TY - GEN A1 - Hennings, Felix A1 - Petkovic, Milena A1 - Streubel, Tom T1 - On the Numerical Treatment of Interlaced Target Values - Modeling, Optimization and Simulation of Regulating Valves in Gas Networks N2 - Due to the current and foreseeable shifts in energy production, the trading and transport operations of gas will become more dynamic, volatile, and hence also less predictable. Therefore, computer-aided support in terms of rapid simulation and control optimization will further broaden its importance for gas network dispatching. In this paper, we aim to contribute and openly publish two new mathematical models for regulators, also referred to as control valves, which together with compressors make up the most complex and involved types of active elements in gas network infrastructures. They provide full direct control over gas networks but are in turn controlled via target values, also known as set-point values, themselves. Our models incorporate up to six dynamical target values to define desired transient states for the elements' local vicinity within the network. That is, each pair of every two target values defines a bounding box for the inlet pressure, outlet pressure as well as the passing mass flow of gas. In the proposed models, those target values are prioritized differently and are constantly in competition with each other, which can only be resolved dynamically at run-time of either a simulation or optimization process. Besides careful derivation, we compare simulation and optimization results with predictions of the commercial simulation tool SIMONE. T3 - ZIB-Report - 21-32 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-85359 SN - 1438-0064 ER - TY - THES A1 - Streubel, Tom T1 - Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks Y1 - 2022 UR - https://edoc.hu-berlin.de/handle/18452/25434 U6 - https://doi.org/http://dx.doi.org/10.18452/24688 ER - TY - JOUR A1 - Hennings, Felix A1 - Petkovic, Milena A1 - Streubel, Tom T1 - On the Numerical Treatment of Interlaced Target Values - Modeling, Optimization and Simulation of Regulating Valves in Gas Networks JF - Optimization and Engineering N2 - Due to the current and foreseeable shifts towards carbon dioxide neutral energy production, which will likely result in balancing fluctuating renewable energy generation by transforming power-to-gas-to-power as well as building a large-scale hydrogen transport infrastructure, the trading and transport operations of gas will become more dynamic, volatile, and hence also less predictable. Therefore, computer-aided support in terms of rapid simulation and control optimization will further broaden its importance for gas network dispatching. In this paper, we aim to contribute and openly publish two new mathematical models for regulators, also referred to as control valves, which together with compressors make up the most complex and involved types of active elements in gas network infrastructures. They provide direct control over gas networks but are in turn controlled via target values, also known as set-point values, themselves. Our models incorporate up to six dynamical target values to define desired transient states for the elements' local vicinity within the network. That is, each pair of every two target values defines a bounding box for the inlet pressure, outlet pressure as well as the passing mass flow of gas. In the proposed models, those target values are prioritized differently and are constantly in competition with each other, which can only be resolved dynamically at run-time of either a simulation or optimization process. Besides careful derivation, we compare simulation and optimization results with predictions of the widely adopted commercial simulation tool SIMONE, serving as our substitute for actual real-world transport operations. Y1 - 2024 U6 - https://doi.org/10.1007/s11081-023-09812-0 VL - 25 SP - 491 EP - 527 PB - Springer Nature ER - TY - JOUR A1 - Streubel, Tom A1 - Tischendorf, Caren A1 - Griewank, Andreas T1 - Piecewise Polynomial Taylor Expansions - The Generalization of Faà di Bruno's Formula JF - Modeling, Simulation and Optimization of Complex Processes HPSC 2018 N2 - We present an extension of Taylor's Theorem for the piecewise polynomial expansion of non-smooth evaluation procedures involving absolute value operations. Evaluation procedures are computer programs of mathematical functions in closed form expression and allow a different treatment of smooth operations or calls to the absolute value function. The well known classical Theorem of Taylor defines polynomial approximations of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential- or differential-algebraic equations, for the construction of solvers for continuous non-linear optimization of finite dimensional objective functions and for root solving of non-linear systems of equations. The long term goal is the stabilization and acceleration of already known methods and the derivation of new methods by incorporating piecewise polynomial Taylor expansions. The herein provided proof of the higher order approximation quality of the new generalized expansions is constructive and allows efficiently designed algorithms for the execution and computation of the piecewise polynomial expansions. As a demonstration towards the ultimate goal we will derive a prototype of a {\$}{\$}k{\$}{\$}k-step method on the basis of polynomial interpolation and the proposed generalized expansions. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-55240-4_3 IS - Modeling, Simulation and Optimization of Complex Processes HPSC 2018 SP - 63 EP - 82 PB - Springer International Publishing ER - TY - JOUR A1 - Griewank, Andreas A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - On the abs-polynomial expansion of piecewise smooth functions JF - Optimization Methods and Software N2 - Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bar d-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|...|$ apart from arithmetic operations and $\bar d$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $\mathcal O({\bar d}^2)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\bar d$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains. Y1 - 2020 U6 - https://doi.org/10.1080/10556788.2020.1817448 PB - Taylor & Francis ER - TY - GEN A1 - Griewank, Andreas A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - On the abs-polynomial expansion of piecewise smooth functions N2 - Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bar d-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|...|$ apart from arithmetic operations and $\bar d$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $\mathcal O({\bar d}^2)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\bar d$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains. T3 - ZIB-Report - 20-14 KW - Nonsmooth Taylor polynomial/series KW - forward mode propagation KW - abs-normal form KW - abs-linear form KW - absolute convergence KW - Moore recurrences KW - quadratic complexity Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78448 SN - 1438-0064 ER - TY - CHAP A1 - Benner, Peter A1 - Grundel, Sara A1 - Himpe, Christian A1 - Huck, Christoph A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - Gas Network Benchmark Models T2 - Applications of Differential-Algebraic Equations: Examples and Benchmarks N2 - 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 power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models. Y1 - 2019 SN - 978-3-030-03718-5 U6 - https://doi.org/10.1007/11221_2018_5 SP - 171 EP - 197 PB - Springer International Publishing ER - TY - GEN A1 - Hasenfelder, Richard A1 - Lehmann, Lutz A1 - Radons, Manuel A1 - Streubel, Tom A1 - Strohm, Christian A1 - Griewank, Andreas T1 - Computational aspects of the Generalized Trapezoidal Rule N2 - In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F:IR^n -> IR^n. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a non-differentiability of F. In such a situation the investigated generalized trapezoidal rule achieves a higher convergence order than the classical method. While the asymptotic behavior of the generalized method was investigated in a previous work, in the present article we develop the algorithmic structure for efficient implementation strategies and estimate the actual computational cost of the latter. Moreover, energy preservation of the generalized trapezoidal rule is proved for Hamiltonian systems with piecewise linear right hand side. T3 - ZIB-Report - 18-23 KW - Algorithmic Differentiation KW - Automatic Differentiation KW - Lipschitz Continuity KW - Piecewise Linearization KW - Nonsmooth KW - Trapezoidal Rule KW - Implementation KW - Computational Cost Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68615 SN - 1438-0064 ER - TY - GEN A1 - Streubel, Tom A1 - Tischendorf, Caren A1 - Griewank, Andreas T1 - Piecewise Polynomial Taylor Expansions – The Generalization of Faà di Bruno’s Formula N2 - We present an extension of Taylor’s theorem towards nonsmooth evalua- tion procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in closed form expression and al- low a different treatment of smooth operations and calls to the absolute value value function. The well known classical Theorem of Taylor defines polynomial approx- imation of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential or differential algebraic equations, for the construction of solvers for the continuous nonlinear op- timization of finite dimensional objective functions and for root solving of nonlinear systems of equations. The herein provided proof is construtive and allow efficiently designed algorithms for the execution and computation of generalized piecewise polynomial expansions. As a demonstration we will derive a k-step method on the basis of polynomial interpolation and the proposed generalized expansions. T3 - ZIB-Report - 18-24 KW - generalized Taylor expansion KW - implicit generation of splines KW - nonsmooth integration of differential algebraic equations (DAE and ODE) KW - multistep methods KW - generalized hermite interpolation KW - algorithmic piecewise differentiation (AD and APD) KW - evaluation procedures KW - treating absolute values (abs, max and min) Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68859 SN - 1438-0064 ER - TY - GEN A1 - Radons, Manuel A1 - Lehmann, Lutz A1 - Streubel, Tom A1 - Griewank, Andreas T1 - An Open Newton Method for Piecewise Smooth Systems N2 - Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. For local bijectivity of the linearization at a root, a radius of quadratic convergence was explicitly calculated in terms of local Lipschitz constants of the underlying PS function. In the present work we relax the criterium of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions f:IR^2 --> IR^2 satisfying the weaker criterium where every neighborhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighborhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given. T3 - ZIB-Report - 18-43 Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-70418 SN - 1438-0064 ER - TY - JOUR A1 - Griewank, Andreas A1 - Hasenfelder, Richard A1 - Radons, Manuel A1 - Lehmann, Lutz A1 - Streubel, Tom T1 - Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation JF - Optimization Methods and Software N2 - In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side F : IR^n -> IR^n based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F. The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third-order interpolation polynomial for the numerical trajectory. In the smooth case, the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented. Y1 - 2018 U6 - https://doi.org/10.1080/10556788.2017.1378653 VL - 33 SP - 1089 EP - 1107 PB - Taylor & Francis ER - TY - JOUR A1 - Griewank, Andreas A1 - Streubel, Tom A1 - Lehmann, Lutz A1 - Radons, Manuel A1 - Hasenfelder, Richard T1 - Piecewise linear secant approximation via algorithmic piecewise differentiation JF - Optimization Methods and Software N2 - It is shown how piecewise differentiable functions F : IR^n -> IR^m that are defined by evaluation programmes can be approximated locally by a piecewise linear model based on a pair of sample points \check x and \hat x. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x - \check x||*||x - \hat x||). As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equalling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods. Y1 - 2017 U6 - https://doi.org/10.1080/10556788.2017.1387256 VL - 33 IS - 4-6 SP - 1108 EP - 1126 PB - Taylor & Francis ER -