A GRADIENT SYSTEM SOLUTION TO POTTS MEAN FIELD EQUATIONS AND ITS ELECTRONIC IMPLEMENTATION

1993 ◽  
Vol 04 (01) ◽  
pp. 27-34 ◽  
Author(s):  
KIICHI URAHAMA ◽  
SHIN-ICHIRO UENO
Keyword(s):  
Solution Method ◽  
Optimization Problems ◽  
Mean Field ◽  
Gradient System ◽  
Field Equations ◽  
Combinatorial Optimization Problems ◽  
System Solution ◽  
Mean Field Equations ◽  
Optimum Solution ◽  
Winner Take All

A gradient system solution method is presented for solving Potts mean field equations for combinatorial optimization problems subject to winner-take-all constraints. In the proposed solution method the optimum solution is searched by using gradient descent differential equations whose trajectory is confined within the feasible solution space of optimization problems. This gradient system is proven theoretically to always produce a legal local optimum solution of combinatorial optimization problems. An elementary analog electronic circuit implementing the presented method is designed on the basis of current-mode subthreshold MOS technologies. The core constituent of the circuit is the winner-take-all circuit developed by Lazzaro et al. Correct functioning of the presented circuit is exemplified with simulations of the circuits implementing the scheme for solving the shortest path problems.

scholarly journals THE EXTENDED VARIATIONAL PRINCIPLE FOR MEAN-FIELD, CLASSICAL SPIN SYSTEMS

2005 ◽  
Vol 17 (10) ◽  
pp. 1209-1239
Author(s):  
E. KRITCHEVSKI ◽  
S. STARR
Keyword(s):  
Variational Principle ◽  
Optimization Problems ◽  
Mean Field ◽  
Spin Systems ◽  
Field Equations ◽  
Convex Optimization Problems ◽  
Mean Field Equations ◽  
Statistical Mechanical ◽  
Thermodynamic Pressure ◽  
Extended Variational Principle

The purpose of this article is to obtain a better understanding of the extended variational principle (EVP). The EVP is a formula for the thermodynamic pressure of a statistical mechanical system as a limit of a sequence of minimization problems. It was developed for disordered mean-field spin systems, spin systems where the underlying Hamiltonian is itself random, and whose distribution is permutation invariant. We present the EVP in the simpler setting of classical mean-field spin systems, where the Hamiltonian is non-random and symmetric. The EVP essentially solves these models. We compare the EVP with another method for mean-field spin systems: the self-consistent mean-field equations. The two approaches lead to dual convex optimization problems. This is a new connection, and it permits a generalization of the EVP.

scholarly journals Service Restoring Reconfiguration for Distribution NetworksConsidering Uncertainty in Available Information

2021 ◽  
Vol 11 (9) ◽  
pp. 4169
Author(s):  
Hirotaka Takano ◽  
Junichi Murata ◽  
Kazuki Morishita ◽  
Hiroshi Asano
Keyword(s):  
Power Distribution ◽  
Solution Method ◽  
Optimization Problems ◽  
Problem Formulation ◽  
Distribution Networks ◽  
Accurate Information ◽  
Power Distribution Networks ◽  
Combinatorial Optimization Problems ◽  
Service Restoration ◽  
Available Information

The recent growth in the penetration of photovoltaic generation systems (PVs) has brought new difficulties in the operating and planning of electric power distribution networks. This is because operators of the distribution networks normally cannot monitor or control the output of the PVs, which introduces additional uncertainty into the available information that operations must rely on. This paper focuses on the service restoration of the distribution networks, and the authors propose a problem framework and its solution method that finds the optimal restoration configuration under extensive PV installation. The service restoration problems have been formulated as combinatorial optimization problems. They do, however, require accurate information on load sections, which is impractical in distribution networks with extensively installed PVs. A combined framework of robust optimization and two-stage stochastic programming adopted in the proposed problem formulation enables us to deal with the PV-originated uncertainty using readily available information only. In addition, this problem framework can be treated by a traditional solution method with slight extensions. The validity of the authors’ proposal is verified through numerical simulations on a real-scale distribution network model and includes a discussion of their results.

Precipitation in small systems—II. Mean field equations more effective than population balance

1996 ◽  
Vol 51 (19) ◽  
pp. 4423-4436 ◽  
Author(s):  
S. Manjunath ◽  
K.S. Gandhi ◽  
R. Kumar ◽  
Doraiswami Ramkrishna
Keyword(s):  
Mean Field ◽  
Population Balance ◽  
Field Equations ◽  
Small Systems ◽  
Mean Field Equations

Solution of the mean field equations for spontaneous fission

1987 ◽  
Vol 35 (3) ◽  
pp. 1007-1027 ◽  
Author(s):  
G. Puddu ◽  
J. W. Negele
Keyword(s):  
Spontaneous Fission ◽  
Mean Field ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean

Mean field equations with probability measure in 2D-turbulence

2014 ◽  
Vol 63 (S1) ◽  
pp. 255-264 ◽  
Author(s):  
Tonia Ricciardi ◽  
Gabriella Zecca
Keyword(s):  
Probability Measure ◽  
Mean Field ◽  
Field Equations ◽  
2D Turbulence ◽  
Mean Field Equations

scholarly journals Distributed Control of Clustered Populations of Thermostatic Loads in Multi-Area Systems: A Mean Field Game Approach

Energies ◽  
2020 ◽  
Vol 13 (24) ◽  
pp. 6483
Author(s):  
Vincenzo Trovato ◽  
Antonio De Paola ◽  
Goran Strbac
Keyword(s):  
Power System ◽  
Frequency Response ◽  
Mean Field ◽  
Cost Effective ◽  
Support Network ◽  
Field Equations ◽  
Mean Field Game ◽  
Flexible Resources ◽  
Mean Field Equations ◽  
The Impact

Thermostatically controlled loads (TCLs) can effectively support network operation through their intrinsic flexibility and play a pivotal role in delivering cost effective decarbonization. This paper proposes a scalable distributed solution for the operation of large populations of TCLs providing frequency response and performing energy arbitrage. Each TCL is described as a price-responsive rational agent that participates in an integrated energy/frequency response market and schedules its operation in order to minimize its energy costs and maximize the revenues from frequency response provision. A mean field game formulation is used to implement a compact description of the interactions between typical power system characteristics and TCLs flexibility properties. In order to accommodate the heterogeneity of the thermostatic loads into the mean field equations, the whole population of TCLs is clustered into smaller subsets of devices with similar properties, using k-means clustering techniques. This framework is applied to a multi-area power system to study the impact of network congestions and of spatial variation of flexible resources in grids with large penetration of renewable generation sources. Numerical simulations on relevant case studies allow to explicitly quantify the effect of these factors on the value of TCLs flexibility and on the overall efficiency of the power system.

TWO-OSCILLATOR BASIS EXPANSION FOR THE SOLUTION OF RELATIVISTIC MEAN FIELD EQUATIONS

2000 ◽  
Vol 09 (06) ◽  
pp. 507-520
Author(s):  
S. V. S. SASTRY ◽  
ARUN K. JAIN ◽  
Y. K. GAMBHIR
Keyword(s):  
Ground State ◽  
Mean Field ◽  
Surface Region ◽  
Binding Energies ◽  
Basis Functions ◽  
Field Equations ◽  
Basis Expansion ◽  
Relativistic Mean Field ◽  
Mean Field Equations ◽  
Spherical Nuclei

In the relativistic mean field (RMF) calculations usually the basis expansion method is employed. For this one uses single harmonic oscillator (HO) basis functions. A proper description of the ground state nuclear properties of spherical nuclei requires a large (around 20) number of major oscillator shells in the expansion. In halo nuclei where the nucleons have extended spatial distributions, the use of single HO basis for the expansion is inadequate for the correct description of the nuclear properties, especially that of the surface region. In order to rectify these inadequacies, in the present work an orthonormal basis composed of two HO basis functions having different sizes is proposed. It has been shown that for a typical case of (A=11) the ground state constructed using two-HO wave functions extends much beyond the second state or even third excited state of the single HO wave function. To demonstrate its usefulness explicit numerical RMF calculations have been carried out using this procedure for a set of representative spherical nuclei ranging from 16 O to 208 Pb . The binding energies, charge radii and density distributions have been correctly reproduced in the present scheme using a much smaller number of major shells (around 10) in the expansion.

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