Sara Grundel

• Doctor of Philosophy
• Max Planck Institute for Dynamics of Complex Technical Systems

Transient gas network simulations can significantly assist in design and operational aspects of gas networks. Models used in these simulations require a detailed framework integrating various models of the network constituents ‐ pipes and compressor stations among others. In this context, the port‐Hamiltonian modelling framework provides an energy‐...

Transient gas network simulations can significantly assist in design and operational aspects of gas networks. Models used in these simulations require a detailed framework integrating various models of the network constituents - pipes and compressor stations among others. In this context, the port-Hamiltonian modelling framework provides an energy-...

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is spanned by snapshots functions over spatial transformations, and we compute our reduced approximation via residual minimization. To speedup the residual mini...

Objective: In the coronavirus disease 2019 (COVID-19) pandemic, child and adolescent psychiatry wards face the risk of severe acute respiratory coronavirus 2 (SARS-CoV-2) introduction and spread within the facility. In this setting, mask and vaccine mandates are hard to enforce, especially for younger children. Surveillance testing may detect infe...

No matter if natural gas, biogas or hydrogen, gas transport needs to be simulated ahead of dispatch to account for volatilities in demand and supply, so denominations are delivered reliably. The emancipation from producing countries alongside the renewable energy transition increases the number of scenarios to be simulated manifold, which in turn r...

Utilizing hydrogen in energy sources plays a crucial step towards a complete transition to renewable energies since production at‐scale is possible. The study works towards developing monitoring and control strategies of proportionally induced hydrogen in existing pipeline infrastructure. With this aim, a scalable model is utilized to compute the f...

In this chapter, we discuss how residential batteries within microgrids (MGs) can be used to provide flexibility to the DSO. On this lowest level of the grid hierarchy we only consider active power demand. In particular, we manipulate the aggregated power demand by charging and discharging residential batteries while neglecting the grid topology.

In this chapter, we present a numerical example of the approach presented in the previous chapters based on modifications of small-scale standard test systems. We emphasise that the sizes of these grids are very small and the power demands very low when compared to real-world grids. For example, active power transmission demand in real-world TSO gr...

In this chapter, we discuss how flexibilities from microgrids conveyed through the distribution level can be utilized in the operation of transmission systems. To this end, we consider both nonlinear (2.2)–(2.7) and semidefinite (2.8)–(2.13) OPF formulations and introduce flexible demand nodes (representing the connections to the DSO level) to anal...

In this chapter, we describe the implementation of the optimal amount of power delivered by the transmission grid in the distribution grid and the microgrids.

In this chapter, we describe how to determine the flexibility of a distribution grid given the flexibility of a set of microgrids (see Chap. 3). Specifically, we compute the minimum, maximum, and optimal active power demand of the entire distribution grid, i.e., the amount of power which must be supplied by the transmission grid.

In this chapter, we present the power-flow equations [5, 9], the optimal power-flow problem, the semidefinite approach for solving optimal power-flow problems [7], and the dynamic structure-preserving power grid model [8] which are relevant to several of the sections in the remainder of the book.

Dissipation of energy — as well as its sibling the increase of entropy — are fundamental facts inherent to any physical system. The concept of dissipativity has been extended to a more general system theoretic setting via port-Hamiltonian systems and this framework is a driver of innovations in many of areas of science and technology. The particula...

To overcome many-query optimization, control, or uncertainty quantification work loads in reliable gas and energy network operations, model order reduction is the mathematical technology of choice. To this end, we enhance the model, solver and reductor components of the morgen platform, introduced in Himpe et al. [J. Math. Ind. 11:13, 2021], and co...

Owing to the ongoing pandemic of COVID-19 an increased interest in epidemiological mathematical modelling arised. Several specific extensions of the classical susceptible-infected-recovered (SIR) modeling approach for the COVID-19 pandemic were developed to make forecasts. However, in all models, parameters have to be fitted on historical data. In...

We are interested in numerical schemes for the simulation of large scale gas networks. Gas transport is described by a simplified Euler equation with a general equation of state for the pressure, including in particular the isentropic as well as the isothermal case. The numerical scheme is based on transformation of conservative variables in Rieman...

We propose a novel framework for model-order reduction of hyperbolic differential equations. The approach combines a relaxation formulation of the hyperbolic equations with a discretization using shifted base functions. Model-order reduction techniques are then applied to the resulting system of coupled ordinary differential equations. On computati...

Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this...

We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term $$\mathbf{f}(\mathbf{E}\mathbf{x}),$$ f ( E x ) , where $$\mathbf{E}$$ E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow...

To overcome many-query optimization, control, or uncertainty quantification work loads in reliable gas and energy network operations, model order reduction is the mathematical technology of choice. To this end, we enhance the model, solver and reductor components of the "morgen" platform, introduced in Himpe et al [J.~Math.~Ind. 11:13, 2021], and c...

To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model redu...

Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this topic using optimization-based control...

We propose a novel framework for model-order reduction of hyperbolic differential equations. The approach combines a relaxation formulation of the hyperbolic equations with a discretization using shifted base functions. Model-order reduction techniques are then applied to the resulting system of coupled ordinary differential equations. On computati...

To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these new circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model...

Modeling and model reduction for gas network simulations.

Triggered by the increasing number of renewable energy sources, the German electricity grid is undergoing a fundamental change from mono to bidirectional power flow. This paradigm shift confronts grid operators with new problems but also new opportunities. In this chapter we point out some of these problems arising on different layers of the grid h...

Modern smart grids are required to transport electricity along transmission lines from the renewable energy sources to the customer’s demand in an efficient manner. It is inevitable that power is lost along these lines due to active as well as reactive power flows. However, the losses caused by reactive power flows can be reduced by optimizing the...

Simulations of the gas network infrastructure play an important role in energy supply and the green energy transition. Especially volatilities induced by renewable energies increase the need for more transient simulations in shorter time-spans.

The world is waiting for a vaccine to mitigate the spread of SARS-CoV-2. However, once it becomes available, there will not be enough to vaccinate everybody at once. Therefore, vaccination and social distancing has to be coordinated. In this paper, we provide some insight on this topic using optimization-based control on an age-differentiated compa...

In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to r...

In this paper, we provide insights on how much testing and social distancing is required to control COVID-19. To this end, we develop a compartmental model that accounts for key aspects of the disease: 1) incubation time, 2) age-dependent symptom severity, and 3) testing and hospitalization delays; the model's parameters are chosen based on medical...

For a sustainable and CO 2 neutral power supply, the entire energy cycles for power, gas and heating grids have to be taken into account simultaneously. Despite rapid progress, the energy industry is insufficiently equipped for the super-ordinate planning, monitoring and control tasks, based on increasingly large and coupled network simulation mode...

For a sustainable and CO 2 neutral power supply, the entire energy cycles for power, gas and heating grids have to be taken into account simultaneously. Despite rapid progress, the energy industry is insufficiently equipped for the super-ordinate planning, monitoring and control tasks, based on increasingly large and coupled network simulation mode...

In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to r...

We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative variables in Riemann invariants and its corresponding numerical dsicretization. A particular, novelty of the pr...

We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is a span of snapshots evaluated on a shifted spatial domain, and we compute our reduced approximation via residual minimization. To speed-up the residual minim...

Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this...

We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term f(Ex), where E is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of...

This book contains articles presented at the 9th Workshop on Differential-Algebraic Equations held in Paderborn, Germany, from 17–20 March 2019. The workshop brought together more than 40 mathematicians and engineers from various fields, such as numerical and functional analysis, control theory, mechanics and electromagnetic field theory. The parti...

The energy transition entails a rapid uptake of renewable energy sources. Besides physical changes within the grid infrastructure, energy storage devices and their smart operation are key measures to master the resulting challenges like, e. g., a highly fluctuating power generation. For the latter, optimization based control has demonstrated its po...

The energy transition entails a rapid uptake of renewable energy sources. Besides physical changes within the grid infrastructure, energy storage devices and their smart operation are key measures to master the resulting challenges like, e.g., a highly fluctuating power generation. For the latter, optimization based control has demonstrated its pot...

State‐space realizations of input‐output systems or control systems are a widely used class of models in engineering, physics, chemistry and biology. For the qualitative and quantitative classification of such systems, the system‐theoretic properties of reachability and observability are essential, which are encoded in so‐called system Gramian matr...

In this paper, we propose new algebraic Gramians for continuous-time linear switched systems, which satisfy generalized Lyapunov equations. The main contribution of this work is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a linear switched system. As a consequence, a simple Gramian-b...

Correction to: Gas Network Benchmark Models

The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired...

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). By introducing the concept of \textit{long pipes}, we can reduce the dimension...

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). By introducing the concept of long pipes, we can reduce the dimension of the al...

In this paper, we propose new algebraic Gramians for continuous-time linear switched systems, which satisfy generalized Lyapunov equations. The main contribution of this work is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a linear switched system. As a consequence, a simple Gramian-b...

We propose an index-aware model order reduction for differential algebraic equations (DAEs) arising from gas transport networks. This approach involves first the automatic decoupling of the DAEs into differential and algebraic parts, then each part can be reduced separately leading to easier-to-simulate reduced-order models (ROMs).

Using isothermal Euler equations and a network graph to model gas flow in a pipeline network is a classical description, and we prove that any direct space discretization results in a system of index 2 nonlinear differential algebraic equations (DAE). Those are hard to simulate, and model order reduction techniques are not very developed for this s...

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchroniza...

In the recent paper (Monshizadeh et al. in IEEE Trans Control Netw Syst 1(2):145–154, 2014. https://doi.org/10.1109/TCNS.2014.2311883), model reduction of leader–follower multi-agent networks by clustering was studied. For such multi-agent networks, a reduced order network is obtained by partitioning the set of nodes in the graph into disjoint sets...

The simulation of gas transportation networks becomes increasingly more important as its use-cases broadens to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fire...

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optima...

We develop a stability preserving model reduction method for linearly coupled linear time-invariant (LTI) systems. The method extends the work of Monshizadeh et al. for multi-agent systems with identical LTI agents. They propose using Bounded Real Balanced Truncation to preserve a sufficient condition for stability of the coupled system. Here, we e...

In this paper, we study a model reduction technique for leader-follower networked multi-agent systems defined on weighted, undirected graphs with arbitrary linear multivariable agent dynamics. In the network graph of this network, nodes represent the agents and edges represent communication links between the agents. Only the leaders in the network...

In this paper, we extend our clustering-based model order reduction method for multi-agent systems with single-integrator agents to the case where the agents have identical general linear time-invariant dynamics. The method consists of the Iterative Rational Krylov Algorithm, for finding a good reduced order model, and the QR decomposition-based cl...

We construct a C 2 multiscale approximation scheme for functions defined on the Riemann sphere. Based on a 3-directional box-spline, a flexible C 2 scheme over a valence 3 extraordinary vertex can be constructed. Such a flexible C 2 subdivision scheme is known to be impossible for arbitrary valences. The subdivision scheme can be used to model sphe...

The chapter focuses on the numerical solution of parametrized unsteady Eulerian flow of compressible real gas in pipeline distribution networks. Such problems can lead to large systems of nonlinear equations that are computationally expensive to solve by themselves, more so if parameter studies are conducted and the system has to be solved repeated...

Modeling and Simulation of fluids in large network is a rather challenging problem. We provide an approach combining techniques in Model Order Reduction (MOR) and implicit-explicit (IMEX) integration to create efficient and stable simulations.

Let b/a be a strictly proper reduced rational transfer function, with a monic. Consider the problem of designing a controller y/x, with deg(y) < deg(x) < deg(a) – 1 and x monic, subject to lower and upper bounds on the coefficients of y and x, so that the poles of the closed loop transfer function, that is the roots (zeros) of ax + by, are, if poss...

The reduced basis method (RBM) generates low-order models of parametrized partial differential equations. These allow for the efficient evaluation of parametrized models in many-query and real-time contexts. We use the RBM to generate low-order models of microscale models under variation of frequency, geometry, and material parameters. In particula...

Subdivision rules for meshes with boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise \(C^{\ell }\)-continuous boundary limit curves and ensure \(C^{\ell }\)-continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359-397, 2000), we present in this paper gen...

Given optimal interpolation points σ 1,…,σ r , the \(\mathcal {H}_{2}\)-optimal reduced order model of order r can be obtained for a linear time-invariant system of order n≫r by simple projection (whereas it is not a trivial task to find those interpolation points). Our approach to linear time-invariant systems depending on parameters \(p\in \math...

We present an approach which creates a linear reduced order model whose transfer function interpolates at certain given points and approximates at others. We describe how to create the corresponding state space system and explain how this method is used for the simulation of large scale parametric systems or several realizations of an uncertain sys...

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k consistent affine constraints on the coefficients. We show that there always exists an optimal polynomi...

We explore the Tractability Index of Differential Algebraic Equations (DAEs) that emerge in the simulation of gas transport networks. Depending on the complexity of the network, systems of index 1 or index 2 can arise. It is then shown that these systems can be rewritten as Ordinary Differential Equations (ODEs). We furthermore apply Model Order Re...

CPU-intensive engineering problems such as networks of gas pipelines can be modelled as dynamical or quasi-static systems. These dynamical systems represent a map, depending on a set of control parameters, from an input signal to an output signal. In order to reduce the computational cost, surrogates based on linear combinations of translates of ra...

The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Varia...

Gradient flows on surfaces play a role in a wide range of applications, for example in biological modelling, computer graphics and shape optimization. We present an approach to compute a discrete gradient flow by using a newly developed C2-smooth subdivision algorithm. We will briefly present the subdivision algorithm and how we can use it to compu...

It is known from a result of Prautzsch and Reif [19] that it is impossible to construct a flexible C 2 scheme over extraordinary vertices unless the regular subdivision scheme (assumed in [19] to be based on polynomial splines) is capable of producing polynomial patches of total degree 8 in the triangle case and bi-degree 6 in the quadrilateral cas...