A
simplified DQ Controller for Single-Phase
Grid-Connected PV Inverters
Abdalbaset M. Mnider, David J. Atkinson, Mohamed Dahidah, Matthew Armstrong
School of Electrical and Electronic Engineering, Newcastle University, Newcastle Upon Tyne, UK
Abdalbaset.mnider@ncl.ac.uk
Abstract- S ynchronous dq-frame controllers are generally
accepted due to their high performance compared to stationary
a Jl -frame ones, as they operate on dc quantities, achieving zero
steady-state error. In single-phase systems, however, PI-based
dq
controllers cannot be directly applied due to the reduced number
of input signals available compared to three-phase systems. The
common
approach
in
single-phase
systems
is
to
create
a
synthesized phase signal orthogonal to the fundamental signal of
the single-phase system so as to obtain dc quantities by means of
(o.p to dq)
transformation.
The
orthogonal
components
in
conventional approaches are usually obtained by phase shifting
the real signals by a quarter of the fundamental period. The
introduction of such delay in the system deteriorates the dynamic
response, which becomes slower and oscillatory. In this paper, an
alternative
dq
dq
controller scheme which referred as to simplified
controller is proposed. The proposed scheme does not require
orthogonal quantities to be generated, making it easier to be
implemented. The simplified
dq
control method is experimentally
compared to the conventional delay-based
shown
to
improve
the
poor
dynamics
dq
control method and
of
the
conventional
approach while not adding excessive complexity to the controller
structure. A single-phase five-level diode-clamped grid-connected
PV inverter is considered as an example in this paper.
Keywords-Diode-clamped Multilevel inverter, single-phase dq
controller, orthogonal system generation (OSG).
I. INTRODUCTION
Single-phase voltage-source converters (VSCs) are widely
utilized as an interface between renewable energy sources and
the utility grid, especially in small scale photovoltaic
applications[l]. Current control of VSCs has seen great
attention from researches in the recent years, and various
approaches have been proposed, including, hysteresis,
predictive, proportional-integral (PI) and proportional­
resonant (PR)-based control strategies [2]-[5].
Simple PI-based a� controllers are the most conventional
approach. However, because of the time-varying nature of
converter, it is difficult to achieve, a zero steady-state error. On
the other hand, PI-based dq control approaches have been
efficiently used to control three-phase grid connected inverters,
due to their great advantages of presenting infmite control gain
at the steady-state operating point, leading to zero steady-state
error [6]-[8]. However, in single-phase grid-connected
inverters, these controllers cannot be applied directly, where a
second quantity in quadrature with the existing physical one
should be created such that the physical and the synthesized
components together form the stationary a� frame. In the
technical literature, many attempts have been reported to obtain
978-1-4673-9768-1I16/$3l.00 ©2016 IEEE
the required orthogonal signal [9]-[11]. In [9], the authors
proposed to phase shifting the circuit variables by a quarter
cycle of the fundamental period. However, phase shifting the
real components to create the orthogonal signals deteriorates
the dynamic response of the system, as the real and fictive axes
do not run concurrently. In [10], the authors used
differentiation to create the imaginary circuit to avoid the
delay. But, the perfonnance of the differentiation approach can
be deteriorated significantly under distorted grid voltage
conditions. In [11], the authors developed a fictive-axis
emulation technique to create the imaginary circuit with a
fictive axis running concurrently with the real circuit, which
helps improving the poor dynamics of the conventional
approaches.
In this paper, an alternative dq current-control scheme is
proposed based on the use of the deference between the desired
and real measured currents in the stationary reference frame to
generate the steady state error signals in the dq frame which are
required for the dq controller. Contrary to the conventional dq
controller, the proposed method does not require producing any
orthogonal signal, making it easy to be implemented with
superior dynamic performance compared to that of the
conventional delay-based approach.
The rest of this paper is organized as follows. In Section II,
a description of the utilized test system, with mathematical
model for the adopted single-phase system is provided. Section
III briefly explains the conventional dq current-control
strategy. Section IV introduces the proposed simplified dq
control scheme with mathematical analyses. In Section V the
performance evaluation of the proposed and the conventional
approaches based on experimental results are presented.
Finally, Section VI concludes this paper.
II. SYSTEM DESCRIPTION AND MODELLING
Fig. 1 shows the schematic diagram of the implemented test
system in which a five-level diode-clamped PV inverter is
connected to the utility grid through an LCL filter. The first
stage is a dc-dc converter used for maximum power point
tracking (MPPT) algorithm of the PV array [12]-[15]. For
simplicity, this stage is replaced by four ideal voltage sources,
neglecting the dc-side dynamics. In this case, capacitor
voltages are balanced and fixed at a desired level. The
parameters of the adopted test system are given in Table I.
In the following, a mathematical model of the single-phase
system in Fig. 1 is described, and a structural diagram of that is
derived, which is adopted for the design of both conventional
and simplified controllers in the dq reference frame.
PV
Generator
Vgrid_s
Five-level
DC Diode-clamped
Inverter
Link
MPPT
LCL Filter
-
Vinv_s
i1
Vdc
DC/DC
MPPT
R1
Vinv
(R1+R2)
Utility
Grid
-
L1
R2
Vcf
is
1
S(L1+L2)
i2
.
Fig.2. Structural diagram of the test system in the αβ reference frame
L2
Cf
Vgrid_d
Vgrid
(R1+R2)
Gating Signals
i1 or i2
Vgrid
DQ Current
Controller
Vinv_d
-
VC_d
-
+
Fig.1. Schematic of the implemented single-phase test system
id
ωff (L1+L2)
Table 1
Hardware experimental test system parameters
-ωff (L1+L2)
Symbol
Description
Value
Unit
Vgrid
P
ωff
fsw
Vdc
Cdc
L1
R1
Cf
L2
R2
Nominal grid voltage (rms)
Rated output power
Nominal grid frequency
Switching frequency
DC-link voltage
DC-link capacitance
LCL-filter inverter-side inductor
LCL-filter inverter -side resistor
LCL-filter parallel capacitor
LCL-filter grid-side inductor
LCL-filter grid-side resistor
90
0.4
2.π.50
20
280
4×1000
1.2
0.113
3.3
0.5
0.103
V
kW
Rad/s
kHz
V
µF
mH
Ω
µF
mH
Ω
(R1+R2)
Vinv = R1 i1 + SL1 i1 + Vcf
Vcf = R 2 i2 + SL2 i2 + Vgrid
⟦
⟧
vcf
i1 = i2 + Cf
S
VC_q
-
-
1
S(L1+L2)
iq
Vgrid_q
Fig.3. Structural diagram of the test system in the RRF
(1)
Vinvd = (R1 + R 2 )id + S(L1 + L2 )id
−ωff (L1 + L2 )iq + Vgrid_d
⟦
⟧
Vinvq = (R1 + R 2 )iq + S(L1 + L2 )iq
(3)
+ωff (L1 + L2 )id + Vgrid_q
The system in the rotating reference frame (RRF) based on
(3) is diagrammatically illustrated in Fig 3, containing both, the
typical compensation and the decoupling terms.
III. CONVENTIONAL SINGLE-PHASE DQ CURRENT CONTROLLER
where, Vinv, Vcf, Vgrid, i1, and i2 represent the inverter terminal
voltage, the capacitor voltage, the utility grid voltage, the
converter-side current and the grid current, respectively.
By neglecting the capacitor’s influence in the current
control design as its value is low, the vector control for the
proposed system takes exactly the same policy as for VSC with
an L filter. Thus, in the (α-β frame), the single-phase grid
connected inverter equation (1) is simplified into (2).
Vinv_α = (R1 + R 2 )iα + S(L1 + L2 )iα + Vgrid_α
⟧
Vinv_β = (R1 + R 2 )iβ + S(L1 + L2 )iβ + Vgrid_β
+
Vinv_q
Based on the system in Fig. 1, the dynamic of the ac side of the
test system can be described as:
⟦
1
S(L1+L2)
(2)
Based on (2), a structural diagram of the system in the
stationary reference frame is drawn as in Fig. 2.
The subscript “s” in Fig. 2 denotes the quantities in the (αβ)
stationary reference frame.
Further, after applying a stationary-to-rotating transformation
to (2) according to xdq = xαβ e−jwt , the dynamic of the ac-side
variables in the rotating-frame (dq frame) is derived as (3)
The conventional dq current control strategy is well known
and widely studied in the literature [6-8].
Adopting (3), in order to achieve a decoupled control of id and
iq, the inverter terminal voltage should be controlled as follows:
Vinv_d = VC_d − ωff (L1 + L2 )iq + Vgrid_d
⟦
⟧
Vinv_q = VC_q + ωff (L1 + L2 )id + Vgrid_q
(4)
where, VC_d and VC_q represent the control signals of the d and
q axes in the RRF respectively, while ωff is the nominal grid
frequency (rad/s).
Substituting Vinv_d and Vinv_q from (4), in (3), the following
decoupled system is deduced:
VC_d = (R1 + R 2 )id + S(L1 + L2 )id
⟦
⟧
VC_q = (R1 + R 2 )iq + S(L1 + L2 )iq
(5)
Therefore, the transfer function of the decoupled system
described in (5) can be derived as
Vgrid_d
i*d
id
-
ɛd
PI
VC_d +
Power Stage
Vdc
Vinv_d
+
i
Five-Level
Diode Clamped
Inverter
(R1+R2) (L1+L2)
Vgrid
Vinv
ωff (L1+L2)
SPWM
-ωff (L1+L2)
iq
PI
VC_q
Vinv_q
+
iα
α-β
-
iq
dq
iβ
α-β
Vinv_β
(6)
in which, the time constant TS = (L1 + L2 )⁄(R1 + R 2 ) and
K S = 1⁄(R1 + R 2 )
It should be noted that, since id and iq respond to VC_d and
VC_q through a simple first-order transfer function, the control
rule of (4) is completed by defining feedback loops with simple
first order PI controllers. Based on (4), the structural diagram
of the current regulator based on PI controllers is shown in Fig.
4 in which the voltage feedforward and the coupling terms are
shown.
The structural diagram of the test system along with the
conventional single-phase dq controller is shown in Fig. 5, in
which the orthogonal imaginary current component (iβ) is
obtained by phase shifting the real component (iα) by a quarter
of the fundamental period. The measured and the shifted
current components are then employed in a (αβ–dq)
transformation, and a conventional dq current controller shown
in Fig. 4, with decoupling strategy is implemented. The
resulting output control signals in the dq-frame (Vinv_d) &
(Vinv_q) are transferred back to the α-β frame to obtain the
corresponding ac control signals. Usually, the α component
(Vinv_α) of the control signal is employed and fed into the pulsewidth modulation (PWM) modulator, while the β component
(Vinv_β) is discarded.
Note that, a single-phase phase-locked loop (PLL) based on
second-order generalized integrator (SOGI) [16] is used, to
generate the reference phase angle (θPLL) for the (αβ–dq) and
the (dq-αβ) transformation as shown in Fig. 5.
This approach is relatively simple and straightforward;
however, phase shifting the current to create the required
orthogonal signal tends to deteriorate the dynamic response of
the system, as the real and fictive components do not run in the
same time. Therefore, any transient in the real physical
component is also experienced in the fictive orthogonal
component a quarter of fundamental period later. Since the
reference current is subject to frequent step changes, delaying
the current deteriorates the dynamics of the system and makes
it slower and oscillatory. To avoid this shortcoming, the
simplified dq current control strategy is proposed in the coming
section.
Vinv_α
sT 4
e
ωPLL
ƟPLL
dq
i*q
Vgrid_q
Fig. 4 Structural diagram of the conventional dq current controller
1
KS
GDe (S) =
=
(R1 + R 2 ) + S(L1 + L2 ) 1 + STS
ɛq
Vinv_d
ɛq
+
Vinv_q
-
- id
ɛd
dq Current
controller
SOGI
PLL
i*q i*d
Vgrid
Control Circuit
Fig. 5 Test system along with the conventional dq current controller
IV. SIMPLIFIED SINGLE-PHASE DQ CURRENT CONTROL
The control strategy of the previous section necessitates a
αβ to dq transformation, which, in single-phase systems, is not
viable because there is only one phase variable available, while
this transformation needs at least two orthogonal variables.
Therefore, to make the aforementioned current-control strategy
applicable to single-phase systems, fictitious component
orthogonal to the existing physical one should be created by
phase shifting the measured real signal such that the physical
and fictitious signals together form the stationary or αβ frame.
The introducing of such delay in the system deteriorates the
dynamic of the system, which becomes slower and oscillatory.
To overcome the aforementioned shortcoming, a simplified
current controller scheme as depicted in Fig 6, is proposed
where the difference between the reference and the measured
currents iα∗ and iα in the stationary reference frame is used to
generate the steady state error signals in the dq frame ƹd and
ƹq .It is worth noting that the proposed scheme does not require
any artificial orthogonal generator by simply assuming that i∗β
= iβ at the steady state [17].
The relationship between stationary and rotating frames is
given by (7) which defines the transformation from stationary
to rotating frame, and (8) from rotating frame to stationary
frame
𝑑
cos(𝜃) sin(𝜃) 𝛼
(7)
[ ]=[
].[ ]
𝑞
−sin(𝜃) cos(𝜃) 𝛽
𝛼
cos(𝜃) −sin(𝜃) 𝑑
[𝛽 ] = [
].[ ]
sin(𝜃)
cos(𝜃) 𝑞
(8)
In single-phase systems, however, this transformation
cannot be applied directly because there is only one variable
available iα . This drawback however, can be solved without
creating a fictitious current iβ as following
Using the estimated phase angle generated by the PLL
(θPLL), two orthogonal reference currents can be generated as
∗
i∗α = i∗ = Im
cos(θPLL )
(9)
Power Stage
Five-Level
Diode Clamped
Inverter
Vdc
Vgrid_d
i
ɛd
(R1+R2) (L1+L2)
id
iq
iα
dq Current
controller
ɛα
-
Vinv_d
ωff (L1+L2)
-ωff (L1+L2)
i*q
i*α
i*d
-
ɛq
PI
cos
Vinv_d
Vinv_q
ɛq
-sin
+
+
SPWM
ɛd
VC_d
i*d -
Vgrid
Vinv
PI
ƟPLL
dq
SOGI
PLL
α-β
VC_q
+
+
Vinv_q
Vgrid_q
Vgrid
Fig. 7 Structural diagram of the simplified dq current controller
Vinv_α
Vinv_β
Control Circuit
Fig. 6 Test system along with the simplified dq current controller
∗
i∗β = Im
sin(θPLL )
(10)
Applying the dq transformation in (7) with (θ=θPLL), two
reference currents in the dq frame can be found as (11) and
(12).
∗ (𝑐𝑜𝑠 2
∗
(11)
i∗d = Im
θPLL + 𝑠𝑖𝑛2 θPLL ) = Im
∗
i∗q = Im
(−cos θPLL sin θPLL + cos θPLL sin θPLL ) = 0
The measured real current iα is defined as in (13)
(13)
iα = i = Im cos(θPLL )
While, the synthesized current iβ is defined as the reference
current i∗β as in (14)
∗
iβ = i∗β = Im
sin(θPLL )
(14)
Applying the assumption that i∗β = iβ at the steady state, a
simplification will result in the current control loop system as
follows
The steady state errors in the dq frame ƹd and ƹq are given by
(15) and (16)
ƹd = id∗ − id
ƹd = [i∗α cos(θPLL ) + i𝛃∗ sin(θPLL )] −
[iα cos(θPLL ) + iβ sin(θPLL )]
ƹd = (i∗α −iα )cos(θPLL ) +
(i∗𝛃 −iβ )sin(θPLL )
ƹd = (i∗α −iα )cos(θPLL )
⟦
⟧
(15)
ƹq = i∗q − iq
ƹq =
⟦
[−i∗α
sin(θPLL ) + i∗𝛃 cos(θPLL )] −
[−iα sin(θPLL ) + iβ cos(θPLL )]
ƹq = −(i∗α −iα )sin(θPLL ) +
(i𝛃∗ −iβ )cos(θPLL )
ƹq = −(i∗α −iα )sin(θPLL )
⟧
Fig. 8. Experimental setup
(12)
(16)
From (15) and (16), both ƹd and ƹq errors were calculated
based on the real current error ƹα and not on the knowledge of
dq currents as in the conventional dq current controller. The
dq current controller based on (15) and (16) which used in the
simplified approach is shown in Fig 7.
V.
PERFORMANCE EVALUATION
The purpose of this section is to experimentally evaluate the
performance of the simplified-dq current control scheme and
also to compare it with that of the conventional delay-based
control strategy. The presented test results show that the
proposed simplified dq controller has the following features: 1)
It is capable of tracking the reference signals with a zero
steady-state error within few milliseconds; 2) has fast
dynamics; and 3) contrary to the conventional approach,
provides practically decoupled d and q current axes.
Adopting the test system in Fig. 1 and the corresponding
parameters in Table I, the single-phase experimental setup
shown in Fig. 8 is implemented based on that. To implement
the control strategies, a Texas Instrument TMS320F28335
digital signal processor (DSP) is used. The control algorithms
are written in C-code and are developed using Code Composer
Studio CCS5.5 software. The sampling frequency has been
fixed to 20 kHz with a dead-time period of 1 μs. The PI
controller gains of the control algorithms are set as follows: kp
=1300 and ki=20000 which are tuned using trial-error method
to obtain the optimal dynamic performance.
To evaluate the performance of both current regulation
schemes experimentally, a reference tracking test is conducted
for each control strategy by stepping –up and stepping-down in
the d-axis reference value while that of the q-axis is kept
constant at zero.
1.5
1) Conventional Delay-Based Controller
Adopting the control strategy of Fig. 5, the inverter initially
injects zero current. At time instant t 0.1s, the reference value
of the d-axis steps up to 0.5 p.u. Moreover, at time instant t
O.2s, the reference value of d-axis steps up to Ip.u, and at t
0.3s it steps down to zero, while the reference value of q-axis is
kept constant at zero p.u during the whole process. As shown
in Fig. 9(a), upon each step change in the d-axis references, the
controller tries to regulate the a current at the desired value;
however, due to excessive transients, there is an overshoot in
the regulated current. Moreover, as shown in Fig. 9(b), the d­
axis of the current experiences non-negligible transients for
about a cycle due to the delay used in the controller. Therefore,
it takes about 20ms for the d component of the current to track
the reference value with a zero steady-state error. The
conducted study demonstrates that the delay-based control
scheme suffers from excessive transients subsequent to any
step change in its d -axis. In Fig. 9(c), although the reference
value of the q-axis is kept constant, however, subsequent to
each step change in the d-axis, the q-axis also experiences a
non-negligible transient, which verifies that the conventional
current control strategy suffers from coupled axes. Note that in
the steady state, the controller can regulate the current with
zero steady-state error. Moreover, the measured total harmonic
distortion (THD) of the grid current in the steady-state is 3.67%
as shown in Fig 10.
=
=
=
�=
I�1111I�
0
.- 0.5
-0
II
-1 .5
=
It
0.1
o
a
ill
"(
0.2
0.4
0.3
(a)
-� i/
-� id
1.5
�
I)
s
.e,
._u
1\
0.5
�
o
if
0.2
0.1
0.4
0.3
(b)
0.6
0.4
0.2
1\
0
Iv'r<!
�
�
r-'!'PTI 'f 'f
'I
-0.2
0.2
0.1
0.4
0.3
(c)
Time(s)
=
=
)11
i
-1
cr-
In order to fairly compare the performance of both
methods, exactly the same test as that of the previous section is
conducted for the simplified-based approach, and the same PI
controllers with the same gains are utilized. The inverter of Fig.
6 initially injects zero current. Keeping the q-axis reference
value constant at zero, at time instant t
O.1s, the reference
value of the d-axis steps up to 0.5 p.u. Moreover, at time
instant t O.2s, the reference value of the d- axis is changed to
1p.u. and finally at t 0.3s, the d-axis reference value is set
back to zero p.u. Subsequent to each change in the d axis
reference value, the controller regulates the current at the
desired level in almost 2ms with zero steady-state error as
shown in Fig. Il(a). Moreover, as shown in Fig. I I(b), the d­
axis of the currents tracks the reference value in almost 2 ms
with zero steady-state error. However, contrary to the
conventional controller, upon each step change in the d-axis,
the q-component of the current experiences very short and
negligible transients as shown in Fig. ll(c) . It should be noted
that similar to the delay-based controller, the controller is
capable of regulating the current with zero steady-state error.
Moreover, the measured THD of the grid current in the steady­
state is 3.76% as shown in Fig 12, which verifies that the
performances of the controllers in the steady state are quite
similar. However, the conducted studies demonstrate that the
proposed simplified control scheme has superior dynamic
performance compared with that of the conventional delay­
based approach.
0.5
:::J
ci.
S
-e,
2) Simplified-Based Controller
-�
--
.-
Fig.9. Experimental results for the transient response of the conventional dq
controller during step changes in d-axis. (a) The grid current and its emulated
orthogonal component. (b) The d-axis of the currents.
(c) The q-axis of the currents.
Fundamental (50Hz) = 5.0!9 , 1HD= 3.67%
2
o
�
0
II
I
10
,1.
1
I ,11.1
.
. . . h, . 11 . .
30
20 .
Harmoruc order
,I. ,,II ••
40
..
50
Fig.!O. Harmonic spectrum of the grid current using the conventional scheme
1.5

1
i


i *, i (p.u)
0.5
0
-0.5
based on experiments, and it was compared with that of
conventional approach. The conducted studies conclude that
the proposed method is characterized by the following. 1) It
can maintain the stability of the system and track reference
values with zero steady error. 2) It is much faster than the
conventional approach. 3) It has superior axis decoupling
capability compared to the conventional approach.
ACKNOWLEDGMENT
A. M. Mnider would like to acknowledge the Libyan
Ministry of Higher Education and Scientific Research, and the
University of Elmergib, for sponsoring his Ph.D. project.
i *

-1
-1.5
0
0.1
0.2
(a)
0.3
0.4
1.5
i *
d
i
id*, i d (p.u)
1
REFERENCES
d
[1]
0.5
[2]
0
-0.5
0
0.1
0.2
(b)
0.3
[3]
0.4
0.6
[4]
iq (p.u)
0.4
0.2
[5]
0
-0.2
-0.4
[6]
0
0.05
0.1
0.15
0.2
0.25
(c)
Time (s)
0.3
0.35
0.4
[7]
Fig.11. Experimental results for the transient response of the simplified dq
controller during step changes in d-axis. (a) The grid current and its
reference. (b) The d-axis of the current. (c) The q-axis of the current.
Mag (% of Fundamental)
Fundamental (50Hz) = 5.019 , THD= 3.76%
[8]
[9]
2
[10]
1.5
[11]
1
0.5
[12]
0
0
10
20
30
Harmonic order
40
50
[13]
Fig.12. Harmonic spectrum of the grid current using the simplified scheme
VI.
CONCLUSION
A simplified dq current control strategy for single-phase
VSCs has been presented. Similar to the conventional current
control strategy, the proposed method can regulate the dq
components of the current at desired reference levels while it
does not impose excessive structural complexity compared to
the conventional approach and can be easily implemented in
digital environments. In addition, the proposed strategy does
not require orthogonal-current component to be generated,
which results in fast and non-oscillatory dynamics. The
performance of the proposed control strategy was evaluated
[14]
[15]
[16]
[17]
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