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Adjoint-Based Control of Model and Discretization Errors for Gas and Water Supply Networks
Abstract
We are interested in the simulation and optimization of gas and water transport in networks. Those networks consist of pipes
and various other components like compressor/pumping stations and valves. The flow through the pipes can be described by different
models based on the Euler equations, including hyperbolic systems of partial differential equations. For the other components,
algebraic or ordinary differential equations are used. Depending on the data, different models can be used in different regions
of the network. We present a strategy that adaptively applies the models and discretizations, using adjoint-based error estimators
to maintain the accuracy of the solution. Finally, we give numerical examples for both types of networks.
... It extends an adaptive multilevel stochastic collocation method recently developed in [22] for elliptic partial differential equations with random data to systems of hyperbolic balance laws with uncertain initial and boundary conditions. We have been developing in-house software tools for fast and reliable transient simulation and continuous optimization of large-scale gas networks over the last decade [7,8,9,10,11]. Exemplarily, here we will investigate the important task of safely driving a stationary running system into a newly desired system defined by uncertain gas nominations at delivery points of the network. ...
... As a rule of thumb, the most complex nonlinear Euler equations (M 1 ) should be used when needed and the simplest algebraic model (M 3 ) should be taken whenever possible without loosing too much accuracy. In a series of papers, we have developed a posteriori error estimates and an overall control strategy to reduce model and discretization errors up to a user-given tolerance [7,8,9,10,11]. A brief introduction will be given next. ...
... A detailed description which would go beyond the scope of our paper is given in [7, Sect. 2.2], see also [9,10]. Polynomial reconstructions in space and time of appropriate orders are used to compute η x,j and η t,j , respectively. ...
In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.
... Such models are commonly used for an optimal control of compressors and compressor stations [4,7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1][2][3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system. Let G = (V, E) a directed graph with vertices V = V + ∪ V − and edges E = E P ∪ E C where V − and V + are the nodes where gas can enter and exit the network respectively and E P and E C being the set of pipes and compressors respectively. ...
... Such models are commonly used for an optimal control of compressors and compressor stations [4,7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1][2][3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system. ...
... We model pipes by a simplification of the isothermal Euler equations [2] ...
One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4,7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1–3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
... Gas transport in a pipe is described by a set of partial differential equations, i.e., a continuation equation, a momentum equation and an energy equation, together with a gas state equation. Under steady-state conditions, the isothermal gas pipe model is simplified into the static algebraic equation, as follows [30][31][32]: ...
... The state of a water pipe is also governed by a set of partial differential equations. In steady state, the water flow through a pipe may be described by the following algebraic equation [32,33]: ...
The increasing linkages and interdependencies between multiple energy systems and the demand for higher overall energy efficiency of these systems put into force their combined management as multi-energy systems (MESs). This study comprised the development of a method for the steady-state analysis of the multi-energy flows of electricity, gas and heating systems, which is a fundamental tool for real operation and operational planning. A sequence iterative method was introduced for sequentially determining the energy flows of each system. This method is a relaxation of an existing uniform Newton’s formulation for handling the negative effect on convergence due to the hybrid quadratic and exponential 1/2 gas equations. Industry-based software for electricity simulation and pipeline programs enabling gas and heating analysis were incorporated into the method. Moreover, the mechanism of a distributed regulating unit scheme was developed for data-exchange between sub-networks under analysis. The applicability of the proposed method was demonstrated by analysis of representative small-scale and large-scale systems.
... So, using error estimation, typically the grid is adapted in space and time and as a new component of the simulation process we will discuss the adaptation of the model within a model hierarchy. We will focus on the pure pipe ow, where the model hierarchy is easily constructed and where it can be used to nd an appropriate trade-o between accuracy and computational complexity, see [15,16,17,18]. ...
... This leaves us with (10a) and (10b), which are referred to as the isothermal algebraic model. A detailed derivation of this model is given in [18]. In the optimization of natural gas networks, this nonlinear algebraic model is often further approximated by piecewise linear functions, see e.g. ...
In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization, and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.
... Since the behaviour of gas and water supply networks may be dynamic, an automatic control of the accuracy of the simulation is beneficial. We address this task by using a goal-oriented adaptive strategy for the simulation, as was derived in [1][2][3]. Besides refinement in space and time, we want to use simplified models in regions of the network with low activity, while sophisticated models are used in very dynamic regions. ...
... We use a heuristic algorithm as described in [3,5] to refine space, time, and/or models to fall below this tolerance, and coarsen if appropriate to reduce the computation time. Note that the applied procedure cannot guarantee strict bounds on the actual error, since the error estimators result from a linearization of the target functional and neglect higher order terms. ...
We consider the simulation and optimisation of transport processes through gas and water supply networks. Using a consistent modeling of the network, adjoint equations for the whole system including initial, coupling and boundary conditions can be derived. These are suitable to compute gradients for optimization tasks but can also be used to estimate the accuracy of models and the discretization with respect to a given cost functional. We show the applicability of an adaptive algorithm that automatically steers the discretization and models while maintaining a given accuracy in an optimisation framework. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
... The relation between the forward error, the backward error, and the condition number can be nicely seen from Eq. (6). Taking the absolute value of the relative change in the solution gives ...
... A concise derivation of these equations is given in Sec. 1.1 and for the isothermal algebraic model in [6]. The Jacobian matrix J f (q) of f (q) is given by ...
The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline. The statistical analysis is performed using both a Monte Carlo Simulation and the Univariate Reduced Quadrature Method and is used to illustrate and confirm the obtained theoretical results.
... The more general aspects of our network simulation tool are described in Chapter 3. More details about the adaptive control of model and discretization errors concerning the theoretical as well as implementational aspects can be found in [20,[22][23][24] and are not part of this work. Concerning the relevance of models based on PDEs in the context of gas and water supply networks, one has to consider that coarse discretizations often lead to similar results as quasi-stationary models without significant additional computational effort. ...
... Here, adjoint-based error estimators with respect to a given target functional are used to adaptively refine or coarsen the discretizations, and additionally, also different models may be applied in each pipe. For more details, we refer to [20,[22][23][24]. ...
In this work, we consider the simulation and optimization of gas and water supply networks. In particular, we are dealing with the daily operation of both types of networks. First, we describe a mathematical model for gas and water supply networks, which consists of partial and ordinary differential equations as well as algebraic equations. Supplemented with initial and boundary conditions, these can be solved by applying appropriate discretization schemes. For this purpose, we develop a software framework for problems on networks, which allows the application of different schemes for each type of equations and which takes care of the correct coupling. A matter of particular importance is the treatment of the underlying hyperbolic partial differential equations. Here, we apply an implicit box scheme as well as explicit central schemes. For both methods, we give a profound stability analysis in a well-known setting. The next step after the solution of simulation tasks is the computation of sensitivity information. Here, we follow an adjoint approach. The resulting gradient information can directly be applied in three state-of-the-art optimization tools, for which we provide interfaces in our software. In practice, the optimization tasks in the daily operation of gas and water supply networks also involve discrete decisions for switching certain network elements on or off. To handle mixed integer problems, we follow two approaches. First, we present a heuristic penalization strategy, and secondly, we develop an adaptive linearization technique for the treatment of the underlying nonlinear problems with mixed integer linear programming methods. Finally, numerical results for a real-life gas and water supply network are presented as well as for a representative test network provided by one of our industry partners.
... In general, those decisions are made for certain time blocks [T k−1 , T k ] within the simulation time [0, T ] and accordingly the output functional (10) is locally evaluated (time integrals over [T k−1 , T k ] instead of [0, T ]) as well as the error estimates. For the details on the computation of the error estimators and the adaptive strategy, we refer to [4,6,7,8]. ...
We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.
In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.
We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.
In this work, the simulation and optimization of transport processes through gas and water supply networks is considered. Those networks mainly consist of pipes as well as other components like valves, tanks and compressor/pumping stations. These components are modeled via algebraic equations or ODEs while the flow of gas/water through pipelines is described by a hierarchy of models starting from a hyperbolic system of PDEs down to algebraic equations. We present a consistent modeling of the network and derive adjoint equations for the whole system including initial, coupling and boundary conditions. These equations are suitable to compute gradients for optimization tasks but can also be used to estimate the accuracy of models and the discretization with respect to a given cost functional. With these error estimators we present an algorithm that automatically steers the discretization and the models used to maintain a given accuracy. We show numerical experiments for the simulation algorithm as well as the applicability in an optimization framework.
This article surveys a general approach to error control and adaptive mesh
design in Galerkin finite element methods that is based on duality principles
as used in optimal control. Most of the existing work on a
posteriori error
analysis deals with error estimation in global norms like the ‘energy norm’
or the L2 norm, involving usually unknown ‘stability constants’. However, in
most applications, the error in a global norm does not provide useful bounds
for the errors in the quantities of real physical interest. Further, their sensitivity
to local error sources is not properly represented by global stability constants.
These deficiencies are overcome by employing duality techniques, as
is common in a
priori error analysis of finite element methods, and replacing
the global stability constants by computationally obtained local sensitivity
factors. Combining this with Galerkin orthogonality, a
posteriori estimates
can be derived directly for the error in the target quantity. In these estimates
local residuals of the computed solution are multiplied by weights which
measure the dependence of the error on the local residuals. Those, in turn,
can be controlled by locally refining or coarsening the computational mesh.
The weights are obtained by approximately solving a linear adjoint problem.
The resulting a
posteriori error estimates provide the basis of a feedback process
for successively constructing economical meshes and corresponding error
bounds tailored to the particular goal of the computation. This approach,
called the ‘dual-weighted-residual method’, is introduced initially within an
abstract functional analytic setting, and is then developed in detail for several
model situations featuring the characteristic properties of elliptic, parabolic
and hyperbolic problems. After having discussed the basic properties
of duality-based adaptivity, we demonstrate the potential of this approach by
presenting a selection of results obtained for practical test cases. These include
problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative
transfer, and optimal control. Throughout the paper, open theoretical
and practical problems are stated together with references to the relevant literature.
The operative planning problem in natural gas distribution networks is addressed. An optimization model focusing on the governing
PDE and other nonlinear aspects is presented together with a suitable discretization for transient optimization in large networks
by SQP methods. Computational results for a range of related dynamic test problems demonstrate the viability of the approach.
We are interested in the simulation and optimization of gas transport in networks. Those networks consist of pipes and various other components like compressor stations and valves. The gas flow through the pipes can be modelled by different equations based on the Euler equations. For the other components, purely algebraic equations are used. Depending on the data, different models for the gas flow can be used in different regions of the network. We use adjoint techniques to specify model and discretization error estimators and present a strategy that adaptively applies the different models while maintaining the accuracy of the solution.
Natural gas is the third most important energy source in the world. Presently, the consumption of natural gas is increasing the most in comparison to other non-renewable energy sources. Therefore, optimization of gas transport in networks poses a very important industrial problem. In this thesis we consider the problem of time-dependent optimization in gas networks, also called Transient Technical Optimization (TTO). A gas network consists of a set of pipes to transport the gas from the suppliers to the consumers. Due to friction with the pipe walls gas pressure gets lost. This pressure loss is compensated by so called compressors. The aim of TTO is to minimize the fuel consumption of the compressors, where the demands of consumers have to be satisfied. Transient optimization of gas transmission is one of the great research challenges in this area. We formulate a mixed integer approach for the problem of TTO which concentrates on time-dependent and discrete aspects. Thereby, the nonlinearities resulting from physical constraints are approximated using SOS (Special Ordered Set) conditions. A branch-and-cut algorithm is developed which guarantees global optimality in dependence on the approximation accuracy. Concerning the nonlinearities, we discuss the quality of approximation grids by calculating approximation errors. The SOS conditions are implicitly handled via a branching scheme, supported by adequate preprocessing techniques. A heuristic approach based on simulated annealing yields an upper bound in our branch-and-cut framework. To improve the lower bound, we incorporate two separation algorithms. The first one results from theoretical studies of the so called switching polytopes which are defined by runtime conditions and switching processes of compressors. Linking of different SOS conditions gives a second separation strategy. We present theoretical investigations of the SOS 2 and SOS 3 polytope. These polytopes arise from the modeling of SOS Type 2 and SOS Type 3 conditions using additional binary variables. The results do not have practical relevance for our solution algorithm, but we characterize facet-defining inequalities providing complete linear descriptions of these polytopes. We evaluate the developed branch-and-cut algorithm using three test networks provided by our project partner E.ON Ruhrgas AG. Two are of artificial nature, as they were developed for test purposes. They contain all important elements of a gas network, but are rather small. The third network characterizes the major part of the Ruhrgas AG network in Western Germany. We test instances from three up to 24 coupled time steps.
We are interested in the simulation and optimisation of gas transport in networks. The gas flow through pipes can be modelled on the basis of the (isothermal) Euler equations. Further network components are described by purely algebraic equations. Depending on the data and the resulting network dynamics, models of different fidelity can be used in different regions of the network. Using adjoint techniques, we derive model and discretisation error estimators. Here, we apply a first-discretise approach. Based on the time-dependent structure of the considered problems, the adjoint systems feature a special structure and therefore allow for an efficient solution. A strategy that controls model and discretisation errors to maintain the accuracy of the solution is presented. We provide (technical) details of our implementation and give numerical results.
Assuming 1D flow in pressurized systems, transient analyses can be performed using a number of well-established models, in the short-term timescale, practical problems are solved using either elastic or rigid models, whereas in the long-term scale a quasi-static model is more convenient. These models can be obtained by simplifying the general equations for flow of an elastic fluid. A brief overview of these models is presented, with the major emphasis being on the use of dimensionless parameters to define the range of their applicability for simple hydraulic systems. Guidelines for applicability are presented in the form of graphs and equations. The effects of resistance, inertia, and elasticity may vary in relative importance under different circumstances. The present analysis provides a unified approach to represent each of these effects using a different parameter.
We investigate the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations. The method is first derived in the case where only model errors arise and then extended to handle simultaneously model and discretization errors. We next present an adaptive model/mesh refinement procedure where both sources of error are equilibrated. Various test cases involving Poisson equations and convection diffusion-reaction equations with complex diffusion models (oscillating diffusion coefficient, nonlinear diffusion, multicomponent diffusion matrix) confirm the reliability of the analysis and the efficiency of the proposed methodology.
The topic of this paper is minimum cost operative planning of pressurized water supply networks over a finite horizon and
under reliable demand forecast. Since this is a very hard problem, it is desirable to employ sophisticated mathematical algorithms,
which in turn calls for carefully designed models with suitable properties. The paper develops a nonlinear mixed integer model
and a nonlinear programming model with favorable properties for gradient-based optimization methods, based on smooth component
models for the network elements. In combination with further nonlinear programming techniques (Burgschweiger et al. in ZIB
Report ZR-05-31, Zuse Institute Berlin, 2005), practically satisfactory near-optimum solutions even for large networks can be generated in acceptable time using standard
optimization software on a PC workstation. Such an optimization system is in operation at Berliner Wasserbetriebe.
Abstract We are interested in simulation,and optimization,of gas transport in networks. Different regions of the network,may,be modelled,by different equations: There are three models,based on the Euler equations that describe the gas flow in pipelines qualitatively different: a nonlinear model, a semilinear model and a stationary also called algebraic model. For the whole network, adequate initial and boundary values as well as coupling,conditions at the junctions are needed.,Using adjoint techniques,one can specify model,error estimators for the simplified models. A strategy to adaptively apply the different models,in different regions of the network,while maintaining,the accuracy,of the solution is presented.
A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.
We consider gas flow in pipeline networks governed by isothermal Euler equations, and introduce a new modeling of compressors in gas networks. Compressor units are modeled as pipe-to-pipe intersections with additional algebraic coupling conditions for compressor behavior. We prove existence and uniqueness of solutions with respect to these conditions, and use the results for numerical simulation and optimization of gas networks.
We investigate the stability and convergence of an implicit box scheme for subsonic flows modelled by scalar conservation
laws with dissipative and possibly stiff source terms. The scheme is proposed for solving transient gas flow problems in pipeline
networks. Such networks are operated in the subsonic flow region and are characterized by pressure losses due to dissipative
friction terms. We verify the properties stated by Kružkov’s theorem (Kružkov, Math. USSR-Sb. 10:217–243, 1970) for the approximate solution and prove its convergence to the entropy solution.
Transiente Technische Optimierung (TTO-Prototyp)
- E Sekirnjak
- Psi
- Ag
Transiente Technische Optimierung (TTO-Prototyp)
- E Sekirnjak
Erwin Sekirnjak. Transiente Technische Optimierung (TTO-Prototyp). Technical report, PSI AG,
November 2000.