Optimized Unsymmetrical Per-Phase Droop for Soft Line Switching of Reconfigurable Unbalanced Inverter-Based Islanded Microgrid

The legacy of power distribution systems is evolving towards more flexible and dynamically reconfigurable microgrids, which substantially increases line-switching actions. This may introduce undesirable transients, poor reliability, deteriorated power quality, and significant wear and tear in switching devices. Frequent line switching is significant in unbalanced inverter-based islanded microgrids (UIBIM). This article proposes an optimized unsymmetrical per-phase droop-controlled approach to mitigate the influence of line switching during UIBIM reconfiguration and partitioning. To determine the unsymmetrical per-phase Optimal Transitional Droop Settings (OTDS), a new mathematical formulation is modeled to minimize power flow at the switching location(s). Given the unbalanced nature of the system, the proposed unsymmetrical droop will be optimized for each phase independently. The activation start and end instants of the OTDS are selected to reduce the UIBIM dynamics due to the transition between the states due to switching. The superiority and effectiveness of applying the proposed unsymmetrical OTDS are assessed via Matlab/Simulink, utilizing case studies implemented on a 6-bus and an IEEE 34-bus unbalanced systems. The simulation results reveal that the proposed approach can independently minimize the current flow in each phase during the switching process by nearly 98%. Furthermore, the transient voltage during the switching process is significantly reduced.

The time period of activating and deactivating the OTDS, respectively.

I. INTRODUCTION
T HE high penetration levels of a wide variety of distributed energy resources (DERs) into existing power distribution networks create a path toward having more microgrids, which could be formed at different voltage levels, i.e., medium and low voltages (MV/LV), and switch back and forth between different modes of operation, e.g., grid-connected and islanded [1].The formation of these microgrids is expected to bring several benefits to the grid operators and customers, such as improvements to system reliability and power quality, a reduction of power losses, and the deferral of grid infrastructure upgrades.Yet, in order to maximize the outcomes of these benefits, the formation of these microgrids needs to be associated with a high degree of system flexibility.The realization of such flexibility could be achieved by different means.Among these, dynamic switching, reconfiguration, and self-healing of these microgrids, i.e., via frequent and automated opening and closing of the system switches over the branches [2], are the most prominent.
Several studies have investigated periodic system reconfiguration in seasonal, weekly, daily, or even hourly schedules for active distribution networks with DERs [3], [4], [5].However, these studies have not considered the formation of microgrids and their ability to operate in islanded mode.Further, several other research works have proposed the dynamic clustering of power distribution networks into microgrids that are able to switch back and forth between grid-connected and islanded modes of operation [6], [7].Yet, the main focus of these works is on the determination of the microgrid boundaries to achieve certain system objectives such as loss minimization, self-adequacy, etc., and thus aspects related to system control and line switching during the transition between different states and/or configurations have not been considered.
Most microgrids are dominated by inverter-based DERs.In inverter-based islanded microgrids (IBIM), these DERs control the microgrid (frequency and voltage) and share the load, usually using the droop-controlled scheme [8].The literature has widely reported that conventional droop control suffers from several deficiencies, including a trade-off between power sharing accuracy and voltage deviation, unbalanced harmonic current sharing, and a high dependency on the line impedance [9].As such, tremendous research efforts have been carried out over the previous two decades to tackle these challenges via improving and/or optimizing the power-sharing among the DERs.For example, reference [10] constructed the droop equation in a nonlinear form to minimize fuel consumption and reactive power-sharing error.The work in [11] proposed coupling compensation and virtual impedance as a vital component of the control scheme to address the lack of the conventional droop of decoupling ability.The works in [12] identified the droop control settings using a multi-objective optimization approach to enhance the loadability of the IBIM during the reconfiguration process in the steady-state scenarios.Yet the authors did not study the transient issues during the switching action.The authors in [13] stained the light on the importance of updating the droop setting after the reconfiguration process in steady-state scenarios to avoid instability issues occurring after changing the IBIM configuration.It is noticed that the works proposed in this part for identifying the droop setting focus on the steady-state scenarios for enhancing the loadability and minimizing the system's power loss while ignoring the system's transient response during the switching actions.Moreover, the vast majority of previous techniques in the control and optimization of IBIMs were developed and/or tested on balanced power distribution systems.In contrast, in practice, MV/LV power distribution networks are characterized as unbalanced systems, including several single-phase and unbalanced three-phase loads.
In this regard, recent papers have been published to tackle the control challenges of droop-based unbalanced IBIM (UIBIM).For example, power-sharing issues in UIBIMs have been addressed in [14], where an online virtual impedance adjustment strategy was conducted to enhance the imbalance of active and reactive power and unbalanced harmonic power sharing.A modified voltage-current droop control was combined with the model predictive control (MPC) technique to underrate the voltage unbalance and system active power overload [15].A mixed-integer linear programming (MILP) model was formulated for unbalanced three-phase droop-based IMG for minimizing the generation cost, and the unsupplied demands [16].The authors in [17] integrated the negative sequence virtual impedance to tackle the conventional droop control drawbacks of inaccurate power sharing among the distributed generators (DGs) due to the feeder/line impedance and the unbalanced loads in UIBIM.Triple-droop control (three inversely symmetrical per-phase droops) was proposed in [18] for managing the real and harmonic power sharing in the LV microgrid under unbalanced and non-linear loads.The work in [19] suggested a symmetrical per-phase power controller for regulating the power-sharing in the grid-connected mode and achieving a smooth transition from grid-tied to islanding operation.It is noted that the above-mentioned control and optimization techniques for both balanced and unbalanced IBIM are specifically developed for normal states of operation (fixed topology) and, thus, their effectiveness hasn't been tested during the transition of IBIM from one state/topology to another via line switching for the purpose of reconfiguring and partitioning the system.
In recent work, the authors in [20] emphasized that careful consideration should be given to the problem of line switching during the transition of low-inertia IBIMs between different states.In particular, frequent line-switching at high current flow in the switched lines (hard switching) causes severe negative effects on IBIMs, including but not limited to: i) poor power quality, shortening of some loads' lifetimes, and harming sensitive electronic loads due to voltage transients (sudden voltage rise or dip) brought on by line-switching processes; ii) triggering the activation of inverter trip signals, which may result in unnecessary system outages, thus, endangering the system's reliability; and iii) wear and tear costs of the line switching devices.Accordingly, one of the promising solutions that could potentially alleviate the prejudicial switching impacts is to minimize the power flow in the lines at which the switching action occurs.With this solution, line switching at low power flows leads to soft switching.Minimizing the power flow in specific lines to achieve soft switching requires changing the power-sharing among the dispatchable DERs.As such, the work in [20] proposed to reduce the current flow over the line at which switching occurs through the deployment of optimal transitional droop settings (OTDS) at the instant of switching.Yet, the application of such an optimization technique is tailored for balanced IBIMs.A simple extension of this technique to unbalanced systems is unlikely, given that UIBIMs might face serious operation challenges when symmetrical three-phase droop control settings are applied.Further, the authors in [20] emphasized only on those cases in which a transient overvoltage occurs due to the imbalance of power-sharing during the line switching, and they paid no attention to the occurrence of a transient undervoltage and the applicability of their proposed control scheme in its mitigation.Indeed, both transient overvoltage and undervoltage could be noticed during the process of line switching and the transition of IBIMs among different states.For this reason, OTDS and control approach must ensure soft switching considering both over-and under-voltage transient occurrences.Most notably, the work in [20] recommends applying the OTDS at the instant of line switching.In this regard, the immediate application of these settings will create a step drop for the power/current flow, which could cause a sudden disturbance, controverting the objective of providing seamless switching.
Aiming to fill in the above-identified research gaps, this paper's contribution is to address the issues associated with the process of line switching in UIBIM-based droop control during reconfiguration and partitioning.To that end, the article proposes the implementation of an optimized per-phase unsymmetrical droop control to force the power-sharing among the DERs in UIBIM systems during the reconfiguration and partitioning processes.Hence, a novel mathematical formulation is introduced to determine the per-phase OTDS needed for the provision of soft line switching while reconfiguring UIBIM.The proposed optimization model is formulated in a generic form, where it is applicable for both balanced and unbalanced IBIM, and it considers any switching action, e.g., opening or closing.In order to achieve a soft transition, the per-phase unsymmetrical OTDS is activated gradually before the instant of line switching.The main contribution of this article is introducing an optimized unsymmetrical per-phase droop-controlled approach in order to minimize the power flow at switching location(s) and, thus, achieve soft line-switching during the reconfiguration and partitioning processes in UIBIMs.To this end, a general mathematical model has been formulated to determine the unsymmetrical per-phase OTDS with consideration of single line-closing, single line-opening, and multiple-line-switching events.

II. LINE SWITCHING TRANSIENT IN UIBIM WITH SYMMETRICAL DROOP SETTINGS
Frequent line switching at high currents may cause undesirable transients and poor power quality for MV/LV microgrids due to their dynamics.The magnitude and impacts of these transients could be exacerbated in systems such as droop-based UIBIM, given their low inertia and high susceptibility to system disturbances.To investigate the transients associated with line switching events for a UIBIM, a 6-bus test system feeding unbalanced three-phase loads, depicted in Fig. 1, is simulated in Matlab/Simulink.The test system has two DGs and a conventional symmetric droop control, with one pair of active and reactive power control settings for each DG, which is considered for controlling the proportional power sharing among the DGs during the switching events.Two switching events are simulated at line L 56 : event 1 (opening o ) and event 2 (closing cl ).The measurement of the three-phase voltages at bus 6 (B6) and the current flow in line L 56 are recorded and displayed in Figs.As shown in Fig. 2, a high voltage rise is observed at B6 during event o because of the switch opening on a high current flow at line L 56 where DG1 becomes lightly loaded.Such a sudden increase in the voltage magnitude may cause a violation of the IEEE standard 1547.7,which states that the voltage magnitude should not exceed 1.2 pu for a period ranging from 3 to 500 (millisecond) [21].During event cl of Fig. 3, and without an update in the droop settings, DG1 is required to immediately share the microgrid load with DG2 based on the normal system configuration.In such a case, a deep undervoltage is observed at B6, and a high transient current (more than 10 pu) is withdrawn, where DG1 injects a high current to the system during the transition towards the normal configuration in order to meet its specified power sharing.After more than 0.5 sec, as depicted in Fig. 3(b), the current levels emigrate to the regular operation.Both issues of severe over and under voltage observed during line switching events in UIBIM could cause degradation in the equipment's performance.Consequently, it is essential to implement appropriate control schemes for DGs and optimize their droop settings during line switching events to mitigate undesirable voltage changes and facilitate soft line switching for UIBIM.

III. PROPOSED UNSYMMETRICAL PER-PHASE DROOP CONTROL SCHEME
Minimizing the flowing of the active and reactive power in switched lines of UIBIM is challenging as the feeding of the line Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.power may be injected from the same or the reverse direction for each phase and with different values, as illustrated in Fig. 4. Accordingly, using the symmetrical conventional droop control is inefficient for handling such a challenge.Therefore, in this work, an unsymmetrical per-phase droop control scheme is implemented to coordinate the injected power of each phase of the droop-controlled DG units in the low-inertia UIBIM separately.The block diagram of the proposed control scheme, shown in Fig. 4, contains a power-sharing control loop, resistive virtual impedance loop, inner voltage, and current control loops.The detailed functions of these loops are described as follows: In the power control loop, droop control is adopted as it is the most popular power-sharing mechanism among DG units in islanded microgrids.Different droop control formulas are proposed in the literature, which mainly vary based on the system X/R ratio and the output impedance of the DGs [20].In the case of low voltage (LV) microgrids, which feature highly resistive lines (i.e., X/R << 1), (V i − P gi and dθ i − Q gi ) droop control is typically applied, where the voltage/active power droop characteristics are recommended because of the coupling between the V i and P gi [22].In contrast, the (dθ i − P gi and (dθ i − P gi − Q gi and V i − Q gi − P gi ) formulas are reported as the most adequate droop representation for medium voltage (MV) microgrids [23].The (dθ i − P gi and V i − Q gi ) droop characteristics, known as the conventional inductive droop, is widely used assuming a highly inductive microgrid property, i.e., the X/R >> 1 [24].Meanwhile, when X/R ≈ 1, both active and reactive powers are participating in the droop characteristics, i.e., (dθ i − P gi − Q gi and V i − Q gi − P gi ) that is known as a complex droop [20].In this work, the unsymmetrical per-phase droop is adopted in a general form represented by (dθ to control the active and reactive power of each phase and tackle the coupling issue between the active and reactive powers.To that end, a binary variable (η i ) is defined to offer a flexible selection between the different control approaches.When (η i ) is set to zero, the control scheme follows the (dθ i − P gi and V i − Q gi ) droop characteristics (X/R >> 1).While η i is set at one for complex droop (where X/R ≈ 1).In the proposed approach, the voltage magnitude and angle of each phase are controlled using the following mathematical formulations, respectively: where i is a counter that refers to the DG number, ω o , V φ o are the nominal angular frequency in rad/sec and nominal per-phase voltage, respectively.The η φ i is a binary variable to assign the type of the implemented droop at each ith DG and phase φ.This variable is added to model a general control scheme where the (dθ φ i − P φ gi and V φ i − Q φ gi ) is implemented for ith DG, and φ th phase when the η φ i is 0, whereas the complex droop As a result, the type of droop control for each DG and phase can be determined.The per-phase droop coefficients are given by nq φ i , mp φ i for each phase φ and ith DG.The voltage angle θ φ i is computed by integrating the droop output (dθ φ i ) of ( 2).Then the average value of the per-phase angles is computed for calculating the instantaneous reference voltage values via the following relation: where θ φ N is the nominal phase angle of specific values (0, −2/3π, 2/3π).The θ m i refers to the average value of the per-phase angles of the ith DG (θ m i = 1 3 φ θ φ i ).The proposed per-phase droop control topology is equipped with a resistive virtual impedance loop to regulate the voltage.The fundamental idea is to simulate a voltage drop inside the controller such that the DG perceives the primary network [25].This target is accomplished by multiplying the resistive virtual impedance (R vir ) by each phase's output current (I φ o ) and subtracting the result from the computed reference voltage of (3).To filter the oscillatory component of the output current, a low-pass filter (LPF) is added to the resistive virtual impedance, as shown in Fig. 4. Hence, an updated per-phase reference voltage is generated based on the following formula: The inner control loop has voltage and current control loops designed to regulate the inverter voltage.In the structured control scheme of Fig. 4, the proportional-resonant controller (PR) is adopted to track the sinusoidal voltage references of each phase of each DG unit, which is brought out of the power control and resistive virtual impedance loops.The transfer function of the PR control can be expressed as follows: where K pV , K iV are the proportional and resonant gain factors, respectively.The ω c , ω o are the cut-off frequency and resonant frequency, respectively.The output current reference made by the voltage control loop is then applied to the proportional controller in the current control loop to generate the a, b, and c reference voltage for producing the pulse width modulation (PWM) signals of the inverter.The proportional controller transfer function is expressed as follows: In the proposed control scheme, the a-b-c reference frame is used in the implementation process.The selection of this frame yields remarkable advantages as it can be recognized without any frame conversion, making the system easier to execute and potentially faster.Implementing the unsymmetrical per-phase droop control scheme to achieve a soft switching requires identifying the OTDS (nq φ i , mp φ i ) for each ith DG and φ th phase.Therefore, the OTDS identification process is a challenging task as it is needed to determine 2 × N × φ variables, where N refers to the total number of DGs in the system, φ is the total number of phases, factor 2 is used because of calculating two droop parameters, i.e., nq, mp, for each DG.For example, when the system contains 2 droop-controlled DG units, it is required to identify 12 variables for the unsymmetrical-phase droop control settings (2 × 2 × 3).Accordingly, modeling a reliable and general mathematical formula helps discover the optimization problem landscape and detect the optimal OTDS for different line-switching scenarios.

IV. FORMULATION OF THE PROPOSED OPTIMIZATION PROBLEM
The aim of this section is to formulate a general mathematical model to determine the OTDS that provide soft line switching via minimization of the power flow in the switched lines of index ij.To that end, two switching types are defined: single-line and multiple-line switching.Fig. 5 shows a schematic diagram for the single-line opening, single-line closing, and multiple-line switching events.As shown in the figure, the proposed model aims to minimize the power flow in the line(s) at which switching occurs.In the single-line opening event depicted in Fig. 5(a), the identified OTDS target the minimization of the pre-switching apparent power flow in each phase (S φ D o ) to avoid the undesirable voltage rise due to a sudden drop in the DG loading.The value of S φ D o is computed by solving for the power flow in line ij considering the pre-switching network configuration (state #1).For the closing event presented in Fig. 5(b), the aim is to close the switch with a smooth increase in the line power flow.Hence, the aim of this event is to minimize the power flow of S φ D cl during the close event of a specific line ij as if the network configuration is in state #2.In the case of multiple-line switching, the system is reconfigured via a number of line-opening and line-closing operations, where pairs of opening and closing operations are simultaneously executed while changing the network topology.Consequently, soft line switching is realized in this case via minimizing the sum of the apparent power flow in these pairs of lines, as illustrated in Fig. 5(c).A planned application of the OTDS has been conducted.As shown in Fig. 5(d), in each line switching event, the UIBIM moves from its current state #1 to a new state #2.The OTDS is activated at instance t 1 while the UIBIM is in state #1 before the occurrence of the line-switching action at t 2 .The OTDS is then deactivated at instance t 3 after the UIBIM reaches its new state #2.It is worth mentioning that the activation and deactivation periods, represented by dt 1 and dt 2 , respectively, are user-defined parameters.These parameters are defined to avoid a steep drop in the line power flow during the switching event, which, in turn, helps achieve soft line switching.The time periods dt 1 and dt 2 , shown in Fig. 5(d), are chosen to provide an adequate time for the UIBIM to readjust the powersharing among its DGs and reach to steady state before and after the transition of the states due to line switching.Such adequate time is selected to be equal to or more than the settling time, the time required for the response to reach the steady state and stay within the specified tolerance bands around the final value (to settle the power flow in the switched lines to zero or closer to zero before the event instance).The settings of dt 1 and dt 2 are thus determined according to the dynamic response characteristic of the studied microgrid system [26].Accordingly, a sudden drop or increase in the power flow in the opening and closing events is avoided.The following formula is modeled as a general objective function to obtain the OTDS: where ( 7) describes a minimization objective function for the apparent power flow in the lines encountering opening and closing events.The variables mp φ i , nq φ i refer to the per-phase droop settings.The symbols of S φ  (7), are singletons that refer to lines influenced by the line-opening and closing operations, respectively.The lines that belong to the opening event do not belong to the closing event at the same time; hence there is no intersection between the opening and closing sets The optimization problem is subjected to a set of constrained as described below: -Subject to the power flow equality constraints: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where f P φ i (x i , x gi ) and f Q φ i (x i , x gi ) are an active and reactive power balance equations for each ith bus and φ th phase.The (x i , x gi ) define the state variables at each ith bus.The θ ij in ( 8)-( 9) is the admittance angle for line (ij) and the δ i denotes the voltage angle at bus i.For each droop-bus i (i ∈ Ω Droop i ).
The (1), ( 2), ( 4) are implemented in the power flow constraints as interpreted in ( 10)-( 12) to model the per-phase droop-based buses, and the virtual impedance loop based on the presented works in [27], [28].For maintaining the phase-shift balance between the three-phase voltage at each droop-based bus, the relations in ( 13)-( 14) are involved in the mismatch equations and are implemented in the control scheme of Fig. 4 by using the reference voltage generator.The power flow equations and constraints have been solved using the Newton trust region presented approach in [27].
-The optimization problem is subjected to additional inequality constraints to set the upper and lower boundaries of the microgrid operation and control parameters, including the system voltage magnitude and angle, system frequency, active and reactive power limits of DGs, and the active and reactive power droop parameters as listed below: It is noteworthy that the values of the upper and lower boundaries for the UIBIM operation and control parameters defined in ( 15)-( 21)are set to ensure that the obtained OTDSs will maintain the operation of the UIBIM within the desired operation limits.For example, the boundaries of the voltage magnitudes and system frequency, defined in ( 15) and ( 16), respectively, are set based on the applicable standards for voltage and frequency operation limits [29], [30].Also, the boundaries of the droop parameters defined in (20) and ( 21) are set to preserve the stability of the microgrid, as illustrated in [31], [32], [33].
For identifying the optimal droop settings that minimize the apparent power flow in the switched lines of ( 7), successful history-based adaptive differential evolution with linear population size reduction (L-SHADE) algorithm [34] is integrated with the power flow equations.L-SHADE was selected to optimize the OTDS as it proved its reliability and superiority in several real optimization problems.It is ranked the top in one of the most comprehensive competitions of IEEE CEC 2014.Hence, this work applies L-SHADE as a reliable optimizer to minimize the modeled objective function of (7).The following pseudo-code 1 summarizes the main steps while identifying the OTDS based on the L-SHADE algorithm.In the first steps in the pseudo-code 1, the L-SHADE parameters, including; search agents (SA), number of iterations (T max ) and control parameters (MF, CMR, CR, F ), upper and lower boundaries of the unknown variables (OTDS) are defined.In the initialization process, a set of initial solutions (OTDS) are generated randomly with a dimension of (SA × dim), where (dim) is the number of the unknown OTDS (for example, dim is 12 for two DGs).For each initial OTDS vector per each search agent (1 × dim), the described UIBIM power flow equations and constraints in ( 1)-( 4), ( 8)-( 21) have been solved using the newton trust region approach in [27] to determine the power flow in the switched line(s).Then the corresponding value of the objective function in (7) has been calculated per each search agent.Throughout the iteration numbers, the initial OTDS sets have been modified based on the L-SHADE structure to fetch the minimal objective function in (7).Since the power flow equations and constraints in (1)-( 4), ( 8)-( 21) corresponding to the updated solutions (OTDs) have been recalculated using the newton trust region approach; then the objective function is evaluated.After reaching the maximum iterations (T max ), the best solution vector (OTDS * ) that corresponds to the minimum objective function is displayed and then applied to the dynamic model of the UIBIM with inverter-based DGs using the control scheme of Fig. 4.

V. CASE STUDIES AND DISCUSSIONS
This section is divided into three subsections to evaluate the performance of the unsymmetrical per-phase control scheme.Subsection A is concerned with providing comparative studies about power-sharing in normal and sudden disturbance conditions.Subsections B and C assess the proposed control scheme and the considered approach for identifying the OTDS in achieving the soft line switching operation.The 6 bus-2DGs system of Fig. 1 is adopted to appraise the performance of the proposed approach for the single-line switching event, while the IEEE 34bus unbalanced system is embraced for evaluating the proposed approach in cases of multiple-line switching.The time-domain simulations are implemented on Matlab/Simulink, and the optimization problem of identifying the OTDS is solved using a Matlab script.The simulations and analyses are conducted on Algorithm 1: The Pseudo Code of the L-SHADE for OTDS.
Define: SA is the total number of search agents and T max is the total number of iterations, UIBIM specifications, Objective function (7).Define: L-SHADE parameters: archives vector for the mutation factor (MF) and crossover operator (MCR) for storing the average values of the mutation operator (F) and the crossover probability (CR); two sets of SF and SCR to store all CR and F values.Generate: the initial OTDS, OT DS = [mp φ i , nq φ i ] for each SA.Compute: Solve the UIBIM power flow equations and the constraints in the ij line using (1), ( 2), ( 8)-( 21) based on the newton trust region approach in [27] to compute the initial values of the objective function described in (7)  OT DS ]. Generate a trial vector based on the following equation [34]: Update the solution (OTDS) using crossover operator [34]: T +1 if r and > CR k Solve the UIBIM power flow equations and the constraints in the ij line using (1), ( 2), ( 8)-( 21) based on newton trust region approach in [27] using the updated solutions.Evaluate the objective function of (7) using the updated solutions.end for Update the SR, SCR, MF.MCR and SA.end for Return OT DS * a personal computer with a Core i7 (2.5 GHz) processor and 16 GB of RAM.

A. Evaluating the Performance of the Proposed Unsymmetrical Per-Phase Droop Control Under Normal and Abnormal Operating Conditions
This subsection investigates the effectiveness of the proposed unsymmetrical per-phase droop control to enhance powersharing in UIBIM compared to the symmetrical droop control under normal and abnormal operating conditions.The connected DGs in the UIBIM schematic diagram of Fig. 1 supply all the unbalanced three-phase loads until t = 1 sec (normal operating condition).To test the performance of the proposed control under abnormal operating conditions, the loads at phase "c" of all nodes are disconnected until t =1.5 sec [18].Afterward, loads of phase "c" are restored, and the system returns to its normal operating condition.Fig. 6 shows the active and reactive power-sharing of the DGs for the unsymmetrical per-phase droop control compared to the conventional symmetric droop control under normal and abnormal conditions.To quantify the performance of the implemented control schemes in the steadystate operation, the percentage of active and reactive power error between the desired shared power of DG i ((P/Q) φ ref i ) and the measured shared power at the ith DG terminal has been calculated using the following formula: For the time horizon of t ≤ 1 at normal operating conditions, the ideal active power dispatched from phases "a,b,c" of DG i are (0.418, 0.354, 0.337) pu, respectively.When the conventional symmetrical droop is applied, Fig. 6(a), the dispatched active power from phases "a,b,c" of DG 1 and DG 2 are (0.388,0.363,0.378)pu and (0.457,0.359,0.311)pu, respectively.Using (22), the percentages of the active power sharing error are (7.177,2.542, 12.166) % and (9.330, 1.412, 7.715)% for DG 1 and DG 2 , respectively.Meanwhile, the measured dispatched active power from phases "a,b,c" of DG 1 and DG 2 are (0.393,0.358, 0.338) pu and (0.419,0.3536,0.333)pu when the unsymmetrical per-phase control scheme is applied as depicted in Fig. 6(c).Accordingly, the percentages of active-power error are (5.981,1.129,0.297)% and (0.239,0.113,1.187)%,respectively.With regard to the reactive power, the percentages of error from phases "a,b,c" of DG 1 and DG 2 extracted from the results of Fig. 6(b) are (15.609,15.012,11.227)%and (16.932,15.012,12.613)%whereas that of Fig. 6(d) are (5.685,12.581,5.664)%and (0.054, 15.199,2.218)%for the symmetrical and unsymmetrical droop, respectively.From these results, it is observed that the implementation of the unsymmetrical per-phase droop control enhances the unbalanced active and reactive power sharing among the DGs in UIBIMs compared to the conventional symmetrical droop control.
For the time of 1 ≤ t ≤ 1.5 at abnormal operating conditions, the dispatched active and reactive power from phase "c" of DG 1 and DG 2 are almost equal to zero while using the unsymmetrical per-phase droop control.In contrast, the symmetrical droop control performs improperly to attenuate the circulating active and reactive powers.Moreover, the zoomed curves of the power of phases "a, b" clarify that changing the power of phase "c" has a notable impact on other phases when the symmetrical conventional droop control is applied as illustrated in Fig. 6(a).On the other hand, slight changes in the power of phases "a, b" are observed in Fig. 6(c) when the unsymmetrical per-phase droop control is applied, where it nearly treats each phase separately.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Single-Line Switching: UIBIM-6 Buses and 2 DGS Based on Per-Phase Droop Control
In this part, the proposed control scheme is appraised while the line between buses 5 and 6 of Fig. 1 encounters two states of switching: an opening event at a time of 2 sec (viewed as o ) (partitioning process), and a closing event at a time of 4 sec (viewed as cl ).The considered droop control scheme is the (dθ i − P gi − Q gi and V i − Q gi − P gi ) as the X/R ratio nearly equals to 1.The L-SHADE algorithm is adopted to solve the optimization problem formulated in the previous section to determine the OTDS corresponding to minimal flowing power in the switched line for achieving soft switching.The identified OTDS is reported in Table I for the o and cl events.For exhibiting the dynamic response of the UIBIM while applying the OTDS, the DGS active and reactive power, the injected power to the switched line (L 56 ), and the voltage at bus 6 of the system are traced during the o and cl events as described below.Fig. 7 displays the UIBIM's dynamic response for the o event at t = 2 sec.The identified OTDS presented in Table I is applied before the switching action to avoid a sudden drop in the line power flow.For example, it has been observed that applying the OTDS one second before the switching action facilitates seamless readjustment of the power-sharing among the droopcontrolled DGs, which in turn, provides smooth decaying of the power flow in the switched line(s).Hence, the OTDS is activated at time 1 sec.as shown in Fig. 7. Fig. 7(a) and (b) illustrate that with activating the OTDS, the generated per-phase active and reactive power from DG1 is reduced gradually; hence, the per-phase active and reactive power flow in line L 56 is reduced to be less than 98% from the initial one as illustrated from Fig. 7(c).To continuously feed the demand to the MG loads, the per-phase active and reactive power of DG2 is increased as shown in Fig. 7(a), (b) to substitute the lack of the injected power from the DG1 side.In consequence, the power flow in line (L 56 ) is diminished gradually until it reaches nearly zero before opening the line, as depicted in Fig. 7(c), meaning the current flow in the line comes to zero as well.By inspecting the voltage profiles of Fig. 7(d) at time 2 sec, it can be noticed that the improvement in the transient response due to the application of unsymmetrical per-phase OTDS is significant as the switching voltage drops at 0.945 pu compared to 1.272 pu in case of applying conventional symmetrical droop control as illustrated in Fig. 2(a).This is due to the fact that opening the line at near-zero current eliminates the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.voltage rise that was observed previously in the system response of Fig. 2. According to the previous observations, it can be seen that unsymmetrical per-phase OTDS controls the power flow in each phase of the switched line; hence, it helps minimize the unbalanced current flow in each phase separately via recoordinating the per-phase injected power into the switched lines from the connected DGs in the microgrid.Furthermore, Fig. 7(e) shows that the system frequency varies between 1.0008 pu and 0.9978 pu during the activation time of the OTDS.These changes in the system frequency, which is in the range of +/-0.02%, are within the acceptable steady-state system frequency deviation limits of 3% [30].The results demonstrate that the process of activating and deactivating the OTDS does not cause issues in the dynamic response of the UIBIM.
For the cl event at t = 4 sec. of Fig. 8, the primary target of applying the OTDS of Table I is achieving a nearly zero flowing of power during the closing event to avoid high transient currents flow.Hence the OTDS is applied pre-closing by 500 (milliseconds) before the switching action.The OTDS is deactivated after one second from the closing event (at t = 5) after reaching a steady state.Then the operation droop parameters are applied at t = 5 sec to return to a UIBIM based 2 DGs configuration.
The and deactivation periods of the OTDS ensure achieving a smoothing power flow until the getaway from the transient state for reaching smooth steady-state operation.Fig. 8 displays the dynamic response of the UIBIM when line L 56 encounters the closing event at t = 4 sec and OTDS is activated and deactivated at t = 3.5, 5 sec, respectively.The DGs' active and reactive power curves of Figs.8(a),(b) at 3.5 < t < 5 reveal the effectiveness of the OTDS in keeping on the DGs shared power that achieves zero power flow in the line until reactivating the operational system droop at t = 5 sec.In this way, the power flow in the switched line at time 4 sec of Fig. 8(c) is still zero after closing action, which means the OTDS keeps zero current flowing in the line to avoid a high dip in the voltage at bus 6 during the closing event as illustrated in Fig. 8(d).The DGs' active and reactive power after 5 sec of Fig. 8(a) and (b) divulge a smooth power-sharing where the injected per-phase power by the lightly loaded generator (DG1) is increased gradually in front of the DG2 power, which is reduced for re-distributing the shared power among the two DGs.The displayed results illustrate the significant improvements that the unsymmetrical per-phase OTDS provides to the dynamic response of the UIBIM compared with the conventional symmetric droop of Fig. 3.The power flow in the switched line is smoothly increased; hence the transient current in the case of using the unsymmetrical per-phase droop is kept to be nearly zero after closing; meanwhile, it reaches > 10 pu while using the symmetrical conventional droop as illustrated in Fig. 3(b).Furthermore, unlike the symmetrical conventional sag, which suffers from severe voltage dip during the close switching event, the unsymmetrical per-phase OTDS can maintain the voltage within its prescribed limits during the transition between different network states.As per the system frequency in Fig. 8(e).it is clear that the defined operation constraints in the optimization problem result in seamless activation and deactivation of the OTDS without causing dynamic response issues to the UIBIM.As illustrated in the figure, the maximum measured frequency is found to be 1.0015 pu during the deactivation of the OTDS, which is less than the acceptable standard deviation of 3% [30] from the nominal frequency.
Based on the previous discussions, it is observed that using the unsymmetrical per-phase OTDS mitigates the transients associated with line switching, and thus soft opening and closing events are realized in the UIBIM with 6 buses and 2 DGS without causing issues in the dynamic response of the UIBIM.

C. Sequential Multiple-Line Switching: UIBIM-34 IEEE Bus Based on Per-Phase Droop Control
In this section, the proposed control scheme and the OTDS identification approach performance are examined for sequential actions of multiple-line switching by adopting the IEEE 34-bus unbalanced benchmark system of Fig. 9.The IEEE 34-bus contains different distributed and spotted loads.The load type includes constant current, constant impedance, and constant power models (single-and two-phase loads, and wye/delta-connected three-phase loads) [35].Three inverter-based unsymmetrical per-phase droop-controlled DGs are connected to the system at buses 808, 832, and 848 through transformers.The depicted configuration of Fig. 9 illustrates the IEEE 34-bus UIBIM with four switches (SW1, SW2, SW3, and SW4).The initial states of the switches are: SW2 and SW4 are closed, while SW1 and SW3 are open.The IEEE 34-bus UIBIM is reconfigured via Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.two sequential switching events: in sequence 1, the initial states of (SW1 and SW2) are changed with keeping on the initials state of (SW3 and SW4).Then, in the second sequence, the initial states of (SW3 and SW4) are changed with considering the (SW1 and SW2) states as in sequence 1 to obtain the final construction of the reconfigured IEEE 34-bus UIBIM.The order of the multiple lines switching sequences is based on the works in Ref. [20], as the sequence of the switching that achieves a minimum apparent power flow in the switched lines is the recommended one.By following this strategy, the switching action of (SW1 and SW2) is selected as a first sequence, then the switching action of (SW3 and SW4) in the following sequence.
The following subsections present the dynamic response of the IEEE 34-bus UIBIM when implementing sequential multipleline switching as described above: 1) Sequence 1: Changing Initial States of SW1 and SW2: In this sequence, a pair of two lines are encountered to switchexchange events simultaneously for providing the first reconfiguration of the IEEE 34-bus UIBIM.For that purpose, the initial states of SW1 and SW2 are changed so that SW1 is closed and SW2 is opened simultaneously at t = 0.7 sec.The SW3 and SW4 are kept in their initial state.Towards achieving soft switching, the L-SHADE algorithm is adopted to optimize the proposed objective of (7) to determine the OTDS corresponding to minimal power flow in the pair switched lines ).The yielded OTDS is listed in Table II.A short period is considered for activating the OTDS before conducting the switching actions to re-distribute the shared power among the DGs and emphasize the soft switching events.Hence, the OTDS is activated at t = 0.2 sec.The dynamic response of the test system before and after conducting the first sequence of the reconfiguration process is depicted in Fig. 10.The DGs' generated active and reactive power before and after the switching process are measured and displayed in Fig. 10(a) and (b).Furthermore, the power on the switched lines is measured and plotted in Fig. 10(c) and (d).Based on the indicated direction of the power flow in Fig. 9, the line power is positive when it flows in the same direction as the arrowhead, and it is negative when flowing in the opposite direction to the arrowhead.It is worth mentioning that before applying the OTDS (t < 0.2 sec.) of Fig. 10(c), the three DGs were sharing the power flow of L2, and hence the measured powers are in positive and negative signs.By inspecting the L2 active and reactive power values at (t < 0.2 sec), it can be observed that the DG1 injects a significant active power for phases (b,c) to the DG2 and DG3 side.On the other hand, DG1 participates by a bit of the active power of phase (a) for the side of DG2 and DG3.For the reactive power of the three phases (a,b,c), DG2 and DG3 feed the line by remarkable values compared to DG1.Given the complexity of power sharing among the DGs in each phase, minimizing the power flow in this scenario cannot be achieved with symmetrical droop control schemes.Thus, appropriate unsymmetrical phase-independent droop control is paramount.The unsymmetrical per-phase complex droop of (1) and ( 2) of Section III is the implemented controller in this section.
The recorded active and reactive power of the DGs of Fig. 10(a) and (b) and the power flow in L2 of Fig. 10(c) after applying the OTDS before reconfiguration process (0.2 < t < 0.7 sec) reveal the effectiveness of using the unsymmetrical perphase droop control in minimizing the line power by controlling the power of each phase separately.In phase (b), for example, before the switching time, the active power flow in L2 comes from the DG1 side; hence it is visualized in positive sign at t < 0.2 sec, as shown in Fig. 10(c); hence the proposed control scheme and OTDS works on reducing the injected power of phase (b) from DG1 in front of increasing the production from DG2 and DG3 side as shown in Fig. 10  is supplied from DG2 and DG3 sides (the power flows from the opposite side to the measurement direction).Accordingly, applying the OTDS minimizes the generation from the DG2 and DG3 side instead of increasing the DG1 generation as exhibited in Fig. 10(b) to reduce the transmitted power through the line L2.Similarly, for phase (c), the active power flow in the line before switching is injected from DG1 side (it is visualized in positive sign as shown in Fig. 10(a)); hence, the OTDS increases the generation from DG2 and DG3 side to minimize the flow from the DG1.For the reactive power of phase (c), applying unsymmetrical per-phase OTDS shows its superiority in minimizing the injected reactive power from DG2 and DG3 side to reduce the negatively measured reactive power in the line in front of increasing the reactive power contribution from DG1.Following the same concept, the OTDS minimizes the reactive power of phase (a).For the active power of phase (a), the OTDS tries to minimize the active power flow of the phase before the switching action, while phase (a) is highly loaded; hence the DG2 and DG3 contribute a bit for this phase to feed the demand before the reconfiguration process continuously.Applying the OTDS of the unsymmetrical per-phase droop minimizes the apparent power in L2 from 0.3 pu at t < 0.2 (at point X of Fig. 10(c)) to be 0.165 pu at t = 0.7 (at point Y of Fig. 10(c)) before the switching action, accordingly the line power is minimized by nearly 50% before the switching action.Furthermore, applying the OTDS before connecting the tie-line (L1) helps in achieving smooth flow for the power through the line until it reaches its steady state value, then the OTDS is deactivated, as illustrated in Fig. 10(d) (at points X, Y, respectively).In the end, this strategy significantly impacts achieving a soft-reconfiguration event at t = 0.7 sec and improves the voltage transients produced by the switching operation, as shown in Fig. 10(e) and (f).The displayed three-phase voltages at buses 850 and 828 are being in the standard limits during/after the reconfiguration process.Based on these observations, the unsymmetrical perphase droop and OTDS prove their ability to control each phase  separately to achieve minimum power flow during switching events.
2) Sequence 2: Changing Initial States of SW3 and SW4: This subsection examines the effectiveness of the proposed optimized unsymmetrical per-phase droop with the second sequence of the multiple line switching of IEEE 34-bus UIBIM.In this sequence, the initial state of SW3 and SW4 are exchanged simultaneously, considering the new topology of the IEEE 34-bus UIBIM after implementing sequence 1.Then, the L-SHADE is executed to identify the OTDS that minimizes the power in the switched lines (L3, L4) using (7) and the updated power flow of the reconfigured IEEE 34-bus UIBIM based on sequence 1.The obtained OTDS is listed in Table III.The dynamic response of the UIBIM before and after implementing sequence (2) of switching is depicted in Fig. 11 to illustrate the DGs active and reactive power, the power in the line L4 and the voltage at the 834 bus.The power flow in line L3 and voltage at bus 824 are not included in the figure because of the limit in the number of pages.The exhibited Fig. 11(a), (b), and (c) emphasize the superiority of the proposed approach in minimizing the power flow in the switched line of each phase individually hence the apparent power of Fig. 11(c) is minimized from 0.71 pu at point X to 0.48 pu at point Y.Using such approach, the displayed three phase voltages at bus 834 are being in the standard limits during/after the reconfiguration process.Accordingly, a soft reconfiguration process of the IEEE 34-bus UIBIM has been achieved by implementing the unsymmetrical per-phase droop and OTDS.

VI. CONCLUSION
Towards seamless reconfiguration and soft line switching in unbalanced inverter-based islanded microgrids, this article proposes an optimized transitional per-phase unsymmetrical droop control.To that end, a general mathematical formula is modeled for optimizing the transitional per-phase droop settings to minimize the power flow in the switched lines.The formula considers single line-closing, single line-opening, and multipleline-switching events.The effectiveness of the proposed soft line switching approach is proven for minimizing the power flow in the switched lines considering the 6-bus system and the IEEE 34-bus benchmark system with different switch locations.Regarding a single-line opening event, a significant reduction in the switched power of more than 98 % is attained.This reduction improves the switching operation's voltage transients.Regarding a single-line closing event, a gradual increase in the power flow is achieved that controls the current transients produced by the switching.Finally, considering a multiple-line switching event, using the unsymmetrical per-phase droop control and OTDS proves their ability to minimize the unbalanced power Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
flow in each phase separately by re-coordinating the per-phase injected power in the switched lines from the connected DGs in the microgrid.

2 and 3
for o and cl events, respectively.The shown results in Figs. 2 and 3 are displayed per unit with 2 MVA and 4.16 KV base values.

Fig. 1 .
Fig. 1.Unbalanced test system to illustrate the significance of line-switching transients in 6-buses UIBIM.

Fig. 2 .
Fig. 2. The dynamic system response before/after the line-opening process o in UIBIM using conventional symmetric droop control scheme (a) per-phase voltage at bus 6, and (b) per-phase current in L 56 .

Fig. 3 .
Fig. 3.The dynamic system response before/after the line-closing process cl in UIBIM using conventional symmetric droop control scheme (a) per-phase voltage at bus 6, and (b) per-phase current in L 56 .

Fig. 4 .
Fig. 4. Schematic diagram of the proposed per-phase droop control of inverter-based DG.
apparent power in the line (ij) during the opening state (o) and the closing state (cl), respectively.The input parameters B o ij , B cl ij have binary values to identify the network switching event as a single line opened (B o ij = 1, B cl ij = 0 ∀ ij ∈ Ω o ij ) Fig. 5(a), single line closed (B cl ij = 1, B o ij = 0 ∀ ij ∈ Ω cl ij ) Fig. 5(b), or the network follows a multiple-line switching event as shown in Fig. 5(c), where (

Fig. 7 .
Fig. 7.The UIBIM dynamic response during opening ( o ) of the line L 56 (a) active power of DG1, DG2, (b) reactive power of DG1, DG2, (c) active and reactive power at L 56 , (d) voltage at bus 6, and (e) ystem frequency while using per-phase droop controlled approach.

Fig. 8 .
Fig. 8.The UIBIM dynamic response during closing ( cl ) of line L 56 (a) active power of DG1, DG2, (b) reactive power of DG1, DG2, (c) active and reactive power at L 56 , (d) voltage at bus 6, and (d) system frequency while using per-phase droop controlled approach.
(a) to minimize the power flow of phase (b) through the line.The reactive power flow of phase (b) in the L2 is measured in a negative sign, meaning it Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Per-phase magnitude of the apparent power flow in the line ij before opening.
by using the initial OTDS.for T ≤ T max do for k ≤ SA do Assign random MF r and MCR r from the archives vectors of MF and MCR.Calculate F k = Gaussian(MF r , 0.1).Calculate CR k = Gaussian(CR r , 0.1).Assign the best solution vector OT DS* T = [mp φ * i , nq φ * i ].Select two rand OT DS (r1,r2) vectors from the total SA.

TABLE I THE
IDENTIFIED UNSYMMETRICAL PER-PHASE OTDS IN CASE OF UIBIM WITH 6 BUSES AND 2 DGS

TABLE II THE
IDENTIFIED UNSYMMETRICAL PER-PHASE OTDS CONSIDERING SEQUENCE 1 OF THE IEEE 34-BUS UIBIM

TABLE III THE
IDENTIFIED UNSYMMETRICAL PER-PHASE OTDS CONSIDERING SEQUENCE 2 OF THE IEEE 34-BUS UIBIM