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Modeling and Control of -Paralleled Grid-Connected Inverters With LCL Filter
Coupled Due to Grid Impedance in PV Plants
Article in IEEE Transactions on Power Electronics · April 2011
DOI: 10.1109/TPEL.2010.2095429 · Source: IEEE Xplore
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IEEE-TPEL
Modelling and Control of N Paralleled Grid-Connected
Inverters with LCL Filter coupled due to Grid Impedance in
PV plants.
r
Fo
Journal:
Manuscript ID:
Date Submitted by the
Author:
30-Jun-2010
Agorreta, Juan; Public University of Navarra, Electrical and
Electronic Engineering
Borrega, Mikel; Ingeteam Energy, Photovoltaic Solar Energy
López, Jesús; Public University of Navarra, Electrical and Electronic
Engineering
Marroyo, Luis; Public University of Navarra, Electrical and Electronic
Engineering
Re
Photovoltaic power systems, Power system modeling, Power system
stability, Pulse width modulated inverters, Current control
ew
vi
Keywords:
Regular Paper
er
Complete List of Authors:
TPEL-Reg-2010-06-0487
Pe
Manuscript Type:
IEEE Transactions on Power Electronics
Page 1 of 26
Modelling and Control of N Paralleled GridConnected Inverters with LCL Filter coupled due to
Grid Impedance in PV plants.
Abstract—Designing adequate control laws for grid-connected inverters with LCL filters is complicated. The power quality
standards and the system resonances burden the task. In order to deal with resonances, system damping has to be
implemented. Active damping is preferred to passive damping so as to improve the efficiency of the conversion,. In addition,
paralleled grid-connected inverters in PV plants are coupled due to grid impedance. Generally, this coupling is not taken into
account when designing the control laws. In consequence, depending on the number of paralleled grid-connected inverters
Fo
and the grid impedance, the inverters installed in PV plants do not behave as expected. In this article the inverters of a PV
plant are modelled as a multivariable system. The analysis carried out enables to obtain an “Equivalent Inverter” which
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describes the totality of inverters of a PV plant. The study is validated through simulation and field experiments. The
coupling effect is described and the control law design of paralleled grid-connected inverters with LCL filters in PV plants is
clarified.
ee
Keywords: LCL filter, grid-connected inverter, active damping, Photovoltaic power systems.
rR
I. INTRODUCTION
Distributed power generation systems based on renewable energy sources are attracting the market and research
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interest as a feasible choice in a sustainable development environment. In this context, grid-connected photovoltaic (PV)
plants are becoming a common technology to generate energy and its penetration level is gradually increasing [1]. These
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plants consist of sets of PV generators and power electronic inverters connected in parallel to the distribution network
through a distribution transformer. Fig. 1 shows the scheme of a PV plant in which the distribution transformer leakage
impedance is integrated into the grid impedance, Zg.
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IEEE-TPEL
i11
Z11
VSC 1
Z21
ig
Zg
i31
v01
v bus_1
i21
vg
Z31
PV 1
i12
Z12
VSC 2
Z22
i32
v02
v bus_2
i22
eg
Z32
PV 2
i1n
Z1n
VSC n
Z2n
…
i3n
v0n
v bus_n
i2n
Z3n
PV n
Fig. 1.
Typical PV power plant scheme.
Fo
The inverters installed in PV plants are generally Voltage Source Converters (VSC) with an output filter. LCL filters
are preferred to L filters because their switching harmonic attenuation with smaller reactive elements is more effective
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[2]-[3]. Thus, the cost and the weight of the inverters are reduced. However, due to the need to damp the resonances, the
filter and the current control design are more complex. Active damping [4]-[14] is preferred to passive damping [2] in
order to improve the efficiency of the conversion.
ee
Either the inverter side current [2]-[11], or the grid side current [12]-[13], of the LCL filter can be controlled. Each
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alternative has its own advantages and drawbacks [15]. Depending on the controlled current, specific active damping
techniques have been proposed.
The quality of the grid injected current is a matter of concern. International standards regulate the connection of PV
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inverters to the grid and limit the harmonic content of the injected current [16]-[18]. Exceeding the harmonic injection
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limits may require the inverters to disconnect from the grid. Thus, the LCL filters are implemented to prevent the grid
from being polluted with switching harmonics, whereas plenty of control algorithms, including repetitive controllers,
integrators in multiple rotating frames, and resonant integrators [16]-[18], have been proposed in order to mitigate the
low order current harmonics.
An added difficulty is that PV inverters are coupled due to the grid impedance, Zg, and influence each other as a result.
All inverters in Fig. 1 share the voltage in the point of common coupling (PCC), vg, and are able to modify this voltage by
injecting their currents. Furthermore, the current of an inverter is able to circulate through a paralleled inverter instead of
coming back through the grid. Notice that, if the grid impedance was ideally considered to be zero, coupling would not
exist since the voltage in the PCC would always be eg. Depending on the number of paralleled inverters, N, and the grid
impedance, Zg, the inverters installed in PV plants might not behave as expected. In fact, resonant behaviour of the
inverters has been detected in many PV plants. This situation exceeds the harmonic regulations and causes the breakage
of many devices such as electronic meters. Fig. 2 shows real waveforms of inverters with resonant behaviour in a PV
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Page 3 of 26
plant of 1400 kW. This figure depicts as well the voltage in the PCC, vg, and the grid side current of an inverter, i2i. The
oscillation occurs at a resonant frequency of 2777 Hz. Other authors have already studied the interaction of inverters and
the distribution network [19]-[20]. These works mention that a resonant behaviour may occur even if all the PV inverters
do individually satisfy the standards. In addition, it is indicated that power quality integration problems are expected as
the penetration level of PV inverters increases.
vg
Fo
i2i
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(a) 100 V/div; 10 A/div; 5 ms/div
ee
vg
i 2i
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(b) 100 V/div; 10 A/div; 1 ms/div
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Fig. 2.
Experimental measures of the voltage in the PCC, vg, and the grid side current, i2i, of an inverter in a PV plant: (a) Distorted waveforms
due to resonant behaviour. (b) Half grid period detail.
Every proposal with regard to active damping strategies and control algorithms reviewed [4]-[18] is very interesting.
However, their analyses are always done for single grid-connected inverters. In consequence, coupling between inverters
due to grid impedance is ignored and the stability and performance of the inverter in a PV plant might be questioned.
The aim of this article is to describe the consequences of this coupling effect and to propose easy guidelines to design
the active damping strategy and the control algorithm for such PV inverters. The N paralleled inverters are modelled as a
multivariable system. As often the case, inverters in a PV power plant are manufactured by the same firm, and have the
same rated power. In addition, industry standards constantly demand smaller tolerances. In consequence, all inverters can
reasonably be assumed to be equal. This analysis enables to obtain an “Equivalent Inverter” which models the N inverters
of a PV plant. Clear criterions to determine the stability, the effectiveness of the active damping strategy, and the
bandwidth of the whole system are exposed.
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In this paper, PI controllers are considered, and the damping technique proposed in [4] is chosen, but the same
methodology can also be applied to other control algorithms or active damping techniques leading to the same
“Equivalent Inverter” model. Although the considered inverters are controlled digitally, the analysis is performed in the
s-domain instead of the z-domain. The article is organized as follows: first, a digital control emulator which maintains the
s-domain analysis with good reliability is presented. Second, the modelling and control of a single grid-connected
inverter is described. Third, the modelling and control of the N paralleled grid connected inverters of a PV plant are
analysed. Fourth, the theoretical study is validated through simulation. Fifth, the experimental waveforms of Fig. 2 are
simulated. Next, some control design guidelines and practical uses are suggested. Finally, conclusions are discussed.
II. CONTINUOUS MODELLING OF THE SAMPLER, THE ZERO ORDER HOLD AND THE COMPUTATION DELAY.
Fo
Simulating digitally controlled systems by s-domain continuous transfer functions is a common practice. This
simplifies the analysis, although some accuracy is lost. Digitally controlled converters have a time delay of one sampling
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period due to the fact that the computation time of the DSP microprocessor relative to the sampling period is not
negligible. Sampling is inherent to the z-domain analysis, but if the aim is to model a discrete system as a continuous
system, the sampler continuous approximation has to be taken into account. Furthermore, the zero order hold fits
ee
perfectly with the PWM pattern [2], [7]-[8]. The continuous block diagram of these three elements [22] is shown in Fig.
3. Ts refers to the sampling period. In the present work, the sampling and the control updating are done at twice the rate of
the PWM carrier frequency, fswitch.
Computation
delay.
Sampler continuous
approximation.
Zero order hold.
e − T s ⋅s
1
Ts
1 − e−Ts ⋅ s
s
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Fig. 3.
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Continuous model of the one sampling period time delay, the sampler and the zero order hold.
The continuous transfer function of the block diagram in Fig 3 is expressed in (1)
Ds =
(
e −Ts ⋅s ⋅ 1 − e −Ts ⋅s
Ts ⋅ s
)
(1)
In this article, the transfer function, Ds, is called “digital control emulator.” Delays, such as the computation delay of
one sampling period or the time delay present in the zero order hold expression of Fig. 3, are difficult to handle because
they make transfer functions non rational. That is why, in power electronics the digital control emulator, Ds, has been
traditionally approximated by the following transfer function [3]-[4]:
Ds ≈ Ds1 =
1
1,5 ⋅ Ts ⋅ s + 1
(2)
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This approximation is useful when tuning PI controllers, since the crossover frequency of the open loop gain is
generally far enough from half the sampling frequency or Nyquist frequency. However, it fails in describing the
behaviour of digitally controlled systems as frequency comes close to the Nyquist frequency. Notice that this issue is
essential if an active damping strategy is to be implemented, since the resonance frequency is usually within the range of
the crossover frequency and the Nyquist frequency. Consequently, the great majority of authors prefer the z-domain
analysis [6]-[8], [11], [15]-[16].
In order to obtain rational transfer functions, delays are usually approximated by poles and zeros. One of the most
popular approximations for delays is the Padé approximation [22]. The first order Padé approximation for a time delay of
one sampling period, Ts, is:
e −Ts ⋅s ≈
Fo
1 − s ⋅ 0,5 ⋅ Ts
= Pade1
1 + s ⋅ 0,5 ⋅ Ts
(3)
In this paper, a different continuous approximation for the digital control emulator, Ds, is proposed. It is obtained
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provided that the first order Padé approximation is replaced in both: the computation delay expression and the zero order
hold expression of Fig. 3. Thus, the digital control emulator, Ds, is approximated as follows:
Ds ≈ Ds 2 =
(4)
rR
1 − 0,5 ⋅ s ⋅ Ts
Pade1 ⋅ (1 − Pade1 )
=
Ts ⋅ s
(1 + 0,5 ⋅ s ⋅ Ts )2
ee
The transfer function in (4) describes with better accuracy than (2) the behaviour of digitally controlled systems such
as PV inverters. It attains the right approximation also within the range of the crossover frequency and the Nyquist
ev
frequency as it will be depicted in Section III. Hence, the approximation Ds2 maintains the s-domain analysis with a fair
agreement between simplicity and accuracy.
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III. MODELLING AND CONTROL OF A SINGLE GRID-CONNECTED INVERTER.
A. System description.
Generally, the control strategy used in PV inverters is a cascaded-loop control [1], [4], [7], [10], [14]. The inner loop
controls the inductor current whereas the outer loop controls the DC-bus voltage. Thereby, the inner loop takes care of
the grid-injected current quality whereas the outer loop is responsible for the maximum power point tracking (MPPT).
Obviously, the inner loop bandwidth has to be much higher in order to maintain the outer loop and the inner loop
decoupled.
In this section, the modelling and control of a single grid-connected inverter with an LCL filter is described. Although
this issue has already been addressed in the available literature [1]-[18], the aim is to get the reader familiarized with the
nomenclature and methodology used along this article. More to the point, the simplifications derived in the following
sections will resemble the single inverter case. Solely the inner current loop is presented so as to simplify. Consequently,
the DC-bus of the inverter is supposed to be an ideal constant voltage source of value vbus.
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The circuit for the current control design of a single grid-connected inverter is shown in Fig. 4 (a). The impedances in
this figure are those of the LCL filter (5). Note that parasitic elements are neglected in order to simplify:
Z1 = L1 ⋅ s; Z 2 = L2 ⋅ s; Z 3 =
1
; Z g = Lg ⋅ s;
C3 ⋅ s
Z1
i1
(a)
(5)
A
i2
Z2
ig
Zg
i3
vZ3
v0
vg
Z3
eg
B
Grid
Converter
Z1
i1
A
ZthAB
(b)
Fo
v0
Fig. 4.
v Z3
B
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(a) Circuit of a single grid-connected inverter. (b) Thévenin-equivalent circuit across the terminals AB.
This paper follows the technique proposed in [4]. Accordingly, the inverter side current, i1, is controlled and the
capacitor voltage, vZ3, is measured in order to implement the active damping technique. The control variable is the
ee
converter voltage, v0. The voltage in the connexion point is vg. The voltage source, eg, represents the grid and behaves as
a disturbance. In this work, only the inductor current tracking reference is discussed, so the disturbance, eg, will be
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considered to be zero when describing the modelling and control of a single grid-connected inverter.
B. Modelling.
A straightforward way to obtain the inverter side inductor current dynamics, i1, in relation to the converter voltage, v0,
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is shown in Fig. 4 (b). A simplified circuit in which the only current is i1 and the only voltage source is v0 is found using
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the Thévenin-equivalent impedance calculated across terminals AB, ZthAB. Thus, the transfer function between i1 and v0 is:
Y1 =
Z3 + Z 2 + Z g
i1
1
(6)
=
=
v0 Z1 + Z thAB Z 3 ⋅ Z 2 + Z g + Z1 ⋅ Z 3 + Z 2 + Z g
(
)
(
)
The plant Y1 has a resonance frequency, ωr, and an anti-resonance frequency, ωa, which are respectively [2], [16]:
ωr =
1
(
C3 ⋅ L1 ⋅ L2 + Lg
)
;
ωa =
(
1
C3 ⋅ L2 + Lg
)
;
(7)
L1 + L2 + Lg
The design procedure of the LCL filter is out of the scope of this paper. For further research the references [2], [3] may
be consulted. The plant Y1 is un-damped since no resistors have been considered. It is depicted in Fig 5 (a) in which
standard LCL filter values for a 10kW inverter (Table I (a)) are chosen.
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(a)
Fig. 5.
(b)
(a) Un-damped plant Y1; (b) Actively damped plant Y1AD; (c) Open Loop OLAD;
TABLE II
CONTROL PARAMETERS VALUES
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TABLE I
LCL FILTER VALUES
L1 (mH)
L2 (mH)
C3 (µF)
Lg (mH)
1
1
1
0.2
13
6
0.1
0.5
Parameter
Value
Parameter
Value
Kp
Ki
2.9
4835
1/(2.π.5000)
1/(2.π.5000)
400 V
Kll
τp/τz
1/(2.π.300)
1/(2.π.800)
10000 Hz
1/(2. fswitch)
τi
τv
ee
(a)
(b)
(c)
Fo
vbus
τz
τp
fswitch
Ts
C. Control strategy.
The scheme of the implemented inner control loop strategy for the inverter side current, i1, is shown in Fig. 6 (a). The
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index i1ref is the inverter side current reference, u0 is the output of the controller, and d is the duty cycle. Fi is the inverter
side current sensor (a low pass filter). A simple PI is chosen as a controller, but repetitive controllers, integrators in
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multiple rotating frames or resonant integrators can be considered as well [11]-[12], [16]-[18]. In [4] a lead-lag element is
suggested. In the present paper it is included in Fv, which contains the voltage capacitor sensor (a low pass filter) in
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addition to the lead-lag element (8).
Fi =
K
K ll ⋅ (τ z ⋅ s + 1)
1
(8)
; PI = K p + i ; Fv =
τi ⋅ s +1
s
(τ v ⋅ s + 1) τ p ⋅ s + 1
(
)
Digital control
emulator
(a)
i1_ref
PI
+
-
u0
+
+
Fv
1
vbu s
Converter model
v0
d
Ds
v Z3
v bus
i1
Y1
Z 3 ⋅ (Z 2 + Z g )
Z3 ⋅ Z 2 + Z g
Fi
(b)
i1_ref
PI
+
u0
1
vbu s
i1
Ds
v bus
Y1A D
Fi
Fig. 6.
(a) Control loop for the inverter side inductor current with feedback of the capacitor voltage for active damping. (b) Corresponding
control loop with the actively damped plant, Y1AD.
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In this work, a full-bridge single phase inverter is supposed, so the gain, vbus, is included in the converter model. This
term behaves as a variable gain and it is compensated through its inverse value in order to decouple the converter model
seen by the controller from the operating-point as in [21].
As it can be observed in Fig. 6 (a), a feedback path of the capacitor voltage for active damping purposes is
implemented. The transfer function between the inverter side current, i1, and the capacitor voltage, vZ3, is easily obtained
from Fig. 4 (b) if i1 is supposed to be an ideal current source:
Z 3 ⋅ (Z 2 + Z g )
vZ 3
= Z thAB =
i1
Z3 ⋅ Z 2 + Z g
(9)
Provided that the feedback path of the capacitor voltage is closed, the control scheme of Fig. 6 (b) is derived.
Accordingly, Y1AD is the actively damped plant, which is obtained as shown in (10). Its bode diagram is plotted in Fig. 5
Fo
(b), where the filter values of Table I (a) and control parameters of Table II are used.
Y1 AD =
(
 Z3 ⋅ Z 2 + Z g
1 − Fv ⋅ Ds ⋅ 
 Z3 + Z 2 + Z g

Z 2 + Z3 + Z g
)  ⋅ Y


=
1
(10)
ee
=
Y1
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(1 − Fv ⋅ Ds ) ⋅ Z 3 ⋅ ( Z 2 + Z g ) + Z1 ⋅ ( Z 3 + Z 2 + Z g )
D. Stability analysis.
In order to discuss the stability of the system of Fig. 6, the open loop transfer function has to be analysed. Three
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different open-loop transfer functions are considered so as to show whether the digital control emulator approximation
proposed in this article (4) performs adequately:
OLADz = PI T ⋅ z −1 ⋅ Y1 ADz ;
OLAD 2 = PI ⋅ Ds 2 ⋅ Y1 AD ⋅ Fi ;
OLAD1 = PI ⋅ Ds1 ⋅ Y1 AD ⋅ Fi ;
(11)
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In expression (11), OLADz is the open-loop transfer function in the z-domain, where PIT is the PI controller discretized
according to the Tustin rule and Y1ADz is the actively damped plant transfer function in the z-domain [6]. OLAD2 is the open
loop transfer function in which the digital control emulator has been approximated by the proposed transfer function (4).
Finally, OLAD1 is the open loop transfer function in which the digital control emulator has been approximated by the
traditional transfer function (2). Note that the digital control emulator appears not only in the open loop transfer function,
but also inside the actively damped plant Y1AD.
The three open-loop transfer functions of (11) are depicted in Fig. 5 (c). OLADz is considered to be the most accurate
since it is performed in the z-domain. It can be observed that the open-loop transfer function OLAD2 is close to OLADZ even
within the range of the crossover frequency and the Nyquist frequency, whereas OLAD1 differs in this range.
Consequently, with the proposed approximation, Ds2, the z-domain analysis might be avoided with good reliability.
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IV. MODELLING AND CONTROL OF N PARALLELED GRID-CONNECTED INVERTERS
A. System description.
Next, a set of N paralleled grid-connected inverters with an LCL filter is described. The dynamics of these inverters is
coupled due to the grid impedance. The circuit for current control design of the N paralleled inverters of Fig. 1 is shown
in Fig. 7, where Z1i (with i=1…N) are the inverter side inductor impedances; Z2i are the inverter grid side inductor
impedances; Z3i are the inverter capacitor impedances; and Zg is the grid impedance. Moreover, i1i are the inverter side
currents; i2i are the grid side currents; i3i are the capacitor currents; ig is the grid-injected current; vZ3i are the capacitor
voltages; v0i are the converter voltages; and vg is the voltage in the PCC. In mathematical expressions and circuits, n is
used instead of N, but both refer to the number of paralleled inverters.
i11
Z11
Fo
i 21
Z21
PCC i
g
Zg
i31
vZ31
v01
rP
vg
Z31
eg
Converter 1
i12
Z12
i 22
Z22
i32
ee
vZ32
v 02
Converter 2
i1n
Z32
rR
Z1n
i2n
Z2n
…
i3n
v0n
vZ3n
Fig. 7.
Circuit of N paralleled grid-connected inverters.
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Z3n
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IEEE-TPEL
In this paper, the technique proposed in [4] is chosen to describe the consequences derived from the grid impedance
coupling. Therefore, the inverter side currents, i1i, are controlled and the capacitor voltages, vZ3i, are measured in order to
implement the active damping strategy. The control variables are the converter voltages, v0i. The voltage source, eg,
represents the grid and behaves as a disturbance. In this work, only the inductor current tracking reference is discussed,
so the disturbance, eg, will be considered to be zero when describing the modelling and control of the N paralleled gridconnected inverters.
B. Modelling.
The dynamics of the N-paralleled grid connected inverters of Fig. 7 can be described with the multivariable control
theory (12):
9
IEEE-TPEL
i 1n = G(s) ⋅ v 0n
 i11   G11 G12
  
G22
i  G
⇔  12  =  21
...
...
...
  
i  G
 1n   n1 Gn 2
... G1n   v01 
  
... G2 n   v02  (12)
⋅
... ...   ... 
  
... Gnn   v0 n 
Thus, i1n is the output vector which contains the controlled variables, i1i; v0n is the input vector representing the control
variables, v0i; and G(s) is the transfer matrix, i.e., the matrix whose product with the input vector, v0n, yields the output
vector, i1n.
Generally, in PV plants all installed inverters have been manufactured by the same firm and are of the same sort. In
addition to this, demanding industrial standards constantly require narrower tolerances. Hence, the inverters in a PV plant
are assumed to be equal. Based on this assumption, the corresponding impedances of the LCL filter of each inverter have
the same value and are represented by Z1, Z2, and Z3:
 Z1 = L1 ⋅ s;
 Z = L ⋅ s;
2
 2
1
⇒ 
Z =
;
 3 C ⋅s
 Z = L3 ⋅ s;
g
 g
rP
Z11 = Z12 = ... = Z1n = Z1
Z 21 = Z 22 = ... = Z12 = Z 2
Z 31 = Z 32 = ... = Z 32 = Z 3
Fo
(13)
If the inverters are equal, the system adopts a characteristic symmetry: all diagonal elements of the transfer matrix,
ee
G(s), will be identical since each converter voltage, v0i, influences its own current, i1i, in an identical way. Therefore, all
diagonal elements of G(s) can be replaced by G11. Likewise, all non-diagonal elements of the transfer matrix, G(s), will
rR
be identical since each converter voltage, v0i, influences another converter current, i1j, (with i≠j) in an identical way.
Hence, all non-diagonal elements of G(s) can be replaced by G12.
... G12 

... G12 
... ... 

... G11 
(14)
iew
 G11 G12

Gii = G11 
G11
G
⇒ G(s) =  12
Gij (i ≠ j ) = G12 
...
...

G
G
12
 12
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Page 10 of 26
The elements G11 and G12 are calculated through the “superposition principle” and Thévenin equivalent circuits. The
diagonal element, G11, might be regarded as the transfer function between the inverter side current, i11, and its own
converter voltage, v01. Accordingly, G11 can be calculated if all the converter voltages, v0i, are supposed to be zero except
v01. Thus, the auxiliary circuit of Fig 8 (a) is derived from Fig 7. A circuit in which the only current is i11 and the only
voltage source is v01 is shown in Fig. 8 (b) where ZthCD is the Thévenin-equivalent impedance calculated across the
terminals CD. The diagonal element, G11, such as the transfer function between i11 and v01, is directly obtained (15):
G11 =
1
i11
=
v01 Z1 + Z thCD
(15)
10
Page 11 of 26
Z1
i11
C
Z2
i21
Zg
(a)
i31
v01
Z2/(n-1)
Z3
Z1/(n-1)
Z 3/(n-1)
D
Converter 1
Z1
i11
ZthCD
C
(b)
v01
D
Converter 1
Fig. 8.
(a) Auxiliary circuit of the N paralleled inverters provided that all the converter voltages v0i are zero except v01. (b) Thévenin-equivalent
circuit across the terminals CD.
The non-diagonal element, G12, might be interpreted as the transfer function between the inverter side current, i11, and
Fo
the converter voltage of a paralleled inverter, v02. Accordingly, G12 can be calculated if all the converter voltages, v0i, are
supposed to be zero except v02. The auxiliary circuit shown in Fig 9 (a) is derived from Fig 7. A circuit in which the only
rP
current is i11 and the only voltage source is v02.ZEF is shown in Fig. 9 (b); the Thévenin-equivalent circuit is calculated
across the terminals EF. The Thévenin-equivalent impedance is ZthEF. Hence, the non-diagonal element G12, which for
ee
instance might be the transfer function between i11 and v02, is directly obtained (16):
G12 =
i11
Z EF
=−
v02
Z1 + Z thEF
rR
(16)
i11
Z1
E
(a)
i21
Z2
i31
Z3
Z1
i22
Z2
i32
v02
iew
F
Converter 1
i12
Zg
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IEEE-TPEL
Z2/(n-2)
Z3
Z3/(n-2)
Z1/(n-2)
Converter 2
i11
Z1
E
ZthEF
(b)
v 02.ZEF
F
Converter 1
Fig. 9.
(a) Auxiliary circuit of N paralleled inverters provided that all the converter voltages, v0i, are zero except v02. (b) Thévenin-equivalent
circuit across the terminals EF.
11
IEEE-TPEL
The expanded expressions of G11 and G12 are rather complicated and are shown in the appendix. G11 and G12
denominators coincide, but not their numerators. This is to be expected since the poles of a multivariable system, though
not the zeros; have to be the same regardless the input [23].
So far, it has been considered that the corresponding impedances of each inverter have the same value, since all
inverters installed in a PV plant are generally of the same sort. Going ahead with the assumption that all the installed
inverters are equal, not only their impedances, but also their hardware, their software, and their PV generators; it is
reasonable to assume that they will react all in the same way. In this scenario, the converter voltages of all inverters may
be considered equal, i.e., v0i=v0. Replacing this in (12), we obtain:
 i1   G11 G12
  
 i1   G12 G11
 ... =  ...
...
  
i  G
G
12
 1   12
... G12   v0 
  
... G12   v0 
⋅
... ...   ... 
  
... G11   v0 
Fo
(17)
rP
Multiplying a G(s) row by vector v0n (17), we derive the simplification Y1eq (19).
i1 = (G11 + (n − 1) ⋅ G12 ) ⋅ v0 = Y1eq ⋅ v0
Y1eq =
(
)
Z3 + Z 2 + n ⋅ Z g
i1
=
v0 Z 3 ⋅ Z 2 + n ⋅ Z g + Z 1 ⋅ Z 3 + Z 2 + n ⋅ Z g
(
)
(
(18)
ee
)
(19)
If the expression of Y1eq (19) is compared with the expanded version of G11 and G12 (shown in the appendix), it is
rR
observed that Y1eq is much simpler. The transfer function Y1eq can be identified with that of the single inverter case (6).
The only difference between both expressions, Y1 and Y1eq, is that in Y1eq the grid impedance is multiplied by n. In fact, if
ev
all converter voltages are equal, v0i=v0, an equivalent single inverter whose grid impedance is N times bigger represents
the N inverters. In other words, an inverter in a PV plant sees a grid impedance N times bigger. This simplification is
iew
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Page 12 of 26
referred to as “Equivalent Inverter” and its corresponding circuit is shown in Fig. 10.
Z1
i1
Z2
i2
n.Zg
ig /n
i3
v0
Fig. 10.
Z3
vg
eg
Circuit of the “Equivalent Inverter” which models the N inverters provided that they are equal.
The resonance frequency, ωreq, and the anti-resonance frequency, ωaeq, of the plant Y1eq depend on the number of
inverters, N. Their expressions are respectively:
ω req =
1
(
C3 ⋅ L1 ⋅ L2 + n ⋅ Lg
)
ωaeq =
(
1
C3 L2 + n ⋅ Lg
)
(20)
L1 + L2 + n ⋅ Lg
12
Page 13 of 26
The bode diagram of Y1eq with the LCL filter values introduced in Table I (a) is plotted in Fig. 12 (a) for different
values of N. Notice that, if the grid inductance in a certain point of the distribution network is Lg=0,1mH,
and the number of inverters is N=1200 (as in Fig. 12 (a)), an inverter in a PV plant will see N.Lg=120mH, whereas a
single inverter would only see Lg=0,1mH. This is the reason why control laws for inverters which have been designed
ignoring the grid impedance coupling do not perform as desired.
The “Equivalent Inverter” proves that the number of inverters installed in a PV plant has a great impact on the
functioning of the inverters due to the grid impedance coupling.
C. Control strategy.
Fig 11 (a) shows the multivariable control loop corresponding to the N paralleled grid-connected inverters with LCL
Fo
filter coupled due to the grid impedance in PV plants of Fig. 7 This is the MIMO (Multiple Input, Multiple Output)
version of the SISO (Single Input, Single Output) control loop of Fig 6 (a).
u0n
-
+
-
+
ee
(a)
…
A(s)
vZ3
Fv(s)
rR
i1n
u0n
i1n_ref
-
J(s)
…
-
ev
PI(s)
-
…
…
(b)
G(s)
…
…
PI(s)
-
i1n
v0n
+
…
rP
i1n_ref
…
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IEEE-TPEL
Fig. 11. (a) Multivariable control loop for the inverter side inductor currents with feedback of the capacitor voltages for active damping. (b)
Corresponding control loop with the actively damped matrix, J(s).
Some of the elements in Fig. 11 (a) have been previously introduced. The unknown elements are listed now: i1n_ref is
the vector containing the inverter side current references; u0n is the vector representing the controllers output; and vZ3 is
the vector which includes the capacitor voltages. Moreover, PI(s) is the matrix which contains the controllers, PI’; Fv(s)
is the matrix which contains the voltage sensors, Fv’; and A(s) is the matrix representing the transfer relation between the
inverter side currents, i1n, and the capacitor voltages, vZ3.
13
IEEE-TPEL
 i11ref

 i12ref
=
...

 i1nref


 u 01 
 vZ 31 






 u02 
 vZ 32 
i 1n_ref
 u 0n =  ...  v Z3n =  ... 





u 
v 

 0n 
 Z 3n 

 Fv′ 0
 PI ′ 0 ... 0 



′
0
P
I
...
0
 0 Fv′


PI(s) = 
Fv (s) = 

... ...
... ... ... ...



 0

0 0
′
0
...
P
I



(21)
0

0
... 

... Fv′ 
...
...
...
The matrices PI(s) and Fv(s) are diagonal because the inverters are completely independent and there is no
communication signal between inverters. Each converter has its own controller in order to control its own inverter side
current, but no decoupling controllers are implemented. In the same way, each converter only has access to its own
capacitor voltage. Since the inverters installed in a PV plant are expected to be equal, the controllers, PI’, and the voltage
Fo
sensors, Fv’, are the same for them all.
If the MIMO loop in Fig. 11 (a) is compared with the SISO loop in Fig. 6 (a), some differences are revealed. In the
rP
MIMO system, the variable gain, vbus, has been suppressed since it is compensated through its inverse value as in [21].
Apparently, in the MIMO system, the inverter side current sensor, Fi, and the digital control emulator, Ds, are not
ee
included. They should be in two independent diagonal matrices but they can be integrated in PI(s) and Fv(s) provided that
their elements are:
rR
PI ′ = PI ⋅ Ds ⋅ Fi ; Fv′ = Fv ⋅ Ds ;
(22)
Based on the assumption that in a PV plant all installed inverters are equal, the transfer matrix A(s), which relates the
ev
inverter side current vector, i1n, and the voltage capacitor vector, vZ3, acquires the characteristic symmetry of this
application. Hence, its diagonal elements are A11 and its non-diagonal elements are A12.
v Z3 = A(s) ⋅ i 1n
 vZ 31   A11

 
v   A
⇔  Z 32  =  12
...
...

 
v   A
 Z 3n   12
A12
A11
...
A12
... A12   i11 
  
... A12   i12  (23)
⋅
... ...   ... 
  
... A11   i1n 
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Page 14 of 26
The elements A11 and A12 are obtained by the “superposition principle” and Thévenin equivalent circuits. In the
following reasoning, all the converter voltages v0i are considered to be zero. The diagonal element, A11, can be regarded
as the transfer function between the capacitor voltage, vZ31, and the inverter side current, i11. Accordingly, A11 can be
obtained if the inverter side currents, i1i, are supposed to be ideal current sources whose value is zero for all of them
except for i11. The corresponding auxiliary circuit has been omitted. The transfer function between vZ31 and i11 is:
A11 =
(
(
v Z 31 Z 3 ⋅ Z 2 + Z 3 ⋅ Z g + Z 2 ⋅ Z 3 + n ⋅ Z g
=
i11
(Z 2 + Z 3 ) ⋅ Z 3 + Z 2 + n ⋅ Z g
(
)
))
(24)
The non-diagonal element, A12, might be regarded as the transfer function between the capacitor voltage, vZ31, and the
inverter side current, i12. A12 can be calculated if the inverter side currents, i1i, are supposed to be ideal current sources
14
Page 15 of 26
whose value is zero for all of them except for i12. The corresponding auxiliary circuit has been omitted. The transfer
function between vZ31 and i12 is:
A12 =
Z 32 ⋅ Z g
vZ 31
=
(Z 2 + Z 3 ) ⋅ Z 3 + Z 2 + n ⋅ Z g
i12
(
)
(25)
Each element of the MIMO loop in Fig. 11 (a) has been described. If the feedback path of the capacitor voltages for
active damping is closed, the control scheme of Fig. 11 (b) is derived from Fig. 11 (a). Note that Fig. 11 (b) is the MIMO
version of Fig. 6 (b). In this sense, J(s) is the actively damped matrix and is obtained as pointed in (26), (27).
i 1n = J(s) ⋅ u 0n
 i11   J11
  
i   J
⇔  12  =  12
...
...
  
i   J
 1n   12
J12
J11
...
J12
... J12   u01 

 
... J12   u02 
⋅
... ...   ... 

 
... J11   u0 n 
Fo
J(s) = [I - G(s) ⋅ Fv (s) ⋅ A(s)]−1 ⋅ G(s)
(26)
(27)
rP
The I element in (27) is the identity matrix. Based on the assumption that in a PV plant all the installed inverters are
equal, all matrices involved in the calculation of J(s) have the characteristic symmetry of this application, i.e., all its
diagonal elements are identical and all non-diagonal elements are also identical but different to the diagonal ones.
ee
Fortunately, the product and inverse operations of this kind of matrices preserve the characteristic symmetry as shown in
rR
the appendix, so the J(s) matrix acquires this symmetry. Hence, its diagonal elements are J11 and its non-diagonal
elements are J12. The expressions of J11 and J12 are rather complicated and are included in the appendix. Clearly, their
denominators are equal but not their numerators.
ev
Since the inverters are expected to be equal, the output of every PI controller may be considered equal, i.e., u0i=u0.
Replacing in (26), we obtain:
 i1   J11
  
 i1   J12
 ... =  ...
  
i   J
 1   12
J12
J11
...
J12
... J12   u 0 
  
... J12   u 0 
⋅
... ...   ... 
  
... J11   u 0 
(28)
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IEEE-TPEL
Multiplying a J(s) row by vector u0n in (28), we derive the simplification Y1ADeq (30).
i1 = ( J11 + (n − 1) ⋅ J 12 ) ⋅ u 0 = Y1 ADeq ⋅ u 0
Y1 ADeq =
Z2 + Z3 + n ⋅ Z g
(1 − Fv′ ) ⋅ Z 3 ⋅ (Z 2 + n ⋅ Z g ) + Z1 ⋅ (Z 3 + Z 2 + n ⋅ Z g )
(29)
(30)
If the expression of Y1ADeq (30) is compared with those of J11 and J12 (shown in the appendix) it is observed that Y1ADeq
is much simpler. Again, the transfer function Y1ADeq can be identified with that of the single inverter case (10). The only
difference between both expressions, Y1ADeq and Y1AD, is that in Y1ADeq the grid impedance appears multiplied by n. In
consequence, if all the outputs of the controllers are equal, u0i=u0, an “Equivalent Inverter” whose grid impedance is N
15
IEEE-TPEL
times bigger (Fig. 10) models the N inverters to which the active damping technique of [4] is applied. The bode diagram
of Y1ADeq with the LCL filter values compiled in Table I (a) and the control parameters gathered in Table II is plotted in
Fig. 12 (b) for different values of N.
(a)
(b)
(c)
Fig. 12.
rP
Fo
Bode diagrams for different values of N: (a) Y1eq; (b) Y1ADeq; (c) OLADeq;
ee
D. Stability analysis.
In order to discuss the stability of the system, the closed loop poles have to be analysed. Hence, the matrix T(s) that
rR
relates the current reference vector, i1n_ref, and the inverter side current vector, i1n, is obtained (31), (32). Based on the
assumption that in a PV plant all the installed inverters are equal, all matrices involved in the calculation of T(s) have the
characteristic symmetry of this application. Since the product and inverse operations of this kind of matrixes preserve the
ev
characteristic symmetry, the closed loop transfer matrix T(s) acquires this symmetry. Hence, its diagonal elements are T11
and its non-diagonal elements are T12.
i 1n = T(s) ⋅ i 1n_ref
 i11   T11 T12
  
 i12   T12 T11
⇔ =
...
... ...
  
 i  T
 1n   12 T12
... T12   i11ref 

 
... T12   i12ref  (31)
⋅
... ...   ... 

... T11   i1nref 
T(s) = [I + J(s) ⋅ PI(s)]−1 ⋅ J(s) ⋅ PI(s)
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(32)
From expression (32), the diagonal element, T11, and the non-diagonal element, T12, are calculated
T11 =
T12 =
((
)
)
2
2
PI ′ ⋅ J 11
− (n − 1) ⋅ J12
⋅ PI ′ + J11 ⋅ (1 + (n − 2 ) ⋅ J12 ⋅ PI ′)
(33)
(1 + (J11 − J12 ) ⋅ PI ′) ⋅ (1 + (J11 + (n − 1)⋅ J12 )⋅ PI ′)
J12 ⋅ PI ′
(1 + (J11 − J12 )⋅ PI ′)⋅ (1 + (J11 + (n − 1) ⋅ J12 )⋅ PI ′)
(34)
These expressions, (33) and (34), lead to too high order transfer functions, difficult to handle. As expected, their
denominators coincide, since the poles of any multivariable system have to be the same regardless the input, but not their
zeros. Notice that the terms in the denominators of T11 and T12 are the closed loop poles of the system. In this work, they
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are classified as “external stability poles” and “internal stability poles” (35). If and only if the external and the internal
stability poles are all in the left half s-plane, the system will be stable:
(1 + (J11 + (n − 1) ⋅ J12 )⋅ PI ′) ⇒ External Stability poles
(1 + (J11 − J12 ) ⋅ PI ′) ⇒ Internal Stability poles
(35)
If the inverters are equal, including their hardware, their software, and their PV generators, they will react all in the
same way. Thus, their inverter side current references will be equal, and so will be the inverter side currents, i1i=i1:
 i1   T11 T12
  
 i1   T12 T11
 ... =  ... ...
  
 i  T
 1   12 T12
... T12   i1ref 

 
... T12   i1ref 
⋅
... ...   ... 

... T11   i1ref 
(36)
Multiplying a T(s) row by vector i1n_ref in (36), we derive the simplification (37).
i1 = (T11 + (n − 1) ⋅ T12 ) ⋅ i1ref =
(J11 + (n − 1) ⋅ J12 ) ⋅ PI ′ ⋅ i
1ref
1 + ( J11 + (n − 1) ⋅ J12 ) ⋅ PI ′
rP
=
Fo
(37)
A simpler transfer function than T11 and T12 is obtained in (37). This is because some elements in the numerators of T11
and (n-1).T12 combine and cancel terms of their denominators. The remaining term corresponds to the external stability
ee
poles, whereas the cancelled term corresponds to the internal stability poles.
Using (29) and replacing in (37), we obtain:
i1 =
Y1 ADeq ⋅ PI ′
⋅ i1ref =
OLADeq = Y1ADeq ⋅ PI ′
OLADeq
1 + OLADeq
⋅ i1ref
(38)
(39)
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1 + Y1 ADeq ⋅ PI ′
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The expression (38) represents the closed-loop transfer function of an “Equivalent Inverter” whose grid impedance is
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N times bigger (Fig. 10) and in which the control strategy of [4] is implemented. Thus, the external stability poles are
related with the “Equivalent Inverter” and might be analyzed representing the bode diagram of the open-loop transfer
function, OLADeq (39), as in Fig. 12 (c). Note that the open-loop transfer function OLADeq not only is useful to determine
the external stability, but also to design the PI controller and to fix the bandwidth and the phase margin of N coupled
inverters. Furthermore, the control law design of N coupled inverters greatly resembles the single grid-connected inverter
case of Section III. The only difference is that a grid impedance N times bigger has to be taken into account.
In order to discuss the internal stability, the term of the T11 and T12 denominator cancelled when all current references
are equal, has to be analysed. Remember that unstable poles cannot be cancelled [22]. Consequently, if the internal
stability term has poles in the right half s-plane, the system will be unstable even if all current references are equal.
If attention is paid to (35), it can be observed that the internal stability poles can be identified with the closed loop
poles of a hypothetical SISO system, H, as expressed in (40).
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H=
(J11 − J12 ) ⋅ PI ′ = OLH
1 + (J 11 − J 12 ) ⋅ PI ′ 1 + OLH
(40)
Thus, representing the open-loop bode of this hypothetical system, OLH, it is possible to know whether the internal
stability term, (1 + OLH ) , has right-half s-plane poles. The plant of this hypothetical system is shown in (41).
J 11 − J12 =
Z2 + Z3
′
(1 − Fv )⋅ Z 3 ⋅ Z 2 + Z1 ⋅ (Z 2 + Z 3 )
(41)
Comparing expressions, we observe that fortunately expression (41) is equal to the Y1ADeq expression if N=0. It is
difficult to find a physical sense to this fact, since N=0 means that any inverter is placed in the PV plant. However the
stated is true:
Y1 ADeq
=
n=0
= ( J11 + (n − 1) ⋅ J12 ) n=0 =
Fo
(42)
Z 2 + Z3
= J −J
(1 − Fv′ ) ⋅ Z3 ⋅ Z 2 + Z1 ⋅ (Z 2 + Z3 ) 11 12
Consequently, equations (40) and (42) confirm that the internal stability can easily be determined by means of the open
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loop of the “Equivalent Inverter,” OLADeq, provided that N=0 (43). Thus, the internal stability can be analyzed in Fig. 12
(c), in which the first bode plot corresponds to OLADeq when N=0.
OLH = OLADeq
n =0
ee
(43)
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Note that the internal stability neither depends on the number of inverters, N, nor on the grid impedance, Zg. Thus, the
internal stability problem is the same whether in a certain PV plant there are many or few inverters, or whether the grid is
stiff or weak, whereas the external stability problem is completely different.
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V. SIMULATION RESULTS
Next, simulation results are analyzed so as to validate the theoretical study of previous sections. The simulations are
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performed with the software PSIM. Two different ways of simulating the N inverters are considered. The first one is to
implement N independent inverters as in Fig. 7. This is the most detailed simulation, since any simplification is done.
However, it is tedious if the number of inverters is large. A second way is to implement the “Equivalent Inverter” of Fig.
10. Only N=3 inverters are considered. They are supposed to be full-bridge single phase inverters of 10kW with unipolar
modulation. All inverters are given the same current reference value, i1ref. A grid effective voltage eg=230 V is supposed
in these simulations.
Firstly, the LCL filter values of Table I (a) and the control parameters of Table II are chosen. In Fig. 13 (a) the open
loop bode diagrams are plotted. The N=3 bode plot shows that the system is externally stable. The N=0 bode plot shows
that the system is internally stable. Consequently, the system is stable since it is both, externally and internally stable. The
simulated waveforms of this system are shown in Fig. 13 (b). The suffix eq is added to the “Equivalent Inverter”
variables in order to distinguish them. First, the inverter side current i11 of one of the N independent inverters and the
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inverter side current, i1_eq, of the “Equivalent Inverter” are shown. Second, the sum of the inverter side currents of the N
independent inverters, i11+i12+i13, and the inverter side current of the “Equivalent Inverter” multiplied by N are shown.
Last, the duty cycles of one of the N independent inverters, d1, and that of the “Equivalent Inverter”, d1_eq, are shown. In
this simulation, a step change occurs in the effective current reference value at t=45ms. At every instant, the N
independent inverter variables and the “Equivalent Inverter” variables are overlapped. This is because the two ways of
simulating are similar provided that the system is internally stable and the same current reference value is given.
(a)
(b)
Fig. 13.
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System externally and internally stable (a). Theoretical analysis, OLADeq (b) Simulated waveforms.
Secondly, the LCL filter values of Table I (b) and the control parameters of Table II are chosen. In Fig. 14 (a) the open
loop bode diagrams are plotted. The N=3 bode plot shows that the system is externally stable. However, the N=0 bode
plot shows that the system is internally unstable since insufficient damping of the resonance frequency is obtained.
Consequently, the system is unstable. The simulated waveforms of this system are shown in Fig. 14 (b). The depicted
variables are the same as in Fig 13 (b). The information of the internal stability poles is lost in the “Equivalent Inverter”
simplification, but not in the N independent inverter simulation. Since the system is internally unstable, the variables of
the N independent inverters will diverge and the duty cycles will saturate. The behaviour of the inverters in saturation is
non-linear and, as a result, the bode diagrams are not valid. In the simulation of Fig. 14 (b) the PWM modulators have
been replaced by ideal voltage sources. Thus, the switching effect is not taken into account but saturation is avoided and
the internal instability can be depicted clearer. If the inverter side current of one of the N inverters, i11, and the inverter
side currents of the “Equivalent Inverter”, i1_eq, are compared, it can be observed that they coincide initially, but at the
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end, current i11 diverges at a resonant frequency due to internal instability. This resonant frequency is that of the N=0
bode plot of Fig. 14 (a). Although not shown in the figure, the currents i12 and i13 also diverge at this resonant frequency.
In contrast, the sum of the currents of the N independent inverters, i11+i12+i13, and the “Equivalent Inverter” current
multiplied by N, n.i1_eq, coincide at every instant. This is because the currents i11, i12, i13 have two components. The first
component, equal for the N independent inverters, is at 50Hz. This component appears in the sum i11+i12+i13 and is
injected in the grid as a result. The second component is at the resonance frequency between inverters which diverges due
to internal instability. This last component goes from the reactive elements of one inverter to the reactive elements of
another inverter and it does not circulate through the grid. That is why it does not appear in the sum i11+i12+i13.
(a)
ee
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Fo
(b)
Fig. 14.
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System externally stable but internally unstable (a). Theoretical analysis, OLADeq (b) Simulated waveforms without saturations.
In Fig. 14 (b), the variables of one of the N independent inverters and the “Equivalent Inverter” coincide initially only
because the same reference value is given. If different current reference values were given to the N independent inverters
in this simulation, the internal stability poles would not tend to cancel and the internal instability would become apparent
from the first instant.
These simulations validate the theoretical results obtained in previous sections. The “Equivalent Inverter” describes the
N inverters of a PV plant. Only the internal stability poles information is lost but it can be studied theoretically in the
bode diagram. The internal stability neither depends on the number of inverters, N, nor on the grid impedance, Zg. In
consequence, only two paralleled inverters have to be programmed in order to simulate the internal stability.
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VI. EXPERIMENTAL RESULTS
The PV inverters which had a resonant behaviour in Fig. 2 were designed ignoring the coupling effect. These inverters
were 5kW full-bridge single phase Ingecon®Sun inverters with unipolar modulation. They satisfied individually all the
standards and performed correctly in other PV plants.
These inverters had an isolation transformer, whose leakage impedances can be neglected, and an LC output filter
instead of an LCL filter. Considering L2=0, we can adapt the study carried out in this article to the LC filter case. Note
that if L2=0 the voltage in the PCC, vg, and the capacitors voltages, vZ3i, coincide. The filter values are shown in Table III.
The total number of installed inverters in the PV plant was 270. The inverters were connected to a three-phase 13kV
network. These single-phase inverters were distributed in three equal phases. In consequence, N=270/3=90 has to be
considered in the “Equivalent Inverter” model. The grid impedance was the leakage impedance of the 1500kW
Fo
distribution transformer. It is calculated according to (44) using the transformer parameters shown in Table III.
Lg =
rP
vcc
v2
0.06
400 2
⋅ 1L =
⋅
= 0.0068 mH
2 ⋅ π ⋅ 50 3 ⋅ PNT 2 ⋅ π ⋅ 50 3 ⋅1500
TABLE III
PV POWER PLANT CHARACTERISTICS
Parameter
PNP
Value
Parameter
PV Plant
1400 kW
N
(44)
ee
Value
270/3=90
PNI
L1
Ingecon®Sun Inverter
5 kW
L2
3 mH
C3
0
6.6 µF
PNT
vcc
Distribution Transformer
1500kW
v1L
0.06
v2L
400 V
13000 V
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The cable impedances of the PV plant and most parasitic impedances were neglected. These parasitic impedances
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mainly affect the high frequency range and are difficult to estimate. Moreover, the resonance frequencies associated to
these parasitic impedances, higher than those of the LCL filter, are correctly damped due to the skin effect and the
magnetic effect [2].
After studying the coupling effect, the resonance problem of the PV plant shown in Fig 2 was identified as an external
stability problem. It can be observed, in Fig. 15 (a), that the whole system was externally stable but near instability. There
was insufficient damping of the resonance and changes in the grid voltage, eg, and in the reference current, i1_ref, excited
the resonance frequency. This exceeded the harmonic regulations and the breakage of many devices such as the
electronic meters. No internal stability problems occurred since LCL filter resonance does not exist provided that L2=0
and N=0. Fig. 15 (b) and (c) shows the corresponding waveforms of the “Equivalent Inverter” simulated according to the
parameters of Table III. First, in Fig. 15 (b), the inverter side current, i1_eq, and its reference value, i1ref_eq are shown.
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Second, the grid side current, i2_eq, and the voltage in the PCC, vg_eq are shown. The current i2_eq is scaled by a factor of 10
in order to make it visible.
(a)
Fo
(b)
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Fig. 15.
PV plant with resonant behaviour (a). Theoretical analysis, OLADeq (b) Simulated waveforms (c) Half grid period detail.
Finally the duty cycle, d1_eq is depicted. Fig. 15 (b) details a half grid period of i2_eq and vg_eq. These simulations
reproduce the measured waveforms of Fig. 2. The resonance oscillation occurs at 2777 Hz. It corresponds to the
resonance frequency, ωreq, of the “Equivalent Inverter” and it is easily obtained by replacing the LC filter values of the
Ingecon®Sun inverter (Table III), the number of inverters, N, and the grid impedance value (44) in expression (45):
ω req =
1
(
C3 ⋅ L1 ⋅ L2 + n ⋅ Lg
)
= 17265 rad/s → 2745 Hz (45)
L1 + L2 + n ⋅ Lg
The “Equivalent Inverter” permitted to describe the resonance problem and to design a suitable solution for the single
phase Ingecon®Sun inverters.
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VII. CONTROL DESIGN GUIDELINES AND PRACTICAL USES OF THE “EQUIVALENT INVERTER”
In this paper, PI controllers are considered, and the damping technique proposed in [4] is chosen, but the methodology
followed in this article can also be applied to other control algorithms or active damping techniques. The authors have
essayed several of the active damping proposals, such as [4]-[14], to control the inverter side current, i1, or the grid side
current, i2, and the same “Equivalent Inverter” model has been derived. In relation to the external stability and the internal
stability problem, analogous results are obtained regardless of the control strategy: the open loop transfer function of the
“Equivalent Inverter” determines the external stability. The internal stability is studied replacing N=0 in the open loop
transfer function expression of the “Equivalent inverter”. In consequence, simplifications derived in this article depend on
the symmetry of the system, but not on the control strategy.
In this study, the number of inverters, N, has been varied to depict the coupling effect (Fig. 12). However, the loop
Fo
characteristics in Fig. 12 do not truly depend on the number of inverters N itself. The really important thing is the product
N.Lg, since all the different values of N and Lg whose product is the same have the same loop characteristics. In other
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words, the loop characteristics are the same whether a PV plant has the values Lg=0,1mH, N=1200, as plotted in Fig. 12,
or the values Lg=0,3mH, N=400, since their products coincide. Fig. 16 displays several curves of constant N.Lg. Points
ee
whose bode diagram is plotted in Fig 12 are marked, but points of the same curve have the same bode plots.
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Fig. 16.
Curves whose points have the same loop characteristics since their products coincide.
The stiffer the grid is, the more paralleled inverters can be placed. In a weak grid, less paralleled inverters have to be
placed in order to obtain the same loop characteristics. Thus, the control law has to be designed to perform in a certain
range of N.Lg. For the external stability requirement, there is a maximum value for N.Lg=NLmax. In order to satisfy the
internal stability requirement, the product value N.Lg=0 has to be taken into account. Due to the parameter uncertainty,
the most sensible is to design to be stable within the whole range N.Lg=[0, NLmax].
The resonance frequencies of the “Equivalent Inverter” (20) depend on the product value N.Lg. If an active strategy is
required, it is essential to be able to estimate the resonance frequency, ωreq. If the product N.Lg increases, ωreq decreases
down to a limited value. As a result, the resonance frequency will always be between its minimum (when N.Lg =∞) and
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its maximum value (when N.Lg=0) as expressed in (46). Thus, the range of variation of the resonance ωreq is well defined.
This eases the implementation of the damping strategy.
1
C3 ⋅ L1
≤ ω req ≤
1
(46)
C3 ⋅ L1 ⋅ L2
L1 + L2
On the contrary, the anti-resonance ωaeq, decreases unlimitedly if the product value N.Lg increases. Therefore, the antiresonance frequency will always be between 0Hz (when N.Lg =∞) and its maximum value (when N.Lg =0) as expressed
in (47). This limits the bandwidth achievable since the crossover frequency is reduced as N.Lg increases (Fig 12 (c)). Note
that, if resonant controllers or similar were used and the coupling effect was not taken into account, the resonant
frequencies of these controllers could surpass the bandwidth of the system. In consequence, the system would be unstable
as pointed by [16].
1
C3 ⋅ L2
rP
0 ≤ ωaeq ≤
Fo
(47)
Different control algorithms and active damping techniques can be tested in simulations without programming the total
N inverters of the plant thanks to the “Equivalent Inverter”. Furthermore, the “Equivalent Inverter” can be given many
ee
experimental uses. The behaviour of a certain control strategy in a real PV inverter can be predicted without having to
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install the N inverters in a PV plant. Simply, a single real inverter has to be connected to the grid through an auxiliary
inductor, Laux, whose value has to be varied within the range of the product N.Lg, under consideration i.e., Laux=[0, NLmax].
The internal stability of a real inverter can be tested if Laux=0, but this may be difficult to meet since in every connexion
ev
point a grid impedance always exists. However, note that only two inverters installed in the same PCC are needed in
order to test the internal stability. This is because the internal stability neither depends on the number of inverters, N, nor
in the grid impedance, Zg.
VIII. CONCLUSIONS
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This paper analyses the modelling and control of N paralleled grid-connected inverters with LCL filter in PV plants.
An “Equivalent Inverter” which models the N inverters of a PV plant is obtained. The internal stability poles information
is lost but it can be studied theoretically through the bode diagram and it can be simulated provided that only two
paralleled inverters are programmed. Thus, the coupling effect due to grid impedance is described and easy control
design guidelines are suggested. Although a specific control strategy is analysed, based on PI control and the active
damping technique proposed in [4], the “Equivalent Inverter” simplification can be obtained regardless of the control
strategy. The study is validated by simulated and experimental results.
APPENDIX
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G11 =
) (
(
(Z 2 ⋅ Z 3 + Z1 ⋅ (Z 2 + Z 3 ))⋅ (Z 3 ⋅ (Z 2 + n ⋅ Z g ) + Z1 ⋅ (Z 3 + Z 2 + n ⋅ Z g ))
(
Z1 ⋅ (Z 2 + Z 3 ) ⋅ Z 2 + Z 3 + n ⋅ Z g + Z 3 ⋅ Z 22 + (n − 1) ⋅ Z 3 ⋅ Z g + Z 2 ⋅ Z 3 + n ⋅ Z g
G12 = −
J 11 =
(
)
(16)
(
b
a
...
b
((1 − Fv ) ⋅ Z 2 ⋅ Z 3 + Z1 ⋅ (Z 2 + Z 3 )) ⋅ ((1 − Fv ) ⋅ Z 3 ⋅ (Z 2 + n ⋅ Z g ) + Z1 ⋅ (Z 3 + Z 2 + n ⋅ Z g ))
))
(1 − Fv )⋅ Z 32 ⋅ Z g
((1 − Fv )⋅ Z 2 ⋅ Z 3 + Z1 ⋅ (Z 2 + Z 3 )) ⋅ ((1 − Fv ) ⋅ Z 3 ⋅ (Z 2 + n ⋅ Z g ) + Z1 ⋅ (Z 3 + Z 2 + n ⋅ Z g ))
...
...
...
...
...
...
...
...
b   e f ... f   g
 
 
b   f e ... f   h
⋅
=
...  ... ... ... ...   ...
 
 
a   f f ... e   h
b

b
...

a 
−1
c

d
=
...

d

d
c
...
d
h
g
...
h
...
...
...
...
...
...
...
...
h

h
 g = a ⋅ e + (n − 1) ⋅ b ⋅ f ;
⇔ 
...
h = a ⋅ f + b ⋅ e + (n − 2) ⋅ b ⋅ f ;

g 
(48)
(49)
(50)
d
a + (n − 2) ⋅ b


c = a 2 + (n − 2) ⋅ a ⋅ b − (n − 1) ⋅ b 2 ;
d

⇔ 
b
...
d = −

;

c 
a 2 + (n − 2) ⋅ a ⋅ b − (n − 1) ⋅ b 2
REFERENCES
ee
[1]
(
rP
a

b
 ...

b

b
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...
b
(15)
(Z 2 ⋅ Z 3 + Z1 ⋅ (Z 2 + Z 3 )) ⋅ (Z 3 ⋅ (Z 2 + n ⋅ Z g ) + Z1 ⋅ (Z 3 + Z 2 + n ⋅ Z g ))
Z1 ⋅ (Z 2 + Z 3 ) ⋅ Z 2 + Z 3 + n ⋅ Z g + (1 − Fv ) ⋅ Z 3 ⋅ Z 22 + (n − 1) ⋅ Z 3 ⋅ Z g + Z 2 ⋅ Z 3 + n ⋅ Z g
J 12 = −
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
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
b

))
Z 32 ⋅ Z g
Fo
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