inductor saturation compensation in three-phase three-wire
TRANSCRIPT
Inductor Saturation Compensation in Three-PhaseThree-Wire Voltage-Source Converters via Inverse
System Dynamics-IIZiya Ozkan, Member, IEEE, and Ahmet M. Hava, Senior Member, IEEE
Abstract—Current waveform quality and current control band-width are the figures of merit for current controllers in three-phase three-wire (3P3W) voltage-source converter (VSC) systems.When saturable inductors are employed as converter side induc-tors due to cost, size, and energy conversion efficiency benefitsin 3P3W VSCs with conventional synchronous frame currentcontrol (CSCC); substantial drawbacks arise by means of thesefigures of merit. In the preceding part of this study, an inversedynamic model based compensation (IDMBC) method has beenproposed to deal with these drawbacks. Complementarily, thispaper presents a thorough analytical investigation of the controlsystem characteristics of CSCC and IDMBC methods that affectthe current waveform quality and current control bandwidth. Theanalyses are verified via dynamic response and waveform qualitysimulations and experiments that employ saturable inductorsreaching 1/9th of the zero-current inductance at full current.The results obtained demonstrate the suitability of the IDMBCmethod for 3P3W VSCs employing saturable inductors.
Index Terms—Bandwidth, current control, disturbance rejec-tion, dynamic response, harmonic distortion, inductor saturation,robust control, three-phase, three-wire, voltage-source converter,waveform quality.
I. INTRODUCTION
INDUCTORS are key constituents of voltage-source con-verters (VSCs) by taking part as element(s) of load (grid)
interface filters. In all filter topologies [1], the converter side isinterfaced by inductors experiencing rectangular-shaped PWMvoltage and associated current. Accordingly, the converter sideinductors have a high impact on the cost, size, weight, and ef-ficiency of the overall VSC system [2], [3]. By means of thesecriteria, the utilization of saturable inductors brings significantimprovement [4]–[10]. The utilization of saturable inductorsin three-phase three-wire (3P3W) VSC systems (shown inFig. 1 realized as well-known two-level topology) can spreadthese benefits due to the wide range and volume energyconversion applications of these systems. On the other hand,such utilization in 3P3W VSC systems with industry-standardconventional synchronous frame current control (CSCC) em-ploying PI compensators introduces unignorable current wave-form quality and bandwidth shrinkage issues as reported in thepreceding part of this study [7].
Probably, the current waveform quality issue is the moredetrimental one introduced by the utilization of saturableinductors in 3P3W VSCs with CSCC. Besides the prohibitiveeffects such as additional losses, electromagnetic interference,and converter derating; certain grid codes such as IEEE1547, VDE-AR-N 4105, and BDEW restrain the total current
Vdc/2
Vdc/2
vanS1
S2
S3
S4
S5
S6
vbn
vcn
o
La
Lb
Lc
Ra
Rb
Rc
ia
ib
ic
nA
BC
vL-c+ -
vL-b+ -
+ -vL-a
Fig. 1. Three-phase three-wire two-level voltage-source converter topology.
harmonic distortion (THDi) and individual current harmonicsinjected to the grid [1], [11]–[13].
The current harmonics associated with the utilization ofsaturable inductors in VSCs with CSCC have various sources.The first and dominant source is the mismatch between thenonlinear physical system and the linear controller [14], [15].In the second, disturbances caused by the converter nonlin-earity, imperfectly compensated dead-time, and measurementerrors contribute to current harmonics not only in VSCsemploying saturable inductors but also in VSCs with linearinductors [16]–[18]. The third source of the harmonics isthe imperfectly decoupled cross-coupling terms [19], [20].Such imperfect decoupling may occur in linear inductor based3P3W VSCs with CSCC due to estimated inductance andactual inductance mismatch. Lessening such effect of themismatch, in [21], a complex vector-based approach is pre-sented. On the other hand, in the case of the VSC systemswith saturable inductors employing CSCC, the cross-couplingdecoupling matrix is indeterminate as the inductance of eachphase is time-varying [7].
Current control bandwidth shrinkage due to the utilizationof saturable inductors in VSCs with CSCC is the other andrelatively scarcely investigated issue in the literature. Beingthe measure of how well an input sinusoid is tracked up tothe frequency where the gain does not drop below -3 dB[22], the bandwidth is highly correlated with the dynamicresponse performance for linear closed-loop systems. In [21],such correlation is demonstrated for 3P3W VSCs exhibitinglinear system characteristics via frequency response functions.On the other hand, for VSCs employing saturable inductors,the dynamic response can still be utilized as an indicator ofthe system bandwidth as a function of time; wherein [8], [23],such a utilization has already been performed and verified forsingle-phase VSCs with saturable inductors. In this study, asimilar approach is adapted to analyze 3P3W VSC systemsemploying saturable inductors.
In the first part of this study [7], an inverse dynamicmodel based compensation (IDMBC) method is presentedto overcome the waveform quality and control bandwidthshrinkage issues of the CSCC method. In accordance withthe preceding part, this paper analytically investigates andcompares the CSCC and IDMBC methods by means of controlsystem characteristics that affect the waveform quality and thecurrent control bandwidth; and verifies these characteristicsand comparisons by means of simulations and experiments.For both methods, the investigation is performed through atwo-phase manner representation. For the reproducibility ofthe results, the s-domain functions related to the analyses aregiven in an explicit manner.
The paper is organized as follows. In Section II, the CSCCand IDMBC methods are summarized in a two-phase mannerand associated analytical convention of the methods for theprogressing sections is provided. In Section III, the control sys-tem characteristics of the methods are elaborated by means ofcommand to output, disturbance rejection, and cross-couplingdecoupling. In Section IV, simulation results regarding theinvestigated control characteristics are provided via dynamicresponse, cross-coupling behaviour, and waveform quality.Similarly, in Section V, experimental results for the samesystem parameters that of the simulations are provided. Thesesimulation and experimental results show the superiority ofthe proposed IDMBC method over the CSCC method andthe viability of the IDMBC method at deep saturation levelsvia saturating the converter-side inductors down to 1/9th
of the zero-current inductance. While the investigation andthe verification are performed for the two-level 3P3W VSCtopology in the paper, the method can be applied to othermultilevel, multiphase VSC configurations also.
II. THE CSCC AND IDMBC METHODS
For the analyses conducted in the subsequent sections,two-phase modelling of CSCC and IDMBC current controlmethods are summarized in this section. In Figs. 2(a) and (b),these current control models are provided in a block diagramrepresentation respectively.
For both of CSCC and IDMBC methods, the load is an L-Rtype with saturable inductors, hence is nonlinear. The distur-bance signal interfered output voltage vσ = [vab vbc]
T is thecontrol input to this load with iσ = [ia ib]
T is the output cur-
rent, and Lσ =
[La −LbLc Lb + Lc
]and Rσ =
[Ra −RbRc Rb +Rc
]are the two-phase nonlinear system inductance and resistancematrices respectively as described in [7].
Considering that an ideal 3P3W VSC is a unity gain ampli-fier (shown in Figs. 2(a) and (b) as 2×2 identity matrices), andassuming the disturbance decoupling is accurately performedand the estimated system parameters are identical to theactual parameters (Lσ = Lσ , Rσ = Rσ), the CSCC andIDMBC methods have the stationary frame dynamic equationsrespectively as
v∗σ = Lσ
d
dtiσ + Rσiσ (1)
wσ = LMd
dtiσ (2)
where LM =
[Lmin 0
0 Lmin
]in which Lmin is the minimum
inductance value to be able to perform designs within themaximum available current control bandwidth.
For the purpose of increasing stability, active dampingtechniques can be incorporated into the systems described by(1) and (2). However, such an approach has a slight effect onthe relative performances of the current controllers and com-plicates the oncoming analyses. Similarly, the zero-sequencevoltage injection ideally does not have any effect on theperformance of the current controllers. Hence such injection isinvolved neither in the block diagram representations in Fig.2 nor in the analyses.
The stationary frame CSCC system equation (eq. (1)) isdependent on the nonlinear (and time varying) system param-eters (Lσ), whereas that of IDMBC method (eq. (2)) is a linearone. The control inputs of (1) and (2) (v∗
σ and wσ respectively)are both generated in synchronous frame and transformed tostationary frame. The synchronous frame to/from stationaryframe transformations are performed differently for CSCC andIDMBC methods as shown in Fig. 2 via different transfor-
mation matrices T(θ) =
[cos(θ) − sin(θ)
cos(θ − 2π/3) − sin(θ − 2π/3)
]and H(θ) = T(θ)−T(θ−2π/3) as discussed in the precedingpart of this study [7].
The error compensation is performed via PI compensatorsin the synchronous frame and is the same for the CSCC andIDMBC methods. The PI compensator coefficients are setto obtain predefined design bandwidth ωBW for each axisby setting Kp = ωBWLmin and Kp = ωBW R [24]. Forthe CSCC method, it is reasonable to employ the minimuminductance value in the setting of the Kp value to preventtime-varying system bandwidth from exceeding maximumavailable system bandwidth restrained by either of the carrierfrequency, sampling method, measurement quality, or theprocessor speed. For the IDMBC method, on the other hand,as the nonlinear plant is linearized to behave as a fictitiouslinear inductor system with a constant inductance value ofLmin, this minimum inductance value is employed to set Kp.Besides, other PI setting approaches such as [25], [26] can alsobe preferred with the consideration of bandwidth run-over.
Similar to the PI compensation, the cross-coupling de-coupling is performed the same for the CSCC and IDMBCmethods in the synchronous frame. The minimum inductancevalue is utilized in the establishment of the cross-couplingdecoupling matrix as
Wcc =
[0 −ωLmin
ωLmin 0
](3)
where ω is the fundamental frequency. The reasoning behindthe use of Lmin for CSCC method is the fictitious linearinductor behaviour of the linearized system with an inductanceof Lmin. In the case of the CSCC method, the same cross-coupling decoupling matrix is employed to establish a fair
iσ
Rσ
Lσ-1
Nonlinear Physical System
vσ*
Kp
idq*
idq
vdqedq
Ki
Ivσ**
vD vD
vσ
Ideal
3P3W VSCDecouplingCompensation
T-1(θ)
H(θ)
Frame
Conversion
Wcc
vσ*
Kp
idq*
idq
vdqedq
Ki
I
Ideal
3P3W VSCCompensation
T-1(θ)
T(θ)
Frame
Conversion
uσLσ
Rσ
Unified Integral IDM
1Lmin
wσ
Wcc
iσ
Rσ
Lσ-1
vD
vσvσ**
vD
Decoupling Nonlinear Physical System
(b)
(a)
Fig. 2. 3P3W VSC nonlinear closed-loop current control systems in two-phase column vector block diagram representation. (a) Conventional synchronousframe current control (CSCC) method. (b) Inverse dynamic model based compensation (IDMBC) method.
comparison basis as it is indeterminate when the inductancesof the three-phase current control system change with time.
III. CONTROL SYSTEM CHARACTERISTICS
In this section, the control system characteristics of theCSCC and IDMBC methods are elaborated. The investigationconsists of command to output, disturbance to output, andcross-coupling characterizations.
The command to output frequency response characteristicsare investigated in synchronous frame as the synchronousframe currents are dc quantities and their dynamic responsesprovide clear verification of the performance of the controlmethods as shown in the following sections via simulationsand experiments. In the case of the disturbance to outputfrequency response investigation, the analyses are performedin the stationary frame as the physical disturbance source andthe output currents are both in the stationary frame.
In the analyses, the VSC system is assumed to operate atthe steady-state where the inductance variation is dominated bythe fundamental frequency large-signal and small magnitudecommand/disturbance applied. Accordingly, the performanceof the controllers is interpreted via frequency responses and ahigh correlation between the interpretations, simulations, andexperiments is observed.
GP(θ,s)
GH(θ,s)
(b)
idq*
GH-dq(θ,s)dσ
GP-dq(θ,s)
(a)
vσ iσ
ddq
vdq idq
Fig. 3. Generic control block diagrams for 3P3W VSC systems for analyzingthe frequency response characteristics of the current controllers. (a) Thegeneric block diagram in synchronous frame for the investigation of commandto output characteristics. (b) The generic block diagram in stationary framefor the investigation of disturbance to output characteristics.
In Fig. 3, the generic current control system block diagramrepresentations are shown for the command to output and thedisturbance to output frequency response analyses. Fig. 3(a)
represents the overall control system for the CSCC and theIDMBC methods in synchronous frame for the command tooutput analyses. In the figure, GP−dq(θ, s) is the physicalplant transfer function matrix whereas GH−dq(θ, s) is theoverall unified form of the other system blocks in the overallcurrent regulation system in the synchronous frame. For thisanalysis, the disturbance signal in the synchronous frame givenby (4) is set to zero.
ddq = vD−dq − vD−dq (4)
where vD−dq and vD−dq are the synchronous frame represen-tations of the estimated and actual disturbances respectively.
The disturbance to output frequency response characteristicsare investigated in the stationary frame based on Fig. 3(b).In the figure, the current command i∗dq is set to zero hencenot shown. The stationary frame disturbance signal is takenas an input to the system wherein GP(θ, s) is the physicalplant transfer function matrix whereas GH(θ, s) is the overallunified form of the other system blocks in the overall currentregulation system in the stationary frame. The followingparagraphs further clarify the obtained frequency responsecharacteristics.
A. Command to Output Characteristics
Considering the two phase modelling in the preceding paperand Fig. 2, equation (5) represents the nonlinear physicalsystem for CSCC and IDMBC methods as
v∗σ + dσ = Lσ
d
dtiσ + Rσiσ. (5)
where dσ = vD − vD. While this nonlinear physical systemrepresentation is common to the CSCC and the IDMBCmethods, the command voltage v∗
σ is generated differentlyfor each method. The representation in (5) with the stationaryframe variables can be expressed in the synchronous frame forCSCC and IDMBC methods respectively as
H(θ)vdq = Lσd
dt(T(θ)idq) + RσT(θ)idq (6)
LσLmin
T(θ)vdq+Rσ
Lmin
∫ t
0
T(θ)vdqdτ
= Lσd
dt(T(θ)idq) + RσT(θ)idq
(7)
by setting the disturbance signal to zero for command to outputfrequency response characterization. The synchronous framecontrol voltage vdq in (6) and (7) is the sum of the output ofthe PI compensation of the current error and the cross-couplingdecoupling term which is given by
vdq = Kpi∗dq− idq+Ki
∫ t
0
i∗dq− idqdτ+Wccidq. (8)
Inserting (8) into (7) and (6) and expanding the derivativeterm one can obtain the closed-loop current control dynamicsfor CSCC and IDMBC methods in s-domain respectively as
H(θ)Kc(s)i∗dq(s)− idq(s)+ Wccidq(s)
=LσX(θ, s) + RσT(θ)
idq(s)
(9)
LσLmin
X(θ, s) +Rσ
LminT(θ)
Kc(s)i∗dq(s)− idq(s)
+ Wccidq(s)
=LσY(θ, s) + RσX(θ, s)
idq(s)
(10)
where Kc(s) =
[Kp +Ki/s 0
0 Kp +Ki/s
]is the s-domain
representation of PI compensator. Further, X(θ, s) = ωP(θ)+sT(θ) and Y(θ, s) = −ω2T(θ) + 2ωsP(θ) + s2T(θ) arethe s-domain representations of first order and second ordertime derivatives of the operator T(θ)(·) respectively in whichP(θ) = T(θ+π/2). The closed-loop current control dynamicsof (9) and (10) can be rearranged in the form given by
idq(s) = GOL−dq(θ, s)i∗dq(s)− idq(s) (11)
where GOL−dq(θ, s), error to output synchronous frameopen-loop transfer function matrix, corresponds to the multi-plication of GH−dq(θ, s)GP−dq(θ, s) shown in Fig. 3. Thesematrices for CSCC and IDMBC methods are obtained as
GCSCCOL−dq(θ, s) =LσX(θ, s) + RσT(θ)−H(θ)Wcc
−1H(θ)Kc(s)
(12)
GIDMBCOL−dq (θ, s) =
LσY(θ, s) + RσX(θ, s)
− LσLmin
X(θ, s) +Rσ
LminT(θ)
Wcc
−1
× LσLmin
X(θ, s) +Rσ
LminT(θ)
Kc(s).
(13)
The frequency response characteristics of CSCC given in(12) describe spatially varying (θ dependent) behaviour inwhich the variation frequency is assumed to be much smallerthan the investigated system dynamics (ω ωBW ). On theother hand, the frequency response characteristics of IDMBCmethod in (13) are spatially invariant as intended [7] whenthe estimated system parameters are identical to the actual
system parameters (Lσ = Lσ , Rσ = Rσ). When the systemis linear (the inductance is constant), the command to outputcharacteristics of CSCC and IDMBC methods also becomethe same and spatially invariant. Accordingly in Fig. 4, thed-axis to d-axis and q-axis to q-axis magnitude and phasecharacteristics of CSCC and IDMBC methods are illustratedfor a linear symmetrical 3P3W L-R current control systemwith L = 500 µH, R = 0.5 Ω, and Kp = 1.57, Ki = 1570to yield ωBW = 2π500 rad/s [24]. As shown in Fig. 4(a)and (b), the d- and q-axis system bandwidths of CSCC andIDMBC methods are the same as the preset design bandwidthωBW , regardless of the phase angle. Similarly, the phasecharacteristics of the d-axis to d-axis and q-axis to q-axisopen-loop frequency responses of the two methods are thesame and constant (−90) yielding a phase margin (PM) of90. The magnitude and phase characteristics of the d- andq-axis show two decoupled single-phase linear L-R controlsystem behaviour as intended.
-20
-10
0
10
20
30
102
103
104
-90
Frequency (rad/s)
-20
-10
0
10
20
30
102
103
104
-90
Frequency (rad/s)
id (s)ed (s)
iq (s)
eq (s)
CSCC, IDMBC CSCC, IDMBC
CSCC, IDMBC CSCC, IDMBC
ωBW ωBW
Mag
nit
ude
[dB
]P
has
e [d
eg]
Mag
nit
ude
[dB
]P
has
e [d
eg]
(a) (b)
(c) (d)
Fig. 4. d-axis to d-axis and q-axis to q-axis open loop magnitude and phasecharacteristics of the frequency responses of CSCC and IDMBC methodsfor symmetrical 3P3W linear VSC system. (a) d-axis to d-axis magnituderesponse. (b) q-axis to q-axis magnitude response. (c) d-axis to d-axis phaseresponse. (d) q-axis to q-axis phase response. These plots are θ independentdue to linear and symmetrical operation. For the IDMBC method withnonlinear inductors, the characteristics are the same as if the system is alinear one due to linearization in the large [7].
In Fig. 5, the case for a nonlinear inductor is elaboratedfor the CSCC method. In the case, the inductance of eachphase decreases from 0.45 mH to 500 µH with zero to fullload current (10 A) having an ESR of 0.5 Ω. The Kp and Ki
values are selected the same as the linear case (Kp = 1.57,Ki = 1570) to yield ωBW = 2π500 rad/s. It should be notedthat the Kp value is calculated in proportion to the minimuminductance Lmin to avoid bandwidth run-over. Consideringthe d-axis to d-axis and q-axis to q-axis magnitude plots inFig. 5 (a) and (b) for θ = 0 and θ = π/3 under full-load operation, the d-axis bandwidth (ωBW−dd) and q-axisbandwidth (ωBW−qq) are observed to be much smaller than thedesign bandwidth ωBW which is called bandwidth shrinkage[8], [23] and elaborated in [7]. Moreover, the phase responsesin Fig. 5(c) and (d) show the PM of d- and q-axis decrease
-120
-110
-100
-90
-110
-105
-100
-95
-90-20
-10
0
10
20
30
102
103
104
Frequency (rad/s)
-20
-10
0
10
20
30
102
103
104
Frequency (rad/s)
id (s)ed (s)
iq (s)
eq (s)CSCC CSCC
ωBW ωBW
Mag
nit
ude
[dB
]P
has
e [d
eg]
Mag
nit
ude
[dB
]P
has
e [d
eg]
(a) (b)
θ=0 [rad/s]
θ=π/3 [rad]
θ=0 [rad]
θ=π/3 [rad]
ωBW-dd
ωBW-qq
θ=0 [rad]
θ=π/3 [rad]
PM
(θ=
0 [
rad])
PM
(θ=π
/3 [
rad])
θ=0 [rad]
θ=π/3 [rad]
PM
(θ=
0 [
rad])
PM
(θ=π
/3 [
rad])
(c) (d)
Fig. 5. d-axis to d-axis and q-axis to q-axis open loop magnitude and phasecharacteristics of the frequency responses of CSCC method for nonlinearinductor case. (a) d-axis to d-axis magnitude response. (b) q-axis to q-axismagnitude response. (c) d-axis to d-axis phase response. (d) q-axis to q-axisphase response. The characteristics of the IDMBC method for the nonlinearinductor case are given in Fig. 4.
down to 62 degrees decreasing the robustness of the stabilityof the closed-loop current control system [22].
In the case of the IDMBC method with nonlinear inductors,the magnitude and phase responses are obtained as the sameshown in Fig. 4 due to the linearization of the nonlinear plantin the large via inverse system dynamics.
B. Disturbance Rejection Characteristics
In 3P3W VSCs, the disturbance voltage is mainly composedof the load voltage (or grid voltage) and the VSC generateddisturbance voltage which can be described in two-phaserepresentation by
vD = vg−σ + vid−σ (14)
where vg−σ is load voltage and vid−σ is inverter generateddisturbance due to dead-time, measurement errors, and switchnonidealities [27]. In (14), the two-phase load voltage candescribed as vg−σ = [(van − vbn) (vbn − vcn)]T in termsof the phase to neutral point load voltages as shown in Fig. 1.The inverter generated component of vD can be expressed in asimilar manner as vid−σ = [(vid−a−vid−b) (vid−b−vid−c)]Twherein vid−a, vid−b, and vid−c denote the inverter generateddisturbance voltage of leg A, B, and C respectively. The sameconvention applies for the estimated disturbance voltage vD.
To decouple the distorting effects of the disturbance voltage,it is conventionally performed that the estimated disturbancevoltage is added to the control system as shown in Fig.2. However, perfect decoupling can not be achieved due tomeasurement and estimation errors. Due to this imperfectdecoupling, depending on the measurement quality and es-timation success, more or less disturbance voltage interferesto the dynamic system as described by (5) so that the currentcontrol systems in Fig. 2 can be interpreted as shown in Fig.
3(b) in the stationary frame with the current command set tozero (i∗dq = 0). With such a setting for disturbance rejectioncharacterization, the current control methods the stationaryframe represented the dynamic system in (5) can be describedin s-domain as
iσ(s) = FD(θ, s)dσ(s) (15)
where FD(θ, s) stands for the disturbance to output transferfunction. Considering Fig. 2 and expanding the commandvoltage v∗
σ in (5) according to the control architectures, thedisturbance to output transfer functions for CSCC and IDMBCmethods can be obtained respectively as
FCSCCD (θ, s) =
sLσ+Rσ−H(θ)Wcc−Kc(s)T(θ)
−1−1
(16)
FIDMBCD (θ, s) =
(sLσ + Rσ)− Lσ
Lmin+
Rσ
sLmin
×T(θ)Wcc −Kc(s)T(θ)−1−1
.
(17)
-150
-100
-50
0
100
105
-150
-100
-50
0
-150
-100
-50
0
-150
-100
-50
0
101
102
103
104
100
105
101
102
103
104
100
105
101
102
103
104
100
105
101
102
103
104
Mag
nit
ud
e [d
B]
Mag
nit
ud
e [d
B]
Frequency [rad/s] Frequency [rad/s]
Frequency [rad/s] Frequency [rad/s]
Mag
nit
ud
e [d
B]
Mag
nit
ud
e [d
B]
ia (s)dab (s)
CSCC
IDMBC
ia (s)dbc (s)
CSCC
IDMBC
ib (s)dbc (s)
ib (s)dab (s)
θ=0 [rad/s]
θ=π/2 [rad/s]
θ=0 [rad/s]
θ=π/2 [rad/s]
θ=0 [rad/s]
θ=π/2 [rad/s] θ=0 [rad/s]
θ=π/2 [rad/s]
CSCC
IDMBC
CSCC
IDMBC
(a) (b)
(c) (d)
Fig. 6. Disturbance to output transfer function magnitude plots of the CSCCand IDMBC methods for nonlinear inductor case at two different angles. (a)dab to ia magnitude response. (b) dbc to ia magnitude response. (c) dab toib magnitude response. (d) dbc to ib magnitude response.
The magnitude plots of the disturbance to output transferfunctions of CSCC and IDMBC methods for two differentangles (θ = 0 and θ = π/2) in Fig. 6 indicate distinctivebehaviours of the current control methods. In the figure theterms dab and dbc represent the elements of the disturbancevector (dσ(s) = [dab(s) dbc(s)]
T ). In the figure, the same non-linear current control system parameters given in the previoussubsection are employed to yield the disturbance to outputmagnitude responses. The obtained magnitude responses inFig. 6(a)-(d) show that the IDMBC method has better distur-bance rejection characteristics than CSCC method especiallyfor low-frequency disturbance signals even for different angles.
The difference of the rejection of low-frequency disturbancesignals between the methods can be as much as 60 dBwhile the disturbance rejection characteristics of two methodsconverge as the frequency of the disturbance increase. Suchcharacteristics of the IDMBC method yield improved currentwaveform quality due to the reduction of the disturbanceeffects on the converter current.
C. Cross-Coupling Characteristics
The cross-coupling decoupling performance of the CSCCmethod in the preceding part of this study [7] has beenfound deficient as the correspondence H(θ)Wcc = ωLσP(θ)is not met when saturable inductors are employed. On theother hand in the IDMBC method, due linearization of thenonlinear system in the large, the system behaves as a constantinductance current control system with unity-gain zero-phasecharacteristics described by (2). Considering this outcome andFig. 2(b), the equivalence
T(θ)Wcc = ωLMP(θ) (18)
holds for Lσ = Lσ . Therefore, contrary to the CSCC method,it is reasonable to expect no cross-coupling problem for theIDMBC method.
IV. SIMULATION RESULTS
In this section, the proposed IDMBC method for 3P3WVSC system simulation results are presented in comparisonwith that of the CSCC method. The current regulator attributesare investigated by means of dynamic response, cross-couplingbehaviour, and steady-state waveform quality. Accordingly, thesimulated system configuration by means of hardware andcontrol is provided.
A. Simulated System Configuration
1) Simulated System Hardware: Simulation studies areconducted for the 3P3W VSC setup to prove the viability ofthe proposed IDMBC control method in comparison with theCSCC method. Two hardware configurations are employed asshown in Fig. 7 for the investigation of the performance ofthe current controllers. For the dynamic response and cross-coupling behaviour investigation, the VSC+inductor terminalsshort-circuited configuration shown in Fig. 7(a) is employedto render the sole characteristics without grid voltage interfer-ence. On the other hand, the waveform quality is investigatedvia grid-connected configuration shown in Fig. 7(b).
In Fig. 7(c), the L-i characteristics of saturable inductorsemployed as converter side inductors are illustrated. Thesecharacteristics are obtained from experimental measurementsas performed in [8] and used for analyses, simulations, andthe implementation of experimental inverse system dynamics.The selected core is deeply saturated so that the inductance isdecreased with the increase in the current down to 1/9th of theinitial value. Such deep saturation levels can be tolerated in3P3W VSC systems by means of PWM ripple due to balancingeffect of the inductors as the phase currents are driven by the
VSC line-to-line voltages and the minimum inductance valuesof any two phases do not overlap in time.
In the grid-connected configuration shown in Fig. 7(b), thefilter capacitor is selected to be 2.2 µF connected in a wyeconfiguration. The grid side inductor is selected to be 2 mH.The LCL filter provides sufficient PWM harmonic attenuationeven for the minimum inductance value of the converter sideinductor within the design constraints provided [28]. Theparameters related to the simulated hardware configuration inthis figure is tabulated in Table-I. These parameters are alsothe same for experimental configuration.
(a)
(b)
La Ra
Vdc
3P VSC
Rb
Rc
Lb
Lc
ia
ib
ic
La vania
n
Ra
Cf
Vdc
3P VSC
vbnRb
vcnRc
Lb
Lc
Lg-a
Lg-b
Lg-c
ib
ic
n
ig-a
ig-b
ig-c
vcc vcavcb
0 2 4 6 8 100.5
1
1.5
2
2.5
3
3.5
4
4.5
5
L [
mH
]
i [A]
LaLbLc
(c)
Fig. 7. Equivalent experimental configurations. (a) VSC+inductor terminalsshort-circuited configuration. (b) Grid-connected configuration. (c) L-i char-acteristics of the converter side inductors employed throughout the study.
TABLE ISIMULATED AND EXPERIMENTAL PARAMETERS FOR 3P3W VSC SYSTEM
Parameter Value
dc-link voltage (Vdc) 350 VRated current 10 Apeak/phaseGrid voltage 115 VRMS/phaseFundamental frequency ω 2π50 rad/sCarrier frequency fc 10 kHzModulation method SVPWMSampling period (Ts) 50 µsInverter dead-time 3 µsLmax/Lmin (converter side) 4.45/0.5 mHLg (grid side) 2 mHR (converter side equivalent resistance/phase) 0.5 Ω
Cf (LCL filter capacitor) 2.2 µF (2.2 %)
2) Simulated System Control: The 3P3W VSC system sim-ulations are performed both for CSCC and IDMBC methodswith a sufficiently small step size of 0.02 µs for the simulationof nonlinear systems involving saturable inductors havingthe characteristics of Fig. 7(c). The synchronous frame PIcontrollers (for the d- and q-axis) in Fig. 2 are tuned toprovide a bandwidth of 2π500 rad/s as if the system is a
linear one (having constant converter side inductance of Lmin)and accordingly the PI gains are tuned as Kp = ωBWLminand Ki = ωBW R for both methods. The cross-couplingdecoupling is performed as the same for both controllers byusing Lmin to form the cross-coupling decoupling matrixWcc. Space vector PWM (SVPWM) method is employed toefficiently utilize the dc-link voltage and to have reduced ripplecurrent. Moreover, such utilization of SVPWM method provesthe compatibility of zero-sequence voltage injection in 3P3WVSC systems with nonlinear inductors. The zero-sequencesignal of SVPWM is generated by selecting the minimumvoltage command magnitude of the three phases, multiplyingby 0.5 and adding to each of the three-phase voltage command[29]. Simulation results are obtained as the compensator gainsare kept the same for each configuration, current controlmethod, and test. With these specifications, dynamic response,cross-coupling behaviour, and waveform quality performancesof the current control methods are investigated.
B. Dynamic Response
The dynamic response characteristics of a current regulatorare one of the figures of merit to evaluate and comparethe controller performances. Further, the dynamic responseperformance is a direct indicator of the system bandwidth atthe operating condition of the converter [8]. The faster theresponse implies the higher the control bandwidth.
0
2
4
6
8
10
12
0 2 4 6 8 10
t [ms]
[A
]
id* CSCC IDMBC
Fig. 8. Simulated d-axis step responses of the CSCC and IDMBC methodsat various d-axis dc-bias currents. Note that q-axis current command is set tozero.
To examine the dynamic response characteristics of thecurrent controllers, simulations are performed by providing1 A step changes on the d-axis current command whereasthe q-axis current command is set to zero. In Fig. 8, the d-axis current command and currents are illustrated for CSCCand IDMBC methods. As the q-axis currents are set to zerothey are not illustrated in the figure. For the CSCC method,the response (or rise) time to a 1 A step increase is highlydependent on the current bias as discussed in the precedingpart of this study [7]. When the current dc-bias is zero, the rise
time is approximately 3-4 ms whereas when the d-axis currentdc-bias level is 10 A, the rise time of the CSCC method isaround 1 ms. This bandwidth shrinkage phenomenon in 3P3WVSCs with CSCC is similar to the case that of single-phasecontrolled VSCs under dc excitation [8], [23].
On the other hand, the response time of the IDMBC methodis the same for all current bias levels (1 ms) exhibitingthat the bandwidth is conserved throughout various operatingconditions of the VSC.
C. Cross-Coupling Behaviour
In addition to the dynamic response performance, it isdesired for a 3P3W VSC current control system to haveindependent control of the d- and q-axis currents without anycoupling effects on each other. To examine the cross-couplingbehaviour of the current controllers, simulations have beenperformed. Briefly, the current command value for one axis ischanged instantly while the actual current of the other axis isobserved.
[A
]
id* id
CSCC idIDMBC iq
CSCC iqIDMBC
[A
]
iq* iq
CSCC iqIDMBC id
CSCC idIDMBC
[A
]
[A
](a) (b)
(c) (d)
idCSCC id
IDMBC iqCSCC iq
IDMBCiq
* iqCSCC iq
IDMBC idCSCC id
IDMBCid*
0-0.5
0
0.5
1
1.5
2
2.5
5 10 15 20t [ms]
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15 20t [ms]
3
3.5
4
4.5
5
5.5
0 5 10 15 20t [ms]
3
3.5
4
4.5
5
5.5
0 5 10 15 20t [ms]
Fig. 9. Experimental d-axis (or q-axis) step responses of the CSCC andIDMBC methods at various d- and q-axis dc-bias currents. (a) i∗d = 1 A→ 2A→ 1 A while i∗q = 0 A. (b) i∗q = 1 A→ 2 A→ 1 A while i∗d = 0 A. (c)i∗d = 4 A→ 5 A→ 4 A while i∗q = 4 A. (d) i∗q = 4 A→ 5 A→ 4 A whilei∗d = 4 A.
In Fig. 9, the waveforms regarding the observations areillustrated. For the CSCC method, when the d-axis or q-axisis excited, the current of the other axis responds to the changewhich is undesirable for a current controller. On the otherhand, in the case of the IDMBC method, the two axes seemdecoupled from each other almost perfectly as when one axiscurrent is changed the other axis current does not exhibitany considerable change as a desirable feature for a high-performance current controller.
D. Waveform Quality
The waveform quality performances of the CSCC andIDMBC methods are investigated at steady-state by means ofsimulations in a grid-connected mode shown in Fig. 7(b).
In Figs. 10(a) and (b), the converter side current, voltagecommand signal, zero-sequence signal, and zero-sequenceadded voltage command signal (the signal provided to trianglecomparator) of phase-A of the CSCC and IDMBC methods areillustrated respectively. With the application of the IDMBCmethod, the converter side current waveform seems to beimproved when compared to the converter side current wave-form of the CSCC method even if these waveforms containPWM current ripple. In Figs. 10(c) and (d), such a conclusioncan be reached through the grid currents. As filtered by theLCL filter, almost all PWM caused high-frequency currentharmonics are cleared and there remained control-fixable low-order harmonics in the grid current especially in the CSCCmethod. When the current control technique is changed tothe IDMBC, one can observe the current waveform qualityimprovement with the employment of the IDMBC method.To quantify, the THDi is reduced from 5.45 % to 1.82 %. Fig.10(c) and (d) also show that at full load current, the inductanceof the converter side inductor decreases almost to 1/9th of itszero current value as expected.
-200
-150
-100
-50
0
50
100
150
200
θ [rad]0 π-π
va** [V]
va* [V]
ia [A] scale:10
v0 [V]
-200
-150
-100
-50
0
50
100
150
200
θ [rad]0 π-π
va** [V]
va* [V]
ia [A] scale:10
v0 [V]
-15
-10
-5
0
5
10
15
20
25
θ [rad]0 π-π
La [mH] scale:5
ig-a [A], THD:5.45 [%]
-15
-10
-5
0
5
10
15
20
25
θ [rad]0 π-π
La [mH] scale:5
ig-a [A], THD:1.82 [%]
CSCC IDMBC
CSCC IDMBC
(a) (b)
(c) (d)
Fig. 10. Simulated steady-state converter side current, voltage commandsignal, zero-sequence signal, and zero-sequence added voltage commandsignal of phase-a: (a) CSCC method. (b) IDMBC method. The converter sideinductance and the grid-current: (c) CSCC method. (d) IDMBC method.
2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Harmonic Order
Har
mo
nic
Mag
nit
ud
e [%
] CSCCIDMBC
Fig. 11. Simulated harmonic spectra of the phase currents of the CSCC andIDMBC methods around low frequency region.
In Fig. 11, the harmonic spectra of the CSCC and IDMBCmethods around the low-frequency region are illustrated.
Based on the spectra, the most dominant harmonics areobserved to be 5th and 7th for both methods. When comparedthe two methods, the same conclusion obtained from the time-domain waveforms can be reached that the IDMBC methodoutperforms the CSCC method by means of current waveformquality.
V. EXPERIMENTAL RESULTS
In order to observe and compare the dynamic response,cross-coupling behaviour, and steady-state performances ofCSCC and IDMBC methods, experiments are conducted.Accordingly, the experimental system configuration by meansof hardware and control implementation is provided.
A. Experimental System Configuration
1) Experimental System Hardware: In the hardware im-plementation for the performance investigation of the currentcontrol methods, VSC+inductor terminals short-circuited andgrid-connected configurations shown in Fig. 7 are employedas in the case of simulation studies. The power modulePM75RL1A120 is utilized in the realization of the VSCpower semiconductors. The dynamic response and cross-coupling performances of the methods are investigated viaVSC+inductor terminals short-circuited configuration to elim-inate the grid interaction. On the other hand, the waveformquality performances of the methods are investigated via grid-connected experiments. The experimental system parameterskept the same as that are tabulated in Table-I. For bothconfigurations in Fig. 7, the dc-link voltage is obtained fromthe grid via a diode rectifier and a variac. In the grid-connectedexperiments, the ac-grid voltage is stepped down to 115VRMS/phase via three single-phase transformers connected ina wye-wye configuration each having a leakage inductanceof 0.25 mH. Supplemental linear filter inductors as gridside have been inserted also to form grid side inductors(Lg−a, Lg−b, Lg−c) together with the transformer leakageinductance. The converter side saturable inductors are woundto sendust powder material toroid cores [30] yielding theinductance characteristics given in Fig. 7(c). Fig. 12 showsthis three-phase grid-connected VSC system hardware.
2) Control Implementation: The control schemes in Fig.2 are implemented for experimental validation of and perfor-mance comparison of the controllers. A 150 MHz floating-point digital signal processor (DSP) is used to realize thecontrol functions in discrete time. The interrupt cycle isselected as 50 µs to yield a double update for a PWM cycleof 100 µs (10 kHz). At every interrupt cycle, the controllerexecutes the grid synchronization via a three-phase vector PLLmethod [31], current control algorithms including anti-windup,software VSC protection, and PWM signal generation. A dead-time of 3 µs is utilized and its compensation is performed. Thebackward Euler method is utilized for the discretization of thecontinuous-time controllers.
All system control parameters are selected the same as inthe case of simulations. The L-i characteristics of the converterside inductors are modelled as second-order polynomials toestablish an inverse dynamic model of the current-control
Saturable
Inductors
Filter
Capacitors
Diode
RectifierCircuit
Breaker
DC-link
Voltage Input
Current
Sensors
Heatsink
IGBT
Module
DSP System
AC Voltage Source
Fig. 12. Three-phase three-wire grid-connected VSC setup.
system as in [8]. As the converter side currents are doublesampled [24], the d- and q-axis currents obtained via the DSPreadings are almost PWM ripple-free.
B. Dynamic Response
In order to observe and compare the dynamic responseperformances of the CSCC and IDMBC methods, 1 A stepchanges are applied to the d-axis current reference for bothmethods, as in the case of simulations. To mitigate thegrid voltage disturbance, the test is carried out for theVSC+inductor terminals are short-circuited configuration. Theassociated dynamic response waveforms are illustrated in Fig.13. The results presented in this figure are very similar tothe one obtained in the simulations (Fig. 8). When the d-axiscurrent is zero, the rise time of the d-axis current is quitelong in the CSCC method when compared that of the IDMBCmethod even if the associated PI controller parameters are thesame aiming the same bandwidth. When the d-axis currentoffset is increased, the rise time of the CSCC method startsto decrease while the dynamic response characteristics of theIDMBC method are the same which is initially designed tobe. This is due to the bandwidth shrinkage phenomenon inthe CSCC method as described in the preceding part of thisstudy [7].
C. Cross-Coupling Behaviour
The experimental cross-coupling behaviour of the CSCCand IDMBC methods are investigated by applying a step
0 2 4 6 8 10
t [ms]
0
2
4
6
8
10
12
[A
]
id* CSCC IDMBC
Fig. 13. Experimental d-axis step responses of the CSCC and IDMBCmethods at various d-axis dc-bias currents. Note that q-axis current is setto zero.
0-0.5
0
0.5
1
1.5
2
2.5
5 10 15 20t [ms]
[A
]
id* id
CSCC idIDMBC iq
CSCC iqIDMBC
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15 20t [ms]
[A
]
iq* iq
CSCC iqIDMBC id
CSCC idIDMBC
3
3.5
4
4.5
5
5.5
6
0 5 10 15 20t [ms]
[A
]
3
3.5
4
4.5
5
5.5
0 5 10 15 20t [ms]
[A
]
(a) (b)
(c) (d)
idCSCC id
IDMBC iqCSCC iq
IDMBCiq
* iqCSCC iq
IDMBC idCSCC id
IDMBCid*
Fig. 14. Experimental d-axis (or q-axis) step responses of the CSCC andIDMBC methods at various d- and q-axis dc-bias currents. (a) i∗d = 1 A→ 2A→ 1 A while i∗q = 0 A. (b) i∗q = 1 A→ 2 A→ 1 A while i∗d = 0 A. (c)i∗d = 4 A→ 5 A→ 4 A while i∗q = 4 A. (d) i∗q = 4 A→ 5 A→ 4 A whilei∗d = 4 A.
change to the current command of an axis and then observingthe actual current of the other axis, as in the case of simula-tions. In Fig. 14(a), the d-axis current command is increasedfrom 1 A to 2 A while the q-axis command is kept at 0 A.In the figure, for the CSCC method, the q-axis current is notstationary but it deviates from its reference value with thechange in the d-axis current indicating the imperfect cross-coupling decoupling. When the d- and q-axis currents areinterchanged (Fig. 14(b)), the same phenomenon again appearsin a similar way but distinct in sign with a reasoning that thecross-coupling terms have opposite signs. On the other hand,for both Figs. 14(a) and (b), the IDMBC method exhibitssuperior cross-coupling decoupling performance as expectedfrom analyses and simulations.
-200
-100
0
100
200
θ [rad]0 π-π
ia [A] scale:10
va** [V](a) CSCC
-200
-100
0
100
200
θ [rad]0 π-π
va** [V]
ia [A] scale:10
(b) IDMBC
ig-a [A] scale:10
vca [V]
-200
-100
0
100
200
θ [rad]0 π-π
(c)CSCC
THDi=9.6 [%]
-200
-100
0
100
200
ig-a [A] scale:10
vca [V]
θ [rad]0 π-π
(d)IDMBC
THDi=2.58 [%]
Fig. 15. Experimental grid-connected 3P3W VSC waveforms for the CSCCand IDMBC methods at full load, zero displacement angle (i∗d = 10 A, i∗q = 0A) , and steady-state. (a) SVPWM output voltage command and converterside current for CSCC method. (b) SVPWM output voltage command andconverter side current for IDMBC method. (c) Output filter capacitor voltageand grid current for CSCC method. (d) Output filter capacitor voltage andgrid current for IDMBC method.
In Figs. 14(c) and (d), the dc-bias levels on both axes areset to 4 A. Then the d- and q-axis current commands areincreased to and decreased from 5 A (shown in Figs. 14(c) and(d) respectively). In these cases, even the 6th order harmonicblurs the decoupling effect it can be still identified that thedecoupling is imperfect when the rising and falling regions ofthe current waveforms are considered. Similar to the cases ofFigs. 14(a) and (b), the cases of Figs. 14(c) and (d) illustratethe superior cross-coupling decoupling performance of theIDMBC method.
D. Waveform Quality
The experimental waveform quality investigation of CSCCand IDMBC current control methods is performed via thegrid-connected configuration shown in Fig. 7(b). The systemparameters specified in Table-I are utilized in the experiments.The inductor L-i characteristics are the same in Fig. 7(c) andthe controller parameters are kept as the same that are usedfor simulations.
In Fig. 15, the grid-connected 3P3W VSC system significantwaveforms are illustrated at full load, zero displacement angle,and steady-state conditions. In Figs. 15(a) and (b), the VSCoutput voltage (with SVPWM), and converter side phase-ainductor currents are depicted for CSCC and IDMBC methodsrespectively. In Figs. 15(c) and (d), the capacitor voltage andthe grid current are shown in the same order. The grid currentwaveforms indicate the effectiveness of the IDMBC method,while it can be inferred that the CSCC method can not beused in grid-connected applications for the current systemparameters as 5 % THDi limit is exceeded that is set by thegrid connection standards such as IEEE 519.
In Fig. 16 the three-phase grid currents of the methods areillustrated for three grid cycles. The THDi value for the CSCCmethod is measured to be 9.6 % and the THDi value of theIDMBC method is measured to be 2.58 %. The measured
-15
-10
-5
0
5
10
15
0 2π 4π 6πθ [rad]
[A]
IDMBCig-a ig-b ig-c
THDi=2.58 [%]
(b)
-15
-10
-5
0
5
10
15
0 2π 4π 6πθ [rad]
[A]
CSCCig-a ig-b ig-c
THDi=9.6 [%]
(a)
Fig. 16. Experimental grid currents of the CSCC and IDMBC methods at fullload for zero displacement angle (i∗d = 10 A, i∗q = 0 A). (a) Grid currentsof the three phases for CSCC method. (b) Grid currents of the three phasesfor IDMBC method.
THDi values in the grid-connected configuration are slightlygreater than their simulated counterparts.
In Fig. 17, the harmonic spectra belonging to the gridcurrents shown in Fig. 15 (and grid currents also shown in Fig.16) are illustrated. In the spectra, the zero-sequence currentharmonics are minimal as there is no return path for theseharmonics to flow. For CSCC method, the 5th and the 7th
harmonics flow in a considerable amount making the CSCCmethod prohibitive for grid-connected applications with thecurrent parameters. As discussed, such spectrum can be at-tributed to imperfect cross-coupling decoupling, measurementerrors and delays, low disturbance rejection characteristics, andload nonlinearity. In the case of IDMBC method, the spectrumhas acceptably low 5th and 7th harmonic current magnitudesdemonstrating the suitability of the method for grid-connectedapplications.
2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
11
Harmonic Order
Har
monic
Mag
nit
ude
[%] CSCC
IDMBC
Fig. 17. Experimental grid-connected system grid side current low frequencyregion harmonic spectra for the CSCC and IDMBC methods.
VI. CONCLUSION
This paper presented detailed analyses of the control sys-tem characteristics of conventional synchronous frame currentcontrol (CSCC) and inverse dynamic model based compen-sation (IDMBC) methods for three-phase three-wire (3P3W)voltage-source converter (VSC) systems employing saturableinductors. The current control methods are evaluated by taking
the current waveform quality and current control bandwidthas the performance criteria. Constituting the determinants forthese criteria; command to output, disturbance to output, andcross-coupling decoupling characteristics of the methods areinvestigated on a two-phase representation basis. Simulationsand experiments are conducted as well for a 3P3W VSCsystem employing saturable inductors reaching 1/9th of thezero-current inductance at full current to verify the analysesand performance comparisons. It is shown via analyses, sim-ulations, and experiments that the CSCC method suffers boththe current waveform quality and current control bandwidth.On the other hand, the IDMBC method exhibits superiorperformance with uniform current control bandwidth and highsteady-state current waveform quality that meets grid codesenabling the utilization of deeply saturable inductors in 3P3WVSC systems.
REFERENCES
[1] R. N. Beres, X. Wang, M. Liserre, F. Blaabjerg, and C. L. Bak, “Areview of passive power filters for three-phase grid-connected voltage-source converters,” IEEE Journal of Emerging and Selected Topics inPower Electronics, vol. 4, pp. 54–69, March 2016.
[2] J. Muhlethaler, M. Schweizer, R. Blattmann, J. W. Kolar, and A. Eck-lebe, “Optimal design of LCL harmonic filters for three-phase pfcrectifiers,” IEEE Transactions on Power Electronics, vol. 28, no. 7,pp. 3114–3125, 2013.
[3] Y. Jiao and F. C. Lee, “Lcl filter design and inductor ripple analysis for3-level npc grid interface converter,” in 2014 IEEE Energy ConversionCongress and Exposition (ECCE), pp. 1911–1918, 2014.
[4] Q. Li, D. Jiang, and Y. Zhang, “Analysis and calculation of currentripple considering inductance saturation and its application to variableswitching frequency pwm,” IEEE Transactions on Power Electronics,vol. 34, no. 12, pp. 12262–12273, 2019.
[5] N. Femia, K. Stoyka, and G. Di Capua, “Impact of inductors saturationon peak-current mode control operation,” IEEE Transactions on PowerElectronics, pp. 1–1, 2020.
[6] L. Milner and G. A. Rincon-Mora, “Small saturating inductors for morecompact switching power supplies,” IEEJ transactions on electrical andelectronic engineering, vol. 7, no. 1, pp. 69–73, 2012.
[7] Z. Ozkan and A. M. Hava, “Inductor saturation compensation in three-phase three-wire voltage-source converters via inverse system dynamics-I,” Submitted Manuscript, 2020.
[8] Z. Ozkan and A. M. Hava, “Inductor saturation compensation withresistive decoupling for single-phase controlled vsc systems,” IEEETransactions on Power Electronics, vol. 35, no. 2, pp. 1993–2007, Feb.2020.
[9] G. Di Capua, N. Femia, and K. Stoyka, “Validation of inductorssustainable-saturation-operation in switching power supplies design,” in2017 IEEE International Conference on Industrial Technology (ICIT),pp. 242–247, 2017.
[10] Z. Ozkan, High performance current control methods for voltage sourceconverters with saturable inductors. PhD thesis, Middle East TechnicalUniversity, June 2019.
[11] “Ieee standard for interconnection and interoperability of distributedenergy resources with associated electric power systems interfaces,”IEEE Std 1547-2018 (Revision of IEEE Std 1547-2003), pp. 1–138,2018.
[12] FNN/VDE, “Power generation systems connected to the low-voltage dis-tribution network–technical minimum requirements for the connectionto and parallel operation with low-voltage distribution networks,” 2011.
[13] W. Bartels, F. Ehlers, K. Heidenreich, R. Huttner, H. Kuhn, T. Meyer,T. Kumm, J. Salzmann, H. Schafer, and K. Weck, “Technical guidelinegenerating plants connected to the medium-voltage network. guilde-line for generating plants’ connection to and parallel operation withthe medium-voltage network,” BDEW Bundesverband der Energie-und Wasserwirtshaft e. V.(German Association of Energy and WaterIndustries), 2008.
[14] L. G. Franquelo, J. Napoles, R. C. P. Guisado, J. I. Leon, and M. A.Aguirre, “A flexible selective harmonic mitigation technique to meet gridcodes in three-level pwm converters,” IEEE Transactions on IndustrialElectronics, vol. 54, pp. 3022–3029, Dec 2007.
[15] F. Wang, J. L. Duarte, M. A. M. Hendrix, and P. F. Ribeiro, “Modelingand analysis of grid harmonic distortion impact of aggregated dginverters,” IEEE Transactions on Power Electronics, vol. 26, pp. 786–797, March 2011.
[16] M. Kang, S. Lee, and Y. Yoon, “Compensation for inverter nonlinearityconsidering voltage drops and switching delays of each leg’s switches,”in 2016 IEEE Energy Conversion Congress and Exposition (ECCE),pp. 1–7, 2016.
[17] K. Wiedmann, F. Wallrapp, and A. Mertens, “Analysis of inverternonlinearity effects on sensorless control for permanent magnet machinedrives based on high-frequency signal injection,” in 2009 13th EuropeanConference on Power Electronics and Applications, pp. 1–10, 2009.
[18] Y. Park and S. Sul, “Implementation schemes to compensate for in-verter nonlinearity based on trapezoidal voltage,” IEEE Transactions onIndustry Applications, vol. 50, no. 2, pp. 1066–1073, 2014.
[19] S. Zhou, J. Liu, L. Zhou, and H. She, “Cross-coupling and decouplingtechniques in the current control of grid-connected voltage sourceconverter,” in 2015 IEEE Applied Power Electronics Conference andExposition (APEC), pp. 2821–2827, 2015.
[20] S. Zhou, J. Liu, L. Zhou, and H. She, “Cross-coupling and decouplingtechniques in the current control of grid-connected voltage sourceconverter,” in 2015 IEEE Applied Power Electronics Conference andExposition (APEC), pp. 2821–2827, 2015.
[21] F. Briz, M. W. Degner, and R. D. Lorenz, “Analysis and designof current regulators using complex vectors,” IEEE Transactions onIndustry Applications, vol. 36, no. 3, pp. 817–825, 2000.
[22] K. Ogata and Y. Yang, Modern control engineering. Prentice Hall,3rd ed., 2002.
[23] Z. Ozkan and A. M. Hava, “Current control of single-phase vscsystems with inductor saturation using inverse dynamic model-basedcompensation,” IEEE Transactions on Industrial Electronics, vol. 66,pp. 9268–9277, Dec. 2019.
[24] S. K. Sul, Control of electric machine drive systems. John Wiley &Sons, 2011.
[25] W. Taha, A. R. Beig, and I. Boiko, “Design of pi controllers for a grid-connected vsc based on optimal disturbance rejection,” in IECON 2015- 41st Annual Conference of the IEEE Industrial Electronics Society,pp. 001954–001959, 2015.
[26] J. Dannehl, C. Wessels, and F. W. Fuchs, “Limitations of voltage-oriented pi current control of grid-connected pwm rectifiers with LCLfilters,” IEEE Transactions on Industrial Electronics, vol. 56, no. 2,pp. 380–388, 2009.
[27] M. P. Kazmierkowski and L. Malesani, “Current control techniques forthree-phase voltage-source pwm converters: a survey,” IEEE Transac-tions on Industrial Electronics, vol. 45, pp. 691–703, Oct 1998.
[28] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCL-filter-based three-phase active rectifier,” IEEE Transactions on IndustryApplications, vol. 41, no. 5, p. 1281, 2005.
[29] A. M. Hava and N. O. Cetin, “A generalized scalar pwm approach witheasy implementation features for three-phase, three-wire voltage-sourceinverters,” IEEE Transactions on Power Electronics, vol. 26, no. 5,pp. 1385–1395, 2010.
[30] “Inductor core datasheet 0077908a7.” [Online] Available:https://www.mag-inc.com/Media/Magnetics/Datasheets/0077908A7.pdfAccessed March 6, 2018.
[31] V. Kaura and V. Blasko, “Operation of a phase locked loop systemunder distorted utility conditions,” IEEE Transactions on Industry Ap-plications, vol. 33, no. 1, pp. 58–63, 1997.