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This paper presents a new method for enhancing Alternating Current Power Flow (ACPF) analysis. The method integrates the Newton-Raphson (NR) method with Enhanced Gradient Descent (GD) and computational graphs. The integration of renewable energy sources in power systems introduces variability and unpredictability, and this method addresses these challenges. It leverages the robustness of NR for accurate approximations and the flexibility of GD for handling variable conditions, all without requiring Jacobian matrix inversion. Furthermore, computational graphs provide a structured and visual framework that simplifies and systematizes the application of these methods. The goal of this fusion is to overcome the limitations of traditional ACPF methods and improve the resilience, adaptability, and efficiency of modern power grid analyses. We validate the effectiveness of our advanced algorithm through comprehensive testing on established IEEE benchmark systems. Our findings demonstrate that our approach not only speeds up the convergence process but also ensures consistent performance across diverse system states, representing a significant advancement in power flow computation. 

Alternative current optimal power flow (ACOPF) problems have been studied for over fifty years, and yet the development of an optimal algorithm to solve them remains a hot and challenging topic for researchers because of their nonlinear and nonconvex nature. A number of methods based on linearization and convexification have been proposed to solve ACOPF problems, which result in near-optimal or local solutions, not optimal solutions. Nowadays, with the prevalence of machine learning, some researchers have begun to utilize this technology to solve ACOPF problems using the historical data generated by the grid operators. The present paper reviews the research on solving ACOPF problems using machine learning and neural networks and proposes future studies. This body of research is at the beginning of this area, and further exploration can be undertaken into the possibilities of solving ACOPF problems using machine learning. 

This manuscript presents a novel approach utilizing computational graph strategies for solving the power flow equations through the synergistic use of Newton-Raphson (NR) and Gradient Descent (GD). As a foundational element for operational and strategic decision-making in electrical networks, the power flow analysis has been rigorously examined for decades. Conventional solution techniques typically depend on second-order processes, which may falter, especially when faced with subpar starting values or during heightened system demands. These issues are becoming more acute with the dynamic shifts in generation and consumption patterns within modern electrical systems. Our research introduces a dual-mode algorithm that amalgamates the principles of first-order operation. This inventive method is adept at circumventing potential local minima traps that hinder current methodologies, thereby reinforcing the dependability of power flow solutions. We substantiate the effectiveness of our advanced algorithm with comprehensive testing on established IEEE benchmark systems. Our findings reveal that our approach not only expedites the convergence process but also ensures consistent performance across diverse system states, signifying a meaningful progression in the realm of power flow computation. 

The computational aspects of power systems have been exploring the steady states of AC power flow for several decades. In this paper, we propose a novel approach to AC power flow calculation using resultants and discriminants for polynomials, which are primarily compiled for quadratic power flow equations. In the case of AC power flow nonlinear systems, it is not possible to determine the number of isolated solutions. However, for polynomial systems, the theorem of Bézout is the primary theorem of algebraic geometry. This study considers a certain multi-homogeneous structure in an algebraic geometry system to demonstrate that the theorem of Bézout is indeed a generalization of the fundamental theorem, among other results. 

It has been decades since category theory was applied to databases. In spite of their mathematical elegance, categorical models have traditionally had difficulty representing factual data, such as integers or strings. This paper proposes a categorical dataset for power system computational models, which is used for AC Optimal Power Flows (ACOPF). In addition, categorical databases incorporate factual data using multi-sorted algebraic theories (also known as Lawvere theories) based on the set-valued functor model. In the advanced metering infrastructure of power systems, this approach is capable of handling missing information efficiently. This methodology enables constraints and queries to employ operations on data, such as multiplicative and comparative processes, thereby facilitating the integration between conventional databases and programming languages like Julia and Python’s Pypower. The demonstration illustrates how all elements of the model, including schemas, instances, and functors, can modify the schema in ACOPF instances. 

This paper discusses a market-based pool strategy for a microgrid (MG) to optimally trade electric power in the distribution electricity market (DEM). The increasing penetration levels of distributed energy resources (DERs) and MGs in distribution system (DS) stress distribution system operator (DSO) and require higher levels of coordinated control strategies. The distribution system operator has limited visibility and control over such distributed resources. To reduce the complexity of the system and improve the efficiency of the electricity market operation, we propose a decentralized pool strategy for an MG to integrate with a distribution system through a market mechanism. A market-based interactions procedure between MGs and DS is developed for MGs as price-makers to find an optimal bidding/offering strategy efficiently. To achieve a market equilibrium among all entities, we initially cast this problem as a bi-level programming problem, in which the upper level is an MG optimal scheduling problem and the lower level presents a DEM clearing mechanism. The proposed bi-level model is converted to a single mix-integer model which is easier to solve. Uncertainties associated with MG's rivals' offers and demands' bids are considered in this problem. The solution results from a modified IEEE 33-Bus distribution system are presented and discussed. Finally, some conclusions are drawn and examined. 

We propose a multiphase distribution locational marginal price (DLMP) model. Compared to existing DLMP models in the literature, the proposed model has three distinctive features: i) It provides a linear approximation of relevant DLMP components which captures the global behavior of nonlinear functions; ii) it decomposes into most general components, i.e., energy, loss, congestion, voltage violations; and iii) it incorporates both wye and delta grid connections along with unbalanced loadings. The developed model is tested on a benchmark IEEE 13-bus unbalanced distribution system with the inclusion of distributed generators (DGs). 

Newly, there has been significant research interest in the exact solution of the AC optimal power flow (AC-OPF) problem. A semideflnite relaxation solves many OPF problems globally. However, the real problem exists in which the semidefinite relaxation fails to yield the global solution. The appropriation of relaxation for AC-OPF depends on the success or unfulflllment of the SDP relaxation. This paper demonstrates a quadratic AC-OPF problem with a single negative eigenvalue in objective function subject to linear and conic constraints. The proposed solution method for AC-OPF model covers the classical AC economic dispatch problem that is known to be NP-hard. In this paper, by combining successive linear conic optimization (SLCO), convex relaxation and line search technique, we present a global algorithm for AC-OPF which can locate a globally optimal solution to the underlying AC-OPF within given tolerance of global optimum solution via solving linear conic optimization problems. The proposed algorithm is examined on modified IEEE 6-bus test system. The promising numerical results are described.