. Instead of Lyapunov functions, a pair of so called paths in the state space and input space play important roles. As applications, our results are utilized for analysis of asymptotic stability and also input-to-state stability on proper cones. The results are illustrated by means of examples.

We present a modified version of the time-marching algorithm (TMA) [van Ruymbeke et al., 2005, 2010] for predicting the linear viscoelasticity of monodisperse symmetric star polymer melts. Several new elements were added to the original TMA model. In particular, the remaining fraction of the initial tube segments is considered as a function of time, while taking into account the past relaxation history of each molecular segment. We validate the TMA model and, for the first time, compare it with the so-called BoB model [Das et al., 2006]. The predictions obtained with the two models are compared to a large set of experimental data, which cover a broad range of arm molecular weights (from 1.5 to 55 entanglements per arm) of different chemistries (polystyrene, polybutadiene, and polyisoprene). We indicate the significant differences between the two approaches, which mainly affect the choice of material parameters of the models. We then point out a systematic deviation of the TMA model in accurately predicting the intermediate regime of relatively short star chains, while the BoB model fails in correctly describing the plateau modulus of these short chains. From our point of view, both disparities have the same origin and, in the case of the TMA, can be solved by removing the high-frequency mode contribution to the contour length fluctuation of the primitive path. An excellent agreement between data and theory is then obtained for all molecular weights of the arms, which indicates the capability of the TMA to provide a quantitative prediction of the rheology of monodisperse star-shaped polymers. A very good agreement is also obtained with the BoB model, despite the expected discrepancy observed with short star chains. This proves the fact that current tube-based models are successful in quantitative predictions of linear rheology of monodisperse star-shaped polymers.

We propose a novel reduction approach for switched linear systems with a fixed mode sequence based on subspaces related to the (time-varying) reachable and unobservable spaces. These subspaces are defined in such a way that they can be used to construct a weak Kalman decomposition, which is then in turn used to define a reduced switched linear system with an identical input-output behavior. The proposed method is illustrated with a low dimensional academic example.

This paper proposes a Real-Time Market (RTM) platform for an aggregator and its corresponding prosumers to participate in the electricity wholesale market. The proposed energy market platform is modeled as a bilevel optimization problem where the aggregator and the prosumers are considered as self-interest agents. The current state-of-the-art Mathematical Programming with Equilibrium Constraints (MPEC) and Mixed-Integer Programming (MIP) based approaches to solve bilevel optimization problems are not satisfactory for real-time applications. The computation time for such approaches grows exponentially as the number of prosumers and decision variables increase. This paper presents a convex optimization problem which can capture a subset of the set of global optima of the bilevel problem as its optimal solution.

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $\Delta$ with Dirichlet boundary conditions and the free Laplace operator $\Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(\Delta)-g(\Delta_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $\Delta_1$ and $\Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(\Delta) - g(\Delta_1) - g(\Delta_2) + g(\Delta_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^\frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation., Comment: 33 pages, 1 figure, small typos corrected

Switch-based adaptive dynamic programming (ADP) is an optimal control problem in which a cost must be minimized by switching among a family of dynamical modes. When the system dimension increases, the solution to switch-based ADP is made prohibitive by the exponentially increasing structure of the value function approximator and by the exponentially increasing modes. This technical correspondence proposes a distributed computational method for solving switch-based ADP. The method relies on partitioning the system into agents, each one dealing with a lower dimensional state and a few local modes. Each agent aims to minimize a local version of the global cost while avoiding that its local switching strategy has conflicts with the switching strategies of the neighboring agents. A heuristic algorithm based on the consensus dynamics and Nash equilibrium is proposed to avoid such conflicts. The effectiveness of the proposed method is verified via traffic and building test cases.

openaire: EC/H2020/856602/EU//FINEST TWINS For platoons under the multiple-predecessor following (MPF) topology, communication delays can compromise both the internal stability and string stability. The most straightforward solution to guarantee stability is by increasing the time headway. However, time headway plays a significant role in road capacity and increasing its value is in contrast with the idea of platooning. In this study, internal stability and string stability of platoons suffering from communication delays are investigated and a lower bound for the time headway is proposed. Using this bound, platoons do not need to massively increase the time headway in order to compensate for the effects of communications delays. Finally, we evaluate the proposed lower bound on the time headway and the simulation results demonstrate its effectiveness.

We discuss a pseudo-parabolic equation modelling two-phase flow in porous media, which includes dynamic effects in the capillary pressure. We extend the results obtained previously for linear higher order terms and investigate the existence of travelling wave solutions in the non-linear and degenerate case. This case leads to non-smooth travelling waves, as well as to a discontinuous capillary pressure. Keywords: Dynamic capillary pressure; Non-linear; Degenerate; Pseudo-parabolic equations; Sharp travelling waves

We show non-asymptotic geometric convergence of Sinkhorn iterates to the Schrödinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^d$. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. More precisely, we provide exponential $\mathrm{L}^1$ and pointwise convergence of the iterates (resp. their gradient and Hessian) to Schrödinger potentials (resp. their gradient and Hessian) for large enough values of the regularization parameter. As a corollary, we establish exponential convergence of Sinkhorn plans and bridges w.r.t. a symmetric relative entropy. Up to the authors' knowledge, these are the first results which establish geometric convergence of Sinkhorn algorithm in a general setting without assuming bounded cost functions or compactly supported marginals. Our results are proven following a probabilistic approach that rests on integrated semiconvexity estimates for Sinkhorn iterates that are of independent interest., 51 pages

We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove $$L^2$$ L 2 and $$L^2-L^{\infty }$$ L 2 - L ∞ observability estimates on unbounded domains $$\omega $$ ω for a restricted class of initial data. This data includes a class of compactly supported piecewise $$C^1$$ C 1 functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from $$\omega ^c=\Omega $$ ω c = Ω , a connected bounded domain, is observable, but data centered over $$\Omega $$ Ω must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.

The multiple-predecessor following (MPF) topology is used in vehicle platoons to make it robustly string stable and reduce the minimum employable time headway. It has been demonstrated that communication imperfections such as time delays coming from wireless communications can affect string stability as well as the minimum time headway required to guarantee string stability. Specifically, it was shown that the larger the time delay, the longer the minimum time headway will be. However, by utilizing on-board vehicle sensors, such as radar, lidar and cameras, the distance and speed of nearby vehicles can be measured almost instantaneously, i.e., with almost no delay. Another effective parameter on string stability and minimum time headway is the heterogeneity of the vehicles. Due to the immense complexity of the MPF topology, string stability analysis of this topology in literature has been confined to homogeneous platoons. In this paper, we consider the case of heterogeneous platoons under the MPF topology with the use of the combination of sensors and wireless communications for receiving information. Following that, we find conditions to guarantee the internal and string stability for the heterogeneous case and propose the minimum time headway required to guarantee string stability. Finally, we provide a table, in which we propose the minimum time headway for two other wireless communication scenarios as well: (i) having no communication delay and (ii) having fully-delayed information, i.e., all information, whether it comes from the ego vehicle or its predecessors, is delayed. In addition to exploring the analysis of string stability for the vehicles with more possible connections (vehicles after the $r^{th}$ vehicle, when information from $r$ immediate vehicles is used), we study the string stability conditions (with which we aim at avoiding collisions) and find the minimum time headway for the first few vehicles (vehicle $r$ and all its predecessors). Numerical results clearly show the effectiveness of the proposed lower bounds.

A fast method is presented for the calculation of the MSD and the MWD of polymers obtained via step-growth polymerization of polyfunctional monomers bearing identical reactive groups (i.e., systems of type "Afi"). Using this method, the complete distribution can be calculated rapidly, not just the statistical averages of the polymer population such as Mn or Mw. The computed MSD and MWD give more insight than these averages and can be compared to similar data measured on actual polymer systems. The low- and intermediate molecular size/weight part of the distribution curves are calculated using a recurrence scheme, while the high-molecular tail (large and very large polymers) of the distributions is derived from an asymptotic approximation of the associated generating functions.

In this paper, we investigate a class of pseudo-parabolic equations. Such equations model two-phase flow in porous media where dynamic effects are included in the capillary pressure. The existence and uniqueness of a weak solution are proved, and error estimates for an Euler implicit time discretization are obtained. Copyright © 2011 John Wiley & Sons, Ltd.

There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing $\Omega$. This domain is sharp in the sense that there is no larger domain for which this is true. If $\Omega$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighborhood of $\mathcal{E}(\Omega)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case., Comment: 16 pages 1 figure, some additional explanations and references added

The concept of dissipativity, as introduced by Jan Willems, is one of the cornerstones of systems and control theory. Typically, dissipativity properties are verified by resorting to a mathematical model of the system under consideration. In this paper, we aim at assessing dissipativity by computing storage functions for linear systems directly from measured data. As our main contributions, we provide conditions under which dissipativity can be ascertained from a finite collection of noisy data samples. Three different noise models will be considered that can capture a variety of situations, including the cases that the data samples are noise-free, the energy of the noise is bounded, or the individual noise samples are bounded. All of our conditions are phrased in terms of data-based linear matrix inequalities, which can be readily solved using existing software packages.