Finite Control Set Model Predictive Control for Parallel Connected Online UPS System under Unbalanced and Nonlinear Loads

Abstract

[...]uninterrupted power supply (UPS) is attracting much attention from both industry and research, so as to find new ways to meet the demand of these devices. [...]it is necessary to fulfill the growing demand without upgrading the existing UPS system. [...]the parallel UPS system operation is the most effective solution to provide continuous power to load, enhance system stability, lower the maintenance cost, and increase system capacity. MPC has attracted the attention of researchers because it can handle a multivariable system, has a fast-dynamic response, and can incorporate system constraints into control law in a simple manner. [...]it can also integrate nested control loops in a single loop, has high delay controlling ability, and shows robustness against system parameter variations [21,22]. Switching Frequency Reduction Scheme In a UPS system, the inverter is used to interface the battery bank with the output load in order to feed the critical load. [...]switching frequency can play a key role in decreasing loss and is associated with low frequency and vice versa.

1. Introduction

With the advancement in modern communication systems, uninterrupted power to supply lifesaving medical equipment and data centers is required for critical load in all grid conditions. In order to operate this sensitive equipment, the reliable power requirement is a major challenge for engineers and researchers. Hence, uninterrupted power supply (UPS) is attracting much attention from both industry and research, so as to find new ways to meet the demand of these devices.

Generally, UPS systems provide regulated sinusoidal output voltage, with low total harmonic distortion, and fast transient response irrespective of the change in a load connected with the system. As per international electrotechnical commission (IEC) standards for UPS systems, all types of UPS system must satisfy the following conditions.

1. THD of inverter’s output voltage is less than 4% under all loading conditions.

2. Voltage drop is not more than 5% under all loading conditions.

3. The inverter provides constant output root mean square (RMS) voltage, irrespective to disturbance or variation in system parameters such as temperature, load current, etc. [1]

The UPS system is classified into three categories, offline, line interactive, and online UPS.

Of these three categories, online UPS systems are famous for both high power and voltage applications. Online UPS systems provide strong tolerance to overcome the variations in grid frequency, system voltage, and power issues. Moreover, online UPS systems also provide isolation to load from the polluted grid. Online UPS consists of a controlled rectifier, an inverter, and a static switch as demonstrated in Figure 1 [2]. Under the normal mode of operation, the rectifier maintains regulated DC-link voltage, whilst also charging the battery bank of the UPS. The inverter converts the DC-link voltage into regulated AC to maintain the load demand. During a grid outage mode of operation, the AC grid is disconnected through the circuit breaker However, the inverter continues to supply power to load from the battery without any interruption. The inverter of the online UPS remains active in both modes of operation.

Usually, UPS systems feed critical loads that cannot afford power failure. Therefore, it is necessary to fulfill the growing demand without upgrading the existing UPS system. Thus, the parallel UPS system operation is the most effective solution to provide continuous power to load, enhance system stability, lower the maintenance cost, and increase system capacity. It is for these reasons that many control schemes have been proposed for parallel operations of the inverter in the literature, such as, centralized control [3], master–slave control [4], average load sharing [5], etc. However, the above-mentioned control schemes require an intercommunication network between the inverters for coordination.

Using a communication network decreases the reliability and increases the complexity of the system. Therefore, communication less control methods, such as droop control, are used to regulate power sharing among UPSs. Several types of droop control are reviewed in [6]. The objectives of droop control in UPS systems are power sharing between parallel-connected inverters in UPS, and stabilizing the voltage and frequency at the AC bus.

Research has been performed on linear control, deadbeat control [7,8], robust control [9,10], slide mode control [11] etc. [12]. Linear control schemes such as proportional-integral (PI), proportional-resonant (PR), and proportional-integral-derivative (PID) are frequently implemented for the inverter in UPS systems. However, this system inherits major practical limitations, such as tuning of parameters, poor transient response, and poor tracking of the sinusoidal reference. In [13], a feedback linearization control was proposed for a three-phase UPS system and the nonlinear behavior of the system was used to achieve low voltage THD. Consequently, it is not easy to implement this technique due to the computation complexities.

In nonlinear controls, The deadbeat controller has widely been implemented for UPS systems [7]. Deadbeat control puts all system poles at zero. Hence, tracking errors are equal to zero in small sampling instants, therefore giving the fastest transient response. However, one major disadvantage includes parameter variations and external disturbance. Further control schemes, based on nonlinear control theory, have been proposed in [14,15,16]. In these, the authors proposed the control scheme for cell divisions in nervous systems as asymmetry and symmetry [14]. In [15] a fuzzy based proportional-integral (PI-FC) controller was developed to control the class of a servo system. A new stable Takagi–Sugeno fuzzy control developed for the process model and matrices gain has been achieved by a swarm optimization algorithm in [16]. On the other hand, digital robust control approaches, such as hysteresis control, can handle system uncertainties and give an optimal tradeoff between performance and stability [10]. However, the mathematical modeling and implementation on digital platforms is quite complex. As compared to a classic controller, slide mode control (SMC) has optimal performance, robustness against parameter variations, and excellent dynamic response under different load operations. However, the practical limitation of SMC includes chattering phenomena, a complex mathematical approach, and high switching losses [17]. The learning based controls, such as neural network control [18], repetitive control [19], and interactive learning control [20] are able to improve the steady-state performance of the controller. Yet, these controllers have slowed dynamic response and consume a significant amount of time in learning algorithm. Therefore, the hybrid control technique is normally used for implementation, as highlighted in [18,20].

Model predictive control (MPC) is a type of non-linear, digital, and advanced control technique. Predictive controllers use the system mathematical model to envisage future responses and generate optimal control action according to predefined optimized standard. MPC has attracted the attention of researchers because it can handle a multivariable system, has a fast-dynamic response, and can incorporate system constraints into control law in a simple manner. Moreover, it can also integrate nested control loops in a single loop, has high delay controlling ability, and shows robustness against system parameter variations [21,22].

In [23] a new algorithm for model predictive direct power control has been proposed for AC/DC active rectifiers. The proposed scheme can control rectifier power and reduce its switching losses. A modified model predictive current control had been proposed in [24]. The researcher of this study attempted to reduce the spread spectrum by adding a filter to the variables that are evaluated by the cost function. The results of this highlighted that such a process helps in reducing the switching losses and lessens system harmonics. Similarly, in [25] the authors proposed predictive power control for a battery storage system and compared thus with PI linear control. MPC was shown to have good results and performance as compared to conventional PI control. In [26,27] the authors developed a tool for the analysis of predictive current control (PCC) based on the locus of a reduced set of performance indices, and highlighted the comparison between the common PCC approach. The issues that occur due to long prediction horizon in FCS–MPC has been comprehensively examined in [28], in terms of harmonic content, but the major complication with long receding horizon is computational complexity. Voltage MPC has been proposed for a four leg inverter but its THD in voltage has been shown to be more than 3%, which is above the legal limit accepted by the some European countries [29]. In [30] an observer-based voltage model predictive control (VMPC) was proposed for UPS applications in which the system cost is reduced but its performance is not adequate in terms of THD and steady-state error. In [31] an implicit MPC was developed for VSI with LC filters in an islanded mode. But this technique showed susceptibility to system external uncertainties and parametric variation. Moreover, this system required more computational burden as compared to other MPC based approaches. It measured the filter and load current instead of estimation, which may have increased the system cost due to the use of sensors. However, the MPC has not been thoroughly investigated for the parallel operation of the inverter in UPS applications.

This paper investigates the FCS–MPC control scheme for the inverter of UPS system and parallel operation of the UPS to feed a common AC load. The proposed control scheme used the system mathematical model to forecast the future behavior of the system for all possible switching states at the corresponding sampling instant. Following this, a cost function was conducted to find the optimal action, which was then applied to the next sampling instant. For the reduction of inverter switching frequency, a two-step prediction horizon-based cost function (CF) was employed. Droop control was also implemented for accurate power sharing among the inverters of the UPS system. Real-time simulation was conducted in MATLAB/Simulink to show the robustness and effectiveness of the controller so as to maintain system power quality under linear and nonlinear loading conditions.

The remaining sections of the paper are organized as follows: In Section 2, the proposed system is described and the comprehensive mathematical model of two-level voltage source inverter (VSI) with linked LC filter is proposed. Following this, a discussion of the required discretization aspects, for FCS–MPC implementation is presented. In Section 3, the operating principle of FCS–MPC and the working of the proposed controller is explained. In Section 4, for the parallel operation of inverters, droop control is proposed, and its limitations are discussed. Detailed simulation results with linear control are discussed and compared in Section 5. Finally, Section 6 provides the conclusion.

2. Description and Mathematical Modelling of System

A cluster of parallel-connected inverters in the UPS system is illustrated in Figure 2. For system modeling, a single inverter, along with its LC filter, is considered, as shown in Figure 3. Accurate mathematical modeling of both inverter and filter are required to achieve good performance of the controller. _vpc is the output voltage at PCC, _i0 is output current of the inverter, _Lf is filter inductance, _Cf is filter capacitance, and _vdc is capacitor DC link voltage. Similarly, _{Sx , Sy , Sz} are the legs of the converter. The inverter considered in this paper, is two-level voltage source inverters (2L-VSI) and its circuit diagram is demonstrated in Figure 3. However, the harmonics generated by 2L-VSI are reduced by the LC filter.

Though modeling of VSI has been shown to be realized in any frame of reference [32], VSI modeling in an αβ frame of reference was performed in this paper. In order to implement the proposed control schemes, the following assumptions were made i.e., the system conditions were balanced, and zero sequence components were neglected. By using the Clark transformation, all the state variables, such as three-phase voltages and currents were transformed into the stationary frame of reference.

v¯=_vα+_jvβ=T¯^{[_vx,_vy,_vz]′}

i¯=_iα+_jiβ=T¯^{[_ix,_iy,_iz]′}

where

T¯=13[^{1ej23π1ej43π}]

The switching arrangements of VSI are presented in Figure 4a. _Sx,_Sy,_Sz are the three gating pulses given to the corresponding legs of VSI. Each leg has two possible states i.e., 0 or 1 and expressed in Equations (4) to (6):

_Sx={1, if _Q1 is ON and _Q4 is OFF0,if _Q1is OFF and _Q4 is ON}

_Sy={1, if _Q2 is ON and _Q5 is OFF0,if _Q2is OFF and _Q5 is ON}

_Sz={1, if _Q3 is ON and _Q6 is OFF0,if _Q3is OFF and _Q6 is ON}

Two level VSI has a possible eight (²³) voltage vectors. The output voltage of inverter legs can be determined by taking the product of the state of the respective leg and dc link voltage.

_{vxN =Sx ·vdcvyN}=_Sy·_vdcvzN=_Sz·_vdc

Common mode voltage _VnN can be found by using Kirchhoff’s voltage law:

_VnN=_vxN+_vyN+_vzN3

Inverter phase voltage can be calculated through the following equation:

_vxn=_vxN−_vnNvyn=_vyN−_vnNvzn=_vzN−_vnN

Table 1, explains the position of all possible voltage vectors in the αβ frame. _v0→ and _v7→ are zero vector and lie at the origin.

2.1. LC Filter Modelling

As shown in Figure 3, a three-phase LC filter was connected at the output terminal of VSI. As previously mentioned, the LC filter was used to attenuate the switching harmonics. Each filter consists of an inductor _Lf and ESR resistance _Rf , with current _if flowing through it. Similarly, a capacitor _Cf with voltage _vpc is connected in parallel to the load. _if and _vpc were used as the state variables in this system. The block diagram of an LC filter model is shown in Figure 4b. Filter model’s dynamics can be explained by Equations (10) and (11), one describes the inductance behavior and other explains the capacitance behavior. Filter current equation is expressed as:

_Lfdifdt=_vt−_vpc−_{if Rf}

where _vt is the possible voltage vectors taken from Table 1. Dynamic response of capacitor voltage is expressed as follows:

_Cfdvpcdt=_if−_i0

where _i0 represents output current. Equation (10) and Equation (11) can be rewritten in state space form as Equation (12):

dd[_ifvpc]=A[_ifvpc]+B[_vti0]

Here,

A=[_−RfLf−1_Cf1_Cf0]

and

B=[1_Lf00−1_Cf]

Equations (12) to (14) present the complete continuous state space (CSS) model of LC filter, which has two inputs i.e., voltage vector _vt ,which is defined in Table 1 and output current _i0 .

2.2. Discrete Time Domain Modelling

For the practical implementation of a CSS model on the digital control system, it is necessary to convert the CSS model into a discrete time model. Among discretization techniques, in this paper, zero-order hold (ZOH) method is deployed because it gives the precise conversion of the CSS model to discrete time state space model at sampling instants of staircase inputs.

It is assumed that the value of _vdc is constant and model in the discrete time domain as expressed in Equation (15):

[_if(_tk+1)_vpc(_tk+1)]=_Ad[_if(_tk)_vpc(_tk)]+_Bd[_{vt (tk})_i0(_tk)]

_Ad=^e_ATs

_Bd=_∫0Ts ^eAτ _{B dτ}

_Ts is the sampling instant. Equation (14) is used to calculate the voltage and current for two step receding horizons.

3. Operating Principle of FCS–MPC

FCS–MPC is proposed in this paper for the voltage control of an inverter in the UPS system as demonstrated in Figure 5. The basic principle of voltage MPC is as follows,

1. In the start of each sampling time, _vpc,_if and _i0 are measured through sensors.

2. The information is sent to the algorithm, it is defined by the initial point from where the algorithm forecasts the future behavior of controlled variables using Equation (14), for all possible voltage vectors.

3. These predicted values are used to find the predefined cost functions (CF) and a voltage vector that has a minimal value of CF as applied to VSI.

A flowchart of the VMPC algorithm is shown in Figure 6. Designing a cost function (CF) in FCS–MPC is fundamental. CF permits to solve multi-objective problems and can be defined as:

_gGen=∑_i=tktk+N−1_{‖vfe(i)‖22}+_hlim(i)+_λu ^sw2(i)

_vfe(i) is predicted tracking error, _hlim(i) is current limit constraint, ^sw2(i) is a term used to reduce the switching frequency and can be controlled by weighting factor _λu . The term used in Equation (18) can be stated as:

_vfe(i)=_vf∗(i)−_vf(i)

_hlim(i)={0,if|_if(i)|≤_imax∞,if|_if(i)|>_imax

sw(i)=∑|u(i)−u(i−1)|

CF used in this paper is defined in Equation (22) and is a generalized form of Equation (18) with N = 2 and has no secondary objective. Its basic function is to minimize the Euclidean distance at each sampling time. In this paper, cost function defines the desired system behavior and is expressed as follows:

_gv=^{(v∗ _(pc)α(_tk+2)−_v(pc)α(_tk+2))2}+^{(v∗ _(pc)β(_tk+2)−_v(pc)β(_tk+2))2}

^v∗ _(pc)α and ^v∗ _(pc)β are real and imaginary parts of output voltage reference at _tk+2 instant, taken from droop control. _v(pc)α and _v(pc)β are predicted output voltage at _tk+2 instant. This cost function delivers a minimum voltage error and it also reduces the switching frequency of VSI, The method used in this paper for switching frequency reduction is found in [33]. There is the possibility to add additional constraints, such as current limitations etc. However, by adding these constraints, the tuning of weight coefficients creates an issue and deteriorates the controller performance.

Switching Frequency Reduction Scheme

In a UPS system, the inverter is used to interface the battery bank with the output load in order to feed the critical load. Thus, switching frequency can play a key role in decreasing loss and is associated with low frequency and vice versa. In this case, a two-step reduction technique was employed in order to obtain the low switching frequency. In a one-step sampling case, there are eight vectors exploited for one sampling period. To evaluate the prediction horizon N = 2, two voltage vectors were considered, such as one voltage vector was applied during the first sampling period, and another vector was applied during second sampling period. The total possible sequences of the voltage vector (72) is shown in Figure 7b. It shows that the total possible voltage vector sequences are 49, which requires a large amount of calculations, hence the increase in the computational burden and control accuracy.

In order to overcome the above-mentioned issue, the two-steps prediction was accomplished. For the first and second sampling period, the same voltage vector was applied [33]. Moreover, in two-step prediction, only seven voltage vectors are employed, as shown in Figure 7c. This technique showed similar performance, but it had less switching frequency. Equation (15) can be used to predict the voltage up to ( _tk+2 ). In other words, for a two sampling instant, all eight possible voltage vector can be used sequentially. This technique also reduced the switching frequency of VSI and voltage ripple.

4. Droop Control

Generally, UPS systems are used to deliver power to sensitive loads, which cannot bear a surge or spike of voltage and power outage. Therefore, it is essential to overcome the growing load demand by upgrading the present UPS system. However, such upgrades increase system cost and the only solution to overcome the above-described problem, is to operate the UPS system in parallel. A parallel connected UPS system increases the system stability in terms of continuous power supply to load, in spite of any disruption and also fulfills system requirements, lowers the cost, and increases the system reliability. In a parallel-connected UPS system, the major task is to achieve accurate and precise active and reactive power sharing amongst the VSI’s. To connect several inverters in parallel without any intercommunication, the droop control is the best choice [34]. Fundamentally, the basic idea of droop control is based on instantaneous power theory. Active power can be calculated by phasor or power angle ∅ and reactive power Q is based on voltage amplitude difference. Normally, a droop approach is used for sharing the load among the parallel operating generators in a conventional power system based on droop curve characteristics as demonstrated in Figure 8, according to their output impedances.

Droop control uses the frequency and amplitude of inverter voltage to regulate the active and reactive power. Droop control can be expressed as follows:

_Vref=_Vnom−_{kq Qcal}

_fref=_fnom−_{kp Pcal}

where, _Vref and _fref are reference voltage amplitude and frequency used to synthesis the _V(ref)αβ . _Vnom and _fref are nominal voltage amplitude and frequency at the point of common coupling. Following equations are used to calculate the _Pcal and _Qcal as:

_Pcal=_{vpc,α i0,α}+_{vpc,β i0,β}

_Qcal=_{vpc,β i0,α}−_{vpc,α i0,β}

Power, as determined by Equations (25) and (26), is fed to low pass filter in order to mitigate the oscillations. However, it shows slow response during load transients. _kq and _kp are voltage and frequency droop coefficients. There is a tradeoff between droop coefficients and voltage regulation i.e., increasing the droop coefficients will result in accurate power-sharing but deprive the voltage regulation. The droop coefficients also determine the slope of droop curves. Droop coefficients are defined as:

_kq=ΔV_Qmax

_kp=Δf_Pmax

ΔV and Δf are the extreme allocated divergence of voltage and frequency. _Qmax and _Pmax are nominal active and reactive power delivered by the system. Usually, inductive based output impedance is considered where larger inductors filter is used. Yet, in a low-voltage system, resistive based droop is used because line impedance is resistive in nature. Comparisons between droop are expressed in the Table 2. Figure 9 shows the block diagram of droop control used in this paper.

5. Results

In order to validate the performance of the proposed controller for UPS system, extensive simulations have been performed for both single and parallel-operated inverters (VSI) using MATLAB/Simulink. Figure 10 presents the system under study in this paper. Parameters for the test system are given in Table 3.

The filter design procedure in an FCS–MPC regulated UPS system has not been clearly developed. Hence, its parameters are found in a similar way as reported in the literature [35].

Different simulation cases have been carried out to show the robustness and performance of the proposed controller. First, steady state simulation for a single inverter was conducted to verify the performance of the proposed controller under sinusoidal voltage reference. Following this, a transient response, step change of load, and system response under nonlinear load was investigated. The results showed that the system maintained high robustness and stability. Finally, two inverters were operated in parallel with the proposed controller. A proper power-sharing model was attained by using droop control and respective power-sharing was verified.

5.1. Steady State Analysis

Figure 11 demonstrates the voltage and output current waveforms of the single inverter in the UPS system under linear RL balanced load. During operation, 18 kW active and 7 kvar reactive power was supplied to the load through an inverter. The reference of voltage amplitude and frequency was set to 318 V and 60 Hz. Output voltage and current waveforms were sinusoidal with low harmonic distortions. The performance of the designed controller, in term of THD, was compared with the standard literature [31]. The output voltage THD, in case of [31], was observed as 2.93%. However, in the proposed FCS–MPC scheme, the THD of the output voltage was 0.95%, which was much less than IEC standards for the UPS system.

5.2. Load Transient Analysis

The load transient analysis was performed to examine the transient recovery time and overshoot of magnitude of current and voltage under load changing from no load to full load and vice versa. Here, an 18-kW active load and the 7 kvar reactive load were connected. Figure 12 shows the transient behavior of the system under step change in load from 100% to 0% and 0% to 100%. At time t = 0.1 s, the total load was disconnected from the system and at t = 0.12 s the full load was connected back to the system. As can be seen from the Figure 12 output current of the system varied according to load. However, output voltage of the system was slightly distorted during transients and returned to steady state within no time. Figure 13 shows the RMS output voltage graph. It shows a voltage spike at time t = 0.1 s when the load was disconnected from AC bus. The maximum value of the spike was nearly 260 V, which was under the specifications of the switches.

The proposed controller was also tested for the nonlinear loading condition as. This test was performed as a UPS system supply power to the nonlinear load in the emergency mode of operation. Figure 14 represents output voltage and load current under nonlinear load. It was observed that the voltage waveform was still sinusoidal with very small harmonic distortion despite the distorted load current. THD of output voltage waveform was 1.42%, which was under the IEC62040-3.

Figure 15 shows controller performance for a step change in load where at t = 0.12 s, the load was instantaneously doubled to the rated load. However, within no time, the voltage became stable and remained sinusoidal with low harmonic distortion.

Figure 16 and Figure 17 show the voltage and current waveform of MPC and PI controller under different loading conditions, respectively. As shown in Figure 16, at t = 0.05 s the unbalanced load was energized, and the proposed controller almost eliminated the imbalance in voltage while in case of PI controller, voltage remained unbalanced. Figure 16a,b illustrate the current and voltage waveform under linear and unbalanced loads when MPC is employed. Table 4 presents the comparison between PI and proposed MPC in term of %THD in voltage under different loads.

At t = 0.05 s, the nonlinear load was energized while the unbalanced load was switched off. It was observed in Figure 18b and Figure 19b that the current becomes distorted due to nonlinear load. Figure 18 shows that the PI controller was ineffective to suppress the harmonics in voltage. But, the proposed MPC removed the harmonic distortion in output voltage as demonstrated in Figure 19a.

Table 5 expresses the comparison among the different control techniques. The proposed controller harmonics rejection capability was quite effective, and its performance was far superior as compared to the other controllers.

5.3. Multiple UPS Systems

Two or more inverters were connected in a UPS system to increase the system reliability and decreases the system cost. Therefore, in this subsection, the proposed controller was tested for the parallel-connected UPS operation. Conventional droop control was used for proper load sharing among parallel inverters of the respective UPS systems. At time t = 0.03 s, the load on the system was instantaneously doubled, both the VSIs increased their power generation to fulfill the load demand. Figure 20 illustrates the output voltage and load current of VSI₁ and VSI₂. It was observed that at t = 0.03 s, increased in demand the load was fulfilled by both inverters and the voltage remained stable. Power sharing between the inverters is shown in Figure 21 where 36 kW active and 16 kvar reactive power were equally shared among the VSI’s.

To verify the proposed system performance during single UPS fault, hot-swap operation was performed, as shown in Figure 22. At t = 0.1 s, VSI₂ was disconnected from the system due to an external fault. It can be noticed that VSI₁ instantly increased its power generation to fulfill the load demand. Quality of voltage and current remained stable despite the transition period when VSI₂ was isolated from the system. Figure 23 represents the active and reactive power provided by the inverter modules in the UPS system.

6. Conclusions

In this paper, an FCS–MPC control scheme is presented for a 2 L three-phase inverter with output LC filter. Furthermore, this paper highlighted the possible implementation of parallel inverters in a UPS system. The proposed controller had no parameters to adjust, requiring a model of the system for calculating predictions of the controlled variables. The gate-drive signals were generated directly by the controller, so a modulator was not needed. The output voltage was directly controlled, without using a cascaded control structure. The performance of the proposed predictive controller was demonstrated by implementing it in the MATLAB/Simulink environment. Simulation results showed that the proposed scheme achieved a good voltage regulation with linear and nonlinear loads. Results were compared with other MPC controllers. The proposed controller showed exemplary results with a considerable reduction in the THD of the system. Droop control was implemented for power sharing among the inverters. It showed accurate power sharing in both steady state and transient conditions. The proposed FCS–MPC showed excellent performance in dynamic steady state, transients conditions, and hot-swap operation of the system.

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Space Vector		Switching Vector	On-State Switch	Vector Placing
Zero vector	v0,7→	[PPP]	Q1,Q3,Q5	v0→=0 v7→=0
		[OOO]	Q4,Q6,Q2
Active vector	v1→	[POO]	Q1,Q4,Q6	v1→=23vdc
	v2→	[PPO]	Q1,Q3,Q6	v2→=13vdc+j33vdc
	v3→	[OPO]	Q2,Q3,Q6	v3→=−13vdc+j33vdc
	v4→	[OPP]	Q2,Q3,Q5	v4→=−23vdc
	v5→	[OOP]	Q2,Q4,Q5	v5→=−13vdc−j33vdc
	v6→	[POP]	Q1,Q4,Q5	v6→=13vdc−j33vdc

Output Impedance	Z=jX (Inductive ∅=90°)	Z=R (Resistive ∅=90°)
Active Power	P=EVXsin∅≅EVX∅	P=EVcos∅−V2R≅VR(E−V)
Reactive Power	Q=EVcos∅−V2X≅VX(E−V)	Q=−EVRsin∅≅−EVR∅
Frequency Droop	ω=ω∗−kPP	ω=ω∗ +kPQ
Amplitude Droop	E=E∗−kQQ	E=E∗−kQP
Droop coefficient	kP=ΔωPN	kP=Δω2QN
Droop coefficient	kQ=ΔE2QN	kQ=ΔEPN

Parameter	Value
DC link Voltage	vdc=1000 V
Sampling Time	Ts=2 μs
LC-filter	Cf=250 μF, Lf=2 mH
Damping Resistance	Rf=0.94 Ω
Linear load	P=18 kW, Q=7 kvar
Non-linear load (diode rectifier)	P=10 kW, Q=4 kvar
Nominal Voltage	Vnom=318 V
Average Switching Frequency	fsw=4.4 kHz
Rated Frequency	fnom=60 Hz
Droop coefficients	kq=0.001, kp=0.001
Proportional Gain	kp = 50
Proportional Integrator	kI=0.15

Load Types	Proposed MPC Voltage THD (%)	PI Voltage THD (%)
Balanced resistive load	0.96	1.22
No Load	0.96	1.22
Non-Linear Load	1.23	4.31

Reference	Control Technique	%Voltage THD Linear Load	%Voltage THD Non-Linear Load	Controller Complexity
[36]	PI	16	42	Low
[7]	Dead Beat	2.1	4.8	Medium
[37]	PR	1.4	4.6	Low
[11]	SMC	-	2.66	High
[29]	MPC	5.1	6.7	High
[30]	MPC (observer based)	2.82	3.8	Medium
[31]	Implicit-MPC	2.93	Not studied	Medium
Proposed	FCS–MPC	0.96	1.23	Medium

Author Contributions

H.S.K. studied the literature review and found the research gap, performed the simulation work, and wrote the paper. M.A. helped in controller stability analysis. A.W. and S.U.A. actively contributed to the arrangement and format of the paper. J.I. helped to revise the paper. M.A. supervised this research work.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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