1 Short-Circuit Constrained Distribution Network Reconfiguration Considering Closed-Loop Operation 1Leonardo H. Macedo1,*, Juan M. Home-Ortiz1, Renzo Vargas2, José R. S. Mantovani1, Rubén Romero1, João P. S. Catalão3 1 Department of Electrical Engineering, São Paulo State University, Avenida Brasil 56, Centro, 15385-000, Ilha Solteira, SP, Brazil 2 Center for Engineering, Modeling and Applied Social Sciences, Federal University of ABC Santo André, Ave- nida dos Estados 5001, Bairro Santa Terezinha, 09210-580, Santo André, SP, Brazil 3 FEUP and INESC TEC Porto, R. Dr. Roberto Frias, 4200-465, Porto, Portugal Abstract is paper presents a novel scenario-based stochastic mixed-integer second-order cone pro- gramming model to solve the problem of optimal reconfiguration of distribution systems with renewable energy sources considering short-circuit constraints. e proposed formulation min- imizes technical losses by modifying the statuses of sectionalizing and tie switches, allowing the operation of distribution networks with radial and closed-loop topologies. Since the for- mation of loops could impact fault current levels, short-circuit constraints are considered in the problem formulation. Numerical experiments are carried out using an 84-node system and the results demonstrate the effectiveness of the proposed formulation to reduce technical losses notably when a closed-loop operation is allowed. Additionally, it is verified that short-circuit constraints prevent the adoption of network configurations with high short-circuit values. Keywords: Closed-loop operation; distribution network reconfiguration; renewable energy sources; short-circuit; stochastic programming. Nomenclature Indices and sets: 𝑖, 𝑗 Indices for nodes 𝑖𝑗, 𝑗𝑖 Indices for branches 𝑐 Index for a short-circuit scenario 𝑠 Index for stochastic scenarios Ω Set of branches Ω Set of branches for short-circuit calculations Ω Set of short-circuit scenarios Ω Set of nodes with distributed generators (DGs) Ω Set of nodes Ω Set of nodes for short-circuit calculations *Principal corresponding author E-mail addresses: leohfmp@ieee.org (Leonardo H. Macedo), juan.home@unesp.br (Juan M. Home-Or- tiz), renzo.vargas@ufabc.edu.br (Renzo Vargas), mant@dee.feis.unesp.br (José R. S. Mantovani), ruben.ro- mero@unesp.br (Rubén Romero), catalao@fe.up.pt (João P. S. Catalão). 2 Ω Set of stochastic scenarios Ω Set of substations (SSs) nodes Parameters: 𝒞 Energy price of a stochastic scenario 𝐼 Current capacity of a branch 𝐼 Short-circuit current limit for a branch on a short-circuit scenario 𝑀 , 𝑀 Big-M parameters 𝑁 Maximum number of basic loops allowed 𝑃 , 𝑄 Active/reactive power demands 𝑃𝐹 , 𝑃𝐹 Power factor limits of a DG 𝑅 , 𝑋 , 𝑍 Resistance, reactance, and impedance of a branch 𝑃 Installed DG capacity 𝑆 Apparent power capacity of a DG 𝑆 Apparent power capacity of a SS 𝑉 , 𝑉 Maximum/minimum voltage magnitude limits 𝑉̃ Estimate of the voltage magnitude ∆ Duration of a stochastic scenario Φ Generation factor of a DG unit Continuous variables: 𝒾 Square of the current magnitude on a branch 𝒾 ̂ , 𝒾 ̂ Real/imaginary parts of the current on a branch on a short-circuit sce- nario 𝒾 ̂ , 𝒾 ̂ Real/imaginary parts of the current injected by a SS on a short-circuit scenario 𝒾 ̂ , 𝒾 ̂ Real/imaginary parts of the short-circuit current at a faulted node in a short-circuit scenario 𝑝 , 𝑞 Active/reactive power flows through a branch 𝑝 , 𝑞 Active/reactive power injected by a DG 𝑝 , 𝑞 Active/reactive power injected by a SS 𝑣 Square of the voltage magnitude at a node 𝑣̂ , 𝑣̂ Real/imaginary parts of the voltage at a node on a short-circuit sce- nario 𝑓 Fictitious flow on a branch 𝑔 Fictitious generation at SS nodes 𝛿 , 𝛿 Slack variable for the real/imaginary parts of the voltage drop on a branch on a short-circuit scenario 𝜃 Voltage angle at a node 𝜆 Slack variable for the voltage drop calculation 𝜆 Slack variable for the angle difference calculation Binary variables: 𝑤 Operational state of a branch 3 1. Introduction Conventionally, electric power distribution networks are planned to have a weakly-meshed structure but are operated with a radial topology [1]. e adoption of radial structures responds to various technical reasons including the simplification of the protection system coordination and the reduction of short-circuit current values [2], [3]. However, in recent years, the closed- loop operation has been considered as an alternative to radial topologies due to operational advantages [4], [5]. For instance, in a normal state, a weakly-meshed configuration reduces technical losses [3], improves the reliability of distribution systems [6], and increases the host- ing capacity of renewable energy sources (RES) in distribution networks [5], [7], [8]. On the other hand, the creation of loops during the restorative state improves the system’s response to permanent faults by rearranging more de-energized loads to adjacent feeders [9]. One of the most common approaches to improve the operation of distribution systems is by performing network reconfiguration. It consists of changing the topology of the network through switching operations for alleviating congestions, reducing losses, and improving the voltage profile while, typically, maintaining a radial configuration for the system [10], [11], [12], [13]. An alternative to the radial operation is to allow the formation of loops in the network. In [4], a genetic algorithm-based approach is presented for solving the optimal power flow prob- lem in a real Czech urban closed-looped distribution network. In [5], the authors present a com- parison between closed-loop alternatives and network reinforcement solutions to increase dis- tributed generation connection to a real French network. Reference [3] presents a mixed-integer nonlinear programming (MINLP) model for the reconfiguration problem allowing a significant decrease in losses with a reduced number of closed-loops. However, the formation of loops in distribution networks should be taken into account in the fault calculation as this type of con- figuration could impact fault current levels [5], [6]. Electric distribution networks are exposed to short-circuit faults that can cause undesirable 4 conditions to consumers and damage to utilities’ equipment. Fault current levels depend on many factors, including network topology, grounding arrangements, and the number of in-ser- vice distributed generators (DGs) [14]. DGs have different contributions to a fault current de- pending on the technology of the generation unit. In general, synchronous and induction gen- erators have high contributions to fault current levels, however inverter-based units’ contribu- tion to a fault could be neglected due to disconnection speed [14]. Short-circuit constraints are considered in this work to avoid network topologies that can present high values of short-circuit currents, such as when different substations (SSs) are interconnected [6]. Note that, even for radial configurations, short-circuit constraints should be considered in the reconfiguration prob- lem. However, this type of constraint is widely ignored in the literature [10], [15]. To the best of the authors’ knowledge, no paper has yet presented a mathematical optimization model tak- ing into account short-circuit constraints in the distribution network reconfiguration problem. In this work, we consider network reconfiguration for reducing technical losses of distribu- tion systems with RES. e proposed approach considers opening sectionalizing switches to provide more flexibility to the network operation. Uncertain parameters, such as renewable generation availability, energy prices, and loads are represented through stochastic scenarios. Moreover, short-circuit constraints are considered in the problem so that the resulting configu- rations can be formed without compromising the isolation levels of equipment. e proposed formulation consists of a new scenario-based stochastic mixed-integer second-order cone pro- gramming (MISOCP) model. To handle the uncertainties of RES, a stochastic scenario-based formulation is used. Tests are performed using an 84-node distribution system. e main contributions of this work are as follows:  From a modeling perspective, a new stochastic programming-based model is proposed to determine the optimal network topology to improve the operation of distribution systems taking into account short-circuit limits of the network’s as- sets. 5  From a methodological perspective, the resulting mixed-integer nonlinear pro- gramming problem is recast in order to obtain a relaxed MISOCP model that is treatable, scalable, and can be effectively solved by off-the-shelf optimization solvers. e remainder of this paper is organized as follows: the mathematical formulation of the problem is presented in Section 2; the tests with the 84-node system and a discussion of the results are shown in Section 3; finally, the conclusions of the work are presented in Section 4. 2. Mathematical Formulation 2.1 Objective Function e objective function of the problem, presented in (1), minimizes the total cost of the energy losses. minimize 𝒞 ∆ 𝑅 𝒾 ∈∀ ∈ (1) Note that (1) considers the values of the losses for all branches in all operating scenarios, multiplied by the corresponding duration of the scenario and the energy price in the scenario. Alternative objectives can also be considered in the formulation, such as improving the voltage regulation of the system, balancing the load among the substations, or increasing the hosting capacity of the network for renewables. 2.2 Power Flow Constraints e ac operation of the distribution system is represented by the power flow equations (2)– (8) [16]. 𝑝 ∈ − 𝑝 + 𝑅 𝒾 ∈ + 𝑝 + 𝑝 = 𝑃 (2) 𝑞 ∈ − 𝑞 + 𝑋 𝒾 ∈ + 𝑞 + 𝑞 = 𝑄 (3) ∀𝑖 ∈ Ω , 𝑠 ∈ Ω 6 𝑣 − 𝑣 + 𝜆 = 2 𝑅 𝑝 + 𝑋 𝑞 + 𝑍 𝒾 (4) 𝑉̃ 𝑉 ̃ 𝜃 − 𝜃 + 𝜆 = 𝑋 𝑝 − 𝑅 𝑞 (5) 𝑣 𝒾 ≥ 𝑝 + 𝑞 (6) 𝜆 ≤ 𝑀 1 − 𝑤 (7) 𝜆 ≤ 𝑀 1 − 𝑤 (8) ∀𝑖𝑗 ∈ Ω , 𝑠 ∈ Ω Constraints (2) and (3) are the active and reactive power balance equations, respectively, that represent the application of Kirchhoff’s current law to the system. Constraints (4)–(8) represent the application of Kirchhoff’s voltage law to the system, in which (7) and (8) are used to calcu- late the slack variables 𝜆 and 𝜆 according with the statuses of the switches. Note that (6) is a second-order cone constraint. Ideally, this constraint should be active in the solution, oth- erwise, the terms 𝑅 𝒾 , 𝑋 𝒾 , and 𝑍 𝒾 in (2), (3), and (4) will be overestimated, re- sulting in higher values of losses. Note, however, that the active power losses are minimized in the objective function (1), leading (6) to remain active. 2.3 Physical and Operational Limits of the System e physical and operational limits of the system are considered in constraints (9)–(13). 0 ≤ 𝒾 ≤ 𝐼 𝑤 ∀𝑖𝑗 ∈ Ω , 𝑠 ∈ Ω (9) 𝑝 ≤ 𝑉 𝐼 𝑤 ∀𝑖𝑗 ∈ Ω , 𝑠 ∈ Ω (10) 𝑞 ≤ 𝑉 𝐼 𝑤 ∀𝑖𝑗 ∈ Ω , 𝑠 ∈ Ω (11) 𝑉 ≤ 𝑣 ≤ 𝑉 ∀𝑖 ∈ Ω , 𝑠 ∈ Ω (12) 𝑝 + 𝑞 ≤ 𝑆 ∀𝑖 ∈ Ω , 𝑠 ∈ Ω (13) 7 Constraint (9) represents the current limits for the branches according to the status of the switches, while (10)–(11) are the active and reactive power limits for the branches, also de- pendent on the status of the switches. Constraint (12) is the voltage magnitude limit for the nodes. Finally, constraint (13) is the apparent power capacity of the SSs. 2.4 Renewable DGs e operational limits of the renewable DGs are shown in (14)–(16). 𝑝 + 𝑞 ≤ 𝑆 ∀𝑖 ∈ Ω , 𝑠 ∈ Ω (14) 0 ≤ 𝑝 ≤ Φ 𝑃 ∀𝑖 ∈ Ω , 𝑠 ∈ Ω (15) −𝑝 tan(cos− (𝑃𝐹 )) ≤ 𝑞 ≤ 𝑝 tan cos− 𝑃𝐹 (16) ∀𝑖 ∈ Ω , 𝑠 ∈ Ω Constraint (14) represents the power generation capacity of the DGs. Constraint (15) limits the active power of the renewable DGs according to the availability of the renewable resource. Finally, constraint (16) limits the power factor of the DGs. 2.5 Topological Constraints e connectivity of the system and the maximum number of loops allowed to be formed are controlled by (17)–(20) through artificial demands that must be attended at all nodes. |Ω | − |Ω | ≤ 𝑤 ∈ ≤ |Ω | − |Ω | + 𝑁 (17) 𝑓 ∈ − 𝑓 ∈ + 𝑔 = 1 ∀𝑖 ∈ Ω (18) 𝑓 ≤ |Ω |𝑤 ∀𝑖𝑗 ∈ Ω (19) 0 ≤ 𝑔 ≤ |Ω | ∀𝑖 ∈ Ω (20) 8 Constraint (17) limits the maximum number of basic loops allowed to be formed in the sys- tem. is constraint operates together with (18)–(20), which ensure the connectivity of the sys- tem, by requiring that there must be a path from each node of the system to a SS. For the load nodes (Ω \Ω ), 𝑔 = 0. 2.6 Short-Circuit Constraints Changes in the network topology due to reconfiguration and the formation of loops affect the voltage profile and the currents of the system in normal operation scenarios, as well as the short-circuit currents in fault scenarios. It must be guaranteed that the reconfiguration of the network will not produce an increase of the short-circuit currents beyond the short-circuit ca- pacity of the protective devices. Moreover, it is desirable to produce only small changes in the short-circuit currents, so that the coordination of the protection is not overly affected. e following considerations are made for the short-circuit analysis: i. e SS is represented by a power source with a voltage of 1∠0° p.u. in series with an equivalent impedance of the upstream network [17]; ii. e inverter-interfaced renewable DG does not contribute to the short-circuit currents [14]; iii. A single fault at a node is considered in each fault scenario, and the faults are symmetrical three-phase short-circuits [17]; iv. Since short-circuit currents are much larger than the load currents in steady-state, the loads are ignored in the calculation of the short-circuit currents [17]. e short-circuit constraints, with considerations (i)–(iv), are presented in (21)–(27). 𝒾 ̂ ∈ − 𝒾 ̂ ∈ + 𝒾 ̂ = 𝒾 ̂ (21) 𝒾 ̂ ∈ − 𝒾 ̂ ∈ + 𝒾 ̂ = 𝒾 ̂ (22) ∀𝑖 ∈ Ω , 𝑐 ∈ Ω 9 𝑣̂ − 𝑣̂ + 𝛿 = 𝑅 𝒾 ̂ − 𝑋 𝒾 ̂ (23) 𝑣̂ − 𝑣̂ + 𝛿 = 𝑋 𝒾 ̂ + 𝑅 𝒾 ̂ (24) 𝛿 ≤ 2𝑉 1 − 𝑤 (25) 𝛿 ≤ 2𝑉 1 − 𝑤 (26) 𝒾 ̂ + 𝒾 ̂ ≤ 𝐼 𝑤 (27) ∀𝑖𝑗 ∈ Ω , 𝑐 ∈ Ω In (21)–(27), the set Ω contains the branches of Ω and the additional branches with the equivalent impedance of the upstream network connected to the SS buses; the set Ω contains the nodes of Ω and the terminal nodes of the additional branches from Ω \Ω ; finally, Ω is the set of fault scenarios. A fault at a single node is considered in each fault scenario. Constraints (21) and (22) are the application of Kirchhoff’s current law to the real and imag- inary components of the short-circuit currents. Constraints (23)–(26) represent the application of Kirchhoff’s voltage law to the system in the fault scenarios. Finally, constraint (27) is the limit for the short-circuit current on a branch according to the fault scenario and the status of the branch. ese values must also be limited by the isolation levels of the equipment installed in the network. It should be noted that, in each fault scenario, the voltage at the faulted node is set to zero in the model. Note that the short-circuit constraints presented in this paper allow changes in the network topology, differently from [17], which considers a short-circuit model for pre-defined fixed to- pologies. Moreover, the formulation (21)–(27) can be applied to networks considering both traditional and modern digital protective devices that can be reparametrized. For the latter, it is possible to obtain more flexibility in the solutions. However, the isolation levels of the equip- ment present in the network must also be taken into account when these devices are considered. In the proposed scenario-based formulation, the objective function (1) is linear, as well as constraints (2)–(5), (7)–(12), and (15)–(26). Constraint (6) is a second-order cone, while (13), 10 (14), and (27) are quadratic constraints. Due to the presence of the binary variable 𝑤 , the resulting formulation is a stochastic MISOCP model, which can be solved by off-the-shelf op- timization solvers. 3. Tests and Results e proposed model is tested using the 84-node system, adapted from [18], shown in Fig. 1, which operates at 11.40 kV. is system has four 600 kW photovoltaic (PV) generation units with inductive and capacitive power factor limits of 0.9 at nodes 9, 24, 42, and 64. e minimum and maximum voltage magnitude limits are 0.93 and 1.05 p.u., respectively. For a radial oper- ation, the system has 13 normally open switches. It is assumed that every branch has a switch. Complete data for the system is available in [19]. e solar irradiations, energy prices, and power demands are represented using a set of 16 stochastic scenarios obtained from historical data and reduced using the k-means method [20]. e proposed formulation was implemented in AMPL [21] and solved with the commercial solver CPLEX v20.1.0 [22] on a computer with a 3.2 GHz Intel® Core™ i7–8700 processor and 32 GB of RAM. 3.1 Study Cases Considering the initial topology presented in Fig. 1, the solution of the proposed model dis- regarding network reconfiguration provides the short-circuit current values of the system con- sidering a fault at each node connected to a substation node, i.e., faults are considered at nodes 1, 11, 15, 25, 30, 43, 47, 56, 65, 73, and 77, one at a time, in each fault scenario. Based on these short-circuit currents, the study cases proposed in Section 3.3 allow controlled variations, from 1% until 25%, in the short-circuit current values, 𝐼 , to find topologies with lower costs of the energy losses. ese short-circuit values are also limited by the isolation levels of the equip- ment. e proposed model is solved considering the two approaches presented below: 11  Case I – without network reconfiguration: Starting with the initial radial config- uration, it is only allowed to close tie (normally-open) switches, while opening sectionalizing (normally-closed) switches is disregarded, as considered in [8];  Case II – with network reconfiguration: e reconfiguration process considers both closing tie switches and opening sectionalizing switches. e following subsection presents the results obtained with the proposed model for both approaches and disregarding the short-circuit constraints, (21)–(27), in the model. 3.2 Results Without Considering Short-Circuit Constraints Disregarding any switching operation, the initial configuration shown in Fig. 1, with the switches of branches 5-55, 7-60, 11-43, 12-72, 13-76, 14-18, 16-26, 20-83, 28-32, 29-39, 34- 46, 40-42, and 53-64 open, has a total annual cost of the energy losses of US$ 104,569.30. Table I summarizes the main results obtained for the problem without considering short- circuit constraints. When disregarding the short-circuit constraints, (21)–(27), of the model, the optimal radial configuration, obtained with 𝑁 = 0, presents a total annual cost of the energy losses of US$ 92,957.95, with 13 switches of branches open. is represents a cost reduction of 11.10% in comparison with the initial configuration. For Case I, only allowing closing switches and disregarding the short-circuit constraints, the optimal configuration, obtained with 𝑁 = 13, i.e., allowing the formation of up to 13 basic loops, presents a total annual cost of the energy losses of US$ 92,070.36, with 4 switches of branches open. is represents a cost reduction of 11.95% in comparison with the initial con- figuration, while 9 loops are formed in the system. For Case II, allowing network reconfiguration and closed-loop operation and disregarding the short-circuit constraints, the optimal configuration, obtained with 𝑁 = 13, presents a total annual cost of the energy losses of US$ 91,375.26, with 8 switches of branches open. is represents a cost reduction of 12.62% in comparison with the initial configuration, while 5 loops are formed in the system. 12 Note that, from the results of Cases I and II, not necessarily an all-closed-switches solution presents the lowest value of losses. Indeed, by closing all switches in the system, its operation becomes infeasible, since the current capacities of branches 43-84 and 44-45 are violated in some operation scenarios. Figure 1: Initial configuration of the 84-node system. Table I: Results without considering short-circuit constraints Case Objective function value (US$) Open switches Number of loops I 104,569.30 5-55, 7-60, 11-43, 12-72, 13-76, 14-18, 16-26, 20-83, 28-32, 29-39, 34-46, 40-42, 53-64 0 II 92,957.95 6-7, 12-13, 33-34, 37-38, 39-40, 62-63, 71-72, 82-83, 5-55, 11-43, 14-18, 16-26, 28-32 0 I 92,070.36 11-43, 14-18, 28-32, 29-39 9 II 91,375.26 12-13, 32-33, 37-38, 39-40, 81-82, 5-55, 11-43, 14-18 5 1 2 3 4 5 6 7 9 8 10 55 54 53 52 51 50 49 48 47 11 12 13 15 16 17 18 19 20 21 23 24 22 25 26 27 28 29 30 31 32 33 34 35 36 37 38 41 42 39 40 43 44 45 46 77787980818283 73747576 6566676869707172 565758596061626364 84 96 43 87 85 14 89 91 93 94 11 86 92 95 90 88 86 84 84 84 84 84 84 84 84 84 84 84 13 e following subsection presents the results obtained with the proposed model for both approaches and considering variations of the maximum value of the short-circuit current limits, obtained from the initial configuration. 3.3 Results Considering Short-Circuit Constraints First, we consider the maximum short-circuit current on a branch in a fault scenario, 𝐼 , the short-circuit current value obtained from the initial configuration. en, we allow maximum deviations, from 1% up to 25% (also limited by the isolation levels of the equipment), in these values for obtaining solutions that comply with the isolation levels of equipment and try to maintain the coordination of the protection. Fig. 2 presents the summary of the solutions obtained for Cases I and II for the values of maximum short-circuit current deviations: Fig. 2(a) presents the annual cost of the energy losses for each solution, while Fig. 2(b) presents the number of basic loops in the corresponding solu- tion. It can be observed, in Fig. 2(a), that by allowing higher values for the maximum value of the short-circuit current deviation, it is possible to obtain solutions with lower values of the annual cost of energy losses for both Cases I and II. Moreover, since Case II allows for more flexibility than Case I, the solutions for Case II will always present lower (or in the worst-case equal) values of the annual cost of energy losses than the solutions for Case I. Moreover, from Fig. 2(b), it is possible to conclude that, by increasing the value of the max- imum short-circuit current deviation, it is possible to form more loops in the system or to obtain different network configurations that allow reducing the annual cost of energy losses. Table II summarizes the main results obtained for the problem considering short-circuit con- straints. For a maximum short-circuit current deviation of 1%, in Case I, the topology of the system does not change and remains equal to the initial configuration. For Case II, the corre- sponding solution has a cost of US$ 101,159.96, with 13 switches of branches open, and pre- sents a radial configuration. is represents a cost reduction of 3.26% in comparison to the solution of Case I. 14 (a) (b) Figure 2: (a) Annual costs of energy losses obtained by changing the maximum short-circuit deviation in relation to the initial configuration and (b) the corresponding numbers of loops in the system for each solution. 0 5 10 15 20 25 Maximum short-circuit current deviation (%) 9 9.5 10 10.5 11 104 With network reconfiguration Without network reconfiguration 0 5 10 15 20 25 Maximum short-circuit current deviation (%) 0 1 2 3 4 5 6 7 8 With network reconfiguration Without network reconfiguration 15 For a maximum short-circuit current deviation of 9%, in Case I, a solution with a total annual cost of the energy losses of US$ 104,255.70 is obtained, in which 11 switches of branches are open, and 2 basic loops are formed in the system. For Case II, the corresponding solution has a cost of US$ 93,041.95, with 11 switches of branches open, and, again, 2 basic loops are formed. e solution for Case II represents a cost reduction of 10.76% in comparison to the solution of Case I. Finally, for a maximum short-circuit current deviation of 25%, in Case I, a solution with a total annual cost of the energy losses of US$ 93,644.96 is obtained, in which 6 switches of branches are open, and 7 basic loops are formed in the system. For Case II, the corresponding solution has a cost of US$ 92,024.57, with 8 switches of branches open, and 5 basic loops formed. e solution for Case II represents a cost reduction of 1.73% in comparison to the solution of Case I. It can be verified that by increasing the values of the maximum deviations in the short-circuit currents, it is possible to obtain solutions with lower energy losses costs. From an optimization problem perspective, this can be explained by the larger feasible region that is obtained with higher maximum short-circuit current values. is allows us to obtain solutions with lower costs of energy losses. From the power system operation perspective, by allowing larger values of short-circuit currents, it is possible to obtain more flexible configurations for the network oper- ation, allowing, for example, the interconnection of substations and the formation of more loops Table II: Solutions considering short-circuit constraints Case Maximum short- circuit current deviation (%) Objective function value (US$) Open switches Number of loops I 1 104,569.30 5-55, 7-60, 11-43, 12-72, 13-76, 14-18, 16-26, 20-83, 28-32, 29-39, 34-46, 40-42, 53-64 0 II 1 101,159.96 6-7, 36-37, 38-41, 54-55, 60-61, 75-76, 11-43, 12-72, 14-18, 16-26, 20-83, 28-32, 34-46 0 I 9 104,255.70 5-55, 7-60, 11-43, 13-76, 14-18, 16-26, 20-83, 28-32, 29-39, 34-46, 53-64 2 II 9 93,041.95 6-7, 12-13, 32-33, 35-36, 38-39, 60-61, 81-82, 5-55, 11-43, 16-26, 28-32 2 I 25 93,644.96 5-55, 11-43, 13-76, 14-18, 16-26, 28-32 7 II 25 92,024.57 12-13, 33-34, 38-39, 82-83, 5-55, 11-43, 14-18, 16-26 5 16 in the system. Note that, a system with more loops may present lower values of normal opera- tion currents on branches and, consequently, lower values of energy losses. However, the short- circuit currents of meshed systems are usually higher, especially when substations are intercon- nected [6]. e results indicate that the approach considered in Case II is capable of providing better solutions than the approach in Case I, reducing the total annual cost of the energy losses in the system while maintaining adequate levels of the short-circuit currents. Note that, by allowing a maximum deviation of the short-circuit current in the system of 9%, it is possible to obtain a solution that is only 0.71% worse than the solution for the problem without considering short- circuit constraints. For the solution obtained without considering short-circuit constraints, shown in the previous subsection for Case II, the maximum deviation of the short-circuit current value is 29.75%. e average computational time to solve the problems in Case I is 46.16 s while for Case II, the average computational time is 1.23 h. e results were validated using a power flow algo- rithm and it was verified that the operation of all of them is feasible. e obtained results indicate that the proposed formulation is capable of obtaining configu- rations that reduce the total annual cost of the energy losses while maintaining adequate levels of short-circuit currents in the system. 3.4 Performance Comparison is subsection presents performance tests of the proposed model according to the number of stochastic and short-circuits scenarios considering a maximum short-circuit current deviation Table III: Performance tests considering a maximum short-circuit current deviation of 7% Number of stochastic scenarios Number of short-circuit scenarios CPU time (s) 16 0 1636 16 3 6261 16 8 7463 16 11 22183 1 11 901 4 11 4184 8 11 14495 12 11 17318 17 of 7%. Table III presents the number of scenarios and the CPU time needed to find the optimal solution to the problem. As expected, the results show that the solution time increases consid- erably when the number of scenarios increases. Note that, each stochastic scenario increases the model size in 2|Ω | + 5|Ω | + 2|Ω | + 2|Ω | variables and in 3|Ω | + 8|Ω | + |Ω | + 4|Ω | constraints, of which 3|Ω | + 7|Ω | + 3|Ω | are linear constraints, |Ω | + |Ω | are quadratic constraints, and |Ω | are second-order cone constraints, as can be verified in (2)–(16), while each short-circuit scenario only increases the model size in 4|Ω | + 4|Ω | + 2|Ω | variables and in 2|Ω | + 5|Ω | constraints, of which 2|Ω | + 4|Ω | are linear con- straints and |Ω | are quadratic constraints, as it can be verified in (21)–(27). 4. Conclusion is work presented a novel mixed-integer second-order cone programming model for the distribution network reconfiguration problem considering short-circuit constraints. e sce- nario-based stochastic formulation accounted for the uncertainty of the demand, renewable gen- eration, and energy prices. Tests were carried out using an 84-node system, and the results indicated that the pro- posed formulation is capable of providing high-quality solutions for the problem while main- taining the short-circuit currents of the system within acceptable ranges, complying with the isolation levels of equipment, and trying to maintain the coordination of the system protection. Future works will consider other objectives in the problem, such as increasing the host- ing capacity of the network. Moreover, a three-phase unbalanced representation of the network operation can be used for both normal operation and the calculation of short-circuit currents. e proposed short-circuit constraints will also be included in the service restoration problem so that the protective devices can adequately actuate in the case of a fault in the network during the restorative state for the configurations provided by the restoration model. 18 5. Acknowledgment is work was supported by the Coordination for the Improvement of Higher Education Personnel (CAPES) – Finance Code 001, the Brazilian National Council for Scientific and Technological Development (CNPq), grants 305852/2017-5, 304726/2020-6, and 408898/2021-6, and the São Paulo Research Foundation (FAPESP), under grants 2015/21972- 6, 2018/20355-1, 2019/01841-5, 2019/23755-3, and 2021/08832-1. J. P. S. 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