analytical approach to circulating current mitigation in...

Research ArticleAnalytical Approach to Circulating Current Mitigation inHexagram Converter-Based Grid-Connected Photovoltaic SystemsUsing Multiwinding Coupled Inductors

Abdullrahman A. Al-Shamma’a ,1,2 Abdullah M. Noman ,1,2 Khaled E. Addoweesh,1

Ayman A. Alabduljabbar,3 and A. I. Alolah1

1Department of Electrical Engineering, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia2Department of Mechatronics Engineering, College of Engineering, Taiz University, Taiz, Yemen3King Abdulaziz City for Science and Technology (KACST), P.O. Box 6086, Riyadh 11442, Saudi Arabia

Correspondence should be addressed to Abdullrahman A. Al-Shamma’a; [email protected]

Received 31 December 2017; Accepted 2 April 2018; Published 7 May 2018

Academic Editor: Francesco Riganti-Fulginei

Copyright © 2018 Abdullrahman A. Al-Shamma’a et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

The hexagrammultilevel converter (HMC) is composed of six conventional two-level voltage source converters (VSCs), where eachVSC module is connected to a string of PV arrays. The VSC modules are connected through inductors, which are essential tominimize the circulating current. Selecting inductors with suitable inductance is no simple process, where the inductance valueshould be large to minimize the circulating current as well as small to reduce an extra voltage drop. This paper analyzes theutilization of a multiwinding (e.g., two, three, and six windings) coupled inductor to interconnect the six VSC modules insteadof six single inductors, to minimize the circulating current inside the HMC. Then, a theoretical relationship between the totalimpedance to the circulating current, the number of coupled inductor windings, and the magnetizing inductance is derived.Owing to the coupled inductors, the impedance on the circulating current path is a multiple of six times the magnetizinginductance, whereas the terminal voltage is slightly affected by the leakage inductance. The HMC is controlled to work undervariable solar radiation, providing active power to the grid. Additional functions such as DSTATCOM, during daytime, are alsodemonstrated. The controller performance is found to be satisfactory for both active and reactive power supplies.

1. Introduction

Recently, photovoltaic (PV) energy systems have gainedmore attention as distributed generation units, as they offerlow cost of generation closer to that of conventional plants,as well as less maintenance and no grid noise [1, 2]. More-over, PV systems can solve multiple typical problems presentin conventional AC power systems. However, PV systemspresent frequency and voltage fluctuations when islandingoperation occurs. Therefore, PV plants should be integratedwith the power system in order to maintain the overallfrequency and voltage at a stable condition. Several studiessuggest interconnection methods of PV systems to the gridthrough voltage source converters (VSCs), because they pro-vide versatile functions enhancing capabilities of the system

[3–5]. The main purpose of the VSC is to connect the PVplant to the grid while guaranteeing power quality (PQ) stan-dards. However, the high-frequency switching of VSCs intro-duces additional harmonic components to the system, hencecreating PQ problems if not implemented accurately [6].

Most of the available VSCs for PV systems are traditionaltwo-level three-phase VSCs with low power capacity [7]. Inthe literature, researchers have suggested numerous multi-level topologies for grid-connected PV plants [8–10]. Multi-level VSCs are considered more attractive than traditionaltwo-level VSCs, as they improve the output voltage qualityand reduce electromagnetic interference, voltage stress onIGBTs, and common-mode voltage. Moreover, multilevelVSCs operate at a low switching frequency and henceincrease system efficiency [10]. Consequently, multilevel

HindawiInternational Journal of PhotoenergyVolume 2018, Article ID 9164528, 22 pageshttps://doi.org/10.1155/2018/9164528

http://orcid.org/0000-0002-5237-0199http://orcid.org/0000-0002-3671-0317https://doi.org/10.1155/2018/9164528

VSCs have been widely used in chemical, oil, and differentkinds of plants, as well as in power plants, transmissionsystems, and PQ compensators [11].

Current research is mainly focused on three specific mul-tilevel VSC topologies, namely, the neutral-point clamped(NPC) [12], flying capacitor (FC) [13] and the CascadedH-bridge (CHB) [14] topologies. The NPC VSC requires ahigh number of clamping diodes to increase the number ofvoltage levels, which can cause problems related to switcheswith different ratings, high-voltage rating in blocking diodes,and capacitor voltage imbalance. However, the FC VSC con-sists of a large number of capacitors, which leads to compli-cations in regulating capacitor voltages. Finally, the CHBVSC provides isolated DC sources, hence being appropriatefor use in PV systems [15]. In addition, other advantages ofthis topology include its modular structure and the reducednumber of components compared to the other multilevelconverters (i.e., NPC and FC). Therefore, the CHB VSCcan reach the same number of voltage levels with a simplerassembly and maintenance. However, this topology presentsthree main drawbacks: (i) high overall component count;(ii) high energy storage requirement because the instanta-neous power related to each H-bridge varies at twice thefundamental frequency, given its single-phase modularstructure; and (iii) difficulty to control the voltage acrossDC-link capacitors.

The concept of interconnecting three traditional three-phase VSCs to produce a multilevel converter was firstproposed in [16] and applied to medium-voltage variable-speed drives. In [17], three three-phase two-level VSCs areinterconnected using three single-phase transformers with a1 : 1 turn ratio. These interconnected transformers increasethe output voltage and suppress the circulating current insidethe converter. The power capacity of the overall converter isthree times the capacity of each interconnected converter,whereas the volt-ampere rating of each intermediate

transformer is equal to that of each interconnected converter.In [18], another topology known as the hexagram multilevelconverter (HMC) is proposed, which combines six two-levelconverters by using six inductors. This topology shares manyadvantages of the CHB but uses fewer switches and reducesthe size of the DC-link capacitor [19]. The advantages ofthe proposed topology can be summarized as follows: (1)only six standard three-phase VSCs are necessary togenerate multilevel output voltage; (2) each VSC module isbalanced in operation, equally loaded, and supplies 1/6 out-put power; (3) modular construction which facilitates systemmaintenance and spare management; (4) only six isolatedDC links with no voltage unbalance problem; (5) low numberof power electronics switches and low DC energy storagerequirement; and (6) the output transformer contributes tohigher output voltage. Figures 1 and 2 are schematic dia-grams illustrating multilevel topologies based on cascadedtwo-level VSCs according to previous research.

However, selecting an inductor with suitable inductanceis no simple process, where the inductance value should beadequate to minimize the circulating current as well as toreduce an extra voltage drop on the inductors that affectsthe terminal voltage. This difficulty can be circumvented byusing a multiwinding coupled inductor. Therefore, this paperanalyzes the utilization of a multiwinding (e.g., two, three,and six windings) coupled inductor to interconnect the sixVSCs instead of six single inductors. Consequently, bothgoals minimize the circulating current and the minimaleffects to the output voltage can be accomplished instanta-neously. Then, an analytical model to calculate the totalinductance imposed to the circulating current path isderived. The equivalent circuit model of the HMC is detailedin the abc reference frame and then transformed into theorthogonal dq0 reference frame. Moreover, to extract themaximum power from the PV arrays, a control algorithmfor maximum power point tracking is also presented.

C

C

b3c3

a3 b3c3

a3

V dc

Vdc

Vdc

V dc

VdcVdc

L

a1b1

c1

a1b1

c1

A+

− AL

Lc2b2 a2 c2

b2 a2B

B

T3

T1

T2

+

−

+−

+

−

+

−

+−

Figure 1: Topologies of cascade two-level converters.

2 International Journal of Photoenergy

The paper is organized as follows: Section 2 describesthe HMC. Section 3 presents the mathematical model fora grid-connected PV system. Section 4 demonstrates thebenefits of the multiwinding coupled inductor inside theHMC. Section 5 describes the control system and modula-tion strategy. Section 6 shows the simulation resultsfollowed by the corresponding discussion. Finally, the con-clusions achieved from the present work are summarizedin Section 7.

2. System Description

The proposed configuration of the three-phase grid-connected PV plants is shown in Figure 3. This configurationis composed of six traditional three-phase two-level VSCmodules, which have a hexagonal interconnection to producehigher voltage levels as shown in Figure 2. The circulatingcurrent in the obtained loop can be suppressed using sixinductors, which present a small impedance at the switchingfrequency. Each VSC module supplies 1/6 of the converteroutput power. In addition, each module has one of its AC ter-minals (i.e., a, b, or c) designated as converter output. Thethree-phase AC output terminals of the HMC are labeled as

follows: A (AC terminal a1 of module 1), B (AC terminalb3 of module 3), and C (AC terminal c5 of module 5). Theremaining three-phase AC output terminals are labeled asfollows: A′ (AC terminal a4 of module 4), B′ (AC terminalb6 of module 6), and C′ (AC terminal c2 of module 2). Theother two AC terminals of each module are, respectively,connected to an adjacent module through an inductor. Forinstance, AC terminal b1 of module 1 is coupled to AC termi-nal b2 of module 2 through an inductor Ld; AC terminal c1 ofmodule 1 is coupled to AC terminal c6 of module 6 throughanother inductor. The remaining modules consist of similarconnections, as shown in Figure 2.

As shown in Figure 3, the PV system based on theHMC has six output terminals. Therefore, when linked tothe three-phase electrical grid, an open-end winding(OEW) transformer is necessary to provide these six ter-minals. The secondary windings of the OEW transformerare connected differentially. In the proposed converter,AC terminals A-A′, B-B′, and C-C′ are used to providephases A, B, and C, respectively. High-voltage windingsare arranged in a star configuration and coupled directlyto the three-phase grid. Each VSC module is supplied witha separate PV string to produce two-level individual

V dc6

Vdc4

Vdc5

Vdc2

V dc3

b2

a2

b1c1

A

+

−

+

−+

−

+

−

+

−

+

−

C′

B

A′

C

B′a3

c3

b4 c4

b5

a5

a6

c6

L d Ld

L d

L dLd

L d

Vdc1

Figure 2: HMC electrical diagram.

3International Journal of Photoenergy

outputs. The proposed converter topology presents severaladvantages, such as the use of fewer power electronicswitches, diodes, and storage capacitors when comparedwith other topologies, making it suitable for renewableenergy and other applications.

3. System Modeling

3.1. PV System Modeling and MPPT Control. Figure 4 showsthe equivalent model of a PV array. The PV array is com-posed of several series-parallel connected solar cells. Thebasic equation of the PV array is given by [20].

I = Ipv − I0 expV + RsIaV t

− 1 − V + RsIRp

, 1

where Ipv =NpIpv,cell is the PV current of the array, Ipv,cellis the current generated by incident light (directly pro-portional to Sun irradiation), Np is the number of cellsconnected in parallel, I0 =NpI0,cell is the saturation cur-rent of the array, I0,cell is the reverse saturation or leakagecurrent of the diode, V t =NskT/q is the thermal voltageof the array, Ns is the number of cells connected inseries, q is the electron charge, k is the Boltzmann con-stant, T is the temperature of the p-n junction, a is thediode ideality constant, Rs is the equivalent series resis-tance of the array, and Rp is the equivalent parallel resis-tance. Table 1 shows the nominal parameters of theKC200GT PV array. Figure 5 shows the I-V and P-Vcurves obtained using (1) of the PV array under variablesolar radiation.

IPV1

IPV2

IPV3

IPV4

IPV5

IPV6

Vdc1

Vdc2

Vdc3

Vdc4

Vdc5

Vdc6

AC

DCA

B

C

Cʹ

Bʹ

Aʹ

Open-end

winding

transformer

IA

IB

IC UCA

UAB

Interface inductorLf

UBC EBC

PCC

EAB

ECO’

EBO’

EAO’

O′

Grid

ECA

Figure 3: Complete HMC for a grid-connected PV system.

Practical PV array

I

VRp

Id

Ipv

Ideal PV cell Rs+

−

Figure 4: Equivalent model of a PV array.


3.2. Maximum Power Point Tracking. The proposed systemconsists of 12 PV modules distributed as six pairs of series-connected modules and coupled directly to a single DC link(see Figure 3). Therefore, the power rating of the proposedconverter is 2.4 kW. The controller is aimed to regulate thevoltage of the PV-module pair at 52.6V, to guarantee maxi-mum power transfer to the grid.

Next, the aim of the MPPT control algorithm used in thispaper is to ensure that under any solar radiation and temper-ature conditions, the maximum power is extracted from thePV modules. This is achieved by matching the PV-arraymaximum power point to the corresponding operating volt-age and current of the converters. The perturb and observe(P&O) algorithm is a commonly used MPPT techniquebecause it is easy to implement [21]. The operating principleof the P&O algorithm is shown in Figure 6. This algorithmmeasures the PV plant voltage and current, then it variesthe operating voltage and compares the power receivedbetween the two voltage values. After each perturbation, thealgorithm compares the output power from the PV beforeand after the perturbation. The direction of a new pertur-bation depends upon the output power: the perturbationwill follow the same direction if higher power is mea-sured when the output voltage varies and the oppositedirection otherwise. These procedures are repeated contin-uously, and the reference voltage is generated and fed tothe converter controller.

The numerical illustration of the P&O algorithm [2] isgiven as follows:

dPpv k

dVpv k=

Ppv k − Ppv k − 1Vpv k − Vpv k − 1

2

Here, Ppv k and Ppv k − 1 stand for current power andprevious measured power, while Vpv k and Vpv k − 1 standfor current PV voltage and previous one.

3.3. Voltage and Current Analysis. Under the symmetricaloperation condition, the fundamental components of thesix VSC modules are the same. The corresponding open-circuit voltage of the HMC is presented in Figure 7.

3.3.1. Current Relations. The phase currents of each VSCmodules fulfill the following expressions:

ia1 + ib1 + ic1ia2 + ib2 + ic2ia3 + ib3 + ic3ia4 + ib4 + ic4ia5 + ib5 + ic5ia6 + ib6 + ic6

= 0 3

Table 1: Parameters of the KC200GT PV array at 25°C, AM1.5, and 1000W/m2.

Imp(A)

Vmp(V)

Pmax(W)

Isc(A)

Voc(V)

Kv(V/K)

K I(A/K)

Nss(No.)

Rp(Ω)

Rs(Ω)

Io,n(A)

a

7.61 26.3 200.143 8.21 32.9 −0.123 0.0032 54 415.405 0.221 9.825× 10−8 1.3

x: 26.3y: 200.1

P-V curve @1000 W/m2P-V curve @800 W/m2

P-V curve @600 W/m2P-V curve @400 W/m2

x: 26.3y: 7.61

x: 0y: 8.211

I-V curve @1000 W/m2I-V curve @800 W/m2

I-V curve @600 W/m2I-V curve @400 W/m2

x: 32.89y: 0.004

0

20

40

60

80

100

120

140

160

180

200

Mod

ule p

ower

(W)

5 10 15 20 25 30 350Module voltage (V)

5 10 15 20 25 30 350Module voltage (V)

0

1

2

3

4

5

6

7

8

9

Mod

ule c

urre

nt (A

)

Figure 5: I-V and P-V curves of the KC200GT PV array.


Start

Measurement of Vpv(k) and Ipv(k)

Calculate Ppv(k), ΔPpv(k), and ΔVpv(k)

Ppv(k) = Vpv(k) × Ipv(k)

ΔPpv(k) = Ppv(k) − Ppv(k−1)

ΔVpv(k) = Vpv(k) − Vpv(k−1)

ΔPpv(k) > 0

ΔVpv(k) > 0 ΔVpv(k) < 0

ΔVpv(k + 1) = Vpv(k) − ΔVpv(k) ΔVpv(k+1) = Vpv(k) + ΔVpv(k) ΔVpv(k + 1) = Vpv(k) − ΔVpv(k) ΔVpv(k + 1) = Vpv(k) + ΔVpv(k)

Yes

YesYes No No

No

Figure 6: Flowchart for P&O MPPT method.

01

O3

05

L d

Ld

06

02

04 AA′

B

B′C

C′

−Va1o1+

+Va2o2−

+ V b1o

1−

+Vc1o1 −

− V b2o

2+

+Vc2o

2−

+Va4o4−

+Va6o6−−Va5o5+

−Vc5o5 +

ic5

i b5

i b2

i b1

i b4

i b3

ic4

ic1

i b6

ic6

ic2

ic3

ia5

ia6

ia2

ia3

ia1ia4

Ld

Ld

Ld

Ld

+ V b5o

5−

− V b4o

4+

−Vc4o4 +

+Vc3o3 −

+ V b3o

3−

−Va3o3+

−Vc6o6 +

− V b6o

6+

Figure 7: Phasor voltage of the HMC.


Assuming that the intermediate coupled inductors havelarge magnetizing inductance, the circulating currents aresuppressed to a low value and can be ignored, hence

ib1 + ia2 + ic3 + ib4 + ia5 + ic6 = 0 4

Because every pair of the six VSC modules is coupled, thecurrent within the proposed converter has the followingexpressions:

ib1

ia2

ic3

ib4

ia5

ic6

= −

ib2

ia3

ic4

ib5

ia6

ic1

5

Next, suppose that the HMC is linked to a three-phase grid. Then, the output currents will satisfy thefollowing equation:

IA

IB

IC

=ia1

ib3

ic5

= −ia4

ib6

ic2

=I 0

I −120I +120

6

Using (2), (3), and (4), the output current of each VSCmodule can be shown to be

ia1

ib1

ic1

=

ia3

ib3

ic3

=

ia5

ib5

ic5

= −

ia2

ib2

ic2

= −

ia4

ib4

ic4

= −

ia6

ib6

ic6

=

IA

IB

IC

=

I 0

I −120

I +120

,

7

where IA, IB, IC T and I are the converter output phasecurrents and their RMS values, respectively. It can beconcluded from (5) that each VSC module within theHMC will have an identical current under the symmetricaloperation condition.

3.3.2. Voltage Relations. Under the symmetrical operationcondition, the fundamental component (RMS) of the phasevoltages at the VSC modules can be described as

Va1o1

Vb1o1

Vc1o1

=

Va3o3

Vb3o3

Vc3o3

=

Va5o5

Vb5o5

Vc5o5

= −

Va2o2

Vb2o2

Vc2o2

= −

Va4o4

Vb4o4

Vc4o4

= −

Va6o6Vb6o6

Vc6o6

=

V 0

V −120

V +120

,

8

V = 12 2

Vdcma, 9

where V , Vdc, and ma are the RMS values of each VSC-module phase voltage, the DC-link voltage of each VSCmodule, and the amplitude modulation index, respectively;subscripts o1, o2, o3, o4, o5, and o6 represent the virtualneutral points of each VSC.

The output voltages of the proposed converter arewritten as

VAA

VBB

VCC

=

Va1a4

Vb3b6

Vc2c5

=

Va1o1 − Va4o4 + Vc4o4 −Vc3o3 +Va3o3 −Va2o2 +Vb2o2 − Vb1o1Vb3o3 − Vb6o6 + Va6o6 − Va5o5 + Vb5o5 − Vb4o4 + Vc4o4 − Vc3o3Vc5o5 − Vc2o2 +Vb2o2 − Vb1o1 + Vc1o1 − Vc6o6 + Va6o6 − Va5o5

− 2Zd

ia1

ib3

ic5

,

10

where impedance Zd = jωLd.Thus, using (5), (6), (7), and (8), the net output voltages

of the HMC under the symmetrical operation condition aregiven by

VAA

VBB

VCC

=Va1a4

Vb3b6

Vc2c5

=6V 0

6V −1206V +120

− 2Zd

IA

IB

IC11

Consequently, (9) demonstrates that the three-phasevoltage of the HMC is approximately six times the phase volt-age of each VSC module. In other words, the voltage stress isreduced by a factor of six.

3.4. Equivalent Circuit Model. Based on Figure 3, the linevoltages of the HMC are expressed by


UAB

UBC

UCA

=

VAA −VBB

VBB −VCC

VCC − VAA

=

6 3V −30

6 3V −150

6 3V +90

− 2jωLd

IA − IB

IB − IC

IC − IA

=

EAB

EBC

ECA

+ jωLf

IA

IB

IC

− jωLf

IB

IC

IA

,

12

where EAB, EBC, ECA T are the RMS values of the line-to-linevoltages in the grid, UAB,UBC,UCA T and IA, IB, IC Tare the RMS values of line voltages and the RMS valuesof the output current, respectively, and Lf is the AC-sidefiltering inductance.

When the conversion powers between the VSC moduleswithin the HMC are balanced, the six DC-link voltages areequal to VDCav, hence

VDCav =Vdc1 =Vdc2 =Vdc3 = Vdc4 =Vdc5=Vdc6 =

Vdc1 +Vdc2 +Vdc3 +Vdc4 +Vdc5 + Vdc66

13

Moreover, the equivalent DC-link voltage can beexpressed as

UDC eq = 6VDC av 14

If each VSC is driven using the same PWM, the relation-ship between the DC-link voltages and the phase voltages canbe expressed as

V = 12 2

UDC eqma 15

From (12) and (14), the HMC can be modeled by

UAB̄

UBC̄

UCĀ

=EAB

EBC

ECA

+jω Lf +2Ld

IA

IB

IC

−jω Lf +2Ld

IB

IC

IA

,

16

where

UAB̄

UBC̄

UCĀ

=6 3V −30

6 3V −150

6 3V +90

17

Using (14) and (17), the HMC can be modeled as a con-ventional three-phase two-level VSC, as shown in Figure 8.

Figure 8 shows the equivalent circuit model of theHMC. The equivalent circuit is composed of an open-circuit voltage source linked in series to the HMC outputimpedance. The open-circuit voltage of the HMC is shownin Figure 7. As shown in Figure 8, the equivalent interfaceinductor of the HMC is 2Ld + Lf . This feature is advanta-geous, because the filtering inductor Lf could be minima-lized or even removed. However, this can be inopportunewhen the line transformer has sufficient inductance for fil-tering the output current. In this case, essentially if theHMC supplies reactive power, the output impedancecauses a voltage drop, decreasing the power capability.Thus, it might be better to minimize the interconnectedtotal inductances. However, the inductances are essentialto suppress the circulating current within the HMC. Theaforementioned decision shows that the interconnectedinductance value selection is not a simple process, wherethe inductance should be large enough to minimize thecirculating current but small enough to avoid an extravoltage drop. In the following section, this feature is inves-tigated using multiwinding coupled inductors.

4. The Role of the Coupled Inductors

The function of the multiwinding coupled inductors withinthe HMC is investigated in this section. The HMC perfor-mance analysis can be demonstrated based on the functionof the coupled inductors. The six VSC modules are intercon-nected to each other, creating a hexagon as explained inSection 3.4. However, instead of using six inductors, multi-winding coupled inductors are used to interconnect theVSC modules.

4.1. HMC with Three Two-Winding Coupled Inductors. Twoinductors with an equal number of turns are coupledtogether; that is, the input to one side will produce an outputon both, as shown in Figure 9. Since the turn ratio of thecoupled inductor is approximately 1 : 1, the self-inductancesof the primary and secondary windings are the same (e.g.,Lp = Ls = LB). Applying the voltage relationships of coupledinductors, the following equations are given:

vb1b2 = L11dib1dt

+ L12dib5dt

, 18

and

vb5b4 = L22dib5dt

+ L21dib1dt

, 19

where L22 = L11 = LB + Lm, Lm = L21 = L12, ib1 = ib1 + icir, andib5 = −ib1 + icir, and where ix x = a, b, c is the line current,LB is the leakage inductance of the inductor windings, whichis expected to be identical for all windings, and Lm is the mag-netizing inductance of the coupled inductor.

It is obvious in Figure 10 that the voltage drop on one ofthe coupled inductors is

v = vb1b2 + vb5b4 20


A

B

DC

ACC

UD

C_eq

PCC

Equivalent interfaceinductor

O′

EAO’

EBO’

ECO’

2LdIA

IB

IC

UA

BU

BC

UCA

EA

BE

BC

ECA

Lf

2Ld Lf

2Ld Lf

Figure 8: Equivalent circuit model of HMC.

Lm

LB LBN : N

Vb1b2 Vb5b4

ib5 + icirb1

b2

b5

b4

ib1 + icir

Figure 9: Equivalent circuit of the two-winding coupled inductor.

LB

LB

L C

L C

c3

Ba3

a2

b2c6

a6

LALA

Vdc

A1

b1c1

Vdc

b4

b5

a5

c4

A′

B′C′

C

Vdc

Vdc

V dc

V dc

+ −

+

+

−

−+

−+

−+

−

Figure 10: The HMC with three two-winding coupled inductors.


Using (18) and (19), the voltage across the coupledinductor can be expressed as

v = LB + Lmd ib1 + icir

dt+ Lm

d −ib1 + icirdt

+ LB + Lmd −ib1 + icir

dt+ Lm

d ib1 + icirdt

,

v = 2LB + 4Lmd icirdt

21

Therefore, the total voltage drop across the three coupledinductors can be expressed as

vtotal = 3 2LB + 4Lmd icirdt

= 6LB + 12Lmd icirdt

22

4.2. HMCwith Two Three-Winding Coupled Inductors. Threeinductors with an equal number of turns are coupledtogether as shown in Figure 11.Writing the voltage equationsof one coupled inductor with winding inductance Z1, the fol-lowing equations are given:

va5a6 = Ladiadt

+ Labdibdt

+ Lacdicdt

,

vb1b2 = Lbdibdt

+ Lbadiadt

+ Lbcdicdt

,

vc3c4 = Lcdicdt

+ Lcadiadt

+ Lcbdibdt

,

23

where

La = Lb = Lc = L + Lm, Lm = Lab = Lba = Lbc = Lcb = Lca = Lac,24

and

ia = ia5 + icir,ib = ib1 + icir,ic = ic3 + icir,

25

where ix x = a, b, c is the line current, L is the leakage induc-tance of the inductor windings, and Lm is the magnetizing

inductance. It is obvious in Figure 12 that the voltage dropon the coupled inductor is

v = va5a6 + vb1b2 + vc3c4 26

Using (23), the voltage drop across one of the coupledinductor can be expressed as

v = L + Lmd ia5 + icir

dt+ Lm

d ib1 + icirdt

+ Lmd ic3 + icir

dt

+ L + Lmd ib1 + icir

dt+ Lm

d ia5 + icirdt

+ Lmd ic3 + icir

dt

+ L + Lmd ic3 + icir

dt+ Lm

d ia5 + icirdt

+ Lmd ib1 + icir

dt,

v = L + Lmd ia5 + ib1 + ic3

dt+ 3 L + Lm

d icirdt

+ 2Lmd ia5 + ib1 + ic3

dt+ 6Lm

d icirdt

27

The voltage drop across the coupled inductor windings is

v = L + 3Lmd ia5 + ib1 + ic3

dt+ 3L + 9Lm

d icirdt

28

Under balanced conditions,

ia5 + ib1 + ic3 = ia + ib + ic = 0 29

Accordingly, by applying (28) and (29), the voltage dropon the coupled inductor windings is

v = 3L + 9Lmd icirdt

30

The total voltage drop across the two coupled inductorscan be expressed as

vtotal = 6LB + 18Lmd icirdt

31

4.3. HMC with One Six-Winding Coupled Inductor. Insteadof using six inductances, one six-winding coupled inductorcan be used to interconnect the VSC modules. Since the turnratio of the coupled inductor is approximately 1 : 1, the self-inductances of all the windings are the same as shown in

ia5 + icir icir + ib1

icir + ic3

L1a5

a6

L1b1

c3

c4

V b1b

2

V a5a

6

V c3c

4

b2L1

Lm

N : N : N

Figure 11: Equivalent circuit of the three-winding coupled inductor.


Figure 13. The HMC with one six-winding coupled inductoris shown in Figure 14. Applying the voltage equations of thecoupled inductor, the following equations are given:

va5a6 = Ladia5dt

+ Lmdib1 + dic3 + dib5 + dia3 + dic1

dt,

vb1b2 = Lbdib1dt

+ Lmdia5 + dic3 + dib5 + dia3 + dic1

dt,

vc3c4 = Lcdic3dt

+ Lmdia5 + dib1 + dib5 + dia3 + dic1

dt,

va3a2 = Ladia3dt

+ Lmdib1 + dic3 + dib5 + dia5 + dic1

dt,

vb5b4 = Lbdib5dt

+ Lmdia5 + dic3 + dib1 + dia3 + dic1

dt,

vc1c6 = Lcdic1dt

+ Lmdia5 + dib1 + dib5 + dia3 + dic3

dt

32

Since,

ia = ia5 = ia3 = ia1 = −ia2 = −ia4 = −ia6,ib = ib5 = ib3 = ib1 = −ib2 = −ib4 = −ib6,ic = ic5 = ic3 = ic1 = −ic2 = −ic4 = −ic6

33

The voltage drop equations across each coupled inductorcan be expressed as

va5a6 = va3a2 = Ladiadt

+ Lmdia + 2dib + 2dic

dt,

vb1b2 = vb5b4 = Lbdibdt

+ Lmdib + 2dia + 2dic

dt,

vc3c4 = vc1c6 = Lcdibdt

+ Lmdic + 2dia + 2dib

dt,

34

where

La = Lb = Lc = L + Lm, 35

and

ia = ia + icir,ib = ib + icir,ic = ic + icir

36

Using (34), the voltage drop across the coupled inductorcan be expressed as

v = va5a6 + vb1b2 + vc3c4 + va3a2 + vb5b4 + vc1c6, 37

v = 2va5a6 + 2vb1b2 + 2vc3c4, 38

Z1

Z2

Z2

Z1

c3

a3

b2c6

a6

Z2Z1

Vdc

A

b1c1

Vdc

b4

b5

a5

c4

C

Vdc

Vdc

V dc

V dc

+ −

+

+

−

−+

−+

−+

−

a2B

A′

C′B′

Figure 12: The HMC with two three-winding coupled inductors.


v = 6L + 36Lmdicirdt

+ 2L + 12Lmd ia + ib + ic

dt39

Under balanced conditions,

ia + ib + ic = 0 40Consequently, by applying (39) and (40), the voltage

drop on the coupled inductor windings is

v = 6L + 36Lmdicirdt

41

It is considered that the magnetizing inductance ismuch higher than the leakage inductance. Therefore,neglecting the leakage inductance, the two-windingcoupled inductor imposes twelve times the magnetizinginductance for the circulating current, while the impedanceto the circulating current using three-winding coupled

inductors is eighteen as much as the magnetizing induc-tance. Thanks to the six-winding coupled inductor, theimpedance on the circulating current path is thirty six timesthe magnetizing inductance.

In a general way if k is the number of the coupled induc-tor windings, the impedance to the circulating current can beexpressed as

Zcir = 2πf 6kLm 42

5. Control Scheme and Modulation Strategy

5.1. Control Scheme. The main function of the controller isto generate reference currents such that the proposed con-verter only provides available active power from the DClinks to the grid at the point of common coupling (PCC)[22, 23]. Using the equivalent circuit model presented in

A′

C′

C B

b4

L L

c4

Vdc

Vdc

V dcVdc

Vdc V dc

c1 b1A1

c3b5

a5

a6

c6 b2

a2a3

− +

−+

−+

−+

−+

−+

B′

L

L

L

L

Figure 14: The HMC with one six-winding coupled inductor.

ib5 + icirib1 + icir

ia5 + icir

ic3 + icir

L

ia3 + icirL

ic1 + icirL

N : N :NL

Lm

L

L

b5

V b5b

4

b4a3

V a3a

4

a4c1

V c1c

6

c6

b1

V b1b

2

b2a5

V a5a

6

a6c3

V c3c

4

c4

Figure 13: Equivalent circuit of the six-winding coupled inductor.


Figure 7 and applying Kirchhoff’s voltage and current lawsat the PCC, the following two equations in the abc framecan be obtained:

EAO′

EBO′

ECO′

=UAŌ

UBŌ

UCŌ

+Lfdddt

IA

IB

IC

+UO′O, 43

CeqdUDC av

dt= SA SB SC

IA

IB

IC

, 44

where Lfd = Lf + Ld is the equivalent interface inductor andSA, SB, and SC represent the switching states of the equivalentcircuit model under balanced conditions.

Assuming that the voltages are balanced and the zero-sequence component is zero, the voltage between the neutralvirtual point of equivalent circuit model (O) and the gridneutral point (O′) is given by

UO O = −UAŌ +UBŌ +UCŌ

3 ,45

UAŌ

UBŌ

UCŌ

=UDC av

SA

SB

SC

46

Substituting (46) into (43) and (44), the following rela-tion is obtained:

ddt

IA

IB

IC

= 1Lfd

EAO

EBO

ECO

−UDCavLfd

SA

SB

SC

−13 SA SB SC

1

1

147

The dynamic model in the abc frame of the HMCequivalent circuit is represented by (47). The switching statefunctions, di (i = A, B, C), are defined as

dA

dB

dC

=SA

SB

SC

−13 SA SB SC

111

48

The dynamic model of the equivalent circuit model in theabc frame is achieved by combining (47) and (48) in thefollowing equation:

Lfdddt

IA

IB

IC

=EAO

EBO

ECO

−UDCav

dA

dB

dC

49

The DC side differential equation can be written as

dUDC avdt

= 1Ceq

Idc = dA dB dC

IA

IB

IC

, 50

sin_cos

sin_c

os

Eq

Eq

Phase-lockedloop

(PLL)

Ed

ECO

EBO

EAO

abc

dq0

abc

dq0

abc

dq0

MPPT

DC link voltage control

Current control1/6

++++++

Vdc

Vdc6Vdc5Vdc4Vdc3Vdc2Vdc1

VDC_REF

VDC_av

IPV

ILqILC

ILB

ILA

IC

IB

IA

𝜔Lfd

𝜔Lfd

ILd

Iq

Iq_REF

Id

+ PI

PIPI

Ed

Ed

Id_REF

VDC_REF

Vd_REFVa_REF

1

1

PSPW

M #

1

Module #1

Module #3

Module #5

Module #2

Module #4

Module #6

PSPW

M #

2

1

‒1

‒1

‒1

Vb_REF

Vc_REF

abc

dq0

Vq_REF−+−

−

+ −++−

+− ×

×÷ × ÷

×÷

Figure 15: HMC control block diagram.


dUDC avdt

= 1Ceq

2dA + dB IA +1Ceq

dA + 2dB IB 51

It can be seen that the model represented by (49) and (51)is time varying. Thus, to facilitate the control algorithmimplementation, the model can be expressed in the synchro-nous reference frame rotating at constant frequency ω. Thecorresponding conversion matrix is

Cabcdq =23

cos θ cos θ − 2π3 cos θ − 4π

3−sin θ −sin θ − 2π3 −sin θ − 4

π

3

,

52

where θ = ωt.Applying the coordinate transformation to (49) we

obtain

Lfdddt

Id

Iq=

Ed

Eq+ Lfω

Iq

−Id−UDCav

dd

dq53

Similarly, applying this transformation to (50) we obtain

CeqdUDC av

dt= ddId + dqIq 54

The obtained model, represented by (53) and (54), isnonlinear owing to the product between the state variables(i.e., Id, Iq, andUDC av) and the inputs (i.e., dd and dq).

Figure 11 shows the control principle of the HMC.Because the proposed configuration is a three-wire system,only two phase currents are required to be measured.

Currents IA and IB are measured and converted to the dq0frame to obtain the corresponding currents Id and Iq.Accordingly, (53) is rewritten as follows:

Lfdud

uq=

Ed

Eq+ Lfdω

Iq

−Id−UDCav

dd

dq, 55

whereud

uq= d/dt

Id

Iq.

Table 2: General data of the proposed system.

Parameter Value

System parameters

Nominal power P = 2.5 kVA

Phase voltage and frequencyVph = 110V (rms), fsys = 50Hz,

fswiching = 2500Hz

Output current Iout = 8.8 A (rms)

DC bus voltagesVdc1 = Vdc2 = Vdc3 = Vdc4= Vdc5 = Vdc6 = 52.6V

Open-end winding transformers Vpri/Vsec = 1, ftran = 50Hz

Current controller parameters Kp = 55, Ki = 0.001

Voltage controller parameters Kp = 3, Ki = 20

Coupled inductor parameters

RMS voltage, current 20V (rms), 8.8 A (rms)

Magnetizing inductance 3.2mH

Leakage inductance 150 μH

Carrier#1 Carrier #3 Carrier #2

120°

+

−

Module #1

Module #3

Module #5

Module #2

Module #4

Module #6

Comparator

+

−

Comparator

Modulation signals

PSPW

M #

1PS

PWM

#2

−1

Carrier #1 Carrier #3 Carrier #2

120°

Figure 16: Proposed PS-PWM diagram.


250

200

150

100

50

0

−50

−100

−150

0.4 0.41 0.42 0.43 0.44 0.45Time (s)

0.46 0.47 0.48 0.49

VAA′VBB′VCC′

0.5

−200

−250

Figure 17: Three-phase output voltage of the HMC.

10

0

−10

10

0

−10

10

0

−10

10

0

−10

IAA′IBB′ICC′

ia1′ib1′ic1′

ia4′ib4′ic4′

ia3′ib3′ic3′

ia6′ib6′ic6′

ia5′ib5′ic5′

ia2′ib2′ic2′

0.4 0.41 0.42 0.43 0.44 0.45Time (s)

0.46 0.47 0.48 0.49 0.5

Figure 18: Currents inside the HMC.


Demonstrated that currents Id and Iq can be controlledseparately by acting on inputs ud and uq, respectively. Hence,the controller is designed using the following expressions:

ud = kpid~+ki id

~ dt,

uq=kpiq~+ki iq

~ dt,56

where id~=Id REF−Id and iq

~=Iq REF−Iq are the current errorsignals, with Id REF and Iq REF being the reference valuesfor currents Id and Iq, respectively.

Using (55), the current control law is given by the follow-ing expression:

dd

dq= LfdωUDC av

Iq

−Id+ 1UDC av

Ed

Eq−

LfdUDCav

ud

uq

57

The d-axis reference current (Id REF) is produced usingthe DC-link voltage controller, while the q-axis referencecurrent (Iq REF) is taken as the load q-axis current (ILq).To aim for a unity power factor, the Iq REF is set to zero(i.e., iq REF = 0). The active power exchange between theDC links and the grid is proportional to the direct-axiscurrent Id and can be expressed as

Pdc =32 EdId + EqIq =

32 EdId 58

6

5

4

3

2

Mag

(% o

f fun

dam

enta

l)

1

00 50 100 150

Fundamental (50 Hz) = 184, THD = 13.57%

Harmonic order200 250 300

Figure 19: THD of the HMC output voltage.

0.4

0.35

0.3

0.25

0.2

Mag

(% o

f fun

dam

enta

l)

0.15

0.1

0.05

0 5 10 15 20 25 30 35 40 45 500

Fundamental (50 Hz) = 8.8, THD = 0.68%

Harmonic order

Figure 20: THD of the HMC output current.


(58) it is shown that direct-axis current Id is responsiblefor maintaining the DC-link voltages at a desired value. Thus,using (54), one deduces that

CeqdUDC av

dt= ddId = udc 59

Consequently, the active current is

Id =udcdd

= udcVDC avddVDC av

60

Assuming that the current loop is perfect and theHMC works under balanced conditions, the followingexpressions hold:

ddVDC av = Ed,

Id =udcdd

= udcVDC avEd

,

Ed

Eq= 32

V̂

0,

61

where V̂ and Ed are the RMS voltage and the direct-axisphase voltage at the PCC, respectively. Thus, the controleffort of the DC link voltage loop is given by

Id REF =udcdd

= 23VDC av

V̂udc 62

To control the DC-link voltage, a PI controller is used,which is expressed as

udc = kpdcvdc~+kidc vdc

~ dt, 63

where vdc~=VDC REF−VDC av is the DC-link voltage error,

whereas VDC REF and VDC av are the DC-link reference andaverage voltages, respectively.

5.2. DC-Link Voltage Controller. With reference to Figure 3,the HMC is based on a symmetric configuration, havingsix converters with identical power capabilities that are sup-plied by six equal PV strings. The PV strings are directlyconnected to each converter, Vpv1 = Vpv2 = Vpv3 = Vpv4 =Vpv5 =Vpv6 =Vpv , the MPPT must be achieved by the con-verters, and the DC-link voltages continuously fluctuate.Because the PV strings are supposed to be identical, beingcreated by a single PV string divided into six identical parts,a single MPPT algorithm can be considered. For this reason,the same DC-link voltage reference for the six convertershas been considered. The DC-link voltage reference is com-pared to the sum of the actual six DC-link voltages, and theerror is passed through a PI controller to determine thecontrol parameter udc.

5.3. Modulation Strategy. To obtain an output voltage withlow total harmonic distortion (THD), a multicarrier phase-shifted PWM (PS-PWM) switching strategy [24, 25] isimplemented to drive each IGBT in the HMC. Optimum har-monic cancellation is accomplished by shifting each carrier

cell by 2πTs/3T in sequence, where Ts is the switching timeand T is the cycle modulation time. A reference signal of50Hz is generated using the control algorithm representedin Figure 15, with the switching frequency fixed at 2500Hz.

Figure 16 shows the relationship between the modulationwaveforms and the three groups of carriers within the HMC.As shown in Figure 16, triangular carriers (i.e., carrier #1, car-rier #2, and carrier #3) are phase-shifted 120° to each otherand directly compared with the modulation signals to drivethe IGBTs within module #1, module #2, and module #3.In order to generate the switching signals used to drive theIGBTs within module #2, module #4, and module #6, themodulation signals are inverted and then compared withthe triangular carriers.

6. Simulation Results

In order to demonstrate the performance of the HMC and itscontrol algorithm, the complete grid-connected PV system

−2.5−2

−1.5−1

−0.50

0.51

1.52

2.53

Circ

ulat

ing

curr

ent (

A)

2-winding coupled inductor3-winding coupled inductor6-winding coupled inductor

Figure 22: Circulating current inside the HMC with multiwindingcoupled inductors.

×10−3

2-winding coupled inductor3-winding coupled inductor6-winding coupled inductor

02468

1012141618

Circ

ulat

ing

curr

ent, I ci

r,pea

k (A

)

2 3 4 5 6 7 8 9 101Magnetizing inductance (H)

Figure 21: Peak circulating current versus magnetizing inductance.


was simulated using the MATLAB/Simulink environment.The technical characteristics of the system parameters andthe coupled inductors are provided in Table 2. In order to val-idate the performance of the HMC, a PS-PWM technique hasbeen implemented, as shown in Figure 16. The output activeand reactive power supplies in response to the fluctuations insolar radiation value are shown in the following subsections.

6.1. Performance Analysis. The seventeen-level phase volt-ages of the HMC are generated at the steady state and shownin Figure 17. It can be seen that the voltages are balanced, asshown in the voltage phasor diagram. Figure 18 demon-strates the currents of modules 1, 3, and 5, and the currentsof modules 2, 4, and 6. Given the configuration of thethree-phase HMC, the line currents of every VSC modulewithin the HMC are symmetrical, hence verifying therelationship in (5).

From this figure, it can also be acknowledged that thedirection of the currents of modules 2, 4, and 6 is reversedfrom those of modules 1, 3, and 5. Moreover, the currentsinside every VSC module within the HMC are the same asthe output currents.

The THD values of the output voltage and current arecalculated using the following equation.

%THDx = 100 〠h≠1

xshxs1

2, 64

where the subscript x indicates the THD in the signal (voltageor current), xs1 is the fundamental component, xsh is thecomponent at the h harmonic frequency.

The harmonic spectra of the output voltage and currentare shown in Figures 19 and 20, respectively. The THD of

the HMC current is 0.68%, which is fewer than 5% and meetsthe power quality standard. Using the suggested modulationtechnique, the highest harmonic family of the phase voltageappears at the band of the 100th harmonic order. Conse-quently, the effective switching frequency of the phasevoltage is two times higher than the switching frequency.

As a comparison, the HMC using two-, three-, and six-winding coupled inductors are simulated under the condi-tions. The magnetizing inductance of each coupled inductoris 3.5mH. According to the current from the equations pro-vided in Section 4, if the voltage of the DC links is unbalancedthe circulating current will be introduced on the converteroutput currents. Therefore, to intentionally produce a circu-lating current, the DC link voltage of Module 3 is decreasedfrom 52.6 to 26.3V. The created loop voltage is computedusing the following equation:

V loop,rms =3

2 252 6 − 26 3 ma 65

Thus, using (42) and (65), the circulating current insidethe HMC is expressed as

Icir,peak =3

24π 52 6 − 26 3maf kLm

66

Investigations with different magnetizing inductancelevels have been carried. Figure 21 shows the theoreticalvalues of the peak circulating currents with different magne-tizing inductance levels. Figure 22 shows the simulationresults with a relatively large magnetizing inductance(3.5mH). The circulating currents are found by measuringthe difference between the output currents of module 1 andmodule 3. As shown in the waveforms, the circulating

2-winding coupled inductor



−0.5

0

0.5

−0.5

0

0.5

Circ

ulat

ing

curr

ent (

A)

−0.5

0

0.5

Figure 23: Circulating current inside the HMC.


currents inside the HMC with two-, three-, and six-windingcoupled inductors are 2.3, 1.5, and 0.76A peaks as calcu-lated in (66). From the waveforms in Figure 22, using a

six-winding coupled inductor, the circulating current isefficiently minimized.

In order to achieve the same circulating currents (e.g.,0.5A), the magnetizing inductance of the two-winding,three-winding and six-winding coupled inductors should be

40

45

50

55

60

DC-

links

vol

tage

(V)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Time (s)

Figure 29: Variations in DC-link voltages in response to thechanges in solar radiation.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Time (s)

−10−5

05

10

Grid

curr

ent (

A)

−200−100

0100200

Grid

vol

tage

(V)

Figure 28: Response to changes in solar irradiance in the hexagramconverter.

0200400600800

1000

Sola

r rad

iatio

n (W

/m2 )

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Time (s)

Figure 27: Step changes in solar radiation.

Table 3: Total harmonic distortion under different solar radiationconditions.

Solar radiationTHD (%) Fundamental value

Voltage Current Voltage (V) Current (A)

1000 (W/m2) 13.64 0.68 184.0 8.80

600 (W/m2) 13.65 1.15 181.8 5.20

400 (W/m2) 12.78 2.80 180.9 3.44

0 (W/m2) 23.94 0.35 148.3 9.98

Icir.Ia1Ia3

−15

−10

−5

0

5

10

15

Curr

ent (

A)

Figure 26: Circulating current with six-winding coupled inductorswith a magnetizing inductance of 0.5mH.

Icir.Ia1Ia3

−20−15−10−5

05

101520

Curr

ent (

A)

Figure 25: Circulating current with three-winding coupledinductors with a magnetizing inductance of 0.5mH.

Icir.Ia1Ia3

−30

−20

−10

0

10

20

30

Curr

ent (

A)

Figure 24: Circulating current with two-winding coupled inductorswith a magnetizing inductance of 0.5mH.


increased to 17.5, 11.6, and 6mH, respectively. As shown inFigure 23, the waveform matches the theoretical analysis inFigure 21. The output current of module 1, module 3, andthe circulating current inside the HMC with two-winding,three-winding, and six-winding coupled inductors under anunbalanced DC link voltage are shown in Figures 24, 25,and 26, respectively. The waveforms show the simulationresult with an extremely small magnetizing inductance of0.5mH, which is 1/7 of that in Figure 22. The simulationsvalidated the quantitative relationship that the circulatingcurrent will be too big to maintain a normal operation ifthe coupled inductors with low magnetizing inductance areused. As shown, the current ia3 nearly doubles the circulatingcurrent, which evidently proves that the HMC is sensitive toan unbalanced DC voltage. The waveforms match thetheoretical analysis in Figure 21.

6.2. Active Power Variation. The system is tested for differentsolar radiation conditions and the results are shown inTable 3. Figure 27 shows the solar radiation variation whichis considered in this study. The output current and gridvoltage variation in response to the changes in solar radiationis shown in Figure 28. The transient behavior of the totalDC-link voltage is presented in Figure 29. The parame-ters are successfully attuned by the controller to keepthe DC-link voltage at the desired level of 52.6V. Fluctua-tions are observed in the DC-links, due to the step variationsin the solar radiation. Nevertheless, the controller brings thevoltage to the reference level within 0.02 s.

The active power supplied by the HMC is directly pro-portional to the magnitude of direct axis, id, as indicated by(38). The solar radiation at all DC links is reduced by 40%at 0.25 s. The fluctuation in solar radiation is imitatedthrough the reduction in the direct-axis current by approxi-mately 40%, in a step. It decreased to 5.216A from the initialvalue of 8.823A, as shown in Figure 30. This reduction in idwas to keep the DC-link voltage at the reference level bydecreasing the output power drawn from the PV. Moreover,

to guarantee maximum utilization of the PV system, thequadrature axis current was kept at zero. At 0.45 s, the solarradiation is further reduced to 400W/m2. The direct-axiscurrent is reduced to 3.267A from the original value of5.216A. Later, the direct-axis current is increased to8.823A, because of the increment in the solar radiation.

6.3. Reactive Power Compensation. The described controlalgorithm permits the HMC to act as DSTATCOM in theabsence of solar radiation. The output reactive power canbe calculated as

Qout = −32 IqVd 67

In this condition, the reactive power is increased by2700VAR in a step, in the absence of solar radiation. Theinfluence, for the DSTATCOMmode operation, is presentedin Figure 31.

The direct-axis current is found to be zero, indicatingthe fact that no active power is being transferred to the gridin the absence of solar radiation. However, due to the stepchange in reactive power, the capacitors consume currentfrom the grid. It is found that the direct-axis current takes0.02 s to stabilize. The nature of the fluctuation in theDC-link voltage is depicted in Figure 29. The DC-link con-troller effectively keeps the DC-link voltage by regulatingthe power flow through the capacitor. Hence, it can bementioned that the HMC effectively operates as DSTAT-COM in the absence of solar radiation. Figure 31 shows thesource voltage and converter output current in DSTATCOMmode. The waveforms show that the output current increasesafter the reactive power command comes at 0.8 s. Moreover,Figure 31 approves that the phase difference of the con-verter output current with grid voltage is 90° in this modeof operation.

3000

Active powerReactive power

2000

1000

0

−1000

−2000

0.2 0.4 0.6Time (s)

0.8 1−3000

(a)

12

10

8

6

4

2

0

−2

Direct-axis currentQuaderature-axis current

0.2 0.4 0.6Time (s)

0.8 1

(b)

Figure 30: Response to changes in solar radiation. (a) Output active and reactive power, and (b) direct and quadrature axis current.


7. Conclusion

The HMC for a grid-connected PV system shares manyadvantages of the CHB but uses fewer switches andreduces the size of the DC-link capacitor. However, theHMC is sensitive to the loop voltage produced by theprobable DC-link voltage unbalance. The unbalanced con-ditions of the DC-link voltages will initiate a line-frequency circulating current. Therefore, to minimize thecirculating current as well as to reduce an extra voltagedrop on the inductors that affects the terminal voltage,the inductance value should be adequate. The multiwind-ing coupled inductors are the key to control the circulat-ing current and ensure the proper operation of theHMC. The equivalent circuit model of the HMC configu-ration is derived to recognize the control scheme. Thetwo-winding coupled inductor imposes twelve times themagnetizing inductance for the circulating current, whilethe impedance to the circulating current using three-winding coupled inductors is eighteen times as much asthe magnetizing inductance. Owing to the six-windingcoupled inductor, the impedance on the circulating currentpath is thirty-six times the magnetizing inductance. Theresults show the good performance of the control algo-rithm in both steady state and transient conditions. More-over, it is interesting to note that in the absence of solarradiation, the controller acts in DSTATCOM mode tosupply reactive power to the grid. The performance of thecontroller under different solar radiation conditions isfound to be satisfactory.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to express their thanks to KingAbdulaziz City for Science and Technology (KACST) forproviding financial and technical support to this study.

References

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200

150

100

50

0

−50

−100

−150

−2000.82 0.84 0.86 0.88 0.9

Time (s)0.940.92 0.96 0.98 1

VAVBVC

IAIBIC

Figure 31: System performance with the increase in the reactive power supply in DSTATCOM mode.


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