Optimal Network Reconﬁguration in Active Distribution Networks with Soft Open Points and Distributed Generation

: In this study, we allocated soft open points (SOPs) and distributed generation (DG) units simultaneously with and without network reconﬁguration (NR), and investigate the contribution of SOP losses to the total active losses, as well as the e ﬀ ect of increasing the number of SOPs connected to distribution systems under di ﬀ erent loading conditions. A recent meta-heuristic optimization algorithm called the discrete-continuous hyper-spherical search algorithm is used to solve the mixed-integer nonlinear problem of SOPs and DGs allocation, along with new NR methodology to obtain radial conﬁgurations in an e ﬃ cient manner without the possibility of getting trapped in local minima. Further, multi-scenario studies are conducted on an IEEE 33-node balanced benchmark distribution system and an 83-node balanced distribution system from a power company in Taiwan. The contributions of SOP losses to the total active losses, as well as the e ﬀ ect of increasing the number of SOPs connected to the system, are investigated to determine the real beneﬁts gained from their allocation. It was clear from the results obtained that simultaneous NR, SOP, and DG allocation into a distribution system creates a hybrid conﬁguration that merges the beneﬁts o ﬀ ered by radial distribution systems and mitigates drawbacks related to losses, power quality, and voltage violations, while o ﬀ ering a far more e ﬃ cient and optimal network operation. Also, it was found that the contribution of the internal loss of SOPs to the total loss for di ﬀ erent numbers of installed SOPs is not dependent on the number of SOPs and that loss minimization is not always guaranteed by installing more SOPs or DGs along with NR. One of the ﬁndings of the paper demonstrates that NR with optimizing tie-lines could reduce active losses considerably. The results obtained also validate, with proper justiﬁcations, that SOPs installed for the management of constraints in LV feeders could further reduce losses and e ﬃ ciently address issues related to voltage violations and network losses.


Introduction
The high penetration of distributed generation (DG) units poses new challenges-power loss increase, harmonic distortion aggregation, equipment overloads, and voltage quality problems-in the planning and operation of power distribution systems. Thus, there is significant room for improvement. Bi-level programming was used to find the optimal allocation of DGs, CBs, and a SOP where the annual costs and power losses were considered as the problem levels.

33-node and 83-node
A simultaneous SOPs and DGs allocation along with NR is proposed. The proposed strategy was tested with/without SOPs loss consideration. Besides, a new NR methodology is proposed to provide resiliency in the distribution system power flow. Moreover, reverse powers are not permitted unlike previous works.
* PS denotes a power system perspective and PE denotes a power electronics perspective.

Proposed Network Reconfiguration
Distribution systems have sectionalizing switches (normally closed switches) that connect line sections and tie switches (normally open switches) that connect two primary feeders, two substation buses, or loop-type laterals. Each line is assumed to be a sectionalized line with a normally closed sectionalized switch. In addition, each normally open tie switch is assumed to be in each tie line. Thus, NR is the change that occurs in the status of tie and sectionalized switches to reconnect distribution feeders to form a new radial structure for a certain operation goal without violating the condition of having a radial structure. In this study, the procedure of NR to generate possible radial configurations in a fast and efficient manner is implemented analytically and is clarified as follows: Step 1: A binary vector X (0) rand = [1 0 0 1 1 . . . 1] 1×N br is initialized with random binary values, in which its length is equal to the number of lines (N br ) with its sectionalized and tie switches. The sectionalized switches are denoted as "1" and the tie switches are denoted as "0".
Step 2: The best reconfiguration vector of the system (X rec best ), which represents the best vector that meets the radiality requirements (described in Step 6) and achieves the desired goal, is initialized with the base configuration of the system.
Step 3: A temporary vector X (0) temp that is equal to X rec best is created. At that point, each element in X (0) temp is compared with the corresponding element in X temp (b) = 1, it means that this bth line is changed to a tie line in the random vector; also if D  Step 4: Starting from the first element in D where j denotes a random line selected from the remaining lines in the system with the condition that b j, a vector check is then checked for radiality described in Step 6. If it is found to be radial, then b is updated so that b = b + 1, and the vector X (1) temp is generated equal to X rec (1) best . It should be mentioned that a set of X (0) check vectors may be generated when b is smaller than or equal to N br , and the vectors found to be radial in this set are evaluated based on their fitness value to offer the best X rec best .
Step 5: The steps terminate when we achieve a very small distance among serial solutions by evaluation of the objective function.
Step 6: The procedure of radiality check is done as follows: • Build an incidence matrix M where its rows and columns represent the lines and nodes of the distribution network, respectively. The nodes of each line are denoted as "1" in M, and the rest of the elements in the row are denoted as "0". • Elements in the rows of each tie line are set to "0". Then, we create a vector S, in which its length is equal to the number of nodes, and each element e in S is equal to the sum of its corresponding e th column in M. If an element in S is equal to "1", it means that this element represents an end node. Further, the row that corresponds to this end node in M is set to "0". • Recalculate S and repeat the former process as soon as an element in S is equal to 1. At that point, calculate the sum of all the elements in M. If the sum is equal to zero, this means that the configuration is radial, otherwise, it is not radial.

SOP Modeling
SOPs were first presented in 2011 [38] to provide resilience between distribution feeders. They can be integrated in distribution networks using three topologies, comprising a back-to-back (B2B) voltage source converter (VSC), static series synchronous compensator, and unified power flow controller [39]. In this work, we used a B2B-VSC as the integration topology for SOPs connected to the studied systems Energies 2019, 12, 4172 9 of 31 because of its flexibility and dynamic capability to enhance the power quality. Figure 1 shows an illustration of SOPs' integration into a distribution system. To model an SOP, the main equations to perfect the flow of power in the network under study are expressed as follows [16]: where P i and Q i are the injected active and reactive powers at the i th node, P L i+1 and Q L i+1 are the active and reactive powers of the connected loads onto node i + 1, |V i | is the magnitude of the i th node voltage, and r i,i+1 and x i,i+1 are the feeder resistance and reactance between nodes i and i + 1. SOPs were first presented in 2011 [38] to provide resilience between distribution feeders. They can be integrated in distribution networks using three topologies, comprising a back-to-back (B2B) voltage source converter (VSC), static series synchronous compensator, and unified power flow controller [39]. In this work, we used a B2B-VSC as the integration topology for SOPs connected to the studied systems because of its flexibility and dynamic capability to enhance the power quality. Figure 1 shows an illustration of SOPs' integration into a distribution system. To model an SOP, the main equations to perfect the flow of power in the network under study are expressed as follows [16]: where and are the injected active and reactive powers at the node, and are the active and reactive powers of the connected loads onto node + 1, | | is the magnitude of the node voltage, and , and , are the feeder resistance and reactance between nodes and + 1. Then, the SOP is integrated using its active and reactive powers injected at its terminals as presented in Figure 1, in which the summation of the injected powers at the SOP terminals and the internal power loss of its converters must equal zero [16], as expressed in (4). Thus: The reactive power limits [16] are given in (5) and the SOP capacity limit [16] is shown in (6). Thus: where is the number of feeders, is the SOP's active power injected to the feeder, is the SOP's active power to the feeder, is the active power loss of the converter Then, the SOP is integrated using its active and reactive powers injected at its terminals as presented in Figure 1, in which the summation of the injected powers at the SOP terminals and the internal power loss of its converters must equal zero [16], as expressed in (4). Thus: The reactive power limits [16] are given in (5) and the SOP capacity limit [16] is shown in (6). Thus: where N f is the number of feeders, P SOP ) and the total SOPs active power loss (P SOP−loss ) are formulated in (7) and (8) as follows [33]: where A SOP loss is the loss coefficient of VSCs, which represents leakage in the transferred power to the total power transferred between feeders [33,40,41].
Mathematically, to represent the SOP variables, first, we can consider a lossless SOP, i.e., P SOP− loss I = 0, ∀I ∈ N f ; hence, a SOP can be represented by its injected active and reactive powers . Thus, it can be independently solved using numerical analysis methods such as Newton's method to find the value of the root (P SOP J ) of (10). Therefore, assuming that A SOP loss is known; an SOP can be represented by its injected active and reactive powers P SOP I , Q SOP I , Q SOP J as the lossless SOP case.

DG Modeling
In this study, we used two types of DGs. The first includes generators with unity power factor and the second is DGs with smart inverters [21] with a reactive power compensation capability within specified limits of the reactive power.
The DGs with unity PF are limited by the maximum capacity limit S DG of the installed DGs as follows: where P DG i is the active DG power injected at the i th node. In the second type of DG, the reactive power varies based on specified PF limits, so that −β min and β min are the minimum leading and lagging PF values.
Energies 2019, 12, 4172 where Q DG i is the reactive DG power injected at the i th node.

PQ Indices
In power distribution systems, apart from the functions that describe the objective and constraints that assess the operational performance, there are other indices that evaluate the impacts of the proposed solution on the PQ performance of the studied systems, such as load balancing index (LBI), and aggregate voltage deviation index (AVDI). The mathematical expressions for these quantities are given as follows: Changing the state of the switches of a distribution system will change its topography. In turn, the loads between the feeders can be distributed to balance the system and avoid the overloading of feeders. In this work, the balancing index (LBI) is used to reflect the loading level of each line in the distribution network [16]. The LBI of the b th line is formulated as follows: where I b is the current flowing in line b and is limited by its rated value I rated b and N br is the number of lines. Hence, the total load balancing index LBI tot is expressed as the sum of the balancing indices of the lines, thus: LBI of a certain line decreases if the total load connected to this line decreases, and hence, the line current decreases. However, line currents may increase in other lines, increasing their LBIs. For that, the LBI tot is calculated for all branches to help determine the overall load balancing of all lines in the distribution network.

Aggregate Voltage Deviation Index (AVDI)
Voltage deviation is a measure of the voltage quality in the system. It is formulated as the summation of voltage deviations at all nodes in the system from a reference value of 1 per unit, and is given as: where i and N n are the node number and total number of nodes, respectively. A system with lower AVDI indicates a secure system with reduced voltage violations.

Objective Function
The main aim of this work is to minimize total power loss (P tot loss ). The objective function P tot loss is divided into two parts, namely the feeder losses due to current flowing in the lines and the SOP's internal power loss (P SOP−loss ) as expressed in (17). where µ = 0 with no SOP losses considered and µ = 1 if SOP losses are considered.

Constraints and Operation Conditions
In addition to the radiality requirements described in Section 2. A, power flow equality given in (4), SOP reactive power limits given in (5), SOP capacity limit given in (6), SOP active power loss given in (8), DG capacity limit given in (11) for the first type and (12) for the second type, and DG reactive power limits given in (13), the following constraints regarding voltage magnitudes, lines thermal capacities and the total reactive power injected by DGs and/or SOPs into the system are expressed, respectively, as follows: where V min and V max represent minimum and maximum voltage limits, respectively, and N DG is the number of connected DGs. It should be noted that the total reactive power injected by DGs and SOPs must not exceed the total demand reactive power, as expressed in (20), to avoid the system's overcompensation, and to maintain the PF to be within higher lagging values [42,43]. In addition, no reverse power flow is permitted in the system, as expressed in (21). Otherwise, further precautions should be taken by network operators to control excessive reverse power flows and the associated problems resulting from high DG penetration levels.
where a equals 1 in the case of node i connected to a DG unit, b equals 1 in the case of node i connected to a SOP through feeder I, and c equals 1 in the case of node i connected to a SOP through feeder J; otherwise, a = b = c = 0.

Search Algorithm
The hyper-spherical search (HSS) algorithm was developed by Karami et al. in 2014 [36] to solve nonlinear functions and was further enhanced in 2016 [37] to consider mixed continuous-discrete decision variables to solve MINLP problems. The DC-HSS has the advantages of fast convergence for optimal/near-optimal solutions and good performance in solving mixed continuous-discrete problems. Therefore, we have used the DC-HSS algorithm to solve our optimization problem.

Continuous HSS
The population is categorized into two types: particles and sphere-centers (SCs). The algorithm searches the inner space of the hyper-sphere to find a new particle position with a better value of objective function as follows: Step 1: Initialization: the algorithm starts by assigning the population size N pop , the distance between the particle, and the sphere-center (r), taking into account random values between [r min , r max ], the number of sphere-centers (N SC ), the number of decision variables (N), the probability of changing the particle's angle Pr angle , and the maximum number of iterations (Max iter ). Then, a vector of decision variables (x i ) is initialized with random values between [X imin , X imax ] by a uniform probability function. A set equal to N pop containing the objective function values is formed for each vector, in which each vector of the decision variables [x 1 , x 2 , . . . , x N ] is named as a particle. Further, the particles are sorted according to their objective function values, and then the best N SC particles with the lowest objective function are selected as the initial sphere-centers. The rest of the particles (N pop − N SC ) are then distributed among the sphere-centers. Finally, a distribution of the (N pop − N SC ) particles among the SCs is performed by the objective function difference (OFD) for each SC, where the OFD is equal to the objective function of SC ( f SC ) subtracted from the maximum objective value of SCs (OFD = f SC − max SCs f ).
The normalized dominance for each SC is defined as: A randomly chosen round D SC × N pop − N SC number of particles is assigned to each SC.
Step 2: Searching: each particle seeks to find a better solution by searching the bounding sphere whose center is the assigned SC. The radius of this sphere is r. The particle parameters (r and θ) are changed to perform the searching procedure. The angle of the particle is changed by ∝, which ranges between (0, 2π) with a probability equal to Pr angle . For each particle, r is changed between [r min , r max ], where r max can be calculated from (23): After the search for particles, if a new particle position has a lower objective function value than that of its SC, both the SC and particle will exchange their roles, i.e., the particle becomes the new SC and the old SC becomes the new particle.
Step 3: Dummy particles recovery: An SC with its particles forms a set of particles. The values of the set objective function (SOF) for each set of particles sort these sets to find the worst sets, in which dummy (inactive) particles are located. The SOF is given by (24).
where γ is scalar. If γ is small, SOF will be biased towards f SC , otherwise, SOF will be biased towards f particles o f SC .
To assign dummy particles to other SCs, two parameters are calculated: the first parameter represents the difference of SOF (DSOF) for each set and the second one represents the assigning probability (AP) for each SC. These parameters are expressed as follows: Further, a preset number of particles N newpar with the worst function values are exchanged with the new generated N newpar particles. Hence, after several iterations, the particles and their SCs become close.
Step 4: Termination: the termination criterion is fulfilled if the number of iterations reaches its Max iter or the difference between the function values of the best SCs is smaller than a pre-set tolerance value.

Discrete HSS
Like the continuous HSS, the discrete HSS starts with the initialization of particles, but with discrete variables. Solutions are then generated randomly from the discrete variables (X id,min , X id,min + 1, . . . , X id,max − 1, X id,max ) with a uniform probability. N SC particles with the lowest function values are assigned as SCs. The rest of the particles are distributed among the SCs. Then, the same searching procedure as the continuous HSS is performed. It should be mentioned that the angle ∝ is not considered in the searching procedure of the discrete HSS and the only parameter used is the radius r d , where r d is selected between (r d,min , r d,min + 1, . . . , r d,max − 1, r d,max ). r d,max is calculated as follows: The other steps will be performed as presented in the continuous HSS algorithm.

Discrete-continuous HSS (DC-HSS)
DC-HSS combines both continuous and discrete HSS algorithms, in which the particles contain both continuous and discrete variables. The procedure for the continuous variables is structured as presented in the continuous HSS formulation, whilst the procedure for the discrete variables is structured as presented in the discrete HSS formulation. To sum up, the optimization parameters of DC-HSS are as follows: N pop =1000, N SC = 100, r min = 0, r max = 1, r d,min = 0, r d,max = 1, N newpar = 5, Pr angle = 75%, and Max iter = 1000. Figure 2 illustrates a comprehensive flowchart for the proposed problem formulation using the DC-HSS algorithm.

Results and Discussion
In this section, the results obtained in the nine scenarios are presented for the IEEE 33-node and 83-node systems under different loading conditions. Further, the contribution of SOP loss to the total active power loss as well as the effect of increasing the number of SOPs connected to the systems are studied. Case studies are carried out on an Intel Core i7 CPU, second generation, at 2.2 GHz and 3 GHz maximum turbo boost speed, with 6 GB of RAM with speed 1333 MHz, 6 MB cache memory and contains SSD hard disk at 550 MB per second.

IEEE 33-Node Distribution System
The IEEE 33-node base configuration consists of 32 sectionalized lines and 5 tie-lines as shown in Figure 3. The number of SOPs that can be installed ranges from 1 to 5, i.e., N SOP ∈ [1,5], where the individual SOP rating (S SOP I = S SOP J ) is 1 MVA and A SOP loss equals 0.02 [33,40,41]. N DG is set to 3, while S DG equals 1 MVA with unity PF. V min and V max values are 0.95 and 1.05 p.u., respectively. Also, I rated b is set to 300 A.   First, the results obtained for the system in the first three scenarios with no SOPs installed are given in Table 2. On the one hand, the results clarify that optimizing the NR and DGs allocation strategies separately cannot satisfy the voltage requirements in either the normal or heavy loading conditions, and only a sub-optimal performance can be achieved in the light loading case. On the other hand, simultaneous NR and DGs allocation can meet the problem limits in light and normal loading conditions only. Hence, one can conclude that the first three scenarios cannot guarantee acceptable performance level of the IEEE 33-node system with loads alteration.
Second, the results obtained for Scenarios 4 to 9 with lossless SOPs installed in the system are presented in Table 3 under the three loading conditions. On the one hand, the results obtained with one SOP installed in the system with or without NR in the case of no DGs connected exhibit poor performance, which can be explained by the lack of an acceptable solution to the problem because of minimum voltage value violation under both the normal and heavy loading conditions, as shown in Scenarios 4 and 5. Therefore, to meet the minimum voltage requirement, the reactive power should be compensated by installing additional SOPs, as presented in Scenario 6, with 3 to 5 SOPs when NR was considered. On the other hand, the results obtained when DGs were connected into the system without NR (Scenario 7) decreased the need for an increasing number of installed SOPs. Further, when NR is enabled, an additional reduction of the number of SOPs is noticed, which will result in reducing the power losses, as revealed by the proposed Scenario 9 because it allows freedom in locating SOPs.
To sum up, the results obtained for Scenario 9 (simultaneous NR with DGs and SOPs allocation) resulted in the best solutions, highlighted in bold in Table 3, with 5 SOPs at the normal and heavy loading levels and 4 SOPs at the light loading level compared to the results obtained by the other scenarios, in which the power losses are reduced by 74.787% at normal, 77.362% at light, and 78.788% at heavy loading levels with respect to the corresponding base system values. Also, the improvement of the voltage profile obtained in Scenario 9 for the system at the normal loading condition is shown in Figure 4.  Table 3, with 5 SOPs at the normal and heavy loading levels and 4 SOPs at the light loading level compared to the results obtained by the other scenarios, in which the power losses are reduced by 74.787% at normal, 77.362% at light, and 78.788% at heavy loading levels with respect to the corresponding base system values. Also, the improvement of the voltage profile obtained in Scenario 9 for the system at the normal loading condition is shown in Figure 4. Thirdly, the results obtained for Scenarios 4 to 9 with the SOPs' internal power losses considered are presented in Table 4 at the three loading levels.  Thirdly, the results obtained for Scenarios 4 to 9 with the SOPs' internal power losses considered are presented in Table 4 at the three loading levels. Regardless of economic aspects, in the lossless SOP scenarios, the system with an increased number of installed SOPs becomes more efficient because of the considerable power loss reduction. However, this is not the case if the SOPs' internal losses are considered, because power loss minimization is considerably affected by the SOPs internal losses. This makes clear that loss minimization is not guaranteed by installing more SOPs. In addition, one cannot simply suppose that increasing the number of installed SOPs will increase the SOPs' internal losses proportionally, as this depends on the power transferred by the SOPs and also on the SOPs' locations, as clarified in Figure 5, with results obtained in Scenario 9 that make clear that choosing an appropriate number of SOPs is a matter of optimization. Moreover, after considering the internal power losses of the SOPs, it is obvious that the results obtained for Scenario 9 are the best results obtained so far compared to the results obtained for the other scenarios, in which the power losses are reduced by 67.374% using two SOPs at normal, 64.374% using two SOPs at light, and 67.184% using five SOPs at heavy loading levels. All values are given with respect to the corresponding base system values. Furthermore, all the considered PQ indices are enhanced using the same scenario by different values as presented in Table 4, which validates the effectiveness of the proposed solution. The improvement of the voltage profile obtained in Scenario 9 for the system at the normal loading condition with the SOPs' power loss considered is shown in Figure 6. A detailed summary of the optimal results obtained for scenarios 4 to 9 at the normal loading condition is given in Tables A1 and A2 in the Appendix A. Also, the IEEE 33-node system after applying Scenario 9 in a normal loading condition is shown in Figure A1 in Appendix A. Finally, optimizing the NR, DGs, and SOPs allocation strategies collectively facilitates collaboration between strategies, which will help achieve the best performance level of the system. the IEEE 33-node system after applying Scenario 9 in a normal loading condition is shown in Figure  A1 in Appendix A. Finally, optimizing the NR, DGs, and SOPs allocation strategies collectively facilitates collaboration between strategies, which will help achieve the best performance level of the system.

83-node Distribution System
In order to validate the effectiveness of Scenario 9 proposed in this work, it was examined on an 83-node balanced distribution system from a power company in Taiwan, in which the 83-node base

83-node Distribution System
In order to validate the effectiveness of Scenario 9 proposed in this work, it was examined on an 83-node balanced distribution system from a power company in Taiwan, in which the 83-node base configuration consisted of 83 sectionalized lines and 13 tie-lines, as shown in Figure 7. The number of SOPs that can be installed ranges from 1 to 5, i.e., N SOP ∈ [1,5], where the individual SOP rating (S SOP I = S SOP J ) is 1.5 MVA and A SOP loss equals 0.02 [33,40,41]. N DG is set to 8 with S DG equal to 3 MVA and PF ranges from 0.95 lagging to unity. The V min and V max values are 0.95 and 1.05 p.u., respectively. Also, I rated b is set to 310 A.  First, the results obtained for the system in the first three scenarios with no SOPs installed in the system are given in Table 5. Once more, the results make it clear that optimizing the NR and DGs allocation strategies separately cannot satisfy the voltage requirements at the heavy loading level, and only a sub-optimal performance can be achieved at the light and normal loading levels. However, First, the results obtained for the system in the first three scenarios with no SOPs installed in the system are given in Table 5. Once more, the results make it clear that optimizing the NR and DGs allocation strategies separately cannot satisfy the voltage requirements at the heavy loading level, and only a sub-optimal performance can be achieved at the light and normal loading levels. However, simultaneous NR and DGs allocation can meet the problem limits considered in the normal and light loading conditions only. Second, the results obtained for Scenarios 4 to 9 with/without SOPs internal losses in the system are presented in Tables 6 and 7 at the three loading levels. From Tables 6 and 7, it can be observed that installing SOPs without NR optimization and DGs allocation (Scenario 4) failed to operate the system within the specified limits, even after increasing the number of SOPs. On the one hand, for the lossless SOPs cases, Scenario 7 succeeded in finding acceptable solutions for the problem, contrary to Scenarios 4, 5, 6, and 8, all of which failed to find an acceptable solution, even with an increased number of SOPs. On the other hand, taking SOPs' losses into account, Scenarios 4 to 8 were not capable of finding an acceptable solution for the problem at a heavy loading level. Still, Scenario 9 remains the most successful scenario as it has the ability to improve the system performance and keep it within the specified limits. The improvement of the voltage profile obtained in Scenario 9 for the system at the normal loading condition with SOPs power loss considered is shown in Figure 8. The contribution of SOPs' losses to the total power losses with different numbers of SOPs is clarified in Figure 9, where the contour plots agree with the conclusions drawn in the IEEE 33-node case study. A detailed summary of the optimal results obtained in Scenarios 5 to 9 at the normal loading condition is given in Tables A3 and A4 in the Appendix A. Also, an 83-node system is shown in Figure A2 in Appendix A after applying Scenario 9 at the normal loading condition. Considering the main point, we conclude that the combination of NR, SOPs, and DGs allocation strategies led to the best solution with minimum losses and noticeably enhanced PQ indices, rather than the sub-optimal solutions provided by individual strategies, particularly at the different loading levels. In addition, a comparison of the results obtained using the proposed algorithm and the results obtained using three conventional optimization algorithms presented in previous works [7]-genetic algorithm (GA), harmony search algorithm (HSA) and modified honeybee mating (MHM)-is conducted to show the effectiveness of the DC-HSS algorithm. The proposed NR methodology is used in these optimization algorithms to find the optimal/near-optimal solutions of the NR problem for both the IEEE 33-node and 83-node distribution systems, as presented in Tables 8 and 9, respectively. It can be noted that the optimal/ near-optimal (best) result is obtained using the other conventional algorithms due to usage of the proposed NR methodology but with a lower computational time to find the best value compared to the other three algorithms, which validate the   In addition, a comparison of the results obtained using the proposed algorithm and the results obtained using three conventional optimization algorithms presented in previous works [7]-genetic algorithm (GA), harmony search algorithm (HSA) and modified honeybee mating (MHM)-is conducted to show the effectiveness of the DC-HSS algorithm. The proposed NR methodology is used in these optimization algorithms to find the optimal/near-optimal solutions of the NR problem for both the IEEE 33-node and 83-node distribution systems, as presented in Tables 8 and 9, respectively. It can be noted that the optimal/ near-optimal (best) result is obtained using the other conventional algorithms due to usage of the proposed NR methodology but with a lower computational time to find the best value compared to the other three algorithms, which validate the In addition, a comparison of the results obtained using the proposed algorithm and the results obtained using three conventional optimization algorithms presented in previous works [7]-genetic algorithm (GA), harmony search algorithm (HSA) and modified honeybee mating (MHM)-is conducted to show the effectiveness of the DC-HSS algorithm. The proposed NR methodology is used in these optimization algorithms to find the optimal/near-optimal solutions of the NR problem for both the IEEE 33-node and 83-node distribution systems, as presented in Tables 8 and 9, respectively. It can be noted that the optimal/near-optimal (best) result is obtained using the other conventional algorithms due to usage of the proposed NR methodology but with a lower computational time to find the best value compared to the other three algorithms, which validate the effectiveness of the proposed NR methodology, regardless of the optimization technique used. Finally, the minimum power losses obtained by applying Scenario 9 for both the IEEE 33-node and 83-node systems are presented in Table 10, compared to the power loss reported in previous works.

Conclusions
This article presents a multi-scenario analysis of optimal reconfiguration and DGs allocation in distribution networks with SOPs. The DC-HSS algorithm was used to solve the MINLP of SOPs and DGs allocation along with NR at different loading conditions to minimize the total power loss in balanced distribution systems. A new NR methodology is proposed to obtain the possible radial configurations from random configurations to minimize the power loss in two distribution systems: the IEEE 33-node and an 83-node balanced distribution system from a power company in Taiwan. Nine scenarios were investigated to find the best solution that provides the lowest power loss while improving the system performance and enhancing the PQ measures. The contribution of SOP losses to total active losses, as well as the effect of increasing the number of SOPs connected to the system, are investigated at different loading conditions to determine the real benefits gained from their allocation. It was clear from the results obtained for Scenario 9 that simultaneous NR, SOP, and DG allocation into a distribution system creates a hybrid configuration that merges the benefits offered by radial distribution systems and mitigates drawbacks related to losses, PQ, and voltage violations, while offering far more efficient and optimal network operation. Also, it was found that the contribution of the internal loss of SOPs to the total loss for different numbers of installed SOPs is not dependent on the number of SOPs and that loss minimization is not always guaranteed by installing more SOPs or DGs along with NR. Finally, SOPs can efficiently address issues related to voltage violations, HC, and network losses to assist the integration of DGs into distribution systems.
From the analysis conducted to identify opportunities and strategies for reducing network losses by improving system design and deploying loss-reduction technologies, it is concluded that integrating both DGs and SOPs along with NR simultaneously successfully increased the integration of DGs rather than other scenarios. One of the interesting findings of the manuscript was demonstrating that NR with optimizing tie-lines could reduce active losses considerably. The modeling also demonstrated that SOPs, installed for the management of constraints in LV feeders, could potentially further reduce losses in modern distribution systems. Further studies will be conducted to integrate that strategy for increasing HC of the distribution systems to accommodate more DGs in balanced and unbalanced distribution systems. It should be noted that a linear power flow formulation can be considered to relax the optimization problem and decrease the computational burden.
Another factor that was beyond the framework of the study, and will be included in future studies, is the cost-benefit analysis using a large-scale multi-objective MINLP model of cost and benefits gained by optimal siting and sizing of SOPs and DGs in the engineering practice for large-scale balanced distribution systems. Further, a probabilistic approach is currently being conducted to discuss the effectiveness of the proposed deterministic approach, while considering seasonality and uncertainty in DGs and demand.

Conflicts of Interest:
The authors declare no conflict of interest. Line resistance and reactance between nodes i and i + 1 r, θ Distance and angle between the particle and the sphere-center r min , r max Minimum and maximum radius of the sphere-center for continuous HSS r d,min , r d,max

Abbreviations
Minimum and maximum radius of the sphere-center for discrete HSS S SOP I Maximum capacity limit of the planned SOP