Triple-harmonic-droop control strategy for accurate harmonic power sharing in low-voltage islanded microgrids

This paper proposes a power sharing control scheme for power electronics interfaced distributed generation to attain accurate harmonic power dispatch as well as unbalanced power sharing. The proposed controller is composed of two main blocks: 1) a harmonic power sharing unit, which leads to an accurate harmonic power sharing, and 2) individual triple-droop control units separately applied to the control unit of each phase, which properly provide unbalanced power sharing. For this purpose, a virtual harmonic conductance concept is developed to control the inherent impedance of the inverter under non-linear load conditions. As a result, the fundamental power sharing performance as well as the harmonic power dispatch accuracy is ensured. Simulation results from an islanded microgrid prototype using the CIGRE low-voltage distribution network are provided to validate the effectiveness of the proposed method.


INTRODUCTION
Non-linear and unbalanced loads in low-voltage (LV) microgrids are the most common loads that affect the regular system operation [1]. Unbalanced and harmonic power sharing between distributed generation (DG) units is a critical issue in islanded LV microgrids, wherein electronic single-phase loads are widely used. This type of loads degrades power quality indexes. Total harmonic distortion (THD) index known as one of the most important power quality indexes is affected by application of non-linear loads. It must be lower than the standard harmonic limits [2].
The conventional droop control strategy accurately works at the fundamental frequency, but it has no proper effect on nonfundamental powers sharing due to frequency-based inverter impedances, the mismatch in the feeder impedances, and local loads [3,4]. In other words, the harmonic powers are distributed among DG units, according to their effective harmonic impedances, i.e. the sums of inverter output impedances and grid impedances. Therefore, the harmonic power sharing issue can be similarly addressed by reshaping voltage-source inverter (VSI) output impedances at selective harmonic frequencies.
Although it has some desired features such as redundancy and flexibility, the droop control method inherently exposes some  [5].
In [5], a central controller employing adaptive regulation of the virtual impedances has been used to share dominant harmonic powers. In [6], a control system in a master/slave microgrid architecture coordinated by a centralised controller has been developed to distribute harmonic currents. Negative virtual impedance techniques were also used to share harmonic power in [2] and [7]. A multi-proportional-resonant (PR) regulator and a virtual negative-sequence impedance technique have been utilised in [8] to share the harmonic and unbalanced currents of non-linear single-phase loads among DG units. In [9], the application of the distributed consensus protocol has been developed by adjusting the virtual impedances to overcome the impacts of mismatched line impedances, which results in poor harmonic and unbalanced power dispatch performance. The method of [10] is superior in compensating the voltage distortion of non-linear loads. In addition, harmonic currents are dispatched from DG units, according to their ratings by extracting the harmonic components of the point of common coupling (PCC) voltage and sending to all DG units via a communication link. In [11], a two-dimensional impedance control scheme using the distributed consensus control algorithm has been proposed to improve the harmonic power sharing accuracy. In [12], a hierarchical control scheme has been used to improve power sharing of microgrids and to reduce the voltage harmonics at the PCC. The virtual impedance at fundamental and harmonic frequencies is regulated by the line current and voltage at the PCC in a complex way.
A different imbalance and harmonics power sharing based on conservative power theory (CPT) and consensus algorithm is presented in [13]. The CPT is used to identify the balanced, unbalanced, and distorted components of the currents and powers in the system, and the control loops are then used to distribute these components between the various converters in the microgrid.
However, all the aforementioned papers require communication links, and in some cases, a microgrid central controller is needed to handle the predetermined low-order harmonic powers among DG units, which increases the cost of the whole microgrid and the risk of communication failure. Besides, these methods are only applicable for a limited range of low-order harmonic frequencies, and also, the power sharing unit of the control system cannot work accurately while facing an unbalanced non-linear load condition. In other words, these methods suffer from some insufficiency when encountering unbalanced non-linear load conditions, which causes power sharing inaccuracy.
To avoid the obstacles of the communication-based methods and to increase the reliability and flexibility of large-scale microgrids, communicationless control systems may be preferable in some cases. In this regard, a virtual output impedance with a magnitude larger than the feeder impedance at the fundamental negative-sequence and harmonic frequencies has been introduced to share the unbalanced and/or harmonic power [14]. In [15], the output impedances of DG units are adaptively handled according to their delivered harmonic power. The concept is to regulate the output virtual impedance of the inverters according to the unbalanced and harmonic power. An adaptive virtual impedance control scheme based on the injection of small AC signal to the output voltage of the DG unit has been proposed in [16]. However, these methods have applied some control blocks to only dispatch a limited range of low-order harmonic frequencies. Obviously, all of these methods cannot work accurately while facing various kinds of non-linear loads generating a wide range of harmonics and inter-harmonics.
It will be more noticeable in the LV microgrid, where the real power sharing is influenced by the P-V droop equation resulting in inaccurate power dispatch. Meanwhile, the circulating current caused by unbalanced loads cannot be properly handled. To mitigate the side effects of this phenomenon, a virtual impedance method is used in [17] to regulate the output impedance of DG units. The authors of [18] apply an adaptive virtual impedance technique at negative sequence. In [19], threephase Q-f droop and single-phase P-V droop equations have been used to handle the unbalanced load current. The method has been developed in [20] to improve the negative-sequence current sharing. In [21], the obstacles of the unbalanced load sharing have been solved. However, the operation of the microgrid under non-linear load conditions has been ignored.
This paper proposes a triple-harmonic-droop control strategy to improve the power sharing accuracy in a typical LV microgrid under non-linear and unbalanced load conditions. The name of the proposed method is derived from the concept that each phase of the converter individually participates in harmonics and unbalanced power sharing. This paper presents a technique to achieve accurate harmonic power sharing without any side effect on dispatching real and reactive powers and develops a triple-droop control strategy to ensure an accurate power sharing under unbalanced load conditions. To achieve this, the proposed technique employs a harmonic power sharing unit as well as a triple-droop control strategy to ensure accurate harmonics and fundamental power sharing under non-linear load conditions. The features are fulfilled by the proposed triple-harmonic-droop control strategy applied in the control system of each phase of the converter. Based on the proposed droop control strategy, the real power sharing as well as the harmonic power sharing accuracy is significantly improved. The main idea of the method is to control the harmonic power sharing based on the remaining capacity of DG units. In this technique, a virtual harmonic conductance (VHC) concept is defined to control the inherent impedance of the inverter under non-linear load conditions. Based on this idea, DG units operating at their maximum generation are not allowed to contribute in the harmonic power sharing. By this technique, the harmonic power is accurately shared among DG units regardless of harmonic degree. Figure 1 illustrates the single-phase diagram of the understudy ac microgrid. The three-phase LV microgrid operates at a 400-V distribution network. A step-up transformer is used to connect the incoming feeder of the radial microgrid to the 20-kV network. The structure and parameters of the microgrid are taken from the CIGRE LV network [22,23]. The four-wire microgrid operates in the islanded mode. Four dispatchable DC-type DG units (including two fuel cell (FC) units and two battery storage systems) and two non-dispatchable DC-type DG units (including photovoltaic (PV) sources) are fully prepared to uninterruptedly supply different kinds of distributed loads. The inverters of the PV sources operate as a unity power factor to only generate the real power. The power converter system used in DG units consists of a typical sinusoidal pulse-width modulation inverter with an LC output filter. The full capacity of the microgrid is 96 kW. The conductors of the main feeder are twisted insulated cables, and the short length outgoing feeders from DG units or to distributed loads are underground cables. The microgrid parameters, including the grid and the load data, are summarised in Table 1. The microgrid loads are categorised into single-phase linear loads, three-phase linear loads, and a three-phase non-linear load. A wide diversity of single-phase commercial and domestic customers is supplied by the microgrid. In such a case, the total demand of the loads is constantly unbalanced and tends to draw low-order harmonic currents from the sources. The dispatchable sources are individually comprised of a DC-type DG unit (FC or battery) and a three-phase inverter equipped with the individual control system. The block diagram of the DG inverter is shown in Figure 2. The objective is to handle proper harmonic power sharing as well as to manage accurate power sharing among available dispatchable DG units. Figure 3 shows the block diagram of the control structure of an inverter-based DG used in the understudy ac microgrid. The control system is comprised of: (1) harmonic power sharing unit, (2) unbalanced power sharing unit, (3) inner current and voltage loops, and (4) a virtual impedance loop. The unbalanced power sharing unit develops the triple-droop control technique to separately control the fundamental power of each phase of DG units [21]. To track the reference value of the fundamental voltage generated by the P-V droop control method, a PR regulator (G V (s)) is adopted. The voltage control loop makes the reference input signals (I * a DGi , I * b DGi , orI * c DGi ) of the current control loop. The PI controller (G I (s)) applied in this loop generates the PWM signals of the inverter. The voltage and current regulators are implemented in the original abc frame and the PR controllers [22,23] G V (s) = k pV + k iV s where k pV and k pI are the proportional gains, k iI is the integral gain factor, and k iV is the resonant gain. 0 and cV are the resonant frequency and cut-off frequency for bandwidth control, respectively. A virtual resistance loop (r vir DGi ) is employed in this method to improve the power sharing accuracy. To achieve this goal, the output current of each phase (I oa DGi , I ob DGi , or I oc DGi ) is multiplied by the virtual resistance and the result subtracted from the ref-

CONTROL STRATEGY OF VSI IN MICROGRID
. Consequently, the virtual resistance is in series with the voltage reference and the physical feeder impedance, as shown in

Triple-droop control structure
In order to develop the triple-droop control scheme, it is necessary to verify the operation of the conventional droop technique under unbalanced load conditions.

Conventional droop control technique
In LV microgrids with highly resistive network, DG units are conventionally managed by the Q-ω and P-V droop controllers as follows: Virtual resistance serialised with feeder impedance where and 0 are the reference and nominal angular frequencies of the DG unit, respectively; V and V 0 are the reference and nominal DG voltage magnitudes, respectively. P and Q are the generated three-phase real and reactive powers of the DG unit, respectively, and m and n are the droop coefficients, which determined as follows: where P max and Q max are the rated real and reactive power capacity of a DG, respectively; Δ and ΔV are the maximum allowed deviation of frequency and voltage, respectively. Since the reference voltage amplitude of all phases is equal, the conventional droop suffers from a considerable error in unbalanced power sharing. Consequently, there are significant circulating currents among DG units. In other words, by individually regulating the reference voltage of each phase, the power generated by that phase is properly handled.

Triple-droop control technique
In the triple-droop method, the purpose is to separately handle the generated powers of each phase; then, the following droop scheme for DG inverters is applied: where V ph DGi , V ph DGi , and n ph DGi are the voltage amplitude and the voltage droop factor of phase ph ( ′′ A", "B", "C" ) of DGi. Accordingly, the real power of the individual phase is managed by the droop structure applied to the control unit of the same phase, and the circulating current among DG units is nearly zero. With this strategy, all three-phase DG units change to three single-phase DG units.

Principles of the proposed triple-harmonic-droop control scheme
Due to the presence of unbalanced non-linear loads in ac microgrids, a triple-harmonic-droop control scheme is proposed to guarantee proper harmonic power sharing as well as accurate unbalanced power sharing. For this purpose, the proposed harmonic power sharing loop is applied to the recent triple-droop control system so that the harmonic power dispatch of each phase of the inverters is accurately determined, in addition to unbalanced power sharing. Figure 3 shows the control structure of the proposed method. The main idea of the method is to control the harmonic power sharing based on the remaining capacity of the DG units. In this technique, the virtual harmonic conductivity (VHC) concept is defined based on the instantaneous remaining capacity of the DG unit according to Figure 5.
By multiplying the VHC by the output harmonic voltage, the reference harmonic current of the DG unit can be determined. This signal is applied to the internal current loop to maintain voltage quality under harmonic conditions. It is evident that the VHC value of the DG units operating at their maximum generation is zero, and then, they are not allowed to contribute in the harmonic power sharing. Accordingly, the corresponding DG units only participate in the fundamental frequency power sharing. By this technique, the harmonic current and consequently harmonic power is accurately shared between the DG units regardless of harmonic degree I h,ph DGi refers to the VHC value that can be obtained as follows: The LPF is low-pass filter by a cut-off frequency of c to determine the average voltage at fundamental frequency as follows: Then, the VHC factor (G h,ph DGi ) is determined as follows: In the above equation, I del is the deliverable current from the DG unit based on the remaining capacity of the same DG unit. When the DG unit is in over-generation status, then I del < 0, and the corresponding DG unit is not allowed to participate in the harmonic current sharing. k DGi g is defined as a coefficient of the remaining capacity of the DG unit, which is correlated with the maximal harmonic voltage allowed in the ac grid where V ph n is the nominal phase voltage of the network and S ph n,DGi refers to the nominal capacity of the DG unit in each phase.
Since the harmonic current signal is applied to the internal current loop, the voltage loop is only responsible for controlling the voltage of the fundamental frequency, which improves the output voltage quality of the inverter. To verify the concept, the output voltage of the closed-loop system is expressed as follows: . (20) Considering these equations, the inverter model under nonlinear load can be drawn, as shown in Figure 5. The inverter output current in the presence of non-linear loads is a combination of the fundamental component (I 1 out ) and the harmonic components (I h out ) of the load current. Obviously, as the proposed method does not apply any harmonic voltage reference to share harmonic components of the load current among DG units, the output voltage of the inverter is almost independent of the harmonics. In this case, the harmonic components of the load current pass through the output impedance of the inverter, as shown in Figure 6(b).
The quality of the output voltage of the inverter under harmonic conditions directly depends on the output impedance of the DG inverter and also the feeder impedance. Since the impedance of the feeder is an inherent parameter of the microgrid network, reducing the amplitude of the inverter output impedance leads to an improvement in the quality of the inverter output voltage. Therefore, in order to lessen the effect of the harmonic current on the quality of the voltage waveform, it is essential that the inverter output impedance has a constant and low amplitude over a wide range of harmonic frequencies (f > 50 Hz). To achieve this, it is necessary to develop the control system so that the inverter output impedance is mostly resistive. In this strategy, by adjusting the inverter output impedance, a precise harmonic current and harmonic power sharing among DG units can be achieved over a wide range of frequencies.   Figure 7 shows the output impedance of the inverter (Z o (s)) in the proposed method. As shown in this figure, the amplitude of inverter output impedance is low (less than 0.09 Ω) and constant over a wide range of frequencies. In addition, the phase angle of the impedance at frequencies above the fundamental component is almost zero, which indicates the resistive behaviour of the output impedance at the harmonic frequencies. Obviously, using the proposed method, the quality of the inverter output voltage in the worst harmonic conditions of the grid is in an acceptable range. It is worth mentioning that in the proposed method, the harmonic current/power sharing is independent of the type and degree of harmonic frequency.

SIMULATION VALIDATION
The microgrid shown in Figure 1 is simulated in PSCAD software to validate the performance of the triple-harmonic-droop method compared with the conventional droop control scheme. In order to investigate the performance of the new strategy, the unbalanced and harmonic power sharing among DG units is evaluated for both the proposed method and the conventional droop scheme in four different steps of load conditions, as shown in Table 2.
A three-phase diode bridge rectifier as a non-linear load with the current waveform of Figure 8 is connected to the microgrid PCC. The low-order harmonic current components of the nonlinear load are taken into consideration according to Table 3.
To investigate the steady-state performance of the proposed method, harmonic power error is determined as where H 0 DGi is the harmonic power generated at the terminal of DGi and H ref DGi is the desired harmonic power sharing of DGi. Four steps are considered to validate the   In the first step, all DG units, including dispatchable and nondispatchable sources, work at their full capacity to supply the existing balanced loads according to Table 1. At t = 2 s, the generation of non-dispatchable DGs (including DG4 and DG5) stops, and the loads of phases "B" and "C" of all feeders are suddenly disconnected. Therefore, the control scheme of DG units has to compensate the power shortage caused by the nondispatchable DG outage as well as to manage the unbalanced power sharing. At t = 3 s, a 5-kW single-phase bridge rectifier is added to feeder F02. After 1 s (t = 4 s), a three-phase bridge rectifier with the nominal power of 12 kW is connected to this feeder in the fourth step. Figure 8 shows the single-phase and three-phase non-linear load currents for Steps 3 and 4 of the simulation. In this figure, a load change happens from a singlephase non-linear load to a combination of single-phase and three-phase non-linear loads at t = 4 s. Table 3 summarises the harmonic components amplitude of the loads current. The load current comprises several harmonic components with a considerable THD of 45%. Figures 9 and 10 illustrate the real and reactive power of DGs during these steps in the conventional and triple-droop methods, respectively. The harmonic power produced by the dispatchable DGs to provide the non-linear loads in both control methods is depicted in Figure 11.

4.1
Step 1 (0 < t < 2 s)-balanced and linear loads The loads at the time of 0-2 s are totally linear and balanced. As derived from Figures 9 and 10, the fundamental powers generated by DG1 and DG2 in two methods are around three times greater than the fundamental power produced by DG3 and DG6 because the rated capacity of DG1 and DG2 is three times higher than the rated capacity of DG3 and DG6.

4.2
Step 2 (2 < t < 3 s)-single-phase linear loads At t = 2 s, the generating power of the PV units quickly drops to zero. At the same time, a massive load outage occurs on phases "B" and "C," and the microgrid consequently suffers a severe unbalanced load condition. Even though this condition may never happen, it is purposely applied for better demonstrating the advantages of the proposed technique comparing with the conventional method. Thus, not only the power shortage must be supplied by the dispatchable DGs, but also the control  scheme of the DG units has to manage the power sharing of each phase. From Figure 9, phase "A" of DG1 and DG2 delivers a real power of 3601 and 2369 W, respectively, whereas the real powers dispatched from DG3 and DG6 are 4730 and 4952 W, respectively. Using (21), the method gives a real power error of 39%, 60%, 142%, and 153% for DG1, DG2, DG3, and DG6, respectively, which implies an improper power sharing under imbalanced microgrid. Also, the reactive power error of DG1, DG2, DG3, and DG6 is about 51%, 33%, 73%, and 82%, respectively. Moreover, phases "B" and "C" of DG2 deliver an undesirable real power of 826 W to be received by DG2, DG3, and DG6, which generates a circulating current between no-load phases of DG units. The same conditions happen for reactive power among DG units.

(< t < 4 s)-single-phase linear and non-linear loads
At t = 3 s, a 5-kW single-phase diode rectifier is connected to feeder F02. After 1 s (t = 4 s), a three-phase bridge rectifier with the nominal power of 12 kW also added to this feeder in the fourth step. Figure 11 shows the harmonic power dispatching from DG units when supplying the non-linear loads in both methods.
In the time interval of 3-4 s, the only non-linear load connected to the microgrid is single phase. Figure 11(b) shows the harmonic power sharing of DG units in conventional droop control method. The harmonic power generated by DG1 and DG2 on phase "A" is 1179 and 1156 VAR, respectively, whereas DG3 and DG6 provide a harmonic power of 1118 and 1019 VAR on the same phase, respectively, whereas the desired harmonic power is 1677 VAR for DG1 and DG2 and 559 VAR for DG3 and DG6. Obviously, the method suffers from a harmonic power sharing error of around 30%, 31%, 100%, and 102% for DG1, DG2, DG3, and DG6, respectively. As a result, the conventional droop technique works poorly to share the non-linear single-phase load power.
In the triple-droop method, as shown in Figure 11(a), the harmonic power generated by DG1, DG2, DG3, and DG6 on phase "A" are 2395, 1933, 676, and 722 VAR with an error of around 11%, 10%, 5%, and 1%, respectively. It is evident that the harmonic power sharing of DGs equipped with the proposed technique under non-linear single-phase load condition is significantly improved.

4.4
Step 4 (4 < t < 5 s)-single-phase and three-phase unbalanced non-linear loads During 4 s < t < 5 s, the dispatchable DG units have to supply a three-phase non-linear load. Figure 11(a) shows the harmonic power sharing curves of DG units in the proposed droop scheme. In this condition, the dispatched harmonic power from phase "A" of DG1 and DG2 is 3368 and 2731 VAR with a percentage error of 10% and 11%, respectively, whereas DG3 and DG6 generate a harmonic power of 998 and 1096 VAR, with a percentage error of 2% and 7%, respectively. The power sharing of DG units in the conventional method never track the predetermined ratio of the droop concept, as shown in Figure 11(b). Tables 4 and 5 summarise the power generated by DGs and the error in both techniques under unbalanced non-linear load conditions. As derived from Table 5, the accuracy of the power sharing in the proposed method is much higher than that in the conventional method.

CONCLUSION
This paper has proposed a triple-harmonic-droop control strategy for DG units to improve the accuracy of the harmonic power dispatch as well as unbalanced power sharing in an islanded microgrid with non-linear and unbalanced loads. In this method, each phase of the converter individually participates in harmonics and unbalanced power sharing independent of other phases. The controller of the inverter is composed of two main blocks: (1) a harmonic power sharing unit, which leads to an accurate harmonic power sharing without any side effect on dispatching real and reactive powers, and (2) individual triple-droop control units separately applied to the control unit of each phase, which leads to a precise power dispatch under unbalanced load conditions. The main idea of the method is to control the harmonic power sharing based on the instantaneous remaining capacity of DG units. For this purpose, a VHC concept has been defined to control the inherent impedance of the inverter under non-linear load conditions. Based on this idea, DG units operating at their maximum generation are not allowed to participate in the harmonic power sharing. These DG units only participate in the fundamental frequency power sharing. Thanks to the proposed method, the accuracy of fundamental power sharing as well as the harmonic power sharing in the LV distribution microgrid is considerably enhanced. Moreover, the voltage loop is only responsible for controlling the voltage of the fundamental frequency, as the harmonic current signal is applied to the internal current loop, which improves the output voltage quality of the inverter.