# MPC-Based Arctan Droop Control Strategy of the Parallel Inverter System in an Islanded AC Microgrid

Authors: J. D. D. Iyakaremye; G. N. Nyakoe; C. W. Wekesa
Journal: Journal of Engineering
 (2021)

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## Abstract

Distributed generations (DG) are one of the upcoming technologies recently used by many electric utilities in all corners of the world. Most of those DG form the microgrid (MG) to serve local loads and can be connected to the grid. This DG’s technology is enabled by utilizing renewable energy sources (REs) that are ecofriendly; however, these REs are intermittent by their nature, so controlling a power electronic device interfaced with them to be connected to the grid is another challenge. Many researchers have worked on the inverters’ control in MG. This study also elaborates on the control strategy for inverters adapted to REs for proper control of voltage and frequency used in an islanded microgrid. The study proposes a hybrid control strategy made of the virtual impedance droop control with arctan function and model predictive control. Extensive simulations have been carried out to validate the proposed control strategy’s effectiveness in terms of rapid transient response and stabilization of voltage, frequency, and power equitability among the microsources in the islanded microgrid.


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## Body

## 1. Introduction

Distributed generations (DG) are seen as alternative solutions to combat pollution and to meet the growing demand for electricity in recent years [1]. A network approach that views the generator and related loads as a module or “microgrid” is a better way of understanding the distributed generation evolving promise [2]. The reduction in the conventional economy of scale in central power generation facilities gives way to a lower, more dispersed generation. A broad range of prime mover technologies, such as internal combustion engines (IC), gas turbines, microturbines, photovoltaic fuel cells, and wind turbines, is used in distributed generation. The bulk of emerging developments such as microturbines, photovoltaics, fuel cells, and wind energy sources are interfaced to the grid with an inverter. These new technologies have lower costs and are theoretically less expensive than conventional systems [2].Power electronics converters are the most critical item for any power conditioning process used in a microgrid to satisfy various sources/loads. Therefore, either a single microsource or parallel microsource is very critical for optimizing their operation. Many control techniques have been proposed for the three-phase voltage source inverters (VSI) [3]. The most widely used are droop control techniques, deadbeat control, predictive control, and cascaded linear control [4–6].Predictive control and droop control occupy a wide variety of controls that have been established to control power converters. To address existing challenges such as nonlinearity, slow response, unequal power sharing, and complexity of the control design, the arctan function droop-based model predictive control technique is used in this study to control the voltage, frequency, and power sharing of the voltage source inverter in a microgrid.Some researchers worked on the droop control in conjunction with model predictive control (MPC) to control the microgrid. Babqi and Etemadi [7] used MPC as a primary control and conventional droop control as a secondary control; both were used to regulate the output voltage and frequency of each DG-based VSI in an autonomous microgrid. Finite control set MPC has been used by Dragicevic [8] as a control method based on precise tracking of derivative’s reference voltage to regulate the voltage at the shared bus, and a simple traditional droop controller was used for power sharing among the VSIs. Chen et al. [9] have proposed an MPC-based power sharing control strategy and the current estimator in the capacitor to minimize the additional current sensor; in this method, a traditional reverse droop controller and virtual resistive concept were used to ensure power sharing.Bouzid et al. [10] proposed the new control method for stable current distribution applied to parallel microgrid inverters systems, based on a decoupled trigonometric saturation regulator (DTS). Introduced with the latest DTS control, energy distribution has been improved with more excellent stability and an efficient energy efficiency level. Active and reactive power dynamic decoupling in the presence of various impedances, also this method, ensures voltage and frequency stabilities; however, the technique utilized the secondary control loop based on proportional integral (PI) for voltage and frequency restoration, which operates at the slow dynamic response and increases the complexity of control strategy design. The new concept utilizing the arctan function for the power frequency droop profile has been introduced in [11, 12]. Using this arctan function improves the small stability of the two inverters’ microgrid signal, offers a natural frequency limit, and is flexible in its application; however, this discussed method presents poor voltage regulation.Zhongwen et al. [13] proposed a fully distributed approach to hierarchical management. It combines secondary and tertiary monitoring into a single control point. Detailed switches’ designs are used instead of standard versions for paralleled VSIs connected to the grid; the method used droop control and virtual impedance for voltage, frequency control, and adequate power sharing. However, the introduction of virtual impedance may result in the voltage drop in the system; thus, a proper selection of virtual impedance value must be considered.Finite control set MPC (FCS-MPC) is based on a radically different concept, as opposed to linear control [14, 15]. Instead of developing loops for each controlled variable individually and then cascading them together, FCS-MPC uses the VSI mathematical model to forecast its future behavior and then evaluate the power converter’s optimum switching status by the specified objective function. Cortés et al. [16] proposed a standard FCS-MPC scheme for a three-phase inverter with an output LC filter. The proposed method achieved effective voltage control with linear loads and also with nonlinear loads.Zheng et al. [17] have recommended that the hardware budget be reduced and system efficiency be enhanced by decreasing existing MPC sensors. FCS-MPC provides the advantages of robustness, outstanding transient properties, and incorporation of nonlinear variables, constraints, and external control objectives; these have proved to be an essential candidate for VSIs regulation [18, 19]. The work on MPC-dependent droop control in the coordinated control of parallel inverters in a microgrid is still immature to the best of the authors’ knowledge. A lot can be performed to increase such control strategy’s capability in hardware simplification, robustness, and transient response. The use of a control method dependent on MPC requires a heavier computational burden; therefore, using a powerful processor is required; this may result in a different budget [20]. With the recent development of modern processors, this can be managed.The conventional droop control strategies dependent on linear droop function have a common drawback where the corresponding output range scaling factor does not affect system dynamics and steady state. Therefore, there is a possibility that the system frequency can largely deviate from the nominal value if the same is determined through the droop control [21].This study introduces a new control method combining the arctan virtual impedance droop control strategy for primary voltage and frequency control and power sharing issues, and FCS-MPC strategy is used for tracking the reference voltage. An advanced FCS-MPC technique for controlling the voltage and choosing appropriate control behavior in the next phase is implemented. With hierarchical linear regulation, the proposed approach can achieve comparable steady state efficiency while enhancing the transitional response. The control design expenditure is reduced concerning current cascaded linear control-based systems as there is no need for voltage and current loops in the control strategy.The study’s sections are arranged as follows: the AC microgrid structure is defined in Section2. Section 3 sets out the existing control theory for VSI. Section 4 describes the proposed control algorithm and its use in VSIs based on an alternating current (AC) microgrid. The results and discussion are presented in Section 5, and finally, Section 6 ends with a conclusion.


## 2. Description of the System

Figure1 displays a VSI-based MG. It is composed of DG units that are parallel linked via the VSI. Since the DGs are usually connected to the electricity storage device to ensure a stable power supply, direct current sources are considered the inverters’ inputs [22]. The load is shared through the same AC bus known as the common coupling point. The MG can be linked to the primary grid or run in the autonomous mode.Figure 1
Typical microgrid-based VSI configuration [23].A control strategy has been designed to control the paralleled VSIs better to assure the MG’s proper control. Therefore, the MPC-based arctan droop controller with virtual impedance is used to ensure voltage and frequency stabilization; moreover, the power equitability among the microsources is assured. Each part of this hybrid control strategy will provide adequate energy management and the MG operation. Thus, it includes the virtual impedance to balance lines impedance. The arctan droop control method used in the proposed control technique eliminates the droop slope of constant frequency and replaces it with an algorithm based on the arctan function. When the arctan-based power profile is utilized, the microgrid operating frequency is still within preset limits [11, 12]. For its excellent transient performance, the FCS-MPC strategy is used to boost tracking performance and reduce the complexity compared to the traditional cascaded linear control method [9].


## 3. The Existing Control Theory for VSI

### 3.1. Arctan Droop Control

The droop functions may be linear or nonlinear. In any linear droop function scenario, droop function blocks may be explicitly substituted by unit gains. The resulting performance range scaling factor has no impact on system dynamics and steady state for linear droop function. The output range of linear droop lies in the range minus infinity to plus infinity. Therefore, any disturbance in the system can lead to large frequency deviations from the nominal values if determined by the same droop control method. An arctan-based droop control mechanism is introduced to prevent this specific issue. With the arctan frequency droop function, the frequency variance spectrum may be controlled within the preset values. The frequency variance spectrum may be further extended or minimized by specifying the output range scaling factor, while the voltage droop feature is always considered linear [21]. A comparison between arctan and the fixed gradient droop control strategy is shown in Figure 2. Li and Kao [24] highlighted that the conventional way to enhance the proportionate power sharing between the VSIs connected in parallel was to boost the droop gradient significantly. The introduction of the arctan function in the droop control method increases the tangential gradient around the power setpoint; therefore, a faster response can be obtained [11]. The arctan function enables voltage and frequency control to be achieved through variation of both the gradient and concavity [10]. The presumption that the arctan droop at frequency limits would have a slower response is the drawback of the arctan droop-based control methods. However, as high-level controls are generally used to change the power operating point dynamically, the frequency constraints’ slower reaction is not a big contention point. It can be advantageous as it reduces the power to go out of limits [11].Figure 2
Comparison of arctan and fixed gradient droop control strategy.Patel et al. [25] highlighted the shortcomings of the traditional droop control approach. An updated virtual droop control approach using the arctan feature for frequency control had been implemented to reduce the impact of impedance mismatch, and the response time has been improved.Figure3 shows the control strategy developed by [25].Figure 3
Arctan virtual impedance droop control strategy [25].In Figure3, Vi and Io are the output voltage and output current for the inverter. ωc represents the low-pass filter cutoff frequency, m and n are the droop coefficients, P is the active output power, Q is the reactive output power, Po is the active reference power, Qo denotes the reactive reference power, and fo and Vo denote the reference frequency and voltage, respectively.Virtual impedance causes a “voltage drop” without impacting the physical loss of exchanged active/reactive power. This method improves the stability of the system and reduces circulating current among the microsources [26]. The implementation of such an approach is shown in the following equation.(1)Vref=Vdroop−ZVs∗Io,where Vdroop denotes the reference voltage provided by the droop method, and ZVS is the virtual output impedance.Guerero et al. [27] have shown how the calculation and design of virtual impedances are performed. In [10], the arctan droop control-based algorithm was used to overcome traditional droop control limitations, which always uses the constant frequency droop to control the frequency within predetermined limits. The control method showed better control concerning voltage and frequency in MG. It has demonstrated better control of the power profile’s concavity and gradient than traditional power frequency droop with zero’s constant concavity. The arctan-based algorithm is a monotonous rise in the magnitude of the preset boundary. The method was chosen because it showed a low complexity and required low computing time. The concavity is controlled by ρ [10], and the bounds are governed by the parameter ap, for example, Rowe et al. [11] have used ap equal to 1, so as to make the frequency bounds to remain in the interval range of 50−0.5,50+0.5. In general, the frequency can vary in the range of fo−ap/2,fo+ap/2.


### 3.2. Finite Control Set Model Predictive Control (FSC-MPC) Control Method

An FCS-MPC approach depends on a mathematical system model to assess the impact of control measures on system response. Therefore, the action expected to minimize a specific objective function is implemented, and the process is repeated sequentially. Appropriate converters and filter models are needed to achieve good control efficiency when using this technique to VSI. The VSI three-phase shown in Figure4 is frequently used in AC microgrids converter topology. It can be seen that the output of VSI is attached to an LC filter that minimizes the switching harmonics.Figure 4
Typical three-phase VSI supplying a load in an MG.The output voltage of VSI before passing the filter, which here isVt, is calculated using the following equation:(2)Vt=23Vdceij−1π/3,j=0,…,7,where Vdc denotes the direct current (DC), and Vt denotes the output voltage of the VSI.By enumerating all potential control behaviors, FCS-MPC anticipates the future conduct of the state component. The dynamic model of the system is thus required. Using Kirchhoff’s laws, we can model the LC filter as follows [28]:(3)dxdt=Ax+B1Vt+B2io.While(4)x=ifvpc,A=−RfLf−1Lf1Cf0,B1=1Lf0,B2=0−1Cf,vpc=vc,equation (2) is used to calculate Vt; whereas, the measured variables are if and vc.Depending on the sampling time (Ts) method, the VSC discrete-time model can be obtained from equation (5) as follows:(5)xk+1=Adxk+B1dVtk+B2diok.While(6)Ad=eATs,B1d=∫0TseAτB1dτ,B2d=∫0TseAτB2dτ.The capacitor’s predictive voltage is given by(7)vPCk+1=vPCk+TsCfifk−iok.The objective function can be represented in the traditional predictive voltage regulation using alpha-beta transformation as(8)gp,i=vpc,α∗−vpc,αk+12+vpc,β∗−vpc,βk+12,where vPC,αk+1 and vPC,βk+1 represent the real and imaginary parts of one-step prediction of the capacitor voltage, respectively. vPC,α∗ and vPC,β∗ represent the real and imaginary parts of the reference voltage, respectively. Equation (8) shows how the minimization of error is obtained between the reference and the capacitor’s predictive voltage.


### 3.3. Power Sharing Conditions in VSIs-Based Microgrids

The proportionate power sharing between the VSIs is obtained when the relationship of any two DGs’ power and impedance in equation (9) is achieved.(9)PiPj=ZjZi,QiQj=ZjZi,equation (9) help us to write(10)S1Z1=S2Z2=⋯=SnZn,where S=P2+Q2. S denotes the apparent power; Zi and Zj represent output line impedances of different voltage source inverters.In other words, this is generally satisfied when(11)m1m2=P2P1=⋯=mkmk+1=Pk+1Pk,n1n2=Q2Q1=⋯=nknk+1=Qk+1Qk,m,n are the active and reactive power droop coefficients, respectively. P,Q are the active and reactive powers delivered by microsources in active and reactive power sharing with exact power ratio and droop coefficients.Equation (11) is not satisfied when using the traditional droop control if the VSIs output impedances are not the same; thus, the introduction of fictitious impedance is required, so that the power sharing is proportionately assured [29].


## 3.1. Arctan Droop Control

The droop functions may be linear or nonlinear. In any linear droop function scenario, droop function blocks may be explicitly substituted by unit gains. The resulting performance range scaling factor has no impact on system dynamics and steady state for linear droop function. The output range of linear droop lies in the range minus infinity to plus infinity. Therefore, any disturbance in the system can lead to large frequency deviations from the nominal values if determined by the same droop control method. An arctan-based droop control mechanism is introduced to prevent this specific issue. With the arctan frequency droop function, the frequency variance spectrum may be controlled within the preset values. The frequency variance spectrum may be further extended or minimized by specifying the output range scaling factor, while the voltage droop feature is always considered linear [21]. A comparison between arctan and the fixed gradient droop control strategy is shown in Figure 2. Li and Kao [24] highlighted that the conventional way to enhance the proportionate power sharing between the VSIs connected in parallel was to boost the droop gradient significantly. The introduction of the arctan function in the droop control method increases the tangential gradient around the power setpoint; therefore, a faster response can be obtained [11]. The arctan function enables voltage and frequency control to be achieved through variation of both the gradient and concavity [10]. The presumption that the arctan droop at frequency limits would have a slower response is the drawback of the arctan droop-based control methods. However, as high-level controls are generally used to change the power operating point dynamically, the frequency constraints’ slower reaction is not a big contention point. It can be advantageous as it reduces the power to go out of limits [11].Figure 2
Comparison of arctan and fixed gradient droop control strategy.Patel et al. [25] highlighted the shortcomings of the traditional droop control approach. An updated virtual droop control approach using the arctan feature for frequency control had been implemented to reduce the impact of impedance mismatch, and the response time has been improved.Figure3 shows the control strategy developed by [25].Figure 3
Arctan virtual impedance droop control strategy [25].In Figure3, Vi and Io are the output voltage and output current for the inverter. ωc represents the low-pass filter cutoff frequency, m and n are the droop coefficients, P is the active output power, Q is the reactive output power, Po is the active reference power, Qo denotes the reactive reference power, and fo and Vo denote the reference frequency and voltage, respectively.Virtual impedance causes a “voltage drop” without impacting the physical loss of exchanged active/reactive power. This method improves the stability of the system and reduces circulating current among the microsources [26]. The implementation of such an approach is shown in the following equation.(1)Vref=Vdroop−ZVs∗Io,where Vdroop denotes the reference voltage provided by the droop method, and ZVS is the virtual output impedance.Guerero et al. [27] have shown how the calculation and design of virtual impedances are performed. In [10], the arctan droop control-based algorithm was used to overcome traditional droop control limitations, which always uses the constant frequency droop to control the frequency within predetermined limits. The control method showed better control concerning voltage and frequency in MG. It has demonstrated better control of the power profile’s concavity and gradient than traditional power frequency droop with zero’s constant concavity. The arctan-based algorithm is a monotonous rise in the magnitude of the preset boundary. The method was chosen because it showed a low complexity and required low computing time. The concavity is controlled by ρ [10], and the bounds are governed by the parameter ap, for example, Rowe et al. [11] have used ap equal to 1, so as to make the frequency bounds to remain in the interval range of 50−0.5,50+0.5. In general, the frequency can vary in the range of fo−ap/2,fo+ap/2.


## 3.2. Finite Control Set Model Predictive Control (FSC-MPC) Control Method

An FCS-MPC approach depends on a mathematical system model to assess the impact of control measures on system response. Therefore, the action expected to minimize a specific objective function is implemented, and the process is repeated sequentially. Appropriate converters and filter models are needed to achieve good control efficiency when using this technique to VSI. The VSI three-phase shown in Figure4 is frequently used in AC microgrids converter topology. It can be seen that the output of VSI is attached to an LC filter that minimizes the switching harmonics.Figure 4
Typical three-phase VSI supplying a load in an MG.The output voltage of VSI before passing the filter, which here isVt, is calculated using the following equation:(2)Vt=23Vdceij−1π/3,j=0,…,7,where Vdc denotes the direct current (DC), and Vt denotes the output voltage of the VSI.By enumerating all potential control behaviors, FCS-MPC anticipates the future conduct of the state component. The dynamic model of the system is thus required. Using Kirchhoff’s laws, we can model the LC filter as follows [28]:(3)dxdt=Ax+B1Vt+B2io.While(4)x=ifvpc,A=−RfLf−1Lf1Cf0,B1=1Lf0,B2=0−1Cf,vpc=vc,equation (2) is used to calculate Vt; whereas, the measured variables are if and vc.Depending on the sampling time (Ts) method, the VSC discrete-time model can be obtained from equation (5) as follows:(5)xk+1=Adxk+B1dVtk+B2diok.While(6)Ad=eATs,B1d=∫0TseAτB1dτ,B2d=∫0TseAτB2dτ.The capacitor’s predictive voltage is given by(7)vPCk+1=vPCk+TsCfifk−iok.The objective function can be represented in the traditional predictive voltage regulation using alpha-beta transformation as(8)gp,i=vpc,α∗−vpc,αk+12+vpc,β∗−vpc,βk+12,where vPC,αk+1 and vPC,βk+1 represent the real and imaginary parts of one-step prediction of the capacitor voltage, respectively. vPC,α∗ and vPC,β∗ represent the real and imaginary parts of the reference voltage, respectively. Equation (8) shows how the minimization of error is obtained between the reference and the capacitor’s predictive voltage.


## 3.3. Power Sharing Conditions in VSIs-Based Microgrids

The proportionate power sharing between the VSIs is obtained when the relationship of any two DGs’ power and impedance in equation (9) is achieved.(9)PiPj=ZjZi,QiQj=ZjZi,equation (9) help us to write(10)S1Z1=S2Z2=⋯=SnZn,where S=P2+Q2. S denotes the apparent power; Zi and Zj represent output line impedances of different voltage source inverters.In other words, this is generally satisfied when(11)m1m2=P2P1=⋯=mkmk+1=Pk+1Pk,n1n2=Q2Q1=⋯=nknk+1=Qk+1Qk,m,n are the active and reactive power droop coefficients, respectively. P,Q are the active and reactive powers delivered by microsources in active and reactive power sharing with exact power ratio and droop coefficients.Equation (11) is not satisfied when using the traditional droop control if the VSIs output impedances are not the same; thus, the introduction of fictitious impedance is required, so that the power sharing is proportionately assured [29].


## 4. Proposed Control Algorithm

The proposed control algorithm uses the arctan droop control with the virtual impedance strategy and the FCS-MPC to gain the advantages of the two control strategies. Arctan droop control has demonstrated the ability to monitor the power profile’s concavity and gradient compared to the conventional power frequency droop with a constant concavity of zero. As it is used with a virtual impedance control loop, it adds another advantage of monitoring the output power of different microsources interfaced with VSIs for different output impedances. Whereas, the FCS-MPC control strategy provides a benefit of hardware simplification as this replaces the voltage and current inner loops in the cascaded traditional droop control, robustness compared to conventional droop control, and track the reference voltage. Figure5 illustrates the proposed control methodology in an MG of one VSI and a simple load.Figure 5
MPC-based arctan droop control strategy.In Figure5, ωnom is the nominal frequency, Vnom is the nominal voltage, Pref is the reference active power, Qref is the reference reactive power, if is the inductor filter current, vo=vc is the output voltage, io is the output current, vref is the output voltage reference droop with virtual impedance, and vdroop denotes the droop voltage.Figures6–8 show the arctan droop control, virtual impedance control, and FSC-MPC control strategies diagrams, respectively.Figure 6
Arctan droop control diagram.Figure 7
Virtual impedance control diagram.Figure 8
FSC-MPC control diagram.TheP/ω arctan droop can now be expressed as a nonlinear function:(12)ω=ωnom−2ap∗ArctanρP−Pref.Or, it can also be expressed as(13)f=fnom−apπ∗ArctanρP−Pref,whereas the Q/V droop is expressed as a linear function as follows:(14)Vdroop=Vnom−mq∗Q−Qref,where ap is the constant for controlling the bounds; in our study, it is chosen to be 1 to control the frequency in the bounds of 50 ± 1% Hz, mq is the droop coefficient for voltage control, and ρ is a constant for controlling the concavity.The reference voltage is given by(15)vref=Vdroop−io∗zv.The virtual impedance is given by(16)zv=Rv∓ωcsLvs+ωc,vzv=io∗zv,where Lv is a virtual inductance, ωc/s+ωc represents the low pass filter, and Rv represents the virtual resistance.The one-step prediction for voltage using FSC-MPC is depicted in Figure8, where the inputs are the output reference voltage after introducing the virtual impedance loop, the measured filter current, and the measured capacitor filter voltage.


## 5. Results and Discussion

All the simulations have been performed using Matlab/Simulink software, and the proposed control strategy has been tested on the MG, which is composed of two microsources and linear loads (RL), as shown in Figure 9.Figure 9
Microgrid network with parallel-connected VSIs.In this study, the analyses include steady-state and load transient; the scenarios are as follows:(a)
Two VSIs with the same power outputs and identical output impedances(b)
Two VSIs with different power outputs and different output impedancesTwo linear loads (RL) have been considered for the study; the system parameters are indicated in Table 1.Table 1
System parameters.
ParameterValueDC bus voltage700 VNominal bus frequency50 HzSampling timeTs = 30 usNominal voltage380 VLine impedance, DG10.528 mH, 1.284 ΩLine impedance, DG20.528 mH, 1.284 ΩLoad 16000 W, 1500 VArLoad 26000 W, 1500 VArDroop coefficients, DG1m1=3e−5V/W, n1=4e−5rad/sVArDroop coefficients, DG2m2=3e−5V/W, n2=4e−5rad/sVArThe allowable frequency range of variation50 Hz ± 1%The allowable voltage range of variation380 V ± 5%In the normal operation mode, all parallel inverters and load 1 are always connected to PCC. At time 0.3 s, load 2 is connected to PCC. A fault is introduced to the system during the time 0.5–0.501 s. These scenarios are shown in Table2.Table 2
Mode of operation.
EventTimeOperationLoad step change0.3 s, the breaker is onAll the two loads are connectedFault0.5–0.501 sThe three-phase to ground fault was introduced to the system


### 5.1. Two VSIs with the Same Power Outputs and Identical Output Impedances

The inverter active and reactive capacity should be DG1 : DG2 = 1 : 1 ratio according to their droop coefficients, and the results showed that beforet = 0.3 s, the system was in steady-state operation, during that period, the voltage drops due to the rise of the load, but the voltage remained the same for all inverters. Their variations were in the range recommended by EN 50160 and IEEE1547 for low voltage networks; Figure 10 illustrates the variations for the two VSIs; as observed, the voltage decreased from 380.8 to 380.2 V. On the side of frequency, the controller behaved as expected while keeping the frequency variation for both VSIs to remain in the set range, and the frequency decreased from 50 Hz to 49.75 Hz during the period 0–0.3 s; this can be observed in Figure 11. The VSIs also delivered active and reactive power equitably as expected, and the results are presented in Figures 12-13. The results reveal that the inverters capacity ratio was P1 : P2 = 3 kW : 3 kW = m2 : m1 = 1 : 1 and Q1 : Q2 = 750 VAr : 750 VAr = n2 : n1 = 1 : 1, which fulfills power sharing conditions.Figure 10
Voltage delivered by VSIs.Figure 11
Frequency variation for VSI.Figure 12
Active power sharing of VSIs.Figure 13
Reactive power sharing of VSIs.Att = 0.3 s, a load step change was introduced, and during the period 0.3–0.5 s, the system remained stable while responding to the load added. The voltage and frequency for VSIs were decreased from 380.2 V to 380 V and from 49.75 Hz to 49.70 Hz, respectively; this can be seen from Figures 10-11. The power sharing was also performed properly during this period, where the ratio was P1 : P2 = 6 kW : 6 kW = m2 : m1 = 1 : 1 and Q1 : Q2 = 1500 VAr : 1500 VAr = n2 : n1 = 1 : 1 as shown in Figures 12-13.At 0.5–0.501 s, a fault was introduced to check the proposed controller’s effectiveness after the occurrence of a transient during operation. According to Figures10–13, the system remained stable after the fault has been cleared; during the fault, there was a reduction of active power and excessive reactive power demand.At 0.501–0.8 s, the voltage and frequency remained stable and the same as in the previous scenario. For power sharing, the VSIs continued to share the power as expected, and the ratio wasP1 : P2 = 6 kW : 6 kW = Q1 : Q2 = 1500 VAr : 1500 VAr = m2 : m1 = n2 : n1 = 1 : 1, as depicted on Figures 12-13.During all scenarios, all the voltage and frequency variations were kept in the preset range values of 380 V ± 5% and 50 Hz ± 1%, respectively; moreover, all the variations for voltages and frequency were kept in the range recommended by the IEEE 1547 standard. It has been noticed that there were no voltage and frequency deviations between inverters; hence, the proportional power sharing between the inverters is assured.


### 5.2. Two VSIs with Different Power Outputs and Different Output Impedances

The proposed strategy has also been tested on the parallel-connected VSIs with different powers and impedance outputs to study for its effectiveness. The parameters considered are given in Table3.Table 3
Tested data for different VSI’s power outputs and impedances.
ParameterValueDC bus voltage700 VNominal bus frequency50 HzNominal voltage380 VSampling timeTs = 30 usLine impedance, DG10.528 mH, 1.284 ΩLine impedance, DG20.352 mH, 0.856 ΩLoad 16000 W, 1500 VArLoad 26000 W, 1500 VArDroop coefficients, DG1ρ1=3e−5V/W, n1=4e−5rad/sVArDroop coefficients, DG2ρ2=2e−5V/W, n2=4e−5rad/sVArVirtual impedanceVirtual resistance of DG1 = 0 ΩVirtual resistance of DG2 = 0 ΩThe virtual inductance of DG1 = 1 mHThe virtual inductance of DG2 = 1.176 mHThe allowable frequency range of variation50 Hz ± 1%The allowable voltage range of variation380 V ± 5%The operation mode remains the same as in the previous case; however, the perfect power sharing ratio isP1 : P2 = m2 : m1 = 2 : 3 and Q1 : Q2 = n2 : n1 = 1 : 1.The proposed controller for VSIs connected in parallel is tested with two linear loads (RL). As highlighted in Figures14-15, the proposed control strategy showed the ability to control the voltage and the frequency even during the load step change and faulty conditions. The voltage decreased during period 0–3 s for both DGs. It is stabilized at V1 = V2 = 380.2 V based on the voltage droop characteristics; when the VSIs supplied a permanent load, the frequency for both inverters decreased from 50 Hz to 49.75 Hz. From t = 0.3–0.5 s, load 2 was connected and the voltage decreased again to V1 = V2 = 379.8 V, and the frequency was f1 = f2 = 49.70 Hz.Figure 14
VSIs voltage outputs.Figure 15
Frequency variation for VSIs.The three-phase fault was introduced from time 0.5 s to 0.501 s; during this period, there was a reduction of active power and excessive reactive power demand; however, the voltage and frequency variation range was 379.5–380.9 V and 49.70–49.73 Hz, respectively. From 0.501 to 0.8, the system remained stable and kept in the same conditions as the previous scenario. The voltages and frequency were kept within the allowable range of variations throughout the operation, as shown in Figures14-15. It has been noticed that the proposed controller continued to minimize deviations for voltage and frequency between the inverters.Power sharing of parallel inverters was examined with a common load ofPload = 6000 W and Qload = 1500 VAR, and at t = 0.3 s, the sudden local load value of Pload = 6000 W and Qload = 1500 VAR is added. Two different line impedances were considered in this study. The MPC-based arctan droop control showed its capability to reduce the effect of line impedance difference on the parallel inverters to ensure power sharing among the inverters. The virtual inductor impedance was used to reduce the line impedance mismatch among the inverters, enhancing power decoupling and realizing the equal distribution of loads. From time 0–0.3 s, P1 = 2500 W, P2 = 3800 W, and Q1 = 748 kVAr, Q2 = 752 kVAr, and when there was load change at t = 0.3–0.5 s, the load 2 was connected to the system, and the power shared is as follows: P1 = 4800 W, P2 = 7200 W and Q1 = 1500 VAr, Q2 = 1500 VAR; the same results hold after the fault was cleared; the power sharing results are highlighted in Figures 16-17. According to the results, the power sharing ratio in all scenarios is as follows: P1 : P2 = m2 : m1 = 2 : 3 and Q1 : Q2 = m2 : m1 = n2 : n1 = 1 : 1, which is following the condition for good power sharing in this study.Figure 16
Active power sharing for VSIs.Figure 17
Reactive power sharing for VSIs.The findings showed the voltage THD evaluation of 1.23% for the studied system, which is below the one recommended by EN 61000-2-2 and IEC62040-3 in low voltage networks; Figure18 shows the voltage THD evaluation. The designed control strategy shows its superiority in voltage THD reduction compared to the other controllers, presented in Table 4.Figure 18
Voltage THD evaluation.Table 4
Comparison of different control strategies based on voltage THD.
ReferenceControl strategyVoltage THD (%)[30]FSC-MPC2.05[31]FSC-MPC with improved droop control3.8[32]Deadbeat control2.8[33]Modulated MPC2.54[9]FSC-MPC with a current observer2.71Proposed control strategyMPC-based arctan droop control1.23


## 5.1. Two VSIs with the Same Power Outputs and Identical Output Impedances

The inverter active and reactive capacity should be DG1 : DG2 = 1 : 1 ratio according to their droop coefficients, and the results showed that beforet = 0.3 s, the system was in steady-state operation, during that period, the voltage drops due to the rise of the load, but the voltage remained the same for all inverters. Their variations were in the range recommended by EN 50160 and IEEE1547 for low voltage networks; Figure 10 illustrates the variations for the two VSIs; as observed, the voltage decreased from 380.8 to 380.2 V. On the side of frequency, the controller behaved as expected while keeping the frequency variation for both VSIs to remain in the set range, and the frequency decreased from 50 Hz to 49.75 Hz during the period 0–0.3 s; this can be observed in Figure 11. The VSIs also delivered active and reactive power equitably as expected, and the results are presented in Figures 12-13. The results reveal that the inverters capacity ratio was P1 : P2 = 3 kW : 3 kW = m2 : m1 = 1 : 1 and Q1 : Q2 = 750 VAr : 750 VAr = n2 : n1 = 1 : 1, which fulfills power sharing conditions.Figure 10
Voltage delivered by VSIs.Figure 11
Frequency variation for VSI.Figure 12
Active power sharing of VSIs.Figure 13
Reactive power sharing of VSIs.Att = 0.3 s, a load step change was introduced, and during the period 0.3–0.5 s, the system remained stable while responding to the load added. The voltage and frequency for VSIs were decreased from 380.2 V to 380 V and from 49.75 Hz to 49.70 Hz, respectively; this can be seen from Figures 10-11. The power sharing was also performed properly during this period, where the ratio was P1 : P2 = 6 kW : 6 kW = m2 : m1 = 1 : 1 and Q1 : Q2 = 1500 VAr : 1500 VAr = n2 : n1 = 1 : 1 as shown in Figures 12-13.At 0.5–0.501 s, a fault was introduced to check the proposed controller’s effectiveness after the occurrence of a transient during operation. According to Figures10–13, the system remained stable after the fault has been cleared; during the fault, there was a reduction of active power and excessive reactive power demand.At 0.501–0.8 s, the voltage and frequency remained stable and the same as in the previous scenario. For power sharing, the VSIs continued to share the power as expected, and the ratio wasP1 : P2 = 6 kW : 6 kW = Q1 : Q2 = 1500 VAr : 1500 VAr = m2 : m1 = n2 : n1 = 1 : 1, as depicted on Figures 12-13.During all scenarios, all the voltage and frequency variations were kept in the preset range values of 380 V ± 5% and 50 Hz ± 1%, respectively; moreover, all the variations for voltages and frequency were kept in the range recommended by the IEEE 1547 standard. It has been noticed that there were no voltage and frequency deviations between inverters; hence, the proportional power sharing between the inverters is assured.


## 5.2. Two VSIs with Different Power Outputs and Different Output Impedances

The proposed strategy has also been tested on the parallel-connected VSIs with different powers and impedance outputs to study for its effectiveness. The parameters considered are given in Table3.Table 3
Tested data for different VSI’s power outputs and impedances.
ParameterValueDC bus voltage700 VNominal bus frequency50 HzNominal voltage380 VSampling timeTs = 30 usLine impedance, DG10.528 mH, 1.284 ΩLine impedance, DG20.352 mH, 0.856 ΩLoad 16000 W, 1500 VArLoad 26000 W, 1500 VArDroop coefficients, DG1ρ1=3e−5V/W, n1=4e−5rad/sVArDroop coefficients, DG2ρ2=2e−5V/W, n2=4e−5rad/sVArVirtual impedanceVirtual resistance of DG1 = 0 ΩVirtual resistance of DG2 = 0 ΩThe virtual inductance of DG1 = 1 mHThe virtual inductance of DG2 = 1.176 mHThe allowable frequency range of variation50 Hz ± 1%The allowable voltage range of variation380 V ± 5%The operation mode remains the same as in the previous case; however, the perfect power sharing ratio isP1 : P2 = m2 : m1 = 2 : 3 and Q1 : Q2 = n2 : n1 = 1 : 1.The proposed controller for VSIs connected in parallel is tested with two linear loads (RL). As highlighted in Figures14-15, the proposed control strategy showed the ability to control the voltage and the frequency even during the load step change and faulty conditions. The voltage decreased during period 0–3 s for both DGs. It is stabilized at V1 = V2 = 380.2 V based on the voltage droop characteristics; when the VSIs supplied a permanent load, the frequency for both inverters decreased from 50 Hz to 49.75 Hz. From t = 0.3–0.5 s, load 2 was connected and the voltage decreased again to V1 = V2 = 379.8 V, and the frequency was f1 = f2 = 49.70 Hz.Figure 14
VSIs voltage outputs.Figure 15
Frequency variation for VSIs.The three-phase fault was introduced from time 0.5 s to 0.501 s; during this period, there was a reduction of active power and excessive reactive power demand; however, the voltage and frequency variation range was 379.5–380.9 V and 49.70–49.73 Hz, respectively. From 0.501 to 0.8, the system remained stable and kept in the same conditions as the previous scenario. The voltages and frequency were kept within the allowable range of variations throughout the operation, as shown in Figures14-15. It has been noticed that the proposed controller continued to minimize deviations for voltage and frequency between the inverters.Power sharing of parallel inverters was examined with a common load ofPload = 6000 W and Qload = 1500 VAR, and at t = 0.3 s, the sudden local load value of Pload = 6000 W and Qload = 1500 VAR is added. Two different line impedances were considered in this study. The MPC-based arctan droop control showed its capability to reduce the effect of line impedance difference on the parallel inverters to ensure power sharing among the inverters. The virtual inductor impedance was used to reduce the line impedance mismatch among the inverters, enhancing power decoupling and realizing the equal distribution of loads. From time 0–0.3 s, P1 = 2500 W, P2 = 3800 W, and Q1 = 748 kVAr, Q2 = 752 kVAr, and when there was load change at t = 0.3–0.5 s, the load 2 was connected to the system, and the power shared is as follows: P1 = 4800 W, P2 = 7200 W and Q1 = 1500 VAr, Q2 = 1500 VAR; the same results hold after the fault was cleared; the power sharing results are highlighted in Figures 16-17. According to the results, the power sharing ratio in all scenarios is as follows: P1 : P2 = m2 : m1 = 2 : 3 and Q1 : Q2 = m2 : m1 = n2 : n1 = 1 : 1, which is following the condition for good power sharing in this study.Figure 16
Active power sharing for VSIs.Figure 17
Reactive power sharing for VSIs.The findings showed the voltage THD evaluation of 1.23% for the studied system, which is below the one recommended by EN 61000-2-2 and IEC62040-3 in low voltage networks; Figure18 shows the voltage THD evaluation. The designed control strategy shows its superiority in voltage THD reduction compared to the other controllers, presented in Table 4.Figure 18
Voltage THD evaluation.Table 4
Comparison of different control strategies based on voltage THD.
ReferenceControl strategyVoltage THD (%)[30]FSC-MPC2.05[31]FSC-MPC with improved droop control3.8[32]Deadbeat control2.8[33]Modulated MPC2.54[9]FSC-MPC with a current observer2.71Proposed control strategyMPC-based arctan droop control1.23


## 6. Conclusion

This study proposes a hybrid control method called MPC-based arctan droop control to control the parallel-connected VSIs in an islanded microgrid. The voltage and frequency were adequately monitored. The proposed control strategy has been tested in an MG with VSIs of the same power ratings and same output impedance and VSIs of different power ratings and different output impedances. The proposed control strategy demonstrated promising results in controlling the MG frequency and voltage according to EN 50160 and IEEE 1547-2018. It also showed the ability to control the VSIs to share power equitably in a microgrid. Various simulation results demonstrate that the proposed control method is feasible and effective.



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*Source: 1870590-2021-03-11.xml*