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With the increase in the investment in the gas-fired electricity generation technology, capturing the operation constraints of the natural gas network in the electricity and natural gas operation problems becomes more crucial. The nonconvexity in the feasibility region formed by natural gas network constraints will impede achieving the global solut...
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... From the flow equation point of view, the Optimal Gas Flow (OGF) problem utilized in the existing literature of coordinated operation problem is best categorized as steady-state and transient models. The coordinated operation problem of IEGS has been extensively investigated in research works such as securityconstrained approach [12], high renewable penetration case [13], and a convex relaxation method [14]. However, these studies consider natural gas steady-state model, which fails to properly model how the natural gas network dynamics respond to changes in the demand side [15]. ...
... In Ref. [123], linear restrictions were introduced using an MC model for both gas flow and hydraulic-thermal [153] flow. 4) The technique of convex relaxation [156][157][158]: Second-order cone (SOC) relaxation was used in Ref. [156] to effectively convexify the Weymouth equation with an unknown gas flow direction. In Ref. [157], the optimal solution for a detailed MES model was obtained using sparse semidefinite programming (SDP) relaxation. ...
... 4) The technique of convex relaxation [156][157][158]: Second-order cone (SOC) relaxation was used in Ref. [156] to effectively convexify the Weymouth equation with an unknown gas flow direction. In Ref. [157], the optimal solution for a detailed MES model was obtained using sparse semidefinite programming (SDP) relaxation. Manshadi [158] provided a new convex relaxation for natural gas network constraints that were both tight and tractable. ...
... From the flow equation point of view, the Optimal Gas Flow (OGF) problem utilized in the existing literature of coordinated operation problem is best categorized as steady-state and dynamic models. The coordinated operation problem of interconnected electricity and natural gas networks has been extensively investigated such as security-constrained approach [12], high renewable penetration case [13], and a convex relaxation method [14]. However, these works consider natural gas steady-state model, which fails to properly model how the natural gas network dynamics respond to changes in the demand side [15]. ...
In electricity networks with high penetration levels of renewable resources, Flexible Ramping Products (FRPs) are among the utilized measures for dealing with the potential fluctuations in the net demand. This paper investigates the impacts of FRPs on the operation of interdependent electricity and natural gas networks. To accurately model and reflect the effects of variations in the natural gas fuel demand on the natural gas network, a dynamic Optimal Gas Flow (OGF) formulation is utilized. The non-convex dynamic model of the natural gas system is represented in a convex form via a tight relaxation scheme. An improved distributed optimization method is proposed to solve the coordinated operation problem in a privacy-preserving manner, where the two infrastructures only share limited information. We introduce the Inexact Varying Alternating Direction Method of Multipliers (IV-ADMM) and show that compared with the classic ADMM, it converges considerably faster and in fewer iterations. Through a comparison of day-ahead and real-time operation planning results, it is concluded that without accounting for natural gas network dynamics, the FRP model is not a trustworthy tool in day-ahead planning against uncertainties.
... In some research works, the authors have approximated the Weymouth model with linearization techniques [3], [9], [17]. Another approach is opting for convex relaxation of the Weymouth formulation [5], [6], [11], [18], [19], with Second-Order Cone Programming (SOCP) [20]- [22] and Semi-definite Programming (SDP) [23] being among the utilized methods. ...
... 1) Limitation of the Weymouth equation in the natural gas short-term operation problem: A short-term scenario with a varying gas load is generated to exhibit the shortcomings of the Weymouth equation in modeling natural gas network dynamics. Then, the OGF equations are formed and solved for both the non-linear OGF problem presented in (4) and the Weymouth model given in (3) as presented in [18]. During the first hour of operation, the renewable units can provide the required demand. ...
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 in the short-term operation problem. 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.
... In some research works, the authors have approximated the Weymouth model with linearization techniques [3], [9], [17]. Another approach is opting for convex relaxation of the Weymouth formulation [5], [6], [11], [18], [19], with Second-Order Cone Programming (SOCP) [20]- [22] and Semi-definite Programming (SDP) [23] being among the utilized methods. ...
... 1) Limitation of the Weymouth equation in the natural gas short-term operation problem: A short-term scenario with a varying gas load is generated to exhibit the shortcomings of the Weymouth equation in modeling natural gas network dynamics. Then, the OGF equations are formed and solved for both the non-linear OGF problem presented in (4) and the Weymouth model given in (3) as presented in [18]. During the first hour of operation, the renewable units can provide the required demand. ...
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
in the short-term operation problem
. 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.
... In fact, due to the unique interplay between two historically-independent energy systems, the electricity-gas coupled system is receiving an ever-growing research interest, especially in the IEEE PES community, under a broader concept known as the "integrated energy system". A large amount of research aims to address the optimal electricitygas flow problem, an extension to the traditional optimal power flow problem in electrical engineering that also considers the optimality of gas flow [1], [2]. More specifically, for optimal gas flow, the major difficulty lies in how to properly address the nonconvex gas flow equations inherent in the natural gas model, for which many algorithms have emerged. ...
In this letter, we propose a data-driven warm start approach, empowered by an artificial neural network, to boost the efficiency of convex relaxations in optimal gas flow. Case studies show that this approach significantly decreases the number of iterations for the convex-concave procedure algorithm, while optimality and feasibility of the solution can still be guaranteed. We also confirm the robustness of this algorithm, and show that this approach can be extended to the optimal dispatch of large-scale electricity-gas coupled integrated energy systems.
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... This modeling of IEGS results in an NP-hard MINLP problem that has a nonconvex continuous relaxation, which is extremely challenging to solve, even to local optimality, using current state-of-the-art MINLP technology. Therefore, tractable alternatives such as linear programming (LP) approximations [2]- [4], second-order cone programming (SOCP) relaxations [5]- [7], semidefinite programming (SDP) relaxations [8], mixed-integer linear programming (MILP) approximations [9]- [16], and mixed-integer second-order cone programming (MISOCP) relaxations [17]- [24], are garnering considerable attention in the research community. ...
... While convex relaxations such as the SOCP, SDP, and the MISOCP are necessary to better understand the structure and complexity of the problem, they generally do not yield feasible solutions. Moreover, existing SOCP relaxations in [5]- [7] can only be applied when the direction of flow in pipeline is known beforehand, and the existing SDP relaxation in [8] can only be applied in steady-state modeling where the square of the pressure terms can be substituted by linear ones. On the other hand, the MISOCP relaxation in the transient domain in [17]- [22], although able to model bi-directional gas flow in pipes, also yields infeasible solutions in general. ...
... k2 ← k2 + 1. 27: end while a certain tolerance ǫ. 8 The solution obtained at the termination of Phase I is likely to be non-integral, i.e., some of the z t mn variables may not be strictly 0 or 1. This solution is therefore used as a warm start for Phase II which is iterative MILPbased algorithm that closely resembles Phase I but with the integrality constraints (29c) instead of the relaxed ones in (28g). ...
In light of the increasing coupling between electricity and gas networks, this paper introduces two novel iterative methods for efficiently solving the multiperiod optimal electricity and gas flow (MOEGF) problem. The first is an iterative MILP-based method and the second is an iterative LP-based method with an elaborate procedure for ensuring an integral solution. The convergence of the two approaches is founded on two key features. The first is a penalty term with a single, automatically tuned, parameter for controlling the step size of the gas network iterates. The second is a sequence of supporting hyperplanes together with an increasing number of carefully constructed halfspaces for controlling the convergence of the electricity network iterates. Moreover, the two proposed algorithms use as a warm start the solution from a novel polyhedral relaxation of the MOEGF problem, for a noticeable improvement in computation time as compared to a cold start. Unlike the first method, which invokes a branch-and-bound algorithm to find an integral solution, the second method implements an elaborate steering procedure that guides the continuous variables to take integral values at the solution. Numerical evaluation demonstrates that the two proposed methods can converge to high-quality feasible solutions in computation times at least two orders of magnitude faster than both a state-of-the-art nonlinear branch-and-bound (NLBB) MINLP solver and a mixed-integer convex programming (MICP) relaxation of the MOEGF problem. The experimental setup consists of five test cases, three of which involve the real electricity and gas transmission networks of the state of Victoria with actual linepack and demand profiles.
... This modeling of IEGS results in an NP-hard MINLP problem that has a nonconvex continuous relaxation, which is extremely challenging to solve, even to local optimality, using current state-of-the-art MINLP technology. Therefore, tractable alternatives such as linear programming (LP) approximations [2]- [4], second-order cone programming (SOCP) relaxations [5]- [7], semidefinite programming (SDP) relaxations [8], mixed-integer linear programming (MILP) approximations [9]- [16], and mixed-integer second-order cone programming (MISOCP) relaxations [17]- [24], are garnering considerable attention in the research community. ...
... While convex relaxations such as the SOCP, SDP, and the MISOCP are necessary to better understand the structure and complexity of the problem, they generally do not yield feasible solutions. Moreover, existing SOCP relaxations in [5]- [7] can only be applied when the direction of flow in pipeline is known beforehand, and the existing SDP relaxation in [8] can only be applied in steady-state modeling where the square of the pressure terms can be substituted by linear ones. On the other hand, the MISOCP relaxation in the transient domain in [17]- [22], although able to model bi-directional gas flow in pipes, also yields infeasible solutions in general. ...
... k2 ← k2 + 1. 27: end while a certain tolerance ǫ. 8 The solution obtained at the termination of Phase I is likely to be non-integral, i.e., some of the z t mn variables may not be strictly 0 or 1. This solution is therefore used as a warm start for Phase II which is iterative MILPbased algorithm that closely resembles Phase I but with the integrality constraints (29c) instead of the relaxed ones in (28g). ...
In light of the increasing coupling between electricity and gas networks, this paper introduces two novel iterative methods for efficiently solving the multiperiod optimal electricity and gas flow (MOEGF) problem. The first is an iterative MILP-based method and the second is an iterative LP-based method with an elaborate procedure for ensuring an integral solution. The convergence of the two approaches is founded on two key features. The first is a penalty term with a single, automatically tuned, parameter for controlling the step size of the gas network iterates. The second is a sequence of supporting hyperplanes together with an increasing number of carefully constructed halfspaces for controlling the convergence of the electricity network iterates. Moreover, the two proposed algorithms use as a warm start the solution from a novel polyhedral relaxation of the MOEGF problem, for a noticeable improvement in computation time as compared to a cold start. Unlike the first method, which invokes a branch-and-bound algorithm to find an integral solution, the second method implements an elaborate steering procedure that guides the continuous variables to take integral values at the solution. Numerical evaluation demonstrates that the two proposed methods can converge to high-quality feasible solutions in computation times at least two orders of magnitude faster than both a state-of-the-art nonlinear branch-and-bound (NLBB) MINLP solver and a mixed-integer convex programming (MICP) relaxation of the MOEGF problem. The experimental setup consists of five test cases, three of which involve the real electricity and gas transmission networks of the state of Victoria with actual linepack and demand profiles.
... In fact, due to the unique interplay between two historically-independent energy systems, the electricity-gas coupled system is receiving an ever-growing research interest, especially in the IEEE PES community, under a broader concept known as the "integrated energy system". A large amount of research aims to address the optimal electricitygas flow problem, an extension to the traditional optimal power flow problem in electrical engineering that also considers the optimality of gas flow [1], [2]. More specifically, for optimal gas flow, the major difficulty lies in how to properly address the nonconvex gas flow equations inherent in the natural gas model, for which many algorithms have emerged. ...
In this letter, we propose a data-driven warm start approach, empowered by artificial neural networks, to boost the efficiency of convex relaxations in optimal gas flow. Case studies show that this approach significantly decreases the number of iterations for the convex-concave procedure algorithm, and optimality and feasibility of the solution can still be guaranteed.
... Recently, convex optimization attracts substantial attention due to its global optimality and computational superiority. Some convex relaxation techniques, such as second-order cone (SOC) relaxation and semidefinite relaxation [26], [27] are introduced to solve OPGF, and especially the SOC relaxation is widely applied to tackle the Weymouth equation [9], [10], [13], [21]- [23]. However, it is non-trivial to guarantee the evincible exactness of the relaxation. ...
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.
