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The exact complexity of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Tomer Kotek, Johann A. Makowsky
So far the complexity of computational problems was measured with respect to one parameter of the input only: the size of the input (e.g., the number of vertices or edges of the input graph). Parameterized complexity measures the complexity of problems in terms of additional parameters of the graph. A problem is fixed-parameter tractable with respect to a parameter k if there is a computable function f, a polynomial p, and an algorithm solving the problem in time f(k)·p(n), where n is the size of the problem, and the function f(k) only depends on k. However, f may grow arbitrarily in k, and indeed is often exponential in k. The set of fixed-parameter tractable decision problems is denoted by FPT.
Optimal structure screening for large-scale multi-state series-parallel systems based on structure ordinal optimization
Published in IISE Transactions, 2021
Yishuang Hu, Yi Ding, Yu Lin, Ming J. Zuo, Donglian Qi
To solve the difficulties of an enormous search space containing multiple systems and maintain the reliability evaluation accuracy in structure screening, an Ordinal Optimization (OO) algorithm is adopted in this article. OO can effectively reduce the large search space into a relatively small selected subset (Lau and Ho, 1997; Ho et al., 2006), which has been widely-used in many optimization problems including power systems (Lee et al., 2019), a flow line system (Horng et al., 2018a) and a pull-type production system (Horng et al., 2017). The proposed OO-based approaches yield an outstanding calculation performance with a higher quality and efficiency (Horng et al., 2018b). As this algorithm can solve the computational problem by considering the “order” rather than the “value” (Tian et al., 2020), an efficient performance approximation method is inevitable. To maximally preserve the performance order (Liu and Huang, 2010), FUGF is applied as an approximate reliability evaluation method in this article, which can group similar system states to reduce the computational complexity and guarantee the accuracy.
Stability limits and consolute critical conditions for liquid mixtures
Published in Chemical Engineering Communications, 2018
Naif A. Darwish, Fahad M. Al Sadoon, Muhammad Qasim
Equation (12) is inconvenient and inappropriate, since it involves derivation by holding μ1 as constant. This computational problem can be eliminated by expressing in the form of lower order Legendre transform as follows (Tester and Modell, 1996):where G is the molar Gibbs energy of the mixture at T, P, x1,…,xn which, as will be shortly shown, is synthesized from the molar excess Gibbs energy GE for the real mixture (and this is provided by the NRTL model), the molar Gibbs energy change of mixing for the corresponding ideal mixture , and the molar Gibbs energy for the pure components at the same temperature and pressure of the real mixture. Equation (12) (or equivalently Eq. 13) represents one external constraint that must be satisfied by a system at its limit of stability. By applying Gibbs phase rule (Eq. 14) for a nonreactive, single-phase, ternary liquid system at the limits of stability, three degrees of freedom are obtained (Smith et al., 2005):where F represents degrees of freedom, N is the number of components in the mixture, π is the number of phases present, and s is the number of special constraints. In general, LLE data are measured at conditions of constant temperature and pressure, a thing which results one degree of freedom, i.e., one independent variable to be fixed. Therefore, Eq. (13) can be solved by specifying the value of x1 and finding the corresponding values of x2 for the ternary mixture to be on its stability limit.
Real Power Loss Minimization Considering Multiple DGs and Battery in Distribution System
Published in Electric Power Components and Systems, 2022
Kamal Chellappan, Vedula Sri Siva Chaitanya Satyanarayana Murty, Narayanan Krishnan, Gulshan Sharma, Tomonobu Senjyu
The battery size that is being optimized for the location using the Genetic Algorithm. Since the battery storage system has the circuit parameters consisting of state of charge, power of the battery and the energy of the battery, the genetic algorithm is preferred for the complex and multidimensional formulation. The purpose of the genetic algorithm used for battery is that it converges faster for such a complex computational problem. The genetic algorithm has three operations which are selection, crossover and mutation [23]. After these operations, the elite member is said to be the optimal solution for the considered problem. The random particle generation is done in a search space. The chromosomes considered here are the bus numbers (locations) that has to be optimized according to the fitness value. The normalized fitness function is used for the algorithm and it is normalized using the base MVA considered. The chromosomes are chosen randomly, and their fitness are evaluated. The evaluated fitness is sorted in descending order such that the minimum loss for the particular generation is at the top. In crossover, the two random chromosomes are chosen and these are converted into bits. The single point crossover is used. In both the strings a random position is chosen as crossover point and the bits after that point are interchanged between each other. The mutation probability is 0.01 and it randomly choses the bit which comes out of the crossover operation and reverses the bit. The elite members from the above three operations are sent to the next generation to check if there are more elite members in the next generations. The genetic algorithm parameters are as follows: Pcr = 0.85, Pmu = 0.01, Elr = 0.05.Maximum Generation: 50.Stopping Criteria: 20 generations unchanged.