(358c) Novel Piecewise Linearization of Rigorous Nonlinear Natural Gas Transportation Model for Optimal Network Operation

Natural gas has become a vital energy source due to its electricity generation efficiency and low emissions relative to other fossil fuels. Currently natural gas accounts for an estimated 22% of the total global energy mix, and conservative estimates in sustainable energy growth predicts the share of natural gas to grow to 25% of the global energy mix by 2040 [1]. The large and growing demand for natural gas makes the operational challenges associated with its transportation ever-important. Operations research has the potential to significantly improve the efficient operation of natural gas transportation systems if accurate solutions can be obtained within practical time constraints. For instance, an estimated 3-5% of all natural gas transported by pipeline is consumed by the compressor stations responsible for maintaining gas flow. It has been estimated that at least a 20% reduction in compressor station fuel consumption can be obtained through global optimization of pipeline operation [2]. However, natural gas transportation system operation is highly dynamic due to frequently changing uncontrollable variables, such as customer demand and available gas supply. As operating conditions change, the optimal operating point also changes, so solution must be attainable relatively rapidly for operations research to be useful for guiding the online operation of natural gas transportation systems.

Modelling the operation of natural gas transportation through pipelines in a rigorous way requires the use of a large set of nonlinear equations to capture gas flow and compressor station physics [3]. Further, integer variables are often required to model equipment states and bidirectional gas flow [4]. As a result, natural gas transportation is most rigorously modelled as a mixed-integer nonlinear program (MINLP). The computational complexity of this problem class means that applying rigorous operations research methods to guide natural gas transportation operation has been limited to small problems which are not representative of those faced in industry.

To enable solution of large-scale problems major simplifications to the rigorous MINLP model have been employed. A common simplification strategy is to assume certain modelling parameters such as the friction factor, gas compressibility factor, gas isentropic exponent, and compressor efficiency to be constants, and their nonlinear relationships to the system operating conditions are ignored. The drawback of this approach is that the error introduced into the model through these simplifications is significant, often resulting in solutions which are found to be infeasible when tested on a rigorous simulation [5].

There are also very few operations research studies that consider the ability for pipelines to act as storage vessels, through what is referred to as linepacking. This capability of pipelines is a valuable tool that can be used as a buffer to mitigate the interaction between variable upstream and downstream operating conditions, enabling both a better security of gas supply and the ability to take advantage of gas market fluctuations [6]. In large gas transportation networks the storage capacity available through linepacking is significant, so the network operators heavily consider linepacking in how they operate the network. Determining an optimal linepacking policy requires accurate modelling of compressor stations and incorporation future operating uncertainty in the transportation model. This additional computation complexity has resulted in the vast majority of operations research to-date ignoring the capability of pipelines to store linepack, and focus simply on the compressor fuel cost minimization problem [6].

By using a model of natural gas transportation that ignores the role of linepacking in gas network operation, and employs the common simplifications found in the literature, the practical usefulness of optimal solutions obtained is severely limited. This work aims to increase the practical usefulness of operations research in guiding gas network operation by increasing the modelling accuracy over the traditional approaches, while ensuring the model can be solved on large-scale networks rapidly. To do so, a novel linearization approach is used to capture the complex nonlinear relationships that are often ignored in the simplified models. The linearization method is an MILP that produces the optimal piecewise-linear function which most closely approximates the relationship being approximated. In our previous work we introduced the optimal piecewise-linear function generation method to approximate the relationships of the gas compressibility factor, gas isentropic exponent, and compressor efficiency with the pipeline operating condition [5]. We demonstrated that a model using the piecewise-linear function approximation obtains optimal solutions that have less than 1% error when compared to a rigorous simulation. Further, the solution time of the novel partially piecewise-linear model is shown to be at least an order of magnitude less than the typical MINLP model considered as the most rigorous [5].

In the previous work the nonlinearity in the gas transportation model was only partially approximated by piecewise-linear functions, and line-packing was not considered. This work fully linearizes the gas transportation model using the same novel piecewise-linear function generating technique employed previously. Additionally, a novel linepack modelling formulation is introduced which uses a two-stage stochastic modelling framework to capture uncertainty in future pipeline operating conditions. A case-study is performed which demonstrates that despite the drastically increased problem size due to the consideration of line-pack, optimal solutions to a large-scale gas network representative of the Norwegian pipeline system are able to be obtained within 1 minute when using the novel linearized model. Further, the optimal solutions from the novel linearized model are compared to a rigorous simulation which demonstrates that they contain less than 1% error. For comparison, the typical MINLP model was also applied in the case study and was only able to find a feasible solution after 12 hours.

The case study results demonstrate that the novel linearized model can substantially reduce solution time and maintain the solution accuracy compared to the typical MINLP model that is currently consider as the most rigorous natural gas optimization model available. Overall, by applying the novel linearization technique to approximate the nonlinear relationships inherent in natural gas transportation modelling the model complexity is drastically reduced, and the model accuracy maintained. The significance of these results is that accurate solutions to problems of practical importance can be obtained within reasonable solution times, thereby improving the usefulness of operations research in guiding online operation of natural gas transportation systems.

References

[1] IEA. World Energy Outlook 2018. International Energy Agency, Paris, 2018.

[2] Schroeder, D. Hydraulic analysis in the natural gas industry. In Advances in Industrial Engineering Applications and Practice I; International Journal of Industrial Engineering: Houston, TX, USA, 1996, 960–965.

[3] Schmidt, M.; Steinbach, M.C.; Willert, B.M. High detail stationary optimization models for gas networks. Optim. Eng. 2015, 16, 131–164.

[4] Wu, S.; Rios-Mercado, R.; Boyd, E.; Scott, L. Model Relaxations for the Fuel Cost Minimization of Steady-State Gas Pipeline Networks. Math. Comput. Model. 2000, 31, 197–220.

[5] Kody Kazda and Xiang Li. Approximating nonlinear relationships for optimal operation of natural gas transport networks. Processes, 2018, 6(10), 198.

[6] Roger Z. Rios-Mercado and Conrado Borraz-Sanchez. Optimization problems in natural gas transportation systems: A state-of-the-art review. Applied Energy, 2015, 147, 536-555.