A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids

Phase balancing is a classical optimization problem in power distribution grids that involve phase swapping of the loads and generators to reduce power loss. The problem is a non-linear integer and, hence, it is usually solved using heuristic algorithms. This paper proposes a mathematical reformulation that transforms the phase-balancing problem in low-voltage distribution networks into a mixed-integer convex quadratic optimization model. To consider both conventional secondary feeders and microgrids, renewable energies and their subsequent stochastic nature are included in the model. The power flow equations are linearized, and the combinatorial part is represented using a Birkhoff polytope B3 that allows the selection of phase swapping in each node. The numerical experiments on the CIGRE low-voltage test system demonstrate the use of the proposed formulation.

Download Full-text

Stochastic Expansion Planning of Various Energy Storage Technologies in Active Power Distribution Networks

Sustainability ◽

10.3390/su13105752 ◽

2021 ◽

Vol 13 (10) ◽

pp. 5752

Author(s):

Reza Sabzehgar ◽

Diba Zia Amirhosseini ◽

Saeed D. Manshadi ◽

Poria Fajri

Keyword(s):

Energy Storage ◽

Power Distribution ◽

Power Flow ◽

Lithium Ion ◽

Distribution Networks ◽

Renewable Energy Resources ◽

Flow Equations ◽

Second Order Cone ◽

Li Ion ◽

The Cost

This work aims to minimize the cost of installing renewable energy resources (photovoltaic systems) as well as energy storage systems (batteries), in addition to the cost of operation over a period of 20 years, which will include the cost of operating the power grid and the charging and discharging of the batteries. To this end, we propose a long-term planning optimization and expansion framework for a smart distribution network. A second order cone programming (SOCP) algorithm is utilized in this work to model the power flow equations. The minimization is computed in accordance to the years (y), seasons (s), days of the week (d), time of the day (t), and different scenarios based on the usage of energy and its production (c). An IEEE 33-bus balanced distribution test bench is utilized to evaluate the performance, effectiveness, and reliability of the proposed optimization and forecasting model. The numerical studies are conducted on two of the highest performing batteries in the current market, i.e., Lithium-ion (Li-ion) and redox flow batteries (RFBs). In addition, the pros and cons of distributed Li-ion batteries are compared with centralized RFBs. The results are presented to showcase the economic profits of utilizing these battery technologies.

Download Full-text

Laplacian Matrix-Based Power Flow Formulation for LVDC Grids with Radial and Meshed Configurations

Energies ◽

10.3390/en14071866 ◽

2021 ◽

Vol 14 (7) ◽

pp. 1866

Author(s):

Zahid Javid ◽

Ulas Karaagac ◽

Ilhan Kocar ◽

Ka Wing Chan

Keyword(s):

Fixed Point ◽

Power Flow ◽

Low Voltage ◽

Mathematical Formulation ◽

Distribution Networks ◽

Load Flow ◽

Banach Fixed Point Theorem ◽

Loading Conditions ◽

Flow Equations ◽

Uniqueness Of The Solution

There is an increasing interest in low voltage direct current (LVDC) distribution grids due to advancements in power electronics enabling efficient and economical electrical networks in the DC paradigm. Power flow equations in LVDC grids are non-linear and non-convex due to the presence of constant power nodes. Depending on the implementation, power flow equations may lead to more than one solution and unrealistic solutions; therefore, the uniqueness of the solution should not be taken for granted. This paper proposes a new power flow solver based on a graph theory for LVDC grids having radial or meshed configurations. The solver provides a unique solution. Two test feeders composed of 33 nodes and 69 nodes are considered to validate the effectiveness of the proposed method. The proposed method is compared with a fixed-point methodology called direct load flow (DLF) having a mathematical formulation equivalent to a backward forward sweep (BFS) class of solvers in the case of radial distribution networks but that can handle meshed networks more easily thanks to the use of connectivity matrices. In addition, the convergence and uniqueness of the solution is demonstrated using a Banach fixed-point theorem. The performance of the proposed method is tested for different loading conditions. The results show that the proposed method is robust and has fast convergence characteristics even with high loading conditions. All simulations are carried out in MATLAB 2020b software.

Download Full-text

Heuristic Methodology for Planning AC Rural Medium-Voltage Distribution Grids

Energies ◽

10.3390/en14165141 ◽

2021 ◽

Vol 14 (16) ◽

pp. 5141

Author(s):

Oscar Danilo Montoya ◽

Federico Martin Serra ◽

Cristian Hernan De Angelo ◽

Harold R. Chamorro ◽

Lazaro Alvarado-Barrios

Keyword(s):

Power Flow ◽

Minimum Spanning Tree ◽

Search Algorithm ◽

Flow Simulation ◽

Mixed Integer ◽

Voltage Distribution ◽

Medium Voltage ◽

Planning Project ◽

Distribution Grids ◽

Selection Of

The optimal expansion of AC medium-voltage distribution grids for rural applications is addressed in this study from a heuristic perspective. The optimal routes of a distribution feeder are selected by applying the concept of a minimum spanning tree by limiting the number of branches that are connected to a substation (mixed-integer linear programming formulation). In order to choose the caliber of the conductors for the selected feeder routes, the maximum expected current that is absorbed by the loads is calculated, thereby defining the minimum thermal bound of the conductor caliber. With the topology and the initial selection of the conductors, a tabu search algorithm (TSA) is implemented to refine the solution with the help of a three-phase power flow simulation in MATLAB for three different load conditions, i.e., maximum, medium, and minimum consumption with values of 100%, 60%, and 30%, respectively. This helps in calculating the annual costs of the energy losses that will be summed with the investment cost in conductors for determining the final costs of the planning project. Numerical simulations in two test feeders comprising 9 and 25 nodes with one substation show the effectiveness of the proposed methodology regarding the final grid planning cost; in addition, the heuristic selection of the calibers using the minimum expected current absorbed by the loads provides at least 70% of the calibers that are contained in the final solution of the problem. This demonstrates the importance of using adequate starting points to potentiate metaheuristic optimizers such as the TSA.

Download Full-text

Black hole optimizer for the optimal power injection in distribution networks using DG

Journal of Physics Conference Series ◽

10.1088/1742-6596/2135/1/012010 ◽

2021 ◽

Vol 2135 (1) ◽

pp. 012010

Author(s):

Oscar Danilo Montoya ◽

Diego Armando Giral-Ramírez ◽

Luis Fernando Grisales-Noreña

Keyword(s):

Black Hole ◽

Power Distribution ◽

Power Flow ◽

Optimal Power Flow ◽

Distribution Networks ◽

Test System ◽

Load Flow ◽

Optimization Strategy ◽

Optimal Power ◽

Distribution Company

Abstract The optimal sizing of Distributed Generators (DG) in electric power distribution networks is carried out through a metaheuristic optimization strategy. To size DG it is proposed an optimal power flow model is formulated by considering that the location of these sources has been previously defined by the distribution company. The solution of the optimal power flow is reached with the Black Hole Optimizer (BHO). A methodology is used master-slave optimization methodology, where the BHO (i.e., master stage) defines the sizes of the DG and the slave stage evaluates the objective function with a load flow algorithm, this work using the triangular-based power flow method. Numerical results in the 33-node and the 69-node test system demonstrates the effectiveness and robustness of the proposed approach when compared with literature results.

Download Full-text

Integration of Electric Vehicles in Low-Voltage Distribution Networks Considering Voltage Management

Energies ◽

10.3390/en13164125 ◽

2020 ◽

Vol 13 (16) ◽

pp. 4125

Author(s):

Miguel Carrión ◽

Rafael Zárate-Miñano ◽

Ruth Domínguez

Keyword(s):

Power Systems ◽

Electric Vehicles ◽

Power Flow ◽

Optimal Power Flow ◽

Low Voltage ◽

Distribution Network ◽

Planning Horizon ◽

Distribution Networks ◽

Mixed Integer ◽

Distribution Grid

The expected growth of the number of electric vehicles can be challenging for planning and operating power systems. In this sense, distribution networks are considered the Achilles’ heel of the process of adapting current power systems for a high presence of electric vehicles. This paper aims at deciding the maximum number of three-phase high-power charging points that can be installed in a low-voltage residential distribution grid. In order to increase the number of installed charging points, a mixed-integer formulation is proposed to model the provision of decentralized voltage support by electric vehicle chargers. This formulation is afterwards integrated into a modified AC optimal power flow formulation to characterize the steady-state operation of the distribution network during a given planning horizon. The performance of the proposed formulations have been tested in a case study based on the distribution network of La Graciosa island in Spain.

Download Full-text

A Mixed-Integer Convex Model for the Optimal Placement and Sizing of Distributed Generators in Power Distribution Networks

Applied Sciences ◽

10.3390/app11020627 ◽

2021 ◽

Vol 11 (2) ◽

pp. 627

Author(s):

Walter Gil-González ◽

Alejandro Garces ◽

Oscar Danilo Montoya ◽

Jesus C. Hernández

Keyword(s):

Power Distribution ◽

Power Flow ◽

Distribution Networks ◽

Global Optimum ◽

Optimal Placement ◽

Mixed Integer ◽

Power Distribution Networks ◽

Distributed Generators ◽

Second Order Cone ◽

Proposed Model

The optimal placement and sizing of distributed generators is a classical problem in power distribution networks that is usually solved using heuristic algorithms due to its high complexity. This paper proposes a different approach based on a mixed-integer second-order cone programming (MI-SOCP) model that ensures the global optimum of the relaxed optimization model. Second-order cone programming (SOCP) has demonstrated to be an efficient alternative to cope with the non-convexity of the power flow equations in power distribution networks. Of relatively new interest to the power systems community is the extension to MI-SOCP models. The proposed model is an approximation. However, numerical validations in the IEEE 33-bus and IEEE 69-bus test systems for unity and variable power factor confirm that the proposed MI-SOCP finds the best solutions reported in the literature. Being an exact technique, the proposed model allows minimum processing times and zero standard deviation, i.e., the same optimum is guaranteed at each time that the MI-SOCP model is solved (a significant advantage in comparison to metaheuristics). Additionally, load and photovoltaic generation curves for the IEEE 69-node test system are included to demonstrate the applicability of the proposed MI-SOCP to solve the problem of the optimal location and sizing of renewable generators using the multi-period optimal power flow formulation. Therefore, the proposed MI-SOCP also guarantees the global optimum finding, in contrast to local solutions achieved with mixed-integer nonlinear programming solvers available in the GAMS optimization software. All the simulations were carried out via MATLAB software with the CVX package and Gurobi solver.

Download Full-text

Optimal Selection of Conductors in Three-Phase Distribution Networks Using a Discrete Version of the Vortex Search Algorithm

Computation ◽

10.3390/computation9070080 ◽

2021 ◽

Vol 9 (7) ◽

pp. 80

Author(s):

John Fernando Martínez-Gil ◽

Nicolas Alejandro Moyano-García ◽

Oscar Danilo Montoya ◽

Jorge Alexander Alarcon-Villamil

Keyword(s):

Objective Function ◽

Power Flow ◽

Phase Distribution ◽

Search Algorithm ◽

Distribution Networks ◽

Discrete Version ◽

Optimal Selection ◽

Flow Method ◽

Three Phase ◽

Selection Of

In this study, a new methodology is proposed to perform optimal selection of conductors in three-phase distribution networks through a discrete version of the metaheuristic method of vortex search. To represent the problem, a single-objective mathematical model with a mixed-integer nonlinear programming (MINLP) structure is used. As an objective function, minimization of the investment costs in conductors together with the technical losses of the network for a study period of one year is considered. Additionally, the model will be implemented in balanced and unbalanced test systems and with variations in the connection of their loads, i.e., Δ− and Y−connections. To evaluate the costs of the energy losses, a classical backward/forward three-phase power-flow method is implemented. Two test systems used in the specialized literature were employed, which comprise 8 and 27 nodes with radial structures in medium voltage levels. All computational implementations were developed in the MATLAB programming environment, and all results were evaluated in DigSILENT software to verify the effectiveness and the proposed three-phase unbalanced power-flow method. Comparative analyses with classical and Chu & Beasley genetic algorithms, tabu search algorithm, and exact MINLP approaches demonstrate the efficiency of the proposed optimization approach regarding the final value of the objective function.

Download Full-text