Stochastic optimal control model for natural gas networks
Introduction
Consider a gas network with links comprising of long pipelines and nodes consisting of junction points and compressors. Gas is withdrawn from the network at a set of demand nodes and make-up gas is brought into the system through a set of supply nodes. In a real-time environment, the system operator must balance the network to satisfy demand flows and delivery pressures at all times. To achieve this balance, compressors are operated to coordinate buildup and release of inventory inside the pipes. This procedure, called “line-pack management” (Rachford & Carter, 2000), consists on determining dynamic operating policies for the compressors to balance supply, inventory, and demand. The policies must respect compression limits and minimize compressor power or fuel. One of the key issues arising in operations is that demand profiles cannot be predicted with full certainty and thus inventory must be built up, in advance, to ensure that enough capacity is available to satisfy a range of possible future scenarios. Uncertainty in gas pipeline operations is becoming an increasing concern as the power grid adopts larger amounts of intermittent weather-driven resources, because gas-fired power plant units are typically used to balance supply at short notice (Liu et al., 2009, Rachford and Carter, 2000).
Optimization of gas networks has been performed in diverse studies. These studies differ in the decision setting and physical models used. Optimization models for mid-term planning and contracting purposes do not require information about line-pack dynamics so steady-state models are appropriate. O’Neill, Williard, Wilkins, and Pike (1979) present a steady-state transmission model. De Wolf and Smeers (2000) develop an extension of the simplex method to solve models with this structure. An optimal design model for pipes diameters is proposed by the same authors in De Wolf and Smeers (1996). Martin, Möller, and Moritz (2006) present a steady-state nonlinear transmission model that allows for hybrid (on/off) decisions (a mixed-integer nonlinear optimization model) and develop strategies to approximate nonlinear terms using piecewise linear functions, thus enabling the use of mixed-integer linear programming solvers.
For real-time operations, system dynamics must be captured in order to ensure feasible and implementable policies. Moritz (2007) presents a mixed-integer optimal control model with detailed conservation and momentum equations, network balances, and hybrid valve and compressor components. As the author acknowledges, however, computational limitations forced her to consider conservation and momentum equations in simplified form, by defining only inlet and output points. This is equivalent to discretizing the underlying partial differential equations (PDEs) using two discretization points placed at the boundary nodes. Ehrhardt and Steinbach (2005) present a nonlinear continuous optimal control model in which compressor policies are optimized to satisfy demands and minimize compressor fuel. A full space-time discretization of the PDEs is performed and a sequential quadratic programming algorithm is used for the solution of the resulting nonlinear programming (NLP) problem. Steinbach (2007) proposed the use of an interior point algorithm to solve the NLP and proposed a strategy to exploit the underlying linear algebra structure. These studies focused on computational performance, with limited modeling and economic performance analysis. Baumrucker and Biegler (2010) present an optimal control formulation allowing for hybrid behavior arising from flow reversals. The authors cast the problem as a mathematical program with equilibrium constraints and analyze the effect of different electricity price structures on economic performance.
None of the real-time optimization models available in the literature accounts for uncertainty, with the exception of the work of Carter and Rachford (2003). In their work, they present a detailed discussion of uncertainties prevailing in real-time operations and discuss the benefits of using stochastic optimization formulations to manage line-pack inventory. The authors provide a sound physical analysis of the resulting optimal policies; however, they do not report the model and the solution strategy used.
In this work, we present a detailed stochastic optimal control model that considers conservation and momentum equations, typical operational constraints, and uncertainty in demands. We perform a degrees-of-freedom (DOF) analysis to verify the consistency of the model and we use this analysis to derive consistent initial conditions and nonanticipacity constraints. In addition, we propose to incorporate a risk metric into the objective function to mitigate cost variance and system volatility. Using a computational study, we demonstrate the benefits obtained with stochastic formulations against deterministic and robust counterparts and we discuss the effects of discretization mesh resolution on economic performance.
The paper is structured as follows. In Section 2 we present the physical model for the pipelines, network, and compressors. In Section 3 we present the DOF analysis to characterize the differential and algebraic equation (DAE) system and provide conditions to achieve model consistency. In Section 4 we formulate the stochastic optimal control model by defining the objective function, operational constraints, initial conditions, and nonanticipativity constraints. In Section 5 we present a computational study to demonstrate the benefits of the stochastic model over a range of different formulations and we discuss computational issues. The paper closes in Section 6 with concluding remarks and directions of future work. The model nomenclature as well as variables and parameter units are presented in Appendix A.
Section snippets
Physical model
In this section, we present the conservation and momentum equations governing the dynamics of each pipeline in the network as well as the equations describing the network interconnections. Nomenclature, physical units, and typical values for all variables and parameters are given in Appendix A.
DOF analysis
In this section, we perform a DOF analysis to determine conditions for consistency of the physical model and to determine appropriate initial conditions. For clarity in the presentation, we consider a discretized version of the conservation, momentum, and network equations. Assume that the PDEs of each link are discretized in space by using a finite difference scheme using Nx points of equal length Δxℓ and discretized in time by using an implicit Euler scheme with Nt points of equal length Δτ (
Stochastic optimal control model
In this section we describe the objective function, physical constraints, and initial conditions of the optimal control model.
Computational study
We consider the network system sketched in Fig. 1. The system comprises nodes, links ( compressors), supply flow located at the first node, and demand flow located at the last node. The system spans 1600 km with 36 in. (914 mm) pipes. The distance between compressors is 100 km. The pressure at the demand node () should be maintained at least at 39 bar (565 psia) with a nominal demand flow of 24×106 SCM/day. The pressure at the supply point is 57 bar (827 psia). The
Conclusions and future directions
We have presented a stochastic optimal control model for natural gas network operations. The consistency of the model has been verified using a degrees of freedom analysis. We have found that the use of stochastic policies can significantly aid robustness in operations compared with deterministic policies. Robustness is particularly critical in gas networks because of the complex physical behavior propagating throughout the system and because of the limited degrees of freedom available for
Acknowledgements
The author acknowledges support from the DOE Office of Science Early Career award. The author would also like to thank Drew Kouri for pointing him to existing work on stochastic optimization of natural gas networks. This work was supported by the U.S. Department of Energy, under contract no. DE-AC02-06CH11357.
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2021, Computers and Chemical EngineeringCitation Excerpt :The study of (Carter and Rachford Jr, 2003) aims at finding an optimal schedule for the line pack under future demand uncertainty. Liu et al. (2020) and Zavala (2014) proposed stochastic optimization models to minimize the expected electricity cost of compressors considering some demand scenarios. The deviations of the final flow rate and pressure profiles from the initial profiles are penalized in the objective function.