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Fast Probabilistic Energy Flow Calculation for Natural Gas Systems: A Convex multiparametric programming Approach

IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 2025年 22卷 6786-6796页

作者： Jia, Wenhao Ding, Tao Yuan, Yi Zhang, Hongji Xi An Jiao Tong Univ Dept Elect Engn Xian 710049 Shaanxi Peoples R China

Probabilistic energy flow (PEF) calculation is a fundamental task for the operation and planning of both natural gas systems (NGSs) and integrated energy systems considering uncertainties. Traditional Monte Carlo simulation (MCS) based PEF approach requires repeated energy flow calculations based on a large number of random samples, leading to a huge computation burden. Hence, this paper proposes a convex multiparametric programming (MPP) based fast PEF calculation method for NGSs. First, we develop an energy function based convex optimization model whose optimal solution is equivalent to the solution of deterministic energy flow equations. Then, we propose a convex MPP model with uncertain boundary conditions (such as the gas injections/loads) as the parameters. A multiparametric quadratic approximation algorithm is further introduced to solve the proposed MPP and obtain the analytical energy flow expression. This analytical expression characterizes the mapping relationship between the energy flow solution and uncertain parameters. Finally, we establish an online PEF calculation framework, in which the repeat energy flow calculations can be efficiently performed by merely substituting the uncertain parameters into the analytical energy flow expression obtained offline. Case studies on multiple NGSs verify the effectiveness of the proposed method. Note to Practitioners-This paper introduces a novel approach for efficient PEF calculation in NGSs that addresses the computational challenges posed by traditional methods. Conventional MCS-based PEF approaches involve numerous energy flow calculations with massive stochastic scenarios, resulting in significant computational burdens. The key contribution of our work is to equivalently reformulate the original PEF problem as an energy function based convex MPP model, which incorporates uncertain boundary conditions (e.g., gas injections/loads) as its parameters. Then, a multiparametric quadratic approximation algorithm is s

关键词： Mathematical models Boundary conditions Natural gas programming Probabilistic logic Iterative methods Computational efficiency Integrated energy systems natural gas systems probabilistic energy flow multiparametric programming

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multiparametric programming based algorithms for pure integer and mixed-integer bilevel programming problems

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COMPUTERS & CHEMICAL ENGINEERING 2010年第12期34卷 2097-2106页

作者： Dominguez, Luis F. Pistikopoulos, Efstratios N. Univ London Imperial Coll Sci Technol & Med Dept Chem Engn Ctr Proc Syst Engn London SW7 2AZ England

This work introduces two algorithms for the solution of pure integer and mixed-integer bilevel programming problems by multiparametric programming techniques. The first algorithm addresses the integer case of the bilevel programming problem where integer variables of the outer optimization problem appear in linear or polynomial form in the inner problem. The algorithm employs global optimization techniques to convexify nonlinear terms generated by a reformulation linearization technique (RLT). A continuous multiparametric programming algorithm is then used to solve the reformulated convex inner problem. The second algorithm addresses the mixed-integer case of the bilevel programming problem where integer and continuous variables of the outer problem appear in linear or polynomial forms in the inner problem. The algorithm relies on the use of global multiparametric mixed-integer programming techniques at the inner optimization level. In both algorithms, the multiparametric solutions obtained are embedded in the outer problem to form a set of single-level (M)(I)(N)LP problems - which are then solved to global optimality using standard fixed-point (global) optimization methods. Numerical examples drawn from the open literature are presented to illustrate the proposed algorithms. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Integer and mixed-integer bilevel programming multiparametric programming Global optimization Reformulation linearization technique

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multiparametric programming-Based Coordinated Economic Dispatch of Integrated Electricity and Natural Gas System

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IEEE SYSTEMS JOURNAL 2023年第2期17卷 2523-2533页

作者： Wu, Chenyu Yang, Lun Lin, Chenhui Southeast Univ Sch Elect Engn Nanjing 210096 Peoples R China Tsinghua Univ Tsinghua Berkeley Shenzhen Inst Tsinghua Shenzhen Int Grad Sch Shenzhen 518055 Peoples R China

Continuous development and application of gas-fired units worldwide have raised concerns about the inherent interdependency between electricity and natural gas systems. The coordinated dynamic economic dispatch for integrated electricity and natural gas systems (IENs) is increasingly essential. Since electricity and natural gas systems belong to different administrative organizations, we build a coordinated economic dispatch model for IENs and propose a decentralized algorithm to realize collaborative distributed control. A distributed algorithm based on multiparametric programming and benders cuts is proposed to crack the optimization problem in a decentralized way. The quadratic exchanging function is built to accelerate the convergence speed, and the auxiliary variables relaxation method is applied to deal with the infeasible subproblems. The global convergence of the proposed methods is proved in detail. Furthermore, from the point of view of both iteration number and computation time, simulation results demonstrate that the computational performance of the proposed method is far superior to the popular decentralized methods.

关键词： Coordinated economic dispatch decentralized optimization integrated power and natural gas system (IEN) multiparametric programming

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A space exploration algorithm for multiparametric programming via Delaunay triangulation

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OPTIMIZATION AND ENGINEERING 2021年第1期22卷 555-579页

作者： Burnak, Baris Katz, Justin Pistikopoulos, Efstratios N. Texas A&M Univ Artie McFerrin Dept Chem Engn College Stn TX 77845 USA Texas A&M Univ Texas A&M Energy Inst College Stn TX 77845 USA

We present a novel parameter space exploration algorithm for three classes of multiparametric problems, namely linear (mpLP), quadratic (mpQP), and mixed-integer linear (mpMILP). We construct subsets of the parameter space in the form of simplices through Delaunay triangulation to facilitate identification of the optimal partitions that describe the solution space. The presented exploration strategy prioritizes identifying volumetrically larger critical regions compared to existing methods. We demonstrate the exploration algorithm on an illustrative example, and compare the volumetrically identified parameter space against existing solvers on randomly generated problems in all three classes.

关键词： multiparametric programming Triangulation Optimization under uncertainty

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Proactive Scheduling of Batch Processes by a Combined Robust Optimization and multiparametric programming Approach

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AICHE JOURNAL 2013年第11期59卷 4184-4211页

作者： Wittmann-Hohlbein, Martina Pistikopoulos, Efstratios N. Univ London Imperial Coll Sci Technol & Med Dept Chem Engn Ctr Proc Syst Engn London SW7 2BY England

We address short-term batch process scheduling problems contaminated with uncertainty in the data. The mixed integer linear programming (MILP) scheduling model, based on the formulation of Ierapetritou and Floudas, Ind Eng Chem Res. 1998;37(11):4341-4359, contains parameter dependencies at multiple locations, yielding a general multiparametric (mp) MILP problem. A proactive scheduling policy is obtained by solving the partially robust counterpart formulation. The counterpart model may remain a multiparametric problem, yet it is immunized against uncertainty in the entries of the constraint matrix and against all parameters whose values are not available at the time of decision making. We extend our previous work on the approximate solution of mp-MILP problems by embedding different uncertainty sets (box, ellipsoidal and budget parameter regulated uncertainty), and by incorporating information about the availability of uncertain data in the construction of the partially robust scheduling model. For any parameter realization, the corresponding schedule is then obtained through function evaluation. (c) 2013 American Institute of Chemical Engineers AIChE J, 59: 4184-4211, 2013

关键词： process scheduling mixed integer programming multiparametric programming robust optimization

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Parameter estimation using multiparametric programming for implicit Euler's method based discretization

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CHEMICAL ENGINEERING RESEARCH & DESIGN 2019年 142卷 62-77页

作者： Mid, Ernie Che Dua, Vivek UCL CPSE Dept Chem Engn London England Univ Malaysia Perlis Sch Elect Engn Syst Arau Malaysia

This work presents a study that aims to compare two discretization methods for solving parameter estimation using multiparametric programming. In our earlier work, parameter estimation using multiparametric programming was presented where model parameters were obtained as an explicit function of measurements. In this method, the nonlinear ordinary equations (ODES) model was discretized by using explicit Euler's method to obtain algebraic equations. Then, a square system of parametric nonlinear algebraic equations was obtained by formulating optimality condition. These equations were then solved symbolically to obtain model parameters as an explicit function of measurements. Thus, the online computation burden of solving optimization problems for parameter estimation is replaced by simple function evaluations. In this work, we use implicit Euler's method for discretization of nonlinear ODEs model and compare with the explicit Euler's method for parameter estimation using multiparametric programming. Complexity of explicit parametric functions, accuracy of parameter estimates and effect of step size are discussed. (C) 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

关键词： Parameter estimation multiparametric programming Implicit Euler's method

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Process scheduling under uncertainty using multiparametric programming

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AICHE JOURNAL 2007年第12期53卷 3183-3203页

作者： Li, Zukui Ierapetritou, Marianthi G. Rutgers State Univ Dept Chem & Biochem Engn Piscataway NJ 08854 USA

In this article, the problem of process scheduling under uncertainty was studied using multiparametric programming method. Based on the uncertainty type (prices, demands, and processing times), the scheduling formulation results in different parametric problems including multiparametric mixed integer linear (mpMILP), quadratic (mpMIQP), and general nonlinear programming (mpMINLP) problem. This article analyzes the solution characteristics and proposes a novel solution framework for specific mpMILP/mpMIQP problems addressing a wide variety of scheduling problems under different types of uncertainty, which are modeled as coefficient in objective function, as coefficient of integer variable and right hand side vector of the constraints. The main idea of the proposed framework is to decompose the problem into a series of smaller subproblems, each of them producing the parametric information around a given parameter value. The parametric solution of every subproblem is retrieved by solving a series of multiparametric linear programming (mpLP) and mixed integer linear/nonlinear programming problems (MILP/MINLP). Several examples were solved to analyze the complexity and the effectiveness of the proposed method. (c) 2007 American Institute of Chemical Engineers AIChE J, 53: 3183-3203, 2007.

关键词： process scheduling uncertainty multiparametric programming

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Combined model approximation techniques and multiparametric programming for explicit nonlinear model predictive control

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COMPUTERS & CHEMICAL ENGINEERING 2012年 42卷 277-287页

作者： Rivotti, Pedro Lambert, Romain S. C. Pistikopoulos, Efstratios N. Univ London Imperial Coll Sci Technol & Med Dept Chem Engn Ctr Proc Syst Engn London SW7 2AZ England

This work presents a methodology to derive explicit multiparametric controllers for nonlinear systems, combining model approximation techniques and multiparametric model predictive control (mp-MPC) algorithms. Particular emphasis is given to an approach that applies a nonlinear model reduction technique, based on balancing of empirical gramians, which generates a reduced order model suitable for nonlinear mp-MPC algorithms. This approach is compared with a recently proposed method that uses a meta-modelling based model approximation technique which can be directly combined with standard multiparametric programming algorithms. The methodology is illustrated for two nonlinear models, of a distillation column and a train of CSTRs, respectively. Crown Copyright (c) 2012 Published by Elsevier Ltd. All rights reserved.

关键词： multiparametric programming Nonlinear MPC Explicit MPC Nonlinear model reduction Model approximation

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A Comparison of the Embedding Method With multiparametric programming, Mixed-Integer programming, Gradient-Descent, and Hybrid Minimum Principle-Based Methods

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 2014年第5期22卷 1784-1800页

作者： Meyer, Richard T. Zefran, Milos DeCarlo, Raymond A. Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA Univ Illinois Dept Elect & Comp Engn Chicago IL 60607 USA

In recent years, the embedding approach for solving switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared with alternative solution approaches. The goal of this paper is thus to compare the embedding approach with multiparametric programming, mixed-integer programming [mixed integer programming (MIP), commercial (CPLEX)], and gradient-descent-based methods in the context of five recently published examples: 1) a spring-mass system;2) moving-target tracking for a mobile robot;3) two-tank filling;dc-dc boost converter;and 5) skid-steered vehicle. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index (PI) for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower PI cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described;guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the PI), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of MIP, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches. Finally, common misconceptions regarding the embedding approach are addressed, including whe

关键词： Boost converter embedding method model predictive control multiparametric programming numerical optimization switched optimal control

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Integrating deep learning models and multiparametric programming

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COMPUTERS & CHEMICAL ENGINEERING 2020年 136卷 106801-000页

作者： Katz, Justin Pappas, Iosif Avraamidou, Styliani Pistikopoulos, Efstratios N. Texas A&M Univ Artie McFerrin Dept Chem Engn College Stn TX 77843 USA Texas A&M Univ Texas A&M Energy Inst College Stn TX 77843 USA

Deep learning models are a class of approximate models that are proven to have strong predictive capabilities for representing complex phenomena. The introduction of deep learning models into an optimization formulation provides a means to reduce the problem complexity and maintain model accuracy. Recently it has been shown that deep learning models in the form of neural networks with rectified linear units can be exactly recast as a mixed-integer linear programming formulation. However, developing the optimal solution of problems involving mixed-integer decisions in online applications remains challenging. multiparametric programming alleviates the online computational burden of solving an optimization problem involving bounded uncertain parameters. In this work, a strategy is presented to integrate deep learning and multiparametric programming. This integration yields a unified methodology for developing accurate surrogate models based on deep learning and their offline, explicit optimal solution. The proposed strategy is demonstrated on the optimal operation of a chemostat. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Deep learning Neural networks Rectified linear units Mixed-integer linear programming multiparametric programming

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