scholarly journals Solving Signal Control Problems with Second-Order Sensitivity Information of Equilibrium Network Flows

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Hsun-Jung Cho ◽  
You-Heng Huang
Keyword(s):  
Control Problem ◽  
Network Flows ◽  
Stackelberg Game ◽  
Nonlinear Approximation ◽  
Second Order ◽  
Signal Control ◽  
Reaction Function ◽  
Lower Level Problem ◽  
Level Problem ◽  
Upper Level

The equilibrium network signal control problem is represented as a Stackelberg game. Due to the characteristics of a Stackelberg game, solving the upper-level problem and lower-level problem iteratively cannot be expected to converge to the solution. The reaction function of the lower-level problem is the key information to solve a Stackelberg game. Usually, the reaction function is approximated by the network sensitivity information. This paper firstly presents the general form of the second-order sensitivity formula for equilibrium network flows. The second-order sensitivity information can be applied to the second-order reaction function to solve the network signal control problem efficiently. Finally, this paper also demonstrates two numerical examples that show the computation of second-order sensitivity and the speed of convergence of the nonlinear approximation algorithm.

Minimization of cost and congestion management using interline power flow controller

2016 ◽  
Vol 35 (5) ◽  
pp. 1495-1512
Author(s):  
Uma Velayutham ◽  
Lakshmi Ponnusamy ◽  
Gomathi Venugopal
Keyword(s):  
Social Welfare ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Real Power ◽  
Content Type ◽  
Optimal Power ◽  
Lower Level Problem ◽  
Level Problem ◽  
Upper Level ◽  
Power Flow Controller

Purpose The purpose of this paper is to optimally locate and size the FACTS device, namely, interline power flow controller in order to minimize the total cost and relieve congestion in a power system. This security analysis helps independent system operator (ISO) to have a better planning and market clearing criteria during any operating state of the system. Design/methodology/approach A multi-objective optimization problem has been developed including real power performance index (RPPI) and expected security cost (ESC). A security constrained optimal power flow has been developed as expected security cost optimal power flow problem which gives the probabilities of operating the system in all possible pre-contingency and post-contingency states subjected to various equality and inequality constraints. Maximizing social welfare is the objective function considered for normal state, while minimizing compensations for generations rescheduling and maximizing social welfare are the objectives in case of contingency states. The proposed work is viewed as a two level problem wherein the upper-level problem is to optimally locate IPFC using RPPI and the lower-level problem is to minimize the ESC subjected to various system constraints. Both upper-level and lower-level problem are solved using particle swarm optimization and The performance of the proposed algorithm is tested under severe line outages and has been validated using IEEE 30 bus system. Findings The proposed methodology shows that IPFC controls the power flows in the network without generation rescheduling or topological changes and thus improves the performance of the system. It is found that the benefit achieved in the ESC due to the installation of IPFC is greater than the annual investment cost of the device. ISO cannot achieve minimum total system cost by merely rescheduling generators. Instead of rescheduling, FACTS devices can be used for compensation by achieving minimum cost. IPFC can be used to compensate the congested lines and transfer cheaper power from generators to consumers. Originality/value Operational reliability, financial profitability and efficient utilization of the existing transmission system infrastructure has been achieved using single FACTS device. Instead of using multiple FATCS devices, if a single FACTS device like IPFC which itself can compensate several transmission lines is used, then in addition to the facility for independently controlled reactive (series) compensation of each individual line, it provides a capability to directly transfer real power between the compensated lines. Hence an attempt has been made in this paper to incorporate IPFC for relieving congestion in a deregulated environment. However, no previous researches have considered incorporating compensation of multi-transmission line using single IPFC in minimizing ESC. Thus, in this paper, the authors indicate how much the ESC is reduced by installing IPFC.

scholarly journals Particle Swarm Optimization Based on Smoothing Approach for Solving a Class of Bi-Level Multiobjective Programming Problem

2017 ◽  
Vol 17 (3) ◽  
pp. 59-74
Author(s):  
Qingping He ◽  
Yibing Lv
Keyword(s):  
Particle Swarm Optimization ◽  
Programming Problem ◽  
Multiobjective Programming ◽  
Particle Swarm ◽  
Swarm Optimization ◽  
Pareto Optimal Front ◽  
Lower Level Problem ◽  
Level Problem ◽  
Multiobjective Programming Problem ◽  
Upper Level

Abstract As a metaheuristic, Particle Swarm Optimization (PSO) has been used to solve the Bi-level Multiobjective Programming Problem (BMPP). However, in the existing solving approach based on PSO for the BMPP, the upper level and the lower level problem are solved interactively by PSO. In this paper, we present a different solving approach based on PSO for the BMPP. Firstly, we replace the lower level problem of the BMPP with Kuhn-Tucker optimality conditions and adopt the perturbed Fischer-Burmeister function to smooth the complementary conditions. After that, we adopt PSO approach to solve the smoothed multiobjective programming problem. Numerical results show that our solving approach can obtain the Pareto optimal front of the BMPP efficiently.

scholarly journals Inexact Derivative-Free Optimization for Bilevel Learning

2021 ◽  
Author(s):  
Matthias J. Ehrhardt ◽  
Lindon Roberts
Keyword(s):  
Approximate Solutions ◽  
Bilevel Optimization ◽  
Worst Case ◽  
Lower Level Problem ◽  
Derivative Free Optimization ◽  
Derivative Free ◽  
Common Strategy ◽  
Regularization Techniques ◽  
Level Problem ◽  
Upper Level

AbstractVariational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by-now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require exact lower-level problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worst-case complexity bound for our approach. We test our proposed framework on ROF denoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as high-accuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).

Non-Probabilistic Based Structural Design Optimization Under External Load Uncertainty With Eigenvalue-Superposition of Convex Models

2012 ◽  
Author(s):  
Xike Zhao ◽  
Hae Chang Gea ◽  
Limei Xu
Keyword(s):  
Design Optimization ◽  
Structural Design ◽  
External Load ◽  
Optimization Problems ◽  
Global Optimum ◽  
Global Optimality ◽  
Lower Level Problem ◽  
Structural Design Optimization ◽  
Level Problem ◽  
Upper Level

The non-probabilistic-based structural design optimization problems with external load uncertainties are often solved through a two-level approach. However there are several challenges in this method. Firstly, to assure the reliability of the design, the lower level problem must be solved to its global optimality. Secondly, the sensitivity of the upper level problem cannot be analytically derived. To overcome these challenges, a new method based on the Eigenvalue-Superposition of Convex Models (ESCM) is proposed in this paper. The ESCM method replaces the global optimum of the lower level problem by a confidence bound, namely the ESCM bound, and with which the two-level problem can be formulated into a single level problem. The advantages of the ESCM method in efficiency and stability are demonstrated through numerical examples.

On Bilevel Optimization with Inexact Follower

2020 ◽  
Vol 17 (1) ◽  
pp. 74-95 ◽  
Author(s):  
M. Hosein Zare ◽  
Oleg A. Prokopyev ◽  
Denis Sauré
Keyword(s):  
Optimization Problems ◽  
Broad Class ◽  
Bilevel Optimization ◽  
Modeling Framework ◽  
Optimization Framework ◽  
Lower Level Problem ◽  
Level Problem ◽  
Potential Impact ◽  
Upper Level ◽  
Optimal Reaction

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

scholarly journals A Computationally Efficient Optimization Method for Battery Storage in Grid-connected Microgrids Based on a Power Exchanging Process

Energies ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 1512 ◽  
Author(s):  
Ping Liu ◽  
Zexiang Cai ◽  
Peng Xie ◽  
Xiaohua Li ◽  
Yongjun Zhang
Keyword(s):  
Energy Management ◽  
Computational Efficiency ◽  
Optimization Method ◽  
Pattern Search ◽  
Battery Storage ◽  
Computationally Efficient ◽  
Lower Level Problem ◽  
Management Scheme ◽  
Level Problem ◽  
Upper Level

Battery storage (BS) sizing problems for grid-connected microgrids (GCμGs) commonly use stochastic scenarios to represent uncertain natures of renewable energy and load demand in the GCμG. Though taking a large number of stochastic scenarios into consideration can deliver a relatively accurate optimal result, it can also highly deteriorate the computational efficiency of the sizing problem. To make an accuracy-efficiency trade-off, a computationally efficient optimization method to optimize the BS capacities based on the power exchanging process of the GCμG is proposed in this paper. According to the imbalanced power of the GCμG, this paper investigates the power exchanging process between the GCμG, BS and external grid. Motivated by the BS dynamics, a forward/backward sweep-based energy management scheme is proposed based on the power exchanging process. A heuristic two-level optimization model is developed with sizing BS as the upper-level problem and optimizing the operational cost of the GCμG as the lower-level problem. The lower-level problem is solved by the proposed energy management scheme and the objective function of the upper-level is minimized by the pattern search (PS) algorithm. To validate the accuracy and computational efficiency of the proposed method, the numerical results are compared with the mixed integer linear programming (MILP) method. The comparison shows that the proposed method shares similar accuracy but is much more time-efficient than the MILP method.

A New Method To Solve Bi-Level Quadratic Linear Fractional Programming Problems

2015 ◽  
Vol 17 (02) ◽  
pp. 1540017
Author(s):  
Sanjeet Singh ◽  
Nivedita Haldar
Keyword(s):  
Objective Function ◽  
Fractional Programming ◽  
Optimal Solution ◽  
Linear Constraints ◽  
New Method ◽  
Linear Fractional Programming ◽  
Single Level ◽  
Lower Level Problem ◽  
Level Problem ◽  
Upper Level

In this paper, we have developed a new method to solve bi-level quadratic linear fractional programming (BLQLFP) problems in which the upper-level objective function is quadratic and the lower-level objective function is linear fractional. In this method a BLQLFP problem is transformed into an equivalent single-level quadratic programming (QP) problem with linear constraints by forcing the duality gap of the lower-level problem to zero. Then by obtaining all vertices of the constraint region of the dual of the lower-level problem, which is a convex polyhedron, the single-level QP problem is converted into a series of finite number of QP problems with linear constraints which can be solved by any standard method for solving a QP. The best among the optimal solutions gives the desired optimal solution for the original bi-level programming (BLP) problem. Theoretical results have been illustrated with the help of a numerical example.

scholarly journals An Evolutionary Algorithm for Solving Bilevel Programming Problems Using Duality Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hecheng Li ◽  
Lei Fang
Keyword(s):  
Evolutionary Algorithm ◽  
Bilevel Programming ◽  
Dual Problem ◽  
Optimization Problems ◽  
Duality Principle ◽  
Single Level ◽  
Nonconvex Functions ◽  
Lower Level Problem ◽  
Level Problem ◽  
Upper Level

Bilevel programming is characterized by two optimization problems located at different levels, in which the constraint region of the upper level problem is implicitly determined by the lower level problem. This paper is focused on a class of bilevel programming with a linear lower level problem and presents a new algorithm for solving this kind of problems by combining an evolutionary algorithm with the duality principle. First, by using the prime-dual conditions of the lower level problem, the original problem is transformed into a single-level nonlinear programming problem. In addition, for the dual problem of the lower level, the feasible bases are taken as individuals in population. For each individual, the values of dual variables can be obtained by taking the dual problem into account, thus simplifying the single-level problem. Finally, the simplified problem is solved, and the objective value is taken as the fitness of the individual. Besides, when nonconvex functions are involved in the upper level, a coevolutionary scheme is incorporated to obtain global optima. In the computational experiment, 10 problems, smaller or larger-scale, are solved, and the results show that the proposed algorithm is efficient and robust.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

scholarly journals Bi-Level Optimization for Electricity Transaction in Smart Community With Modular Pump Hydro Storage

2020 ◽  
Author(s):  
Yang Chen ◽  
Xiao Kou ◽  
Mohammed Olama ◽  
Helia Zandi ◽  
Chenang Liu ◽  
...  
Keyword(s):  
Stackelberg Game ◽  
Cost Minimization ◽  
Necessary Optimality Conditions ◽  
Local Energy ◽  
Energy Prices ◽  
Profit Distribution ◽  
Level Model ◽  
Upper Level ◽  
Smart Community ◽  
Best Responses

Abstract Grid integration of the increasing distributed energy resources could be challenging in terms of new infrastructure investment, power grid stability, etc. To resolve more renewables locally and reduce the need for extensive electricity transmission, a community energy transaction market is assumed with market operator as the leader whose responsibility is to generate local energy prices and clear the energy transaction payment among the prosumers (followers). The leader and multi-followers have competitive objectives of revenue maximization and operational cost minimization. This non-cooperative leader-follower (Stackelberg) game is formulated using a bi-level optimization framework, where a novel modular pump hydro storage technology (GLIDES system) is set as an upper level market operator, and the lower level prosumers are nearby commercial buildings. The best responses of the lower level model could be derived by necessary optimality conditions, and thus the bi-level model could be transformed into single level optimization model via replacing the lower level model by its Karush-Kuhn-Tucker (KKT) necessary conditions. Several experiments have been designed to compare the local energy transaction behavior and profit distribution with the different demand response levels and different local price structures. The experimental results indicate that the lower level prosumers could benefit the most when local buying and selling prices are equal, while maximum revenue potential for the upper level agent could be reached with non-equal trading prices.

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