An Unbalance and Power Controller Allowing Smooth Islanded Transitions in Three-Phase Microgrids

This article introduces a power controller for three-phase inverters in microgrids that can be used in three-phase three-wire and three-phase four-wire systems. The controller enables active and reactive power tracking and unbalanced current control during grid-tied operation, while also allowing seamless transitions into islanded operation. The proposal is motivated by the compelling need in forthcoming power-electronics-dominated grids to provide ancillary services for the main grid and to support grid-forming functionalities for the microgrid, especially in case of islanded operation. The control is developed on the symmetrical components framework. Power tracking is achieved by dedicated control loops that, exploiting <inline-formula><tex-math notation="LaTeX">$P$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$-$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$f$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$Q$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$-$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$V$</tex-math></inline-formula> droop laws applied on positive-sequence powers, accommodate output-power control during grid-tied operation as well as grid-forming capabilities during islanded operation. The controller also includes synchronous <inline-formula><tex-math notation="LaTeX">$dq$</tex-math></inline-formula>-frame control for negative-sequence current regulation for providing unbalanced current compensation. The proposed solution addresses the challenge of simultaneously providing concurrent grid-tied control features, such as output power tracking, during grid-tied operation as well as grid-forming capabilities during islanded operation. The related challenge stems from the intrinsically different control actions involved in the two modes of operation, namely, grid tied and islanded. The proposal is verified on an experimental setup composed of converters rated 3 kW.

Abstract-This article introduces a power controller for three-phase inverters in microgrids that can be used in three-phase three-wire and three-phase four-wire systems.The controller enables active and reactive power tracking and unbalanced current control during grid-tied operation, while also allowing seamless transitions into islanded operation.The proposal is motivated by the compelling need in forthcoming power-electronics-dominated grids to provide ancillary services for the main grid and to support grid-forming functionalities for the microgrid, especially in case of islanded operation.The control is developed on the symmetrical components framework.Power tracking is achieved by dedicated control loops that, exploiting P −f and Q−V droop laws applied on positive-sequence powers, accommodate output-power control during grid-tied operation as well as grid-forming capabilities during islanded operation.The controller also includes synchronous dq-frame control for negative-sequence current regulation for providing unbalanced current compensation.The proposed solution addresses the challenge of simultaneously providing concurrent grid-tied control features, such as output power tracking, during grid-tied operation as well as grid-forming capabilities during islanded operation.The related challenge stems from the intrinsically different control actions involved in the two modes of operation, namely, grid tied and islanded.The proposal is verified on an experimental setup composed of converters rated 3 kW.Index Terms-Droop control, grid-tied inverter, microgrid, symmetrical sequences, unbalanced voltage.

I. INTRODUCTION
T HE rapidly evolving energy scenario and the related chal- lenges foster the development of power electronic based The authors are with the Department of Management and Engineering, University of Padova, 36100 Vicenza, Italy (e-mail: andrea.lauri@studenti.unipd.it;tommaso.caldognetto@unipd.it;davide.biadene@unipd.it; paolo.mattavelli@unipd.it).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TIE.2023.3347836.
Digital Object Identifier 10.1109/TIE.2023.3347836 power systems [1].Microgrids play a pivotal role in this respect [2] since they represent a paradigm for the effective integration of distributed energy resources, such as renewables and energy storage systems, and the management of loads interfaced to the ac distribution system using electronic power converters (EPCs).Flexible power control is a crucial feature of EPCs in microgrids.It allows us to react to power references issued by controllers for local power flow optimizations [3], to provide demand-response services requested to the microgrid, or to participate in transactive energy markets [4], [5].A second crucial feature is the availability of suitable EPC controllers endowing the microgrid capability of operating islanded from the main grid, which is valuable in several modern circumstances [6], [7].
Last, unbalanced current control should be included considering the control of fundamental quantities at the output of EPCs for power quality enhancements.This control flexibility can be exploited to compensate unbalanced currents measured at the connection with the upstream grid in three-phase systems populated by unbalanced or single-phase loads, which is a common issue in low-voltage grids.The outlined scenario is schematically represented in Fig. 1.
The analysis of the literature reveals that the cohesive integration of these features presents a relevant challenge.The main difficulty stems from the fundamentally different control requirements for grid-tied operation and islanded operation.In grid-tied operation, the goal of reference tracking must be achieved regardless of grid conditions.On the other hand, if disconnected from the mains, inverters must adapt to grid conditions to properly operate in parallel with other units and support the local grid voltage.
Droop control is widely used to implement grid-forming converters and achieve islanded operation capabilities.However, when droop control is applied, output power tracking is not automatically obtained, and controlling unbalanced output currents requires dedicated provisions.While current controllers [8] allow the most flexible control during grid-tied operation, the islanded operation is not supported.An example is provided in [9], in which a control for grid-tied converters is proposed that is capable of injecting unbalanced and harmonic currents, but it does not allow islanded operation.On the other hand, droop control approaches like [10] allow islanded operation but do not support unbalanced compensation.Similar limitations can be found in [11] and [12].In [13], a compensation method is proposed that allows to share the unbalanced power so that the total power on each phase is distributed among converters in proportion to their nominal power.Still, the additional flexibility of directly controlling the converters' contributions based on given references may be required for network optimizations [3], [14].Several works discuss the concept of virtual synchronous machines, for example, [15], [16], [17].In this case too, the capability of supporting phase-by-phase power control, to provide, for example, unbalanced compensation, coupled with the grid-forming function is typically not achieved.In [18] and [19], the virtual-oscillator control technique is modified to enhance the operation in three-phase unbalanced grids.Operation with distinct power references for each phase has not been shown.
Preliminary solutions to couple power control and islanded operation are documented in [20], where a total output power controller is proposed that also allows seamless transitions toward the islanded operation.As a prosecution of this work, output power control performed on the three-phases (i.e., perphase) aiming at unbalance power compensation is introduced in [21] and [22].In these papers, per-phase active and reactive power control in three-phase four-wire systems, the former, and three-wire systems, the latter, are tackled.The approach in these two articles, however, assumes that the unbalance compensation is performed providing unbalanced power, as shown in [23].However, symmetrical-components give a more natural approach to unbalance compensation, as shown in [24], and allow a unique controller implementation for three-phase EPCs with or without the neutral wires.Remarkably, three-wire systems with the neutral wire and also without the neutral wire are both relevant in low-voltage networks [25], and the controller proposed herein is compatible with both configurations.
Approaching the problem through a positive-and negativesequence power controller would be ineffective.While, during normal operation, positive-sequence voltage has a welldefined and bounded amplitude value, this is not true for negative-sequence voltage.The negative-sequence power loopgain would vary with the negative-sequence voltage amplitude, and may reduce to zero in case of perfectly balanced threephase voltages.For this reason, droop control on negativesequence powers cannot be performed as commonly done for positive-sequence powers.The solution proposed here controls positive-sequence powers, for active and reactive output power tracking, and injects negative-sequence currents, to provide unbalanced compensation, by generating a suitable negativesequence voltage component.In the control, the active-power control loop is used to synchronize the generated voltages with the grid, exploiting the feature discussed in [26], and to derive the negative-sequence instantaneous phase used to regulate the injected negative-sequence currents.In summary, the proposed method includes the following key features: 1) active and reactive power tracking; 2) unbalance current regulation; 3) grid-forming capability; 4) parallel operation with multiple grid-forming units; 5) smooth and seamless transition toward islanded operation.The rest of this article is organized as follows.Section II briefly reviews the main power flow relations, used in the following sections, in ac single-and three-phase systems; Section III describes the operating principles of the proposed approach and derives the small-signals loop gains; Section IV illustrates a basic design guideline for the controller parameters; Sections V and VI show, respectively, simulation and experimental results validating the control approach.Finally, Section VII concludes this article.

II. FUNDAMENTAL MODELING OF AC POWER TRANSFER
This section recalls the main power transfer relations between two voltage sources, representing the ac main grid and a voltage-controlled grid-tied inverter, in both single-phase and three-phase systems.It is assumed that the line impedance connecting the sources is mainly inductive; a related remark on this assumption is reported at the end of this section.
For a single-phase inverter connected to the ac main grid by an inductive line, the power flow equations can be written as [27], [28] where V i ∠ϕ i and V g ∠0 are, respectively, the inverter and the grid-voltage phasors and ω g is the grid frequency.Assuming small ϕ i and small ΔV i V i − V g , by linearizing (1) it yields Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where γ p V 2 g ω g L and γ q V g ω g L .For three-phase four-wires connections (i.e., three-phase with neutral line), ( 1), ( 2) hold independently for each of the three phases of the inverter.However, when the neutral line is absent, as displayed in Fig. 2, the additional constraint i a + i b + i c = 0 must be satisfied, thus preventing an independent power control of the three phases, namely, a per-phase power control.In [22], the implications of this constraint on the per-phase power flow are reported.Herein, the focus is on positive-sequence power flow; in this case, ( 1) and ( 2) can be written as These relations are considered in the following sections to describe the proposed controller.
It is worth remarking that the power-flow relations (4), holding with inductive interconnection impedances, allow the use of P −f and Q−V droop laws [28].These laws are preferred, over other alternatives, because they allow accurate active power sharing, and they preserve the desirable active-power versus frequency relations characteristic of synchronous generators [29].If necessary, the inductive behavior is commonly enforced through impedance emulation, by a suitable design of the EPC output voltage regulator, or by adding a dedicated virtual-impedance loop as done, for example, in [29], [30], and [31].

III. PROPOSED CONTROL TECHNIQUE
The proposed power controller allows the following: 1) the control of direct-sequence active and reactive powers; 2) the injection of negative-sequence currents for unbalanced load supply; 3) smooth and seamless transition to islanded operation.Power control is achieved via droop control, which assumes in this case inductive line impedance.As discussed in Section II, this is an assumption commonly verified.Moreover, it is possible to enforce this condition via impedance emulation techniques.

A. Controller Structure
The controller can be divided into three sections, as shown in Fig. 3 and explained in the following.: first, the active power controller is composed of an inner loop that implements the classical P −f droop characteristic

1) Active and Reactive Power Controllers
where ω 0 and ω are the reference and actual grid frequencies, k p is the droop coefficient, P * is the power reference, and P + is the direct-sequence measured active-power.Then, an outer control loop adjusts P * to track the reference signal and achieve This loop is responsible for the synchronization with the instantaneous phase of the voltage at the point of connection of the EPC with the grid [26].
Similarly, the reactive power controller is composed of an inner loop that implements the Q−V droop characteristic where V 0 and V are the reference and the actual grid voltages, respectively, k q is the droop coefficient, Q * is the power reference, and Q + is the measured direct-sequence reactive power.Then, an outer control loop adjusts Q * to track the given reference signal and achieve Then, the positive-sequence voltage reference is defined by the instantaneous phase given by the active power controller and the voltage amplitude given by the reactive power controller.
The small-signals block diagrams for the active and reactive power loops are shown in Fig. 4(a) and (b).The loop gains can be written as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where 3γ p and 3γ q , as defined in (4), are plant parameters describing the inverter response to phase-shift and voltage amplitude variations, while H pm (s) models the dynamics of the power measurement.
2) Negative-Sequence Currents Controller: This controller adds a negative-sequence voltage component to the reference produced by the power controllers, in order to enforce the flow of a set negative-sequence current that can be exploited for unbalance control.Assuming inductive line impedances, the voltages v − d and v − q , together with the components v − d,pcc and v − q,pcc at the point of common coupling (PCC), determine the negative-sequence current, as represented in Fig. 4(b) and (c).Herein, a clockwise rotating instantaneous phase, where d-axis is leading the q-axis, is used.From the scheme, considering the PCC voltage as an exogenous input, the loop gains of the negative-sequence current controller are where 1/(ω g L g ) represents the plant, that is, the dq impedance seen by the converter when injecting negative-sequence components.

3) Reference Voltage Generation:
The generation of the voltage reference v ref abc is built by adding the contribution from the active and reactive power controller and the negative-sequence current controller.The former contribution constitutes the positive sequence component of the voltage reference; it presents amplitude V and instantaneous phase ϑ + .The latter contribution constitutes the negative sequence component of the voltage reference and it is derived from the components v − d,ref and v − q,ref .
It is worth remarking that the final reference provided by the proposed controller is a voltage reference for the EPCs, which are always controlled as voltage sources.

B. Grid-Tied Operation
During grid-tied operation EPC output power regulation is physically possible, because the presence of the main grid ensures the presence of a slack node that can supply the mismatch among the power absorbed by the loads and the generation by the distributed EPCs [32].The integrators in the active and reactive power loops will adjust the reference values P * and Q * in order to set the output power to the reference values This is needed in case the grid-voltage frequency and amplitude are differ from the nominal value.
The negative-sequence instantaneous phase is computed using a phase-locked loop (PLL) fed with the output capacitor voltages v a , v b , v c and providing the phase ϑ PLL , as shown in Fig. 3.In fact, the phase ϑ + may differ from the phase of the actual output voltages due to the implemented virtual output impedances and the nonideality of the voltage regulator.
Active and reactive power reference values, together with negative-sequence current ones, are provided externally with respect to this controller, as foreseen in the scenario represented in Fig. 1.

C. Islanded Transition and Operation
When the main grid disconnects, the conditions highlighted at the beginning of Section III-B decay, and the generated power within the microgrid must automatically match loads consumption.Being power regulation no more possible, the power regulation error increases, leading the integrators in the power regulation loop to saturate.Eventually, the system enters islanded operation with fixed droop characteristics which allows automatic load sharing among the EPCs connected to the islanded microgrid.The transition, from the point of view of the output voltage, happens in a smooth and seamless way.When at least one integrator reaches saturation, a reset signal is sent to the negative-sequence current controller, disabling it.Remarkably, the island detection is performed independently by each inverter, so that no coordination is needed, neither among converters nor toward a centralized controller.

IV. CONTROLLER DESIGN
The main equations for the design of the controller are derived in this section, and the design process is discussed, starting from the choice of the saturation thresholds, through the design of the power regulation loops, to the negative-sequence current loops design.

A. Droop Coefficients and Saturation Thresholds
Droop coefficients can be chosen in various ways, and in general there is a tradeoff between the error in the drooped quantities and the speed of the dynamic response of the system.A simple design procedure is to consider the nominal power of the inverter S N (assuming to be both nominal absorbed and supplied output power) and the width of the interval in which ω and V can vary, then as it is commonly done.More sophisticated methods exist, allowing the investigation of small-signals behavior of multiple parallel-connected droop-controlled ac sources, like, for example, [33], [34].
Once droop coefficients are determined, it is possible to define the saturation values for the integrators in the power loops.Starting with the active power, from (5) it is possible to write which is the droop characteristic reference value needed to generate a particular value P + of active power when the actual grid frequency is ω.By knowing the minimum and maximum ω value of the tolerable grid frequency and the maximum and minimum P + that the inverter can generate (or absorb), it is possible to find Similarly for Q * , from ( 6) by knowing V min , V max , Q min , and Q max .It is worth remarking that, in order to track all the references for the reactive power, it may be necessary to extend the considered thresholds, as voltage amplitude is not the same along the whole microgrid.

B. Power Regulation Loops
The active-power loop and the reactive-power loop have different dynamic behavior, because the former includes an additional pole in the origin due to the integrator computing the instantaneous-phase signal ϑ.
From (7), the value of h p i can be chosen based on desired specifications of control bandwidth and phase margin.Neglecting the power measurement dynamics H pm (s), initial licit solutions are h p i = 3γ p k p/4, to achieve a critically damped (i.e., coincident real poles) closed-loop transfer function, or h p i = 3γ p k p/2, to achieve a second-order Butterworth poles placement (i.e., ξ For what concerns the reactive-power loop, it is possible to achieve wider control bandwidths, which are limited only by the power measurement block while the process to be regulated is static.Considering (8), and neglecting the dynamics of the power measurement, one can write and then choose h q i = ω * /A Q + where ω * is the target bandwidth, which, for example, may be set to match the active-power loop bandwidth.

C. Negative-Sequence Current Loop
The negative-sequence current regulation is performed by decomposing the currents into symmetrical components, using a reference frame synchronized with the instantaneous phase ϑ − provided by the PLL.As discussed in the previous section, i − d is regulated adding a v − q component, while i − q by adding a v − d component.From ( 9), and neglecting the current decomposition dynamics, the design is straightforward.For example, the target bandwidth ω * can be set first and, subsequently, choose

D. Poles Trajectories
Fig. 5 shows the poles trajectories of the closed-loop system with respect to the design of the outer integral regulator.In particular, Fig. 5 shows the closed-loop pole locations of the active-power loop (7) as h p i is varied.The same is shown in Fig. 5(b) for the reactive-power loop (8), varying h q i .Similarly, the location of the poles for negative-sequence current-loops are shown, assuming the same regulator gain h − i for both loops (9).The power measurement and current decomposition dynamics are considered to be first-order filters, as follows:

V. SIMULATION RESULTS
The proposed controller has been validated by simulation results, discussed in this section, and experimental results, discussed in Section VI.The simulation results are carried out in order to validate the parallel operation of two inverters implementing the proposed control method.
The test is devised to validate all the capabilities of the proposed method, namely, the following: a) grid-tied operation; b) active power reference tracking; c) reactive power reference tracking; d) unbalance current compensation at the PCC; e) transition into islanded operation; f) islanded operation.
In the test, two inverters are connected in parallel and initially operating in grid-tied mode.The two inverters have identical control parameters, reported in Table II.Two different loads are connected at the PCC, namely, an unbalanced three-phase Y resistive load, composed of two 10 Ω and one 30 Ω resistors, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.7), (b) for reactive-power loop (8), and (c) for negative-sequence current loops (9).The power measurement and current decomposition dynamics are assumed to be first-order low-pass filters, with cut-off frequencies, respectively, ω pm and ω im , as shown in (14).

TABLE I INVERTER AND GRID PARAMETERS
and a balanced three-phase Y capacitive load, composed of three 100 μF capacitors.The test scenario corresponds to the one depicted in Fig. 1, assuming n = 2.
Fig. 6 reports the main quantities of the EPCs along the performed test and a table that describes the succession of events.From top to bottom, the left side of Fig. 6 displays active powers, reactive powers, and the current amplitudes at the PCC; the right side of Fig. 6 displays the grid frequency, the voltage measured at the output of an EPC and the instantaneous PCC voltage across the disconnection from the main grid disconnection, which happens at t = 15 s.Steps of power references are applied during intervals b), considering active power, and c), considering reactive power.Interval d) shows the unbalanced current control capability of the proposed controller exploited with the goal of compensating the unbalanced current at the PCC.To this end, the negative-sequence currents at the PCC are measured and then used as reference for the signals i − q,ref and i − d,ref .The reference signals are set at the same value for EPC 1 and EPC 2 , equal to half of the negative-sequence currents initially measured at the PCC, in order to evenly share the compensation effort; remarkably, the measured current during this interval d) are balanced, as desired and expected based on the set references.In e), at t = 15 s, the main grid is disconnected.The two inverters start the transition to islanded operation, which is completed at the beginning of f), at t = 22.2 s.

VI. EXPERIMENTAL RESULTS
The proposed controller has been validated experimentally under different operating conditions.The results reported in the following were obtained using the experimental setup displayed in Fig. 7.It includes two EPCs, implemented by using a rapid prototyping system that is described in detail in [35].Each EPC embeds an ImperixBBoard Pro digital controller and three halfbridge power boards based on 600-V, 50-mΩ GaN FETs by Texas Instruments (LMG341xR050).Each EPCs is supplied by a dedicated power supply Keysight RP7962 A. The upstream main grid is emulated by a Cinergia ac voltage source, model GE/EL+20 vAC/DC.Data acquisition is performed by Dewesoft SIRIUSi digital acquisition systems.The parameters of the EPCs as defined in Fig. 1 are listed in Table I, and control parameters are listed in Table II.
In the following, three test configurations are reported, namely, Section VI-A power regulation in grid-connected operation with balanced load plus a transition toward islanded operation, Section VI-B negative-sequence current regulation for supplying an unbalanced three-phase load plus a transition toward islanded operation, Section VI-C power and unbalance regulation of two parallel-connected inverters connected to a grid with unbalanced voltages plus a transition toward islanded operation.The first two configurations are meant to show the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.principle of operation of the controller in a simple application example with a single EPC.The third configuration shows the operation of the controller in a more realistic condition and considers two EPCs.

A. Power Regulation With a Single Inverter
In this test, a three-phase inverter, namely, EPC 1, is tied to the three-phase ac grid; EPC 2 is disconnected.A three-phase balanced 27 Ω resistive load is locally connected.The main waveforms are shown in Fig. 8, which reports the direct-sequence active and reactive powers, the rms values of output voltage and current, and the frequency.8 shows that the system correctly tracks the given power reference values.At t = 19.6 s, the grid is disconnected by opening the circuit breaker CB in Fig. 7 and the EPC automatically initiates its transition to the island operation.Being some active power absorbed by the local load, the output frequency decreases until the integrator in the active-power regulation loop saturates.Similarly, a small value of reactive power is absorbed by the system, thus, the integrator in the reactive power loop saturates too toward its upper limit.After t ≥ 23 s, the system operates with a fixed droop characteristic.

B. Unbalance Regulation With a Single Inverter
In this test, an unbalanced three-phase load is connected to the grid.EPC 1 is connected, while EPC 2 is disconnected.The load is constituted of a 108 Ω resistive load connected between phase b and phase c only.Fig. 9 shows the negative sequence d-axis and q-axis currents for the grid and for the EPC, and the grid rms current values.Initially, the unbalanced load is supplied by the grid, both in balanced and unbalanced components: the positive-sequence power and negative-sequence current references of the EPC are set to zero.At t = 14.5 s, a step-change in the negative-sequence current references is performed, in order to compensate the negative-sequence current absorbed by the load.Remarkably, for 15 s ≤ t ≤ 25 s, the grid current is mainly a positive sequence current; in fact, the EPC supplies only negative-sequence components, whilst the positive-sequence power is still supplied by the grid.At t = 25.5 s, the grid is disconnected, and the load is entirely supplied by the EPC.The EPC performs its transition toward island operation as in the previous test.

C. Multiple Inverters Operating in an Unbalanced Scenario
In this test, an unbalanced three-phase load is connected to the grid.The load is constituted of a 108 Ω resistive load connected between phase b and phase c only.The grid voltage is unbalanced, with an unbalance factor equal to 2.5%, defined as the magnitude ratio of negative-sequence to positive-sequence voltage.Both EPC 1 and EPC 2 are connected and controlled with the proposed method.The experimental waveforms are shown in Figs. 10 and 11.
Initially, both the inverters are set to zero output power and unbalanced current.At t = 5 s, a step of active power reference is applied to both the EPCs, such that P + ref,1 : 0 W → 600 W, and P + ref,2 : 0 W → 900 W. The EPCs track the given power references and, as expected, the provided phase currents are all balanced.At t = 8 s, the negative-sequence current references are set to compensate the unbalanced current absorbed by the grid.This means i − q,ref,1 : 0 A → −0.84 A, and i − q,ref,2 : 0 A → −0.6 A. Remarkably, even in an unbalanced scenario, the controller is capable of regulating the positive-sequence power and negative-sequence currents with the same dynamical performances as in the balanced case.At t ≈ 16 s, the main grid    is disconnected, by opening the circuit breaker CB displayed in Fig. 7.The inverters are involved in the transition toward islanded operation, which is completed at around t ≈ 17.5 s when the integrators in the power loops reach saturation levels.At this point, the negative-sequence current loop is disabled.This creates a transient visible on the inverters rms currents in the time interval 17.5 s ≤ t ≤ 18.5 s.Subsequently, the system reaches a steady state and the transition can be considered complete.Voltage waveforms taken around t = 20 s are shown in Fig. 9, to prove the stability among multiple grid-forming inverters connected together.

VII. CONCLUSION
A controller capable of regulating positive-sequence powers and negative-sequence currents at the output of grid-tied inverters, also allowing seamless transitions into islanded operation, was proposed.The control approach conjugates flexible power control and unbalance compensation, during grid-tied operation, with the capability to adapt to the microgrid needs, during islanded operation.A simple yet effective design procedure was reported, which can be further tailored to comply with specific control requirements.The approach was tested considering an experimental power electronic system composed of two inverters.Simulation and experimental results demonstrated the control performance of the proposed controller and its capability to smoothly transition into islanded operation, even in the presence of unbalanced loads, unbalanced grid voltages, and multiple converters operating in parallel with the proposed controller.To summarize, the proposed solution: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
1) integrates power control and unbalanced currents control with seamless islanded transitions; 2) showed proper operation in unbalanced grid conditions; 3) was demonstrated experimentally to support islanded operation when applied to multiple parallel-connected grid-forming converters.The solution finds applications in microgrid inverters controllers to implement active and reactive power control, unbalanced power control, and islanded operation capability of distributed three-phase inverters.Remarkably, the approach is applicable to both three-wire and four-wire systems; besides, in the latter case, it is possible to add a further loop for controlling the output homopolar current of the EPC.

Fig. 4 .
Fig. 4. Small-signals block schemes of a grid-tied converter controlled with the proposed method.(a) and (b) Active and reactive power loops, (c) and (d) negative-sequence current loops.

Fig. 5 .
Fig. 5. Trajectories of the closed-loop poles varying the design of the integral regulators.(a) Trajectories for active-power loop (7), (b) for reactive-power loop(8), and (c) for negative-sequence current loops(9).The power measurement and current decomposition dynamics are assumed to be first-order low-pass filters, with cut-off frequencies, respectively, ω pm and ω im , as shown in(14).

Fig. 6 .
Fig.6.Simulation test with two parallel-operating inverters, initially operating in grid-tied condition.Unbalanced three-phase Y load is connected at the PCC, composed of two 10 Ω and one 30 Ω resistors, and three 100 μF capacitors.Nomenclature as in Fig.1.The sequence of events (a)-(f) is described in the table on the bottom.

Fig. 9 .
Fig. 9. Experimental waveforms showing unbalanced compensation capabilities of the proposed controller.(a) i are set to unbalanced current flow at PCC; (b) disconnection from the main grid.Bottom figure shows clean voltage waveforms at (b).