A battery energy management system to improve the financial, technical, and environmental indicators of Colombian urban and rural networks

A battery energy management system to improve the financial, technical, and environmental indicators of Colombian urban and rural AC networks L. F. Grisales-Nore˜ naa,∗, Brandon Cort´ es-Caicedob, Oscar Danilo Montoyac,d, J.C Hernand´ eze, G. Alcal´ ae aDepartment of Electrical Engineering, Faculty of Engineering, Universidad de Talca, Campus Curic´o 3340000, Chile bFacultad de Ingenier´ıa, Instituto Tecnol´ogico Metropolitano, Campus Robledo, Medell´ın 050036, Colombia cGrupo de Compatibilidad e Interferencia Electromagn´etica, Facultad de Ingenier´ıa, Universidad Distrital Francisco Jos´e de Caldas, Bogot´a 110231, Colombia dLaboratorio Inteligente de Energ´ıa, Facultad de Ingenier´ıa, Universidad Tecnol´ogica de Bol´ıvar, Cartagena 131001, Colombia eDepartment of Electrical Engineering, Universidad de Ja´en, Ja´en, 23071, Spain fCentro de Investigaci´on en Recursos Energ´eticos y Sustentables, Universidad Veracruzana, Coatzacoalcos, Veracruz 96535, Mexico Abstract This paper addresses the problem of managing battery energy in urban and rural alternating current networks, aiming at improving their financial, technical, and environmental indicators. To this end, a mathematical model was formulated that proposes as objective functions the optimization of energy operational costs of the grid, the minimization of power losses associated with energy transport, and the reduction in CO2emissions related to the production of energy. This model also considers the set of constraints involved in the operation of an alternating current network in an environment with distributed energy resources (Photovoltaic generators + Batteries storage systems). A master–slave strategy that combines a parallel version of the Vortex Search Algorithm (VSA) and an Hourly Power Flow method based on Successive Approximations (HPFSA) was proposed as the solution methodology. To create the test scenarios, urban and rural electrical networks documented in the specialized literature were adapted using generation and demand data of an average day of operation in Medell´ ın-Antioquia (Urban) and Capurgan´ a-Choc´ o (Rural). These data represent the energy behavior of an urban and a rural network within the Colombian territory. In relation to the distributed energy devices, the integration of three Photovoltaic Distributed Generators (PV DGs) and three lithium-ion batteries of different types into the electrical networks was considered. In addition, the energy production costs and CO2emissions of the local network and the diesel fuel in Colombia were determined, as well as the maintenance costs associated with the batteries and the PV DGs. With the purpose of validating the effectiveness of the proposed method in terms of solution, repeatability, and processing times, two comparison methods reported in the specialized literature were employed to solve the problem addressed in this paper. In the urban and rural networks under study, the proposed solution methodology achieved the best results in terms of solution quality, repeatability, and processing time. Keywords: urban networks, rural networks, parallel processing tool, battery storage systems, optimization techniques, energy cost optimization, energy loss minimization, environmental impact reduction. Nomenclature f1Objective function in charge of optimizing the microgrid operating costs using batteries. ∆tTime interval when the generator injects a fixed power level during a given period. In this case, it corresponds to 1 hour. ϕB iCharge/discharge coefficient of the battery located at node i. θi,hVoltage angle at node i at hour h. θj,hVoltage angle at node j at hour h. ∗Corresponding author Email addresses: [email protected] (L. F. Grisales-Nore˜ na ), [email protected] (Brandon Cort´ es-Caicedo), [email protected] (Oscar Danilo Montoya), [email protected] (J.C Hernand´ ez), [email protected] (G. Alcal´ a) φi j Admittance angle of the line that interconnects nodes i and j. CB iNominal capacity in kW of the battery located at node i. CB i,O&MMaintenance cost per kW of the battery located at node i. Cgd i,O&MMaintenance cost per kW of the photovoltaic generator located at node i. costscg i,hCost of purchasing or producing energy from the conventional generator located at node i at hour h. fCCost of generating or purchasing energy from conventional generators. fMCost of maintaining the batteries and photovoltaic generators installed within the microgrid. Preprint submitted to RI November 27, 2022 Ii j,hCurrent flowing through line ij at hour h. Imax i j Maximum current allowed in line ij. Pcg,max iMaximum active power to be injected by the conventional generator located at node i. Pcg,min iMinimum active power to be injected by the conventional generator located at node i. Pgd,max iMinimum active power to be injected by the distributed generator located at node i. Pgd,min iMaximum active power to be injected by the distributed generator located at node i. Pchargmax B,iMaximum charging power of the battery located at node i. Pdischgmax B,iMaximum discharge power of the battery located at node i. pB i,hPower generated/stored by the battery located at node i at hour h. Pd i,hActive power demanded at node i at hour h. Pcg i,hPower generated by the conventional generator located at node i at hour h. pcg i,hActive power injected by the conventional generator located at node i at hour h. pgd i,hActive power injected by the distributed generator located at node i at hour h. Qcg,max iMaximum reactive power to be injected by the conventional generator located at node i. Qcg,min iMinimum reactive power to be injected by the conventional generator located at node i. Qd i,hReactive power demanded at node i at hour h. qcg i,hReactive power injected by the conventional generator located at node i at hour h. SOCB i,hState of charge of the battery located at node i at hour h. SOC0 iInitial state of charge of the battery located at node i. SOCf iFinal state of charge of the battery located at node i. tcB iCharging time of the battery located at node i. tdB iDischarge time of the battery located at node i. vi,hVoltage magnitude at node i at hour h. Vmax iMaximum voltage allowed at node i. Vmin iMinimum voltage allowed at node i. vj,hVoltage magnitude at node j at hour h. Yi j Admittance magnitude of the line that interconnects nodes i and j. 1. Introduction 1.1. General context Electrical networks are transforming their traditional ways of operating as well as their main generators and loads into more dynamic configurations that favor the intelligent interaction of their devices to improve their technical, financial, and environmental conditions [1]. These transformations are aimed at optimizing the use of the existing energy resources and improving the quality of life of the users connected to the networks. All this is possible thanks to the development of non-conventional energy sources based on renewable resources and the advances in power electronics over the last decades [2]. These technological developments have led to the integration of a less polluting energy resource that can be more flexibly managed by controlling the power of the generation sources, the loads, and the energy storage systems that make up the networks [3]. Alternating current networks have traditionally been employed for more than a century and are now the most common type of network worldwide, given that they can transport large amounts of power, with low losses, and at different voltage levels [4]. However, these systems have changed in recent years as a result of the integration of Distributed Generators (DGs) and energy storage systems, aiming to offer more resilient and autonomous networks that maximize the benefits of a smart energy management [5]. Said benefits are related to optimized energy purchase costs, minimized power losses during energy transport, reduced system loadability, low emission of greenhouse gases, improved voltage profiles, among other factors [6]. Obtaining such benefits requires the implementation of energy management strategies that control the power of the system. The positive effects on the technical, financial, and environmental indicators proposed by the network operator or owner, as well as the processing times required [7], are directly linked to the energy management strategy. Consequently, new methodologies are proposed every day in the specialized literature to improve the impact of these strategies on the operation of the electrical network in terms of solution quality and repeatability, always seeking the shortest processing times that allow the system to respond efficiently to generation and demand variations during regular operation. Today, this can be done in urban and rural networks independently, but there is a need to promote strategies that work in both types of networks with excellent performance indices. 1.2. State-of-the-art review Multiple strategies have been proposed in the literature to solve the problem of energy management in alternating current networks [8]. Such strategies focus on DGs, mainly on those based on solar energy, aiming to improve different technical, financial, and environmental indicators of the network [9]. In addition, they implement energy storage elements to control the variability of the renewable generation sources and take advantage of the fluctuations in energy demand and production costs within the electrical network. Moreover, while 2 controlling the power of the distributed energy resources, these strategies ensure compliance with the technical and operational constraints of the electrical devices that integrate the network [10]. Said constraints include generator power limits, battery state-of-charge limits, battery charge and discharge power limits, initial and final states of charge, line current limits, nodal voltage profiles, among others [11]. In the specialized literature, from the perspective of management strategies, the problem of operating energy storage devices in alternating current networks has been discussed in financial, technical, and environmental terms. That is precisely the focus of this study; therefore, some of the papers on this subject are presented below. The authors of [12] presented an optimization model for the joint operation of PV generators and batteries. The integrated PV–battery system was operated by means of a mixed-integer linear programming model, which was solved using the Cplex solver of GAMS. The main objective of the study was to minimize the operating costs of the network. The numerical results in the proposed test scenarios showed the efficiency of the model, considering the processing time it takes for the Cplex solver to reach the solution. However, the authors did not use comparison methodologies to demonstrate the robustness of the model when solving the problem. In [13], the authors solved the problem of operating and configuring energy storage devices in electrical networks following a convex optimization method and using CVXGEN. The main objective of the study was to minimize the configuration and operation costs of the energy storage devices. To demonstrate the efficiency of the proposed methodology, this study compared its results with those of other solution methods reported in the literature, such as the SDPT3 solver. The authors also analyzed the time required for the method to find a solution, thus demonstrating its efficiency in terms of solution and processing times. A second-order cone programming model was employed in [14] to determine the optimal operation of energy storage devices in alternating current distribution networks. The main objective of this study was to minimize both the costs associated with energy losses and the emission of greenhouse gases for one day of operation in a 33-node urban network. The authors used MATLAB CVX to solve the model and different simulation scenarios to demonstrate the efficiency of the proposed methodology. However, they did not compare their results with those of other methodologies documented in the literature, nor did they evaluate the processing times. In [15], the authors employed second-order cone programming and semidefinite programming models to solve the problem of managing energy of storage devices in three-phase distribution networks. The purpose of this study was to minimize the power losses and energy purchase costs at the substation node. The first model was solved using GAMS, while the second one was solved with YALMIP/MOSEK. The numerical results obtained in different test scenarios for the IEEE 4-, 37-, and 123-node test systems revealed the numerical stability and accuracy of the proposed models. This work did not analyzed the impact of the solution methodologies used in processing times. The authors of [16] described a nonlinear programming model for the intelligent operation of energy storage devices, which was solved employing GAMS. The main goal of this study was to minimize greenhouse gas emissions and energy losses for one day of operation of the 33and 69-node test systems (corresponding to urban networks). The numerical results proved the effectiveness of the proposed methodology; however, they were not compared with those of other methods documented in the specialized literature. The processing times required by the solution methodology were not analyzed either. Through the optimal operation of energy storage devices, the authors of [17] sought to minimize the total installation costs, considering the maximum power demand, energy purchase costs, and battery lifetime. To this end, they built an optimization model based on multi-integer linear programming and solved it using GAMS and MATLAB. The numerical results demonstrated the effectiveness of the proposed methodology. However, the authors did not implement comparison methodologies to validate their simulation scenarios, nor did they analyze the processing times. One of the main characteristics of the methodologies described above is that they use commercial software to improve the networks’ technical, financial, and environmental indicators and achieve excellent results in terms of solution. However, this type of software requires investment, in addition to extensive and sometimes complex formulations to solve the mathematical models that represent the problems under study [7]. For this reason, the optimization techniques based on sequential programming have become relevant solution strategies because they can be developed in free software, thus offering an efficient solution as regards response quality, processing times, and repeatability. Consequently, they can be used to operate energy storage devices in an intelligent way, with low implementation costs and few operating difficulties. Moreover, they are replicable in any platform. For example, in [18], the authors proposed the coyote optimization algorithm to minimize power losses in alternating current distribution networks through the implementation and operation of energy storage devices. The authors employed the 48-node test system to demonstrate the efficiency of the proposed methodology and compared their strategy with metaheuristic algorithms reported in the specialized literature. However, they did not perform a statistical analysis, nor did they examine the processing times, which is key to validate the robustness and repeatability of the solution methodology. The authors of [19] presented a master–slave strategy to solve the problem of integrating and operating energy storage devices in alternating current distribution systems. In both stages, the Chu & Beasley genetic algorithm was employed as the optimization technique: a binary version for the location and a continuous for the operation of batteries systems. The main objective of this study was to minimize the energy losses of the 69-node test system. The numerical results for the different simulation scenarios demonstrated the efficiency of the methodology; however, the authors did not use other metaheuristic techniques as comparison methods. The 3 statistics and processing times were not analyzed; therefore, the robustness of the proposed solution methodology could not be validated. For their part, the authors of [20] suggested an optimization methodology to find the optimal dispatch of energy storage systems in coordination with wind generators and capacitors. The main purpose of this study was to simultaneously minimize the energy losses and costs in the 33and 108-node test systems. To solve this optimization problem, the authors used the non-sorting genetic algorithm, along with the technique for order of preference by similarity to ideal solution. The numerical results showed the efficiency of the solution methodology; however, the authors did not perform statistical or comparative analyses, nor did they analyze the processing times, which makes it impossible to validate the robustness of the solution strategy. In [21], the authors proposed the gravitational search algorithm and a hybrid methodology (particle swarm optimizer/genetic algorithm) to solve the problem of integrating and operating energy storage devices in electrical networks. The objective of this study was to minimize the operating costs, voltage profile deviation, and greenhouse gas emissions in the 30-node test system. The authors used different simulation scenarios to validate and compare the proposed methodologies, results, and processing times. However, they did not perform any statistical analysis to determine repeatability and robustness. The genetic algorithm was used in [22] to minimize the energy losses of an urban alternating current distribution network by optimizing the operation of energy storage devices. The authors proposed three simulation scenarios to validate the numerical results obtained for one day of operation in a real system located in northern Romania. The results were not compared with those of other methodologies available in the specialized literature, nor were they analyzed in terms of statistical data and processing times. In [7], the researchers proposed a parallel version of the Particle Swarm Optimization (PSO) algorithm to solve the battery management problem in a 22-node urban network under an environment of PV DGs operating at maximum power. This methodology sought to minimize the costs of purchasing energy from the conventional generator. The results demonstrated the efficiency of the solution methodology when compared with three methods found in the literature: a black hole optimization method, a black hole optimization method, continuous version of the genetic algorithm and a parallel version of the particle swarm optimization. The results also revealed the effectiveness of the proposed methodology in terms of solution quality, standard deviation, and processing times. The authors of [23] described a strategy based on modes of operation to manage energy in a standalone network composed of PV DGs, batteries, and loads. The purpose of this strategy was to ensure the operational and technical constraints of the network. In the results, the authors emphasized the effectiveness of the proposed methodology; however, they did not compare their results with those of other studies. From the state-of-the-art review, we can conclude that many authors have made a remarkable effort to propose intelligent algorithms based, to a greater extent, on sequential programming methods. This type of algorithm avoids the implementation of commercial software and favors the improvement of technical, financial, and environmental indicators of the networks. However, there is still a need to devise successful strategies that allow researchers to reach efficient solutions and repeatability within reduced processing times [7, 19], which can be achieved using parallel processing methods [24, 25]. In addition, such strategies should offer the possibility of improving the main indicators of both connected and standalone networks in rural and urban areas. Most of the efforts have been made in connected urban networks; however, standalone rural networks present the greatest inconveniences from the energetic point of view due to the implementation of fossil fuels. Furthermore, the proposed methodologies should also guarantee that both the mathematical model that represents the problem and the solutions obtained meet the technical and operational constraints of the alternating current networks in an environment of distributed energy resources. Lastly, the solutions should operate efficiently in both urban and rural systems—the latter being associated with a network standalone from the national interconnection system within the Colombian context addressed in this document. 1.3. Proposed solution and main contributions Considering the problems and needs identified in the state-of-the-art review, this study was aimed at solving the problem of energy management in urban and rural alternating current networks. The proposed strategy was controlling the charging and discharging actions of the batteries located in the network in a scenario of distributed generation based on solar energy and operating at the maximum power. As test scenarios, we used standalone and connected electrical networks from a rural and an urban area in Colombia. In this country, urban networks are traditionally connected to a national interconnected system, while rural networks usually operate independently based on fossil energy resources (diesel in most cases). For each network, it was considered the power generation and demand behavior of the corresponding region: Capurgan´ a (Choc´ o) for the rural network and Medell´ ın (Antioquia) for the urban network. We also took into account the integration of three lithium-ion batteries of different technical characteristics and three PV generation devices that make use of the energy potential of the region. The problem to be solved was the management of the battery energy within the urban and rural networks to minimize the energy losses, CO2emissions, and costs associated with the purchase of energy from conventional generators and the maintenance cost of the distributed energy resources (PV DGs+Batteries). As solution methodology, it was proposed a master–slave strategy that combines a parallel version of the Vortex Search Algorithm (VSA) and an Hourly Power Flow method based on Successive Approximations (HPFSA). The former was selected for its high efficiency to solve optimal power flow problems in alternating current networks under distributed energy resource 4 environments [26–28]. The latter has excellent performance in terms of convergence and processing times [29]. To validate this methodology, it was used as test systems the urban and rural networks described above. In addition, it was employed two methods proposed in the specialized literature as comparison methods, reported to solve the problem under study. The selection of these was based on excellent performance in terms of solution quality, repeatability, and processing times [7, 19]. Said methods were a continuous version of the Chu & Beasley genetic algorithm and a parallel version of the PSO algorithm. Some of the main contributions of this study are listed below: •The development of a comprehensive model of the energy management problem that includes financial, technical, and environmental objective functions; integrates the cost of maintenance of the devices; and considers all the constraints of an alternating current network in a distributed generation scenario with telescopic topology. •The implementation of two test systems based on the energy generation and demand conditions of two Colombian regions, that is, a urban network connected to the national system and an standalone rural network that uses diesel. •The introduction of a new solution methodology based on the VSA that takes advantage of a parallel processing strategy, thus maximizing the performance of this optimization algorithm. •The development of a solution methodology to the problem of battery management in urban and rural alternating current networks, which offers the best results in financial, technical, and environmental terms and shows excellent repeatability and processing times. 1.4. Structure of the paper This paper consists of six sections. Section 1 describes the problem addressed, current needs, and benefits of the proposed solution methodology. Section 2 introduces the mathematical formulation of the energy management problem in alternating current networks, using as objective functions some technical, financial, and environmental indicators and considering all the constraints of these systems under a distributed generation environment. Section 3 explains the proposed solution methodology. Section 4 presents the urban and rural networks used, as well as the energy conditions and main considerations of the regions where they are located. Section 5 analyzes the simulation results obtained. Finally, Section 6 draws the main conclusions of this study and proposes future research lines. 2. Mathematical Formulation This section describes the financial, technical, and environmental objective functions proposed for managing battery energy within an alternating current system, considering the whole set of constraints of this network in an environment of distributed energy resources. This model includes the injection of power from DGs but does not control its level. In other words, the power is injected hourly according to the type of distributed generation technology installed at each node, which, in this case, is a PV generation system operating at maximum power. Said injection is performed through the implementation of generation curves associated with each technology, which take into account the energy behavior and type of technology used for solar generation. In addition, when this energy resource is not required, this model can perform null power injections. The proposed mathematical formulation was based on one day of operation (24 hours) discretized in 1-hour periods—which is the way energy generation and demand data are analyzed by network operators in Colombia [30]. This with the objective of assessing the impact of the proposed energy management strategy on an average day of operation of both urban and rural networks. The meaning of each of the acronyms and symbols used in the mathematical model proposed in this study can be found at the beginning of this document. 2.1. Objective functions In this study, we analyze the financial, technical, and environmental impacts of managing the energy of the batteries integrated into an urban and a rural network. The objective functions for each of the indices employed is formulated below: f1=min (fC+fM)(1) fC=        X h∈ΩHX i∈Ωcg costscg i,hPcg i,h∆t         ,{∀i∈Ωcg,∀h∈ΩH}(2) fM=P h∈HP i∈ΩBpB i,hCB i,O&M∆t+P h∈HP i∈Ωgd pgd i,hCgd i,O&M∆t, {∀i∈ΩB,∀h∈ΩH} (3) Equation (1) seeks to reduce the operating costs of the microgrid by optimally operating the batteries located in the electrical system on an average day of operation, using data from one day of PV generation and energy demand from the users connected to the system. This equation uses fC, formulated in Equation (2), which evaluates the costs of purchasing/producing energy from conventional generators to ensure the operation of the microgrid during one day of operation. It also employs fM, formulated in Equation (3), which measures the costs of maintaining the batteries and PV DGs installed in the microgrid. Such costs depend on the power operated by these devices during the period analyzed. f2=min P h∈ΩHP i∈ΩNP j∈ΩN Yi jvi,hvj,hcos θi,h−θj,h−φi j∆t!, n∀i∈ΩN,∀h∈ΩHo(4) 5 f3=min      P h∈ΩHP i∈Ωcg Pcg i,hCEcg i∆t+P h∈ΩHP i∈Ωgd Pgd iCgd hCEgd i∆t     , {∀i∈Ωcg,∀i∈Ωgd,∀h∈ΩH} (5) Lastly, Equations (4) and (5) assess the effect on energy losses and CO2emissions of the power managed in the battery systems installed in the electrical network. 2.2. Constraints Equations (6) to (19) represent the set of constraints of the problem of operating batteries in an alternating current network under a distributed generation environment. Pcg i,h+pgd i,h±PB i,h−Pd i,h=vi,hP j∈ΩN Yi jvj,hcos θi,h−θj,h−φi j, n∀i∈ΩN,∀h∈ΩHo(6) qcg i,h−Qd i,h=vi,hP j∈ΩN Yi jvi,hsin θi,h−θj,h−φi j, n∀i∈ΩN,∀h∈ΩHo(7) Pcg,min i≤pcg i,h≤Pcg,max i,n∀i∈Ωcg,∀h∈ΩHo(8) Qcg,min i≤qcg i,h≤Qcg,max i,n∀i∈Ωcg,∀h∈ΩHo(9) Pgd,min i≤pgd i≤Pgd,max i,n∀i∈Ωgd,∀h∈ΩHo(10) Pchargmax B,i≤pB i,h≤Pdischmax B,i,{∀i∈ΩB,∀h∈ΩH}(11) Equation (6) ensures the balance of active power in the electrical system, considering the injection of power from the conventional and distributed generators, the demand from users, and the injection and absorption of active power from the energy storage elements. Equation (7), for its part, guarantees the balance of reactive power. This equation does not consider the power managed by the battery or the DGs because, within this paper, they only operate with active power. However, it does consider the conventional generators and loads that compose the system. Equations (8), (9), (10), and (11) ensure that conventional and distributed generators operate within the allowed ranges of active and reactive power. Equation (11) establishes the charge and discharge power limits of the battery located within the power network by evaluating the power level at each hour of operation. Pdischmax B,i=CB i tdB i ,{∀i∈ΩB}(12) Pchargmax B,i=−CB i tcB i ,{∀i∈ΩB}(13) S OCB i,h=S OCB i,h−1−ϕB iPB i,h∆t,{∀i∈ΩB,∀h∈ΩH}(14) ϕB i=1 tdB iPdischargmax B,i =1 tcB iPchargmax B,i ,{∀i∈ΩB,∀h∈ΩH}(15) SOCB i,h=0=SOC0 i,{∀i∈ΩB}(16) SOCB i,h=24=SOCf i,{∀i∈ΩB}(17) Vmin i≤vi,h≤Vmax i,n∀i∈ΩN,∀h∈ΩHo(18) Ii j,h≤Imax i j n∀i j ∈ΩN,∀h∈ΩHo(19) To calculate the charge and discharge power limits, we propose Equations (12) and (13), which depend on the nominal capacity of the battery and the charge and discharge times, assuming the battery charge power as negative and the discharge power as positive. Equation (14) calculates the effect of the power supplied or stored on the state of charge of the battery at each hour of operation. This equation is a function of the current state of charge of the battery and the charge coefficient of the battery located at node i. Equation (15) calculates the charge coefficient of the different batteries located in the electrical network, which depends on the battery charge and discharge time and power data provided by the technical data sheet of each device [7]. Note that, to calculate the state of charge of the batteries at each hour of operation, we need to know the initial state of charge at hour 0, which is described in Equation (16). If a level of charge is required at the end of the battery operation (i.e., hour 24), it must be established in Equation (17). In addition, the state of charge of the battery must always be within its maximum and minimum limits, for which Equation (17) is defined in this mathematical model. After guaranteeing all the equations related to the operation of the energy generation and storage elements installed in the electrical network, Equations (18) and (19) establish the nodal voltage limits and line current limits, based on the data provided by the network operator and the construction constraints of the microgrid. Most of the line current limits in this document are different because we employed telescopic networks, whose gauge reduces as they get closer to the end user. This type of network is typically found in conventional electrical networks. 2.3. Fitness function To guarantee each of the technical and operational constraints mentioned in the previous formulation, we propose the implementation of the fitness function described in Equation (20) to penalize the objective function in the event that a 6 solution to the problem incurs in violations of any kind. In this equation, firepresents the objective function, where i can take a value of 1, 2, and 3, indicating the selection of any of the technical, financial, or environmental functions discussed above. Meanwhile, FP and βrepresent the penalty value obtained and a normalization constant that regulates the impact of FP on FF, according to the level of violation. In this study, we heuristically found a value of β=1x103to adjust FF to the different objective functions proposed. Employing a fitness function within the optimization algorithms enables the solution strategies to explore infeasible zones that accelerate, in most cases, the convergence of the algorithms and reduce processing times [31]. FF =fi+βPF (20) PF =                                                                      max (0,P i∈N (pB i,h−Pchargmax B,i)) + min (0,P i∈N pB i,h−Pdischgmax B,i)) max (0,P i∈N (S OCB i,h−S OCmax i)) + min (0,P i∈N (S OCB i,h−S OCmin i)) +max (0,P i∈N (vi−Vmax i)) + min (0,P i∈N (vi−Vmin i)) +max (0,P i∈N P j∈N (Ii j −Imax i j ))                                                                      (21) 3. Proposed Methodology To solve the problem of managing battery energy in alternating current networks, it was used the codification shown in Figure 1 [7]. This figure describes a vector codification of 1x(|ΩB|x24) size, where the number of columns represents the number of batteries installed in the network over the operation time horizon (24 hours in this case). The figure presents the state of charge of the battery at the different hours of operation, showing Battery 1 in green, Battery 2 in gray, and Battery k in blue. In the proposed operation, all the batteries must start and end the day of operation (i.e., hour 0 and hour 24, respectively) with 50% of the state of charge. Therefore, the charge and discharge actions to improve the technical, financial, and environmental conditions according to the objective function can be performed from hour 1 to hour 23. This initial state of charge and discharge is recommended in the specialized literature because it makes a better use of the power of the batteries located in the electrical network [32]. After presenting the codification of the problem, we propose the implementation of a master–slave strategy that combines a parallel version of the VSA [33] and an HPFSA [29]. This strategy allows us to operate the batteries using the parallel optimization algorithm and employ the hourly power flow to assess the fitness function described in the previous section. 3.1. Parallel vortex search algorithm This subsection presents a parallel version of the VSA. To clearly and accurately explain this methodology, we first describe the VSA and then the parallel processing strategy that allows us to enhance this algorithm. 3.1.1. Vortex search algorithm The VSA is a metaheuristic optimization technique originally proposed in [33], which is based on the behavior of vortices created in stirred fluids. For its evolution, this algorithm uses non-concentric hyperspheres whose radius at iteration t (rt) and center (µt) represent the size of the solution space and the position of the current best solution, respectively. As the iterations proceed, the center of the hypersphere changes its position and the radius is reduced until converging to the optimal solution [33]. The process of exploration and exploitation of the solution space takes on a vortex shape, hence the name of the algorithm. Based on the codification presented in Figure 1, we can calculate the initial center of the hypersphere (µ0), as shown in Equation (22). This equation uses the maximum and minimum values of each variable to center the first solutions to the problem at a midpoint. This is achieved by implementing the vectors ymin and ymax, which contain the maximum and minimum limits of the decision variables associated with the problem. In this particular case, they contain the limits of the state of charge and power allowed for the batteries installed in the network, as well as the initial and final state of charge established for each battery, as explained in Section 2 of this document. µ0=(ymax +ymin) 2(22) In the VSA, the population of individuals is generated randomly using a Gaussian distribution around the center of the hypersphere, as shown in Equation (23). Pt m=1 p(2π)Nv|Σ|)exp (−1 2(x−µ)TΣ−1(x−µ)),(23) Where xis a vector of random variables and Σis a covariance matrix. This matrix can be simplified by taking the elements of the diagonal as the variance (σ0) and the off-diagonal elements as zero values, as shown in Equation (24) [34]. σ0= max {ymax}−min nymino 2(24) It is worth mentioning that σ0is taken as the initial radius of the hypersphere (r0). Once the population of individuals has been generated, the FF is evaluated for each of them (see Equation (20)), which allows the algorithm to identify and select the best individual (i.e., incumbent). Subsequently, the incumbent is 7 0.5 0.7 …. 0.3 0.5 0.5 0.35 … 0.7 0.5 …. 0.5 0.4 … 0.4 0.5 h=1 h=2 … . h=23 h=24 h=1 h=2 …. h=23 h=24 h=1 h=2 … h=23 h=24 Battery 1 Battery 2 Battery k Figure 1: Codification for managing battery energy in electrical networks. taken as the new center of the hypersphere and its radius is reduced in size as shown in Equation (25). rt=σ0γ−1(w,1−t tmax ) (25) Note that the radius reduction uses the inverse incomplete gamma function, which can be solved using MATLAB [35]. Here the maximum number of iterations (tmax) and the parameter ware selected at the time of tuning the algorithm. Once the radius has been reduced, new individuals are created around the new center of the hypersphere in order to continue with the iterative process. As the iterations proceed, the center changes if a better incumbent is found, and the radius is reduced to enclose this solution. 3.1.2. Parallel processing The Parallel Vortex Search Algorithm (PVSA) is created by evaluating the FF of each individual in the population by means of a parallel processing strategy that reduces the processing times associated with this task [32]. This strategy simultaneously evaluates the objective function of a group of individuals in the population. The number of individuals that can be evaluated simultaneously is equal to the number of workers available in the computer (W). Therefore, Bnumber of processes is required to evaluate the entire population, Bbeing set by the largest integer in the relation between the population size Pand W(see Equation (26)): B=CEIL(n/W) (26) NPP =B·MTRP (27) Finally, the time required for parallel processing is equal to multiplying Bby the Maximum Processing Time Required when evaluating the Particles (MTRP), as shown in Equation (27). As the number of Wincreases in this equation, the evaluation time of the FF of the population of individuals within the VSA is shorter, which improves the efficiency of the algorithm. 3.2. Hourly power flow method based on successive approximations This study on the management of energy of batteries located in electrical networks seeks to identify the battery operation that provides the best technical, financial, and environmental benefits in one average day of operation. Consequently, an hourly power flow is necessary to update the values of power demanded by the users, generated by the PV DGs, and stored or injected by the batteries. This in order to calculate the effect of these configurations on each hour of operation and quantify, at the end of the day, the overall effect on the indicators mentioned above. To that end, we employed an adaptation of the power flow method based on successive approximations documented in [29, 36]. This method uploads, hour by hour, the values of power generated and demanded within the system by the loads, generators, and batteries from h=1 to h=24. Subsequently, it calculates the effect of one day of operation, as well as the impact on the fitness function of the power configuration proposed for each battery by the PVSA, considering the generation and demand behavior of the PV DGs and electric loads connected to the network. Algorithm 1 describes the process to properly implement the HPFSA. Data: Read the electrical system’s data and the parameters of the HPFSA; for h=1 : 24 do Load the power demanded by the loads during hour h; Load the power generated by the PV DGs during hour h; Load the power supplied or demanded by the batteries during hour h; Solve the AC power flow for the hour husing the successive approximation method in Equation (28); Calculate the objective function associated with the hour h; Evaluate the FF by using Eq. PF for the hour h; end Add the FF values obtained for each hour in order to estimate the overall value of the objective function used; Return the total FF to the PVSA; Algorithm 1: Pseudo-code proposed for the HPFSA. Equation (28) represents the power flow based on successive approximations. Vt+1 d=−Zdd[YdsVs+diag−1(Vt,∗ d)S∗ d] (28) 8 This method iteratively solves the equation until reaching a predefined number of iterations or convergence error; in this case, the values identified heuristically were 1000 iterations and a convergence error of 1x10−10. In this equation, Vddescribes a vector that contains the voltages at the demand nodes; Vs denotes the voltage at the main generator or slack node; Zdd is associated with the inverse of the admittance matrix at the demand nodes, known in the literature as the nodal impedance matrix. Furthermore, Yds, a sub-component of the admittance matrix, represents the admittance values that interconnect the demand nodes with the slack node. Lastly, S∗ dindicates the final power demanded at the system nodes after balancing the power generated and demanded. This vector can take positive or negative values for the different nodes of the system, assuming negative load values as generation in this particular case. Data: Read the electrical system’s data, the power generation and demand curves, and the parameters of the PVSA; if t=1then Define µ0and r0of the hypersphere; Generate the population of candidate solutions (Pt m) using Equation (23); Evaluate the FF for all the individuals in the population using the HPFSA →Parallel processing; Find the incumbent of the population; end for t=2 : tmax do Update the center (µt+1) with the incumbent; Calculate the new radius (rt+1); Generate the new population of solution candidates (Pt+1 m) using Equation (23); Evaluate the FF for all the individuals in the population using the HPFSA →Parallel processing; Find the incumbent of the population; if t≥tmax then Select µt+1as the solution to the problem; Return µt+1and FF associated with this; Break; end end Algorithm 2: Pseudo-code proposed for the PVSA. Finally, Algorithm 2 summarizes the master–slave strategy that combines the PVSA and the HPFSA. The initial step reads the electrical system’s data, the power generation and demand curves related to the PV DGs and the users, and the algorithm parameters, which were tuned using the PSO algorithm reported in [7]. This same method was used to tune the comparison methods, aiming to achieve the best possible performance for each algorithm in terms of solution quality. The parameters of all the optimization algorithms are listed in Table 5. In the first iteration of the algorithm, the values of µ0 and r0are defined using the minimum and maximum state of charge values allowed for each battery located in the system. Subsequently, the population of individuals is generated within the vortex employing Equation (23). This equation generates N individuals that represent multiple solutions to the battery energy management problem and improve the selected objective function. Each individual generated by the population is made feasible according to the state of charge limits, which are between 10% and 90% for lithium-ion batteries. This process of making individuals feasible also guarantees that the hourly charge and discharge power values respect the maximum and minimum limits established for each battery. Afterwards, the FF is evaluated for each individual using the parallel processing strategy and the HPFSA. Within the FF, the improvement of the technical, financial, or environmental indicator must be selected (see the mathematical model described in Section 2). Finally, the individual that achieves the best solution within the population is identified and classified as the incumbent. From iteration 2 and until reaching the maximum number of iterations (tmax), the algorithm updates both the center of the vortex (µt+1) employing the solution related to the incumbent and the radius of the vortex (rt+1) using Equation (25). The purpose of this updating process is that the new population of individuals explore a region close to the incumbent and find new solutions of good quality. Subsequently, Pt+1 mis also updated using Equation (23); the FF of each individual is evaluated employing parallel processing; and the incumbent of the problem is finally updated. This process is repeated until reaching the maximum number of iterations of the algorithm. Once the algorithm ends, the FF and the solution that generates it are stored in the incumbent. That is, therefore, the solution to the problem. 4. Test Scenarios and Considerations In this study, we used two test systems with the objective of evaluating one urban network connected to the Colombian national interconnection system and one rural network standalone from it. The urban network [37] corresponds to Medell´ ın (Antioquia) and is operated by Empresas P´ ublicas de Medell´ ın (EPM) [38]. The rural network corresponds to Capurgan´ a (Choc´ o) [39] and its operation is based on diesel. This network is standalone from the national system; therefore, it is regulated by Instituto de Planificaci´ on y Promoci´ on de Soluciones Energ´ eticas para Zonas no Interconectadas (IPSE) [40]. The following are the circuit diagrams and the power generation and demand curves and elements employed for these two networks. 4.1. Urban network We employed the 33-node test system as the urban electrical network [32]. Figure 2 shows the electrical diagram, which consists of 32 lines, 33 nodes, and a single generator. In addition, the system uses base values of 12.66 kV and 100 kW. The technical parameters and line current limits are detailed in 9 0.0608 0.39916 2.2074 0.0856 0.0525 0.3954 2.4551 0.0905 0.0086 0.0728 0.1273 0.0007 0.0091 0.0376 1.5954 0.0067 Fixed Ecost Var Ecost Eloss Emissions Fixed Ecost Var Ecost Eloss Emissions 100 Minimum and average reductions in objective functions (%) CGA PPSO 0.8671 77.6271 36.3046 32.4710 69.6764 70.6713 70.1105 70.7658 99.8475 73.4684 98.8518 15.7793 56.8048 55.156 56.4487 54.0512 Fixed Ecost Var Ecost Eloss Emissions Fixed Ecost Var Ecost Eloss Emissions 100 Reductions in STD and average time (%) Minimum reductions in objective functions Average reductions in objective functions Reductions in STD Reductions in average processing times Figure 9: Reductions achieved by the PVSA with respect to the comparison methods in the urban network. seconds, which is a reduced time to manage a one-day operation of the batteries. Consequently, the proposed methodology is highly effective in solving the battery energy management problem in urban and rural electrical networks. Based on the financial, technical, and environmental analyses carried out in this study, it is possible concluded that the management of energy in rural networks can lead to considerable reductions in technical, environmental and financial indicators. However, there is a need to employ variable-cost scenarios that make energy management more dynamic in terms of operational costs. When an electrical network operates with fixed costs—as is the case of standalone rural networks in Colombia—it can only save a few dollars in a day of operation, which makes the integration and operation of batteries financially unattractive. This situation highlights the importance of promoting new variable-cost policies for Colombian standalone networks in order to optimize the costs of the system. In this work was identified similar behaviors in urban networks regarding the improvement of technical and environmental conditions. However, given that this type of network can operate under fixedand variable-cost scenarios, we observed a reduction of 97.01% in the energy purchase costs when using variable costs. This demonstrates the importance of implementing variable costs within the national interconnection system for all types of users, so that the distributed devices, in this case the batteries, can manage energy intelligently and improve the financial indicators of the network. Future studies could adopt more robust methodologies to solve the same model, which would maximize the impact on the financial, technical, and environmental conditions of the network and reduce the processing times of the algorithms, thus obtaining better solutions in shorter times. These studies could also address the management of energy using PV DGs located in the microgrid, with the objective of improving the management dynamics within the electrical network and enhancing the impact on the indicators used as objective functions. Other studies could employ reactive devices such as static reactive-power compensators, ultracapacitors, fuel cells, and other devices that help to improve the operating conditions of the electrical network. CRediT Author Statement All authors of this manuscript worked on the conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing-original draft, writing-review & editing, visualization, supervision, project administration, and funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This research was supported by Universidad de Talca (Chile), in collaboration with Minciencias (Colombia) through the Fondo Nacional de Financiamiento para la Ciencia, la Tecnolog´ıa y la Innovaci´on, Fondo Francisco Jos´e de Caldas; Instituto Tecnol´ogico Metropolitano; Universidad Nacional de Colombia; and Universidad del Valle under the project entitled Estrategias de dimensionamiento, planeaci´on y gesti´on inteligente de energ´ıa a partir de la integraci´on y la optimizaci´on de las fuentes no convencionales, los sistemas de almacenamiento y cargas el´ectricas, que permitan 16 la generaci´on de soluciones energ´eticas confiables para los territorios urbanos y rurales de Colombia, which is part of the research program entitled Estrategias para el desarrollo de sistemas energ´eticos sostenibles, confiables, eficientes y accesibles para el futuro de Colombia. References [1] D. Lee, C.-C. 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