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2664
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
Single-Phase Single-Stage Transformer less
Grid-Connected PV System
Bader N. Alajmi, Khaled H. Ahmed, Senior Member, IEEE, Grain Philip Adam, and Barry W. Williams
Abstract—In this paper, a single-phase, single-stage current
source inverter-based photovoltaic system for grid connection is
proposed. The system utilizes transformer-less single-stage conversion for tracking the maximum power point and interfacing
the photovoltaic array to the grid. The maximum power point is
maintained with a fuzzy logic controller. A proportional-resonant
controller is used to control the current injected into the grid. To
improve the power quality and system efficiency, a double-tuned
parallel resonant circuit is proposed to attenuate the second- and
fourth- order harmonics at the inverter dc side. A modified carrierbased modulation technique for the current source inverter is proposed to magnetize the dc-link inductor by shorting one of the
bridge converter legs after every active switching cycle. Simulation
and practical results validate and confirm the dynamic performance and power quality of the proposed system.
Index Terms—Current source inverter (CSI), grid-connected,
maximum power point tracking (MPPT), photovoltaic (PV).
I. INTRODUCTION
UE to the energy crisis and environmental issues, renewable energy sources have attracted the attention of researchers and investors. Among the available renewable energy
sources, the photovoltaic (PV) system is considered to be a most
promising technology, because of its suitability in distributed
generation, satellite systems, and transportation [1]. In distributed generation applications, the PV system operates in two
different modes: grid-connected mode and island mode [2]–[6].
In the grid-connected mode, maximum power is extracted from
the PV system to supply maximum available power into the
grid. Single- and two-stage grid-connected systems are commonly used topologies in single- and three-phase PV applications [7], [8]. In a single-stage grid-connected system, the PV
system utilizes a single conversion unit (dc/ac power inverter)
to track the maximum power point (MPP) and interface the PV
system to the grid. In such a topology, PV maximum power
is delivered into the grid with high efficiency, small size, and
low cost. However, to fulfill grid requirements, such a topology
D
Manuscript received May 15, 2012; revised August 12, 2012 and September
25, 2012; accepted October 25, 2012. Date of current version December 7, 2012.
Recommended for publication by Associate Editor V. Agarwal.
B. N. Alajmi, G. P. Adam, and B. W. Williams are with the Department of Electronic and Electrical Engineering, University of Strathclyde,
Glasgow G1 1XQ, U.K. (e-mail: bader.alajmi@eee.strath.ac.uk; grain.adam@
eee.strath.ac.uk; barry.williams@eee.strath.ac.uk).
K. H. Ahmed is with the Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria 21526, Egypt (e-mail:
khaledh20@yahoo.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2012.2228280
requires either a step-up transformer, which reduces the system efficiency and increases cost, or a PV array with a high dc
voltage. High-voltage systems suffer from hotspots during partial shadowing and increased leakage current between the panel
and the system ground though parasitic capacitances. Moreover,
inverter control is complicated because the control objectives,
such as MPP tracking (MPPT), power factor correction, and harmonic reduction, are simultaneously considered. On the other
hand, a two-stage grid-connected PV system utilizes two conversion stages: a dc/dc converter for boosting and conditioning
the PV output voltage and tracking the MPP, and a dc/ac inverter
for interfacing the PV system to the grid. In such a topology, a
high-voltage PV array is not essential, because of the dc voltage
boosting stage. However, this two-stage technique suffers from
reduced efficiency, higher cost, and larger size.
From the aforementioned drawbacks of existing gridconnected PV systems, it is apparent that the efficiency and
footprint of the two-stage grid-connected system are not attractive. Therefore, single-stage inverters have gained attention,
especially in low voltage applications. Different single-stage
topologies have been proposed, and a comparison of the available interface units is presented in [8], [9]. The conventional
voltage source inverter (VSI) is the most commonly used interface unit in grid-connected PV system technology due to
its simplicity and availability [10]. However, the voltage buck
properties of the VSI increase the necessity of using a bulky
transformer or higher dc voltage. Moreover, an electrolytic capacitor, which presents a critical point of failure, is also needed.
Several multilevel inverters have been proposed to improve the
ac-side waveform quality, reduce the electrical stress on the
power switches, and reduce the power losses due to a high
switching frequency [11]–[14]. However, the advantages are
achieved at the expense of a more complex PV system. Moreover, a bulky transformer and an unreliable electrolytic capacitor
are still required.
The current source inverter (CSI) has not been extensively
investigated for grid- connected renewable energy systems [15].
However, it could be a viable alternative technology for PV distributed generation grid connection for the following reasons:
1) the dc input current is continuous which is important for
a PV application;
2) system reliability is increased by replacing the shunt input
electrolytic capacitor with a series input inductor;
3) the CSI voltage boosting capability allows a low-voltage
PV array to be grid interface without the need of a transformer or an additional boost stage.
Grid-connected PV systems using a CSI have been proposed.
The three-phase CSI for PV grid connection proposed in [16],
0885-8993/$31.00 © 2012 IEEE
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
successfully delivered PV power to the grid, without sensing
the ac output current, with a total harmonic distortion (THD)
of 4.5%. However, an ac current loop is essential in the gridconnected application in order to limit the current and quickly
recover the grid current variation during varying weather conditions. A dynamic model and control structure for a single-stage
three-phase grid-connected PV system using a CSI is proposed
in [17]. The current injected into the grid has a low THD and
unity power factor under various weather conditions. However,
the controller consists of only current loops, which affect system
reliability.
Unlike the three-phase grid-connected CSI, the single-phase
system has even harmonics on the dc side, which affect MPPT,
reduce the PV lifetime, and are associated with odd-order harmonics on the grid side [9], [18]. Therefore, eliminating the
even harmonics on the dc side is essential in PV applications.
Various techniques have been proposed to reduce the even harmonic effects in CSI PV applications. The conventional solution
to the dc current oscillation is to use a large inductor, which is
capable of eliminating the even-order harmonics. Practically,
the CSI inverter produces high dc current [17]; therefore, an
inductor with a large value is usually bulky and large in size.
Thus, this technique is practically unacceptable. To eliminate
the harmonics without using large inductance, two solutions
have been proposed in the literature, namely feedback current
control and hardware techniques. Specially designed feedback
current controllers intended to eliminate the odd harmonics on
the ac side without using large inductance are proposed in the
literature. In [19], the oscillating power effect from the grid
is minimized by employing a tuned proportional resonant controller at the third harmonic. Nonlinear pulsewidth modulation
(NPWM) has been proposed in [20] to improve harmonic mitigation. NPWM is based on applying computational operations,
such as a band-pass filter, a low-pass filter, a phase-shifter block,
and various division operations to extract the second-order harmonic component from the dc-link current. In [21], the power
oscillating effect is mitigated by using a modification of the carrier signal on pulse amplitude modulation (PAM). The carrier
signal is varied with the second-order harmonic component in
the dc-link current to eliminate its effect on the grid current.
These techniques [19]–[21] are not suitable for a single-stage
grid-connected PV system, because the dc current oscillation
is large, which causes high system losses and reduces its lifetime. In the hardware solution proposed in [22], second-order
harmonics are eliminated by using an additional parallel resonant circuit on the dc-side inductor. Even though the hardware
solution adds costs, losses, and size, it is considered to be a practical solution for CSI-based PV systems. Usually, the impact of
second-order harmonics in the dc-side current can significantly
affect the ac-side current. Additionally, the fourth-order harmonic in the dc-side current could affect the ac-side current at
high modulation indices.
In this paper, a single-stage single-phase grid-connected PV
system-based on a CSI is proposed. A doubled-tuned parallel
resonant circuit is proposed to eliminate the second- and fourthorder harmonics on the dc side. Moreover, a modified carrierbased modulation technique is proposed to provide a continuous
Fig. 1.
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Single-phase grid connected current source inverter.
path for the dc-side current after each active switching cycle.
The control structure consists of MPPT, an ac current loop, and a
voltage loop. To demonstrate the effectiveness and robustness of
the proposed system, computer-aided simulation and practical
results are used to validate the system.
II. SYSTEM DESCRIPTION
A grid-connected PV system using a single-phase CSI is
shown in Fig. 1. The inverter has four insulated-gate bipolar
transistors (IGBTs) (S1–S4) and four diodes (D1–D4). Each
diode is connected in series with an IGBT switch for reverse
blocking capability. A doubled-tuned parallel resonant circuit
in series with dc-link inductor Ldc is employed for smoothing
the dc link current. To eliminate the switching harmonics, a C–L
filter is connected into the inverter ac side.
III. DOUBLE-TUNED RESONANT FILTER
In a single-phase CSI, the pulsating instantaneous power of
twice the system frequency generates even harmonics in the
dc-link current. These harmonics reflect onto the ac side as loworder odd harmonics in the current and voltage. Undesirably,
these even harmonics affect MPPT in PV system applications
and reduce the PV lifetime. In order to mitigate the impact of
these dc-side harmonics on the ac side and on the PV, the dclink inductance must be large enough to suppress the dc-link
current ripple produced by these harmonics. Practically, large
dc-link inductance is not acceptable, because of its cost, size,
weight, and the fact that it slows MPPT transient response.
To reduce the necessary dc-link inductance, a parallel resonant
circuit tuned to the second-order harmonic is employed in series
with the dc-link inductor. The filter is capable of smoothing
the dc-link current by using relatively small inductances. Even
though the impact of the second-order harmonic is significant
in the dc-link current, the fourth-order harmonic can also affect
the dc-link current, especially when the CSI operates at high
modulation indices. Therefore, in an attempt to improve the
parallel resonant circuit, this paper proposes a double-tuned
parallel resonant circuit tuned at the second- and fourth-order
harmonics, as shown in Fig. 2
In order to tune the resonant filter to the desired harmonic
frequencies, the impedance of C1 and the total impedance of
L1 , L2 , and C2 should have equal values of opposite sign. For
simplicity, assume component resistances are small, and thus
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2666
Fig. 2.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
Proposed double-tuned resonant filter.
Fig. 4. Characteristics of the double-tuned resonant filter: the resonant capacitances (C 1 and C2 ) as a function of the resonant inductances L1 and L2 .
Fig. 3. Impedance versus frequency measurement for the doubled-tuned parallel resonant circuit.
Fig. 5.
can be neglected in the calculation
From (4), to avoid complex numbers in the solution, the relationship between L1 and L2 should be
ZC 1 + Zt = 0.
(1)
From (1), the capacitances are represented by the following
equations:
C1 =
C2 =
L2 C2 − ω12
ω 2 L1 L2 C2 − L1 − L2
(2)
−L2
1
+ 2
2
ω L2
− ω L1 L2
(3)
L2
C1
where C1 and C2 are the resonant filter capacitances, L1 and
L2 are the resonant filter inductances, ZC 1 is C1 impedance, Zt
is the total impedance of L1 , L2 , and C2 , and ω is the angular frequency of the second- or fourth-orders harmonics. After
selecting the inductance values, which are capable of allowing
the maximum di/dt at rated current, the angular frequency of the
second harmonics in (2) and the angular frequency of the fourth
harmonic in (3) are used. The desired capacitances are calculated
by numerically solving (2) and (3). Fig. 3 shows the impedance
versus frequency measurement for the doubled-tuned parallel
resonant circuit. The filter is capable of eliminating both the
second- and fourth-order harmonics.
In order to obtain the relationship between the resonant inductances (L1 and L2 ), (1) and (2) are solved for C1 , (4) as
shown at the bottom of the page.
C1 =
Proposed double-tuned resonant filter for eliminating n harmonics.
L2 ≤ 1.778L1 .
(5)
To select the optimum values for the proposed filter components, the effects of varying resonant circuit inductance are
analyzed. Fig. 4 shows resonant capacitance (C1 and C2 ) as a
function of the resonant inductances L1 and L2 . It can be shown
that C1 is not significantly affected when varying L1 and L2 ,
whereas C2 is affected mainly by L2 . As L2 decreases, the value
of C2 increases. Therefore, increasing the capacitance leads to
reduced overall system weight and size by reducing the dc-link
inductance.
The proposed filter concept can be extended to eliminate any
number of harmonics by employing the cascaded circuit shown
in Fig. 5. In order to compute the filter passive components, (6)
is numerically solved for C1 to Cn
1
+ Zt = 0
ω1 C1
1
+ Zt = 0
L1 ω2 +
ω2 C1
L1 ω1 +
..
.
L1 ωn +
1
+ Zt = 0
ωn C1
L1 (L1 ω14 + L1 ω24 − 2L1 ω12 ω22 − 4L2 ω12 ω22 ) + L1 ω12 + L1 ω22
2L21 ω12 ω22 + 2L1 L2 ω12 ω22
(6)
(4)
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
Fig. 6.
2667
Proposed carriers-based PWM along with switching sequence for one fundamental frequency.
where Zt is the total impedance of the series–parallel circuit
components L2 to Ln and C2 to Cn and n is the harmonic
order. To clarify the proposed filter design for the mitigation
of more harmonics, an example of eliminating three harmonics
is outlined. To eliminate the second-, fourth- and sixth-order
harmonics, (6) is rewritten, as (7)–(9) shown at the bottom of
the page.
By numerically solving (7)–(9), and (9), the capacitances that
eliminate the desired harmonics can be computed.
IV. MODIFIED CARRIER-BASED PULSEWIDTH MODULATION
Modified carrier-based pulsewidth modulation (CPWM) is
proposed to control the switching pattern for the single-phase
grid-connected CSI. In order to provide a continuous path for the
C1 =
L1 ω22
+
L2 ω22
dc-side current, at least one top switch in either arm and one bottom switch must be turned ON during every switching period.
In conventional sinusoidal pulsewidth modulation (SPWM), the
existence of overlap time as the power devices change states
allows a continuous path for the dc current. However, the overlap time is insufficient to energize the dc-link inductor, which
results in increased THD. Therefore, CPWM is proposed to
provide sufficient short-circuit current after every active switching action. CPWM consists of two carriers and one reference.
Fig. 6 shows the reference and carrier waveforms, along with
the switching patterns. The carrier with the solid straight line
shown in Fig. 6 is responsible for the upper switches, while
the dashed line carrier is responsible for the lower switches and
is shifted by 180◦ . To understand the switching patterns of the
proposed CPWM, Fig. 6 is divided into ten regions (t1 − t10 ),
−(C2 L2 ω22 + C2 L3 ω22 + C3 L3 ω22 − C2 C3 L2 L3 ω24 − 1)
+ L3 ω22 − C2 L1 L2 ω24 − C2 L1 L3 ω24 − C3 L1 L3 ω24 − C3 L2 L3 ω24 + C2 C3 L1 L2 L3 ω26
(7)
C2 =
−(C1 L1 ω42 + C1 L2 ω42 + C1 L3 ω42 + C3 L3 ω42 − C1 C3 L1 L3 ω44 − C1 C3 L2 L3 ω44 − 1)
L2 ω42 + L3 ω42 − C1 L1 L2 ω44 − C1 L1 L3 ω44 − C3 L2 L3 ω44 + C1 C3 L1 L2 L3 ω46
(8)
C3 =
C1 L1 ω62 + C1 L2 ω62 + C1 L3 ω62 + C2 L2 ω62 + C2 L3 ω62 − C1 C2 L1 L2 ω64 − C1 C2 L1 L3 ω64 − 1
C1 L1 L3 ω64 − L3 ω62 + C1 L2 L3 ω64 + C2 L2 L3 ω64 − C1 C2 L1 L2 L3 ω66
(9)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
TABLE I
SWITCHING COMBINATION SEQUENCE
By neglecting inverter losses, the PV output power is equal
to the grid power
VPV IPV = 1/2 Ig ,p eak Vg ,p eak cos θ
(10)
where θ is the phase angle, Vpv and Ipv are the PV output voltage
and current, respectively, while Vg ,p eak and Ig ,p eak are the grid
peak voltage and current, respectively. The grid current is equal
to the PV output current multiplied by the inverter modulation
index M
Ig ,p eak = M IPV .
(11)
Substituting (11) into (10), assuming unity power factor, the
equation describing the relationship between the PV output voltage and the grid voltage is
VPV =
Fig. 7. Comparison between the proposed CPWM and conventional SPWM:
(a) CSI output current using CPWM and (b) CSI output current using
SPWM.
and each region represents one carrier frequency period. Table I
shows the switch combination for each of the ten regions. As
shown in Fig. 6 and Table I, CPWM operates in two modes,
a conductive mode and a null mode, and the switching action
of each IGBT is equally distributed during every fundamental
period. To validate the proposed CPWM, simulation results of
a CSI operated by both CPWM and SPWM are shown in Fig. 7.
The CSI is operated in an island mode and has the following
specification.
The dc voltage Vdc is 50 V; in the double-tuned resonant filter
L1 = 10 mH, L2 = 5 mH, C1 = 125 μF, and C2 = 250 μF,
the capacitor on the ac side is 20 μF, the inductance is 1 mH,
the resistive load is 50 Ω, the output voltage is 110 V, and the
switch frequency is 4 kHz.
Fig. 7 shows that the proposed CPWM generates lower
switching frequency harmonics in the ac output current when
compared with conventional SPWM. The THD is 1% with the
proposed CPWM and 4.4% with conventional SPWM.
V. PROPOSED SYSTEM CONTROL TECHNIQUE
To design a grid-connected PV system using a CSI, the relationship between the PV output voltage and the grid voltage is
derived as follows.
1
M Vg ,p eak .
2
(12)
Therefore, in order to interface the PV system to the grid
using a CSI, the PV voltage should not exceed half the grid
peak voltage.
The CSI is utilized to track the PV MPP and to interface the
PV system to the grid. In order to achieve these requirements,
three control loops are employed, namely MPPT, an ac current
loop, and a voltage loop.
To operate the PV at the MPP, MPPT is used to identify
the optimum grid current peak value. Any conventional MPPT
technique can be used. However, to prevent significant losses in
power, the tracking technique should be fast enough to handle
any variation in load or weather conditions. Therefore, a fuzzy
logic controller (FLC) is used to quickly locate the MPP.
The inputs of the FLC are
ΔP = P (k) − P (k − 1)
ΔIPV = IPV (k) − IPV (k − ‘1)
(13)
(14)
and the output equation is
ΔIg ,ref = Ig ,ref (k) − Ig ,ref (k − 1)
(15)
where ΔP and ΔIPV are the PV array output power and current
change, ΔIg ,ref is the grid current amplitude change reference,
Ig ,ref is the grid current reference, and k is the sample instant.
The variable inputs and output are divided into four fuzzy subsets: PB (Positive Big), PS (Positive Small), NB (Negative Big),
and NS (Negative Small). Therefore, the fuzzy algorithm requires 16 fuzzy control rules; these rules are based on the regulation of the hill climbing algorithm, where the fuzzy rules are
shown in Table II. To operate the fuzzy combination, Mamdani’s
method with Max–Min is used.
From the behavior of the controller inputs and output, the
shapes and fuzzy subset partitions of the membership function
in both the inputs and output are shown in Fig. 8. A center of area
algorithm (COA) is used in the defuzzification stage to convert
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
2669
TABLE II
FUZZY LOGIC RULES
Fig. 10.
Equivalent circuit of the CSI ac side.
controller transfer function is expressed as
es
y = Kp e + K i 2
s + ωo2
(17)
where Kp is the proportional gain, Ki is the integral gain, e is
the signal error, and ωo is the fundamental angular frequency.
The transfer function of the PR controller is digitized using
the following derivation.
Let
s
e(s).
(18)
z(s) = Ki 2
s + ωo2
Rearrange (18) as
Fig. 8.
sz(s) +
Membership function: (a) input ΔP , (b) input ΔI, and (c) output ΔD.
ωo2
z(s) = Ki e(s).
s
(19)
Let
ωo2
z(s).
s
By taking the derivative of (19) and (20) we have
w(s) =
Fig. 9.
Block diagram of the FLC-based MPPT.
the fuzzy subset duty cycle changes into real numbers [6]
n
ΔIg ,ref =
i
μ(Ig ,ref ,i )Ig ,ref ,i
n
i μ(Ig ,ref ,i )
(16)
where ΔIg ,ref is the fuzzy controller output and Ig ,ref ,i is
the center of max–min composition at the output membership
function.
To ensure synchronization between the grid current and voltage, a sinusoidal signal generated by a phase-locked-loop (PLL)
is multiplied by the MPPT output. Fig. 9 shows a block diagram
of the MPPT structure.
For precise control of the single-phase inverter, proportionalresonant (PR) control is employed in the voltage and current
loop controllers. The basic principle of the PR controller is to
introduce an infinite gain at a selected resonant frequency in
order to eliminate steady-state error at that frequency. The PR
(20)
dw(t)
= ωo2 z(t)
(21)
dt
dz(t)
= Ki e − w(t)
(22)
dt
where z and w are auxiliary control variables used to facilitate
the control design. In the following section, the subscripts i and
v will be used with these control variables to signify the current
and voltage controllers, respectively.
From (17), (21), and (22), the output of the PR controller can
be rewritten as
y = kp e(t) + z(t).
(23)
In order to compute the controller output (21), (22), and (23)
are solved numerically as
W (k + 1) = W (k) + Ts ωo2 Z(k)
(24)
Z(k + 1) = Z(k) + Ts (Ki e(k) − W (k + 1))
(25)
y(k + 1) = Kp e(k) + Z(k + 1).
(26)
From the equivalent circuit of the CSI ac side, which is shown
in Fig. 10 and the PR controller equations, the ac current and
voltage loops are designed, where Iin is the CSI output current,
Cf is the filter capacitor, Lf is the filter inductor, R is the
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
Fig. 11.
AC current and voltage loops.
TABLE III
DESIGN SPECIFICATIONS AND CIRCUIT PARAMETERS
By taking the derivative of (31) and (32)
dzi (t)
= ki (Ig ,ref (t) − Ig (t)) − wi (t)
dt
dwi (t)
= ωo2 zi (t).
dt
(33)
(34)
From (30), (33), and (34), the state- space model of the
current-loop controller is obtained as
inductor internal resistor, Ic is the current passing through the
capacitor, Ig is the grid current, and Vg is the grid voltage.
The differential equation that describes the ac-side dynamic
voltage is
R
dIg
Vc − Vg
= − Ig +
.
dt
L
L
(27)
u = Vc − Vg .
(28)
⎛
−(R + Kpi ) 1
I (t)
⎜
d ⎝ g
L
L
zi (t) ⎠ = ⎜
⎝
−Kii
0
dt
wi (t)
0
ωo2
⎛
⎞
Kpi
⎜ L ⎟
⎟
+⎜
⎝ Kii ⎠ Ig ,ref (t).
0
⎛
⎞
dzi (t)
+ ω02
dt
(36)
λ = Iin − Ig .
(37)
(29)
From (36) and (37), by feeding the grid current error into the
PR controller, the value of u is obtained from (23)
(30)
λ = kpv (Vc,ref (t) − Vc (t)) + zv (t).
Substituting (29) into (27)
where
Iin − Ig
dVc
=
.
dt
C
Let
From (27) and (28), by feeding the grid current error to the
PR controller, the value of u is obtained from (23) as
dig
1
= (kpi Ig ,ref (t) − kpi Ig (t) − RIg (t) + zi(t))
dt
L
(35)
Similarly, the differential equation that describes the ac-side
dynamic current is
Let
u(t) = kpi (Ig ,ref (t) − Ig (t)) + zi (t).
⎞⎛
⎞
0 ⎟ Ig (t)
⎟
⎟⎜
−1 ⎠ ⎝ zi (t) ⎠
wi (t)
0
(38)
Substituting (38) into (36)
zi (t)dt = kii (Ig ,ref (t) − Ig (t))
and
(31)
1
dVc (t)
= (kpv Vc,ref − kpv Vc (t) + zv (t))
dt
C
wi (t) =
ωo2 zi (t)dt
(32)
where kpi is the current controller proportional gain and kii is
the current controller integral gain.
(39)
where
dzv (t)
+ ω02
dt
zv (t)dt = kiv (Vc,ref (t) − Vg (t))
(40)
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
Fig. 12.
2671
(a) PV output power, (b) PV output current, (c) grid voltage (factored by 10) and current, (d) CSI output current, and (e) grid active and reactive power.
and
From (39), (42), and (43), the state- space model of the voltage
loop controller is
wv (t) =
ωo2 zv (t)dt.
(41)
where kpv is the voltage controller proportional gain and kiv is
the voltage controller integral gain.
By taking the derivative of (40) and (41)
dzv (t)
= kiv (Vc,ref − Vc (t)) − wv (t)
dt
dwv (t)
= ωo2 zv (t).
dt
(42)
(43)
⎛
⎞
⎛
−Kpv
⎜
C
d ⎜
⎟
⎝ zv (t) ⎠ = ⎜
−K
⎝
iv
dt
wv (t)
0
⎛
Kpv
⎜ L
+⎜
⎝ Kiv
Vc (t)
1
C
0
ωo2
⎞
0
⎞⎛
Vc (t)
⎞
⎟⎜
⎟
⎟
−1 ⎠ ⎝ zv (t) ⎠
wv (t)
0
⎟
⎟ Vc,ref (t).
⎠
(44)
0
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
Fig. 13. Simulation results of the proposed system under two radiation levels: 500 and 1000 W/m2 . (a) PV output power, (b) PV output current, (c) grid voltage
(factored by 10) and current, (d) CSI output current, and (e) grid active and reactive power.
To obtain the overall state-space model of the controllers, (28)
is rewritten as
Vc,ref (t) = u(t) + Vg (t) = kpi (Ig ,ref (t) − Ig (t))
+ zi (t) + Vg (t).
By substituting (45) into (39)
(45)
dVc (t)
1
= (−kpv Vc (t) + zv (t) − kpv Kii Ig (t) + zi (t)
dt
C
(46)
+ kpv Kii Ig ,ref + kpv Vg (t))
dzv (t)
= −kiv Vc (t) − kiv kpi Ig (t) + kiv kpi Ig ,ref (t)
dt
(47)
+ kiv zi (t) + kiv Vg (t) − wv (t)
dwv (t)
= ωo2 zv (t).
dt
Fig. 14.
Proposed system efficiency as a function of PV output power.
(48)
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
Fig. 15.
(a) Test rig photograph and (b) hardware diagram.
From (30), (33), (34), (46), (47), and (48), the system statespace model is
⎛
2673
⎞
⎛ −kpv
1
C
0
0
kpv kpi
C
kiv kpi
kpv
C
kiv
Vc (t)
⎜ C
⎜ z (t) ⎟ ⎜ −k
−1
iv
⎟ ⎜
⎜ v
⎟ ⎜
⎜
2
w (t) ⎟ ⎜
ωo
0
0
0
d ⎜
⎜ v ⎟=⎜ 0
⎜
⎟
⎜
−k
1
dt ⎜ Ig (t) ⎟ ⎜
pi
0
0
⎟ ⎜ 0
⎜
L
L
⎝ zi (t) ⎠ ⎜
⎝ 0
0
0
−kii
0
wi (t)
0
0
0
0
ωo2
⎛k k
kpv ⎞
pv pi
⎛
⎞
Vc (t)
⎜ C
C ⎟
⎜ z (t) ⎟ ⎜
⎟
⎜ v
⎟ ⎜ kiv kpi kiv ⎟
⎜
⎟
⎟ ⎜
⎜ wv (t) ⎟ ⎜ 0
Ig ,ref (t)
0 ⎟
⎟
⎟+⎜
×⎜
.
⎜ I (t) ⎟ ⎜ kpi
⎟
Vg (t)
⎜ g
⎟
⎟ ⎜
0
⎜
⎟
⎟ ⎜
⎟
⎝ zi (t) ⎠ ⎜ L
⎝ kii
0 ⎠
wi (t()
0
0
0
⎞
⎟
0 ⎟
⎟
⎟
0 ⎟
⎟
⎟
0 ⎟
⎟
⎟
−1 ⎠
Equation (49) can be rewritten in short form as
dx(t)
= Ax(t) + Bu(t).
(50)
dt
Using the first-order Euler approximation, (50) can be written
in discrete form as
x(k + 1) = (1 + Ts A)x(k) + Ts Bu(k).
(51)
Therefore, (49) can be expressed in discrete form as (52)
shown at the bottom of the next page, where Ts is the control
sample time, which is selected to be to the reciprocal of the
PWM switching frequency.
The overall control structure is shown in Fig. 11.
0
VI. SIMULATIONS
(49)
Ten series-connected PV modules with a total rated power of
500 W are tested in the proposed system, for which the design
specification and circuit parameters are given in Table III. The
ac-side filter capacitor is selected to attenuate high frequency
harmonics that are associated with switching frequencies and
their sidebands, taking into account the rated ac and dc currents. So that at rated power, the converter is able to supply the
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2674
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013
Fig. 16. Comparison of even harmonic affects on CSI: (a) CSI with a large inductance at M = 0.5, (b) CSI with a resonant filter at M = 0.5, (c) CSI with a
large inductance at M = 0.9, and (d) CSI with a resonant filter at M = 0.9.
ac-side active and reactive power demands, and compensate for
filter capacitor and inductor reactive power, without sliding into
overmodulation. The parameters of the dc-side tuned filter are
selected as described in Section III.
To validate the performance of the proposed system, simulations are performed using MATLAB/Simulink. The results
of the proposed system under normal weather conditions are
shown in Fig. 12. The PV maximum power is extracted in a
relatively short time with small oscillation in steady state, as
shown in Fig. 12(a). Moreover, MPPT successfully locks the dc
current to the optimum value, as shown in Fig. 12(b). On the ac
side, the PV maximum power is successfully injected into the
grid with low THD, high efficiency, and unity power factor. The
grid voltage and current are shown in Fig. 12(c), where both
are synchronized and the THD of the grid current is only 1.5%.
⎡
kpv
Vc (k + 1)
⎢ 1 − Ts C
⎢ Z (k + 1) ⎥ ⎢
−Ts kiv
⎢ v
⎥ ⎢
⎢
⎥ ⎢
⎢ Wv (k + 1) ⎥ ⎢
0
⎢
⎥ ⎢
⎢ I (k + 1) ⎥ = ⎢
⎢
⎢ g
⎥ ⎢
0
⎢
⎥
⎣ Zi (k + 1) ⎦ ⎢
⎢
0
⎣
Wi (k + 1)
0
⎡
⎤
1
Ts
C
1
2
ω0 Ts
−Ts
1
0
0
0
0
0
0
0
kpv kpi
Ts
C
Ts kiv kpi
0
kpi
1 − Ts
L
Ts kii
0
Fig. 12(d) shows that the CSI output current is not violated,
by allowing switching of one current level at the same time. In
addition, the grid active power and reactive power are shown in
Fig. 12(f). The total system efficiency is 95%, and the power
factor is almost unity.
To demonstrate the effectiveness of the proposed system under varying weather conditions, a simulation is carried out using
two radiation levels, 500 W/m2 and 1000 W/m2 . As shown in
Fig. 13(a), the PV MPP is extracted in a relatively short time,
has a small oscillation around the MPP during steady state,
and the new MPP is correctly extracted during varying weather
conditions. Also the MPPT maintains the PV output current at
its optimum value during both weather conditions, Fig. 13(b)
shows the current on the dc side. Fig. 13(c) shows that the grid
current has a small THD and unity power factor under both
kpv
Ts
C
Ts kiv
0
1
Ts
L
1
2
ω0 T
⎤
⎡ k k
pv pi
0 ⎥ ⎡ Vc (k) ⎤
Ts
⎢
C
⎥⎢
⎢
Zv (k) ⎥
0 ⎥
⎥ ⎢ Ts kiv kpi
⎥⎢
⎢
⎥ ⎢
Wv (k) ⎥ ⎢
0 ⎥
0
⎥⎢
⎢
⎥+⎢
⎥⎢
⎢
⎥ ⎢ Ig (k) ⎥
⎥ ⎢ Ts kpi
0 ⎥⎢
⎢
L
⎥ ⎣ Z (k) ⎥
⎦ ⎢
i
⎥
⎣ Ts kii
−Ts ⎦
Wi (k)
0
1
kpv
C
Ts kiv
0
Ts
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥ Ig ,ref (k)
⎥
⎥ Vg (k)
⎥
⎥
⎥
⎦
(52)
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ALAJMI et al.: SINGLE-PHASE SINGLE-STAGE TRANSFORMER LESS GRID-CONNECTED PV SYSTEM
2675
B. Grid Connection Validation
I–V curves are programmed into the PV source simulator to
test the experimental system. The results of the proposed gridconnected CSI are shown in Fig. 17. The optimum PV current
is attained in a relatively short time and has a small steady-state
oscillation. Also, the CSI successfully injects the PV current
into the grid with low total harmonics distortion.
VIII. CONCLUSION
Fig. 17.
Experimental results of the proposed grid connected system.
weather conditions. Fig. 13(d) shows that the CSI output current is not violated by allowing switching of one current level
at the same time on both radiation levels. Moreover, Fig. 13(f)
shows the active power and reactive power under both weather
conditions. From Fig. 13(a) and (f), the total system efficiency
is about 95% for each radiation level. For further validation of
the proposed system efficiency, Fig. 14 illustrates the efficiency
as function of the input power under different radiation levels.
VII. EXPERIMENTAL RESULTS
The performance of the proposed grid-connected CSI is verified experimentally with the hardware shown in Fig. 15. The
experimental setup consists of an Agilent modular solar array
simulator to emulate PV system operation; a CSI with a 4-kHz
switching frequency to boost the output voltage, track the maximum power point, and interface the PV system to the grid; and
a single-phase auto-transformer to emulate the power grid. An
Infineon TriCore TC1796 is used to generate the PWM signals
and realize the proposed feedback loop controllers.
A. Practical Validation of the Proposed Filter
To validate the effectiveness of the proposed system, the results of the CSI with the double-tuned resonant filter are compared with those from the CSI with large dc- link inductance, L
= 300 mH. Fig. 16 shows the input dc current and the output ac
voltage and current of the CSI under both test conditions. From
Fig. 16(a) and Fig. 16(b), the double-tuned resonant filter and
the large inductance filter successfully eliminate even harmonic
effects at low modulation indices. The THD at the ac side for the
CSI with a resonant filter is 1.29%, whereas the THD for the CSI
with large link inductance is 1.92%. On the other converter side,
Fig. 16(c) and (d) shows the harmonic effects of both cases with
a high modulation index. The proposed double-tuned resonant
filter successfully eliminates the even-order harmonics in the dc
current, reducing the THD on the ac side to 2.73%. However,
the CSI with large inductance reduces the THD to 6.16%, which
does not meet the IEEE-519 harmonics standard. Since PV applications operate over a wide range of modulation indices to
track the MPP, the proposed double-tuned filter system is better
suited for PV applications.
A single-stage single-phase grid-connected PV system using
a CSI has been proposed that can meet the grid requirements
without using a high dc voltage or a bulky transformer. The
control structure of the proposed system consists of MPPT, a
current loop, and a voltage loop to improve system performance
during normal and varying weather conditions. Since the system
consists of a single-stage, the PV power is delivered to the grid
with high efficiency, low cost, and small footprint. A modified
carrier-based modulation technique has been proposed to provide a short circuit current path on the dc side to magnetize the
inductor after every conduction mode. Moreover, a double-tuned
resonant filter has been proposed to suppress the second- and
fourth-order harmonics on the dc side with relatively small inductance. The THD of the grid- injected current was 1.5% in the
simulation and around 2% practically. The feasibility and effectiveness of the proposed system has been successfully evaluated
with various simulation studies and practical implementation.
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Bader N. Alajmi received the B.Sc. and M.Sc. degrees from California State University, Fresno, in
2001 and 2006, respectively. He is currently working toward the Ph.D. degree in electrical engineering from the Department of Electrical and electronic,
Strathclyde University, Glasgow, U.K.
His research interests include digital control
of power electronic systems, microgrids and distributed generation, photovoltaic inverters, and dc–dc
converters.
Khaled H. Ahmed (SM’12) received the B.Sc. (first
class Hons.) and M.Sc. degrees from the Faculty
of Engineering, Alexandria University, Alexandria,
Egypt, in 2002 and 2004, respectively, and the Ph.D.
degree in electrical engineering from the University
of Strathclyde, Glasgow, U.K., in 2008.
Since 2009, he has been a Lecturer with Alexandria University, Alexandria, Egypt. He has authored
or coauthored more than 50 technical papers in refereed journals and conferences. His research interests
include digital control of power electronic systems,
power quality, microgrids, distributed generation, dc–dc converters, and HVDC.
Dr. Ahmed is a Reviewer for the IEEE TRANSACTIONS and several
conferences.
Grain Philip Adam received the Ph.D. degree in
power electronics from the University of Strathclyde,
Glasgow, U.K., in 2007.
He is a Research Fellow with the Institute of
Energy and Environment, University of Strathclyde.
His research interests include fault tolerant voltage
source multilevel converters for HVDC systems; control of HVDC transmission systems and multiterminal HVdc networks; voltage source converter-based
FACTS devices; and grid integration issues of renewable energies. He has authored or coauthored several
technical reports, journal, and conference papers in the area of multilevel converters and HVdc systems, and grid integration of renewable power.
Dr. Adam has contributed in reviewing process for several IEEE and IET
TRANSACTIONS, and journals and conferences.
Barry W. Williams received the M.Eng.Sc. degree
from the University of Adelaide, Adelaide, S.A., Australia, in 1978, and the Ph.D. degree from Cambridge
University, Cambridge, U.K., in 1980.
After seven years as a Lecturer at Imperial College, University of London, London, U.K., he became
a Chair of Electrical Engineering at Heriot–Watt University, Edinburgh, U.K, in 1986. He is currently a
Professor at Strathclyde University, Glasgow, U.K.
His teaching covers power electronics (in which he
has a free internet text) and drive systems. His research interests include power semiconductor modeling and protection, converter topologies, soft switching techniques, and application of ASICs and
microprocessors to industrial electronics.
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