Space vector-based model predictive current controller for grid-connected converter under unbalanced and distorted grid without a phase-locked loop

Space vector-based model predictive current controller for grid-connected converter under unbalanced

and distorted grid without a phase-locked loop

Item Type Article

Authors El-Nagar, Mohammed;Elattar, Omar;Ahmed, Khaled;Hamdan, Eman;Abdel-Khalik, Ayman S.;Hamad, Mostafa S.;Ahmed, Shehab Citation El-Nagar, M., Elattar, O., Ahmed, K., Hamdan, E., Abdel-Khalik,

A. S., Hamad, M. S., & Ahmed, S. (2023). Space vector-based model predictive current controller for grid-connected converter under unbalanced and distorted grid without a phase-locked loop. Alexandria Engineering Journal, 77, 265–281. https://

doi.org/10.1016/j.aej.2023.06.092 Eprint version Publisher's Version/PDF

DOI 10.1016/j.aej.2023.06.092

Publisher Elsevier BV

Journal Alexandria Engineering Journal

Rights Archived with thanks to Alexandria Engineering Journal under a Creative Commons license, details at: http://

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Link to Item http://hdl.handle.net/10754/693277

ORIGINAL ARTICLE

Space vector-based model predictive current controller for grid-connected converter under unbalanced and distorted grid without a phase- locked loop

Mohammed El-Nagar

^a,*

, Omar Elattar

^a

, Khaled Ahmed

^b

, Eman Hamdan

^c

, Ayman S. Abdel-Khalik

^a

, Mostafa S. Hamad

^d

, Shehab Ahmed

^e

aElectrical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria, Egypt

bDepartment of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1XQ, UK

cMarine Engineering Technology Department, Arab Academy for Science, Technology and Maritime Transport, Alexandria 1029, Egypt

dResearch and Development Center, Arab Academy for Science, Technology and Maritime Transport, Al-Alamein 5060305, Egypt

eCEMSE Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Received 2 May 2023; revised 12 June 2023; accepted 28 June 2023

KEYWORDS Distorted grid;

Harmonic mitigation;

Power quality;

Extended complex Kalman filter;

Phase locked loop problems;

Model predictive current controller;

Space vector pulse width modulation;

Voltage source converter

Abstract This paper presents an improved current control strategy for a three-phase voltage source converter connected to distorted grid. The proposed controller can successfully compensate the induced harmonic currents, while suppressing possible active power ripple. To manage various objectives and deal with tough restrictions, model predictive control can offer a promising solution.

However, a modified extended complex Kalman filter is introduced to estimate the positive and neg- ative fundamental components of the grid voltages which are used with the real and reactive power references to generate pure sinusoidal reference currents without the need for a phase-locked loop (PLL), avoiding its challenges under abnormal grid conditions. An improved space vector-based model predictive control (SVM-MPC) is proposed to regulate the real and reactive power compo- nents. The main target of this controller is to maintain harmonic free grid currents, even in unbal- anced and distorted grid conditions, with enhanced performance in terms of fast dynamic response and good steady-state behavior. The proposed SVM MPC uses Pythagoras theorem to determine duty factors, avoiding trigonometric functions, which improves the computational burden and implementation complexity. In addition, fixed switching frequency can be ensured. Simulation

* Corresponding author.

E-mail address:Mohammed.elnagar@alexu.edu.eg(M. El-Nagar).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

H O S T E D BY

Alexandria University

Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

https://doi.org/10.1016/j.aej.2023.06.092

1110-0168Ó2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

and experimental results are carried out to validate the effectiveness and the practical feasibility of the proposed controller under distorted grid conditions.

Ó2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

The process of integrating distributed energy sources into elec- trical grid through power electronic converters commonly leads to unwanted harmonics in voltage and current wave- forms. The limits of grid current harmonics are defined accord- ing to the standards of IEEE 519[1]. However, the dominant harmonics of distorted grid voltage are typically 5th, 7th,11th, and 13th harmonics[2]. According to IEEE Std. 519-2014, the maximum allowable percentage share of grid current harmon- ics is given inTable 1.

The current controller for the three-phase grid-connected voltage source converter is expected to use harmonic free cur- rent references under unbalanced and distorted grid condi- tions. Furthermore, the control strategy is expected to avoid active power oscillations in case of unbalanced and distorted grid conditions.

Several control methods have been used in grid connected applications. In vector current control, the real power compo- nentidand the reactive power componentiqare controlled sep- arately in rotatingdqframe[3]. Therefore, a phase locked loop (PLL) is necessary to synchronize with grid[4]. Under unbal- ance and distorted grid conditions, the design of PLL is a chal- lenging task, where, PLL dynamics and improper bandwidth could interface with the current control and lead to instability problem[5–8]. The PLL topologies proposed for unbalanced and distorted grids are highly complex[9], and need large com- putational burden[10]. In addition, the linear controllers as multiple proportional-integral (PI)[11,12]or multiple propor- tional resonant (PR) controllers[13]used in control implemen- tation considering harmonics require difficult tuning to achieve acceptable dynamic response[14,15]and to ensure stability in discrete time domain[15,16]. However,[17]presents compara- tively a modified adaptive bandpass filter ABPF-based PLLs as an attempt to improve the transient performance of cas- caded delayed signal cancellation CDSC-based PLL and mov- ing average filter MAF-based PLL. However, it still experiences slow dynamic response under harmonic conditions and is sensitive to frequency deviations. Several proposals are introduced in[18,19]to avoid PLL and the rotating d-q frame.

In[20], dual PR controllers with constant reference frequency are applied in the stationaryabframe to achieve the positive and negative sequence decomposition. However, it is still not applied in grid connected applications. In conclusion, VCC techniques have a number of difficult issues to deal with, such

as their reliance on recent measurements rather than future ones, which leads to relatively poor transient behavior, the need for linearization in nonlinear systems, which leads to a complex control structure, and complex and challenging PLL techniques that operate on separate positive and negative con- trol tracks.

Nonlinear controllers, on the other hand, such as sliding mode control[21], partial feedback linearization based control [22], back-stepping control [23], and others provide fast dynamic response because they use the dynamic model of the grid-connected converter to obtain their control input, elimi- nating the need for multiple parallel nonlinear controllers for tracking multi frequency reference currents. Model predictive controllers provide the same benefit and, in addition, it fea- tures a simple real-time hardware implementation, higher sta- bility margin, an ability to handle multiple objectives and nonlinear constraints, and the ability to handle nonlinear and complicated linear systems[24–28].

The performance of MPC under unbalanced and distorted grid conditions has been evaluated in [10,29–33]. However, none of these studies has addressed the trade-off between active power oscillation and current THD. Moreover, the finite control set (FCS) predictive current controller in[32]is intro- duced with the MAF-based PLL to obtain sinusoidal current references, where, MAF-based PLL demerits are discussed above. In [34], a model-free predictive current control (MFPCC) method is proposed based on an extended state observer (ESO) to reject active disturbance; however the CDSC used in current reference calculations makes it unattrac- tive for situations with a high current control bandwidth. Esti- mation of sequence components is performed using neural network in[35]where, trigonometric calculations are required and the learning rate must be tuned carefully. Although several control techniques have been introduced to deal with unbal- anced grid issues, same interest has not been given to distorted conditions. Sliding mode observer for unbalanced grid volt- ages is introduced in[36], where, the requirement for frequency locked loop (FLL) with its challenges is necessary [37]. The state observers are employed for unbalanced grid in [38,39];

however, the complexity of system and tuning gains to ensure stability still represent a burden.

In[40], extended complex Kalman filter (ECKF) is used to estimate the positive sinusoidal signal in grid with white noise.

Ref. [41], [42] and [43] have been proposed to calculate the power frequency in distorted grid by estimating their symmet-

Table 1 Maximum Harmonic Current Distortion in Percent of Current for Odd Harmonics (ORDER h) by IEEE Std. 519-2014.

h <11 11h17 17h23 23h35 35h

% 4.0 2.0 1.5 0.6 0.3

rical components. The idea to compensate the DSP calculation delay using modified Kalman filter is discussed in [44]. The ECKF proposed in [45] is used to estimate the symmetrical components from unbalanced grid voltages without PLL. So that, the tuning and delay issues have been avoided and supe- rior noise rejection has been achieved compared with existing techniques. Furthermore, the grid voltages are estimated two step in advance where, the estimated values are used in the computation time compensation with no need to Lagrange extrapolation. The introduced predictive current controller, however, failed to handle distorted grids. Although, the state space model in [46] provides ECKF considering harmonics that can be used for voltage prediction in case of distorted grid, it requires the fundamental frequency to be provided. In addi- tion, complementing it into a model predictive current con- troller is not discussed thus far. Model modification can be made to be applied in the stationaryab frame with no need for the fundamental frequency. However, complexity of tuning the covariance matrices along with significant computational burden are critical challenges. In this paper, EKCF is intro- duced to estimate fundamental and harmonic sequence com- ponents of distorted grid to be used in the proposed model predictive equations. Since it is common practice to estimate positive and negative fundamental components using PR con- trollers, a comparison between the modified ECKF and the PR based estimator given in[20]is carried out in this paper.

The FCS-MPC is frequently used to control grid-connected VSI, and only the best voltage vector from among those avail- able is applied at each sampling time to reduce the cost func- tion. However, the FCS has two main drawbacks: its reliance on the accuracy of the system modelling[47]and its variable switching frequency, which can result in a spread switching harmonic spectrum interfaced with the accompanied filter, making filter design challenging and, as a result, lack of robustness against grid distortions [48,49]. Alternately, the FCS-MPC technique was improved such that MPC-DSVM is introduced in [50]. In MPC-DSVM algorithm, a discrete space vector modulator (DSVM) is used where, extra virtual voltage vectors have been defined. Although only 12 new vec- tors could be used, the computation time is greatly increased by the requirement to evaluate these virtual voltage vectors extensively. Numerous MPC techniques, e.g. modulated MPC (MMPC)[51]and SVM MPC[52], have been proposed to address the drawbacks of FCS-MPC. These techniques typ- ically choose two active vectors, one zero vector, and a modu- lator exchanges between vectors to achieve a deadbeat response with a fixed switching frequency and to provide good steady state behavior in comparison to existing MPC tech- niques[51,52], but they do not perform well in case of harmon- ics and/or during the transition between different modulation zones. To enhance the dynamic response, smooth the transi- tion duration, and lessen the computation overhead for grid- connected converter, MMPC and SVM MPC were extended into the overmodulation region[47]. Moreover, the proposed MMPC in[45]uses the direction of the current reference vector to select the active vectors, reducing the computational burden with no degradation in the performance. For MMPC in[45], the duty cycles are calculated based on the estimated currents at optimum and second best voltage vectors. This entails more computational time to determine the optimum and second optimum voltage vectors and hence use their values based on the current predictive equation to estimate the currents two

step in advance. Unlike, MMPC in [45], the proposed SVM MPC uses the direction of the voltage reference vector to select active vectors in the linear and overmodulation zones. In addi- tion, duty cycles are calculated based on the voltage active vec- tors. However, the duty cycles calculation in SVM MPC proposed in[47]uses the optimum voltage vectors directly, it still uses trigonometric functions in controller to determine duty factors. On contrary, the proposed SVM MPC uses Pythagoras theorem to remove trigonometric functions from the controller and hence less computational burden is achieved. Moreover, there is no need to provide the best and second-best voltage vectors where, there is no need to split each of the six known sectors into two subsectors in order to calculate duty cycles. In addition, the proposed SVM MPC suggests a discretization method, by which, only one step in advance calculations are needed to determine the reference converter voltage at instantkþ1. A summary of the previous MPC techniques and the solution presented in the proposed MPC is introduced inTable 2.

This paper proposes a modified SVM MPC, which offers the following advantages:

Less computational burden compared with the existing techniques.

Enhanced steady state operation and better transient per- formance under unbalanced and distorted grid conditions thanks to the modified ECKF to estimate the positive and negative fundamental and harmonic components of grid voltages with better noise immunity. To investigate the modified ECKF, a comparison with PR based estimator is carried out.

The estimated fundamental components of grid voltages are used to generate the harmonic-free current references one step in advance without applying Lagrange extrapolation.

The proposed SVM MPC is verified based on Lyapunov stability theory. Moreover, sensitivity analysis is carried out to investigate steady state performance of the proposed SVM in case of parameter mismatch and in case of current reference change. In addition, the proposed SVM MPC behaves a fast transient performance with overmodulation extension.

2. System overall

In this section, the mathematical modelling of a grid connected VSC interfaced using anL filter is discussed. The grid con- nected converter is shown inFig. 1, where,v_ca;v_cbandv_ccare the three phase converter voltages,vga;vgbandvgcare the three phase grid voltages, andia;ibandicrepresent the output three phase currents, andRandLare the resistance and inductance of the interfacing filter, respectively.

The voltage equations of the grid-connected converter in the stationaryabreference frame are presented by (1).

v_ca v_cb

¼ v_ga v_cb

þR i_a i_b þL

di_a dt di_b dt

" #

ð1Þ

Using backward Euler discretization method, the current derivative is given by (2).

di

dt¼iki_k1

T_s ð2Þ

Substituting (2) into (1) yields the discrete model given by (3).

V_{cað Þk} V_{cbð Þk}

¼ V_{gað Þk} V_{cbð Þk}

þR i_{að Þk} i_{bð Þk}

þL

i_að Þki_aðk1Þ Ts i_{bð Þk}i_bðk1Þ

Ts

" #

ð3Þ The relevant computations take a time by which the control action may be out of. To compensate this delay, the converter voltages one sample in advance are predicted at instant k + 1 as given by (4).

V_caðkþ1Þ V_cbðkþ1Þ

¼ V_gaðkþ1Þ V_cbðkþ1Þ

þR i_aðkþ1Þ i_bðkþ1Þ

þL

i_að_kþ1Þiað Þk Ts i_bðkþ1Þibð Þk

Ts

" # ð4Þ

3. Proposed control system

In this section, the proposed control scheme for the distorted grid-connected converter is introduced.

3.1. Proposed control scheme

Fig. 1shows the block diagram of the proposed SVM-MPC applied for distorted grid-connected converter. The modified

ECKF is proposed to decompose the measured grid voltages into fundamental and harmonic components. The ECKF is suggested to estimate the positive and negative fundamental components of grid voltages in order to generate an adequate current reference that satisfies the control objectives. The harmonic-free current reference is generated depending on the estimated fundamental voltage components and used to track the reference active and reactive power components P^r and Q^r. In addition, the fundamental one step in advance and the harmonic content of the grid voltages are estimated to be used in model predictive equations. The proposed SVM-MPC is introduced in this section and is used to obtain two active and zero vectors and the corresponding duty cycles each sample. Then, the switching signals are generated by the PWM modulator and applied to the converter switches.

3.2. Modified extended complex Kalman filter

The proposed nonlinear recursive filter based on extended complex Kalman filter (ECKF) is introduced to estimate com- plex fundamental positive and negative voltages one step in advance from noisy distorted measurements.

The noisy distorted signal zk of m positive and negative sinusoids is given by (5).

Table 2 Overview of model predictive control techniques.

Technique Description Advantages Disadvantages Solutions in Proposed SVM

MPC FCS-

MPC [26,32]

Only the best voltage vector among eight vectors is applied at each sampling time

No need for modulator and simplicity

Variable switching frequency Fixed switching frequency

MPC – DSVM [50]

FCS-MPC with extra virtual vectors and discrete space vector modulator

Fixed switching frequency and improved power quality

Computation time is greatly increased by the requirement to evaluate these virtual voltage vectors extensively

Small computational burden

MMPC [51]SVM MPC[52]

Two active vectors, one zero vector, and a modulator exchanges between vectors to achieve a deadbeat response

Fixed switching frequency and good steady state behavior

Bad performance in case of harmonics and/or during the transition between different modulation zones (Bad transient behavior)

Extension into overmodulation zone operation leads to good dynamic response

MMPC [47]

Duty cycles are calculated based on the estimated currents at optimum and second best voltage vectors with extension the operation in overmodulation zone

Fixed switching frequency, good steady state behavior, enhanced dynamic response and smoothed transient behavior

It still uses trigonometric functions in controller to determine duty factors and as a result large

computational burden

Direct use of voltage reference direction to select active vectors in the linear and overmodulation zones and hence duty cycles are calculated based on the voltage active vectors. In addition, Pythagoras theorem is used. The result is small computational burden

MMPC [45]

The same as MMPC in[47]but Pythagoras theorem is used.

Same advantages as MMPC in [47], with smaller computa- tional burden compared with it.

Duty cycles are still calculated based on the estimated currents at optimum and second best voltage vectors

Direct use of voltage reference direction to select active vectors, which leads to smaller

computational burden SVM

MPC[47]

Direct use of voltage reference direction to select active vectors in the linear and overmodulation zones and hence duty cycles are calculated based on the voltage active vectors.

Fixed switching frequency, good steady state behavior, enhanced dynamic response, smoothed transient behavior and less computational burden compared with MMPC in[45]

It still uses trigonometric functions in controller to determine duty factors

In addition to SVM MPC in [47], proposed SVM MPC uses Pythagoras theorem to deter- mine duty factors. In addition, proposed discretization method provides less computational burden

z_k¼X^m

i¼1

a_ie^{jð2pfiTsKþ/iÞ}þb_ie^{jð2pfiTsKþ/iÞ}þNoise ð5Þ where,a_i;b_i;f_i;/iare positive sequence amplitude, negative sequence amplitude, frequency, and phase angle of each sinu- soid respectively.Tsis the sampling time.

To model the observation signaly_k, the state space repre- sentation is given by (6) and (7).

x_kþ1¼f xð Þ ¼k Axk ð6Þ

y_k¼Hx_kþv_k ð7Þ

where, (6) and (7) are given by (8) and (9) as follows, c

x1ðkþ1Þ

x2ðkþ1Þ

2 64

3 75¼

1 0 0

0 c 0

0 0 c¹ 2

64

3 75

c x1ð Þk

x2ð Þk

2 64

3

75 ð8Þ

y_k

½ ¼½0 1 1 c x1ð Þk

x_{2ð Þk} 2 64

3

75þ½ vk ð9Þ

where,

c¼expðj2pf₁TsÞ ð10Þ

x_{1ð Þk} ¼a₁e^{jð2pf1TsKþ/1Þ} ð11Þ x_{2ð Þk} ¼b₁e^{jð2pf1TsKþ/1Þ} ð12Þ

The termcrepresents an artificial state which is a function of the fundamental frequencyf₁.x1ð Þk andx2ð Þk are the positive and negative fundamental sequences of distorted grid voltages.

By applying ECKF algorithm ((13)-(17)) as follows, then [40]:

Kk¼Pb_kjk1H^H HPb_kjk1H^HþR

₁

ð13Þ

b

x_kjk¼xb_kjk1þKky_kHbx_kjk1

ð14Þ Pb_kjk¼ðIK_kHÞPb_kjk1 ð15Þ b

x_kþ1jk¼Abx_kjk ð16Þ

Pb_kþ1jk¼FPb_kjkF^HþQ ð17Þ

where,

F¼df xð Þ_k dxk

xk¼b^xkjk

¼

1 0 0

b

x1ðkjkÞ bc 0 bc²xb2ðkjkÞ 0 bc¹ 2

64

3

75 ð18Þ

where,F^HandH^Hare the Hermitian of the linearized tran- sition and measurement matricesFandH, respectively.Rand Qare the covariance of observation noise and the covariance of process noise, respectively. The termxdenotes the estimated state, whereasPdenotes the state error covariance matrix, and Kkis the Kalman gain.

Fig. 1 Complete Proposed SVM MPC control system.

It is worth noting that the increase in Kalman gain Kk

places more weight on the estimated current state, whereas more weight on the recent measurements if it is increased.

The values ofRandQare tuned experimentally to achieve best performance.

The modified ECKF is expected to estimate the positive and negative alpha–beta components of the distorted noisy grid voltage at instant k + 1. The advance of states by one step can be given by (19).

b

x_kþ1¼Axb_k ð19Þ

Noting that.

x1ð Þk ¼V^þ_{gað Þk} þjV^þ_{gbð Þk} ð20Þ

x2ð Þk ¼V_{gað Þk} þjV_{gbð Þk} ð21Þ Hence,

x1ðkþ1Þ¼V^þ_gaðkþ1ÞþjV^þ_gbðkþ1Þ ð22Þ x2ðkþ1Þ¼V_gaðkþ1ÞþjV_gbðkþ1Þ ð23Þ According to (20) – (23), the estimated fundamental ab components of grid voltages at instant k + 1 are.

V¹_gaðkþ1Þ V¹_gbðkþ1Þ

" #

¼ Re x1ðkþ1Þ

Im x 1ðkþ1Þ

" #

þ Re x2ðkþ1Þ

Im x 2ðkþ1Þ

" #

ð24Þ

On the other hand, as the sampling time is much less than the period time of the waveforms of grid voltage harmonics (5th and 7th are the dominant harmonics), it may be assumed that the grid voltage harmonics at instant k + 1 are the same as the current step. In that case, the estimated harmonicab components of grid voltages at instant k + 1 are.

V^h_gaðkþ1Þ V^h_gbðkþ1Þ

" #

ffi V^h_{gað Þk} V^h_{gbð Þk}

" #

¼ V^m_{gað Þk} V^m_{gbð Þk}

" #

Re x 1ð Þk Im x 1ð Þk

" #

Re x 2ð Þk Im x 2ð Þk

" #

ð25Þ where, the V^m_gajk and V^m_gbjk are the ab components of the measured grid voltages at instant k.

The future ab values of the estimated fundamental grid voltages are used to generate the harmonic-free current refer- ence as described in the coming subsection.

3.3. Current reference generation

The current references are calculated to prevent oscillation of the active power. To obtain harmonic free grid currents under unbalanced and distorted conditions, the current references should contain only the fundamental positive and negative voltage components extracted by voltage ECKF at instant k + 1 ((22) and (23)), theabcomponents of the current refer- ence can be given by (26)–(31) as follows[53]:

i^r_aðkþ1Þ¼i^þ_aðkþ1Þþi_aðkþ1Þ ð26Þ

i^r_bðkþ1Þ¼i^þ_bðkþ1Þþi_bðkþ1Þ ð27Þ

where, i^þ_aðkþ1Þ i^þ_bðkþ1Þ

" #

¼

V^þ_gaðkþ1Þ A

V^þ_gbðkþ1Þ B V^þ

gbðkþ1Þ

A ^Vþgb_A^ðkþ1Þ 2

4

3 5 P^r

Q^r

ð28Þ

i_aðkþ1Þ i_bðkþ1Þ

" #

¼ ^Vga_A^{ðkþ1Þ Vgb}_A^{ðkþ1Þ Vgb}_A^{ðkþ1Þ Vga}_A^ðkþ1Þ 2

4

3 5 P^r

Q^r

ð29Þ

A¼V^þ_gaðkþ1Þ²þV^þ_gbðkþ1Þ²V_gaðkþ1Þ²V_gbðkþ1Þ² ð30Þ

B¼V^þ_gaðkþ1Þ²þV^þ_gbðkþ1Þ²þV_gaðkþ1Þ²þV_gbðkþ1Þ² ð31Þ

3.4. Proposed space vector modulation-based model predictive controller

The proposed SVM MPC controller supposes that the refer- ence voltage can be deduced depending on the deadbeat con- trol as used in[47]. However, the fundamental and harmonic currents should be independently controlled.

Assuming that the measured grid currents are decomposed into fundamental and the harmonicabcomponents at instant k. Hence, the measured grid currents are given by (32).

i^m_{að Þk} i^m_{bð Þk}

" #

¼ i¹_{að Þk} i¹_{bð Þk}

" # þ i^h_{að Þk}

i^h_{bð Þk}

" #

ð32Þ where,m;1, and hdenote the measured value, the funda- mental and the harmonic components respectively.

3.4.1. Fundamental predictive equation

Based on (4) and considering that the output current follows exactly the reference current, it is possible to assume that:

i_aðkþ1Þ i_bðkþ1Þ

¼ i^r_aðkþ1Þ i^r_bðkþ1Þ

" #

ð33Þ

Substituting (24), (32) and (33) into (4), the fundamentalab components of converter reference voltage at instant k + 1 (one step in advance) are given by (34).

V¹_ca V¹_cb

" #

¼ V¹_gaðkþ1Þ V¹_gbðkþ1Þ

" #

þR i^r_aðkþ1Þ i^r_bðkþ1Þ

" # þL

i^r aðkþ1Þi¹_{að Þk}

Ts i^r_bðkþ1Þi¹_{bð Þk}

Ts

2 4

3

5 ð34Þ

3.4.2. Predictive harmonic compensation

To eliminate the harmonic reference components of grid cur- rents, the alpha beta harmonic reference currents should be set to zero.

Under this condition and substituting for (25) and (32) into (4), the harmonic abcomponents of converter reference volt- age at instant k + 1 can be given by (35).

V^h_ca V^h_cb

" #

¼ V^h_gaðkþ1Þ V^h_gbðkþ1Þ

" # L

i^h_{að Þk} Ts i^h_{bð Þk}

Ts

2 4

3

5 ð35Þ

3.4.3. Converter reference voltage

The decomposed fundamental and harmonic components in (34) and (35) are summed to get the converter reference voltage in (36).

V^r_ca V^r_cb

" #

¼ V¹_ca V¹_cb

" # þ V^h_ca

V^h_cb

" #

ð36Þ

Substituting for (34) and (35), the converter reference volt- age is given by (37), then (38).

V^r_ca V^r_cb

" #

¼ V¹_gaðkþ1Þ V¹_gbðkþ1Þ

" #

þ V^h_gaðkþ1Þ V^h_gbðkþ1Þ

" #

þR i^r_aðkþ1Þ i^r_bðkþ1Þ

" #

þL ^iraðkþ1Þ i¹_{að Þk}þi^h_{að Þk}

Ts

i^r_bðkþ1Þ i¹_{bð Þk}þi^h_{bð Þk}

Ts

ð37Þ

V^r_ca V^r_cb

" #

¼ V¹_gaðkþ1Þ V¹_gbðkþ1Þ

" #

þ V^h_gaðkþ1Þ V^h_gbðkþ1Þ

" #

þR i^r_aðkþ1Þ i^r_bðkþ1Þ

" #

þL

i^r_aðkþ1Þi^m_{að Þk} Ts i^r

bðkþ1Þi^m_{bð Þk} Ts

2 4

3 5

ð38Þ The reference voltage is created by switching between two adjacent active vectors Vi and Vj. A zero voltage vectors may be used to complement the switching period. The two active vectors V^r_ca,V^r_cb are selected according to the angle of the reference voltagehr that determines the sector where, it is located in. To calculate the duty cycles of the two active and zero vectors d_i;d_j;andd_o using the fact that the vector of the reference voltage should equal the sum of the contribution of the two adjacent vectors. Hence, their values can be calculated by solving the matrix equation in (39);

V^r_ca V^r_cb

" #

¼d_i V_ia V_ib

þd_j V_ja V_jb

ð39Þ

Using (39), the values ofdi,djanddocan be given by (40), (41), and (42), respectively.

di¼V^r_cbV_jaV^r_caV_jb

V_jaV_ibV_iaV_jb ð40Þ

d_j¼V^r_cbV_iaV^r_caV_ib

V_iaV_jbV_jaV_ib ð41Þ

do¼1 diþdj

ð42Þ By referring toFig. 2, modulation zones are classified into two zones: linear and over modulation zones. In the linear zone, the reference voltage lies inside the hexagon asV^r_c1, where diþdj<1. During operation intervals, the reference voltage may lie outside the linear zone (zone I or zone II) where, the above equations yield an infeasible solutiondiþdj>1.

Over modulation zone is divided into two subzones. For the reference vectors in Zone I (e.g.V^r_c2), the relationjd_id_jj<1 is achieved. In this case, two active vectors with no additive zero vectors are considered. The duty cycles of the two active vectors can be calculated based on a new reference voltage located on the boundary line of the linear zoneV^r_c2 and are given by (43) and (44).

di¼ ri

Eij

ð43Þ

dj¼ rj

Eij

ð44Þ where,riandrjcan be obtained using the Pythagorean the- orem as follows;

rj ¼ Eij j jri ð45Þ

r_^a«

j j²¼j jEi² rj

2¼j jEi²Eij j jri2

ð46Þ r^a«

j j²¼ Ej

2j jri² ð47Þ

Fig. 2 Geometrical representation of the proposed controller for the linear-modulation and overmodulation zones.

From (46) and (47), the value ofri is.

r_i¼ E_j ²j jE_i²þ E_ij²

2E_ij ð48Þ

From (45) and (48), the value ofrj is.

rj¼j jEi² Ej 2þ Eij

2

2Eij

ð49Þ

From (43), (44), (48), and (49), the values ofd_i andd_jare.

d_i¼ E_j²j jE_i²þ E_ij² 2Eij

2 ð50Þ

dj¼j jEi² Ej 2þ Eij

2

2Eij

2 ð51Þ

where,Ei;EjandEijare given by (52), (53) and (54).

E_i¼V^r_c2V_i ð52Þ

Ej¼V^r_c2Vj ð53Þ

Eij¼EjEi ð54Þ

Furthermore, the second over modulation subzone is Zone II where,jd_id_jj>1. Only one active vector is applied while the other active vector contributes by zero duty cycle as the case ofV^r_c3. The duty cycles depend on the position angle of the reference voltagehr as given by (55) and (56).

di¼1;dj¼0; if hr<30 d_i¼0;d_j¼1; if hr>30

It is worth mentioning that the two adjacent vectorsV_iand Vjare odd and even, respectively, if the reference voltage lies in sectors one, three, or five. Otherwise,Viis even andVjis odd in cases of second, fourth and sixth sectors. In addition, duty cycles build the references compared with the triangular carrier to provide the switching signals. The switching pattern is showed inFig. 3in case of sector one where,i¼1 andj¼2.

A simplified flowchart of the proposed SVM MPC algorithm with extended overmodulation is shown inFig. 4.

4. Stability analysis

From (29), the grid current will follow exactly the reference value if and only if the converter voltage generated at instant k + 1 is the same as the reference value of the converter volt- ageV^r_c. However, this is not possible as a result of the quanti- zation error between the actual terminal voltage of the converter (V_{c kþ1ð Þ}Þ and the ideal terminal voltage (V^r_{c kþ1ð Þ}Þ which leads to zero error. In addition, some error existed between the actual grid voltageV_{g kþ1ð Þ}and the estimated value V_{g kþ1ð Þ} which is the sum of the fundamental and harmonic components extracted by using ECKF.

V_{c kþ1ð Þ} ¼V^r_{c kþ1ð Þ} þDV_{c kþ1ð Þ} ð57Þ

Fig. 3 Proposed SVM MPC switching pattern in case of sector one.

i j

i

i j

Fig. 4 Simplified flowchart of proposed SVM MPC with extended overmodulation.

V_{g kþ1ð Þ}¼V_{g kþ1ð Þ} þDV_{g kþ1ð Þ} ð58Þ where, DV_{c kþ1ð Þ}u uð >0Þ, DV_{g kþ1ð Þ}d dð >0Þ, u, d are the constants representing the upper limits of the errors associated with the converter quantization and the discretiza- tion process of grid voltage, respectively.

The Lyapunov stability theory is employed to prove that the steady state current error converges to zero under the equa- tion of the proposed model predictive current controller. From (4), this equation can be represented in the vector form by (59).

i_kþ1

½ ¼ 1

Rþ_T^L_s V_{c kþ1ð Þ} V_{g kþ1ð Þ} þ

L Ts

Rþ_T^L_s½ i_k ð59Þ Similarly, for the ideal case, the reference current with zero steady-state error can be stated as:

i^r_kþ1

¼ 1

Rþ_T^L_s V^r_{c kþ1ð Þ} V_{g kþ1ð Þ}

h i

þ

L Ts

Rþ_T^L_s½ i_k ð60Þ The current tracking error at instant k + 1 can be given by:

Di_kþ1¼i_kþ1i^r_kþ1 ð61Þ

Dikþ1¼ 1

Rþ_T^L_s V_{c kþ1ð Þ} V_{g kþ1ð Þ} þ

L Ts

Rþ_T^L_s½ ik

( )

1

Rþ_T^L_s V^r_{c kþ1ð Þ} V_{g kþ1ð Þ}

h i

þ

L Ts

Rþ_T^L_s½ i_k

( )

ð62Þ

Dikþ1¼ 1

Rþ_T^L_s V_{c kþ1ð Þ} V^r_{c kþ1ð Þ}

h i

þ V_{g kþ1ð Þ}V_{g kþ1ð Þ}

h i

n o

ð63Þ The control Lyapunov function must satisfy the following stability criteria[45]:

VðDi_kÞ A₁jDi_kj^m8Di_k2G ð64Þ VðDikÞ A2jDikj^m8Dik2C ð65Þ VðDi_kþ1Þ VðDikÞ<A3jDikj^mþA4 ð66Þ where, A1, A2,A3, and A4 are positive constants, 1m, GRⁿrepresents a positive control invariant set,CGrepre- sents a compact set, andDikis the current error at instant k.

The Lyapunov functionVðDi_kÞis proposed as:

VðDikÞ ¼1

2Di^T_kDik ð67Þ

The change of the Lyapunov function is given by (68).

DVðDikÞ ¼VðDi_kþ1Þ VðDikÞ ð68Þ Substituting for (52) and (56), the change in the Lyapunov function can be expressed as:

DVðDi_kÞ ¼1 2

1 Rþ_T^L_s

V_{c kþ1ð Þ} V^r_{c kþ1ð Þ}

h i

þ V_{g kþ1ð Þ}V_{g kþ1ð Þ}

h i

8>

<

>:

9>

=

>; 0

B@

1 CA

T

1 Rþ_T^L_s

V_{c kþ1ð Þ} V^r_{c kþ1ð Þ}

h i

þ V_{g kþ1ð Þ}V_{g kþ1ð Þ}

h i

8>

<

>:

9>

=

>; 0

B@

1 CA1

2Di^T_kDik

ð69Þ

Here, the converter voltageV_{c kþ1ð Þ} is bounded by the avail- able dc-link voltage. Moreover, the grid voltages and the grid currents are bounded. In addition, errors are delimited by the upper limitsuandd. The future current tracking error is sub- sequently limited to a compact set (Dik2CRⁿ), where,Cis the compact set which is related to the converter voltages and the inverter current reference (bounded). Thus, (69) can be written as:

DVðDikÞ ¼1 2

1 Rþ_T^L_s

!2

uþd ð Þ²1

2Di^T_kDik ð70Þ Therefore, the stability conditions set out in (64)–(66) are satisfied by the following constants:

A1¼1;A2¼1;A3¼1

2 ð71Þ

A4¼1 2

1 Rþ_T^L_s

!2

uþd

ð Þ² ð72Þ

This implies the convergence of the system to zero in the sense of Lyapunov and that the current control error con- verges to a compact set as:

r¼ DijDi ffiffiffiffiffiffi A₄ A3

r

ð73Þ

5. Sensitivity analysis

The proposed MPC method is analyzed in order to develop the sensitivity analysis of the effect of a mismatch between the parameter values and the real system. The actual resistance and inductance valuesRandLin the real system are varied within ± 50 % of their modeled parametersRo andLo. The total harmonic distortion (THD) and the magnitude of the total steady state error (SSE) between the reference current and the measured output current of the proposed MPC are considered as performance indexes.

Fig. 5shows the THD results as the inductance and resis- tance are varied by the same ratio. The THD has relatively high values for under-estimated inductance and resistance val- ues, with a THD of 3.52 % when the values are 50 % of the modeled values. The THD decreases as the values increase, even above the modeled ones.

For simplicity to study SSE given by (74), only one param- eter is variated at a time. It is observed in Fig. 6 that the

Fig. 5 THD% for different combinations of modeled (Ro, Lo) and real (R,L) parameters.

SSE % grows quasi-symmetrically as the inductance deviation increases. Furthermore, any deviation in resistance by the increase or the decrease leads to higher SSE values.

SSE%¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

N X^N

k¼1

i^r_kik

2

vu

ut 100%;N,number of samples

ð74Þ

The minimum SSE is expected when the modeled values are the same as the real ones. However, the results show that the minimum SSE occurs when the actual inductanceLhas smal- ler values than the modeled valueL_o. This is due to the effect of the proposed grid voltage discretization method given by (4). Ref. [45] proves that the forward Euler discretization method provides a good accuracy compared to the method in which the grid voltage discretization is compensated exactly.

A comparison is carried out between the proposed dis- cretization method and the forward Euler method presented in[47]where, SVM MPC algorithm is also used. The results are outlined throughFig. 7. For the over-estimation of actual inductance values, the two methods behave the same SSE. On the other hand, the SSE values of the proposed method is slightly lower than the SSE values of the forward Euler method in the range of L>0:8L_o. For the values of L<0:8L_o, the SSE values of the proposed provide a very small increase than the other method. However, the minimum SSE is expected to occur closer to the modeled inductance value Lo in case of the proposed discretization method.

For both the proposed and the forward discretization meth- ods, the THD is plotted versus the variation of the reference current in Fig. 8. It is observed that the THD decreases as the grid current increases. However, the proposed controller provides very slightly lower THD. As compared to the forward discretization method, the proposed method presents a smaller number of blocks.

Fig. 6 SSE when the resistance and inductance of the real system are varied from 0.5 to 1.5 p.u.

Fig. 7 Steady state current error when the inductance of the real system are varied from 0.5 to 1.5 of the model inductance value which is kept constant (the resistance of the real system is the same as modeled value) in case of the proposed discretization method (blue) and the forward Euler discretization method (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 THD versus reference current in case of the proposed discretization method (blue) and the forward Euler discretization method (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

6. Transient performance

The transient responses of the proposed controller and the SVM MPC given in [47], which is the most up-to-date con- troller according toTable 2, are studied during sudden changes in active power reference (P^rÞ.Fig. 9shows the sudden change in grid current (reference and output) and in the real and reac- tive power (P and Q) as well.

In addition, the harmonic spectrums of the output current are studied inFig. 10where, they provide results close to each other. In addition, the harmonics are distributed around the switching frequency for both methods as the conventional SVM technique. Both controllers present fast dynamic response and enhanced current and power qualities as a result of their overmodulation capability. Although, the results for both methods are approximately the same, the proposed con- troller provides less computational burden as will be discussed in subsection 7.4.

7. Experimental validation 7.1. Experimental setup

The proposed space vector based model predictive current con- troller with optimized overmodulation was tested in the labo-

ratory to verify the performance of the control system in case of distorted and unbalanced grid conditions. The experimental test setup is shown inFig. 11where, real-time power grid dig- ital simulator (OP4510 RT-LAB-RCP/HIL SYSTEMS) intro- duced from Opal RT is used with a clock frequency of 2.1 GHz. The OP4510 accepts any combination of four differ- ent I/O modules such as analog input, analog output, digital input and digital output. A fully isolated two-level three phase power module (IKCM30F60GD) is used with TrenchStopÒ IGBTs and anti-parallel diodes combined with an optimized SOI gate driver for excellent electrical performance. The volt- age measurement is carried out using LV 25-P, while LA 25- NP hall effect sensors are used for current measurements. A 300 V programable supply is used to provide the converter dc-link. The figure shows that the grid voltages connected to high nonlinear load following distorted behavior in which the voltages convert from balanced to unbalanced condition by using series resistance to increase the voltage of phase a (vg_a).

The system parameters are shown inTable 3.

7.2. Experimental results (ECKF versus PR based estimator) To investigate the experimental results of the modified ECKF, theabpositive and negative components of the grid voltage are

Fig. 9 Transient response in grid current (reference (red) and output (blue)) and corresponding real (red) and reactive (blue) power components; the top is the proposed SVM MPC and the bottom is the old SVM MPC in[47]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10 Current Harmonic spectrum of (a) proposed MPC and (b) SVM MPC in[47].