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At least 55 records · Page 3

Next Generation Aircraft Design Considering Airline Operations and Economics

Traditional approaches to design and optimization of a new system often use a system-centric objective and do not take into consideration how the operator will use this new system alongside other existing systems. When the new system design is incorporated into the broader group of systems, the performance of the operator-level objective can be sub-optimal due to the unmodeled interaction between the new system and the other systems. Among the few available references that describe attempts to address this disconnect, most follow an MDO (Multidisciplinary Design Optimization)-motivated sequential decomposition approach of first designing a very good system and then providing this system to the operator who, decides the best way to use this new system along with the existing systems. This paper addresses this issue by including aircraft design, airline operations, and revenue management "subspaces"; and presents an approach that could simultaneously solve these subspaces posed as a monolithic optimization problem rather than the traditional approach described above. The monolithic approach makes the problem an expensive Mixed Integer Non-Linear Programming problem, which are extremely difficult to solve. To address the problem, we use a recently developed optimization framework that simultaneously solves the subspaces to capture the "synergy" in the problem that the previous decomposition approaches did not exploit, addresses mixed-integer/discrete type design variables in an efficient manner, and accounts for computationally expensive analysis tools. This approach solves an 11-route airline network problem consisting of 94 decision variables including 33 integer and 61 continuous type variables. Simultaneously solving the subspaces leads to significant improvement in the fleet-level objective of the airline when compared to the previously developed sequential subspace decomposition approach.

Roy, Satadru↗

Efficient Trajectory Options Allocation for the Collaborative Trajectory Options Program

The Collaborative Trajectory Options Program (CTOP) is a Traffic Management Initiative (TMI) intended to control the air traffic flow rates at multiple specified Flow Constrained Areas (FCAs), where demand exceeds capacity. CTOP allows flight operators to submit the desired Trajectory Options Set (TOS) for each affected flight with associated Relative Trajectory Cost (RTC) for each option. CTOP then creates a feasible schedule that complies with capacity constraints by assigning affected flights with routes and departure delays in such a way as to minimize the total cost while maintaining equity across flight operators. The current version of CTOP implements a Ration-by-Schedule (RBS) scheme, which assigns the best available options to flights based on a First-Scheduled-First-Served heuristic. In the present study, an alternative flight scheduling approach is developed based on linear optimization. Results suggest that such an approach can significantly reduce flight delays, in the deterministic case, while maintaining equity as defined using a Max-Min fairness scheme.

Traffic Management Initiative (TMI)↗

Decomposing a renewable energy design and dispatch model

We address a mixed-integer linear programming model which selects a cost-minimizing set of available technologies with which to design a renewable energy system and prescribe their associated dispatch decisions. Realistically sized instances of such models pose computational challenges. To this end, we develop a Lagrangian heuristic based on a decomposition methodology which partitions the model into blocks and optimizes these more manageable, smaller subproblems. It also provides a lower bound to assess solution quality. In conclusion, we apply this methodology to the National Renewable Energy Laboratory's Renewable Energy Integration and Optimization (REopt TM ) model to generate near-optimal solutions to realistic instances containing, on average, approximately 300,000 variables and at least as many constraints, with a mean 30% optimality gap improvement using a five-minute solution time limit, compared to directly solving the original monolith.

97 MATHEMATICS AND COMPUTING↗

Heuristic solutions to the single depot electric vehicle scheduling problem with next day operability constraints

This study focuses on the single depot electric vehicle scheduling problem (SDEVSP) within the broader context of the vehicle scheduling problem (VSP). By developing an effective scheduling model using mixed-integer linear programming, we generate bus blocks that accommodate electric vehicles (EVs), ensuring successful completion of each block while considering recharging requirements between blocks and during off-hours. Next day operability constraints are also incorporated, allowing for seamless repetition of blocks on subsequent days. The SDEVSP is known to be computationally complex, deriving optimal solutions unattainable for large-scale problems within reasonable timeframes. To address this, we propose a two-step solution approach: first solving the single depot VSP, and then addressing the block chaining problem (BCP) using the blocks generated in the first step. The BCP focuses on optimizing block combinations to facilitate recharging between consecutive blocks, considering operational constraints. Further, a case study conducted reveals that nearly 100% electrification for Chicago, IL and Austin, TX transit buses is viable yet requires 1.6 EVs at 150-mile range per diesel vehicle.

33 ADVANCED PROPULSION SYSTEMS↗

Solving the Grid Optimization Competition Challenge 3 Problem

The Grid Optimization Competition Challenge 3 Problem posed a multiperiod security-constrained unit commitment problem with base-case AC power flow. The problem formulation includes binary unit commitment decisions, nonlinear AC power flow and balance, dispatchable loads, and linearized contingency real power flow, among other features. This talk will present a modified consensus ADMM algorithm, which splits the problem into mixed-integer linear and nonlinear components, as a heuristic solution method for this large-scale mixed integer nonlinear program. We will present some computational results from the competition for our implementation and reflect on the challenges of participating the grid optimization competition.

AC power flow↗

Multi-parametric analysis for mixed integer linear programming: An application to transmission upgrade and congestion management

Upgrading the capacity of existing transmission lines is essential for meeting the growing energy demands, facilitating the integration of renewable energy, and ensuring the security of the transmission system. This study focuses on the selection of lines whose capacities and by how much should be expanded from the perspective of the Independent System Operators (ISOs) to minimize the total system cost. We employ advanced multi-parametric programming and an enhanced branch-and-bound algorithm to address complex mixed-integer linear programming (MILP) problems, considering multi-period time constraints and physical limitations of generators and transmission lines. To characterize the various decisions in transmission expansion, we model the increased capacity of existing lines as parameters within a specified range. This study first relaxes the binary variables to continuous variables and applies the Lagrange method and Karush-Kuhn-Tucker (KKT) conditions to obtain optimal solutions and identify critical regions associated with active and inactive constraints. Moreover, we extend the traditional branch-and-bound (B&B) method by determining the problem’s upper and lower bounds at each node of the B&B decision tree, helping to manage computational challenges in large-scale MILP problems. Here, we compare the difference between the upper and lower bounds to obtain an approximate optimal solution within the decision-makers’ tolerable error range. In addition, the first derivative of the objective function on the parameters of each line is used to inform the selection of lines for easing congestion and maximizing social welfare. Finally, the capacity upgrades are selected by weighing the reductions in system costs against the expense of upgrading line capacities. The findings are supported by numerical simulations and provide transmission-line planners with decision-making guidance.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Cooperative Transmission Expansion Planning Experiment Data and Results

GO WEST is an open-source power grid modeling framework for U.S. Western Interconnection, which allows users to tailor the model depending on their research study and science questions. It covers 28 balancing authorities (BA) and 12 states in U.S. Western Interconnection. GO WEST allows users to select different number of nodes and come up with a simplified network by utilizing 10,000 nodal topology of U.S. Western Interconnection created by Texas A&M University. Users can try and select different number of nodes, mathematical formulations (linear programming vs. mixed-integer linear programming), transmission line limit scaling factors, and hurdle rate scaling factors. GO WEST offers a unit commitment and economic dispatch (UC/ED) module to simulate grid operations on an hourly scale. In this sense, users can calibrate and validate their model versions by comparing model outputs to historical datasets. TEP is an open-source transmission capacity expansion model, built on GO WEST framework. It utilizes linear programming to optimize transmission capacity addition investment on existing lines within GO WEST framework. In this sense, TEP model only increases the thermal capacity of existing transmission lines and does not add new lines to the system, which leaves the topology preserved. TEP minimizes the total cost of the system which comprises the operational cost of satisfying electricity demand (i.e., generation cost), cost of loss of load (i.e., unserved energy), cost of power flow, and cost of new transmission capacity additions (i.e., investment cost). In order to use TEP model, users need to create scenarios with GO WEST framework. In this analysis, outputs from several models are used to create future inputs to GO WEST and TEP models, including GCAM-USA, TELL, CERF and reV. This dataset includes experiment inputs and outputs from three different transmission expansion scenarios (cooperative, intermediate, and individual) for 2019 and 2059. For 2019, a base scenario to illustrate the default (i.e., historical) power grid operations is also included. This study utilizes rcp45hotter_ssp3 scenario from a previous version of GCAM-USA simulations. Sources of the shapefiles in supplementary data are HIFLD Open and U.S. Energy Atlas. Please see the README file for a detailed description of the main and supplementary data.

Capacity Expansion Model↗

IM3 Projected U.S. Western Interconnection Grid Stress Dataset

This dataset provides projected grid stress and reliability results (including all model inputs and outputs from GO WEST and TEP) for Integrated Multisector, Multiscale Modeling (IM3) Phase 2 simulations across eight different scenarios for the U.S. Western Interconnection through 2055. The scenarios include combinations of two Shared Socioeconomic Pathways (SSP3 and SSP5) with four high-resolution climate projections specific to the United States from a set of Thermodynamic Global Warming (TGW) simulations. These climate projections include "hotter" and "cooler" variants for two Representative Concentration Pathways (RCP4.5 and RCP8.5). The resulting eight simulations are: rcp45cooler_ssp3 rcp45cooler_ssp5 rcp45hotter_ssp3 rcp45hotter_ssp5 rcp85cooler_ssp3 rcp85cooler_ssp5 rcp85hotter_ssp3 rcp85hotter_ssp5 GO WEST is an open-source power grid modeling framework for the U.S. Western Interconnection, which allows users to tailor the model depending on their research study and science questions. It covers 28 balancing authorities (BAs) and 12 states in U.S. Western Interconnection. GO WEST allows users to select different number of nodes and come up with a simplified network by utilizing 10,000 nodal topology of the U.S. Western Interconnection (ACTIVSg10k). Users can select different number of nodes, mathematical formulations (linear programming vs. mixed-integer linear programming), transmission line limit scaling factors, and hurdle rate scaling factors. GO WEST offers a unit commitment and economic dispatch (UC/ED) module to simulate grid operations on an hourly scale. In this sense, users can calibrate and validate their model versions by comparing model outputs to historical datasets. TEP is an open-source transmission capacity expansion model, built on the GO WEST framework. It utilizes linear programming to optimize transmission capacity addition investment on existing lines within the GO WEST framework. The TEP model only increases the thermal capacity of existing transmission lines and does not add new lines to the system, which leaves the topology preserved. In order to use TEP model, users need to create scenarios with the GO WEST framework. Please refer to README file for a detailed description of the dataset including individual files and references.

Capacity Expansion Model↗

Optimization of orbital assignment and specification of service areas in satellite communications

The mathematical nature of the orbital and frequency assignment problem for communications satellites is explored, and it is shown that choosing the correct permutations of the orbit locations and frequency assignments is an important step in arriving at values which satisfy the signal-quality requirements. Two methods are proposed to achieve better spectrum/orbit utilization. The first, called the delta S concept, leads to orbital assignment solutions via either mixed-integer or restricted basis entry linear programming techniques; the method guarantees good single-entry carrier-to-interference ratio results. In the second, a basis for specifying service areas is proposed for the Fixed Satellite Service. It is suggested that service areas should be specified according to the communications-demand density in conjunction with the delta S concept in order to enable the system planner to specify more satellites and provide more communications supply.

Wang, Cou-Way↗

Optimal Economic Dispatch and Load-Following Strategies for Nuclear Integrated Energy Systems

The need for distributed and adaptable energy resources that can handle the growing unpredictability in both supply and demand is rising as the power system continues to modernize. In order to satisfy those needs and maintain grid resilience, nuclear power plants can dynamically control their output, despite typically being used as baseload generators. By incorporating energy storage and renewable energy sources, nuclear integrated energy systems are designed to satisfy the electrical and thermal demands of different end-user applications while ensuring flexible power operation. These systems generate revenue by participating in both wholesale and ancillary services electricity markets, as well as commodity markets for various byproducts generated from coupled industrial processes. This study addresses the economic dispatch efficiency of a tightly coupled nuclear integrated energy system comprising a gigawatt-scale light water reactor, commercialized in the U.S., a high-temperature steam electrolysis unit, a district heating network, and specified electrical loads. To demonstrate the nuclear power plant’s flexibility within the day-ahead unit commitment and economic dispatch framework, while maintaining equilibrium even during periods of refueling outages, this paper develops a mixed-integer linear programming framework that models the subsystems and components of its nuclear steam supply system. A systematic comparative analysis of flexible versus baseload nuclear power plant operation under varying levels of renewable energy integration indicates that flexible operation enhances system profitability by more than 18% while also increasing energy storage utilization, improving reactor responsiveness to load fluctuations, and allowing for greater participation across numerous electricity markets.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

Near-Optimal Solutions for Day-Ahead Unit Commitment

Given the difficulty and the time pressure of solving unit commitment problems, near -optimal solutions (those with 0.1 or 0.001% optimality gaps) are often used in practice. The choice in which of the near -optimal solutions is used, however, is random. We investigate the impact of solution choice on the revenues obtained by generator owners across a variety of pricing schemes and problem instances.

market-clearing↗

Assessing the Reliability Benefits of Energy Storage as a Transmission Asset

Utilizing energy storage solutions to reduce the need for traditional transmission investments has been recognized by system planners and supported by federal policies in recent years. This work demonstrates the need for detailed reliability assessment for quantitative comparison of the reliability benefits of energy storage and traditional transmission investments. First, a mixed-integer linear programming expansion planning model considering candidate transmission lines and storage technologies is solved to find the least-cost investment decisions. Next, operations under the resulting system configuration are simulated in a probabilistic reliability assessment which accounts for weather-dependent forced outages. The outcome of this work, when applied to TPPs, is to further equalize the consideration of energy storage compared to traditional transmission assets by capturing the value of storage for system reliability.

co-optimization↗

An optimization model for energy generation and distribution in a dynamic facility

An analytical model is described using linear programming for the optimum generation and distribution of energy demands among competing energy resources and different economic criteria. The model, which will be used as a general engineering tool in the analysis of the Deep Space Network ground facility, considers several essential decisions for better design and operation. The decisions sought for the particular energy application include: the optimum time to build an assembly of elements, inclusion of a storage medium of some type, and the size or capacity of the elements that will minimize the total life-cycle cost over a given number of years. The model, which is structured in multiple time divisions, employ the decomposition principle for large-size matrices, the branch-and-bound method in mixed-integer programming, and the revised simplex technique for efficient and economic computer use.

Lansing, F. L.↗

Extreme-scale EV charging infrastructure planning for last-mile delivery using high-performance parallel computing

Here, this paper addresses stochastic charger location and allocation problems under queue congestion for last-mile delivery using electric vehicles (EVs). The objective is to decide where to open charging stations and how many chargers of each type to install, subject to budgetary and waiting-time constraints. We formulate the problem as a mixed-integer non-linear program, where each station-charger pair is modeled as a multiserver queue with stochastic arrivals and service times to capture the notion of waiting in fleet operations. The model is extremely large, with billions of variables and constraints for a typical metropolitan area; even loading the model in solver memory is difficult, let alone solving it. To address this challenge, we develop a Lagrangian-based dual decomposition framework that decomposes the problem by station and leverages parallelization on high-performance computing systems, where the subproblems are solved by using a cutting plane method and their solutions are collected at the master level. We also develop a three-step rounding heuristic to transform the fractional subproblem solutions into feasible integral solutions. Computational experiments on data from the Chicago metropolitan area with hundreds of thousands of households and thousands of candidate stations show that our approach produces high-quality solutions in cases where existing exact methods cannot even load the model in memory. We also analyze various policy scenarios, demonstrating that combining existing depots with newly built stations under multiagency collaboration substantially reduces costs and congestion. These findings offer a scalable and efficient framework for developing sustainable large-scale EV charging networks.

Capacity allocation↗

Localization of Ad-Hoc Lunar Constellations in Communication Failure Modes for Distributed Spacecraft Autonomy

As lunar missions increase in complexity inspired by NASA’s Artemis Program, they will require reliable and sufficient capability of the Position, Navigation, and Timing (PNT) system to support their scientific objectives. In addition, NASA's Commercial Lunar Payload Services (CLPS) program initiates the proliferation of public and private exploration partnerships using small satellites from commercial and private organizations, expanding traditionally confined low Earth orbit to be used for missions beyond geosynchronous orbit (Zucherman et al., 2022). Therefore, the Lunar PNT system is also required to provide navigation services compatible with the smaller platforms being sent by the public and private sectors, like CubeSats. However, traditional approaches to deep space missions’ navigation based on ground radio facilities have difficulties in providing sufficient support for the increasing number of users and communication at a distance from the Earth (Kaplev et al., 2022). In particular, the existing Lunar navigation technologies such as weak signal global positioning system (GPS) and deep space network (DSN) are not able to ensure operations of the upcoming small-scale Lunar missions due to their limitations in localization performance as well as capacity aspects. Another way to provide Lunar PNT service is to create a dedicated Lunar global navigation satellite system (GNSS) constellation, like GNSS systems on Earth. Space agencies like NASA, ESA, and JAXA are now developing the lunar communications relay and navigation systems (LCRNS) and Lunar navigation satellite systems (LNSS). In their systems, satellites will be deployed in moon orbits to provide the communication, positioning, navigation, and timing (CPNT) service at the lunar south pole region where the Artemis base camp will be expected (Murata et al., 2022). Meanwhile, common challenges considered in lunar PNT research arise from poor geometry of the terrestrial GNSS satellites when seen from the lunar user, highly perturbed lunar orbits, and limitations in power, size, and cost of the equipment on lunar satellites (Iiyama et al., 2023). It is also not clear if there will be enough Lunar users to support the cost and resources this would require as the Low-cost surface missions may not be able to support the large power, mass, and weight requirements that these navigation solutions entail (Niemoeller et al., 2022). As an alternative, existing Lunar science and exploration assets could be used to create a low-cost, autonomous, ad-hoc, and on-demand mission-centric Lunar PNT swarm capable of providing PNT services to these low-cost lunar missions (Hagenau et al., 2021). Introducing the non-dedicated and ad-hoc Lunar navigation constellation gives a way to provide PNT services on-demand. The non-dedicated swarm assets of Lunar constellations are designed to localize themselves with minimal interaction with Earth by adding cooperative autonomous localization to lunar missions, freeing up valuable bandwidth and ground segment resources. An autonomous localization of Lunar constellations is based on the concept of the decentralized PNT system with a distributed extended Kalman filter (DEKF) approach to state estimation for minimal onboard operating costs. In the distributed data processing algorithm, computation is broken down and assigned to each satellite, resulting in a considerably decreased computational amount while maintaining the accuracy of the orbit ephemeris and clock offsets as the result of centralized data processing (Wen et al., 2019). The DEKF requires spacecraft to perform two-way ranging operations with each other to communicate simultaneously, leveraging neighbor two-way intersatellite link (ISL) measurements such as pseudoranges to, and relative velocities between, visible satellites as sensor values (Frank et al., 2021). The Lunar autonomous PNT simulation (LAPS) demonstrated the feasibility of orbital asset localization among ad-hoc Lunar small-sat constellations based on the DEKF in Hagenau et al. (2021) and evaluated the matching algorithm proposed by Frank et al. (2021) in scheduling position estimation updates. In previous papers, all assets and measurements are assumed to be always available without consideration of the impact of intermittent and permanent communication failure. This study presents localization performance with increasing levels of network degradation for swarm assets and users to demonstrate the robustness of the decentralized Lunar PNT service in more realistic scenarios. Main issues arising from communication failure include spacecraft permanent or transient loss, antenna failures, message delays, etc. We tested four possible reasons for network degradation for 7 days in 21 satellites frozen with an altitude of 5500 km, evenly spaced around 3 circular, 40 inclination orbital planes where each spacecraft has two directional antennas. As anchor nodes with an independent estimate of their position are required in the DEKF approach, two ground nodes in each pole and one node in the gateway were implemented in the simulation. First, the most probable failure scenario involves the loss of a single spacecraft due to solar interference and technical malfunctions of the assets. Losing the availability of a single spacecraft means losing the two-way ISL measurement of the asset in the DEKF update. In order to provide the best possible quality of PNT service with limited time and resources, the distributed Lunar constellations must schedule the communication activities. The scheduler leverages mixed-integer linear programming (MILP) for the coordination and scheduling of the desired “as-needed” localization service (Niemoeller et al., 2022). We assume the scheduler has completely excluded the spacecraft information before the DEKF update in the failure scenario. When a random spacecraft has been turned off at a specific time, the robustness of the autonomous Lunar PNT system is evaluated. The simulation results give an 11.5% degradation in median position accuracy compared to the idealized performance excluding the asset loss. Second, a large number of assets may vanish due to major hardware problems or meteor strikes around the moon. A multiple spacecraft loss can degrade the localization performance very fast by losing the communication ability to do cross-plane measurements and in-plane measurements in a 3-plane constellation. When the matching-based scheduler is aware of ISL availability, we investigate a large number of in-plane and cross-plane asset vanishments both in close proximity and equally spaced throughout the orbital plane. According to the simulations, the loss of in-plane measurements gives 40.2% degradation while cross-plane measurements degrade 50.5% of asset localization performance among available assets. Therefore, it is concluded that cross-plane measurements are more important in improving the position estimation accuracy. Third, spacecraft failure information can be lost due to the internal message delay, resulting in the DEKF update scheduler to solve the matching problem with unavailable assets. The DEKF update cycle is comprised of network setup, communication, and computations where a global broadcast network and a 2-way ISL network setup take 6 minutes in total (Frank et al., 2021). Once the broadcast network successfully transmits and receives information, a random spacecraft may lose its availability right before solving the matching problem. This means the matching solution is no longer optimal, resulting in degradation in the localization performance. A numerical assessment shows the matching-based scheduler with knowing failure holds 11.5% of position accuracy degradation, whereas the scheduler without knowing failure gives 34% degraded localization performance without asset loss. Fourth, a transient loss of a single or multiple spacecraft may occur due to their antenna outages. After losing the two-way ISL availability for a few DEKF update cycles, the availability of spacecraft can easily be recovered as their states have been independently updated using measurements from anchor nodes. It is likely that the longer failure will result in worse localization performance. We have tested the transient failure of a random single asset for 30 min in the simulation, which is losing 3 update cycles in the DEKF system. From the simulation results, the position accuracy has been degraded to 4.84% which is better than the degraded localization performance of 11.5% from the permanent loss scenario among available assets. In conclusion, the autonomous Lunar PNT system based on the DEKF approach shows the ability to maintain resilience and robustness in the possible communication failure scenarios, ensuring that localization accuracy is preserved across various network degradation and outages. Future studies on investigating user localization performance near the South Pole and the broadcast network system will be continued in the following months.

Yeji Kim↗

A Decomposition-Based Learn-To-Optimize Approach with Feasibility Layer Assistance for Sub-Hourly Unit Commitment

Sub-hourly unit commitment (UC) with 15-min intervals is gaining significant attention as a way to respond rapidly to the fluctuations in electricity supply and demand introduced by renewable resources. However, the increased temporal resolution and complex inter-temporal dependencies pose substantial computational challenges for traditional optimization methods. To this end, this paper explores a decomposition-based learn-to-optimize approach. Building on recent advances in machine learning, our method revisits the long- overlooked Lagrangian relaxation framework, which is a classical decomposition technique that enables tractable subproblem solving. These smaller subproblems are inherently well-suited for machine learning, as their reduced dimensionality and structural regularity allow predictive models to efficiently learn and generalize solution patterns. We thus propose a generic predictive model, which embeds Gated Recurrent Units (GRUs) and Attention in the encoder-decoder structure, and integrate a rule-based feasibility layer to capture temporal dependencies, reduce training effort, and improve feasibility w.r.t. unit-level constraints. Our method has been validated on the IEEE 118-bus system, demonstrating promising performance in solving sub-hourly UC problems efficiently and feasibly.

97 MATHEMATICS AND COMPUTING↗

Relaxations of the steady optimal gas flow problem for a non-Ideal gas

Natural gas ranks second in U.S. primary energy consumption. Because most production sites are remote, gas must be transported through pipeline networks equipped with compressors, valves, and other components. For both economic efficiency and system reliability, it is desirable to operate these networks optimally. The governing physics across pipeline components entails nonlinear, non-convex equality and inequality constraints, and the most general steady-flow operations problem is a Mixed-Integer Nonlinear Program (MINLP).This work focuses on one such steady-flow problem-the Optimal Gas Flow (OGF) for a natural gas pipeline network-which minimizes production cost subject to the steady-flow physics. For day-to-day operations, the ability to quickly compute a globally optimal solution and a strong lower bound for varying demand profiles is crucial. A promising strategy is to build tight relaxations of the OGF’s nonlinear constraints. However, many nonlinearities arising from non-ideal equations of state either lack relaxations or have relaxations that do not scale to realistic network sizes. We address this gap by combining recent advances in polyhedral relaxations for univariate functions to construct tight, computationally efficient relaxations of the OGF with a non-ideal equation of state. These relaxations solve within seconds on a standard laptop. In conclusion, we demonstrate their quality through extensive numerical experiments on very large-scale test networks from the literature and find that the proposed approach proves optimality in 92% of tested instances.

03 NATURAL GAS↗