Adaptive Control for
Parallel-Connected Energy Storage
Converters with Line Parameter
Mismatch
Cheng Wang 1*, Hongting Hua 1, He Peng 1 and Ze Wang 2
1Intelligent Power Conversion Laboratory, Department of Electrical Engineering, Nanjing University of Science and Technology,
Nanjing, China, 2Control and Learning for Robotics and Autonomy Laboratory, Department of Mechanical and Energy
Engineering, Southern University of Science and Technology, Shenzhen, China Cheng Wang 1*, Hongting Hua 1, He Peng 1 and Ze Wang 2
1Intelligent Power Conversion Laboratory, Department of Electrical Engineering, Nanjing University of Science and Technology,
Nanjing, China, 2Control and Learning for Robotics and Autonomy Laboratory, Department of Mechanical and Energy
Engineering, Southern University of Science and Technology, Shenzhen, China Power electronics interfaced microgrid has become a major trend for modern power
systems. In this paper, a three-phase microgrid system formed by multiple distributed
energy storages (DES) converters is presented. To improve the reliability and ﬂexibility,
DESs are connected in parallel to feed the critical loads, and meanwhile, inject real power
to the medium voltage (MV) distributed grid and the local dc bus through an interfacing
smart transformer (ST). The ST is working at current control mode while the ac bus voltage
is regulated in a sharing way through the DESs. However, due to the physical differences
and different environment conditions, ﬁlter parameters are not likely to be matched for all
the DES modules. Accurate load sharing is therefore the ﬁrst challenging target. Furthermore, both DES module and ST can fail in event of single-point fault. Fast bus
voltage restoring thus becomes the second challenging target. In this paper, the parameter
mismatch is considered within the system modeling process. A control framework is then
presented in form of a virtual current controller and a backstepping adaptive voltage
controller. Adaptive laws are designed to fully compensate for the unknown dynamics
within the system transient response. Convergence and boundedness of the signals in the
closed-loop control system are demonstrated through rigorous Lyapunov-based stability
analysis. Simulation
results
and
experimental
results
are
presented
and
have
demonstrated the effectiveness of the proposed algorithm in terms of both fast bus
voltage restoring and accurate power sharing. Edited by: Edited by:
Liansong Xiong,
Nanjing Institute of Technology (NJIT),
China Reviewed by:
Zhen He,
Hangzhou Dianzi University, China
Zhigang Liang,
Shanghai Huawei Technology Co.,
Ltd., China *Correspondence:
Cheng Wang
chw714@njust.edu.cn *Correspondence:
Cheng Wang
chw714@njust.edu.cn Specialty section:
This article was submitted to
Process and Energy Systems
Engineering,
a section of the journal
Frontiers in Energy Research
Received: 03 September 2021
Accepted: 20 September 2021
Published: 20 October 2021 INTRODUCTION Microgrids are considered as the key building blocks of energy internets (Xiong et al., 2016; Harmon
et al., 2018; Marzal et al., 2019; Liu et al., 2021; Wang et al., 2021) and have aroused great attentions in
the last decade for their potential and the impact they may have in the coming future. Industrial/
residential Microgrids are deﬁned as low voltage systems comprised of loads, distributed generation
units and storage devices, which are connected to the mains at a single Point of Common Coupling
(PCC) (Sivarasu et al., 2015). Transformer is a key component to interface industrial/residential
microgrids with the medium-voltage (MV) distributed grid. A smart transformer (ST) can achieve
power factor correction, voltage sag ride through and harmonic elimination which are not available Keywords: paralleled-inverter, microgrids, parameter mismatch, adaptive control, energy storage ORIGINAL RESEARCH
published: 20 October 2021
doi: 10.3389/fenrg.2021.770372 Citation: Wang C, Hua H, Peng H and Wang Z
(2021) Adaptive Control for Parallel-
Connected Energy Storage Converters
with Line Parameter Mismatch. Front. Energy Res. 9:770372. doi: 10.3389/fenrg.2021.770372 October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 1 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 1 | Conﬁguration of ST enabled microgrids. FIGURE 1 | Conﬁguration of ST enabled microgrids. modules. Due to the stochastic nature of both the renewable
sources and the power consumed by the load, the inclusion of an
advanced control is highly recommended in order to improve the
system stability and its performance. To realize effective control of the
parallel distributions, there are two major control objectives. The ﬁrst
objective is from system point of view, which is to regulate the output
voltage to perfect sinusoid under nonlinear and fast-changing loading
conditions. The second objective is from distributions point of view,
which is to minimize the load current sharing errors among the DESs. As a system’s emergent power supply, good voltage regulation (root-
mean-square/RMS voltage and frequency) is necessary. The control
objective of voltage control can be realized by setting the voltage
control reference to a sinusoidal signal. Circulating currents are
current components ﬂowing among the DESs. Ideally, all currents
generated from DESs ﬂow to the loads. Not well controlled system
will have large circulating currents, which will degrade the energy
efﬁciency and cause unbalanced load sharing even system or module
failure in some conditions. The control objective on circulating
current suppression can be realized by controlling the output
currents to have the same proﬁle other than the magnitude as the
load current. During system upgrade, distributions of different
capacity and parameters might be used. In addition, parameters of
the real-world system might deviate from the nominal value and
change slowly over time. Control under these unbalanced and
unknown operating conditions is challenging. To meet the control
objectives, the controller must be very fast and accurate so as to
provide desired static and transient control performance. with a traditional line-frequency transformer (Huang, 2016;
Huber and Kolar, 2019; Milczarek and Malinowski, 2020). Frontiers in Energy Research | www.frontiersin.org Citation: _voa 

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−
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N
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N
j1Cjc
−
iloadc

N
j1Cjc
_ija  −1
Lja
voa −Rj
Lja
ija + 1
Lja
vja
_ijb  −1
Ljb
vob −Rj
Ljb
ijb + 1
Ljb
vjb
_ijc  −1
Ljc
voc −Rj
Ljc
ijc + 1
Ljc
vjc, j  1, 2, ..., N
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−
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N
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−
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N
j1ijc

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−
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⎩ Droop control is the most popular distributed control solutions
for power system applications (Yao et al., 2011). By adjusting the local
control reference based on predeﬁned droop characteristics,
coordination of multiple subsystems can be realized. Major
advantages
of
droop
control
include
no
inter-module
communication, easy to design and implement. However, droop
controls also suffer from various problems, such as unbalanced load
sharing, voltage/frequency control deviation, etc. Thus, additional
control modules are needed to correct these problems. To
signiﬁcantly improve control performance, advanced distributed
control algorithms are needed. Citation: ⎧
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⎩
_voα 
1
MCαcom

n
j1
ijα +
1
MCdiff

n
j1
ijβ −
1
MCαcom
iloadα −
1
MCdiff
iloadβ
_voβ 
1
MCdiff

n
j1
ijα +
1
MCβcom

n
j1
ijβ −
1
MCdiff
iloadα −
1
MCβcom
iloadβ
_ijα  −
1
MLαcom
voα +
1
MLdiff
voβ +
1
MRαcom
ijα +
1
MRdiff
ijβ +
1
MLαcom
vjα −
1
MLdiff
vjβ
_ijβ 
1
MLdiff
voα +
1
MLβcom
voβ +
1
MRdiff
ijα +
1
MRβcom
ijβ −
1
MLdiff
vjα −
1
MLβcom
vjβ
(2) ⎧
⎪
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_voα 
1
MCαcom

n
j1
ijα +
1
MCdiff

n
j1
ijβ −
1
MCαcom
iloadα −
1
MCdiff
iloadβ
_voβ 
1
MCdiff

n
j1
ijα +
1
MCβcom

n
j1
ijβ −
1
MCdiff
iloadα −
1
MCβcom
iloadβ
_ijα  −
1
MLαcom
voα +
1
MLdiff
voβ +
1
MRαcom
ijα +
1
MRdiff
ijβ +
1
MLαcom
vjα −
1
MLdiff
vjβ
_ijβ 
1
MLdiff
voα +
1
MLβcom
voβ +
1
MRdiff
ijα +
1
MRβcom
ijβ −
1
MLdiff
vjα −
1
MLβcom
vjβ
(2) (2) The rest of the paper is organized as follows. The dynamic
model of the Microgrid is introduced in System Conﬁguration and
Problem Description Section. The designed distributed control
algorithm is presented in Distributed Adaptive Controller Design
for DGs Section. The experimentation results are provided in
Simulation and Experiment Studies Section. Concluding remarks
are provided in Conclusion Section. where, MCαcom and MCβcom are virtually deﬁned as the total
capacitors for the α and β phases respectively. Similarly,
MLαcom and MLβcom are the total virtual inductors, MRαcom and
MRβcom are the total virtual resistors. And MCdiff is virtually deﬁned as the differential capacitor for
the α and β phases respectively. Similarly, MLdiff is the differential
virtual inductor, MRdiff is the differential virtual resistor. Their
expressions are listed in the appendix part. Citation: The
potentiality to use the ST in electric distribution as the enabling
technology for Microgrids functionalities is much higher: the ST
is supposed to replace the standard low-frequency transformer,
connecting the MV grid to the LV grid and offer dc-connectivity,
and services to Low-voltage and Medium Voltage grids
(Andresen et al., 2017; Kumar et al., 2018; Ye et al., 2018; Zou
et al., 2018; He et al., 2021). Therefore, the ST enabled ac
industrial/residential microgrid as shown in Figure 1 is
preferred in this paper. For
traditional
distribution
networks,
frequent
tap/switch
changing of the reactive components like transformers, capacitors
etc. is necessary to achieve fast response to power ﬂuctuation and
voltage deviation, however, will also reduce component’s lifetime
dramatically. As a promising alternative, the power electronics
converters (PEC) interfaced distributed generators are more
ﬂexible (Wang et al., 2019; Zhang et al., 2019; Pugliese et al.,
2020; Qian et al., 2020; Xiong et al., 2020). DES is an essential
PEC enabler for modern industrial/residential Microgrids, to convert
the ac or dc power produced by generators to ac powers at various
voltage levels and frequencies and DC powers at various voltage
levels, needed for supplying loads. In this way, distributed generations
are able to be inserted into or bypassed from the industrial/residential
bus. For high capacity applications, it is sometimes needed to use
multiple modular distributed units due to the limitation of energy
storage technology and the intimidating cost of a single large energy
storage (Zhang et al., 2021). To coordinate the control activities of multiple distributions,
centralized solutions are good choices due to their easy access of
global operating condition. In addition to the proportional-
integration
or
proportional-resonant
(PI/PR)
based
control
algorithms that are widely used in power electronics, other types
of control algorithms are also designed, including model predictive
control (Karami et al., 2021), robust model-following control (Pascual
et al., 2008), and sliding mode control (Darvishzadeh et al., 2012). The Using multiple distribution units is more ﬂexible and less
susceptible against single-point failures. This is also a cost-effective
solution for demands that grow through time. In this way, the parallel
system can generate higher current to meet the large load demand. Citation: Despite the advantages, effective control of the parallel system is
challenging due to the dynamic loading conditions, physical
differences of the distributions, and the interactions of multiple October 2021 | Volume 9 | Article 770372 2 Parallel-Connected Converters Parameter Mismatch Control Wang et al. solar power system and wind power system), ST and variable
loads are considered connected to a common ac bus. The model
shown in Eq. 1 represents the dynamics of the R-L-C ﬁlters
connecting the N DESs to the point of common coupling. algorithms are evaluated through either simulation or hardware
implementation. The
algorithms
have
different
kinds
of
limitations, such as requiring known load proﬁles and matched
subsystem
modules. In
addition,
these
algorithms
are
not
extensively
tested
under
various
loading
conditions,
linear/
nonlinear, static/dynamic. It is known that centralized control
algorithms are inﬂexible and susceptible to single-point-failures. Distributed control has been shown as a compromise of fully
decentralized and centralized control to coordinate multiple
devices especially under sparse communication infrastructure
which is the usual case in today’s power distribution networks. Citation: Designing such control algorithms
require integrating multidisciplinary expertise especially in control
and power electronics, where big gaps exist in-between. Once such
control algorithms are designed and tested through ofﬂine
simulation, it is necessary to further test it through hardware
experimentation to evaluate its performance that are not foreseen
under ideal model-based simulation studies. (1) where iloadx x  a, b, c is the current including the current
injected to the legacy grid through the ST and the current
consumed by the load in the x (x  a,b or c) phase, vo is the ac
bus voltage. Taking the #j DG, for example, ijx is the ﬁlter inductor
current, and vjx is the control input, Rjx, Ljx, Cjx are the line
impedance and ﬁlter parameters. For a three-phase-three-wire
system, the control degrees are limited within two of the three
phases. By using the abc- αβ transformation, the system dynamics
can be expressed as In this paper, a distributed backstepping based control algorithm
is presented for the Microgrid system. Convergence and boundedness
of the signals in the closed-loop control system are demonstrated
through rigorous Lyapunov-based stability analysis. The designed
algorithm is distributed, works for unmatched DES modules and can
realize predeﬁned load sharing. The algorithm is implemented with
multiple DSP-based control boards and tested under various
operating
conditions,
including
balanced
and
unbalanced
distributions, linear and nonlinear loading conditions, constant
and variable loading conditions, etc. Experimentation results
demonstrate the effectiveness of the proposed control algorithm. The success of the designed control algorithm demonstrates that
signiﬁcantly improved control performance can be achieved by
introducing advanced control techniques to power electronics. SYSTEM CONFIGURATION AND PROBLEM
DESCRIPTION An
equivalent
model
of
the
paralleled-inverter
based
microgrid (taking one module for example), given in Figure 3,
can be developed according to Eq. 2. As seen from the model,
ﬁlter currents and ac bus voltages in phase-α are coupled with the
ones in phase-β. The coupling parts are represented by dependent
current sources in the circuit. Figure 2 illustrates the simpliﬁed conﬁguration of a ST enabled
Microgrid, where the line impedances are incomparable with the
output ﬁlter impedances. Multiple distributed generators (i.e. October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 3 Parallel-Connected Converters Parameter Mismatch Control Wang et al. FIGURE 2 | The simpliﬁed conﬁguration of a ST enabled parallel-connected DES based microgrid. FIGURE 2 | The simpliﬁed conﬁguration of a ST enabled parallel-connected DES based microgrid. modules should be regulated as little as possible, i.e., to
minimize ioi/ni −ioj/nj for all i, j  1, 2, . . . , n, j ≠i, where nj is
the load sharing ratio for module #j, satisfying 	n
j1 nj  1, and ioj
is the output current of module #j. Note
that,
more
modules
require
more
interfacing
line
parameters. In practical, it is impossible to obtain the exact values
of circuit parameters. Further, parameter drifts are very likely to
happen with aging of inductors or capacitors. This is also the reason
why each LCR ﬁlter parameter are assigned own variables in the
modeling process. In the following, an adaptive algorithm is proposed
for online tuning the control coefﬁcients, which can reduce the
impact of the parameter measurement or estimation error on the
system performance in a wide operating range. Moreover, such an
adaptive algorithm can improve system dynamic performance due to
its direct backsteping design, thus can realize fast voltage restoring
and load power sharing. The system performance coping with single-
point failure, i.e., DES module fault or ST fault can be thus
guaranteed. Virtual Current Control for Power Sharing
In αβ coordinates, the tracking error for the ac bus voltage is
ﬁrstly deﬁned as evα  voα −vref α
evβ  voβ −vref β
(3) (3) Then, taking the time derivative of Eq. 3 yields Then, taking the time derivative of Eq. SYSTEM CONFIGURATION AND PROBLEM
DESCRIPTION 3 yields _evα 
1
MCαcom

n
j1
ijα −
1
MCαcom
iloadα +
1
MCdiff

n
j1
ijβ −
1
MCdiff
iloadβ −_vref α
_evβ 
1
MCβcom

n
j1
ijβ −
1
MCβcom
iloadβ +
1
MCdiff

n
j1
ijα −
1
MCdiff
iloadα −_vref β
(4) (4) Frontiers in Energy Research | www.frontiersin.org DISTRIBUTED ADAPTIVE CONTROLLER
DESIGN FOR DGS Consider the following Lyapunov function candidate Consider the following Lyapunov function candidate Traditionally, the ac bus voltage is regulated by the ST. Direct
power control or direct current control is selected for DES
modules (She et al., 2013). Such a way is straightforward and
easy to implement. However, one of the main drawbacks is that
the failure of the ST is not able to be tolerated. In this paper, the
ST is working at current control mode while the ac bus voltage is
regulated in a sharing way through the DES modules. In this way,
each DES module is capable of regulating the bus voltage. The
voltage restoring is faster once DES module/ST failure happens. In this section, a distributed adaptive controller is designed to
maneuver the output voltage to track the reference trajectory vref . On the other hand, the load current sharing error between ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
V1α  MCαcom
2
e2
vα
V1β  MCβcom
2
e2
vβ
(5) (5) The time derivative of Eq. 5 is written as ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
_V1α  −ke2
vα −evαiloadα + evα⎛
⎝kevα + 
n
j1
ijα −MCαcom _vref α + rCα
N
j1
icjβ⎞⎠
_V1β  −ke2
vβ −evβiloadβ + evβ⎛
⎝kevβ + 
n
j1
ijβ −MCβcom _vref β + rCβ 
N
j1
icjα⎞⎠ (6) Frontiers in Energy Research | www.frontiersin.org 4 Parallel-Connected Converters Parameter Mismatch Control Wang et al. FIGURE 3 | Equivalent circuit of a DES module in αβ stationary reference frame. FIGURE 3 | Equivalent circuit of a DES module in αβ stationary reference frame. where icjα, icjβ are the capacitance currents. And for simplicity
fake, we deﬁned two ratios of rCα and rCβ as, where icjα, icjβ are the capacitance currents. And for simplicity
fake, we deﬁned two ratios of rCα and rCβ as, _M Cαcom  γCProjevα_evα, MC
αcom, MCαcom
_M Cβcom  γCProjevβ_evβ, MC
βcom, M Cβcom
_r dα  γdProj⎛
⎝evα 
N
j1
icjβ, rd
α, rdα⎞⎠
_rdβ  γdProj⎛
⎝evβ 
N
j1
icjα, rd
β, rdβ⎞⎠
(10) ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
rCα  MCαcom
MCdiff
rCβ  MCβcom
MCdiff
(7) (7) (10) According to the backstepping principle, ijα and ijβ can be taken
as the virtual control inputs for Eq. DISTRIBUTED ADAPTIVE CONTROLLER
DESIGN FOR DGS 8 which is regarded as a
subsystem of Eq. 1. where γC and γd are adaption gains. The projection algorithm
Proj(•): R × R × R →R is given as where γC and γd are adaption gains. The projection algorithm
Proj(•): R × R × R →R is given as ⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
_voα 
1
MCαcom

n
j1
ijα +
1
MCdiff

n
j1
ijβ −
1
MCαcom
iloadα −
1
MCdiff
iloadβ
_voβ 
1
MCdiff

n
j1
ijα +
1
MCβcom

n
j1
ijβ −
1
MCdiff
iloadα −
1
MCβcom
iloadβ
(8) Projx, y, y 
⎧
⎨
⎩
x,
 y < y < y or y  y and x ≥0 ory  y and x ≤0
0,
y  y and x < 0 or y  y and x > 0
(11) (11) (8) with y  Proj(x, y, y). According to the property of the projection
function, it can be veriﬁed that for any initial value y ≤y(0) ≤y,
y ≤y(t) ≤y holds for all time. Moreover, the following inequality
holds An ideal virtual controller idj can be deﬁned as follows An ideal virtual controller idj can be deﬁned as follows ⎧
⎨
⎩
idjα  nj−kevα + MCαcom _vref α −rCαicjβ
idjβ  nj−kevβ + MCβcom _vref β −rCβicjα
(9) (9) y −yx −Projx, y, y ≤0
(12) (12) Deﬁne
MCαcom,
MCβcom, rCα and rCβ, as the estimates of
MCαcom, MCβcom, rCα and rCβ, respectively. In the following
analysis, we will utilize the error deﬁnition of (• )  (•) −(•),
i.e. MCα com  MCα com −MCα com. The update laws of MCαcom,
MCβcom, rCα and rCβ are designed as, for any y satisfying y ≤y ≤y. Note that, the current controllers of
phase-α and phase-β are coupled with each other. Recalling the
system dynamics in Eq. 2, voltages/currents of phase-α and
phase-β are also coupled with each other. These coupled terms
reﬂected in the virtual controller are intended for decoupling the October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 4 | Control block diagram of a DES module. FIGURE 4 | Control block diagram of a DES module. TABLE 1 | System parameters in the simulation. DISTRIBUTED ADAPTIVE CONTROLLER
DESIGN FOR DGS Item
Speciﬁcations
The ﬁlter parameters are balanced
Output inductors (L1a∼L4a, L1b∼L4b, L1c∼L4c)
0.3 mH
Line impedance (R1a∼R4a, R1b∼R4b, R1c∼R4c)
0.5Ω
Output capacitors (C1a∼C4a, C1b∼C4b, C1c∼C4c)
25 µF
DC link voltages
550 V
Output AC voltage (phase voltage)
220V/50 Hz (Amplitude)
Linear load
3.75 Ω
Nonlinear load
Resistance (RL)
5 Ω
Inductance (LL)
0.1 mH
Capacitance (CL)
10 µF
The ﬁlter parameters are unbalanced
(The ﬁlter parameters are unbalanced.)
0.3 mH
Output inductors (L2a∼L4a, L1b∼L4b, L1c∼L4c)
Output inductors (L1a)
0.1 mH The error between the actual input ij and the desired one idj is
deﬁned as TABLE 1 | System parameters in the simulation. ejα  ijα −idjα
ejβ  ijβ −idjβ
(14) (14) Thereafter, the following augmented Lyapunov function V3 is
introduced as V3α  V1α +
M
2
Cαcom
2γC
+
M
2
Cdiff
2γd
+ 1
2ρ 
N
j1
MLαcome2
jα
V3β  V1β +
M
2
Cβcom
2γC
+
M
2
Cdiff
2γd
+ 1
2ρ 
N
j1
MLβcome2
jβ
(15) (15) where MCαcom and MCβcom are the estimate errors, ρ is a positive
constant. To step back to the second subsystem of Eq. 1, taking
the time derivative of ej yields where MCαcom and MCβcom are the estimate errors, ρ is a positive
constant. To step back to the second subsystem of Eq. 1, taking
the time derivative of ej yields ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
MLαcom_ejα  −voα −MRαcom
MLαcom
ijα + vjα −MLαcom_idjα + MLαcomΔβj
MLβcom_ejβ  −voβ −MRβcom
MLβcom
ijβ + vjβ −MLβcom_idjβ + MLβcomΔαj
(
) initial current dynamics. It is actually a cross cancelation
decoupling in advance of the voltage controller design. initial current dynamics. It is actually a cross cancelation
decoupling in advance of the voltage controller design. Adaptive Voltage Control Adaptive Voltage Control (16) (16) For simplicity consideration, Δαi, Δβi are deﬁned to represent the
cross-coupling elements between _ijα and _ijβ, i.e., Recalling the property of the projection algorithm in Eq. 11 and
an assumption that lower and upper bounds of Rj, Lj, Cj are
known, i.e., R ≤Rj ≤R, L ≤Lj ≤L and C ≤Cj ≤C, where R, R, L,
L, C and C are known positive constants (such assumption is
deﬁned as Assumption 1), the derivative of Eq. 15 with respect to
time yields Recalling the property of the projection algorithm in Eq. 11 and
an assumption that lower and upper bounds of Rj, Lj, Cj are
known, i.e., R ≤Rj ≤R, L ≤Lj ≤L and C ≤Cj ≤C, where R, R, L,
L, C and C are known positive constants (such assumption is
deﬁned as Assumption 1), the derivative of Eq. 15 with respect to
time yields Recalling the property of the projection algorithm in Eq. 11 and
an assumption that lower and upper bounds of Rj, Lj, Cj are
known, i.e., R ≤Rj ≤R, L ≤Lj ≤L and C ≤Cj ≤C, where R, R, L,
L, C and C are known positive constants (such assumption is
deﬁned as Assumption 1), the derivative of Eq. 15 with respect to
time yields Recalling the property of the projection algorithm in Eq. 11 and
an assumption that lower and upper bounds of Rj, Lj, Cj are
known, i.e., R ≤Rj ≤R, L ≤Lj ≤L and C ≤Cj ≤C, where R, R, L, ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Δαj 
1
MLdiff
voα +
1
MRdiff
ijα −
1
MLdiff
vjα
Δβj 
1
MLdiff
voβ +
1
MRdiff
ijβ −
1
MLdiff
vjβ
(13) j
j
j
L, C and C are known positive constants (such assumption is
deﬁned as Assumption 1), the derivative of Eq. 15 with respect to
time yields (13) October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org Frontiers in Energy Research | www.frontiersin.org Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 5 | Three-phase simulation results by using proposed control including scenarios of 1) start up (from 0 to 0.5 s), 2) module without mismatch #1 case (from
0.2 to 0.4 s), 3) module failure #2 case (from 0.4 to 0.6 s) and 4) module-mismatch #3 case. Adaptive Voltage Control p
g
y
Firstly, an voltage observer is proposed to obtain _voα and €voα,
and is given as (taking phase-α for example) yv1α  _ξv1α  kv1α(voα −ξv1α)
yv2α  _ξv2α  kv2α(ξv1α −ξv2α)
(21) (21) (19) where yv1α and yv2α are estimate values of _voα and €voα,
respectively, kv1α and kv2α are the voltage observer gains. In
addition, current observers are developed to obtain _icjα: where yv1α and yv2α are estimate values of _voα and €voα,
respectively, kv1α and kv2α are the voltage observer gains. In
addition, current observers are developed to obtain _icjα: With the help of Assumption 1 and further assume that the
load current iloadα, iloadβ is bounded (such assumption is
deﬁned as Assumption 2), i.e., |iloadα| ≤dm and
iloadβ
 ≤dm,
where dm is an unknown positive constant, the time derivative
of Eq. 19 yields ycjα  _ξcjα  kcjαicjα −ξcjα
(22) (22) for j  1, 2, . . . , n, where ycjα is the estimate value of _icjα, and kcjα
is the current observer gain. Deﬁne the voltage observer output
errors as _Vα ≤−(k −εα)e2
vα −
n
j1
kj
ρ e2
jα + d2
mα
4εα
_Vβ ≤−k −εβe2
vβ −
n
j1
kj
ρ e2
jβ +
d2
mβ
4εβ
(20) for j  1, 2, . . . , n, where ycjα is the estimate value of _icjα, and kcjα
is the current observer gain. Deﬁne the voltage observer output
errors as errors as (20) ev1α  yv1α −_voα
ev2α  yv2α −€voα
(23) ev1α  yv1α −_voα
ev2α  yv2α −€voα
(23) (23) along with the current observer output error along with the current observer output error with 0 < εα < k, 0 < εβ < k being a constant. with 0 < εα < k, 0 < εβ < k being a constant. ecjα  ycjα −_icjα
(24) (24) Adaptive Voltage Control FIGURE 5 | Three-phase simulation results by using proposed control including scenarios of 1) start up (from 0 to 0.5 s), 2) module without mismatch #1 case (from
0.2 to 0.4 s), 3) module failure #2 case (from 0.4 to 0.6 s) and 4) module-mismatch #3 case. ⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
vjα  −kjejα −ρevα + voα + Mrαcomijα
+ MLαcom_idjα −mLβvoβ −mRβijβ + mLβvjβ
vjβ  −kjejβ −ρevβ + voβ + Mrβcomijβ
+ MLβcom_idjβ −mLαvoα −mRαijα + mLαvjα −Mrβcomijβ + vjβ −MLβcom_idjβ + mLαvoα + mRαijα −mLαvjα
⎪
⎪
⎩ where kj is a user-deﬁned positive constant, Rj and Lj are the
estimates of the unknown parameters Rj and Lj, respectively. The
update laws are then given in the similar way to MCα com,
MCβ com, rCα and rCβ. The details can be found in the
appendix. Finally, deﬁne the overall Lyapunov function as Where,
MRαcom
MLαcom  Mrα,
MRβcom
MLβcom  Mrβ, mLβ  MLαcom
MLdiff , mRβ  MRαcom
MRdiff ,
mLα  MLβcom
MLdiff and mRα  MLβcom
MRdiff . Next, the actual control law vj can be derived as October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org Parallel-Connected Converters Parameter Mismatch Control Wang et al. FIGURE 6 | The zooming waveform for each scenario transition for the proposed control. FIGURE 6 | The zooming waveform for each scenario transition for the proposed control. FIGURE 6 | The zooming waveform for each scenario transition for the proposed control. Vα  V3α +
	n
j1 M
2
rαcom
2γRρ
+
	n
j1 M
2
Lαcom
2γLρ
+
	n
j1 m2
Rβ
2γRρ
+
	n
j1 m2
Lβ
2γLρ
Vβ  V3β +
	n
j1 M
2
rβcom
2γRρ
+
	n
j1 M
2
Lβcom
2γLρ
+
	n
j1 m2
Rα
2γRρ
+
	n
j1 m2
Lα
2γLρ
(19) output is measurable in most practical systems. Therefore, an
output feedback based control is investigated for the system. Observer Design for j  1, 2, . . . , n. The stability and the boundedness of the
observer output errors are provided in the following lemma
(Behtash, 1990; Ge et al., 1999). for j  1, 2, . . . , n. The stability and the boundedness of the
observer output errors are provided in the following lemma
(Behtash, 1990; Ge et al., 1999). In previous subsection, the information of _voα, €voα and _icjα (also
_voβ, €voβ and _icjβ) are supposed to be available for feedback. This,
however, limits the application of this approach, since only October 2021 | Volume 9 | Article 770372
8 October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 8 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 7 | Three-phase simulation results by using PI-based control including scenarios of 1) start up (from 0 to 0.5 s), 2) module without mismatch #1 case (from
0.2 to 0.4 s), 3) module failure #2 case (from 0.4 to 0.6 s) and 4) module-mismatch #3 case (from 0.6 to 0.8 s). FIGURE 7 | Three-phase simulation results by using PI-based control including scenarios of 1) start up (from 0 to 0.5 s), 2) module without mismatch #1 case (from
0.2 to 0.4 s), 3) module failure #2 case (from 0.4 to 0.6 s) and 4) module-mismatch #3 case (from 0.6 to 0.8 s). FIGURE 8 | Simulation waveforms comparison facing a step load change. FIGURE 8 | Simulation waveforms comparison facing a step load change. FIGURE 8 | Simulation waveforms comparison facing a step load change. October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 9 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 9 | Simulation waveforms of phase-a voltage tracking error and circulating currents under linear load condition. FIGURE 9 | Simulation waveforms of phase-a voltage tracking error and circulating currents under linear load condition. Observer Design With
the designed observer, the virtual controller becomes yv1β  _ξv1β  kv1βvoβ −ξv1β
yv2β  _ξv2β  kv2βξv1β −ξv2β
(30)
ycjβ  _ξcjβ  kcjβicjβ −ξcjβ
(31) (30) idjα  nj−kevα −MCαcomyv1α + MCαcom _vref α + icjα
(25) (25) (31) The virtual controllers and the update law for phase- β using
designed observer can be ﬁgured out as expressed by The virtual controllers and the update law for phase- β using
designed observer can be ﬁgured out as expressed by Then replacing _voα, €voα and _icjα with their observed values in Eq
21, 22 yields Then replacing _voα, €voα and _icjα with their observed values in Eq
21, 22 yields idjβ  nj−kevβ −MCβcomyv1β + MCβcom _vref β + icjβ
(32)
idjβ  nj−k + MCβcomyv1β + k + MCβcom_vref β −MCβcomyv2β + MCβcom€vref β + ycjβ
(33)
_MCβcom  γCProjevβyv1β −_vref β, MC-βcom, M
C-
βcom
(34)
vjβ  −kjejβ −ρevβ + voβ
+ Mrβcomijβ + MLβcomidjβ −mLαvoα −mRαijα + mLαvjα (35)
_MLβcom  γLProj−ejβidjβ, NL β, NLβ
(36) idjβ  nj−kevβ −MCβcomyv1β + MCβcom _vref β + icjβ
(32)
idjβ  nj−k + MCβcomyv1β + k + MCβcom_vref β −MCβcomyv2β + MCβcom€vref β + ycjβ
(33) idjα  nj−k + MCαcomyv1α + k + MCαcom_vref α −MCαcomyv2α + MCαcom€vref α + ycjα
(26) (32) where idjα is the estimate of _idjα, and _MCβcom  γCProjevβyv1β −_vref β, MC-βcom, M
C-
βcom
(34)
vjβ  −kjejβ −ρevβ + voβ
+ Mrβcomijβ + MLβcomidjβ −mLαvoα −mRαijα + mLαvjα (35)
_MLβcom  γLProj−ejβidjβ, NL β, NLβ
(36) _MCαcom  γCProjevαyv1α −_vref α, MC-αcom, M
C-
αcom
(27)
Th
f
h
f
db
k b
d
ll
i
d
d (27) Thereafter, the output-feedback based controller is updated as vjα  −kjejα −ρevα + voα
+ Mrαcomijα + MLαcomidjα −mLβvoβ −mRβijβ + mLβvjβ (28)
Replacing _idjα with idjα, the update law of MLαcom turns to vjα  −kjejα −ρevα + voα vjα  −kjejα −ρevα + voα
+ Mrαcomijα + MLαcomidjα −mLβvoβ −mRβijβ + mLβvjβ (28) (36) + Mrαcomijα + MLαcomidjα −mLβvoβ −mRβijβ + mLβvjβ (28) Replacing _idjα with idjα, the update law of MLαcom turns to The stability analysis of the closed-loop system under output-
feedback controller is shown in Theorem 2 included in the
Supplementary Appendix. The stability analysis of the closed-loop system under output-
feedback controller is shown in Theorem 2 included in the
Supplementary Appendix. Observer Design In a similar way, Firstly, the voltage observer and estimate values
for phase- β are given as In a similar way, Firstly, the voltage observer and estimate values
for phase- β are given as Lemma 1: Consider the observers designed in Eq 21, 22, then the
following facts hold: 1) all signals of the observers are bounded; 2) the
observer output errorsEq 23, 24 can be tuned arbitrarily small by chosen
sufﬁcient large observer gains kv1α, kv2α and kcjα, j  1, 2, ... , n. With
the designed observer, the virtual controller becomes Lemma 1: Consider the observers designed in Eq 21, 22, then the
following facts hold: 1) all signals of the observers are bounded; 2) the
observer output errorsEq 23, 24 can be tuned arbitrarily small by chosen
sufﬁcient large observer gains kv1α, kv2α and kcjα, j  1, 2, ... , n. Observer Design _MLαcom  γLProj−ejαidjα, NL α, NLα
(29) (29) MLαcom  γLProj−ejαidjα, NL α, NLα October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 10 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 10 | Simulation waveforms of phase-a voltage tracking error and circulating current under nonlinear load condition. FIGURE 10 | Simulation waveforms of phase-a voltage tracking error and circulating current under nonlinear load condition. FIGURE 11 | Three-phase simulation results between proposed control and PI-based control suffering from nonlinear load. FIGURE 11 | Three-phase simulation results between proposed control and PI-based control suffering from nonlinear load. FIGURE 11 | Three-phase simulation results between proposed control and PI-based control suffering from nonlinear load. SIMULATION AND EXPERIMENT STUDIES Figure 4 illustrates the control block diagram of a local
inverter unit. The developed adaptive controller only requires
the local measurements. Only a synchronization signal and no
more interconnection communications among DGs are
required. Simulation Veriﬁcations Simulation Veriﬁcations In order to explore the performance of the proposed decoupling
and adaptive control techniques, the simulation on a MATLAB/ October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 11 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 12 | Experimental setup. the total active power consumed by the load, increases to
5 kW. And the load current magnitude, Iload increases to
60.6 A. The transition lasts for a very short time interval. The voltage drops are acceptable [the maximum drop value is
16 V (0.073 p.u.), occurring in phase-b]. Afterwards, the load
currents have good qualities, i.e., the THD is 0.85%. And the
circulating currents are nearly zero, as shown in Figure 9
(before 0.4 s). TABLE 2 | Experimental parameters. Item
Speciﬁcations
Output inductors (L1, L2, L3)
2 mH
Output capacitors (C1, C2, C3)
25 µF
DC link voltages
175 V
Output AC voltage
110 Vrms
Linear load
25 Ω
Nonlinear load
Resistance (RL)
50 Ω
Inductance (LL)
2.5 mH
Capacitance (CL)
235 µF At 0.4 s, #1, #2 and #3 modules keep power injection but the
one of #4 module is bypassed by a solid switch due to a sudden
failure. The module failure shows little impact on the regulating
of ac bus voltage, i.e. the power is balanced very well (Circulating
currents are small enough). And the voltage quality is still good as
expected (THD is not larger than 0.81%). At 0.6 s, the phase-a line
inductance
of
#1
module
changes
to
0.1 mH
(0.33p.u.). Unsymmetrical
line
parameters
degrade
the
operation
performance i.e., the circulating current (i1a-i2a) is increased. But the value is 5 A (0.082 p.u.), which is acceptable. Simulink platform is conducted. A microgrid system formed by
parallel-connected energy storage converters is applied in the
simulation. Parameters of the system are summarized in Table 1. Figure 5 shows the three-phase simulation results including
three scenarios, i.e., Case 1∼Case 3. The waveforms from top to
bottom are respectively about ac bus voltages, load currents,
active power injected to the load from different modules, output
currents
of
DES
module
#1∼#4. Zoomed-in
three-phase
waveforms of the rating steady state are shown at the right
side of Figure 5. Note that, the simulation results of the conventional PI-based
control (PI-based control scheme with a centralized structure) are
also supplied in Figure 7 for comparisons. Frontiers in Energy Research | www.frontiersin.org Simulation Veriﬁcations Simulation parameters
are same as listed in Table 1. The waveforms from top to bottom
are also respectively about ac bus voltages, load currents, active
power injected to the load from different modules, output
currents of DES module #1∼#4, And zoomed-in three-phase
waveforms of the rating steady state are also provided at the
right side of Figure 7. During the whole simulation interval, the control proposed
in Distributed Adaptive Controller Design for DGs Section
takes effect on the coordination for the four DES modules. The zooming waveforms for each scenario transition in
Figure 5 are shown in Figure 6. Start process lasts for 0.2 s
(from 0 to 0.2 s). At 0.2 s, the active power injected by the four
modules, change from 0 to 5 kW (1.0p.u.). Accordingly, Focally, Figure 8 presents the comparison between the
proposed control and the PI-based control facing a step load
change. For conventional PI based control, the maximum drop of October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 12 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 13 | Experimental results of the low bandwidth synchronization: (A) Zooming-in of the experiment results when without synchronization and (B) Zooming-
in of the experiment results when with synchronization. FIGURE 13 | Experimental results of the low bandwidth synchronization: (A) Zooming-in of the experiment results when without synchronization and (B) Zooming-
in of the experiment results when with synchronization. FIGURE 13 | Experimental results of the low bandwidth synchronization: (A) Zooming-in of the experiment results when without synchronization and (B) Zooming-
in of the experiment results when with synchronization. non-linear load condition. Moreover, for PI-based control, the THD
of the ac voltage is high as 5.34% under rating non-linear load (see
Figure 11 after 0.2 s). However, the value of the proposed method is
only 0.81%. Obviously, this conventional way cannot handle the
non-linear load very well. ac bus voltage is large as 38 V. In contrast, for the proposed
control, the value is only 16 V. Furthermore, Waveforms of the ac
bus voltage tracking error and the circulating currents for
comparison between the proposed adaptive control and PI-
based control are intentionally added in Figure 9. The voltage
tracking errors are nearly equal to each other, but the circulating
current (i1a-i2a) in the PI-based control is larger than that of the
proposed control (13 vs. 5 A). Frontiers in Energy Research | www.frontiersin.org Simulation Veriﬁcations Lastly, it should be noted that, if a droop stage is employed, the
PI-based method would need average power calculation. The
sensed load current is a high frequency one, therefore requires
high resolution sensors, wide bandwidth analog signal processing
circuit, and high performance analog to digital converter for
digital control. Furthermore, the calculation of the power is also
not easy due to the complex integration operation, in which the
calculation accuracy may be low. In this condition, the
performance of the controller may not be guaranteed in the
real hardware setup. By using the proposed control method, no In order to better evaluate the performance of the proposed
control solution, the non-linear load condition is considered. And it
is also compared with the widely used PI-based control scheme. Figure 10 shows the simulation waveforms of phase-a voltage
tracking error and circulating current between proposed control
and PI-based control when connecting nonlinear load. The similar
voltage tracking error/circulating currents conclusion applies to this October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 13 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 14 | Linear load under balanced condition. FIGURE 15 | Nonlinear load under balanced condition. FIGURE 14 | Linear load under balanced condition. FIGURE 14 | Linear load under balanced condition. FIGURE 15 | Nonlinear load under balanced condition. FIGURE 15 | Nonlinear load under balanced condition. connected in parallel to share the load that can be switched
between linear load and nonlinear load. The photo of LC ﬁlters
and loads is shown at the right bottom of Figure 12. The
conﬁguration of nonlinear load is illustrated in the below. Parameters of the system are presented in Table 2. The
designed
adaptive
controller
is
implemented
on
a
TSM320F28346 DSP. In this control scheme, each controller is
used to control its own DG. The adaptive controller’s parameters
are chosen as k  2400, ks  2400, ρ  1/152, γ  2 × 10–10, kλ 
0.01, L  2 × 10–3 and C  3.5 × 10–5. additional current sensor is needed and high performance can be
achieved. Experiment Results for a Single-Phase
Microgrid The experiment veriﬁcation is based on a setup shown in
Figure 12, which consists of three DES sub-systems. Every
sub-system has its own control platform and monitoring
interfaces, i.e., control boards, external digital to analog
converter
(DAC)
boards,
personal
computers
and
one
oscilloscope. Each DG is composed by one DC power supply,
one H-bridge inverter and one LC ﬁlter. And the three DGs are In reality, even the same batch of components has different
parameters that deviate slightly from the nominal value. For control
implementation with DSP control board, one major issue is with the October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 14 Wang et al. Parallel-Connected Converters Parameter Mismatch Control FIGURE 16 | Linear load under unbalanced condition (L1  1 mH, L2  L3  2 mH). FIGURE 17 | Nonlinear load under unbalanced condition (L1  1mH, L2  L3  2 mH). FIGURE 16 | Linear load under unbalanced condition (L1  1 mH, L2  L3  2 mH). FIGURE 16 | Linear load under unbalanced condition (L1  1 mH, L2  L3  2 mH). FIGURE 16 | Linear load under unbalanced condition (L1  1 mH, L2  L3  2 mH). FIGURE 17 | Nonlinear load under unbalanced condition (L1  1mH, L2  L3  2 mH). FIGURE 17 | Nonlinear load under unbalanced condition (L1  1mH, L2  L3  2 mH). FIGURE 17 | Nonlinear load under unbalanced condition (L1  1mH, L2  L3  2 mH). experiment, the switching noises will cause random losses
of sync pulse signals. Since the time step is synchronized to
the successfully delivered pulses, unreliable synchronization
signal will seriously deteriorate the control performance. To improve synchronization reliability, a low-bandwidth
synchronization technique is designed and implemented as
introduced below. The synchronization signal is carried
with a fundamental signal. The rising edge will initiate
synchronization and the duration of the signal will be used
for phase shift evaluation. During implementation, the
synchronization signal can be generated with one control
board and then broadcasted to neighboring control boards
one
after
another. In
addition
to
the
leader-follower synchronization of the control activities. Even if the control signals
are periodically synchronized, the control activities between
synchronization cannot be guaranteed due to inaccuracy and
ﬂuctuation of DSP clocks. CONCLUSION In this paper, a backstepping based adaptive controller has been
proposed for a microgrid formed by parallel-connected energy
storage converters. The parameter mismatch is considered
within
the
system
modeling
process. Furthermore,
the
unknown
system
dynamics
have
been
compensated
by
designed adaptive laws. According to the standard Lyapunov
synthesis, the hierarchical control system including a virtual
current
controller
and
a
backstepping
adaptive
voltage
controller has been proved to be bounded with reasonable
assumptions. Both simulation and experimental results were
presented to show the high performance of the developed
controller in terms of voltage tracking (i.e., a low THD even
in the nonlinear load case) and load power sharing (i.e., little
circulating current even with unbalanced line parameters). In
addition, when coping with single-point DES module failure,
the effectiveness of the proposed algorithm for fast voltage
restoring has been veriﬁed. Through the low bandwidth synchronization, only local
information is used for each adaptive controller in experiments. In Figure 14, the results of linear load in balanced system parameter
case (L1  L2  L3, C1  C2  C3) are given out. A good tracking
performance of the output voltage is achieved by the adaptive
controller under the balanced condition. Although the FFT results
of the output voltage have harmonic components, the amplitudes of
3rd, 5th, 7th etc. harmonics are less than 2.5 V which is
approximately 1.6% of fundamental frequency component. The
total harmonic distortion (THD) is less than 1.8%. Moreover, since
the differences between modules currents (i1-i2, i2-i3, i3-i1) are
almost zero as shown in Figure 14, the load current is well
shared. Results of nonlinear load case is presented in Figure 15. In this case, the load is also equivalently shared by modules. Although the output voltage performance is degraded compared
to the linear case, the THD is just 3.2% which satisﬁes related
standards. DATA AVAILABILITY STATEMENT The original contributions presented in the study are included in
the article/Supplementary Material, further inquiries can be
directed to the corresponding author. FUNDING Project supported by Natural Science Foundation of Jiangsu
Province (BK20190461). AUTHOR CONTRIBUTIONS Conceptualization and original draft preparation, CW; writing
and simulation validation, HH; resources and supervision, ZW;
review and experiment veriﬁcation, HP. All authors have read
and agreed to the published version of the manuscript. In the previous two cases, the system parameters are chosen as
presented in Table 2. However, the practical parameters of
different
LC
ﬁlters
cannot
be
exactly
same. Even
the
inductance and capacitance will be varied slowly during the
operation. Thence, unbalanced cases are considered. The
inductance of LC ﬁlter of module #1 is reduced to 1 mH. Figures 16, 17 depict the corresponding results. As shown, the
output voltage performances of both the linear load case and
nonlinear load case under unbalanced ﬁlter parameters condition
are almost the same as the ones under balanced condition. Therefore, the proposed control method can achieve a good
voltage tracking performance with ﬁler parameter differences. However, the current sharing performance is degraded especially
in the nonlinear load case. The current differences between DGs Experiment Results for a Single-Phase
Microgrid Even though clock speed cannot be
adjusted, control signal synchronization update can be realized by
synchronizing the timing of control update with a common/
accurate reference signal. Accurate synchronization is critical to
achieve expected performance. Although the synchronization
function provided by manufacture, such as the ePWM sync
pulse signal of TI TMS320C28346, is easy to implement,
experimental studies show that the high-frequency signal is
susceptible to switching noises. When power circuit works at a
relatively high voltage level i.e., 170 V DC link voltage in the October 2021 | Volume 9 | Article 770372 Frontiers in Energy Research | www.frontiersin.org 15 Wang et al. Parallel-Connected Converters Parameter Mismatch Control communication scheme, centralized or GPS-based pseudo-
distributed schemes can also be implemented. are smaller than 0.5 A which is acceptable for a distributed
control scheme with unknown system parameters. Figure
13
shows
the
results
of
the
low
bandwidth
synchronization. Before the trigging of the synchronization,
each calculation cycle is determined by dividing the DSP
system clock with a ﬁxed ratio. The system clock of each DSP
is conﬁgured as 300 MHz. Thus the initial frequency dividing
ratio is set as 25,000 to get 12 kHz sampling and switching time-
step. Since the DSP clocks are mismatched with each other, the
shifting of calculation cycles is introduced, which is shown in
Figure 13A. Through activating the synchronization, the sub-
system # 1 sends fundamental square signals to sub-system #2
and #3. Inside each sub-system, the frequency dividing ratio is
then adjusted by using the loop shown in Figure 4. The steady
results with synchronization are shown in Figure 13B, where
three cycles are synchronized with each other well. Moreover, the
closed loop in Figure 4 regulates the periods of cycles
dynamically inside every calculation cycle to guarantee the
subsystems are always well synchronized for the imposing of
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distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the
original author(s) and the copyright owner(s) are credited and that the original
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