# Control Strategy of Three-Phase Photovoltaic Inverter under Low-Voltage Ride-Through Condition

Authors: Xianbo Wang; Zhixin Yang; Bo Fan; Wei Xu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The new energy promoting community has recently witnessed a surge of developments in photovoltaic power generation technologies. To fulfill the grid code requirement of photovoltaic inverter under low-voltage ride-through (LVRT) condition, by utilizing the asymmetry feature of grid voltage, this paper aims to control both restraining negative sequence current and reactive power fluctuation on grid side to maintain balanced output of inverter. Two mathematical inverter models of grid-connected inverter containing LCL grid-side filter under both symmetrical and asymmetric grid are proposed. PR controller method is put forward based on inverter model under asymmetric grid. To ensure the stable operation of the inverter, grid voltage feedforward method is introduced to restrain current shock at the moment of voltage drop. Stable grid-connected operation and LVRT ability at grid drop have been achieved via a combination of rapid positive and negative sequence component extraction of accurate grid voltage synchronizing signals. Simulation and experimental results have verified the superior effectiveness of our proposed control strategy.


---

## Body

## 1. Introduction

In recent years, the development and utilization of new energies, such as solar energy, wind energy, and hydrogen energy, are booming in Europe, US, and China, where the distributed photovoltaic power generation technologies are highly concerned. In distributed grid-connected power generation system, most of electric energy generated by energy conversion device will be converted into alternating current (AC) with the same frequency and phase with grid voltage through grid-connected inverter and then transmitted to grid [1–3]. Power system is dynamic, and its dynamic stability can be affected by many factors, such as setting of generator output limit, grid fault, grid resonance, and nonlinear load. Grid-connected inverter plays an essential interface role between renewable energy conversion device and grid and becomes an extremely important component of distributed power generation system [4]. With the increasing utilization of distributed power generation system in public grid, more and more new energy power converters are connected to grid, and thus rational control of converters is a key factor of efficient and safe utilization of new energy. However, grid is not a constant, stable, and balanced system and is often affected by grid fault, resonance, overload and nonlinear load, and so forth, making design of converter control system more difficult [5]. In recent years, under grid fault especially grid voltage sag, strict requirements of LVRT ability and reactive power injection for grid-connected converter in grid code increase the complexity of control. Design of photovoltaic grid-connected inverter will ensure its reliable and stable operation under normal state of grid and continuous operating ability under grid fault, and another requirement is LVRT ability of distributed power generation system grid-connected converter; that is, [6–8] such grid-connected inverter will ensure that operating parameters of each phase will not be disconnected to grid because of ultra-limit triggered protection action under grid fault and provide maximum voltage and reactive power support for grid.Under normal or balanced grid voltage, most of existing photovoltaic grid-connected inverters can operate normally, and rapid non-static-error control of inverter’s output current can be obtained byP
I controller [9, 10]. However, when grid voltage is unbalanced or seriously distorted, grid has unbalanced voltage and current, that is [11]. There are lots of negative sequence components in grid voltage; when carrying out phase lock of grid voltage with phase-locked loop by d
q conversion, axis q would have second harmonic fluctuation which is difficult to restrain, while traditional PI controller could only get high-performance control effects when control objects are DC variables. Therefore, when grid voltage is unbalanced or seriously distorted, grid-connected inverter with traditional PI controller would have deteriorated operating performance [12, 13]. Besides, when grid voltage is unbalanced, second harmonic fluctuation generated on axis q would also affect calculation of grid phase angle by phase-locked loop [5]. Although second component of axis q can be of filtering processing through a specific second wave trap, the detection and computing delay deserve concern. When facing grid voltage flicker, low grid voltage, and unbalanced grid, the grid company requires medium- and high-voltage inverters to detect fluctuations of grid voltage rapidly and make necessary response as required in relevant national standards, in order to carry out rapid support for grid voltage and avoid a larger scale of grid fault [14–16].According to grid code, medium- and high-voltage inverters of large photovoltaic power stations will have certain ability to tolerate abnormal voltage, in order to avoid separation under abnormal grid voltage and lead to unstable grid voltage. According to technical suggestion, photovoltaic inverter can be separated from grid when grid voltage drops to below curve 1 as is shown in Figure2. Abnormal grid voltage is reflected as voltage drop of grid connection points in photovoltaic grid-connected system, and such voltage drop can be divided into three-phase symmetrical drop and three-phase asymmetrical drop according to grid voltage drop categories. As is shown in Figure 1, there are “Q-GDW 617-2011” and “GB/T 19964-2012” standards issued in 2011 and 2012, respectively. The differences are that both standards define different grid voltage drop depth, response time of reactive power output under grid fault, active power recovery rate when grid recovers, and so forth [17, 18].Figure 1
LVRT ability requirement of PV power stations.
(a)
Q-GDW 617-2011 technical requirements for connecting photovoltaic power station to power system (China)
(b)
GB/T 19964-2012 technical requirements for connecting photovoltaic power station to power system (China)Figure 2
Reactive power generation.According to requirements, photovoltaic inverter maintains operation without disconnection and meets the following conditions during LVRT:(1)
At the moment of grid voltage drop, maintain continuous grid-connected operation while protecting photovoltaic inverter to be safe and the time interval from the occurrence of grid drop to generation of reactive current will be less than 30 ms.
(2)
During the interval of LVRT, maintain stable grid-connected operation of inverter, provide reactive support for grid according to grid connection rules, output range of reactive power will follow grid code requirements, and reactive output range will follow requirements in Figure2.
(3)
Upon recovery of grid voltage, active power output will recover to the value before grid fault situation with change rate of at least0.3
P
n
/
s.Figure3 is German grid’s required index of LVRT of wind power generation system [19]. Area 1 in black bold line has power fluctuation ratio per second of 5
%
≤
d
P
/
d
t
≤
20
%, and wind power generation system will have at least 150 ms operating ability when grid voltage drops to 0; the running time of wind power generation system will increase gradually when grid voltage drops to above 0% of rated voltage; the area that will remain connected above Area 1 means that wind power generation system needs 150 ms operating ability when power fluctuation ratio meets 20
%
≤
d
P
/
d
t
≤
40
%, and grid voltage drops to 45% of rated voltage, and operating ability increases with the decrease of drop degree [20].Figure 3
LVRT ability requirement (Germany).Denmark regulates that inverters will maintain continuous operation without disconnection for 10 s after nominal voltage recovers from 25% to 75% under three-phase fault and will generate active power-up to rated level within 10 seconds when grid voltage recovers back to0.9
U
n again. During grid voltage drop, active power of grid connection points will meet the following conditions [21, 22]:(1)
P
current
>
k
p
P
t
=
0
U
current
U
t
=
0
2
,where P
current and U
current are current power and voltage, P
t
=
0 and U
t
=
0 are power and voltage under grid fault, and k
p is active power control factor. When voltage recovers to 0.9
U
n, the output behaviour of inverter will meet reactive power exchanging requirement with grid within 10 s. During grid voltage drop, ensure that inverter will generate reactive current which is equivalent to rated current under normal grid situation.In addition, Danish grid code also requires that power station could respond to dual voltage dip fault and requires another new 100 ms short circuit with an interval of 300 ms upon two-phase short circuit 100 ms later, and power station has no shutdown. Under another new 100 ms voltage drop with an interval of 1 s upon single-phase short circuit 100 ms later, no shutdown is allowed, and voltage drop curve is shown in Figure4.Figure 4
LVRT of double voltage drops (Denmark).
(a)
U
a (two-phase short circuit)
(b)
U
a (one-phase short circuit)
(c)
U
b (two-phase short circuit)
(d)
U
b (one-phase short circuit)
(e)
U
c (two-phase short circuit)
(f)
U
c (one-phase short circuit)EnergyNet.dk standards also regulate LVRT under some special situations, such as 100 ms three-phase short circuit, 100 ms two-phase short circuit, and new 100 ms short circuit with an interval of 300 ms~500 ms; besides, enough energy will be reserved to respond to at least 6 single-phase and three-phase or two-phase short circuit with an interval of 5 minutes, as is shown in Figure5.Figure 5
LVRT requirement of Danish grid under asymmetrical fault.Besides, countries around the world formulate corresponding network access LVRT technology requirements, and LVRT requirements of UK, US, Spain, and Italy are shown in Figure6.Figure 6
LVRT requirements of countries around the world.
(a)
England
(b)
America
(c)
Spain
(d)
ItalyAccording to active and reactive power decoupling control strategy under synchronous rotating coordinate system, control output of outer DC voltage is the given value of active current on axisd, and stable DC voltage represents a balanced relationship between input DC power and output AC power. Under grid fault including single-phase grounding short circuit, two-phase short circuit, two-phase short circuit grounding, and three-phase short circuit grounding fault, positive sequence grid voltage will decrease, which will greatly reduce AC output power. At this time, if input DC power remains unchanged, DC voltage loop will increase given value of active current to maintain constant DC voltage. Therefore, output current will significantly increase and exceed maximum current limit of inverter, which will inevitably trigger overcurrent protection and disconnect inverter from grid. Another noteworthy issue is that, under asymmetrical drop of grid, phase lock method of traditional PI controller cannot eliminate second harmonic fluctuation on axis q, and delay of phase lock would bring lag of controller-driven waveforms and would often lead to greater delay if grid is in serious unbalanced situation and would result in overcurrent accident if there is no phase compensation, and thus LVRT requirement cannot be reached.To solve the above problems, the academic circle carries out relevant researches; literatures [23, 24] put forward an improved current control algorithm, that is, dual d
q conversion and PI controller. It decomposes grid voltage and current under positive and reverse synchronous rotating coordinate systems, respectively, and then obtains positive sequence components under positive coordinate system and negative sequence components under reverse coordinate system, both components being DC variables, controls these components by two PI controllers, and thus realizes respective control of positive and negative sequence components of current under asymmetrical grid fault, in order to ensure asymmetrical fault ride-through (FRT) operating ability of inverter. However, this control method will conduct positive and negative sequence decomposition of inner loop feedback current, which will bring delay and error that cannot be ignored at current loops. Under small fault of asymmetrical stable grid state, such as 2% voltage asymmetry, delay has little influence on system operating performance. However, under large fault of transient asymmetry, such as 25% voltage asymmetry, there is a higher requirement for inverter’s control performance because of bad transient process, and thus delay and error brought by positive and negative sequence decomposition process are bound to reduce transient regulation performance of current controller and thus affect transient operating performance of grid-connected inverter. Literatures [25, 26] use active damping control method which adds capacitive current at LCL end to modulation signal, and an advantage of this control method lies in convenient design of controller. However, under different degrees of unbalanced grid, capacitive current feedforward coefficient has great influence on final inverter ride-through effects. Besides, this control method will add three capacitive current detection sensors in terms of hardware design, resulting in an increase of hardware costs. Papers [27–29] introduce PR (Proportion plus Resonant) controller to grid current loop controller; because the controlled objects are AC variables, this method does not require a
b
c
/
d
q coordinate conversion of current. This control algorithm can control positive and negative sequence components of output current simultaneously under α
β stationary coordinate system directly, without necessity of decomposing positive and negative sequence components of current under positive and reverse synchronous speed rotating coordinate systems, and thus eliminate delay of current control loop and improve dynamic control performance of grid-connected inverter under large asymmetrical fault. However, with this control method, there is a series of digital control delay, such as “detection → filtering → control computation → sending driving waveforms” delay, and thus delay and parameter design of controller will be taken into consideration during design of controller. Besides, as there is no d
q conversion process and control variables are AC ones, only AC variables to be converted need filtering. Control complexity is simplified during digital design of system.


## 2. Mathematical Model

### 2.1. Mathematical Model of Three-Phase Photovoltaic Inverter under Balanced Grid State

Single-stage photovoltaic grid-connected power generation system is generally consisting of solar cell module array, convergence device, inverter, low-voltage power distribution device, isolation boost equipment, and so forth [30, 31]. Figure 7 is the topological structure of three-phase non-midline single-stage photovoltaic grid-connected inverter, which mainly uses DC voltage capacitance, IGBT three-phase bridge, LCL filter, and so forth.Figure 7
Topological structure of three-phase single-stage inverter of LCL filter.Low-frequency mathematical model of three-phase photovoltaic inverter is obtained by ignoring high-frequency harmonics related to switching frequency and analyzing fundamental components of inverter. Circuit structure of three-phase photovoltaic inverter as shown in Figure7 is used to build a low-frequency mathematical model according to basic circuit theorems (KCL and KVL). Before building the model, hypotheses include the following:(a)
All switching devices are equivalent to ideal ones; that is, ignore switching loss.
(b)
Grid-side power is ideal three-phase symmetrical voltage source.
(c)
Three inductances at the AC side are identical and linear regardless of saturation.
(d)
To facilitate description of two-way energy transfer, load is equivalent to load resistanceR
L and load electromotive force e
L in series.Combine equivalent resistanceR
S of power bridge loss with equivalent resistance R
L of filter inductance at AC side, set R
=
R
S
+
R
L, and build circuit voltage current equation of three-phase photovoltaic inverter by KVL and KCL theorems as follows:(2)
L
d
i
a
d
t
+
R
i
a
=
e
a
-
v
dc
s
a
+
v
NO
,
L
d
i
b
d
t
+
R
i
b
=
e
b
-
v
dc
s
b
+
v
NO
,
L
d
i
c
d
t
+
R
i
c
=
e
c
-
v
dc
s
c
+
v
NO
.Consider that three-phase symmetrical system has the following features:(3)
∑
k
=
a
,
b
,
c
i
k
=
0
,
∑
k
=
a
,
b
,
c
e
k
=
0
.Integrate formula (2) into (3):(4)
v
NO
=
-
v
dc
3
∑
i
=
a
,
b
,
c
s
i
.Besides, apply KCL to capacitance at DC side:(5)
c
d
v
dc
d
t
=
i
a
s
a
+
i
b
s
b
+
i
c
s
c
-
v
dc
-
e
l
R
L
.Combine formulas (2)~(5), introduce state variable X
=
i
a
,
i
b
,
i
c
,
v
dc
T, and state variable expression of static mathematical model of three-phase photovoltaic inverter described by single-pole logical switch state value is as follows:(6)
Z
X
˙
=
A
X
+
B
E
,where(7)
A
=
-
R
0
0
-
S
a
-
∑
i
=
a
,
b
,
c
s
i
0
-
R
0
-
S
b
-
∑
i
=
a
,
b
,
c
s
i
0
0
-
R
-
S
c
-
∑
i
=
a
,
b
,
c
s
i
0
0
0
-
1
R
L
,
Z
=
L
L
L
C
,
B
=
1
1
1
-
1
R
L
,
E
=
e
a
e
b
e
c
e
l
T
.Convert mathematical model under three-phasea
b
c static coordinate system to two-phase rotating d
q coordinate system, in order to convert system parameters from AC to DC ones and facilitate controller design and computation of control parameters. Set d
q rotating coordinate system to rotate with angular frequency of grid voltage fundamental wave, analyze with the example of three-phase symmetrical current at AC side, and the relationship between d
q rotating coordinate system and a
b
c coordinate system is shown in Figure 8.Figure 8
Vector relationship betweend
q rotating coordinate system and a
b
c coordinate system.According to equivalent conversion vector relationship,(8)
i
d
=
2
3
i
a
sin
⁡
θ
+
i
b
sin
⁡
θ
-
120
∘
+
i
c
sin
⁡
θ
+
120
∘
,
i
q
=
2
3
i
a
cos
⁡
θ
+
i
b
cos
⁡
θ
-
120
∘
+
i
c
cos
⁡
θ
+
120
∘
.a
b
c
/
d
q converted matrix can be obtained by the following simplification:(9)
C
3
s
/
2
r
=
cos
⁡
θ
cos
⁡
θ
-
120
°
cos
⁡
θ
+
120
°
sin
⁡
θ
sin
⁡
θ
-
120
°
sin
⁡
θ
+
120
°
.Mathematical model underd
q coordinate system is shown in the following formula:(10)
L
·
d
i
d
d
t
=
v
d
-
e
d
+
ω
L
·
i
q
,
L
·
d
i
q
d
t
=
v
q
-
e
q
+
ω
L
·
i
d
,where e
d and e
q denote components on axis d
q of three-phase grid voltage, respectively, i
d and i
q denote components on axis d
q of three-phase grid-connected current, respectively, and v
d and v
q denote components on axis d
q of three-phase inverter output voltage, respectively.


### 2.2. Mathematical Model of Three-Phase Photovoltaic Inverter under Unbalanced Grid State

When grid voltage is unbalanced, grid-side voltage and current have positive and negative sequence components, and three-phase voltage cannot be simply converted into DC variables througha
b
c
/
d
q conversion, making design of control system quite complex. This section conducts a
b
c
/
d
q coordinate conversion for unbalanced three-phase voltage and obtains grid-side mathematical model of three-phase inverter under unbalanced grid according to coordinate conversion principle. When three-phase grid voltage is unbalanced, if we only consider fundamental components, expression of grid voltage is as follows:(11)
e
a
e
b
e
c
=
V
P
cos
⁡
ω
t
+
Ψ
P
cos
⁡
ω
t
+
Ψ
P
-
2
π
3
cos
⁡
ω
t
+
Ψ
P
+
2
π
3
+
V
N
cos
⁡
ω
t
+
Ψ
N
cos
⁡
ω
t
+
Ψ
N
+
2
π
3
cos
⁡
ω
t
+
Ψ
N
-
2
π
3
+
V
0
cos
⁡
Ψ
0
cos
⁡
Ψ
0
cos
⁡
Ψ
0
.In formula (11), V
P denotes positive sequence component amplitude of grid voltage, V
N is negative sequence component amplitude of grid voltage, and V
0 is zero sequence component amplitude of grid voltage. ω denotes angular frequency of fundamental wave, Ψ
P denotes initial phase angle of positive sequence component, Ψ
N denotes initial phase angle of negative sequence component, and Ψ
0 denotes initial phase angle of zero sequence component. Conduct a
b
c
/
α
β conversion of the above formula, and then(12)
e
α
e
β
=
V
P
cos
⁡
ω
t
+
Ψ
P
sin
⁡
ω
t
+
Ψ
P
+
V
N
cos
⁡
ω
t
+
Ψ
N
-
sin
⁡
ω
t
+
Ψ
N
.Formula (12) denotes that rotating vector E of three-phase voltage can be seen as synthesized vector of positive sequence component E
P and negative sequence component E
N under unbalanced grid voltage, as is shown in the following formula:(13)
E
=
E
P
+
E
N
,where(14)
E
P
=
V
P
e
j
ω
t
+
Ψ
P
,
E
N
=
V
N
e
-
j
ω
t
+
Ψ
N
.As is shown in Figure9, positive sequence component E
P rotates anticlockwise with angular frequency ω of fundamental wave, while negative sequence component E
N rotates clockwise with angular frequency ω of fundamental wave. Then, rotating vector E of three-phase voltage is no longer a space vector that rotates with a fixed angular frequency and amplitude of E changes with time as well. To simplify computation, express formula (14) as a vector under α
β coordinate system:(15)
E
α
β
=
E
α
β
P
+
E
α
β
N
,
E
α
β
P
=
e
α
P
+
j
e
β
P
,
E
α
β
N
=
e
α
N
+
j
e
β
N
.Figure 9
Coordinate change of rotating vector under asymmetrical voltage.
(a)
3
s
/
2
s transition
(b)
2
s
/
2
r transitionIn formula (15), E
α
β is the synthesized vector of rotating vector E under α
β coordinate system, E
α
β
(
P
) is the synthesized vector of positive sequence component E
P under α
β coordinate system, and E
α
β
(
N
) is the synthesized vector of negative sequence component E
N under α
β coordinate system.Conductα
β
/
d
q conversion of three-phase grid voltage, multiply matrix in formula (13) by terms in formula (15) successively, and then(16)
E
d
q
=
E
d
q
P
+
E
d
q
N
,
E
d
q
P
=
V
P
cos
⁡
ω
t
+
Ψ
P
-
θ
d
+
j
V
P
sin
⁡
ω
t
+
Ψ
P
-
θ
d
,
E
d
q
N
=
V
N
cos
⁡
ω
t
+
Ψ
N
-
θ
d
+
j
V
N
sin
⁡
ω
t
+
Ψ
N
-
θ
d
,where E
d
q is the synthesized vector of rotating vector E under d
q coordinate system, E
d
q
(
P
) is the synthesized vector of positive sequence component E
P under d
q coordinate system, and E
d
q
(
N
) is the synthesized vector of negative sequence component E
N under d
q coordinate system. Combine (11) with (16); then(17)
e
d
=
V
P
cos
⁡
Ψ
P
-
Ψ
d
+
V
N
cos
⁡
2
ω
t
+
Ψ
N
+
Ψ
d
,
e
q
=
V
P
sin
⁡
Ψ
P
-
Ψ
d
-
V
N
sin
⁡
2
ω
t
+
Ψ
N
+
Ψ
d
.It can be seen from formula (17) that the projection of positive sequence component E
P on d
q coordinate is DC variable, while the projection of negative sequence component E
N on d
q coordinate is doubled frequency AC variable.Assume ad
q rotating coordinate system n, rotate clockwise with angular frequency of ω, and then the angle between axis d and axis α is(18)
θ
N
=
-
ω
t
+
Ψ
d
N
.Based on (16) and (18),(19)
e
d
=
V
P
cos
⁡
2
ω
t
+
Ψ
P
+
Ψ
d
N
+
V
N
cos
⁡
Ψ
N
-
Ψ
d
N
,
e
q
=
V
P
sin
⁡
2
ω
t
+
Ψ
P
-
Ψ
d
N
-
V
N
sin
⁡
Ψ
N
-
Ψ
d
N
.From formula (19), the projection of positive sequence component E
P on this d
q coordinate is doubled frequency AC variable, while the projection of negative sequence component E
N on this d
q coordinate is DC variable. To differentiate, d
q coordinate system that rotates anticlockwise with angular frequency of ω is called positive d
q rotating coordinate system, while rotating coordinate system that rotates clockwise is called negative d
q rotating coordinate system. According to the above analysis, when three-phase voltage is asymmetrical, the projection of positive sequence component E
P on positive d
q rotating coordinate system is DC variable, and the projection on negative d
q rotating coordinate system is doubled frequency AC variable, while the projection of negative sequence component E
N on positive d
q rotating coordinate system is doubled frequency AC variable, and projection on negative d
q rotating coordinate system is DC variable.Conducta
b
c
/
α
β conversion of grid-side mathematical model under static a
b
c coordinate system, ignore parasitic resistance on inductance, and then(20)
V
α
β
=
E
α
β
+
L
d
I
α
β
d
t
,where V
α
β, E
α
β, and I
α
β are synthesized vectors of rotating vectors V, E, and I under α
β coordinate system.If we conductα
β
/
d
q conversion of formula (20) directly, according to analysis in the last section, projection values of three-phase voltage and current under a single d
q rotating coordinate system are sums of DC variables and doubled frequency AC variables, but the projection values of positive sequence and negative sequence components on positive and negative d
q rotating coordinate systems, respectively, are DC variables. This paper takes dual d
q rotating coordinate system control and corresponds it to positive and negative sequence components of three-phase voltage and current under positive and negative d
q rotating coordinate systems, respectively, to control DC variables and simplify design of control system and controller. Divide three-phase variables into positive sequence and negative sequence parts, and then(21)
V
α
β
=
V
α
β
P
+
V
α
β
N
,
V
α
β
P
=
E
α
β
P
+
L
d
I
α
β
P
d
t
,
V
α
β
N
=
E
α
β
N
+
L
d
I
α
β
N
d
t
,where V
α
β
(
P
), E
α
β
(
P
), and I
α
β
(
P
) are synthesized vectors of positive sequence components of positive sequence rotating vectors V
P, E
P, and I
P under α
β coordinate system and V
α
β
(
N
), E
α
β
(
N
), and I
α
β
(
N
) are synthesized vectors of negative sequence components of rotating vectors V
N, E
N, and I
N under α
β coordinate system. Combine formula (15) with (21) and sort and obtain mathematical model of three-phase grid-connected inverter under positive and negative d
q rotating coordinate systems under asymmetrical grid voltage as follows:(22)
v
+
d
P
=
L
·
d
i
+
d
P
d
t
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
L
·
d
i
+
q
P
d
t
+
e
+
q
P
-
ω
L
·
i
+
d
P
,
v
-
d
N
=
L
·
d
i
-
d
N
d
t
+
e
-
d
N
-
ω
L
·
i
-
q
N
,
v
-
q
N
=
L
·
d
i
-
q
N
d
t
+
e
-
q
N
-
ω
L
·
i
-
d
N
.Formula (22) denotes grid-side mathematical models of positive and negative sequence components of three-phase voltage and current under positive and negative d
q rotating coordinate systems, respectively, under the situation of unbalanced three-phase grid voltage. It can be seen that there are mutual coupling phenomena of components on axes d and q under two d
q rotating coordinate systems, which makes current control more difficult. Thus, current loops under these two coordinate systems are of decoupling control, respectively, to realize independent regulation of axes d and q. v
+
d
(
P
), e
+
d
P, and i
+
d
P are components on axis d of positive sequence components V
P, E
P, and I
P under positive d
q coordinate system, respectively; v
+
q
(
P
), e
+
q
P, and i
+
q
P are components on axis q of positive sequence components V
P, E
P, and I
P under positive d
q coordinate system. v
-
d
(
N
), e
-
d
N, and i
-
d
N are components on axis d of positive sequence components V
P, E
P, and I
P under negative d
q coordinate system, respectively; v
-
q
(
N
), e
-
q
N, and i
-
q
N are components on axis q of positive sequence components V
P, E
P, and I
P under negative d
q coordinate system; under asymmetrical fault of grid voltage in static α
β coordinate system, grid voltage and current have both positive sequence components that rotate positively at synchronous speed ω
1 and negative sequence components that rotate reversely at -
ω
1. Consider(23)
F
g
α
β
=
F
+
g
α
β
+
F
-
g
α
β
=
F
+
g
d
q
+
e
j
ω
1
t
+
F
-
g
d
q
-
e
-
j
ω
1
t
.Under positive synchronous rotating coordinate system, there is(24)
F
g
d
q
+
=
F
+
g
d
q
+
+
F
-
g
d
q
+
=
F
+
g
d
q
+
e
j
ω
1
t
+
F
-
g
d
q
-
e
-
j
ω
1
t
.In formula (24), F denotes voltage and current, superscripts + and − denote positive sequence and negative sequence components, respectively, and subscripts + and − denote positive and reverse synchronous rotating coordinate systems, respectively. Under positive synchronous rotating coordinate system, voltage and current have positive sequence DC variables and negative sequence AC variables with doubled frequency fluctuation. Besides, with asymmetrical fault of grid voltage, under positive and reverse synchronous rotating coordinate systems, mathematical model of three-phase grid-connected inverter can be expressed in forms of positive and negative sequence components under each coordinate system:(25)
V
+
g
d
q
+
=
U
+
g
d
q
+
+
L
·
d
I
+
g
d
q
+
d
t
+
j
ω
1
L
I
+
g
d
q
+
,
V
-
g
d
q
-
=
U
-
g
d
q
-
+
L
·
d
I
-
g
d
q
-
d
t
-
j
ω
1
L
I
-
g
d
q
-
.Under asymmetrical fault of grid voltage, active power output and reactive power output of grid-connected inverter are(26)
p
g
=
1.5
Re
⁡
s
=
1.5
Re
⁡
U
g
d
q
+
I
∗
g
d
q
+
,
q
g
=
1.5
Im
⁡
s
=
1.5
Im
⁡
U
g
d
q
+
I
∗
g
d
q
+
.Substitute formula (24) with formula (26), and then power model under asymmetrical fault of grid voltage can be obtained as follows:(27)
p
g
=
p
g
0
+
p
g
cos
⁡
2
·
cos
⁡
2
ω
1
t
+
p
g
sin
⁡
2
·
sin
⁡
2
ω
1
t
,
q
g
=
q
g
0
+
q
g
cos
⁡
2
·
cos
⁡
2
ω
1
t
+
q
g
sin
⁡
2
·
sin
⁡
2
ω
1
t
,where(28)
p
g
0
=
1.5
u
+
g
d
+
i
+
g
d
+
+
u
+
g
q
+
i
+
g
q
+
+
u
-
g
d
-
i
-
g
d
-
+
u
-
g
q
-
i
-
g
q
-
,
p
g
cos
⁡
2
=
1.5
u
+
g
d
+
i
-
g
d
-
+
u
+
g
q
+
i
-
g
q
-
+
u
-
g
d
-
i
+
g
d
+
+
u
-
g
q
-
i
+
g
q
+
,
p
g
sin
⁡
2
=
1.5
u
+
g
d
+
i
-
g
q
-
-
u
+
g
q
+
i
-
g
d
-
-
u
-
g
d
-
i
+
g
q
+
+
u
-
g
q
-
i
+
g
d
+
,
q
g
0
=
1.5
u
+
g
q
+
i
+
g
d
+
-
u
+
g
d
+
i
+
g
q
+
+
u
-
g
q
-
i
-
g
d
-
-
u
-
g
d
-
i
-
g
q
-
,
q
g
cos
⁡
2
=
1.5
u
-
g
q
-
i
+
g
d
+
-
u
-
g
d
-
i
+
g
q
+
+
u
+
g
q
+
i
-
g
d
-
-
u
+
g
d
+
i
-
g
q
-
,
q
g
sin
⁡
2
=
1.5
u
+
g
d
+
i
-
g
d
-
-
u
+
g
q
+
i
-
g
q
-
-
u
-
g
d
-
i
+
g
d
+
-
u
-
g
q
-
i
+
g
q
+
.In the above formula,p
g
0 and q
g
0 denote average components of active and reactive transient power, respectively, p
g
cos
⁡
2 and p
g
sin
⁡
2 denote doubled frequency component of active transient power, and q
g
cos
⁡
2 and q
g
sin
⁡
2 denote doubled frequency component of reactive transient power.If we take positive sequence axisd grid voltage vector oriented control strategy, that is, u
+
g
q
+
=
0, given command values of positive and negative sequence current components can be obtained, respectively, in order to eliminate doubled frequency fluctuation of active power:(29)
i
+
g
d
+
∗
=
p
g
0
∗
1.5
u
+
g
d
+
1
-
k
2
d
d
-
k
2
q
d
,
i
+
g
q
+
∗
=
-
q
g
0
∗
1.5
u
+
g
d
+
1
+
k
2
d
d
+
k
2
q
d
,
i
-
g
d
-
∗
=
-
k
d
d
·
i
+
g
d
+
∗
-
k
d
q
·
i
+
g
q
+
∗
,
i
-
g
q
-
∗
=
k
d
d
·
i
+
g
q
+
∗
-
k
d
q
·
i
+
g
d
+
∗
.In the above formula,k
d
d and k
q
d meet the following relations:(30)
k
d
d
=
u
-
g
d
-
u
+
g
d
+
,
k
q
d
=
u
-
g
q
-
u
+
g
d
+
.


## 2.1. Mathematical Model of Three-Phase Photovoltaic Inverter under Balanced Grid State

Single-stage photovoltaic grid-connected power generation system is generally consisting of solar cell module array, convergence device, inverter, low-voltage power distribution device, isolation boost equipment, and so forth [30, 31]. Figure 7 is the topological structure of three-phase non-midline single-stage photovoltaic grid-connected inverter, which mainly uses DC voltage capacitance, IGBT three-phase bridge, LCL filter, and so forth.Figure 7
Topological structure of three-phase single-stage inverter of LCL filter.Low-frequency mathematical model of three-phase photovoltaic inverter is obtained by ignoring high-frequency harmonics related to switching frequency and analyzing fundamental components of inverter. Circuit structure of three-phase photovoltaic inverter as shown in Figure7 is used to build a low-frequency mathematical model according to basic circuit theorems (KCL and KVL). Before building the model, hypotheses include the following:(a)
All switching devices are equivalent to ideal ones; that is, ignore switching loss.
(b)
Grid-side power is ideal three-phase symmetrical voltage source.
(c)
Three inductances at the AC side are identical and linear regardless of saturation.
(d)
To facilitate description of two-way energy transfer, load is equivalent to load resistanceR
L and load electromotive force e
L in series.Combine equivalent resistanceR
S of power bridge loss with equivalent resistance R
L of filter inductance at AC side, set R
=
R
S
+
R
L, and build circuit voltage current equation of three-phase photovoltaic inverter by KVL and KCL theorems as follows:(2)
L
d
i
a
d
t
+
R
i
a
=
e
a
-
v
dc
s
a
+
v
NO
,
L
d
i
b
d
t
+
R
i
b
=
e
b
-
v
dc
s
b
+
v
NO
,
L
d
i
c
d
t
+
R
i
c
=
e
c
-
v
dc
s
c
+
v
NO
.Consider that three-phase symmetrical system has the following features:(3)
∑
k
=
a
,
b
,
c
i
k
=
0
,
∑
k
=
a
,
b
,
c
e
k
=
0
.Integrate formula (2) into (3):(4)
v
NO
=
-
v
dc
3
∑
i
=
a
,
b
,
c
s
i
.Besides, apply KCL to capacitance at DC side:(5)
c
d
v
dc
d
t
=
i
a
s
a
+
i
b
s
b
+
i
c
s
c
-
v
dc
-
e
l
R
L
.Combine formulas (2)~(5), introduce state variable X
=
i
a
,
i
b
,
i
c
,
v
dc
T, and state variable expression of static mathematical model of three-phase photovoltaic inverter described by single-pole logical switch state value is as follows:(6)
Z
X
˙
=
A
X
+
B
E
,where(7)
A
=
-
R
0
0
-
S
a
-
∑
i
=
a
,
b
,
c
s
i
0
-
R
0
-
S
b
-
∑
i
=
a
,
b
,
c
s
i
0
0
-
R
-
S
c
-
∑
i
=
a
,
b
,
c
s
i
0
0
0
-
1
R
L
,
Z
=
L
L
L
C
,
B
=
1
1
1
-
1
R
L
,
E
=
e
a
e
b
e
c
e
l
T
.Convert mathematical model under three-phasea
b
c static coordinate system to two-phase rotating d
q coordinate system, in order to convert system parameters from AC to DC ones and facilitate controller design and computation of control parameters. Set d
q rotating coordinate system to rotate with angular frequency of grid voltage fundamental wave, analyze with the example of three-phase symmetrical current at AC side, and the relationship between d
q rotating coordinate system and a
b
c coordinate system is shown in Figure 8.Figure 8
Vector relationship betweend
q rotating coordinate system and a
b
c coordinate system.According to equivalent conversion vector relationship,(8)
i
d
=
2
3
i
a
sin
⁡
θ
+
i
b
sin
⁡
θ
-
120
∘
+
i
c
sin
⁡
θ
+
120
∘
,
i
q
=
2
3
i
a
cos
⁡
θ
+
i
b
cos
⁡
θ
-
120
∘
+
i
c
cos
⁡
θ
+
120
∘
.a
b
c
/
d
q converted matrix can be obtained by the following simplification:(9)
C
3
s
/
2
r
=
cos
⁡
θ
cos
⁡
θ
-
120
°
cos
⁡
θ
+
120
°
sin
⁡
θ
sin
⁡
θ
-
120
°
sin
⁡
θ
+
120
°
.Mathematical model underd
q coordinate system is shown in the following formula:(10)
L
·
d
i
d
d
t
=
v
d
-
e
d
+
ω
L
·
i
q
,
L
·
d
i
q
d
t
=
v
q
-
e
q
+
ω
L
·
i
d
,where e
d and e
q denote components on axis d
q of three-phase grid voltage, respectively, i
d and i
q denote components on axis d
q of three-phase grid-connected current, respectively, and v
d and v
q denote components on axis d
q of three-phase inverter output voltage, respectively.


## 2.2. Mathematical Model of Three-Phase Photovoltaic Inverter under Unbalanced Grid State

When grid voltage is unbalanced, grid-side voltage and current have positive and negative sequence components, and three-phase voltage cannot be simply converted into DC variables througha
b
c
/
d
q conversion, making design of control system quite complex. This section conducts a
b
c
/
d
q coordinate conversion for unbalanced three-phase voltage and obtains grid-side mathematical model of three-phase inverter under unbalanced grid according to coordinate conversion principle. When three-phase grid voltage is unbalanced, if we only consider fundamental components, expression of grid voltage is as follows:(11)
e
a
e
b
e
c
=
V
P
cos
⁡
ω
t
+
Ψ
P
cos
⁡
ω
t
+
Ψ
P
-
2
π
3
cos
⁡
ω
t
+
Ψ
P
+
2
π
3
+
V
N
cos
⁡
ω
t
+
Ψ
N
cos
⁡
ω
t
+
Ψ
N
+
2
π
3
cos
⁡
ω
t
+
Ψ
N
-
2
π
3
+
V
0
cos
⁡
Ψ
0
cos
⁡
Ψ
0
cos
⁡
Ψ
0
.In formula (11), V
P denotes positive sequence component amplitude of grid voltage, V
N is negative sequence component amplitude of grid voltage, and V
0 is zero sequence component amplitude of grid voltage. ω denotes angular frequency of fundamental wave, Ψ
P denotes initial phase angle of positive sequence component, Ψ
N denotes initial phase angle of negative sequence component, and Ψ
0 denotes initial phase angle of zero sequence component. Conduct a
b
c
/
α
β conversion of the above formula, and then(12)
e
α
e
β
=
V
P
cos
⁡
ω
t
+
Ψ
P
sin
⁡
ω
t
+
Ψ
P
+
V
N
cos
⁡
ω
t
+
Ψ
N
-
sin
⁡
ω
t
+
Ψ
N
.Formula (12) denotes that rotating vector E of three-phase voltage can be seen as synthesized vector of positive sequence component E
P and negative sequence component E
N under unbalanced grid voltage, as is shown in the following formula:(13)
E
=
E
P
+
E
N
,where(14)
E
P
=
V
P
e
j
ω
t
+
Ψ
P
,
E
N
=
V
N
e
-
j
ω
t
+
Ψ
N
.As is shown in Figure9, positive sequence component E
P rotates anticlockwise with angular frequency ω of fundamental wave, while negative sequence component E
N rotates clockwise with angular frequency ω of fundamental wave. Then, rotating vector E of three-phase voltage is no longer a space vector that rotates with a fixed angular frequency and amplitude of E changes with time as well. To simplify computation, express formula (14) as a vector under α
β coordinate system:(15)
E
α
β
=
E
α
β
P
+
E
α
β
N
,
E
α
β
P
=
e
α
P
+
j
e
β
P
,
E
α
β
N
=
e
α
N
+
j
e
β
N
.Figure 9
Coordinate change of rotating vector under asymmetrical voltage.
(a)
3
s
/
2
s transition
(b)
2
s
/
2
r transitionIn formula (15), E
α
β is the synthesized vector of rotating vector E under α
β coordinate system, E
α
β
(
P
) is the synthesized vector of positive sequence component E
P under α
β coordinate system, and E
α
β
(
N
) is the synthesized vector of negative sequence component E
N under α
β coordinate system.Conductα
β
/
d
q conversion of three-phase grid voltage, multiply matrix in formula (13) by terms in formula (15) successively, and then(16)
E
d
q
=
E
d
q
P
+
E
d
q
N
,
E
d
q
P
=
V
P
cos
⁡
ω
t
+
Ψ
P
-
θ
d
+
j
V
P
sin
⁡
ω
t
+
Ψ
P
-
θ
d
,
E
d
q
N
=
V
N
cos
⁡
ω
t
+
Ψ
N
-
θ
d
+
j
V
N
sin
⁡
ω
t
+
Ψ
N
-
θ
d
,where E
d
q is the synthesized vector of rotating vector E under d
q coordinate system, E
d
q
(
P
) is the synthesized vector of positive sequence component E
P under d
q coordinate system, and E
d
q
(
N
) is the synthesized vector of negative sequence component E
N under d
q coordinate system. Combine (11) with (16); then(17)
e
d
=
V
P
cos
⁡
Ψ
P
-
Ψ
d
+
V
N
cos
⁡
2
ω
t
+
Ψ
N
+
Ψ
d
,
e
q
=
V
P
sin
⁡
Ψ
P
-
Ψ
d
-
V
N
sin
⁡
2
ω
t
+
Ψ
N
+
Ψ
d
.It can be seen from formula (17) that the projection of positive sequence component E
P on d
q coordinate is DC variable, while the projection of negative sequence component E
N on d
q coordinate is doubled frequency AC variable.Assume ad
q rotating coordinate system n, rotate clockwise with angular frequency of ω, and then the angle between axis d and axis α is(18)
θ
N
=
-
ω
t
+
Ψ
d
N
.Based on (16) and (18),(19)
e
d
=
V
P
cos
⁡
2
ω
t
+
Ψ
P
+
Ψ
d
N
+
V
N
cos
⁡
Ψ
N
-
Ψ
d
N
,
e
q
=
V
P
sin
⁡
2
ω
t
+
Ψ
P
-
Ψ
d
N
-
V
N
sin
⁡
Ψ
N
-
Ψ
d
N
.From formula (19), the projection of positive sequence component E
P on this d
q coordinate is doubled frequency AC variable, while the projection of negative sequence component E
N on this d
q coordinate is DC variable. To differentiate, d
q coordinate system that rotates anticlockwise with angular frequency of ω is called positive d
q rotating coordinate system, while rotating coordinate system that rotates clockwise is called negative d
q rotating coordinate system. According to the above analysis, when three-phase voltage is asymmetrical, the projection of positive sequence component E
P on positive d
q rotating coordinate system is DC variable, and the projection on negative d
q rotating coordinate system is doubled frequency AC variable, while the projection of negative sequence component E
N on positive d
q rotating coordinate system is doubled frequency AC variable, and projection on negative d
q rotating coordinate system is DC variable.Conducta
b
c
/
α
β conversion of grid-side mathematical model under static a
b
c coordinate system, ignore parasitic resistance on inductance, and then(20)
V
α
β
=
E
α
β
+
L
d
I
α
β
d
t
,where V
α
β, E
α
β, and I
α
β are synthesized vectors of rotating vectors V, E, and I under α
β coordinate system.If we conductα
β
/
d
q conversion of formula (20) directly, according to analysis in the last section, projection values of three-phase voltage and current under a single d
q rotating coordinate system are sums of DC variables and doubled frequency AC variables, but the projection values of positive sequence and negative sequence components on positive and negative d
q rotating coordinate systems, respectively, are DC variables. This paper takes dual d
q rotating coordinate system control and corresponds it to positive and negative sequence components of three-phase voltage and current under positive and negative d
q rotating coordinate systems, respectively, to control DC variables and simplify design of control system and controller. Divide three-phase variables into positive sequence and negative sequence parts, and then(21)
V
α
β
=
V
α
β
P
+
V
α
β
N
,
V
α
β
P
=
E
α
β
P
+
L
d
I
α
β
P
d
t
,
V
α
β
N
=
E
α
β
N
+
L
d
I
α
β
N
d
t
,where V
α
β
(
P
), E
α
β
(
P
), and I
α
β
(
P
) are synthesized vectors of positive sequence components of positive sequence rotating vectors V
P, E
P, and I
P under α
β coordinate system and V
α
β
(
N
), E
α
β
(
N
), and I
α
β
(
N
) are synthesized vectors of negative sequence components of rotating vectors V
N, E
N, and I
N under α
β coordinate system. Combine formula (15) with (21) and sort and obtain mathematical model of three-phase grid-connected inverter under positive and negative d
q rotating coordinate systems under asymmetrical grid voltage as follows:(22)
v
+
d
P
=
L
·
d
i
+
d
P
d
t
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
L
·
d
i
+
q
P
d
t
+
e
+
q
P
-
ω
L
·
i
+
d
P
,
v
-
d
N
=
L
·
d
i
-
d
N
d
t
+
e
-
d
N
-
ω
L
·
i
-
q
N
,
v
-
q
N
=
L
·
d
i
-
q
N
d
t
+
e
-
q
N
-
ω
L
·
i
-
d
N
.Formula (22) denotes grid-side mathematical models of positive and negative sequence components of three-phase voltage and current under positive and negative d
q rotating coordinate systems, respectively, under the situation of unbalanced three-phase grid voltage. It can be seen that there are mutual coupling phenomena of components on axes d and q under two d
q rotating coordinate systems, which makes current control more difficult. Thus, current loops under these two coordinate systems are of decoupling control, respectively, to realize independent regulation of axes d and q. v
+
d
(
P
), e
+
d
P, and i
+
d
P are components on axis d of positive sequence components V
P, E
P, and I
P under positive d
q coordinate system, respectively; v
+
q
(
P
), e
+
q
P, and i
+
q
P are components on axis q of positive sequence components V
P, E
P, and I
P under positive d
q coordinate system. v
-
d
(
N
), e
-
d
N, and i
-
d
N are components on axis d of positive sequence components V
P, E
P, and I
P under negative d
q coordinate system, respectively; v
-
q
(
N
), e
-
q
N, and i
-
q
N are components on axis q of positive sequence components V
P, E
P, and I
P under negative d
q coordinate system; under asymmetrical fault of grid voltage in static α
β coordinate system, grid voltage and current have both positive sequence components that rotate positively at synchronous speed ω
1 and negative sequence components that rotate reversely at -
ω
1. Consider(23)
F
g
α
β
=
F
+
g
α
β
+
F
-
g
α
β
=
F
+
g
d
q
+
e
j
ω
1
t
+
F
-
g
d
q
-
e
-
j
ω
1
t
.Under positive synchronous rotating coordinate system, there is(24)
F
g
d
q
+
=
F
+
g
d
q
+
+
F
-
g
d
q
+
=
F
+
g
d
q
+
e
j
ω
1
t
+
F
-
g
d
q
-
e
-
j
ω
1
t
.In formula (24), F denotes voltage and current, superscripts + and − denote positive sequence and negative sequence components, respectively, and subscripts + and − denote positive and reverse synchronous rotating coordinate systems, respectively. Under positive synchronous rotating coordinate system, voltage and current have positive sequence DC variables and negative sequence AC variables with doubled frequency fluctuation. Besides, with asymmetrical fault of grid voltage, under positive and reverse synchronous rotating coordinate systems, mathematical model of three-phase grid-connected inverter can be expressed in forms of positive and negative sequence components under each coordinate system:(25)
V
+
g
d
q
+
=
U
+
g
d
q
+
+
L
·
d
I
+
g
d
q
+
d
t
+
j
ω
1
L
I
+
g
d
q
+
,
V
-
g
d
q
-
=
U
-
g
d
q
-
+
L
·
d
I
-
g
d
q
-
d
t
-
j
ω
1
L
I
-
g
d
q
-
.Under asymmetrical fault of grid voltage, active power output and reactive power output of grid-connected inverter are(26)
p
g
=
1.5
Re
⁡
s
=
1.5
Re
⁡
U
g
d
q
+
I
∗
g
d
q
+
,
q
g
=
1.5
Im
⁡
s
=
1.5
Im
⁡
U
g
d
q
+
I
∗
g
d
q
+
.Substitute formula (24) with formula (26), and then power model under asymmetrical fault of grid voltage can be obtained as follows:(27)
p
g
=
p
g
0
+
p
g
cos
⁡
2
·
cos
⁡
2
ω
1
t
+
p
g
sin
⁡
2
·
sin
⁡
2
ω
1
t
,
q
g
=
q
g
0
+
q
g
cos
⁡
2
·
cos
⁡
2
ω
1
t
+
q
g
sin
⁡
2
·
sin
⁡
2
ω
1
t
,where(28)
p
g
0
=
1.5
u
+
g
d
+
i
+
g
d
+
+
u
+
g
q
+
i
+
g
q
+
+
u
-
g
d
-
i
-
g
d
-
+
u
-
g
q
-
i
-
g
q
-
,
p
g
cos
⁡
2
=
1.5
u
+
g
d
+
i
-
g
d
-
+
u
+
g
q
+
i
-
g
q
-
+
u
-
g
d
-
i
+
g
d
+
+
u
-
g
q
-
i
+
g
q
+
,
p
g
sin
⁡
2
=
1.5
u
+
g
d
+
i
-
g
q
-
-
u
+
g
q
+
i
-
g
d
-
-
u
-
g
d
-
i
+
g
q
+
+
u
-
g
q
-
i
+
g
d
+
,
q
g
0
=
1.5
u
+
g
q
+
i
+
g
d
+
-
u
+
g
d
+
i
+
g
q
+
+
u
-
g
q
-
i
-
g
d
-
-
u
-
g
d
-
i
-
g
q
-
,
q
g
cos
⁡
2
=
1.5
u
-
g
q
-
i
+
g
d
+
-
u
-
g
d
-
i
+
g
q
+
+
u
+
g
q
+
i
-
g
d
-
-
u
+
g
d
+
i
-
g
q
-
,
q
g
sin
⁡
2
=
1.5
u
+
g
d
+
i
-
g
d
-
-
u
+
g
q
+
i
-
g
q
-
-
u
-
g
d
-
i
+
g
d
+
-
u
-
g
q
-
i
+
g
q
+
.In the above formula,p
g
0 and q
g
0 denote average components of active and reactive transient power, respectively, p
g
cos
⁡
2 and p
g
sin
⁡
2 denote doubled frequency component of active transient power, and q
g
cos
⁡
2 and q
g
sin
⁡
2 denote doubled frequency component of reactive transient power.If we take positive sequence axisd grid voltage vector oriented control strategy, that is, u
+
g
q
+
=
0, given command values of positive and negative sequence current components can be obtained, respectively, in order to eliminate doubled frequency fluctuation of active power:(29)
i
+
g
d
+
∗
=
p
g
0
∗
1.5
u
+
g
d
+
1
-
k
2
d
d
-
k
2
q
d
,
i
+
g
q
+
∗
=
-
q
g
0
∗
1.5
u
+
g
d
+
1
+
k
2
d
d
+
k
2
q
d
,
i
-
g
d
-
∗
=
-
k
d
d
·
i
+
g
d
+
∗
-
k
d
q
·
i
+
g
q
+
∗
,
i
-
g
q
-
∗
=
k
d
d
·
i
+
g
q
+
∗
-
k
d
q
·
i
+
g
d
+
∗
.In the above formula,k
d
d and k
q
d meet the following relations:(30)
k
d
d
=
u
-
g
d
-
u
+
g
d
+
,
k
q
d
=
u
-
g
q
-
u
+
g
d
+
.


## 3. Existing Control Strategies

By deducing grid-side current model and active/reactive power model under unbalanced grid, during LVRT, inverter control algorithm will solve three major problems.Question 1.
How to eliminate doubled frequency fluctuation of active power during grid voltage dip?Question 2.
How to eliminate doubled frequency fluctuation of reactive power during grid voltage dip?Question 3.
How to eliminate negative sequence injection current during grid voltage dip?The academic circle has put forward some control strategies and mainly studies on these three points. For comparison, single axisd
q
P
I control method will be added to the following control algorithms under symmetrical grid as follows.


### 3.1. Control Strategy Based on Current Loop under Single Axisd
q Rotating Coordinate System

The basic idea of using singled
q
P
I controller method is to conduct a
b
c
/
d
q conversion of IGBT inverter-side current feedback signal, obtain i
d and i
q, and send to PI controller. This control method does not conduct positive and negative sequence decomposition of current, and the control block diagram is shown in Figure 10.Figure 10
Single axisd
q PI control method.This control method will lead to deterioration of control effects during grid unbalance or drop, as it considers feedback current as a whole and does not restrain doubled frequency fluctuation of negative sequence current and active/reactive power. Figure11 is about waveform of active/reactive power under single-phase drop (drop) of grid. Grid voltage has single-phase drop at 0.1 s, and active power and reactive power have single-phase drop 0.1 s later and lead to doubled frequency fluctuation due to absence of responsive reactive power control method.Figure 11
Active/reactive power output waveform under single-phase grid drop.


### 3.2. Control Strategy Based on Current Loop under Duald
q Rotating Coordinate System

Current loop control method under duald
q rotating coordinate system is used to control positive and negative sequence components of current under different rotating coordinate systems, respectively. Therefore, each rotating coordinate system realizes control of DC variables, and thus PI controller could achieve very good stable state and dynamic performance [14, 15]. Under unbalanced three-phase grid voltage, three-phase voltage and positive and negative sequence components of current are in grid-side mathematical models under positive and negative d
q rotating coordinate systems, respectively. It can be seen that there are mutual coupling phenomena of components on axes d and q under two d
q rotating coordinate systems, which makes current control more difficult. Thus, current loops under these two coordinate systems are of decoupling control, respectively, to realize independent regulation of axes d and q.With the example of current regulation under positived
q rotating coordinate system, when grid voltage is unbalanced, positive sequence components of grid-side inverter output voltage under positive d
q rotating coordinate system are shown in the following formula:(31)
v
+
d
P
=
L
·
d
i
+
d
P
d
t
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
L
·
d
i
+
q
P
d
t
+
e
+
q
P
-
ω
L
·
i
+
d
P
.Make approximation, and(32)
Δ
v
+
d
P
=
L
·
d
i
+
d
P
d
t
,
Δ
v
+
q
P
=
L
·
d
i
+
q
P
d
t
.Formula (31) can be expressed as(33)
v
+
d
P
=
Δ
v
+
d
P
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
Δ
v
+
q
P
+
e
+
q
P
-
ω
L
·
i
+
d
P
.To eliminate error under stable state, current loop takes PI regulation, and controller can be expressed as(34)
v
+
d
P
=
k
p
·
i
+
d
P
∗
-
i
+
d
P
+
k
i
·
∫
i
+
d
P
∗
-
i
+
d
P
d
t
,
v
+
q
P
=
k
p
·
i
+
q
P
∗
-
i
+
q
P
+
k
i
·
∫
i
+
q
P
∗
-
i
+
q
P
d
t
.In the above formula,i
+
d
P
∗ and i
+
q
P
∗ are positive sequence current reference components on axis d
q and k
p and k
i are proportion and integral coefficient of PI controller, respectively. Cross decoupling control block diagram of positive sequence components of grid-side current on axis d
q under positive synchronous rotating coordinate system can be obtained according to formula (33) and (34), as is shown in Figure 12.Figure 12
Axisd
q decoupling control block diagram.Similarly, use PI controller to control negative sequence components of current and controller:(35)
v
-
d
N
=
k
p
·
i
-
d
N
∗
-
i
-
d
N
+
k
i
·
∫
i
-
d
N
∗
-
i
-
d
N
d
t
,
v
-
q
N
=
k
p
·
i
-
q
N
∗
-
i
-
q
N
+
k
i
·
∫
i
-
q
N
∗
-
i
-
q
N
d
t
.Assume that there are positive and negatived
q rotating coordinate systems that are symmetrical about axis α with rotating angular frequency of ω and initial angle of ±
Ψ
d. Rotating vector V corresponding to output voltage of grid-side inverter can be seen as the vector sum of positive and negative sequence components; that is(36)
V
=
V
+
d
q
P
·
e
j
θ
d
+
V
+
d
q
N
·
e
-
j
θ
d
,where(37)
θ
d
=
ω
t
+
Ψ
d
,
V
+
d
q
P
=
v
+
d
P
+
j
v
+
q
P
,
V
+
d
q
N
=
v
+
d
N
+
j
v
+
q
N
.Combine formula (34) with (35), and obtain double current loop control block diagram of inverter under positive and negative d
q coordinate systems, as is shown in Figure 13.Figure 13
Block diagram of three-phase grid-connected inverter system control.According to Figure14, positive sequence component V
+
d
q that inverter outputs is current loop output corresponding to positive sequence current, while negative sequence component V
-
d
q that inverter outputs is current loop output corresponding to negative sequence current. Through d
q
/
α
β conversion, V
α
β
P and V
α
β
N are obtained, and output voltage vector V
α
β of inverter is obtained through vector synthesis. Vector V
α
β would be input of space vector pulse width modulation (SVPWM) to control turning on/off of switching devices like IGBT and so forth and finally achieve grid-connected current control objective.Figure 14
Block diagram of LCL active damping control under unbalanced control strategy.


### 3.3. Active Damping Compensation Feedback Control Strategy

In practical application, in order to conduct real-time detection of inverter-side current to facilitate safety protection, place current sensors at the output side of inverter. Meanwhile, to obtain grid synchronous signals, place voltage sensors at the grid side. As inverter-side output voltage is pulse width modulation (PWM) wave and is inconvenient to measure, in order to use the known measurement of grid voltage and obtain accurate command current, convert formula (31) and obtain the relationship between inverter-side voltage and grid voltage under fundamental waves as follows:(38)
u
P
inv
d
=
e
P
d
-
ω
L
inv
i
P
inv
q
-
ω
L
g
i
P
g
q
,
u
P
inv
q
=
e
P
q
-
ω
L
inv
i
P
inv
d
-
ω
L
g
i
P
g
d
,
u
N
inv
d
=
e
P
d
-
ω
L
inv
i
N
inv
q
-
ω
L
g
i
N
g
q
,
u
N
inv
q
=
e
P
q
-
ω
L
inv
i
N
inv
d
-
ω
L
g
i
N
g
d
.Substitute formula (38) with formula (28) and obtain [16, 17]:(39)
P
0
Q
0
P
cos
⁡
2
P
sin
⁡
2
=
3
2
e
P
d
e
P
q
e
N
d
e
N
q
e
P
q
-
e
P
d
e
N
q
-
e
N
d
e
N
d
e
N
q
e
P
d
e
P
q
e
N
q
-
e
N
d
-
e
P
q
e
P
d
i
P
inv
d
i
P
inv
q
i
N
inv
d
i
N
inv
q
+
3
2
-
ω
L
i
P
inv
q
+
i
P
g
q
ω
L
i
P
inv
d
+
i
P
g
d
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
ω
L
i
P
inv
d
+
i
P
g
d
ω
L
i
P
inv
q
+
i
P
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
N
inv
q
+
i
N
g
q
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
P
inv
q
+
i
P
g
q
ω
L
i
P
inv
d
+
i
P
g
d
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
P
inv
d
+
i
P
g
d
-
ω
L
i
P
inv
q
+
i
P
g
q
i
P
inv
d
i
P
inv
q
i
N
inv
d
i
N
inv
q
.Formula (39) is the relationship between known detection and inverter output power, to eliminate negative sequence current (i
N
inv
(
d
)
=
i
N
inv
(
q
)
=
0). Considering full power operation, to simplify computation under designed operating parameters, simplify the above formula and given value of current reference is (40)
i
P
∗
inv
d
=
2
e
P
d
P
0
+
e
P
q
Q
0
3
e
P
d
2
+
e
P
q
2
,
i
P
∗
inv
q
=
2
e
P
q
P
0
-
e
P
d
Q
0
3
e
P
d
2
+
e
P
q
2
,
i
N
inv
d
=
0
,
i
N
inv
q
=
0
,
(41)
Δ
Q
=
ω
C
e
P
c
d
2
+
e
P
c
q
2
+
e
N
c
q
2
+
e
N
c
d
2
.Under stable operating state, setΔ
Q
=
Q
0 and substitute with formula (41), and obtain given value of inner loop current under unit power factor grid-connected operation. When conducting process of LVRT, carry out real-time modification of the value of Q
0 to regulate reactive power output according to detected grid voltage drop depth. To ensure that grid-connected current does not exceed output protection value, active and reactive power output will meet:(42)
i
P
∗
inv
d
≤
i
max
⁡
2
-
i
P
∗
inv
q
2
.In the formula,i
max
⁡ denotes the maximum amplitude limit of output current. Stable control and reactive power support after voltage drop are realized by dynamic change of current command value on axis d
q, and select appropriate controller parameters to ensure system stability. As positive and negative sequence dual d
q current loop control method would separate voltage and current into positive and negative sequence components under unbalanced situation while filter capacitive current i
C is mainly high-frequency component, commonly used positive and negative sequence separation method cannot obtain positive and negative sequence components under high frequency. Thus, active damping control method is taken under balanced situation but cannot realize damping control under unbalanced situation and even affect system stability. Add active damping method of filter capacitive current feedforward to modulation signal, and current loop control block diagram is shown in Figure 14. Dashed block on the left side is the control block diagram of unbalanced control algorithm, and dashed block on the right side is the equivalent mathematical model of LCL. Feedforward method shown in Figure 14 avoids positive and negative sequence separation feedforward variable i
C effectively, and regulation of coefficient k
C does not affect transfer function of active damping algorithm in Figure 5 and can restrain resonance of LCL filter effectively.


## 3.1. Control Strategy Based on Current Loop under Single Axisd
q Rotating Coordinate System

The basic idea of using singled
q
P
I controller method is to conduct a
b
c
/
d
q conversion of IGBT inverter-side current feedback signal, obtain i
d and i
q, and send to PI controller. This control method does not conduct positive and negative sequence decomposition of current, and the control block diagram is shown in Figure 10.Figure 10
Single axisd
q PI control method.This control method will lead to deterioration of control effects during grid unbalance or drop, as it considers feedback current as a whole and does not restrain doubled frequency fluctuation of negative sequence current and active/reactive power. Figure11 is about waveform of active/reactive power under single-phase drop (drop) of grid. Grid voltage has single-phase drop at 0.1 s, and active power and reactive power have single-phase drop 0.1 s later and lead to doubled frequency fluctuation due to absence of responsive reactive power control method.Figure 11
Active/reactive power output waveform under single-phase grid drop.


## 3.2. Control Strategy Based on Current Loop under Duald
q Rotating Coordinate System

Current loop control method under duald
q rotating coordinate system is used to control positive and negative sequence components of current under different rotating coordinate systems, respectively. Therefore, each rotating coordinate system realizes control of DC variables, and thus PI controller could achieve very good stable state and dynamic performance [14, 15]. Under unbalanced three-phase grid voltage, three-phase voltage and positive and negative sequence components of current are in grid-side mathematical models under positive and negative d
q rotating coordinate systems, respectively. It can be seen that there are mutual coupling phenomena of components on axes d and q under two d
q rotating coordinate systems, which makes current control more difficult. Thus, current loops under these two coordinate systems are of decoupling control, respectively, to realize independent regulation of axes d and q.With the example of current regulation under positived
q rotating coordinate system, when grid voltage is unbalanced, positive sequence components of grid-side inverter output voltage under positive d
q rotating coordinate system are shown in the following formula:(31)
v
+
d
P
=
L
·
d
i
+
d
P
d
t
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
L
·
d
i
+
q
P
d
t
+
e
+
q
P
-
ω
L
·
i
+
d
P
.Make approximation, and(32)
Δ
v
+
d
P
=
L
·
d
i
+
d
P
d
t
,
Δ
v
+
q
P
=
L
·
d
i
+
q
P
d
t
.Formula (31) can be expressed as(33)
v
+
d
P
=
Δ
v
+
d
P
+
e
+
d
P
-
ω
L
·
i
+
q
P
,
v
+
q
P
=
Δ
v
+
q
P
+
e
+
q
P
-
ω
L
·
i
+
d
P
.To eliminate error under stable state, current loop takes PI regulation, and controller can be expressed as(34)
v
+
d
P
=
k
p
·
i
+
d
P
∗
-
i
+
d
P
+
k
i
·
∫
i
+
d
P
∗
-
i
+
d
P
d
t
,
v
+
q
P
=
k
p
·
i
+
q
P
∗
-
i
+
q
P
+
k
i
·
∫
i
+
q
P
∗
-
i
+
q
P
d
t
.In the above formula,i
+
d
P
∗ and i
+
q
P
∗ are positive sequence current reference components on axis d
q and k
p and k
i are proportion and integral coefficient of PI controller, respectively. Cross decoupling control block diagram of positive sequence components of grid-side current on axis d
q under positive synchronous rotating coordinate system can be obtained according to formula (33) and (34), as is shown in Figure 12.Figure 12
Axisd
q decoupling control block diagram.Similarly, use PI controller to control negative sequence components of current and controller:(35)
v
-
d
N
=
k
p
·
i
-
d
N
∗
-
i
-
d
N
+
k
i
·
∫
i
-
d
N
∗
-
i
-
d
N
d
t
,
v
-
q
N
=
k
p
·
i
-
q
N
∗
-
i
-
q
N
+
k
i
·
∫
i
-
q
N
∗
-
i
-
q
N
d
t
.Assume that there are positive and negatived
q rotating coordinate systems that are symmetrical about axis α with rotating angular frequency of ω and initial angle of ±
Ψ
d. Rotating vector V corresponding to output voltage of grid-side inverter can be seen as the vector sum of positive and negative sequence components; that is(36)
V
=
V
+
d
q
P
·
e
j
θ
d
+
V
+
d
q
N
·
e
-
j
θ
d
,where(37)
θ
d
=
ω
t
+
Ψ
d
,
V
+
d
q
P
=
v
+
d
P
+
j
v
+
q
P
,
V
+
d
q
N
=
v
+
d
N
+
j
v
+
q
N
.Combine formula (34) with (35), and obtain double current loop control block diagram of inverter under positive and negative d
q coordinate systems, as is shown in Figure 13.Figure 13
Block diagram of three-phase grid-connected inverter system control.According to Figure14, positive sequence component V
+
d
q that inverter outputs is current loop output corresponding to positive sequence current, while negative sequence component V
-
d
q that inverter outputs is current loop output corresponding to negative sequence current. Through d
q
/
α
β conversion, V
α
β
P and V
α
β
N are obtained, and output voltage vector V
α
β of inverter is obtained through vector synthesis. Vector V
α
β would be input of space vector pulse width modulation (SVPWM) to control turning on/off of switching devices like IGBT and so forth and finally achieve grid-connected current control objective.Figure 14
Block diagram of LCL active damping control under unbalanced control strategy.


## 3.3. Active Damping Compensation Feedback Control Strategy

In practical application, in order to conduct real-time detection of inverter-side current to facilitate safety protection, place current sensors at the output side of inverter. Meanwhile, to obtain grid synchronous signals, place voltage sensors at the grid side. As inverter-side output voltage is pulse width modulation (PWM) wave and is inconvenient to measure, in order to use the known measurement of grid voltage and obtain accurate command current, convert formula (31) and obtain the relationship between inverter-side voltage and grid voltage under fundamental waves as follows:(38)
u
P
inv
d
=
e
P
d
-
ω
L
inv
i
P
inv
q
-
ω
L
g
i
P
g
q
,
u
P
inv
q
=
e
P
q
-
ω
L
inv
i
P
inv
d
-
ω
L
g
i
P
g
d
,
u
N
inv
d
=
e
P
d
-
ω
L
inv
i
N
inv
q
-
ω
L
g
i
N
g
q
,
u
N
inv
q
=
e
P
q
-
ω
L
inv
i
N
inv
d
-
ω
L
g
i
N
g
d
.Substitute formula (38) with formula (28) and obtain [16, 17]:(39)
P
0
Q
0
P
cos
⁡
2
P
sin
⁡
2
=
3
2
e
P
d
e
P
q
e
N
d
e
N
q
e
P
q
-
e
P
d
e
N
q
-
e
N
d
e
N
d
e
N
q
e
P
d
e
P
q
e
N
q
-
e
N
d
-
e
P
q
e
P
d
i
P
inv
d
i
P
inv
q
i
N
inv
d
i
N
inv
q
+
3
2
-
ω
L
i
P
inv
q
+
i
P
g
q
ω
L
i
P
inv
d
+
i
P
g
d
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
ω
L
i
P
inv
d
+
i
P
g
d
ω
L
i
P
inv
q
+
i
P
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
N
inv
q
+
i
N
g
q
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
P
inv
q
+
i
P
g
q
ω
L
i
P
inv
d
+
i
P
g
d
-
ω
L
i
N
inv
d
+
i
N
g
d
-
ω
L
i
N
inv
q
+
i
N
g
q
-
ω
L
i
P
inv
d
+
i
P
g
d
-
ω
L
i
P
inv
q
+
i
P
g
q
i
P
inv
d
i
P
inv
q
i
N
inv
d
i
N
inv
q
.Formula (39) is the relationship between known detection and inverter output power, to eliminate negative sequence current (i
N
inv
(
d
)
=
i
N
inv
(
q
)
=
0). Considering full power operation, to simplify computation under designed operating parameters, simplify the above formula and given value of current reference is (40)
i
P
∗
inv
d
=
2
e
P
d
P
0
+
e
P
q
Q
0
3
e
P
d
2
+
e
P
q
2
,
i
P
∗
inv
q
=
2
e
P
q
P
0
-
e
P
d
Q
0
3
e
P
d
2
+
e
P
q
2
,
i
N
inv
d
=
0
,
i
N
inv
q
=
0
,
(41)
Δ
Q
=
ω
C
e
P
c
d
2
+
e
P
c
q
2
+
e
N
c
q
2
+
e
N
c
d
2
.Under stable operating state, setΔ
Q
=
Q
0 and substitute with formula (41), and obtain given value of inner loop current under unit power factor grid-connected operation. When conducting process of LVRT, carry out real-time modification of the value of Q
0 to regulate reactive power output according to detected grid voltage drop depth. To ensure that grid-connected current does not exceed output protection value, active and reactive power output will meet:(42)
i
P
∗
inv
d
≤
i
max
⁡
2
-
i
P
∗
inv
q
2
.In the formula,i
max
⁡ denotes the maximum amplitude limit of output current. Stable control and reactive power support after voltage drop are realized by dynamic change of current command value on axis d
q, and select appropriate controller parameters to ensure system stability. As positive and negative sequence dual d
q current loop control method would separate voltage and current into positive and negative sequence components under unbalanced situation while filter capacitive current i
C is mainly high-frequency component, commonly used positive and negative sequence separation method cannot obtain positive and negative sequence components under high frequency. Thus, active damping control method is taken under balanced situation but cannot realize damping control under unbalanced situation and even affect system stability. Add active damping method of filter capacitive current feedforward to modulation signal, and current loop control block diagram is shown in Figure 14. Dashed block on the left side is the control block diagram of unbalanced control algorithm, and dashed block on the right side is the equivalent mathematical model of LCL. Feedforward method shown in Figure 14 avoids positive and negative sequence separation feedforward variable i
C effectively, and regulation of coefficient k
C does not affect transfer function of active damping algorithm in Figure 5 and can restrain resonance of LCL filter effectively.


## 4. Control Methods Proposed in This Paper

### 4.1. Phase Lock Control Strategy

Under unbalanced or distorted grid, axisq has secondary fluctuation, and thus phase lock method under unbalanced grid is not applicable. When three-phase grid is unbalanced or distorted, the influence of negative sequence components and harmonic components will be considered. For asymmetrical grid three-phase voltage,(43)
e
a
e
b
e
c
=
V
p
cos
⁡
θ
u
cos
⁡
θ
u
-
2
π
3
cos
⁡
θ
u
+
2
π
3
+
∑
n
=
-
∞
n
≠
-
1
∞
V
n
cos
⁡
n
θ
u
+
ϕ
n
cos
⁡
n
θ
u
+
ϕ
n
-
2
π
3
cos
⁡
n
θ
u
+
ϕ
n
+
2
π
3
+
∑
n
=
1
∞
V
n
cos
⁡
n
θ
u
+
ϕ
0
n
cos
⁡
n
θ
u
+
ϕ
0
n
cos
⁡
n
θ
u
+
ϕ
0
n
.The first term on the right side of equation in formula (43) denotes positive sequence fundamental voltage in three-phase grid, the second term denotes negative sequence voltage (n
=
-
1) and harmonic voltage (n
≠
±
1), and the third term denotes zero sequence voltage.For voltage expression of three-phase balanced system,(44)
e
a
e
b
e
c
=
V
p
cos
⁡
θ
u
cos
⁡
θ
u
-
2
π
3
cos
⁡
θ
u
+
2
π
3
,where e
a, e
b, and e
c denote A, B, and C three-phase voltage, V
p is voltage amplitude, and θ
u is A-phase angle.Replace formula (44) with formula (43), and compute new phase error:(45)
e
q
k
=
V
p
sin
⁡
θ
ref
k
-
θ
u
k
+
∑
n
=
-
∞
n
≠
-
1
∞
V
n
sin
⁡
θ
ref
k
-
n
θ
u
k
-
ϕ
n
.When locking phase of positive sequence fundamental components of grid voltage, that is,θ
u
=
θ
ref, phase error can be written in the following form:(46)
e
q
k
=
∑
n
=
-
∞
n
≠
-
1
∞
V
n
sin
⁡
1
-
n
θ
u
k
-
ϕ
n
.From formula (46), negative sequence voltage in three-phase grid would have doubled frequency oscillation signal on phase error e
q; n times of harmonic voltage would generate n
-
1 or n
+
1 times of frequency oscillation signal, and thus e
q cannot express accurately phase error between reference phase θ
ref and actual positive sequence fundamental voltage phase θ
u and thus could not conduct accurate phase lock. To sum up, to accurately realize phase lock of grid-side positive sequence voltage, all oscillation signals in e
q will be filtered. This paper uses sliding Goertzel filter to filter oscillation signals in phase error, and its implementation principle is as follows: compute value of the nth harmonic through digital Fourier Transform. Sliding Goertzel filter used in this paper is to compute with assistance of a sliding window with length of N
SG based on Goertzel filter.According to digital Fourier Transform, for continuous time functionx
t with period of T
1, value of the nth harmonic is(47)
Y
n
f
1
=
∑
k
=
0
N
SG
-
1
x
k
T
s
·
e
-
j
2
π
f
1
·
k
T
s
·
n
,where T
s is sampling period, N
SG is the number of sampling points in a function period, and(48)
f
s
=
1
T
s
=
f
1
·
N
SG
.Substitute formula (48) with formula (47), and(49)
Y
n
f
1
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
1
·
e
-
j
2
π
k
n
/
N
SG
.It is quite complex to compute each harmonic value through (49) directly, and each computation of harmonic value takes a function period. In filter algorithm based on sliding DFT used in this paper, filtering value of previous moment is known, and filtering value of later moment can be obtained through simple recursive operation. This filtering method is simple to compute and has important practical significance. For periodic function x
t, formula (50) is corresponding to discrete Fourier expressions of this periodic function at times q and q
+
1 and the nth harmonic values of this periodic function at times q and q
+
1:(50)
Y
q
,
n
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
1
·
e
-
j
2
π
k
n
/
N
SG
,
Y
q
+
1
,
n
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
2
·
e
-
j
2
π
k
n
/
N
SG
.Thus, the relationship between bothnth harmonics of periodic function x
t at times q and q
+
1 is(51)
Y
q
+
1
,
n
=
Y
q
,
n
+
x
q
+
1
-
x
q
-
N
SG
+
1
·
e
j
2
π
k
n
/
N
SG
.According to formula (51), filter with a sliding Goertzel filter, and derive harmonic value of later moment according to harmonic value of previous moment, which is relatively easy and conducive to digitalization. This paper uses sliding Goertzel filter to filter all oscillation signals in phase error and only keep DC variables. Take n
=
0, simplify formula (51), and obtain(52)
Y
k
=
Y
k
-
1
+
x
k
-
x
k
-
N
SG
.Transfer function of sliding Goertzel filter in domainZ is shown in the following formula:(53)
G
SG
=
1
-
z
-
N
SG
1
-
z
-
1
.Principle block diagram of phase-locked loop applied to three-phase asymmetrical grid is shown in Figure15.Figure 15
Principle block diagram of phase-locked loop.This three-phase phase-locked loop uses sliding Goertzel filter to filter disturbance generated by negative sequence components and harmonic components in three-phase grid, and the system can lock phase of positive sequence fundamental voltage very well. Expression of A-phase angleθ
u and reference phase angle θ
r
e
f in domain Z is(54)
θ
u
z
=
ω
z
-
1
,
θ
ref
z
=
2
π
/
N
pll
z
-
1
.Combine formula (53) and (54) with control principle diagram of phase-locked loop, and mathematical model of phase-locked loop in domain Z can be obtained as shown in Figure 16.Figure 16
Nonlinear mathematical model of phase-locked loop in domainZ.In Figure16, G
SG
z denotes Z-domain transfer function corresponding to Goertzel filter, G
C
z is Z-domain transfer function corresponding to controller, K
G
C
-
T
s
z is a proportion function from controller output to sampling period T
s, and ω is angular frequency of grid. Considering that e
∗
q
=
0 and when phase error is very small and formula (46) is met, Figure 16 can be simplified into Figure 17, and we finally obtain a linear mathematical model in domain Z that is easy to achieve.Figure 17
Linear mathematical model of phase-locked loop in domainZ.This phase-locked loop can eliminate the influence of unbalanced grid voltage and has good dynamic tracking features. Simulation results indicate that this phase-locked loop can lock positive sequence fundamental components in three-phase grid very well under unbalanced three-phase grid voltage, voltage distortion, and sudden frequency changes and has advantages such as good dynamic performance, short dynamic response time, and high precision of stable state.


### 4.2. Take PR Controller for Grid Current Loop

Under positive and reverse synchronous rotating coordinate systems, although duald
q and PI current controller can meet control demand of grid-connected inverter system under small fault of asymmetrical stable grid voltage, it has poor response to dynamic transient process under large fault of asymmetrical transient grid voltage. According to Figure 14, dual d
q and PI controller need to conduct positive and negative sequence decomposition of feedback current in current control loop under positive and reverse synchronous rotating coordinate systems. Therefore, wave trap link is introduced to inner current loop, while delay brought by wave trap will affect dynamic performance of system and thus decrease FRT ability of grid-connected inverter. Therefore, new control algorithm will be used to eliminate delay of inner current loop brought by positive and negative sequence decomposition. Then, this paper puts forward an active damping control algorithm based on PR current controller combined with capacitive current feedback.Mathematical model of grid-connected inverter inα
β static coordinate system is(55)
V
α
β
=
U
α
β
+
L
d
I
α
β
d
t
.Here, introduce PR current controller, that is, proportional resonant controller [4–8], and set(56)
U
∗
g
α
β
=
d
I
g
α
β
d
t
=
F
PR
s
I
∗
g
α
β
-
I
g
α
β
,where F
P
R
s denotes transfer function of PR current controller, and the expression is(57)
F
PR
s
=
K
p
+
R
s
=
K
p
+
K
r
s
s
2
+
ω
1
2
.In formula (57), K
p and K
r denote proportional coefficient and harmonic coefficient of PR controller, respectively and K
p plays the same role with traditional PI controller and is used to regulate dynamic performance of system. R
s is the transfer function of harmonic controller, harmonic angular frequency of which is ±
ω
1. In the situation of grid voltage asymmetrical fault under α
β static coordinate system, positive and negative sequence components of grid voltage and current rotate at synchronous angular speed of ω
1 and -
ω
1, respectively. Thus, positive and negative sequence components of output current can be controlled synchronously by a harmonic current controller. Transfer function of PR controller shown in formula (57) is an s-domain function and will be discrete to domain Z to conduct digital control of grid-connected inverter output current by PR controller. Use Tustin conversion method to make transfer function of PR controller discrete, and then(58)
F
PR
z
=
K
p
+
K
r
R
z
,where R
z can be expressed as(59)
R
z
=
U
z
E
z
=
b
0
+
b
1
z
-
1
+
b
2
z
-
2
1
+
a
1
z
-
1
+
a
2
z
-
2
,where(60)
b
0
=
2
T
s
ω
1
2
T
s
2
,
b
1
=
0
,
b
2
=
2
T
s
4
+
ω
1
2
T
s
2
,
a
1
=
2
ω
1
2
T
s
2
-
8
4
+
ω
1
2
T
s
2
,
a
2
=
1
.In the above formula,T
s is the sampling period of control system.According to formula (58)~formula (60), discrete difference equation of PR controller can be obtained as follows:(61)
u
k
=
K
p
e
k
-
a
1
u
k
-
1
-
u
k
-
2
+
b
0
K
r
e
k
+
b
2
K
r
e
k
-
2
,where u
k is output value at the kth sampling time, e
k is output error at the kth sampling time, u
k
-
1 is output value at the (k
-
1)th sampling time, u
k
-
2 is output value at the (k
-
2)th sampling time, and e
k
-
2 is output error at the (k
-
2)th sampling time.Block diagram of grid-connected inverter system control that uses PR current controller is shown in Figure19. Use PR current controller, set according to control objective and power, decompose positive and negative sequence components of three-phase grid voltage, and calculate and obtain given command values of positive and negative sequence components of output current. In control block diagram, current control loop has not any delay or error brought by such decomposition. Therefore, dynamic regulation performance of control system under grid voltage asymmetrical fault is improved, and thus FRT ability of grid-connected inverter is improved.


### 4.3. Grid Voltage Feedforward

At the moment of voltage drop, current shock during switching between stable grid-connected operation and LVRT operating state will be controlled to ensure safe access of photovoltaic inverter to operating interval after drop. For transient process, literature [18] puts forward a feedforward control method with respect to single-phase grid-connected inverter with L-shaped filter; literature [19] uses full voltage feedforward based on LCL filter to restrain the influence of grid voltage harmonic on grid-connected current but could not restrain transient current shock when grid voltage drops. Therefore, this paper, based on the selected inverter grid-connected system, puts forward a simplified grid voltage feedforward control algorithm that retrains transient current shock in terms of transient state of voltage drop. In Figure 14, system input variables are i
P
∗
inv
(
d
q
) and i
N
∗
inv
(
d
q
) and output variable is i
g. To analyze the influence of disturbance variable e on system stability, simplify Figure 13 into control structure shown in Figure 20, where k
e is grid voltage feedforward coefficient.In Figure20, closed-loop transfer function of the system is(62)
I
g
=
G
i
s
G
1
s
G
2
s
1
+
G
i
s
G
1
s
G
2
s
·
i
∗
inv
-
G
2
s
k
e
G
1
s
-
1
1
+
G
i
s
G
1
s
G
2
s
·
E
.In the above formula,E denotes grid voltage, where(63)
G
1
s
=
G
inv
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
,
G
1
s
=
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
L
inv
L
g
s
3
+
G
inv
G
i
s
-
k
c
G
inv
L
g
C
f
s
2
+
L
+
L
g
s
.According to formula (63), control output variable I
g and input variable i
∗
inv and grid voltage E are all related. Regard E as disturbance term, and grid current will increase proportionally at the moment of voltage drop. Therefore, to eliminate the influence of disturbance variable on grid-connected current, introduce grid voltage feedforward, as is shown in dashed part of Figure 24. If disturbance term is completely eliminated, it can be obtained that(64)
k
e
=
1
G
1
s
=
1
G
inv
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
.Disturbance variablee can be completely restrained by using feedforward coefficient k
e of formula (64); that is, distortion of e will not affect input of I
g. Differential term and 2nd-order differential term in formula (64) will restrain high-frequency components, and proportional term affects low-frequency components. For grid voltage drop especially transient one, the existence of differential variable will make feedforward signal go to infinity and result in unstable system. At the moment of grid voltage drop, to restrain grid-connected current transient shock, protect safe operation of grid-connected device, and meet requirement of LVRT, simplify formula (64), ignore differential term and 2nd-order differential term in feedforward coefficient, and then obtain(65)
k
e
=
1
G
inv
.The introduction of feedforward variables would play a good retraining role in transient current shock caused by grid voltage drop. Simplify feedforward coefficient into a proportional link, and add a low-pass filter to collection channel, in order to eliminate the influence of high-frequency variables in feedforward signale on grid-connected output current waveform.


## 4.1. Phase Lock Control Strategy

Under unbalanced or distorted grid, axisq has secondary fluctuation, and thus phase lock method under unbalanced grid is not applicable. When three-phase grid is unbalanced or distorted, the influence of negative sequence components and harmonic components will be considered. For asymmetrical grid three-phase voltage,(43)
e
a
e
b
e
c
=
V
p
cos
⁡
θ
u
cos
⁡
θ
u
-
2
π
3
cos
⁡
θ
u
+
2
π
3
+
∑
n
=
-
∞
n
≠
-
1
∞
V
n
cos
⁡
n
θ
u
+
ϕ
n
cos
⁡
n
θ
u
+
ϕ
n
-
2
π
3
cos
⁡
n
θ
u
+
ϕ
n
+
2
π
3
+
∑
n
=
1
∞
V
n
cos
⁡
n
θ
u
+
ϕ
0
n
cos
⁡
n
θ
u
+
ϕ
0
n
cos
⁡
n
θ
u
+
ϕ
0
n
.The first term on the right side of equation in formula (43) denotes positive sequence fundamental voltage in three-phase grid, the second term denotes negative sequence voltage (n
=
-
1) and harmonic voltage (n
≠
±
1), and the third term denotes zero sequence voltage.For voltage expression of three-phase balanced system,(44)
e
a
e
b
e
c
=
V
p
cos
⁡
θ
u
cos
⁡
θ
u
-
2
π
3
cos
⁡
θ
u
+
2
π
3
,where e
a, e
b, and e
c denote A, B, and C three-phase voltage, V
p is voltage amplitude, and θ
u is A-phase angle.Replace formula (44) with formula (43), and compute new phase error:(45)
e
q
k
=
V
p
sin
⁡
θ
ref
k
-
θ
u
k
+
∑
n
=
-
∞
n
≠
-
1
∞
V
n
sin
⁡
θ
ref
k
-
n
θ
u
k
-
ϕ
n
.When locking phase of positive sequence fundamental components of grid voltage, that is,θ
u
=
θ
ref, phase error can be written in the following form:(46)
e
q
k
=
∑
n
=
-
∞
n
≠
-
1
∞
V
n
sin
⁡
1
-
n
θ
u
k
-
ϕ
n
.From formula (46), negative sequence voltage in three-phase grid would have doubled frequency oscillation signal on phase error e
q; n times of harmonic voltage would generate n
-
1 or n
+
1 times of frequency oscillation signal, and thus e
q cannot express accurately phase error between reference phase θ
ref and actual positive sequence fundamental voltage phase θ
u and thus could not conduct accurate phase lock. To sum up, to accurately realize phase lock of grid-side positive sequence voltage, all oscillation signals in e
q will be filtered. This paper uses sliding Goertzel filter to filter oscillation signals in phase error, and its implementation principle is as follows: compute value of the nth harmonic through digital Fourier Transform. Sliding Goertzel filter used in this paper is to compute with assistance of a sliding window with length of N
SG based on Goertzel filter.According to digital Fourier Transform, for continuous time functionx
t with period of T
1, value of the nth harmonic is(47)
Y
n
f
1
=
∑
k
=
0
N
SG
-
1
x
k
T
s
·
e
-
j
2
π
f
1
·
k
T
s
·
n
,where T
s is sampling period, N
SG is the number of sampling points in a function period, and(48)
f
s
=
1
T
s
=
f
1
·
N
SG
.Substitute formula (48) with formula (47), and(49)
Y
n
f
1
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
1
·
e
-
j
2
π
k
n
/
N
SG
.It is quite complex to compute each harmonic value through (49) directly, and each computation of harmonic value takes a function period. In filter algorithm based on sliding DFT used in this paper, filtering value of previous moment is known, and filtering value of later moment can be obtained through simple recursive operation. This filtering method is simple to compute and has important practical significance. For periodic function x
t, formula (50) is corresponding to discrete Fourier expressions of this periodic function at times q and q
+
1 and the nth harmonic values of this periodic function at times q and q
+
1:(50)
Y
q
,
n
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
1
·
e
-
j
2
π
k
n
/
N
SG
,
Y
q
+
1
,
n
=
∑
k
=
0
N
SG
-
1
x
k
+
q
-
N
SG
+
2
·
e
-
j
2
π
k
n
/
N
SG
.Thus, the relationship between bothnth harmonics of periodic function x
t at times q and q
+
1 is(51)
Y
q
+
1
,
n
=
Y
q
,
n
+
x
q
+
1
-
x
q
-
N
SG
+
1
·
e
j
2
π
k
n
/
N
SG
.According to formula (51), filter with a sliding Goertzel filter, and derive harmonic value of later moment according to harmonic value of previous moment, which is relatively easy and conducive to digitalization. This paper uses sliding Goertzel filter to filter all oscillation signals in phase error and only keep DC variables. Take n
=
0, simplify formula (51), and obtain(52)
Y
k
=
Y
k
-
1
+
x
k
-
x
k
-
N
SG
.Transfer function of sliding Goertzel filter in domainZ is shown in the following formula:(53)
G
SG
=
1
-
z
-
N
SG
1
-
z
-
1
.Principle block diagram of phase-locked loop applied to three-phase asymmetrical grid is shown in Figure15.Figure 15
Principle block diagram of phase-locked loop.This three-phase phase-locked loop uses sliding Goertzel filter to filter disturbance generated by negative sequence components and harmonic components in three-phase grid, and the system can lock phase of positive sequence fundamental voltage very well. Expression of A-phase angleθ
u and reference phase angle θ
r
e
f in domain Z is(54)
θ
u
z
=
ω
z
-
1
,
θ
ref
z
=
2
π
/
N
pll
z
-
1
.Combine formula (53) and (54) with control principle diagram of phase-locked loop, and mathematical model of phase-locked loop in domain Z can be obtained as shown in Figure 16.Figure 16
Nonlinear mathematical model of phase-locked loop in domainZ.In Figure16, G
SG
z denotes Z-domain transfer function corresponding to Goertzel filter, G
C
z is Z-domain transfer function corresponding to controller, K
G
C
-
T
s
z is a proportion function from controller output to sampling period T
s, and ω is angular frequency of grid. Considering that e
∗
q
=
0 and when phase error is very small and formula (46) is met, Figure 16 can be simplified into Figure 17, and we finally obtain a linear mathematical model in domain Z that is easy to achieve.Figure 17
Linear mathematical model of phase-locked loop in domainZ.This phase-locked loop can eliminate the influence of unbalanced grid voltage and has good dynamic tracking features. Simulation results indicate that this phase-locked loop can lock positive sequence fundamental components in three-phase grid very well under unbalanced three-phase grid voltage, voltage distortion, and sudden frequency changes and has advantages such as good dynamic performance, short dynamic response time, and high precision of stable state.


## 4.2. Take PR Controller for Grid Current Loop

Under positive and reverse synchronous rotating coordinate systems, although duald
q and PI current controller can meet control demand of grid-connected inverter system under small fault of asymmetrical stable grid voltage, it has poor response to dynamic transient process under large fault of asymmetrical transient grid voltage. According to Figure 14, dual d
q and PI controller need to conduct positive and negative sequence decomposition of feedback current in current control loop under positive and reverse synchronous rotating coordinate systems. Therefore, wave trap link is introduced to inner current loop, while delay brought by wave trap will affect dynamic performance of system and thus decrease FRT ability of grid-connected inverter. Therefore, new control algorithm will be used to eliminate delay of inner current loop brought by positive and negative sequence decomposition. Then, this paper puts forward an active damping control algorithm based on PR current controller combined with capacitive current feedback.Mathematical model of grid-connected inverter inα
β static coordinate system is(55)
V
α
β
=
U
α
β
+
L
d
I
α
β
d
t
.Here, introduce PR current controller, that is, proportional resonant controller [4–8], and set(56)
U
∗
g
α
β
=
d
I
g
α
β
d
t
=
F
PR
s
I
∗
g
α
β
-
I
g
α
β
,where F
P
R
s denotes transfer function of PR current controller, and the expression is(57)
F
PR
s
=
K
p
+
R
s
=
K
p
+
K
r
s
s
2
+
ω
1
2
.In formula (57), K
p and K
r denote proportional coefficient and harmonic coefficient of PR controller, respectively and K
p plays the same role with traditional PI controller and is used to regulate dynamic performance of system. R
s is the transfer function of harmonic controller, harmonic angular frequency of which is ±
ω
1. In the situation of grid voltage asymmetrical fault under α
β static coordinate system, positive and negative sequence components of grid voltage and current rotate at synchronous angular speed of ω
1 and -
ω
1, respectively. Thus, positive and negative sequence components of output current can be controlled synchronously by a harmonic current controller. Transfer function of PR controller shown in formula (57) is an s-domain function and will be discrete to domain Z to conduct digital control of grid-connected inverter output current by PR controller. Use Tustin conversion method to make transfer function of PR controller discrete, and then(58)
F
PR
z
=
K
p
+
K
r
R
z
,where R
z can be expressed as(59)
R
z
=
U
z
E
z
=
b
0
+
b
1
z
-
1
+
b
2
z
-
2
1
+
a
1
z
-
1
+
a
2
z
-
2
,where(60)
b
0
=
2
T
s
ω
1
2
T
s
2
,
b
1
=
0
,
b
2
=
2
T
s
4
+
ω
1
2
T
s
2
,
a
1
=
2
ω
1
2
T
s
2
-
8
4
+
ω
1
2
T
s
2
,
a
2
=
1
.In the above formula,T
s is the sampling period of control system.According to formula (58)~formula (60), discrete difference equation of PR controller can be obtained as follows:(61)
u
k
=
K
p
e
k
-
a
1
u
k
-
1
-
u
k
-
2
+
b
0
K
r
e
k
+
b
2
K
r
e
k
-
2
,where u
k is output value at the kth sampling time, e
k is output error at the kth sampling time, u
k
-
1 is output value at the (k
-
1)th sampling time, u
k
-
2 is output value at the (k
-
2)th sampling time, and e
k
-
2 is output error at the (k
-
2)th sampling time.Block diagram of grid-connected inverter system control that uses PR current controller is shown in Figure19. Use PR current controller, set according to control objective and power, decompose positive and negative sequence components of three-phase grid voltage, and calculate and obtain given command values of positive and negative sequence components of output current. In control block diagram, current control loop has not any delay or error brought by such decomposition. Therefore, dynamic regulation performance of control system under grid voltage asymmetrical fault is improved, and thus FRT ability of grid-connected inverter is improved.


## 4.3. Grid Voltage Feedforward

At the moment of voltage drop, current shock during switching between stable grid-connected operation and LVRT operating state will be controlled to ensure safe access of photovoltaic inverter to operating interval after drop. For transient process, literature [18] puts forward a feedforward control method with respect to single-phase grid-connected inverter with L-shaped filter; literature [19] uses full voltage feedforward based on LCL filter to restrain the influence of grid voltage harmonic on grid-connected current but could not restrain transient current shock when grid voltage drops. Therefore, this paper, based on the selected inverter grid-connected system, puts forward a simplified grid voltage feedforward control algorithm that retrains transient current shock in terms of transient state of voltage drop. In Figure 14, system input variables are i
P
∗
inv
(
d
q
) and i
N
∗
inv
(
d
q
) and output variable is i
g. To analyze the influence of disturbance variable e on system stability, simplify Figure 13 into control structure shown in Figure 20, where k
e is grid voltage feedforward coefficient.In Figure20, closed-loop transfer function of the system is(62)
I
g
=
G
i
s
G
1
s
G
2
s
1
+
G
i
s
G
1
s
G
2
s
·
i
∗
inv
-
G
2
s
k
e
G
1
s
-
1
1
+
G
i
s
G
1
s
G
2
s
·
E
.In the above formula,E denotes grid voltage, where(63)
G
1
s
=
G
inv
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
,
G
1
s
=
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
L
inv
L
g
s
3
+
G
inv
G
i
s
-
k
c
G
inv
L
g
C
f
s
2
+
L
+
L
g
s
.According to formula (63), control output variable I
g and input variable i
∗
inv and grid voltage E are all related. Regard E as disturbance term, and grid current will increase proportionally at the moment of voltage drop. Therefore, to eliminate the influence of disturbance variable on grid-connected current, introduce grid voltage feedforward, as is shown in dashed part of Figure 24. If disturbance term is completely eliminated, it can be obtained that(64)
k
e
=
1
G
1
s
=
1
G
inv
L
inv
C
f
s
2
+
G
inv
G
i
s
-
k
c
G
inv
C
f
s
+
1
.Disturbance variablee can be completely restrained by using feedforward coefficient k
e of formula (64); that is, distortion of e will not affect input of I
g. Differential term and 2nd-order differential term in formula (64) will restrain high-frequency components, and proportional term affects low-frequency components. For grid voltage drop especially transient one, the existence of differential variable will make feedforward signal go to infinity and result in unstable system. At the moment of grid voltage drop, to restrain grid-connected current transient shock, protect safe operation of grid-connected device, and meet requirement of LVRT, simplify formula (64), ignore differential term and 2nd-order differential term in feedforward coefficient, and then obtain(65)
k
e
=
1
G
inv
.The introduction of feedforward variables would play a good retraining role in transient current shock caused by grid voltage drop. Simplify feedforward coefficient into a proportional link, and add a low-pass filter to collection channel, in order to eliminate the influence of high-frequency variables in feedforward signale on grid-connected output current waveform.


## 5. Simulation and Test Waveforms

In order to verify the proposed control strategy, this paper applies the simulations and experimental tests to compare response performances by using various methods. The study utilizes Matlab/Simulink for simulation analysis of control strategy and tests on experimental platform. Regarding setup of tests, the situational part is carried out under standard test condition (STC), while the experimental part is carried out under nominal operating cell temperature (NOCT). Table2 shows the differences of these two conditions. The experimental platform includes the PV power model, inverter, grid simulator, and switch gears. The PV model consists of combination of PV arrays. The rated power of inverter in this paper is 500 kW. Table 1 shows the parameters of inverter. The function of grid simulator is to produce the various grid voltage drops. Regarding the realization solutions of the grid voltage drops, there are generally two methods: the first is reactor combination simulator and the second is converter simulator. In contrast, the advantage of the first one is that the test condition is more close to the real circumstance of grid voltage drops. However, the cost of the first one is much more expensive than the second method. There is no need to use the first solution if not to do certification of the inverter. This paper utilizes the second solution and takes the converter simulator as the source of voltage drops.Table 1
Parameters of photovoltaic inverter system.
Category
Name
Parameter
System parameters
Maximum DC voltage
1000V
d
c
Maximum DC power
560 kWp
DC voltage range with MPPT
480~880V
d
c
Maximum current output
1167 A
Startup DC voltage
520V
d
c
Rated power output
500 kW
Rated grid voltage
315V
a
c
Total harmonic distortion of current
<2% (rated power output)
DC component of current
<0.5% (rated current output)
Filter parameters
Inductance value ofL
1
100 uH
Inductance value ofL
2
50 uH
Capacitance values forC
2
200 uF (Δ-connected)
The resonant frequency of LCL
1125 Hz
Other parameters
Capacitance values forC
1
25.12 mF
Switch frequency of IGBT
3000 Hz
Sample frequency
10000 HzTable 2
Parameters of photovoltaic cells.
Parameters
Description
Unit
Value
Standard test condition (STC): 1000 W/m2 in irradiance, battery temperature of 25°C, and air quality of AM1.5, according to EN 60904-3
p
m
a
x
Peak power
Watt
260
Δ
P
m
a
x
Power tolerance
Watt
0/5
η
m
Component efficiency
Percentage
15.9
V
m
p
p
Peak power voltage
Volt
30.9
I
m
p
p
Peak power current
Ampere
8.41
V
o
c
Open circuit voltage
Volt
38.9
I
s
c
Short circuit current
Ampere
8.98
Nominal operating cell temperature (NOCT):800 W/m2 in irradiance, ambient temperature of 20°C, and temperature in the open state of the components under the conditions of 1 m/s wind speed.
P
m
a
x
Peak power
Watt
188.3
V
m
p
p
Peak power voltage
Volt
28.1
I
m
p
p
Peak power current
Ampere
6.7
V
o
c
Open circuit voltage
Volt
35.9
I
s
c
Short circuit current
Ampere
7.27Table2 shows electric parameters of single-block solar cell modules, and a photovoltaic array consists of 20 modules in series.To verify control effects of this algorithm under balanced grid voltage drop and unbalanced grid voltage drop, set two compared groups, respectively, and standard during LVRT will follow Figure1(b). In order to obverse the simulation waveform conveniently, this paper sets t
1
=
0.1 s as grid voltage drop time point and sets t
2
=
0.2 s as grid voltage recover time point. Figure 21 shows three-phase voltage and current when voltage drops to 5% Un under balanced voltage drop condition. Before the voltage drops, grid current keeps high sinusoidal to stabilize grid-connected operation. At the time point of t
1, there is no current shock which benefits from the fast grid voltage feedforward as shown in Figure 20. During the interval of grid voltage drop, the reactive current takes the whole proportion of the total current output. It is easy to find that the reactive current entered the stable state in a time cycle. As required in grid code, the value of reactive current should be equal to the reactive current in 100%P
n in normal grid voltage state. After the grid voltage drop, negative sequence current after regulation is rapidly restrained down. The inverter recovers balanced three-phase current output and certain active power output. The recovery time is less than 20 ms.As required in [6], when grid voltage drops to 5%U
n, photovoltaic inverter will output reactive current at 1.05 times of rated current. In order to observe active and reactive current fluctuation during the interval of voltage drop, the paper gives active and reactive current waveforms after a
b
c
/
d
q conversion. Figure 22 shows the compared waveforms of active/reactive power, respectively, in the four different control strategies conditions. It is clear to find that active current and reactive current have not occurred during asymmetrical voltage drop. Furthermore, there is no double frequency fluctuation during the voltage fault-mode interval. Besides, in order to decrease the shock of transient current, this study adds the soft start solution in all compared control strategies. Regarding the response time of current, Figure 22(a) takes up nearly 55 ms (2.75 time cycles) at the beginning of voltage drop and 10 ms (0.5 time cycles) at the recovery process. However, this method brings the most serious current shock when voltage begins to recover at t
2 time point. Figures 22(b) and 22(d) show a good response performance at the two time points t
1 and t
2. Both response times of current are less than 10 ms (0.5 time cycles). Figure 22(c) has a rapid response at the recovery time point but fails to regulate the value of reactive current of which demerit is similar to Figure 22(a).As discussed above, if asymmetrical grid drop condition occurs (as shown in Figure18, where only two phases B and C drop), positive and negative sequence decomposition of grid voltage conduct phase lock of grid voltage very well with rapid response. Figure 23 gives waves the active and reactive power under four various control strategies. It can be concluded that the first three strategies result in power fluctuation inevitably. Figure 22(d) shows clearly that both active current and reactive current during LVRT interval have not doubled frequency fluctuation. So, the active power and reactive power in Figure 23(d) are quite stable. As it is an algorithm simulation, the requirement that grid voltage response speed will be within 30 ms in literature [6] cannot be truly reflected in simulation model.Figure 18
The wave ofU
q.
(a)
The wave ofU
q under one-phase grid voltage drop
(b)
The wave ofU
q under three-phase grid voltage dropFigure 19
Current control scheme for the GCIs based on PR regulator under grid voltage unbalance.Figure 20
Simplified block diagram of system control.Figure 21
Diagrams of grid current and voltage when grid drops to 30%.Figure 22
Active current and reactive current waveform when grid voltage dropped at a single phase. Note: 1: reactive current/A; 2: active current/A.
(a)
Single axisd
q
P
I control method
(b)
Duald
q rotating coordinate controller
(c)
Active damping compensation feedback control strategy
(d)
PR controllerFigure 23
Active power and reactive power when grid dropped at a single phase. Note: 1: reactive power/kvar; 2: active power/kW.
(a)
Single axisd
q
P
I control method
(b)
Duald
q rotating coordinate controller
(c)
Active damping compensation feedback control strategy
(d)
PR controllerFigure 24
Grid current and grid voltage.
(a)
Current output when grid dropped ats single-phase
(b)
Current output when grid dropped at three-phaseFigure24 shows experimental waveforms under 500 kW full power. Figure 24(a) shows triphase current under symmetrical voltage drop circumstances. When the triphase voltage decreases to 5%U
n, the triphase output current is regulated into a new value rapidly to response the grid changes. When the triphase voltage recovers to the normal degree, the output triphase current is regulated into the formal value. It can be judged that the inverter has ridden through the low-voltage state successfully. Figure 24(b) shows triphase current under asymmetrical voltage drop circumstances. The voltage of A phase drops to 30%U
n, while the two other phases are stable. We can find that the total Thd of the triphase current is less than 15% and recover time is less than 30 ms, which proves the effectiveness of the proposed strategy.


## 6. Conclusion

This paper proposes and analyzes the novel mathematical models of photovoltaic inverter under balanced and unbalanced grid. We focus on photovoltaic inverter control strategy to address the complex issues under unbalanced condition. A systematic framework based on PR controller is proposed to enable the grid code control of photovoltaic inverter under LVRT condition. With the PR based control strategy, the system can eliminate negative sequence components of grid current under unbalanced grid voltage and improve doubled sequence fluctuation of output current and power. The general performance of the proposed control method is superior to the existing representative control approaches. In phase lock link of grid voltage, the sliding Goertzel filter is adopted to purify the disturbance generated by negative sequence and harmonic components in three-phase grid. The system has achieved very good performance in lock phase of positive sequence fundamental voltage. In order to eliminate shaking of grid voltage and asymmetrical influence, grid-side voltage feedback link is added to the front end of output modulation wave. It is found that the introduction of feedforward variable would play a very good restraining role in transient current shock caused by grid voltage drop. In the implemented test platform, a simplified feedforward coefficient is embedded into a proportional link, and a low-pass filter is inserted into the collection channel in order to eliminate effectively the influence of high-frequency variables in voltage feedforward signal on grid-connected output current waveform. The proposed mathematical models and principles are verified using intensive simulations and experiments. The results have indicated that grid-side current could fully track the positive sequence components of grid voltage under unbalanced grid voltage. Moreover, the system could sustain a stable and balanced output of inverter under both restraining negative sequence current and reactive power fluctuation on grid side.



---

*Source: 790584-2015-12-02.xml*