Weymouth Equation Research Articles

Exploring Carbon Equilibrium in Integrated Electricity–Hydrogen System

Global carbon emissions from electricity and hydrogen production have grown rapidly in recent years. Some existing literature has proposed the method of reducing carbon emissions by power-to-hydrogen (P2H) technology. However, due to the fossil-based energy structure of electricity system, excessive electricity to hydrogen may lead to an increase in carbon emissions. To reduce carbon emissions effectively, a low carbon-driven optimal capacity configuration method for P2Hs is proposed in this paper to explore carbon equilibrium in integrated electricity-hydrogen system. A joint economic dispatch model for integrated electricity-hydrogen system is formulated. The electricity system model is a security-constrained economic dispatch model with P2H constraints. The hydrogen system model is a hydrogen supply chain model with pipeline pack constraints and Weymouth equation, whose sources are fossil fuel reforming and electrolytic hydrogen. We propose a second-order cone programming (SOCP)-based solution to transform the hydrogen system model into a mixed integer second order cone programming (MISCOP), which can be readily solved by commercial solvers. To preserve the internal information of each system, a decentralized algorithm based on optimality condition decomposition (OCD) is developed. Case study based on modified IEEE 30-bus and Belgium 20-node integrated energy system demonstrates the effectiveness of the proposed strategy.

Multi-objective robust optimization for optimal operation of gas-electricity networks with the penetration of integrated P2G and electric vehicle parking lots

Energy is considered one of the most important human needs. Therefore, various energy resources are exploited to meet human needs. Among all available resources, electricity and natural gas have accounted for the largest share in meeting these needs. Accordingly, management of these resources is one of the most important issues in different societies. This paper presents a multi-objective optimization model for optimal management and operation of the power distribution network and natural gas network as well as reducing carbon emissions. In this modeling, the natural gas network and distribution power network are connected through two electric vehicle parking lots (EVPLs) and two gas-fired units (GFUs), and these two EVPLs and GFUs exchange energy with each other. The EVPLs are equipped with various facilities including power-to-gas (P2G) systems and natural gas storage that facilitate energy exchange between the two networks. The modeling presented in this article is based on mixed integer linear programming, so the nonlinear equations including power flow equations, the Weymouth equation and the cost functions of GFUs are also linearized using various methods. It should be noted that the price of purchased electricity poses uncertainty which is modeled using robust optimization techniques. The proposed model is tested on the 33-bus power distribution network and the 20-node natural gas network. The results show that the presence of EVPLs significantly reduces the operational costs of the natural gas network and power distribution network and carbon emissions.

Optimization of Interconnected Natural Gas and Power Systems Using Mathematical Programs with Complementarity Constraints

The demand for thermal power generation from natural gas has increased globally due to its cleaner burning properties compared to other fossil fuels. Optimizing the gas flow through the network to meet this demand is challenging due to the nonconvex Weymouth equation constraining gas flow and nodal pressures in pipelines. Traditional methods for addressing this nonconvexity lead to significant approximation errors or high operational costs. This study poses the Weymouth constraint as a Mathematical Programming with Complementarity Constraints (MPCC) for an optimal gas flow problem. The complementarity constraints reformulate the discontinuous sign function using binary-behaving continuous variables. This MPCC-based approach avoids solving mixed-integer programming problems while enhancing the accuracy of conventional linear and second-order approximations. Testing the approach on various interconnected systems, including Colombia’s national gas transportation grid, demonstrated significant reductions in Weymouth approximation errors, thereby supporting effective optimization for interconnected networks.

MPNG: A MATPOWER-Based Tool for Optimal Power and Natural Gas Flow Analyses

MPNG is an open-source simulation package for solving Optimal Power and Natural Gas Flow (OPNGF) problems. It is coded in <small>Matlab</small> and developed using the widely-used tool for power systems research and education, <small>Matpower</small>. MPNG assembles a nonlinear steady state OPNGF formulation for analyzing the integration of electricity and natural gas networks with order-of-priority classes of consumers and gas nodal price-controlled storage facilities. This paper presents the details of the nonlinear optimization model behind MPNG, as well as several features that were designed into it for the purpose of achieving robustness. These include an approximation of the Weymouth equation for handling bi-directional gas flows in pipelines, a per-unit representation for the natural gas network, and penalized slack variables for power and gas balances. Simulation results using a test case and the Colombian power and natural gas networks are presented to highlight MPNG&#x0027;s capabilities to solve OPNGF problems.

Optimal Resiliency‐Oriented Scheduling Framework for Integrated Power‐Gas‐Transportation Networks Based on Model Predictive Control Approach to Increase Electric Load Supply

Today’s societies need a sustainable supply of energy more than ever, whereas unseen events challenge the electric power supply. The coordinated operation of a power‐gas‐transportation system can decrease the effects of such faults in the distribution level. This paper proposes a model predictive control approach for dynamic load restoration in an integrated energy system. The coordination of electric and gas networks besides the mobile energy storage (MES) units provides a novel resilient alternative to serve active and reactive loads. First, the location and time of drastic events are identified as the initialization phase in this paper using a vulnerability analysis by a master–slave problem to find the weak points of the electric grid. Then, a rolling horizon model predictive control‐based approach is proposed to make corrective decisions to serve loads. Meanwhile, a new linearization approach for the Weymouth equation has been proposed based on the binary expansion method. The proposed linear resiliency‐oriented scheduling problem is tested on IEEE 33‐bus and IEEE 69‐bus integrated with a seven‐node gas distribution grid and a six‐station transportation railway. The analysis revealed that the coordination between electric and gas resources minimizes the not‐supplied load; MES units act as certain emergency tools to serve the critical loads.

Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

Optimal operation of coordinated multi-carrier energy hubs for integrated electricity and gas networks

"In the field of energy supply, coordinated multi-carrier systems, which include natural gas and electric energy, offer unique opportunities to enhance energy efficiency and flexibility. Nevertheless, the interdependence between electricity and natural gas networks poses various challenges related to the flow of electricity and gas in feeders, pipes, and connection points. To address these challenges, the concept of an energy hub emerges as an essential component in multi-carrier energy systems, serving as a connection point between the electricity grid and gas grids. This study introduces an optimal operation method for coordinating gas and electricity distribution networks through interconnected energy hubs. The infrastructure of the proposed energy hub comprises a combined heat and power unit, a boiler, an electric energy storage system, a heat pump, and a gas-burning unit, effectively meeting heating and electrical load requirements. The model formulation follows a stochastic approach based on a two-stage scenario. Considering uncertainties related to wind energy, electric load, and real-time energy prices, the primary objective of this study is to minimize the total operating cost of the energy system. The optimization problem is solved using Wall's optimization algorithm. To meet its energy requirements, the integrated energy system presented in this paper actively participates in the real and daily electric energy markets, as well as the natural gas market, providing the necessary energy resources. Additionally, the model incorporates realistic descriptions of electric power and gas flow in feeders and gas pipelines, considering AC load distribution and the Weymouth equation. With consideration of connectivity constraints, the proposed model offers a comprehensive representation of the integrated electricity and gas networks. The simulations conducted on the integrated energy system, comprising a 33-bus electric network and a 6-node gas network with interconnected energy hubs, highlight a remarkable 12 % decrease in electricity purchase costs and a substantial 18 % reduction in natural gas procurement expenses, underscoring the cost-efficiency and effectiveness of the stochastic planning approach.

Congestion management for coordinated electricity and gas grids in the presence of multi-energy hubs: A risk-based optimal scheduling

Due to the development of cost-effective and high-efficient infrastructures in electricity and natural gas carriers, the need for optimal operation of multi-carrier energy systems is essential. On the other side, with the increasing penetration of renewable energy resources (RERs), and power demand, the congestion management (CM) of power systems is turning out to be a significant challenge for system operators. Motivated by these, this paper focuses on the congestion management problem for a multi-energy hub (MEH) system incorporated by RERs, flexible energy resources (FERs), and different energy conversion technologies to supply electrical, heat, and cooling demands, aiming to minimize the total operating costs. In current practice, the steady-state Weymouth equation and AC-power flow method for natural gas and electricity networks are implemented to investigate the interdependency of power and gas networks. The renewable energy generation variations are mitigated by the scenario-based stochastic framework. Also, to manage the risk associated with random variables, the downside risk constraint (DRC) model is employed to analyze the changes in the expected cost per several values of the risk parameters. The numerical results demonstrate that the proposed model can significantly utilize the energy hub’s flexibility to alleviate gas and power congestion while decreasing the total operation cost.

Optimization of integrated energy system considering transmission and distribution network interconnection and energy transmission dynamic characteristics

The traditional hierarchical scheduling and operational model of transmission and distribution networks has failed to fully utilize the adjustable resources of the entire grid, consequently limiting the integration of renewable energy. To fully excavate the potential of transmission and distribution (T&D) coordination in enhancing the level of renewable energy assimilation and operational efficiency, in this paper, transmission grid and distribution network in the framework of synergetic unit commitment (TDS-UC) considering the dynamic characteristics of the electricity-gas-heat integrated energy system is proposed. The framework developed in this paper allows for the integration of multiple energy sources, including electricity, gas, and heat, into the dispatch strategy of the TG. To improve scheduling flexibility, the dynamic characteristics of the natural gas and heating networks are also considered in the TDS-UC model. A coordinated interactive optimization model for T&D is formulated, taking into account the multi-energy coupling relationship of the IES. To solve the TDS-UC model, a decentralized analytical target cascading algorithm is used to obtain an independent TG optimization problem and multiple ADNs local optimization problems that can be solved in parallel. The proposed model is transformed into an easy-to-solve mixed-integer linear programming (MILP) problem using an improved one-dimensional approximation piecewise linearization method to deal with the non-convexity nonlinear natural gas Weymouth equation. Numerical experiments are conducted on both a T6D2 and a T118D10 test system to demonstrate the efficiency and practicality of the proposed methodology.© 2017 Elsevier Inc. All rights reserved.

A successive linearization based optimal power-gas flow method

Optimal power-gas flow (OPGF) problem is a fundamental coordinated operation problem. However, the nonlinear AC power flow model and nonconvex gas flow model pose a huge challenge to OPGF calculation. The common treatments in the existing literature adopt simplified modeling of power and gas systems to obtain a relatively tractable model, which may produce an inaccurate and insecure solution (e.g., violation of voltage limits). Therefore, this paper considers the nonlinear AC power flow modeling and nonconvex gas flow modeling in the OPGF problem. To address the nonconvexities arising from the nonlinear AC power flow model, nonconvex Weymouth equations, and compressors’ gas consumption equations, the concept of successive linearization is introduced and then a successive mixed-integer quadratic programming (SMIQP) method is developed to solve the nonconvex OPGF problem. Numerical results demonstrate the effectiveness of the proposed SMIQP-based OPGF method in terms of high accuracy and high computational efficiency.

Natural Gas Short-Term Operation Problem With Dynamics: A Rank Minimization Approach

Natural gas-fired generation units can hedge against the volatility in the uncertain renewable generation, which may occur during very short periods. It is crucial to utilize models capable of correctly capturing the natural gas network dynamics induced by the volatile demand of gas-fired units. The Weymouth equation is commonly implemented in literature to avoid dealing with the mathematical complications of solving the original governing differential equations of the natural gas dynamics. However, it is shown in this paper that this approach is not reliable <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in the short-term operation problem</i> . Here, the merit of the non-convex transient model is compared with the simplified Weymouth equation, and the drawbacks of employing the Weymouth equation are illustrated. The results demonstrate how changes in the natural gas demand are met by adjustment in the pressure within pipelines rather than the output of natural gas suppliers. This work presents a convex relaxation scheme for the original non-linear and non-convex natural gas flow equations with dynamics, utilizing a rank minimization approach to ensure the tightness. The proposed method renders a computationally efficient framework that can accurately solve the non-convex non-linear gas operation problem and accurately capture its dynamics. Also, the results suggest that the proposed model improves the solution optimality and solution time compared to the original non-linear non-convex model. Finally, the scalability of the proposed approach is verified in the case study.

Optimal dispatching strategy for power and natural gas system in the distribution level with generator using residual differential pressure energy

Natural-gas fired units (NGFU) occupy an increasingly important position in the power system, leading to a closer connection between the power system and natural gas system, especially in the distribution level. To utilize the residual differential pressure energy and reduce operation costs, the conversion model of the generator using residual differential pressure energy is formulated, and then the cooperation model of the power system and natural gas system is built in this paper. Due to the original optimization model of power system and natural gas system in the distribution level is non-convex, the second-order cone programming (SOCP) based on a modified convex concave procedure (CCP) algorithm is used to handle the NP-hard problem. The simulation results show operation costs are reduced via the proposed strategy, and the SOCP based on the modified CCP algorithm achieves good performance in fast solution with the constraint of Weymouth equation.

Risk‐based optimal operation of coordinated natural gas and reconfigurable electrical networks with integrated energy hubs

This paper elaborates on optimal scheduling of coordinated power and natural gas (NG) networks in the presence of interconnected energy hubs considering reconfiguration as a flexibility source. With regard to the energy hub system consisting of several generation units, storage and conversion technologies, as well as natural gas-fired units, the high interdependency between gas and electricity carriers should be captured. The hourly reconfiguration capability is developed for the first time in a multi-energy system to enhance the optimal power dispatch and gas consumption pattern. The realistic interdependency of electrical and NG grids is investigated by employing the steady-state Weymouth equation and AC-power flow model for power and gas networks, respectively. Furthermore, to handle the risk associated with strong uncertainty of wind power, load, and real-time power price, the conditional value at risk approach is employed. The proposed model is implemented on the integrated test system and simulation results are presented for different cases. The impact of the risk aversion level on operating cost and optimal scheduling of controllable units is examined. Numerical results demonstrate that reconfigurable capability reduces the operational cost up to 7.82%.

Data-Driven Chance-Constrained Optimal Gas-Power Flow Calculation: A Bayesian Nonparametric Approach

This paper proposes a data-driven chance-constrained optimal gas-power flow (OGPF) calculation method without any prior assumption on the distribution of uncertainties of wind power generation. The Gaussian mixture model is employed to fit the uncertainty distribution, where the Bayesian nonparametric Dirichlet process is adopted to tune the component number. To facilitate the online application of the proposed methods, an online-offline double-track distribution construction approach is established, where the frequency of training the relatively time-consuming Dirichlet process Gaussian mixture model can be reduced. On account of the quadratic gas consumption expression of gas-fired generators as well as the linear decision rule based uncertainty mitigation mechanism, the chance constraints would become quadratic ones with quadratic terms of uncertainties, which makes the proposed model more intractable. An equivalent linear separable counterpart is then provided for the quadratic chance constraints, after which the intractable chance constraints could be converted into traditional linear ones. The convex-concave procedure is used to crack the nonconvex Weymouth equation in the gas network and the auxiliary quadratic equalities. Simulation results on two test systems validate the effectiveness of the proposed methods.

Decentralized Robust Energy Management of Multi-area Integrated Electricity-gas Systems

This paper proposes a fast and decentralized solution methodology for the robust operation of multi-area integrated electricity-gas systems (M-IEGSs). A deterministic reformulation is obtained for the two-stage robust model by applying the linear decision rule based electrical reserve utilization scheme as well as regulating the distributed gas storages. Two linear approximations are developed for the nonconvex Wey-mouth equation in the gas network to determine the gas flow directions. The penalty convex-concave procedure (P-CCP) is then adopted to refine a feasible local optimum for the nonconvex model with an acceleration strategy. The decentralized deci-sion-making is enabled by the alternating direction multipliers method (ADMM). The convergence as well as computation performance of the overall solution procedure can be guaranteed as only convex optimizations are solved. Simulation results <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">va</sup> lidate the effectiveness of the proposed methods as well as the benefits of the proposed convex programing based solution procedure.

Optimal Operation of Integrated Electrical and Natural Gas Networks with a Focus on Distributed Energy Hub Systems

Coordinated multi-carrier energy systems with natural gas and electricity energies provide specific opportunities to improve energy efficiency and flexibility of the energy supply. The interdependency of electricity and natural gas networks faces multiple challenges from power and gas flow in corresponding feeders and pipes and connection points between two infrastructures’ points of view. However, the energy hub concepts as the fundamental concept of multi-carrier energy systems with multiple conversion, storage, and generation facilities can be considered as a connection point between electricity and gas grids. Hence, this paper proposes an optimal operation of coordinated gas and electricity distribution networks by considering interconnected energy hubs. The proposed energy hub is equipped with combined heat and power units, a boiler, battery energy storage, a heat pump, and a gas-fired unit to meet the heating and electrical load demands. The proposed model is formulated as a two-stage scenario-based stochastic model aiming to minimize total operational cost considering wind energy, electrical load, and real-time power price uncertainties. The proposed integrated energy system can participate in real-time and day-ahead power markets, as well as the gas market, to purchase its required energy. The AC-power flow and Weymouth equation are extended to describe power and gas flow in feeders and gas pipelines, respectively. Therefore, a realistic model for the integrated electricity and gas grids considering coupling constraints is satisfied. The proposed model is tested on the integrated energy system and consists of a 33-bus electrical network and a 6-node gas grid with multiple interconnected energy hubs, where the numerical results reveal the effectiveness of the proposed model.

Exploiting flexibility of combined-cycle gas turbines in power system unit commitment with natural gas transmission constraints and reserve scheduling

With the integration of large-scale renewable energy including wind power into the power grid, efficient flexible resources are required for wind power accommodation. The combined-cycle gas turbines (CCGTs) are widely used in the power system with flexible operation characteristics, which leads to enhanced interdependencies between the power grid and the natural gas network. In this paper, a unit commitment (UC) model with natural gas transmission constraints and reserve scheduling considering the flexibility of CCGT with quick startup-shutdown and flexible operation characteristics is proposed, to reduce wind power curtailment and high startup and shutdown cost of conventional generation units in the power system. Considering the complex operating conditions of CCGT and the non-convexity of Weymouth equation in natural gas network, this paper adopts an improved mode-based model (MBM) for modeling of CCGT and converts the Weymouth equation into a second-order cone relaxation (SOCR) formulation with penalty term through a concave-convex process (CCP) to make the relaxation tight, thus transforming the UC model into an iterative mixed-integer second-order cone programming (MISOCP) problem to be efficiently solved. The effectiveness of the proposed model on reducing wind power curtailment and startup-shutdown cost is demonstrated through case studies.

Urban multi-energy network optimization: An enhanced model using a two-stage bound-tightening approach

The ever-growing interdependence among power distribution, natural gas networks and transportation infrastructure entails efficient and coordinated optimization techniques. This paper proposes an operational optimization model of urban multi-energy networks that encompasses the synergies among the above components, energy converters and electric vehicles. The proposed model defines and encapsulates three energy flows in a convex optimization problem. In the distribution network, the conic relaxation-based branch flow equation is employed to characterize the alternating current power flow. The static characteristics of gas flow are described via the Weymouth equation with convex envelop relaxations. In the transportation network, the traffic flow patterns associated with the interplay among routing and charging behaviors are studied. Since these behaviors are amenable to the Wardrop principle such that the travel cost cannot be reduced by unilaterally altering travel choice, a mixed user equilibrium model is established to describe the related traffic flows. Although the relaxation quality has been improved by deriving the tighter bounds of the variables, to date, bound-tightening approaches have not yet been effectively applied to multi-energy network problems. To strengthen convex relaxations, we enhance the proposed model with a two-stage approach. The first stage improves the variable bounds through sequential optimality-based bound contraction. The second stage iteratively and successively solves the model with a dynamic bound-tightening algorithm. Based on case studies, the interdependence among energy networks is discussed. In addition, the numerical experiments corroborate the solution quality and computational efficiency benefits of the proposed approach.

A convex optimization and iterative solution based method for optimal power-gas flow considering power and gas losses

Optimal power-gas flow (OPGF) is an important problem for the coordinated operation of an integrated electricity and natural gas system (IEGS). However, it is a challenge to solve the OPGF problem due to its non-convexity. This paper proposes a new OPGF method considering both power losses and gas losses based on convex optimization to balance modeling accuracy and tractability. As for power system modeling, a DC power flow model with power losses embedded as equivalent loads is used to approximate the nonlinear AC power flow model. As for gas system modeling, (1) a second-order cone (SOC) relaxation method is proposed to address the non-convex Weymouth equations, and the tightness of the SOC relaxation is guaranteed by introducing a penalty term and an iterative tightening procedure based on the convex envelope. (2) Gas losses of compressors are also modeled as equivalent loads and two more penalty terms on compressors are introduced to improve the accuracy of gas loss modeling. The OPGF problem is finally formulated as a mixed-integer convex programming model with unknown equivalent loads, and an iterative solution approach is then developed to solve the problem. Numerical results validate the effectiveness of the proposed OPGF method.

Risk-Loss Coordinated Admissibility Assessment of Wind Generation for Integrated Electric-Gas Systems

Due to the widely deployment of gas-fired units and the emerging power-to-gas (P2G) technologies, the interdependencies between power systems and gas systems have been enhanced, yet the impacts on the ability of accommodating the uncertain and variable wind generation are unclear. On one hand, the regulation capability of gas-fired units might be restricted due to pipeline congestion or gas supply shortage; on the other hand, the P2G facilities could operate as interruptible electrical loads to enhance the operational flexibility of the power network. This paper aims to revisit the wind generation admissibility assessment problem in view of integrated electric-gas systems. The overall model is established according to the two-stage robust optimization framework. By introducing the stepwise penalty coefficients for power shortage and surplus, the risk preference of the operator can be reflected. The mixed integer linear programming-based approximation methods is adopted to tackle the computational challenging Weymouth equation in the gas network, which requires an additional column-and-constraint (C&CG) loop to solve the second-stage max-min problem, other than the common one to coordinate the first- and second-stage problems, resulting in a nested C&CG algorithmic structure. The effectiveness of the proposed model and solution methods are validated by the simulation results.