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PIDR Sliding Mode Current Control with Online Inductance Estimator for VSC MVDC System Converter Stations under Unbalanced Grid Voltage Conditions .pdf



Original filename: PIDR Sliding Mode Current Control with Online Inductance Estimator for VSC-MVDC System Converter Stations under Unbalanced Grid Voltage Conditions.pdf
Title: PIDR Sliding Mode Current Control with Online Inductance Estimator for VSC-MVDC System Converter Stations under Unbalanced Grid Voltage Conditions
Author: Weipeng Yang, Hang Zhang, Jungang Li, Aimin Zhang, Yunhong Zhou and Jianhua Wang

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energies
Article

PIDR Sliding Mode Current Control with Online
Inductance Estimator for VSC-MVDC System
Converter Stations under Unbalanced Grid
Voltage Conditions
Weipeng Yang 1 , Hang Zhang 1, *, Jungang Li 2 , Aimin Zhang 1 , Yunhong Zhou 1 and
Jianhua Wang 1
1

2

*

School of Electrical Engineering, Xi’an Jiaotong University, No.28, West Xianning Road, Xi’an 710049, China;
wpyang@stu.xjtu.edu.cn (W.Y.); zhangam@mail.xjtu.edu.cn (A.Z.); wsad8246zyh@stu.xjtu.edu.cn (Y.Z.);
jhwang@mail.xjtu.edu.cn (J.W.)
Xuji Group Corporation, State Grid Corporation of China, No.1298, Xuji Road, Xuchang 461000, China;
yzilong2006@stu.xjtu.edu.cn
Correspondence: zhangh@mail.xjtu.edu.cn; Tel.: +86-29-8266-8665 (ext. 167)

Received: 30 August 2018; Accepted: 26 September 2018; Published: 29 September 2018




Abstract: This study aims to present a novel proportional-integral-derivative-resonant law-based
sliding mode current control strategy with online inductance estimator (PIDR-SMCC-OIE) for voltage
source converter medium voltage direct current (VSC-MVDC) system converter stations under
unbalanced grid voltage conditions. A generalized current reference calculation method, by which
the ratio of the amplitude of the active power ripple to that of the reactive power ripple can be
continuously controlled without current distortion is presented. A dynamic model of the current
control errors in the positive sequence synchronous reference frame is developed, and a PIDR
law-based sliding mode current controller is designed, where derivatives of the current references
are obtained by simple algebraic operations. An OIE adopting the dynamic filtering method and
gradient algorithm is proposed to further improve system robustness. In this OIE, the converter
pole voltages are obtained by computation utilizing the gate signals of the switching devices and
the DC bus voltage, so that no additional voltage sensors are needed. To verify effectiveness of the
PIDR-SMCC-OIE strategy, simulation studies on a two-terminal VSC-MVDC system are conducted
in PSCAD/EMTDC. The results show it can provide satisfactory performance over a wide range of
operating conditions.
Keywords: VSC-MVDC; current control; proportional-integral-derivative-resonant control; sliding
mode control; grid voltage unbalance; online inductance estimator

1. Introduction
Owing to obvious advantages over its AC counterpart in terms of flexibility of power control,
transmission capacity, integration of distributed generations (DG) etc., the voltage source converter
medium voltage direct current (VSC-MVDC) system-based distribution network (DN) has been
attracting ever-increasing interests in recent years [1–5]. The operation of the VSC-MVDC system
depends on highly controllable power electronic converters. However, the operating conditions of the
DN often change violently, parameter uncertainties and different kinds of disturbance always exist,
three-phase voltage unbalance is a common phenomenon [6,7]. Therefore, the robustness as well as
performance of the control strategies for VSC-MVDC system converter stations (CS) under unbalanced
grid voltage (UBGV) conditions deserve serious concern.
Energies 2018, 11, 2599; doi:10.3390/en11102599

www.mdpi.com/journal/energies

Energies 2018, 11, 2599

2 of 20

Current control is one of the most commonly adopted strategies for VSC control. Generally, it is
realized in the d-q synchronous reference frame (SRF) by use of the vector-oriented control method [8,
9]. However, the performance will be decreased under UBGV conditions [10]. In order to obtain
satisfactory performance in these situations, the currents can be controlled in the positive sequence (PS)
SRF or in the static two-phase reference frame (STP RF). In references [11,12], current control strategies
in the PS SRF with proportional-integral (PI) plus resonant (PIR) controllers are proposed for modular
multi-level converter high voltage DC (MMC-HVDC) system and doubly fed induction generator
(DFIG) respectively. In references [13,14], current control strategies in the STP RF are proposed for
VSC-HVDC system and MMC respectively, where proportional-resonant (PR) controllers are adopted
to realize current tracking function. In references [15,16], current control strategies in the PS SRF
with PI plus vector resonant (PIVR) controllers are proposed for DFIG and MMC-HVDC systems
respectively. Under general conditions, these linear controllers can provide satisfactory performances.
However, the performances are related to the operating points and affected by the nonlinearity feature
of VSC. Moreover, as the non-ideal resonant law is often used [17,18], high gains are required to
eliminate the steady state errors, which will reduce the stability margins.
Owing to excellent dynamic performance, deadbeat control (DBC) and model predictive control
(MPC) have been studied extensively in the control of VSC. In reference [19], a DBC-based current
control strategy with current predictive calibration is proposed for grid-connected VSC inverter (VSI).
In reference [20], a multistep MPC-based current control strategy is proposed for cascaded H-bridge
inverters. The DBC and MPC can eliminate the steady state errors within several control beats.
However, their performances depend heavily on the accuracy of the model and is significantly affected
by the time delays in the control loops. To overcome these drawbacks, extra compensation measures
have to be taken [21,22], which complicates the controller structure. Besides, generally the DBC and
MPC-based control strategies obtain the voltage vector by look-up table. Consequently, the switching
frequency is not fixed, which increases the difficulty of the AC side filter design.
To reduce the adverse influences of parameter uncertainty and nonlinearity, many nonlinear
control methods have been studied for the control of VSC. In reference [23], a direct Lyapunov control
(DLC)-based current control strategy is proposed for integration of DG into the grid. In reference [24],
a DLC-based controller with a DC-side voltage regulator in a hierarchical primary control structure is
presented for an islanded micro grid. Adoption of DLC ensures asymptotic stability of the system,
and in the meantime makes it robust against parameter uncertainty and disturbance. In references [25,
26], current control strategies based on differential flatness control (DFC) are proposed for VSC
rectifier (VSR) and MMC, respectively. Through design of appropriate flat outputs and planning of
the reference trajectories, the control objectives are achieved. At the same time, system robustness is
guaranteed by feedback of the control errors and their integrals. An interconnection and damping
assignment passivity-based controller is proposed for VSC-HVDC system in reference [27]. With this
controller, influence of the equivalent resistance of the DC side is eliminated, and as a result dynamic
performance of the system is remarkably improved. A droop-passivity-based controller is proposed
for grid-connected single-phase VSI to achieve high performance in the presence of nonlinear loads in
reference [28]. By design of the power capability curves and the current reference generation scheme,
precise power sharing and harmonic compensation are achieved. However, in order to achieve
accurate current tracking control under UBGV conditions, the resonant law is necessary. Using the
above nonlinear methods will probably bring difficulties in constructing the Lyapunov function and
design of the controllers.
As a matured nonlinear control method, sliding mode control (SMC) can simplify the system
design and is robust against parameter uncertainty and disturbance. As a result, the SMC has
been extensively applied the control of power electronic converters [29–32]. In references [6,29] and
reference [9], SMC-based direct power control (DPC) strategies and a current control strategy are
presented for DFIG and VSC-HVDC system, respectively. In references [4,30], integral SMC-based
DPC and current control strategies are presented for VSC-MVDC system and static synchronous

Energies 2018, 11, 2599

3 of 20

compensator respectively, where the integral actions are included to achieve accurate power and
current control. As the output active and reactive power should contain sinusoidal components to
obtain non-distorted AC currents under UBGV conditions, in reference [31] an integral plus resonant
SMC-based direct power controller is presented for VSC-HVDC system, to fully eliminate the steady
state errors. In reference [32], a controller including three sub-loops with a combined utilization of
the passive-injection, DLC and SMC is presented for the MMC-HVDC system. The simulation results
show that satisfactory performance is achieved, and at the same time, the circulating currents are
significantly reduced and balanced capacitor voltages for the sub-modules are obtained.
In practice, parameters of the controller are the trade-offs between system robustness and the
equivalent control bandwidth. One effective way to improve system robustness without obvious
dynamic performance reduction is to eliminate the adverse effect of parameter uncertainty. To this
purpose, adaptive control-based current control strategies are studied. In references [8,33], adaptive
current control strategies for VSC-HVDC system and VSI are proposed respectively. However,
the control law and the adaptation law are coupled together. This not only entails a large amount of
parameter tuning work, but system performance depends heavily on the behavior of the adaptation
law. In reference [34], an MPC-based current control strategy with online disturbance observer (ODO)
is proposed for VSR. The ODO is used to identify the lumped effects of parameter uncertainty and
disturbance; however, its realization is complex. In reference [35], a predictive control-based sensorless
current control strategy for VSR is proposed. However, appropriate excitation signals should be
injected to obtain accurate inductance parameters, which complicates the implementation.
On the other hand, the control objectives are important for the operation of VSC-MVDC system
under UBGV conditions. In the reported current control strategies, the following four objectives are
often applied [8,36], i.e., (1) eliminating the active power ripples at the point of common coupling
(PCC); (2) eliminating the reactive power ripples at the PCC; (3) obtaining balanced AC current; and (4)
eliminating DC bus voltage ripples. Under general circumstances, these four objectives can meet related
requirements. However, they are not sufficient for system operation optimization. In reference [36],
a current control strategy to realize soft switching between the first three objectives is presented for DG.
However, the oscillating amplitude ratio (OAR) of the active power ripple to the reactive power ripple
cannot be continuously controlled. In reference [13], an optimal current control strategy under UBGV
conditions is proposed for VSC-HVDC system. In reference [37], a flexible current control strategy to
realize coordinated control of the power oscillations and the current quality is proposed. Although the
OARs presented in references [13,37] can be controlled continuously, current distortion happens.
This study aims to present a novel proportional-integral-derivative-resonant law-based sliding
mode current control strategy with online inductance estimator (PIDR-SMCC-OIE) for VSC-MVDC
system converter station under UBGV conditions. The three main contributions of this paper are as
follows. First, a generalized current reference calculation (GCRC) method, by which the OAR of the
active power ripple to the reactive power ripple can be continuously controlled, is proposed. Second,
a PIDR law-based sliding mode controller is proposed for the CS to achieve accurate current control,
where the derivatives of the current references are obtained by simple algebraic operations. Third,
to further improve system robustness, an OIE based on the dynamic filtering method and the gradient
algorithm is presented. This OIE utilizes the gate signals of the switching devices and the DC bus
voltage to compute the converter pole voltages, and thus no additional voltage sensors are needed.
Finally, simulation studies on a two-terminal VSC-MVDC system are performed in PSCAD/EMTDC
to verify the effectiveness of the PIDR-SMCC-OIE strategy.
The remainder of the paper is arranged as follows. In Section 2, a mathematic model of the CS
under UBGV conditions is developed, power flow analysis is conducted, and the GCRC method is
derived. In Section 3, the PIDR-SMCC controller and the OIE is designed. In Section 4, simulation
results on a two-terminal VSC-MVDC system are presented, and in Section 5 conclusions are drawn.

Energies
Energies 2018,
2018, 11,
11,xx FOR
FOR PEER
PEER REVIEW
REVIEW

44 of
of 20
20

Energies 2018, 11, 2599

4 of 20

EEg1g1

IIg1g1
PPg1g1++jQ
jQg1g1

20
20kV
kV

++

PCC2
PCC2

YYgg

--

CS1
CS1

IIg2g2
PPg2g2++jQ
jQg2g2

CS2
CS2

EEg2g2

DGs
DGs

Loads
Loads

TT11

--

RRg2g2==84
84mΩ

LLg2g2==12
12mH
mH

15
15MVA
MVA
110/10
110/10kV
kV
RRTT=0.003
=0.003
XXTT=0.08
=0.08

TT22

Filter
Filter
200kVA
kVA
200

++

=1mF
CC11=1mF

RRg1g1==84
84mΩ

LLg1g1==12
12mH
mH

YYgg

Filter
Filter
200kVA
kVA
200

AC
AC
B
System
System11 B11

10
10km
km
RRdd=0.12
=0.12Ω/km
Ω/km
LLdd=0.1
=0.1mH/km
mH/km

=1mF
CC22=1mF

~~

15
15MVA
MVA
110/10
110/10kV
kV
RRTT=0.003
=0.003
XXTT=0.08
=0.08

110
110kV
kV
50
50Hz
Hz
1500
MVA
1500 MVA

~~

110
110kV
kV
50
50Hz
Hz
1500
MVA
1500 MVA

PCC1
PCC1

2.
2. Mathematic
Mathematic Model
Model of
of the
the Studied
Studied VSC-MVDC
VSC-MVDC System
System
2. Mathematic Model of the Studied VSC-MVDC System
2.1.
2.1. Schematic
Schematic of
of the
the VSC-MVDC
VSC-MVDC System
System Studied
Studied
2.1. Schematic of the VSC-MVDC System Studied
The
The single-line
single-line diagram
diagram of
of the
the two-terminal
two-terminal VSC-MVDC
VSC-MVDC system
system studied
studied is
is shown
shown in
in Figure
Figure 1.
1.
The
mainly
stations
(CS1
CS2),
AC
inductor
and
The single-line
diagramof
theconverter
two-terminal
VSC-MVDC
studied
is shown
in Figure
1.
The system
system
mainly consists
consists
ofoftwo
two
converter
stations
(CS1 and
andsystem
CS2), the
the
AC filtering
filtering
inductor
and
DC
capacitor
each
and
the
To
lines,
there
The
system
mainlyfor
consists
of two
stations (CS1 lines.
and CS2),
theDC
AC transmission
filtering
inductor
and
DC
DC bus
bus
capacitor
for
each CS,
CS,
andconverter
the DC
DC transmission
transmission
lines.
To the
the
DC
transmission
lines,
there
may
be
loads
and
DGs.
figure,
are
listed.
Those
with
bus capacitor
for each
CS,
and
the In
DC
transmission
lines.
To the DCparameters
transmission
there
may
be
may
be connected
connected
loads
and
DGs.
In the
the
figure, the
the main
main technical
technical
parameters
arelines,
listed.
Those
with
aaconnected
unit
the
physical
values
and
without
aa unit
denote
per-unit
loads denote
and
DGs.
the figure,
the main
technical
parameters
listed.
Thosethe
with
a unit
unit provided
provided
denote
theInactual
actual
physical
values
and those
those
withoutare
unit
denote
the
per-unit
values.
provided
denote
the
actual
physical
values
and
those
without
a
unit
denote
the
per-unit
values.
values.

AC
AC
BB22 System
System22

Figure
Figure 1.
1. Single-line
Single-line diagram
diagram of
of the
the two-terminal
two-terminal voltage
voltage source
source converter
converter medium
medium voltage
voltage direct
direct
Single-line
diagram
current
(VSC-MVDC) system
system studied.
studied.
current (VSC-MVDC)

Theschematic
schematicof
ofthree-phase
three-phaseVSC
VSCfor
foreach
each
CS
shown
Figure
2.
In
figure,
egcgc
,and
egc ,
The
CS
is
shown
in
Figure
2.
figure,
eegaga,,eega
gb
,, and
gb
The
schematic
of
three-phase
VSC
for
each
CS
isis
shown
inin
Figure
2. In
In the
thethe
figure,
gb,,, ee
i
,
i
,
i
denote
the
grid
voltage
and
current,
respectively;
u
,
u
,
u
denote
the
converter
pole
iiand
ga
,
i
gb
,
i
gc
denote
the
grid
voltage
and
current,
respectively;
u
ca
,
u
cb
,
u
cc
denote
the
converter
pole
ga
gc
ca
cc
gb
cb
ga, igb, igc denote the grid voltage and current, respectively; uca, ucb, ucc denote the converter pole
voltage,
R
andLLLgggdenote
denotethe
theresistance
resistanceand
andinductance
inductance of
of
the
AC
filtering
denote
voltage,
dc
voltage, R
Rggg and
and
denote
the
resistance
and
inductance
of the
the AC
AC filtering
filtering inductor;
inductor; CC and
and uudc
dc denote
denote
the DC
DC bus
bus capacitor
capacitor
and
voltage
respectively.
the
the
DC
bus
capacitor and
and voltage
voltage respectively.
respectively.
T1
T1
eegaga
eegbgb
eegcgc

~~
~~
~~

RRgg

LLgg i
igaga

RRgg

LLgg i
igbgb

RRgg

LLgg i
igcgc

T3
T3

T5
T5

uucaca

++
CC
uudcdc

uucbcb
uucccc
T2
T2

T4
T4

--

T6
T6

Figure
Figure 2.
2. Schematic
Schematic of
of three-phase
three-phase VSC
VSC for
for each
each converter
converter station.
station.

The
CSare
areidentical,
identical,so
controllers are
The structures
structures of
of the
the current
current control
control loops
loops for
for each
each CS
CS
are
identical,
so that
that their
their controllers
are
designed
designed under
under aa unified
unified framework.
framework. In
Inaddition,
addition,assume
assumethe
the grid
grid voltage
voltage on
on each
each side
side contains
contains the
the
fundamental
components only
only and
and that
that the
the grid
grid frequency
fundamental components
frequency is
is 50
50 Hz
Hz ifif not
not otherwise
otherwise specified.
specified.
2.2. Mathematical
MathematicalModel
Model of
of the
the CS
CS
2.2.
2.2.
Mathematical
Model
of
the
CS
Underunbalanced
unbalancedgrid
gridvoltage
voltageconditions,
conditions,
the
space
vectors
of
grid
voltage,
current
Under
the
space
vectors
of
grid
voltage,
current
and
the
Under
unbalanced
grid
voltage
conditions,
the
space
vectors
of the
thethe
grid
voltage,
current
andand
the
the
converter
pole
voltage
can
be
decomposed
into
the
PS
and
negative
sequence
(NS)
components.
converter
pole
voltage
can
be
decomposed
into
the
PS
and
negative
sequence
(NS)
components.
converter pole voltage can be decomposed into the PS and negative sequence (NS) components. In
In
In Figure
3, the
space
vector
of
grid
voltage
in
and
SRF
is shown
as
example.
Figure
3,
space
vector
of
grid
voltage
in
PS
and
NS
SRF
is
as
example.
Figure
3, the
the
space
vector
of the
thethe
grid
voltage
in the
thethe
PSPS
and
NSNS
SRF
is shown
shown
as an
anan
example.

Energies 2018, 11, 2599
Energies 2018, 11, x FOR PEER REVIEW

5 of 20
5 of 20

β

q+

ω
ω

q−
−ω
Eg++
Eg

ω

θ
−θ

−ω

Eg−−

−ω

d+

α
d−

Figure 3.
3. Space vector of the gird voltage in the positive sequence (PS) and negative sequence (NS)
synchronous
synchronous reference
reference frame
frame (SRF).
(SRF).

In
denote the
the PS
denote
In Figure
Figure 3,
3, the
the superscript
superscript++ and
and−
− denote
PS and
and NS
NS SRF,
SRF, and
andthe
thesubscript
subscript++and
and−
− denote
the
PS
and
NS
voltage
and
current
components,
respectively.
We
can
see
from
the
figure
the
PS
voltage
the PS and NS voltage and current components, respectively. We can see from the figure the PS
components
rotate in rotate
the counter-clockwise
direction direction
with the synchronous
angular speed
ω, the
NS
voltage components
in the counter-clockwise
with the synchronous
angular
speed
voltage
components
rotate
in
the
clockwise
direction
with
an
angular
speed
of

ω.
ω , the NS voltage components rotate in the clockwise direction with an angular speed of −ω .
In
In the
the STP
STP RF,
RF, the
thespace
spacevectors
vectorsof
ofthe
thegrid
gridvoltage
voltageEgαβ
Egαβ ,, grid
grid current
current Igαβ
I gαβ ,, and
and the
the converter
converter
pole voltage Ucαβ can be expressed as:
pole voltage U cαβ can be expressed as:


= E+ ++ ejωt
+ Egdq
e−jωt


 Egαβ
 Egαβ =gdq
Egdq+ e jωt + Egdq
e−− jωt

+

 = Igdq
Igαβ
ejωt + I−gdq
e−jωt ,
(1)

+ + jω t
− jω t

,
e
(1)
 I gαβ = +I gdq+ e jωt + I gdq −



jωt
Ucαβ
−−ejω t
 = Ucdq++ e jωt + Ucdq

U cαβ = U cdq+ e + U cdq − e
+
+
+



where Egdq
, Igdq
, Ucdq
and Egdq
, Igdq
, U−cdq
− denote the PS and NS space vectors of the grid
+ +
+ +
+ +
− −
− −
I
U
E
I
U
,
,
and
,
,
denote the PS and NS space vectors of the grid
where Egdq
+
+
+



gdq
cdq
gdq
gdq
cdq
voltage, current and the converter pole voltage, respectively, with:
voltage, current and the converter pole voltage, respectively, with:
+
+
+



Egdq
(2)
+egd+ +
+ jegq++, Egdq
− − = e−gd− +−jegq− ,
+ =
Egdq+
= egd
(2)
+ + jegq+ , Egdq − = egd − + jegq − ,
+
+
+



= −igd
Igdq
jigq
++, Igdq
+ = +igd+ +
− +− jigq− ,
− −
I gdq+ = igd+ + + jigq+
, I gdq
=
i
+
j
i

gd −
gq − ,
+
+
+



Ucdq
=
u
+
ju
,
U
=
u
cq+
+
cd+
cdq−
cd− + jucq− ,

(3)
(3)
(4)

+
+
+



(4)
+
+
+
+
+ U cdq+
+ = ucd + + jucq+ , U cdq − = ucd − + jucq − ,
and egd
,
i
,
u
and
e
,
i
,
u
denote
the
d
and
q-axis
PS
components of the grid voltage,
gq
+
gq
+
cq
+
+ gd+ cd+






+
+
+
+
+
current
and+ the converter
pole
voltage
in the PS SRF, respectively; egd
and
egq
of−the
grid
voltage,
and egd
− , igq− ,
− , igd− , ucd
+ , igd + , ucd + and egq + , igq + , ucq + denote the d and q-axis PS components


− converter


u
and q-axispole
NS components
of the
currenteand
the
pole− voltage
cq− denote
current
and the
thedconverter
voltage in the
PS grid
SRF,voltage,
respectively;
gd − , igd − , ucd − and egq − , igq − ,
in−the NS SRF, respectively.
ucq − denote the d and q-axis NS components of the grid voltage, current
and the converter
pole
+
+
Therefore, in the PS SRF, the space vectors of the grid voltage Egdq
, grid current Igdq
and the
voltage in the NS SRF, respectively.
+
converter pole voltage Ucdq
can be expressed as:
+
+
Therefore, in the PS SRF, the space vectors of the grid voltage Egdq
, grid current I gdq
and the
+
+ be expressed
+
+ as: +

−j2ωt
can
converter pole voltage U cdq
Egdq
= egd
+ jegq
= Egdq+ + Egdq
,
−e

(5)

+
+
+
+

− j2ω t
,
+ Egdq+= egd +
+ jegq =+Egdq+ + E
−gdq − e −j2ωt
Igdq
= igd + jigq
= Igdq+ + Igdq
e
,


(5)
(6)

++
− − − j2ω−
t j2ωt
+
+ +
++
ji=
= cdq
I gdq++ ++IU
Ucdq
= Iugdq
+igd+ju+cq
,
gq U
gdqcdq
−e −e ,
cd =

(7)
(6)

+ +
+
+ , i + , u+ denote the d and q-axis components of the grid voltage, current and
where egd
, igd , ucd
and egq
+
+

− j2ω t
gq cq
U cdq
= ucd+ + jucq+ = U cdq
,
(7)
+ + U cdq − e
the converter pole voltage in the PS SRF, respectively, and the term e−j2ωt denotes the clockwise
+
+
+
+
rotational
operator
twiceegq
the
frequency.
, igd+ , ucd+ ofand
, igrid
where egd
gq , ucq denote the d and q-axis components of the grid voltage,

current and the converter pole voltage in the PS SRF, respectively, and the term e − j2ωt denotes the
clockwise rotational operator of twice the grid frequency.
In the STP RF, the dynamic model of VSC in space vector from is:

Energies 2018, 11, 2599

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In the STP RF, the dynamic model of VSC in space vector from is:
Lg

dIgαβ
= Egαβ − Rg Igαβ − Ucαβ ,
dt

(8)

Combine Equations (1) and (5) to Equations (7) and (8), mathematic model of the VSC under
UBGV conditions in the PS SRF can be obtained as:

+
 L digd = e+ − R i+ + ωL i+ − u+
g dt
g gd
g gq
gd
cd
,
(9)
+
 L digq
+
+
+
+
= e − R i − ωL i − u
g dt

g gq

gq

g gd

cq

When the grid voltage is balanced, the NS components of the grid voltage, current and the
converter pole voltage are null, which means the dynamic equation in this situation is a special case of
the one when the voltage is unbalanced.
2.3. Instantaneous Power Flow Analysis
When the grid voltage is unbalanced, instantaneous power at the PCC can be calculated as:
Sg =



3 + +
3 +

+
−j2ωt I −
Egdq+ + e−j2ωt Egdq
Egdq Igdq =
I
+
e

gdq
gdq− ,
2
2

(10)

+
denotes the complex conjugate of the current vector.
where Igdq
Substitute Equations (2) and (3) into Equation (10) and through some mathematical manipulations,
the active power and reactive power at the PCC can be obtained as:

Sg



+



+
+
j2ωt E+
−j2ωt E−
= Pg + jQg = 23 Egdq
gdq+ Igdq− + e
gdq− Igdq+
+ Igdq+ + Egdq− Igdq− + e


,
= Pg0 + Pg cos 2 cos(2ωt) + Pg sin 2 sin(2ωt) + j Qg0 + Qg cos 2 cos(2ωt) + Qg sin 2 sin(2ωt)

(11)

where Pg0 and Qg0 denote the average components, Pgcos2 , Pgsin2 and Qgcos2 , Qgsin2 denote the ripple
components of the active power and reactive power respectively.
For convenience, the power terms can be expressed in the following compact form:


Pg0
 P
 g cos 2
 P
 g sin 2

 Qg0

 Qg cos 2
Qg sin 2





+
egd
+

 −
 egd−





3
3
 egq−

 = Mdq Idq =  +

2
2  egq+


 e−

 gq−

−egd



egd


+
egq
+

egq


−egd

+
−egd
+

−egd


−egq


+
egd
+

+
−egq
+

egq

+
egq
+
+
egd
+


egq




+
egq
+












+
egd
+


−egd

+
−egd
+
+
egq
+

+
igd
+
+
igq
+

igd


igq





,



(12)

From Equation (12) we can see that it is the interaction between the PS (NS) voltage and the NS
(PS) current that produces the active and reactive power ripples.
Similarly, the active power ripple terms at the AC side of VSC can be calculated as:
"

Pccos2
Pcsin2

#

3
=
2

"


ucd


ucq



ucq


−ucd


+
ucd
+
+
−ucq
+

+
ucq
+
+
ucd+

#
Idq ,

(13)

where Pccos2 and Pcsin2 denote the instantaneous active power ripples, which are sinusoidal signals of
twice the grid frequency.

Energies 2018, 11, 2599

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Similarly, the apparent power SL , active power PL and the reactive power QL , consumed in the
AC filtering inductor can be calculated as:
SL


+
dIgdq
+
Igdq
= PL + jQL = 32 Lg dt
h



i
+

+

+

+

= 3ωLg Igd
I

I
I
cos
2ωt

I
I
+
I
I
sin
2ωt
+ ,
(
)
(
)
gq+ gd−
gq+ gq−
+ gq−
gd+ gd−
h
i
3ωLg +2
+2
−2
−2
j 2 Igd+ + Igq
+ − Igd− − Igq−

(14)

It can be seen from Equation (14) when the AC current is unbalanced, the instantaneous reactive
power consumed in the inductor is constant, while the instantaneous active power it consumed is not
zero but sinusoidal components of twice the grid frequency.
2.4. Current Reference Calculation
2.4.1. Conventional Current Reference Calculation Method
For convenience, the conventional current control objectives (OBJ) without current distortion and
the corresponding current reference calculation methods are directly given below.
OBJ 1: Eliminating the active power ripples at the PCC
To achieve this objective, the current references are calculated as:

+∗

igd

+



 +∗
igq+
−∗

 igd




 i−∗
gq−

h

i
∗ / 1.5e+
2 − k2
= Pg0
1

k
qd i
h gd+ dd
+
2

= − Qg0 / 1.5egd+ 1 + kdd + k2qd
+∗
+∗
= −kdd igd
+ − k qd igq+
+∗
+∗
= kdd igq+ − kqd igd+

,

(15)

,

(16)


+

+
where kdd and kqd are: kdd = egd
− /egd+ , k qd = egq− /egd+ .

OBJ 2: Eliminating the reactive power ripples at the PCC
Similarly, the current references of this objective are:

+∗

igd

+



 +∗
igq+
−∗


igd




 i−∗
gq−

h

i
∗ / 1.5e+
= Pg0
1 + k2dd + k2qd
gd
+
h

i
∗ / 1.5e+
2 − k2
= − Qg0
1

k
dd
qd
gd+
+∗
+∗
= kdd igd
+ + k qd igq+
+∗
+∗
= −kdd igq
+ + k qd igd+

OBJ 3: Obtaining three-phase balanced current
The current references for this objective are:

+∗

igd

+


 i+∗
gq+
−∗

i
 gd−


 i−∗
gq−

∗ /1.5e+
= Pg0
gd+
∗ /1.5e+
= − Qg0
gd+
,
=0
=0

(17)

OBJ 4: Eliminating DC bus voltage ripples
The active power ripples pass through the VSC should be eliminated to achieve constant DC bus
voltage. To this purpose, we can have:

Energies 2018, 11, 2599

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Pg0
Qg0
Pc cos 2
Pc sin 2





+
egd
+

 +

3  egq

+
= 

2

u
 cd−

ucq


+
egq
+
+
−egd
+

ucq


−ucd



egd



egq



egq





−egd

+
ucq
+
+
ucd
+

+
ucd
+
+
−ucq
+

+
igd
+

 +
  igq+

  i−
  gd−

igq





,



(18)

Then, the current references for this control objective can be obtained as:


+∗
igd
+

 +∗
 igq+

 i−∗
 gd−
−∗
igq






+
egd
+


 +


 = 2  egq+


3

 ucd−

ucq


+
egq
+
+
−egd
+

ucq


−ucd



egd


egq


+
ucd
+
+
−ucq
+


egq


−egd

+
ucq
+
+
ucd
+

 −1 
P∗
  g0
  Q∗
  g0
  0

0




,


(19)

which indicates the NS current components are necessary to eliminate the DC bus voltage ripples.
2.4.2. The Proposed GCRC Method
The previous four control objectives can meet the related requirements for many applications.
However, they are not sufficient for system optimization. In this paper, a GCRC method is presented
to make the OAR of the active power ripple to the reactive power ripple continuously controllable,
namely, OBJ 5. The current references of this GCRC method are calculated as:

+∗

igd

+



 +∗
igq+
−∗


igd




 i−∗
gq−

h


i
∗ / 1.5e+
= Pg0
1 − ksk k2dd + k2qd
gd
+
h


i
∗ / 1.5e+
2 + k2
= − Qg0
1
+
k
k
sk
dd
qd
gd+
+∗
+∗
= −ksk kdd igd
+ − k sk k qd igq+
+∗
+∗
= ksk kdd igq
+ − k sk k qd igd+

,

(20)

where ksk denotes the slack coefficient.
Substitute Equation (20) into Equation (12), we have:


, Qg0 = Qg0
,
Pg0 = Pg0

(21)






kdd Pg0
kqd Qg0



,
Pg cos 2 = (1 − ksk )
2
2
2
2
1 − ksk kdd + kqd
1 + ksk kdd + kqd




kqd Pg0
kdd Qg0

+

,
Pg sin 2 = (1 − ksk )
1 − ksk k2dd + k2qd
1 + ksk k2dd + k2qd




kdd Qg0
kqd Pg0

+

,
Qg cos 2 = (1 + ksk )
1 − ksk k2dd + k2qd
1 + ksk k2dd + k2qd




−kdd Pg0
kqd Qg0
+
,


Qg sin 2 = (1 + ksk )
1 − ksk k2dd + k2qd
1 + ksk k2dd + k2qd

(22)

(23)

(24)

(25)

From Equation (21) we can see Pg0 and Qg0 are not affected by ksk . When ksk = 1, Pgsin2 and
Pgcos2 are zero, Qgsin2 and Qgcos2 are at their peaks Qgsin2m and Qgcos2m . When ksk = −1, Qgsin2 and
Qgcos2 are zero, Pgsin2 and Pgcos2 are at their peaks Pgsin2m and Pgcos2m . As kdd and kqd are very small
in practice, the amplitude of the active power ripples and that of the reactive power ripples change
almost linearly with ksk , and moreover, if the influences of the square terms of kdd and kqd in the

Energies 2018, 11, 2599

9 of 20

denominators
neglected,
it can be inferred from Equation (22) to Equation (25) Pgcos2m = − Qgsin2m
Energies 2018, 11, are
x FOR
PEER REVIEW
9 of 20,
Pgsin2m = Qgcos2m .
In Figure
Figure4,4,an
anexample
exampleisisgiven
given
show
relationships
between
Pgsin2
, Pgcos2
, gsin2
Qgsin2, ,QQgcos2
gcos2 and
and
In
toto
show
thethe
relationships
between
Pgsin2
, Pgcos2
,Q
*∗
*∗
−0.2
p.up.u.
sk. In
, kkqd
0.006
, P
, QQg0g0==
.
= −−
0.13
kksk
In the
the figure,
figure, kkdd
0.13,
=−−
0.006,
Pg0g0==0.9
0.9p.u
p.u,
−0.2
qd =
dd =

Figure
4. An
example
to to
show
thethe
relationships
between
Pgsin2
, P, gcos2
Qgcos2
andkskk.sk .
gcos2and
Figure
4. An
example
show
relationships
between
Pgsin2
Pgcos2,,QQgsin2
gsin2,, Q

Suppose
, Qgsin2m
and
Pgsin2m
, Qgcos2m
AmBand
Bm respectively,
thenhave:
we have:
Suppose PPgcos2m
gcos2m, Qgsin2m
and
Pgsin2m
, Qgcos2m
are Aare
m and
m respectively,
then we
AmA
B Bm
Pg cos 2P=
= (1 − k )m , ,
(1 − kskk)sk ) ,m P, gPsin
g cos 2 = (1 −
g sin22 = (1 − ksk )sk
22
2 2

(26)
(26)

Bm
Am
Qg cos 2 = (1 + ksk ) B, mQg sin 2 = −(1 + kskA)m ,
(27)
Qg cos 2 = (1 + ksk 2)
, Qg sin 2 = − (1 + ksk )
2,
(27)
2
2
The amplitudes of the active power ripples Prplam and the reactive power ripples Qrplam can be
The as:
amplitudes of the active power ripples Prplamqand the reactive power ripples Qrplam can be
obtained
(1 − ksk )
obtained as:
A2m +B2m ,
(28)
Prplam =
2
q
(1(1+−kksksk )) A 22 +B2 2,
(28)
Prplam
Qrplam
==
Amm +Bm m ,
(29)
22
From Equations (28) and (29) we can see the amplitudes of the reactive power ripples decrease
(1 + ksk ) ripples
from the maximum to zero and that of the
active
(29)
Qrplam
= power
A 2m +B2m ,increase from zero to the maximum,
2
as ksk changes from 1 to −1. When ksk = 0, the amplitudes of the active power ripples and the reactive
power
ripples
equal, which
corresponds
to see
the the
situation
of obtaining
balancedpower
AC currents.
From
Equations
(28) and
(29) we can
amplitudes
of the reactive
ripples decrease
from the maximum to zero and that of the active power ripples increase from zero to the maximum,
3.
Design
asPIDR-SMCC-OIE
ksk changes from Controller
1 to −1. When
ksk = 0 , the amplitudes of the active power ripples and the

reactive
power
equal,
which
corresponds
the situation of obtaining balanced AC currents.
3.1.
Dynamics
of ripples
the Current
Control
Errors
in the PSto
SRF
+∗
+∗ , and the respective current control errors
Define the d and Controller
q-axis current
references as igd
and igq
3. PIDR-SMCC-OIE
Design
+
+
+
+
as iged and igeq respectively. Taking iged and igeq as the state variables, the dynamics of the current
3.1. Dynamics
of thebeCurrent
Control
control
errors can
obtained
as: Errors in the PS SRF

+*
+
+∗ respective current control
i +*+, anddigd
Define the d and q-axis
references
the
diged
+
+as igd and
 Lcurrent
+ − gq
=
e

R
i
+
ωL
i
u

g
g
g
gq
dt
dt
gd
gd
cd
+
+
+
,
and igeq
respectively.
Taking iged
and i + as the state +∗
variables,
the dynamics of(30)
the
errors as iged
+
 L digeq
+ − R i + − ωLgeqi + − u+ − digq
=
e
g dt
g gq
g gd
gq
cq
dt
current control errors can be obtained as:
+
where these include the derivatives
current references.
 ofdithe
digd+*
ged
+
+
+
+
L
e
R
i
L
i
u
=

+
ω



g
gd
g gdthe PSg SRF
gq
cd
The NS current references can
expressed
in
as:
 be
dt
dt
,

"
# di"+
#" di +* #
+∗ 
−∗
geq
gq
+
+
+
+
igd− Lg
cos
egq(2ωt
igd −)ucq − igd−
=
− Rg)igq −sin
ω L(g2ωt

,
t
d−∗
+∗  = dt − sin(2ωt ) cos(2ωt )
igq−
igq

where these include the derivatives of the current references.
+∗
+∗
whereThe
igd
igq
the d and
NS current
references
NSand
current
references
canq-axis
be expressed
in the
PS SRF in
as:the PS SRF, respectively.
− are


igd+*−   cos ( 2ωt ) sin ( 2ωt )  igd−*− 
 +*  = 
  −*  ,
igq −   − sin ( 2ω t ) cos ( 2ω t )  igq − 

where igd+*− and igq+*− are the d and q-axis NS current references in the PS SRF, respectively.

(30)
(31)

(31)

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The current in the PS SRF can be expressed as:
"

+∗
igd

#

"

=

+∗
igq

+∗
+∗
igd
+ + igd−

#

+∗
+∗
igq
+ + igq−

,

(32)

+∗
+∗
As igd
+ and igq+ are DC components, then we have:
+∗
digd

+∗
+∗
+∗
digd
digq
− digq

=
,
=
,
dt
dt
dt
dt

(33)

Combine Equation (31) to Equation (33), the derivatives of the d- and q-axis current references in
the PS SRF can be obtained as:
+∗
digd

dt

h
i
−∗
−∗
= 2ω −igd
sin
2ωt
+
i
cos
2ωt
(
)
(
)
gq−

+∗
= 2ωigq


+∗
digq

dt

,

(34)

h
i
−∗
−∗
= 2ω −igd
cos
2ωt

i
sin
2ωt
(
)
(
)
gq−


(35)

+∗
= −2ωigd


which means that the derivatives of the current references can be obtained through simple algebraic
operations, which simplifies the implementation of the controller.
3.2. PIDR-SMCC Current Controller Design
Transform the current control error equations Equation (30) into the following compact form:
dx
dxr
= F(x) + G(x)u + δ(t, x) −
,
dt
dt
+
where x = [ iged

T

+
igeq
] denotes the state vector, xr = [ xrd

T

xrq ] denotes the current reference

T

vector, u = [ ud+ uq+ ] denotes the control input vector, and F(x) = [ f d+ (x)
nonlinear functions in the dynamic model, with:


 f+ =
d

 f q+ =
G(x) =

h

+
ggd
(x)

δ(t, x) =

+
egd
Lg
+
egq
Lg




+ (x)
ggq

h

Rg +
+
Lg igd + ωigq
Rg +
+
Lg igq − ωigd

iT

+
δgd
(t, x)

T

f q+ (x) ] denotes the

,

(37)

= −diag(udc , udc )/Lg ,

(38)

+ ( t, x)
δgq

iT

,

(39)

+
where diag(a1 , a2 ) denotes a diagonal matrix with a1 and a2 as the main diagonal elements, δgd
(t, x)
+ ( t, x ) denote the lumped uncertainties in the d- and q-axis current control sub-systems.
and δgq
In the following, the arguments of various functions will not be written for convenience.
+
+ satisfy:
Assume δgd
and δgq



+
+
δgd ≤ ρmgd , δgq
≤ ρmgq ,
with ρmgd and ρmgq denoting the upper bounds of the respective uncertainties.

Energies 2018, 11, 2599

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Design the sliding manifold (SM) for the d- and q-axis current control subsystems as:
"
s=

sd
sq

#

"

=

#

+
iged

" R

+
iged

R

+
igeq

+ KPI

+
igeq

#

"

+ KR HGI

+
iged

#!

+
igeq

,

(40)

with:


KPI = diag KPId , KPIq , KR = diag KRd , KRq ,
!
2ωc p
2ωc p
,
,
HGI = diag
p2 + 2ωc p + ωR2 p2 + 2ωc p + ωR2

(41)
(42)

where p denotes the Laplace operator, ωR = 2ω denotes the resonant frequency, ωc denotes the
bandwidth parameter of the resonant law, KPId , KPIq , KRd and KRq are positive real numbers.
The derivative of s is:
"
# "
#
"
#
"
#!
ds
dsd /dt
dx1 /dt
x1
x1
=
=
+ KPI
+ KR HDGI
,
(43)
dt
dsq /dt
dx2 /dt
x2
x2
with:
HDGI = diag

!
2ωc p2
2ωc p2
,
,
p2 + 2ωc p + ωR2 p2 + 2ωc p + ωR2

(44)

Utilizing V = sT s/2 as the Lyapunov function candidate, then the derivative of V along system
state trajectories can be obtained as:

dV
ds
= sT
= sT [F + Gu + KPI x + KR HDGI (x) + δ],
dt
dt

(45)

Design the control law u as:
h
i
u = −G−1 F + KPI x + KR HDGI (x) + Ks sT + us ,
with:
h
us = η sgn(sd )

h
iT
= diag ηd , ηq sgn(sd ) sgn sq
,

Ks = diag Ksd , Ksq ,

sgn sq

iT

(46)

(47)
(48)

where ηd > ρmd , ηq > ρmq and Ksd > 0, Ksq > 0.
Then, we have:



dV
≤ −sT Ks s − ηd − ρmgd |sd | − ηq − ρmgq sq ≤ 0,
dt

(49)

which indicates that the system is stable.
From the definition of the SM and Barbalat0 s Lemma it can be inferred that the DC components of
the current control errors will converge to zero. In the following, we will prove convergence of the
sinusoidal components to zero using the method of reduction to absurdity.
+
+ will not converge to zero. Then, there will
Suppose the sinusoidal components of iged
and igeq
be sinusoidal components in sd and sq , and consequently, V will fluctuate in a sine-wave pattern.
From Equation (49) we know that V is non-increasing, which contradicts the assumption. Therefore,
+
+ will converge to zero.
iged
and igeq

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Energies 2018, 11, x FOR PEER REVIEW

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3.3.
3.3. Online
Online Inductance
Inductance Estimator
Estimator Design
Design
The
The grid
grid voltage,
voltage, current
current and
and the
the converter
converter pole
pole voltage
voltage always
always satisfy
satisfy the
the Kirchhoff’s
Kirchhoff’s voltage
voltage
law.
This
feature
can
be
used
for
AC
line
parameter
estimation.
The
AC
voltages
of
the
VSC
(u
law. This feature can be used for AC line parameter estimation. The AC voltages of the VSC (ucabc
cabc)) is
is
required
required for
for the
the OIE
OIE design.
design. In
In this
this paper
paper we
we utilize
utilize the
the gate
gate signals
signals of
of the
the switching
switching devices
devices and
and the
the
DC
voltage to
to compute
computeuucabc
, instead
measuring
that
additional
sensors
needed.
cabc
DC bus
bus voltage
, instead
ofof
measuring
it, it,
soso
that
nono
additional
sensors
areare
needed.
To
To
this
purpose,
we
firstly
change
the
non-ideal
converter
bridge
into
an
ideal
one
as
shown
in
Figure
5.
this purpose, we firstly change the non-ideal converter bridge into an ideal one as shown in Figure 5.

~

egabc

Lg igabc

Rt

ucabc

C

+

-

udc

ideal switch.
switch.
Figure 5. Schematic of VSC system with ideal

In the
the figure,
figure, R
Rtt represents the overall
overall resistance
resistance of each
each AC
AC line
line branch,
branch, including
including the effects
effects
introduced
drop
of of
each
arm,
ucabc
represents
thethe
converter
polepole
voltage
of this
introducedby
bythe
theforward
forwardvoltage
voltage
drop
each
arm,
ucabc
represents
converter
voltage
of
ideal
VSC.VSC.
Meanings
of the
symbols
are the
as those
defined
in Figure
2. 2.
this ideal
Meanings
ofother
the other
symbols
aresame
the same
as those
defined
in Figure
3.3.1.
3.3.1. Inductance
Inductance Parameter
Parameter Estimation
Estimation Model
Model
In
In the
the a-b-c
a-b-c RF,
RF, the
the current
currentdynamic
dynamicmodel
modelis:
is:
Lg

digk digk
,
L = −=R−gtRigtgkig+
− uucck
= a,b,c
a, b, c,
gk −
k +ee
k , , kk =
gk
dt g dt

(50)

!
1
= sk − 1 ∑ s j  udc ,
uck =  sk −3 j=
a,b,c s j  udc ,
3 j = a,b,c 


(51)
(51)

with:
with:
uck

and sk denotes the switching function.
switching function.
and Denote
s k denotes
y = ithe
gk and z = egk − uck , then Equation (50) can be expressed as:

Denote y = igk and z = egk − uck , then Equation (50) can be expressed as:
Rgt
1
dy
= − R y + z,
(52)
dt dy
Lggt
L1g
=−
y+
z,
(52)
Lg
Lg
dt
Filtering each side of Equation (52) with a first-order filter Mf , i.e., multiply each side by 1/( p + λf )
sidepositive
of Equation
(52) with
a first-order
with Filtering
λf being aeach
known
real constant.
Then,
we have: filter M f , i.e., multiply each side by
1(p + λf) with λf being a known positive real constant. Then, we have:
y = yf ( λf − h1 ) + zf h2 ,
y = yf ( λf − h1 ) + zf h2 ,
where h1 = Rgt /Lg , h2 = 1/Lg , and:
where h1 = Rgt Lg , h2 = 1 Lg , and:
y
z
, z =
,
yf =
yf f
z
s
+
λ
s
+
yf =
, zf = λf ,
s + λf
s + λf
Express Equation (53) as y = Wha with:
Express Equation (53) as y = Wha with:
h
iT h
iT
W = [ ω1 ω2 ] = [ yf zf ], ha = h1a h2a T = λ f − h1T h2
,
W = [ω1 ω2 ]=[ yf zf ], ha = [ h1a h2a ] = λ f − h1 h2  ,
where ha denotes the true parameter values that vary sufficiently slow.
where ha denotes the true parameter values that vary sufficiently slow.

(53)
(53)

(54)
(54)

3.3.2. The Gradient Algorithm
Define hp = [h1p

h2p ]T and hd = [h1d

h2d ]T = hp − ha as the estimation results at time t and the

errors between the true values and the estimated values, respectively.

Energies 2018, 11, 2599

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3.3.2.
Algorithm
EnergiesThe
2018,Gradient
11, x FOR PEER
REVIEW

13 of 20

T

T

Define
hp predicted
= [ h1p houtput
hd =t is
[ hy1d = Wh
= hp −yhpaa is
asthe
theerror
estimation
results
at time t
h2d . ]Assume
2p ] and
Then, the
at time
between
the predicted
p
p
and the errors between the true values and the estimated values, respectively.
output and the actual output, i.e.,
Then, the predicted output at time t is yp = Whp . Assume ypa is the error between the predicted
ypa = yp − y =Whd ,
output and the actual output, i.e.,
(55)
ypa = yp − y = Whd ,
(55)
Choose the adaptation law as:
Choose the adaptation law as:
dhddhd = −γ WTT ypa ,
(56)
= −γW ypa ,
(56)
dt dt
where
denotesthe
theadaptation
adaptationgain
gainmatrix.
matrix.
diag ((γγ11, ,γγ2 2) )>>
0 0denotes
where γγ== diag
Substituting
Equation
(55)
into
Equation
(56)
yields:
Substituting Equation (55) into Equation (56) yields:
dhddhd
T
=−
= γW
−γ W TWh
Whdd ,
dt dt

(57)

Utilizing
theLyapunov
Lyapunov function candidate,
of of
Va along
the
Utilizing VVaa == hddTTγγ−1−h1d hdasasthe
candidate,then
thenthe
thederivative
derivative
Va along
state
trajectories
can
be
obtained
as:
the state trajectories can be obtained as:
dVadVa
T TT
= 2h
−2TdhW
Whdd ≤
≤ 00,,
=−
d W Wh
dt dt

(58)

which means
means this
this OIE
OIE is
is always
always stable
stable and
and the
the parameter
parameter estimation
estimation errors
errorskeep
keepdecreasing.
decreasing.
In reference
convergence
of the
results
is proved.
As the process
much involved,
reference[38],
[38],
convergence
of estimation
the estimation
results
is proved.
As the is
process
is much
only
the results
are given
If the input
meetsignals
the persisting
excitation
condition,
convergence
involved,
only the
resultshere:
are given
here: signals
If the input
meet the
persisting
excitation
condition,
of
the estimation
is ensured.
the linear
in Equation
m sinusoidal
signals in
convergence
of theresults
estimation
resultsFor
is ensured.
Forsystem
the linear
system in(53),
Equation
(53), m sinusoidal
the
input
accurate estimation
of at least 2m
[39]. For this
OIE,
is at there
least
signals
in guarantees
the input guarantees
accurate estimation
of atparameters
least 2m parameters
[39].
Forthere
this OIE,
the
in w2 , so in
that
of two parameter
estimation
results results
can be
is atgrid
leastvoltage
the gridcomponents
voltage components
w2,convergence
so that convergence
of two parameter
estimation
ensured,
i.e., hp converges
to ha isto
proved.
can be ensured,
i.e., hp converges
ha is proved.
3.4. Block Diagram of the PIDR-SMCC-OIE Strategy
Block diagram of the PIDR-SMCC-OIE current control strategy under UBGV conditions is as
shown
6. In
PLL denotes
the phase-locked
loop, SCEloop,
denotes
sequence
shown ininFigure
Figure
6. the
In figure,
the figure,
PLL denotes
the phase-locked
SCE
denotescomponent
sequence
extraction,
SVPWMand
denotes
space
vectorspace
pulsevector
width pulse
modulation.
componentand
extraction,
SVPWM
denotes
width modulation.
eg

Rg

ig
egabc

igabc

PLL

e

jω t

e

ωt

egαβ

iαβ

e

jω t

Online
Inductance
Estimator
−*
igdq


+
egdq+

SCE of the
Grid Voltage

Converter
Pole Voltage
egabc
igabc

+
igdq

− jω t


gdq −

e

uc

udc

3s/2s
3s/2s

Lg

Current Refs.
Calculation

+*
gdq+

i

-

+

udc

sa,sb,sc

Lest

e j2ωt

C

SVPWM

PIDR-SMCC
Controller

+*
igdq
-

+
+ igdq

6. Block diagram of the
the proportional-integral-derivative-resonant
proportional-integral-derivative-resonant law-based sliding mode
Figure 6.
control strategy
strategy with
with online
online inductance
inductance estimator
estimator (PIDR-SMCC-OIE).
(PIDR-SMCC-OIE).
current control

We can see from the figure that the control structure is simple, and moreover, the PIDR-SMCC
controller and the OIE are decoupled, so that they can be designed independently.

Energies 2018, 11, 2599

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We can see from the figure that the control structure is simple, and moreover, the PIDR-SMCC
controller and the OIE are decoupled, so that they can be designed independently.
4. Simulation
Energies 2018,Study
11, x FOR PEER REVIEW

14 of 20

4. Simulation
Study
4.1. General
Configuration

To
effectiveness of the PIDR-SMCC-OIE strategy, simulation studies are conducted
4.1.verify
Generalthe
Configuration
on the two-terminal VSC-MVDC system in PSCAD/EMTDC. In the simulations, CS1 and CS2 are
To verify the effectiveness of the PIDR-SMCC-OIE strategy, simulation studies are conducted
operated as rectifier and inverter respectively, the loads and DGs connected to the DC transmission
on the two-terminal VSC-MVDC system in PSCAD/EMTDC. In the simulations, CS1 and CS2 are
lines are
takenasasrectifier
lumped
active
power
disturbances.
operated
and
inverter
respectively,
the loads and DGs connected to the DC transmission
The
main
parameters
of
the
PIDR-SMCC-OIE
controller are listed in Table 1.
lines are taken as lumped active power disturbances.
The main parameters of the PIDR-SMCC-OIE controller are listed in Table 1.
Table 1. Main parameters of the PIDR-SMCC-OIE controller.
Table 1. Main parameters of the PIDR-SMCC-OIE controller.

Parameters
Parameters
KPId , KPIq
KPIdK, K,PIqK
Sq
Sd
KSd
K,SqKRq
K,Rd
KRd, KωRqc
ωc γ 1
γ1 γ2
γ2

Value
Value
20
20
1500
1500
7
7
10
10
5000
2500
5000
2500

The per-unit values are used in the simulation results. The AC and DC voltage bases are 10 kV
The per-unit values are used in the simulation results. The AC and DC voltage bases are 10 kV
and 20 kV respectively. The power base is 15 MVA and the inductance base is 12 mH. To maintain
and 20 kV respectively. The power base is 15 MVA and the inductance base is 12 mH. To maintain
the DC
voltage along the transmission line within specified range, the DC bus voltage on CS1 side is
the DC voltage along the transmission line within specified range, the DC bus voltage on CS1 side is
controlled
at 1.025
p.u.p.u.
In addition,
ininorder
thewaveform
waveform
display,
simulation
controlled
at 1.025
In addition,
orderto
tofacilitate
facilitate the
display,
thethe
simulation
datadata
in in
PSCAD/EMTDC
are
imported
into
MATLAB
(R2016a)
for
plotting.
PSCAD/EMTDC are imported into MATLAB (R2016a) for plotting.
The symbols
usedused
in the
simulation
results
meaningsare
arelisted
listed
Table
The symbols
in the
simulation
resultsand
and their
their meanings
in in
Table
2. 2.
Table
2. Symbols
used
ininthe
resultsand
andtheir
theirmeanings.
meanings.
Table
2. Symbols
used
thesimulation
simulation results
Symbol
Symbol

udc1, udc2
udc1 , uEdc2
g1, Eg2
Eg1 , EIg2
g1, Ig2
IPg1
, Ig2g1, Pg2, Qg2
g1, Q
Pg1 , Qg1 , PLg2
Qg2g2
g1,,L

Lg1 , Lg2

Meaning
Meaning
DC bus voltage of CS1 and CS2
DC grid
bus voltage
voltageon
ofCS1
CS1and
andCS2
CS2
Three-phase
side
Three-phaseAC
grid
voltage
on CS1
andside
CS2 side
Three-phase
current
on CS1
and CS2
Three-phase
AC
current
on
CS1
and
CS2
sideCS2
Active and reactive power exchange at the PCC of CS1 and
Active and
reactive
power
exchange
at
the
PCC
of
The estimated inductance for CS1 and CS2 CS1 and CS2

The estimated inductance for CS1 and CS2

4.2. Simulation Results

4.2. Simulation
Results
For convenience,
define the following specified operating condition (SOC), under which most
of the
simulation studies
theoperating
grid voltage
is three-phase
balanced.
0.1 s,most
For
convenience,
defineare
theconducted.
followingInitially,
specified
condition
(SOC),
under At
which
single-phase to ground faults happen to phase A of AC bus B1 and phase C of AC bus B2. At 0.2 s,
of the simulation studies are conducted. Initially, the grid voltage is three-phase balanced. At 0.1 s,
both the faults are cleared and the grid voltages restore to the initial states. The voltage waveforms at
single-phase to ground faults happen to phase A of AC bus B1 and phase C of AC bus B2 . At 0.2 s,
PCC1 and PCC2 are as shown in Figure 7. Besides, the inductance of the AC inductors for both CS
both the
faults are cleared and the grid voltages restore to the initial states. The voltage waveforms at
are set to 75% of the nominal values if not otherwise specified, to investigate robustness of the
PCC1system.
and PCC2 are as shown in Figure 7. Besides, the inductance of the AC inductors for both CS are
set to 75% of the nominal values if not otherwise specified, to investigate robustness of the system.

(a)

(b)

7. grid
The grid
voltage
waveforms
of the
specified
operatingcondition
condition(SOC)
(SOC)for
for CS1
CS1 (a) and
FigureFigure
7. The
voltage
waveforms
of the
specified
operating
and CS2
CS2 (b).
(b).

Energies 2018, 11, 2599

15 of 20

Energies 2018, 11, x FOR PEER REVIEW
Energies 2018, 11, x FOR PEER REVIEW

15 of 20
15 of 20

In Figure 8 the responses of CS1 and CS2 to the SOC for OBJ 1 and OBJ 2 respectively are
In Figure
8 the responses of CS1 and CS2 to the SOC for OBJ 1 and OBJ 2 respectively are shown.
In Figure 8 the responses of CS1 and CS2 to the SOC for OBJ 1 and OBJ 2 respectively are
shown.
shown.

(a)
(b)
(a)
(b)
Figure
8. Responses
of CS1
andCS2
CS2(b)
(b)to
to the
the SOC
SOC for
OBJ
2 respectively.
Figure
8. Responses
of CS1
(a)(a)
and
forOBJ
OBJ1 1and
and
OBJ
2 respectively.
Figure 8. Responses of CS1 (a) and CS2 (b) to the SOC for OBJ 1 and OBJ 2 respectively.

Figure
we can
activepower
powerripples
ripples at
at PCC1
power
ripples
at at
FromFrom
Figure
8 we8 can
seesee
thethe
active
PCC1and
andthe
thereactive
reactive
power
ripples
From
Figure
8 we can without
see the active
power
rippleswith
at PCC1
and the reactive
powerMoreover,
ripples at
PCC2
are
fully
eliminated
current
distortion,
the
PIDR-SMCC
controller.
PCC2 are fully eliminated without current distortion, with the PIDR-SMCC controller. Moreover,
PCC2
are fully
eliminated for
without
current
distortion, with
the PIDR-SMCC
controller.
the transient
performances
both CS
are satisfactory.
In Figure
9, responses of
CS1 and Moreover,
CS2 to the
the transient
performances
forfor
both
CSCSare
satisfactory.
In Figure
Figure9,
9,responses
responses
of
CS1
to the
the
transient
performances
both
are
satisfactory.
In
of
CS1
and.and
CS2CS2
to the
SOC for the control objectives of OBJ 5 are shown. For CS1, ksk = 0.5 ; for CS2, ksk = −0.5
SOC for
the
objectives
of OBJ
5 are
shown.
0.5;
forCS2,
CS2,k ksk
SOC
forcontrol
the control
objectives
of OBJ
5 are
shown.For
For CS1,
CS1, kksk ==0.5
; for
= −=
0.5−. 0.5.
sk

(a)
(a)

sk

(b)
(b)

Figure 9. Responses of CS1 (a) and CS2 (b) to the SOC for OBJ 5.
Figure
9. Responses
CS1(a)
(a)and
and CS2
CS2 (b)
OBJ
5. 5.
Figure
9. Responses
ofofCS1
(b) to
tothe
theSOC
SOCforfor
OBJ
From Figure 9 we can see the expected control objectives are achieved with the combined
From
9 wesee
can
thethe
expected
control
objectives
areOAR
achieved
with
the
combined
From
Figure
9 we
can
thesee
expected
control
objectives
are achieved
with
theiscombined
function
ofFigure
the
GCRC
method
and
PIDR-SMCC
controller.
The
for CS1
about
1/3 function
and
function
of
the
GCRC
method
and
the
PIDR-SMCC
controller.
The
OAR
for
CS1
is
about
1/3
and
that
for
CS2
is
about
3,
which
is
consistent
with
the
theoretical
analysis.
Figure
10
shows
the for
of the GCRC method and the PIDR-SMCC controller. The OAR for CS1 is about 1/3 and that
that
for CS2
is about
3, which
isSOC
consistent
withcontroller
the theoretical
analysis.
Figure
10 shows The
the
responses
of
CS1
and
CS2
to
the
with
PIVR
and
PIR
controller,
respectively.
CS2 is about 3, which is consistent with the theoretical analysis. Figure 10 shows the responses of
responses
of of
CS1
and
CS2
to thefor
SOC
with
controller
and
PIR the
controller,
respectively.
inductances
the
ACwith
inductors
both
CS PIVR
are
nominal
values,
control objectives
areThe
the of
CS1 and
CS2 to the
SOC
PIVR controller
andthe
PIR
controller,
respectively.
The inductances
inductances
of
the
AC
inductors
for
both
CS
are
the
nominal
values,
the
control
objectives
are
the
same
as
those
in
Figure
8.
The
proportional
and
integral
gains
of
the
two
controllers
are
1500
and
the AC
inductors
for both CS
are
the nominal and
values,
the control
objectives
are the same
as those
in
same
those in Figure
The proportional
integral
gains
of the
two controllers
are the
1500PIVR
and
30,000asrespectively.
The8.resonant
gains are 18,000
for the
PIR
controller
and 40 for
Figure30,000
8. Therespectively.
proportional
and
integral
gains
of
the
two
controllers
are
1500
and
30,000
respectively.
The resonant gains are 18,000 for the PIR controller and 40 for the PIVR
controller.
The resonant
gains are 18,000 for the PIR controller and 40 for the PIVR controller.
controller.

Energies 2018, 11, 2599
Energies
Energies 2018,
2018, 11,
11, xx FOR
FOR PEER
PEER REVIEW
REVIEW

(a)
(a)

16 of 20
16
16 of
of 20
20

(b)
(b)

Figure
10.
Responses
(a)
CS2
the
SOC
with
plus
vector
Figure
of of
CS1
(a) and
CS2 (b)
the to
SOC
proportional-integral
plus vector
resonant
Figure10.
10.Responses
Responses
of CS1
CS1
(a) and
and
CS2to(b)
(b)
to
thewith
SOC
with proportional-integral
proportional-integral
plus
vector
resonant
(PIVR)
and
proportional-integral
plus
resonant
(PIR)
controllers
respectively.
(PIVR)
and(PIVR)
proportional-integral
plus resonant
controllers
respectively.
resonant
and proportional-integral
plus(PIR)
resonant
(PIR) controllers
respectively.

We
from
Figure
10 10
thethe
steady
state
performances
of both
CS are
with the
PIVR
We
can
see
from
Figure
steady
state
performances
of
CS
are
with
the
Wecan
cansee
see
from
Figure
10
the
steady
state
performances
of both
both
CSsatisfactory
are satisfactory
satisfactory
with
the
and
PIR
controllers,
respectively.
However,
these
are
obtained
by
high
controller
gains,
which
will
PIVR
and
PIR
controllers,
respectively.
However,
these
are
obtained
by
high
controller
gains,
which
PIVR and PIR controllers, respectively. However, these are obtained by high controller gains, which
reduce
the stability
margins.
To investigate
robustness
of theof
responses
of CS1
will
the
margins.
To
robustness
the
two
responses
of
CS1
will reduce
reduce
the stability
stability
margins.
To investigate
investigate
robustness
oftwo
thecontrollers,
two controllers,
controllers,
responses
ofand
CS1
CS2
theto
SOC
AC inductances
are shown
in Figure
In this
grid the
voltage
and
CS2
the
SOC
with
AC
are
in
Figure
11.
In
figure,
andto
CS2
to
thewith
SOCreduced
with reduced
reduced
AC inductances
inductances
are shown
shown
in 11.
Figure
11.figure,
In this
thisthe
figure,
the grid
grid
conditions
and the control
are objectives
the same asare
those
forsame
Figureas
and the
inductances
the AC
voltage
and
the
the
for
Figure
the
voltage conditions
conditions
and objectives
the control
control
objectives
are
the
same
as10those
those
for
Figure 10
10ofand
and
the
inductors
areof
setthe
to AC
80%
of the nominal
CS.
inductances
are
to
of
the
values
inductances
of
the
AC inductors
inductors
are set
setvalues
to 80%
80%for
ofboth
the nominal
nominal
values for
for both
both CS.
CS.

(a)
(a)

(b)
(b)

Figure
Figure
11.
Responses
of
CS1
(a)
and
CS2
(b)
to
the
SOC
by
the
PIVR
and
PIR
controllers
with
20%
Figure11.
11.Responses
Responsesof
ofCS1
CS1(a)
(a)and
andCS2
CS2(b)
(b)to
tothe
theSOC
SOCby
bythe
thePIVR
PIVRand
andPIR
PIRcontrollers
controllerswith
with20%
20%
reduced
inductances
respectively.
reduced
inductances
respectively.
reduced inductances respectively.

From
Figure 11a
we can
see that
the performances
of both
CS getget
worse when
the inductances
of
From
From Figure
Figure 11a
11a we
we can
can see
see that
that the
the performances
performances of
of both
both CS
CS get worse
worse when
when the
the inductances
inductances
the
AC
inductors
are
reduced
by
20%.
This
is
because
the
reduction
of
the
inductances
is
equivalent
of
of the
the AC
AC inductors
inductors are
are reduced
reduced by
by 20%.
20%. This
This is
is because
because the
the reduction
reduction of
of the
the inductances
inductances is
is
to
increase of
the
controlofgains.
As a result,
theAs
system
is approaching
the
stability boundary
as the
equivalent
to
increase
the
control
gains.
a
result,
the
system
is
approaching
the
stability
equivalent to increase of the control gains. As a result, the system is approaching the stability
inductance
decreases.
In Figure 12, the responses
of CS1
and CS2 to the
SOC with
the function
of the
boundary
boundary as
as the
the inductance
inductance decreases.
decreases. In
In Figure
Figure 12,
12, the
the responses
responses of
of CS1
CS1 and
and CS2
CS2 to
to the
the SOC
SOC with
with
PIDR-SMCC-OIE
controller
are
shown.
The
inductances
of
the
AC
inductors
are
0.7
p.u
for
both
CS,
the
function
of
the
PIDR-SMCC-OIE
controller
are
shown.
The
inductances
of
the
AC
inductors
the function of the PIDR-SMCC-OIE controller are shown. The inductances of the AC inductors are
are
and
the control
objectives
for CS1
and CS2 are tofor
eliminate
the active
power
ripples at PCC1
and to
0.7
0.7 p.u
p.u for
for both
both CS,
CS, and
and the
the control
control objectives
objectives for CS1
CS1 and
and CS2
CS2 are
are to
to eliminate
eliminate the
the active
active power
power
obtain
balanced
three-phase
currents, respectively.
ripples
ripples at
at PCC1
PCC1 and
and to
to obtain
obtain balanced
balanced three-phase
three-phase currents,
currents, respectively.
respectively.

Energies 2018, 11, 2599

17 of 20

Energies 2018, 11, x FOR PEER REVIEW

17 of 20

Energies 2018, 11, x FOR PEER REVIEW

17 of 20

(a)

(b)

12. Responses
of CS1
and
CS2(b)
(b)totothe
theSOC
SOC with
with the
thethe
PIDR-SMCC-OIE
FigureFigure
12. Responses
of(a)
CS1
(a)(a)
and
CS2
thefunction
functionof(b)
of
PIDR-SMCC-OIE
controller
for
OBJ
1
and
OBJ
3
respectively.
controller
for
OBJ
1
and
OBJ
3
respectively.
Figure 12. Responses of CS1 (a) and CS2 (b) to the SOC with the function of the PIDR-SMCC-OIE
controller for OBJ 1 and OBJ 3 respectively.

Figure
12can
we see
can both
see both
transient
and
steady
stateperformances
performances of
of CS1
CS1 and
FromFrom
Figure
12 we
the the
transient
and
steady
state
and CS2
CS2 are
are satisfactory
with
the
PIDR-SMCC-OIE
controller,
and
the
inductance
estimation
results
From
Figure
12
we
can
see
both
the
transient
and
steady
state
performances
of
CS1
and
CS2
satisfactory with the PIDR-SMCC-OIE controller, and the inductance estimation results converge to the
converge
to the true
values
both balanced and unbalanced
grid
voltage conditions.
In Figure
are satisfactory
with
the under
PIDR-SMCC-OIE
and conditions.
the
inductance
estimation
results of
true values
under both
balanced
and unbalancedcontroller,
grid voltage
In Figure
13, responses
13,
responses
of
CS1
and
CS2
to
active
power
step
changes
and
the
SOC
under
UBGV
conditions
converge
to
the
true
values
under
both
balanced
and
unbalanced
grid
voltage
conditions.
In
Figure
CS1 and
CS2 to active
power
step changes
anddemand
the SOCfor
under
UBGV
conditions
are figure
shown.
The initial
are
Theofinitial
average
power
CS2 isand
−0.6the
p.u.
At 0.1
s, this
steps up
13, shown.
responses
CS1 and
CS2 active
to active
power step changes
SOC
under
UBGV conditions
average
active
demand for
CS2tois−0.6
−0.6
p.u.
Ats.0.1
s, this figure steps
up
toinductors
−0.8 p.uare
and then
to
p.upower
and
p.u
at 0.2
AC
are−0.8
shown.
The then
initialstepped
averageback
active
power
demand
forThe
CS2inductances
is −0.6 p.u. of
At the
0.1 s,
this
figure steps 0.7
up
stepped
back
to

0.6
p.u
at
0.2
s.
The
inductances
of
the
AC
inductors
are
0.7
p.u
and
the
control
p.u
andp.u
the and
control
eliminate
DCs.bus
voltage
ripples,offor
to −0.8
thenobjectives
stepped are
backtoto
−0.6 p.uthe
at 0.2
The
inductances
theboth
ACCS.
inductors are 0.7
objectives
arethe
to eliminate
the DCare
bus
CS.ripples, for both CS.
p.u and
control objectives
to voltage
eliminateripples,
the DC for
busboth
voltage

(a)

(b)

Figure 13. Responses(a)
of CS1 (a) and CS2 (b) to active power step changes (b)
under unbalanced grid
voltage
(UBGV)
conditions
for
obtaining
constant
DC
bus
voltage.
Figure 13. Responses of CS1 (a) and CS2 (b) to active power step changes under unbalanced grid

Figure 13. Responses of CS1 (a) and CS2 (b) to active power step changes under unbalanced grid
voltage
(UBGV)
conditions
obtaining
constantDC
DCbus
busvoltage.
voltage.
voltage
(UBGV)
conditions
for for
obtaining
constant
From Figure 13 we can see that the DC bus voltage ripples on both sides are fully eliminated.
The inductance
estimation
results
deviate
from
the trueripples
values for
both CS,are
during
the power
Figure
13 can
we can
that
DCbus
busvoltage
voltage
fully
eliminated.
FromFrom
Figure
13 we
seesee
that
thetheDC
rippleson
onboth
bothsides
sides are
fully
eliminated.
step-up.
However,
they
are
still
accurate
with
relative
errors
no
greater
than
±1%.
In
Figure
the
The inductance estimation results deviate from the true values for both CS, during the 14,
power

The inductance estimation results deviate from the true values for both CS, during the power step-up.
step-up. However, they are still accurate with relative errors no greater than ±1%. In Figure 14, the
However, they are still accurate with relative errors no greater than ±1%. In Figure 14, the responses
of both CS to inductance parameter step changes under UBGV conditions are shown to investigate the
tracking performance of the OIE.

Energies 2018, 11, x FOR PEER REVIEW

18 of 20

responses
of 2599
both
Energies
2018, 11,

CS to inductance parameter step changes under UBGV conditions are shown
to
18 of 20
investigate the tracking performance of the OIE.

(a)

(b)

Figure
ResponsesofofCS1
CS1
(b)inductance
to inductance
parameter
step changes
Figure 14.
14. Responses
(a) (a)
andand
CS2CS2
(b) to
parameter
step changes
under under
UBGV
UBGV
conditions.
conditions.

In
Inthis
thiscase,
case,the
thegrid
gridvoltages
voltages are
arethe
thesame
sameas
asthose
thosein
inFigure
Figure13
13and
andthe
thecontrol
controlobjectives
objectivesare
are
the
same
as
those
in
Figure
8,
for
both
CS.
Initially,
the
inductances
of
the
AC
inductors
the same as those in Figure 8, for both CS. Initially, the inductances of the AC inductors are
are 1.0
1.0 p.u;
p.u;
at
at0.05
0.05s,s,they
theyare
arestepped
steppedto
to0.5
0.5p.u
p.uand
andthen
thenstepped
steppedback
backtoto1.0
1.0p.u
p.uatat0.2
0.2s.s.
We
can
see
from
Figure
14
that
the
OIE
tracks
rapidly
and
accurately
the the
inductance
changes
all
We can see from Figure 14 that the OIE tracks rapidly and accurately
inductance
changes
through
the
process.
As
a
result,
the
steady
state
performances
of
CS1
and
CS2
are
satisfactory
with
all through the process. As a result, the steady state performances of CS1 and CS2 are satisfactory
the
PIDR-SMCC-OIE
controller
even in the
presence
the combined
of voltage
disturbances
with
the PIDR-SMCC-OIE
controller
even
in the ofpresence
of theeffects
combined
effects
of voltage
and
inductance
parameter
step
changes,
although
it
takes
longer
time
for
both
CS
to
settle
down.
disturbances and inductance parameter step changes, although it takes longer time for both CS to

settle down.
5. Conclusions
5. Conclusions
In this paper, a PIDR law-based sliding mode current control strategy with online inductance
estimator
(i.e.,
the PIDR-SMCC-OIE
strategy)
presented
forcontrol
VSC-MVDC
system
converter
station,
In this
paper,
a PIDR law-based
sliding is
mode
current
strategy
with online
inductance
under
unbalanced
voltage conditions.
Theismain
conclusions
are as the following.
First, through
estimator
(i.e., the grid
PIDR-SMCC-OIE
strategy)
presented
for VSC-MVDC
system converter
station,
adding
a
slack
coefficient
to
the
conventional
current
reference
calculation
equations,
the
ratio
of the
under unbalanced grid voltage conditions. The main conclusions are as the following. First,
through
active
power
ripple
amplitude
to
that
of
the
reactive
power
ripple
can
be
continuously
controlled
adding a slack coefficient to the conventional current reference calculation equations, the ratio of the
without
currentripple
distortion.
Second,
adoption
the SMCpower
method
simplifies
controller design
and
active power
amplitude
to that
of theofreactive
ripple
can bethe
continuously
controlled
guarantees
closed-loop
stability
of
the
system
in
the
Lyapunov
sense
in
the
presence
of
parameter
without current distortion. Second, adoption of the SMC method simplifies the controller design and
uncertainty
disturbance.
Moreover,
derivatives
ofLyapunov
the currentsense
references
obtained
simple
guarantees and
closed-loop
stability
of the system
in the
in theare
presence
of by
parameter
algebraic
operations,
which
facilitate
the
implementation
of
the
controller.
Third,
the
OIE
presented
is
uncertainty and disturbance. Moreover, derivatives of the current references are obtained by simple
easy
to
implement
with
no
additional
sensors
needed,
and
it
can
provide
accurate
estimates
under
both
algebraic operations, which facilitate the implementation of the controller. Third, the OIE presented
0 lemma and the
balanced
unbalanced
voltage conditions.
Fourth, and
application
of the Barbalat
is easy toand
implement
withgrid
no additional
sensors needed,
it can provide
accuratesestimates
under
reduction
to absurdity
method proves
convergence
of theFourth,
currentapplication
control errors
to zero.
Theoretical
both balanced
and unbalanced
grid voltage
conditions.
of the
Barbalat′s
lemma
analysis
simulation
studies in
PSCAD/EMTDC
verify theofeffectiveness
superiority
the
and the and
reduction
to absurdity
method
proves convergence
the current and
control
errors toofzero.
proposed
PIDR-SMCC-OIE
Theoretical
analysis and strategy.
simulation studies in PSCAD/EMTDC verify the effectiveness and
superiority
of the proposed
PIDR-SMCC-OIE
strategy.
Author
Contributions:
W.Y. conceived
and performed
the research; J.L. designed the control law; A.Z. designed
parameters of the model; Y.Z. built the model; W.Y. and H.Z. wrote the paper, J.W. revised the manuscript. All the
Authorwere
Contributions:
W.Y. conceived
and performed the research; J.L. designed the control law; A.Z.
authors
involved in preparing
this manuscript.
designed
parameters
of
the
model;
Y.Z.
builtbythe
W.Y.Key
andR&D
H.Z.Program
wrote the
revised
the
Funding: This work was supported in part
themodel;
National
of paper,
China J.W.
(grant
number:
manuscript.
All
the
authors
were
involved
in
preparing
this
manuscript.
2016YFB0900500).
Conflicts of Interest: The authors declare no conflict of interest.

Energies 2018, 11, 2599

19 of 20

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