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As multi-microgrids become readily available, some limited models have been proposed that study operational and power quality constraints with local energy markets independently. This paper proposes a convex optimization model of an energy management system with operational and power quality constraints and interactions in a Local Energy Market (LEM) for unbalanced microgrids (MGs). The LEM consists of a pre-dispatch step and an energy transactions step (ETS). The ETS combines the MGs' objectives while considering two strategies: minimize the cost of buyers or maximize the revenue of sellers. Our proposed model considers harmonic distortion and voltage limit power quality constraints in both steps. Moreover, we model operational constraints such as power flow, power balance, and distributed energy resources behaviors and capacities. We numerically evaluate the proposed model using three unbalanced MGs with residential, industrial, and commercial load profiles, where each microgrid manages its resources locally. Furthermore, we create two groups of cases to analyze the interactions in the local energy market. In the first group, the price of the DSO energy and the surplus from MGs to DSO are the same. The numerical results show that using the increasing revenue strategy promotes MGs to interact more while encouraging them to have high energy prices. When the reducing cost strategy is used, fewer energy interactions occur, and the price of MGs energy is encouraged to be lower., Comment: Applied Energy Journal, 21 pages

This article presents an optimization model for tertiary control in three-phase unbalanced microgrids. This model considers 24h operation and includes renewable energy sources, energy storage devices, and grid code limitations. Power flow equations are simplified using a recently developed approximation based on Wirtinger's calculus. The proposed model is evaluated both theoretically and practically. From the theoretical point of view, the model is suitable for tertiary control since it is convex; hence, global optimum, uniqueness of the solution, and convergence of the interior point method are guaranteed. From the practical point of view, the model is simple enough to be implemented in a small single-board computer with low time calculation. The latter is evaluated by implementing the model in a Raspberry-Pi board with the CIGRE low voltage benchmark; the model is also evaluated in the IEEE 123-nodes test system for power distribution networks., Comment: Paper submitted to Electric Power Systems Research

This paper presents a non-linear stability analysis for dc-microgrids in both, interconnected mode and island operation with primary control. The proposed analysis is based on the fact that the dynamical model of the grid is a gradient system generated by a strongly convex function. The stability analysis is thus reduced to a series of convex optimization problems. The proposed method allows to: i) demonstrate the existence and uniqueness of the equilibrium ii) calculate this equilibrium numerically iii) give conditions for global stability using a Lyapunov function iv) estimate the attraction region. Previous works only address one of these aspects. Numeric calculations performed in cvx and simulations results in Matlab complement the analysis and demonstrate how to use this theoretical results in practical problems.

This short paper presents a Wirtinger's-Calculus based load-flow methodology for power distribution grids. This approach allows to obtain an algorithm which works directly on the complex domain maintaining some useful symmetries and a compact representation. The paper aims to introduce Wirtinger's Calculus as a suitable tool for power systems analysis and simulations; therefore, it is presented as a tutorial, playing especial attention on the implementation in modern scrip languages such as Matlab/Octave, which allow easy representation of complex-arrays with fast calculations. A Newton's-based method is proposed in which the Jacobian is replaced by Wirtinger's derivatives obtaining a compact representation. Simulation results complement the analysis. Despite being a mature theory, Wirtinger's-Calculus has not been applied before in this type of problems., Comment: preprint paper presented in ANDESCON 2018

The power flow is a non-linear problem that requires a Newton's method to be solved in dc microgrids with constant power terminals. This paper presents sufficient conditions for the quadratic convergence of the Newton's method in this type of grids. The classic Newton method as well as an approximated Newton Method are analyzed in both master-slave and island operation with droop controls. Requirements for the convergence as well as for the existence and uniqueness of the solution starting from voltages close to 1pu are presented. Computational results complement this theoretical analysis., Comment: 7 pages, 4 figures

This tutorial presents a simple yet accurate transient stability analysis for Type-A wind turbines. The paper is presented in a tutorial way and therefore it includes some scripts in Matlab/muPad which demonstrate the main concepts., Comment: 5 pages

Tensor analysis has been a widely studied in physics applications including circuit theory and electric machines. This paper reviews some of the main features of this type of representation for unbalanced power distribution systems and discusses its implementation in modern programming languages. A tensor representation simplifies the nomenclature and calculations in three phase unbalanced grids, reduces the storage requirement and improves the computation speed. A fixed point algorithm is proposed for the power flow solution showing the flexibility of the tensor approach. Simulations in Matlab/Octave demonstrate the advantages of this representation.

This paper investigates the existence of an equilibrium point in multiterminal HVDC (MT-HVDC) grids, assesses its uniqueness and defines conditions to ensure its stability. An offshore MT-HVDC system including two wind farms is selected as application test case. At first, a generalized dynamic model of the network is proposed, using hypergraph theory. Such model captures the frequency dependence of transmission lines and cables, it is non-linear due to the constant power behavior of the converter terminals using droop regulation, and presents a suitable degree of simplifications of the MMC converters, under given conditions, to allow system level studies over potentially large networks. Based on this model, the existence and uniqueness of the equilibrium point is demonstrated by returning the analysis to a load-flow problem and using the Banach fixed point theorem. Additionally, the stability of the equilibrium is analyzed by obtaining a Lyapunov function by the Krasovskii's theorem. Computational results obtained for the selected 4 terminals MT-HVDC grid corroborate the requirement for the existence and stability of the equilibrium point., Comment: 8

Power flow in a low voltage direct current grid (LVDC) is a non-linear problem just as its counterpart ac. This paper demonstrates that, unlike in ac grids, convergence and uniqueness of the solution can be guaranteed in this type of grids. The result is not a linearization nor an approximation, but an analysis of the set of non-linear algebraic equations, which is valid for any LVDC grid regardless its size, topology or load condition. Computer simulation corroborate the theoretical analysis., Comment: Under consideration

This paper presents a quadratic approximation for the optimal power flow in power distributions systems. The proposed approach is based on a linearized load flow which is valid for power distribution systems including three-phase unbalanced operation. The main feature of the methodology is its simplicity. The accuracy of the proposed approximation is compared to the non-linear/non-convex formulation of the optimal power flow using different optimization solvers. The studies indicate the proposed approximation provides a very accurate solution for systems with a good voltage profile. Results over a set of 1000 randomly generated test power distribution systems demonstrate this solution can be considered for practical purposes in most of the cases. An analytical solution for the unconstrained problem is also developed. This solution can be used as an initialization point for a more precise formulation of the problem., Comment: 15 pages, 5 figures