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Research ArticleControl Strategy of Three-Phase Photovoltaic Inverter underLow-Voltage Ride-Through Condition

Xianbo Wang,1 Zhixin Yang,1 Bo Fan,2 and Wei Xu1

1Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macau2Information Engineering College, Henan University of Science and Technology, Luoyang, China

Correspondence should be addressed to Zhixin Yang; [email protected]

Received 19 June 2015; Accepted 1 September 2015

Academic Editor: Xinggang Yan

Copyright © 2015 Xianbo Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

In recent years, the development and utilization of newenergies, such as solar energy, wind energy, and hydrogenenergy, are booming in Europe, US, and China, wherethe distributed photovoltaic power generation technologiesare highly concerned. In distributed grid-connected powergeneration system, most of electric energy generated byenergy conversion device will be converted into alternatingcurrent (AC) with the same frequency and phase with gridvoltage through grid-connected inverter and then transmit-ted to grid [1–3]. Power system is dynamic, and its dynamicstability can be affected by many factors, such as settingof generator output limit, grid fault, grid resonance, andnonlinear load. Grid-connected inverter plays an essentialinterface role between renewable energy conversion deviceand grid and becomes an extremely important component ofdistributed power generation system [4]. With the increasingutilization of distributed power generation system in publicgrid, more and more new energy power converters areconnected 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 systemand is often affected by grid fault, resonance, overload andnonlinear load, and so forth, making design of convertercontrol system more difficult [5]. In recent years, undergrid fault especially grid voltage sag, strict requirements ofLVRT ability and reactive power injection for grid-connectedconverter in grid code increase the complexity of control.Design of photovoltaic grid-connected inverter will ensure itsreliable and stable operation under normal state of grid andcontinuous operating ability under grid fault, and anotherrequirement is LVRT ability of distributed power generationsystem grid-connected converter; that is, [6–8] such grid-connected inverter will ensure that operating parameters ofeach phase will not be disconnected to grid because of ultra-limit triggered protection action under grid fault and providemaximum voltage and reactive power support for grid.

Under normal or balanced grid voltage, most of existingphotovoltaic grid-connected inverters can operate normally,and rapid non-static-error control of inverter’s output currentcan be obtained by PI controller [9, 10]. However, when

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 790584, 23 pageshttp://dx.doi.org/10.1155/2015/790584

2 Mathematical Problems in EngineeringPh

otov

olta

ge p

ower

of c

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LVRT remain connected

Normal operation

0 0.15 1 20.625 3 4Time (s)

1.21.11

0.90.80.70.60.50.40.30.20.10

Voltage sags caused by fault grid voltage condition

(a) Q-GDW 617-2011 technical requirements for connecting photovoltaicpower station to power system (China)

Normal operation

0 0.15 1 20.625 3 4Time (s)

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remain connected

Voltage sags caused by fault grid-voltage condition1.21.11

0.90.80.70.60.50.40.30.20.10

(b) GB/T 19964-2012 technical requirements for connecting photovoltaicpower station to power system (China)

Figure 1: LVRT ability requirement of PV power stations.

grid voltage is unbalanced or seriously distorted, grid hasunbalanced voltage and current, that is [11]. There are lotsof negative sequence components in grid voltage; whencarrying out phase lock of grid voltage with phase-lockedloop by 𝑑𝑞 conversion, axis 𝑞 would have second harmonicfluctuation which is difficult to restrain, while traditional PIcontroller could only get high-performance control effectswhen control objects are DC variables. Therefore, whengrid voltage is unbalanced or seriously distorted, grid-connected inverter with traditional PI controller would havedeteriorated operating performance [12, 13]. Besides, whengrid voltage is unbalanced, second harmonic fluctuationgenerated on axis 𝑞 would also affect calculation of gridphase angle by phase-locked loop [5]. Although secondcomponent of axis 𝑞 can be of filtering processing througha specific second wave trap, the detection and computingdelay deserve concern. When facing grid voltage flicker, lowgrid voltage, and unbalanced grid, the grid company requiresmedium- and high-voltage inverters to detect fluctuations ofgrid voltage rapidly andmake necessary response as requiredin relevant national standards, in order to carry out rapidsupport for grid voltage and avoid a larger scale of grid fault[14–16].

According to grid code, medium- and high-voltageinverters of large photovoltaic power stations will havecertain ability to tolerate abnormal voltage, in order toavoid separation under abnormal grid voltage and lead tounstable grid voltage. According to technical suggestion,photovoltaic inverter can be separated from grid when gridvoltage drops to below curve 1 as is shown in Figure 2.Abnormal grid voltage is reflected as voltage drop of gridconnection points in photovoltaic grid-connected system,and such voltage drop can be divided into three-phase sym-metrical drop and three-phase asymmetrical drop accordingto grid voltage drop categories. As is shown in Figure 1, thereare “Q-GDW 617-2011” and “GB/T 19964-2012” standardsissued in 2011 and 2012, respectively. The differences arethat 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].

0 1

0.31

−0.31

Q (pu)

P (pu)

Figure 2: Reactive power generation.

According to requirements, photovoltaic inverter main-tains operation without disconnection and meets the follow-ing conditions during LVRT:

(1) At the moment of grid voltage drop, maintain con-tinuous grid-connected operation while protectingphotovoltaic inverter to be safe and the time intervalfrom the occurrence of grid drop to generation ofreactive current will be less than 30ms.

(2) During the interval of LVRT, maintain stable grid-connected operation of inverter, provide reactivesupport for grid according to grid connection rules,output range of reactive power will follow grid coderequirements, and reactive output range will followrequirements in Figure 2.

(3) Upon recovery of grid voltage, active power outputwill recover to the value before grid fault situationwith change rate of at least 0.3𝑃n/𝑠.

Figure 3 is German grid’s required index of LVRT of windpower generation system [19]. Area 1 in black bold line haspower fluctuation ratio per second of 5% ≤ 𝑑𝑃/𝑑𝑡 ≤ 20%,and wind power generation system will have at least 150msoperating ability when grid voltage drops to 0; the runningtime of wind power generation systemwill increase graduallywhen grid voltage drops to above 0% of rated voltage; thearea that will remain connected aboveArea 1means that wind

Mathematical Problems in Engineering 3

1UL

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Will remainconnected

3-phase short circuit or symm. volt. dips

Limit line 2

Area 1 Will be disconnected by the automatic systemselective disconnection based on the

generation condition

Figure 3: LVRT ability requirement (Germany).

power generation systemneeds 150ms operating ability whenpower fluctuation ratio meets 20% ≤ 𝑑𝑃/𝑑𝑡 ≤ 40%, and gridvoltage drops to 45% of rated voltage, and operating abilityincreases with the decrease of drop degree [20].

Denmark regulates that inverters will maintain continu-ous operation without disconnection for 10 s after nominalvoltage recovers from25% to 75%under three-phase fault andwill generate active power-up to rated level within 10 secondswhen grid voltage recovers back to 0.9𝑈n again. During gridvoltage drop, active power of grid connection points willmeetthe following conditions [21, 22]:

𝑃current > 𝑘𝑝𝑃𝑡=0 (𝑈current𝑈𝑡=0

)

2

, (1)

where 𝑃current and 𝑈current are current power and voltage,𝑃𝑡=0 and 𝑈𝑡=0 are power and voltage under grid fault, and𝑘𝑝 is active power control factor. When voltage recovers to0.9𝑈n, the output behaviour of inverter will meet reactivepower exchanging requirement with grid within 10 s. Duringgrid voltage drop, ensure that inverter will generate reactivecurrent which is equivalent to rated current under normalgrid situation.

In addition, Danish grid code also requires that powerstation could respond to dual voltage dip fault and requiresanother new 100ms short circuit with an interval of 300msupon two-phase short circuit 100ms later, and power stationhas no shutdown. Under another new 100ms voltage dropwith an interval of 1 s upon single-phase short circuit 100mslater, no shutdown is allowed, and voltage drop curve isshown in Figure 4.

EnergyNet.dk standards also regulate LVRT under somespecial situations, such as 100ms three-phase short circuit,100ms two-phase short circuit, and new 100ms short circuitwith an interval of 300ms∼500ms; besides, enough energywill be reserved to respond to at least 6 single-phase andthree-phase or two-phase short circuit with an interval of 5minutes, as is shown in Figure 5.

Besides, countries around the world formulate corre-sponding network access LVRT technology requirements,and LVRT requirements of UK, US, Spain, and Italy areshown in Figure 6.

According to active and reactive power decoupling con-trol strategy under synchronous rotating coordinate system,control output of outer DC voltage is the given value ofactive current on axis 𝑑, and stable DC voltage represents abalanced relationship between input DC power and outputACpower.Under grid fault including single-phase groundingshort circuit, two-phase short circuit, two-phase short circuitgrounding, and three-phase short circuit grounding fault,positive sequence grid voltage will decrease, which willgreatly reduce AC output power. At this time, if input DCpower remains unchanged, DC voltage loop will increasegiven value of active current to maintain constant DCvoltage. Therefore, output current will significantly increaseand exceed maximum current limit of inverter, which willinevitably trigger overcurrent protection and disconnectinverter from grid. Another noteworthy issue is that, underasymmetrical drop of grid, phase lock method of traditionalPI controller cannot eliminate second harmonic fluctuationon axis 𝑞, and delay of phase lock would bring lag ofcontroller-driven waveforms and would often lead to greaterdelay if grid is in serious unbalanced situation and wouldresult in overcurrent accident if there is no phase compen-sation, and thus LVRT requirement cannot be reached.

To solve the above problems, the academic circle carriesout relevant researches; literatures [23, 24] put forwardan improved current control algorithm, that is, dual 𝑑𝑞conversion and PI controller. It decomposes grid voltageand current under positive and reverse synchronous rotatingcoordinate systems, respectively, and then obtains positivesequence components under positive coordinate system andnegative sequence components under reverse coordinatesystem, both components being DC variables, controls thesecomponents by two PI controllers, and thus realizes respec-tive control of positive and negative sequence componentsof current under asymmetrical grid fault, in order to ensureasymmetrical fault ride-through (FRT) operating ability ofinverter. However, this control method will conduct positiveand negative sequence decomposition of inner loop feedbackcurrent, which will bring delay and error that cannot beignored at current loops. Under small fault of asymmetricalstable grid state, such as 2% voltage asymmetry, delay haslittle influence on system operating performance. However,

4 Mathematical Problems in Engineering

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vol

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1.0 1.1 1.20.0 0.1 0.4 0.5Time (s)

(f) 𝑈𝑐 (one-phase short circuit)

Figure 4: LVRT of double voltage drops (Denmark).

under large fault of transient asymmetry, such as 25% voltageasymmetry, there is a higher requirement for inverter’s con-trol performance because of bad transient process, and thusdelay and error brought by positive and negative sequencedecomposition process are bound to reduce transient reg-ulation performance of current controller and thus affecttransient operating performance of grid-connected inverter.Literatures [25, 26] use active damping controlmethodwhichadds capacitive current at LCL end to modulation signal,and an advantage of this control method lies in convenientdesign of controller. However, under different degrees ofunbalanced grid, capacitive current feedforward coefficienthas great influence on final inverter ride-through effects.Besides, this control method will add three capacitive currentdetection sensors in terms of hardware design, resulting inan increase of hardware costs. Papers [27–29] introduce PR(Proportion plus Resonant) controller to grid current loopcontroller; because the controlled objects are AC variables,

this method does not require 𝑎𝑏𝑐/𝑑𝑞 coordinate conversionof current. This control algorithm can control positive andnegative sequence components of output current simul-taneously under 𝛼𝛽 stationary coordinate system directly,without necessity of decomposing positive and negativesequence components of current under positive and reversesynchronous speed rotating coordinate systems, and thuseliminate delay of current control loop and improve dynamiccontrol performance of grid-connected inverter under largeasymmetrical fault. However, with this control method,there is a series of digital control delay, such as “detection→ filtering → control computation → sending drivingwaveforms” delay, and thus delay and parameter design ofcontroller will be taken into consideration during designof controller. Besides, as there is no 𝑑𝑞 conversion processand control variables are AC ones, only AC variables to beconverted need filtering. Control complexity is simplifiedduring digital design of system.


1

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00

100Time (ms)

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ULF

UHF

100

Full-load voltage range

Will remain connected

Two-phase

Three-phase

x

Figure 5: LVRT requirement of Danish grid under asymmetricalfault.

2. Mathematical Model

2.1. Mathematical Model of Three-Phase Photovoltaic Inverterunder Balanced Grid State. Single-stage photovoltaic grid-connected power generation system is generally consistingof 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 structureof three-phase non-midline single-stage photovoltaic grid-connected inverter, which mainly uses DC voltage capaci-tance, IGBT three-phase bridge, LCL filter, and so forth.

Low-frequency mathematical model of three-phase pho-tovoltaic inverter is obtained by ignoring high-frequencyharmonics related to switching frequency and analyzingfundamental components of inverter. Circuit structure ofthree-phase photovoltaic inverter as shown in Figure 7 is usedto build a low-frequency mathematical model according tobasic circuit theorems (KCL and KVL). Before building themodel, hypotheses include the following:

(a) All switching devices are equivalent to ideal ones; thatis, ignore switching loss.

(b) Grid-side power is ideal three-phase symmetricalvoltage source.

(c) Three inductances at the AC side are identical andlinear regardless of saturation.

(d) To facilitate description of two-way energy transfer,load is equivalent to load resistance 𝑅𝐿 and loadelectromotive force 𝑒𝐿 in series.

Combine equivalent resistance 𝑅𝑆 of power bridge losswith equivalent resistance 𝑅𝐿 of filter inductance at AC side,set 𝑅 = 𝑅𝑆 + 𝑅𝐿, and build circuit voltage current equation ofthree-phase photovoltaic inverter by KVL and KCL theoremsas follows:

𝐿𝑑𝑖𝑎

𝑑𝑡+ 𝑅𝑖𝑎 = 𝑒𝑎 − (Vdc𝑠𝑎 + VNO) ,

𝐿𝑑𝑖𝑏

𝑑𝑡+ 𝑅𝑖𝑏 = 𝑒𝑏 − (Vdc𝑠𝑏 + VNO) ,

𝐿𝑑𝑖𝑐

𝑑𝑡+ 𝑅𝑖𝑐 = 𝑒𝑐 − (Vdc𝑠𝑐 + VNO) .

(2)

Consider that three-phase symmetrical system has thefollowing features:

∑

𝑘=𝑎,𝑏,𝑐

𝑖𝑘 = 0,

∑

𝑘=𝑎,𝑏,𝑐

𝑒𝑘 = 0.

(3)

Integrate formula (2) into (3):

VNO = −Vdc3

∑

𝑖=𝑎,𝑏,𝑐

𝑠𝑖. (4)

Besides, apply KCL to capacitance at DC side:

𝑐𝑑Vdc𝑑𝑡

= 𝑖𝑎𝑠𝑎 + 𝑖𝑏𝑠𝑏 + 𝑖𝑐𝑠𝑐 −Vdc − 𝑒𝑙𝑅𝐿

. (5)

Combine formulas (2)∼(5), introduce state variable 𝑋 =

[𝑖𝑎, 𝑖𝑏, 𝑖𝑐, Vdc]𝑇, and state variable expression of static mathe-

matical model of three-phase photovoltaic inverter describedby single-pole logical switch state value is as follows:

𝑍�� = 𝐴𝑋 + 𝐵𝐸, (6)

where

𝐴 =

[[[[[[[[[[[[[[[[

[

−𝑅 0 0 −(𝑆𝑎 − ∑

𝑖=𝑎,𝑏,𝑐

𝑠𝑖)

0 −𝑅 0 −(𝑆𝑏 − ∑

𝑖=𝑎,𝑏,𝑐

𝑠𝑖)

0 0 −𝑅 −(𝑆𝑐 − ∑

𝑖=𝑎,𝑏,𝑐

𝑠𝑖)

0 0 0 −1

𝑅𝐿

]]]]]]]]]]]]]]]]

]

,

𝑍 =[[[

[

𝐿

𝐿

𝐿

𝐶

]]]

]

,

𝐵 =

[[[[[[[

[

1

1

1

−1

𝑅𝐿

]]]]]]]

]

,

𝐸 = [𝑒𝑎 𝑒𝑏 𝑒𝑐 𝑒𝑙]𝑇.

(7)

Convert mathematical model under three-phase 𝑎𝑏𝑐

static coordinate system to two-phase rotating 𝑑𝑞 coordinatesystem, in order to convert system parameters from AC toDC ones and facilitate controller design and computationof control parameters. Set 𝑑𝑞 rotating coordinate system torotate with angular frequency of grid voltage fundamentalwave, analyze with the example of three-phase symmetrical


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1

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0.750.9

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Time (s)

(d) Italy

Figure 6: LVRT requirements of countries around the world.

PVS1 S3 S5

S4 S6 S2Ic

I1 I2

C1

C2

L1 L2R1 R2 Usa

UsbUsc

Idc

Udcarray

Figure 7: Topological structure of three-phase single-stage inverterof LCL filter.

current at AC side, and the relationship between 𝑑𝑞 rotatingcoordinate system and 𝑎𝑏𝑐 coordinate system is shown inFigure 8.

According to equivalent conversion vector relationship,

𝑖𝑑 =2

3[𝑖𝑎 sin 𝜃 + 𝑖𝑏 sin (𝜃 − 120

∘) + 𝑖𝑐 sin (𝜃 + 120

∘)] ,

𝑖𝑞 =2

3[𝑖𝑎 cos 𝜃 + 𝑖𝑏 cos (𝜃 − 120

∘) + 𝑖𝑐 cos (𝜃 + 120

∘)] .

(8)

𝑎𝑏𝑐/𝑑𝑞 convertedmatrix can be obtained by the followingsimplification:

𝐶3𝑠/2𝑟 = [cos 𝜃 cos (𝜃 − 120∘) cos (𝜃 + 120∘)sin 𝜃 sin (𝜃 − 120∘) sin (𝜃 + 120∘)

] . (9)

c

0

b

a

I

q

𝜃u

id

iq

𝜔

Figure 8: Vector relationship between 𝑑𝑞 rotating coordinatesystem and 𝑎𝑏𝑐 coordinate system.

Mathematical model under 𝑑𝑞 coordinate system isshown in the following formula:

𝐿 ⋅𝑑𝑖𝑑

𝑑𝑡= V𝑑 − 𝑒𝑑 + 𝜔𝐿 ⋅ 𝑖𝑞,

𝐿 ⋅𝑑𝑖𝑞

𝑑𝑡= V𝑞 − 𝑒𝑞 + 𝜔𝐿 ⋅ 𝑖𝑑,

(10)

where 𝑒𝑑 and 𝑒𝑞 denote components on axis 𝑑𝑞 of three-phasegrid voltage, respectively, 𝑖𝑑 and 𝑖𝑞 denote components on


𝜔

𝜔

𝛽Ep

EN

E

𝛼𝜃u

b

0

c

a

(a) 3𝑠/2𝑠 transition

𝜔

𝜔

𝛽 Ep

EN

E

𝛼𝜃u

d

𝜃

q

0

(b) 2𝑠/2𝑟 transition

Figure 9: Coordinate change of rotating vector under asymmetrical voltage.

axis 𝑑𝑞 of three-phase grid-connected current, respectively,and V𝑑 and V𝑞 denote components on axis 𝑑𝑞 of three-phaseinverter output voltage, respectively.

2.2. Mathematical Model of Three-Phase Photovoltaic Inverterunder Unbalanced Grid State. When grid voltage is unbal-anced, grid-side voltage and current have positive and neg-ative sequence components, and three-phase voltage cannotbe simply converted into DC variables through 𝑎𝑏𝑐/𝑑𝑞

conversion, making design of control system quite complex.This section conducts 𝑎𝑏𝑐/𝑑𝑞 coordinate conversion forunbalanced three-phase voltage and obtains grid-side math-ematical model of three-phase inverter under unbalancedgrid according to coordinate conversion principle. Whenthree-phase grid voltage is unbalanced, if we only considerfundamental components, expression of grid voltage is asfollows:

(

𝑒𝑎

𝑒𝑏

𝑒𝑐

) = 𝑉𝑃(

cos (𝜔𝑡 + Ψ𝑃)

cos(𝜔𝑡 + Ψ𝑃 −2𝜋

3)

cos(𝜔𝑡 + Ψ𝑃 +2𝜋

3)

)

+ 𝑉𝑁(

cos (𝜔𝑡 + Ψ𝑁)

cos(𝜔𝑡 + Ψ𝑁 +2𝜋

3)

cos(𝜔𝑡 + Ψ𝑁 −2𝜋

3)

)

+ 𝑉0(

cos (Ψ0)

cos (Ψ0)

cos (Ψ0)

) .

(11)

In formula (11),𝑉𝑃 denotes positive sequence componentamplitude of grid voltage,𝑉𝑁 is negative sequence componentamplitude of grid voltage, and𝑉0 is zero sequence componentamplitude of grid voltage. 𝜔 denotes angular frequency offundamental wave, Ψ𝑃 denotes initial phase angle of positive

sequence component,Ψ𝑁 denotes initial phase angle of nega-tive sequence component, and Ψ0 denotes initial phase angleof zero sequence component. Conduct 𝑎𝑏𝑐/𝛼𝛽 conversion ofthe above formula, and then

(𝑒𝛼

𝑒𝛽

) = 𝑉𝑃 (cos (𝜔𝑡 + Ψ𝑃)

sin (𝜔𝑡 + Ψ𝑃))

+ 𝑉𝑁(cos (𝜔𝑡 + Ψ𝑁)

− sin (𝜔𝑡 + Ψ𝑁)) .

(12)

Formula (12) denotes that rotating vector𝐸 of three-phasevoltage can be seen as synthesized vector of positive sequencecomponent 𝐸𝑃 and negative sequence component 𝐸𝑁 underunbalanced grid voltage, as is shown in the following formula:

𝐸 = 𝐸𝑃 + 𝐸𝑁, (13)

where

𝐸𝑃 = 𝑉𝑃𝑒𝑗(𝜔𝑡+Ψ𝑃),

𝐸𝑁 = 𝑉𝑁𝑒−𝑗(𝜔𝑡+Ψ𝑁).

(14)

As is shown in Figure 9, positive sequence component𝐸𝑃 rotates anticlockwise with angular frequency 𝜔 of funda-mental wave, while negative sequence component 𝐸𝑁 rotatesclockwise with angular frequency 𝜔 of fundamental wave.Then, rotating vector 𝐸 of three-phase voltage is no longera space vector that rotates with a fixed angular frequencyand amplitude of 𝐸 changes with time as well. To simplifycomputation, express formula (14) as a vector under 𝛼𝛽coordinate system:

𝐸𝛼𝛽 = 𝐸𝛼𝛽(𝑃) + 𝐸𝛼𝛽(𝑁),

𝐸𝛼𝛽(𝑃) = 𝑒𝛼(𝑃) + 𝑗𝑒𝛽(𝑃),

𝐸𝛼𝛽(𝑁) = 𝑒𝛼(𝑁) + 𝑗𝑒𝛽(𝑁).

(15)

In formula (15), 𝐸𝛼𝛽 is the synthesized vector of rotatingvector 𝐸 under 𝛼𝛽 coordinate system, 𝐸𝛼𝛽(𝑃) is the syn-thesized vector of positive sequence component 𝐸𝑃 under


𝛼𝛽 coordinate system, and 𝐸𝛼𝛽(𝑁) is the synthesized vectorof negative sequence component 𝐸𝑁 under 𝛼𝛽 coordinatesystem.

Conduct 𝛼𝛽/𝑑𝑞 conversion of three-phase grid voltage,multiply matrix in formula (13) by terms in formula (15)successively, and then

𝐸𝑑𝑞 = 𝐸𝑑𝑞(𝑃) + 𝐸𝑑𝑞(𝑁),

𝐸𝑑𝑞(𝑃) = 𝑉𝑃 cos (𝜔𝑡 + Ψ𝑃 − 𝜃𝑑)

+ 𝑗𝑉𝑃 sin (𝜔𝑡 + Ψ𝑃 − 𝜃𝑑) ,

𝐸𝑑𝑞(𝑁) = 𝑉𝑁 cos (𝜔𝑡 + Ψ𝑁 − 𝜃𝑑)

+ 𝑗𝑉𝑁 sin (𝜔𝑡 + Ψ𝑁 − 𝜃𝑑) ,

(16)

where𝐸𝑑𝑞 is the synthesized vector of rotating vector𝐸 under𝑑𝑞 coordinate system, 𝐸𝑑𝑞(𝑃) is the synthesized vector of pos-itive sequence component 𝐸𝑃 under 𝑑𝑞 coordinate system,and 𝐸𝑑𝑞(𝑁) is the synthesized vector of negative sequencecomponent 𝐸𝑁 under 𝑑𝑞 coordinate system. Combine (11)with (16); then

𝑒𝑑 = 𝑉𝑃 cos (Ψ𝑃 − Ψ𝑑) + 𝑉𝑁 cos (2𝜔𝑡 + Ψ𝑁 + Ψ𝑑) ,

𝑒𝑞 = 𝑉𝑃 sin (Ψ𝑃 − Ψ𝑑) − 𝑉𝑁 sin (2𝜔𝑡 + Ψ𝑁 + Ψ𝑑) .

(17)

It can be seen from formula (17) that the projection ofpositive sequence component𝐸𝑃 on𝑑𝑞 coordinate isDCvari-able, while the projection of negative sequence component𝐸𝑁 on 𝑑𝑞 coordinate is doubled frequency AC variable.

Assume a 𝑑𝑞 rotating coordinate system 𝑛, rotate clock-wise with angular frequency of𝜔, and then the angle betweenaxis 𝑑 and axis 𝛼 is

𝜃𝑁 = − (𝜔𝑡 + Ψ𝑑(𝑁)) . (18)

Based on (16) and (18),

𝑒𝑑 = 𝑉𝑃 cos (2𝜔𝑡 + Ψ𝑃 + Ψ𝑑(𝑁))

+ 𝑉𝑁 cos (Ψ𝑁 − Ψ𝑑(𝑁)) ,

𝑒𝑞 = 𝑉𝑃 sin (2𝜔𝑡 + Ψ𝑃 − Ψ𝑑(𝑁))

− 𝑉𝑁 sin (Ψ𝑁 − Ψ𝑑(𝑁)) .

(19)

From formula (19), the projection of positive sequencecomponent 𝐸𝑃 on this 𝑑𝑞 coordinate is doubled frequencyAC variable, while the projection of negative sequencecomponent 𝐸𝑁 on this 𝑑𝑞 coordinate is DC variable. Todifferentiate, 𝑑𝑞 coordinate system that rotates anticlockwisewith angular frequency of 𝜔 is called positive 𝑑𝑞 rotatingcoordinate system, while rotating coordinate system thatrotates clockwise is called negative 𝑑𝑞 rotating coordinatesystem. According to the above analysis, when three-phasevoltage is asymmetrical, the projection of positive sequencecomponent 𝐸𝑃 on positive 𝑑𝑞 rotating coordinate system isDC variable, and the projection on negative 𝑑𝑞 rotating coor-dinate system is doubled frequency AC variable, while theprojection of negative sequence component𝐸𝑁 onpositive𝑑𝑞

rotating coordinate system is doubled frequency AC variable,and projection on negative 𝑑𝑞 rotating coordinate system isDC variable.

Conduct 𝑎𝑏𝑐/𝛼𝛽 conversion of grid-side mathematicalmodel under static 𝑎𝑏𝑐 coordinate system, ignore parasiticresistance on inductance, and then

𝑉𝛼𝛽 = 𝐸𝛼𝛽 + 𝐿𝑑 (𝐼𝛼𝛽)

𝑑𝑡, (20)

where 𝑉𝛼𝛽, 𝐸𝛼𝛽, and 𝐼𝛼𝛽 are synthesized vectors of rotatingvectors 𝑉, 𝐸, and 𝐼 under 𝛼𝛽 coordinate system.

If we conduct 𝛼𝛽/𝑑𝑞 conversion of formula (20) directly,according to analysis in the last section, projection values ofthree-phase voltage and current under a single 𝑑𝑞 rotatingcoordinate system are sums of DC variables and doubledfrequency AC variables, but the projection values of positivesequence and negative sequence components on positiveand negative 𝑑𝑞 rotating coordinate systems, respectively, areDC variables. This paper takes dual 𝑑𝑞 rotating coordinatesystem control and corresponds it to positive and negativesequence components of three-phase voltage and currentunder positive and negative 𝑑𝑞 rotating coordinate systems,respectively, to control DC variables and simplify design ofcontrol system and controller. Divide three-phase variablesinto positive sequence and negative sequence parts, and then

𝑉𝛼𝛽 = 𝑉𝛼𝛽(𝑃) + 𝑉𝛼𝛽(𝑁),

𝑉𝛼𝛽(𝑃) = 𝐸𝛼𝛽(𝑃) + 𝐿𝑑 (𝐼𝛼𝛽(𝑃))

𝑑𝑡,

𝑉𝛼𝛽(𝑁) = 𝐸𝛼𝛽(𝑁) + 𝐿𝑑 (𝐼𝛼𝛽(𝑁))

𝑑𝑡,

(21)

where 𝑉𝛼𝛽(𝑃), 𝐸𝛼𝛽(𝑃), and 𝐼𝛼𝛽(𝑃) are synthesized vectors ofpositive sequence components of positive sequence rotatingvectors 𝑉𝑃, 𝐸𝑃, and 𝐼𝑃 under 𝛼𝛽 coordinate system and𝑉𝛼𝛽(𝑁), 𝐸𝛼𝛽(𝑁), and 𝐼𝛼𝛽(𝑁) are synthesized vectors of negativesequence components of rotating vectors 𝑉𝑁, 𝐸𝑁, and 𝐼𝑁under 𝛼𝛽 coordinate system. Combine formula (15) with(21) and sort and obtain mathematical model of three-phase grid-connected inverter under positive and negative𝑑𝑞rotating coordinate systems under asymmetrical grid voltageas follows:

V+𝑑(𝑃) = 𝐿 ⋅𝑑𝑖

+

𝑑(𝑃)

𝑑𝑡+ 𝑒

+

𝑑(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑞(𝑃),

V+𝑞(𝑃) = 𝐿 ⋅𝑑𝑖

+

𝑞(𝑃)

𝑑𝑡+ 𝑒

+

𝑞(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑑(𝑃),

V−𝑑(𝑁) = 𝐿 ⋅𝑑𝑖

−

𝑑(𝑁)

𝑑𝑡+ 𝑒

−

𝑑(𝑁) − 𝜔𝐿 ⋅ 𝑖−

𝑞(𝑁),

V−𝑞(𝑁) = 𝐿 ⋅𝑑𝑖

−

𝑞(𝑁)

𝑑𝑡+ 𝑒

−

𝑞(𝑁) − 𝜔𝐿 ⋅ 𝑖−

𝑑(𝑁).

(22)

Formula (22) denotes grid-side mathematical modelsof positive and negative sequence components of three-phase voltage and current under positive and negative 𝑑𝑞


rotating coordinate systems, respectively, under the situationof unbalanced three-phase grid voltage. It can be seen thatthere are mutual coupling phenomena of components onaxes 𝑑 and 𝑞 under two 𝑑𝑞 rotating coordinate systems,which makes current control more difficult. Thus, currentloops under these two coordinate systems are of decouplingcontrol, respectively, to realize independent regulation of axes𝑑 and 𝑞. V+𝑑(𝑃), 𝑒

+

𝑑(𝑃), and 𝑖+

𝑑(𝑃) are components on axis𝑑 of positive sequence components 𝑉𝑃, 𝐸𝑃, and 𝐼𝑃 underpositive 𝑑𝑞 coordinate system, respectively; V+𝑞(𝑃), 𝑒

+𝑞(𝑃),

and 𝑖+𝑞(𝑃) are components on axis 𝑞 of positive sequencecomponents 𝑉𝑃, 𝐸𝑃, and 𝐼𝑃 under positive 𝑑𝑞 coordinatesystem. V−𝑑(𝑁), 𝑒

−

𝑑(𝑁), and 𝑖−

𝑑(𝑁) are components on axis 𝑑 ofpositive sequence components 𝑉𝑃, 𝐸𝑃, and 𝐼𝑃 under negative𝑑𝑞 coordinate system, respectively; V−𝑞(𝑁), 𝑒

−

𝑞(𝑁), and 𝑖−

𝑞(𝑁)

are components on axis 𝑞 of positive sequence components𝑉𝑃, 𝐸𝑃, and 𝐼𝑃 under negative 𝑑𝑞 coordinate system; underasymmetrical fault of grid voltage in static 𝛼𝛽 coordinatesystem, grid voltage and current have both positive sequencecomponents that rotate positively at synchronous speed 𝜔1and negative sequence components that rotate reversely at−𝜔1. Consider

𝐹𝑔𝛼𝛽 = 𝐹+

𝑔𝛼𝛽 + 𝐹−

𝑔𝛼𝛽 = 𝐹+

𝑔𝑑𝑞+𝑒𝑗𝜔1𝑡 + 𝐹

−

𝑔𝑑𝑞−𝑒−𝑗𝜔1𝑡. (23)

Under positive synchronous rotating coordinate system,there is

𝐹𝑔𝑑𝑞+ = 𝐹+

𝑔𝑑𝑞+ + 𝐹−

𝑔𝑑𝑞+

= 𝐹+

𝑔𝑑𝑞+𝑒𝑗𝜔1𝑡 + 𝐹

−

𝑔𝑑𝑞−𝑒−𝑗𝜔1𝑡.

(24)

In formula (24), 𝐹 denotes voltage and current, super-scripts + and − denote positive sequence and negativesequence components, respectively, and subscripts + and− denote positive and reverse synchronous rotating coor-dinate systems, respectively. Under positive synchronousrotating coordinate system, voltage and current have positivesequence DC variables and negative sequence AC variableswith doubled frequency fluctuation. Besides, with asymmet-rical fault of grid voltage, under positive and reverse syn-chronous rotating coordinate systems, mathematical modelof three-phase grid-connected inverter can be expressed informs of positive and negative sequence components undereach coordinate system:

𝑉+

𝑔𝑑𝑞+ = 𝑈+

𝑔𝑑𝑞+ + 𝐿 ⋅𝑑𝐼

+𝑔𝑑𝑞+

𝑑𝑡+ 𝑗𝜔1𝐿𝐼

+

𝑔𝑑𝑞+,

𝑉−

𝑔𝑑𝑞− = 𝑈−

𝑔𝑑𝑞− + 𝐿 ⋅𝑑𝐼

−

𝑔𝑑𝑞−

𝑑𝑡− 𝑗𝜔1𝐿𝐼

−

𝑔𝑑𝑞−.

(25)

Under asymmetrical fault of grid voltage, active poweroutput and reactive power output of grid-connected inverterare

𝑝𝑔 = 1.5Re (𝑠) = 1.5Re(𝑈𝑔𝑑𝑞+

∗

𝐼𝑔𝑑𝑞+) ,

𝑞𝑔 = 1.5 Im (𝑠) = 1.5 Im(𝑈𝑔𝑑𝑞+

∗

𝐼𝑔𝑑𝑞+) .

(26)

Substitute formula (24) with formula (26), and thenpower model under asymmetrical fault of grid voltage can beobtained as follows:

𝑝𝑔 = 𝑝𝑔0 + 𝑝𝑔 cos 2 ⋅ cos 2𝜔1𝑡 + 𝑝𝑔 sin 2 ⋅ sin 2𝜔1𝑡,

𝑞𝑔 = 𝑞𝑔0 + 𝑞𝑔 cos 2 ⋅ cos 2𝜔1𝑡 + 𝑞𝑔 sin 2 ⋅ sin 2𝜔1𝑡,(27)

where

𝑝𝑔0 = 1.5 (𝑢+

𝑔𝑑+𝑖+

𝑔𝑑+ + 𝑢+

𝑔𝑞+𝑖+

𝑔𝑞+ + 𝑢−

𝑔𝑑−𝑖−

𝑔𝑑−

+ 𝑢−

𝑔𝑞−𝑖−

𝑔𝑞−) ,

𝑝𝑔 cos 2 = 1.5 (𝑢+

𝑔𝑑+𝑖−

𝑔𝑑− + 𝑢+

𝑔𝑞+𝑖−

𝑔𝑞− + 𝑢−

𝑔𝑑−𝑖+

𝑔𝑑+

+ 𝑢−

𝑔𝑞−𝑖+

𝑔𝑞+) ,

𝑝𝑔 sin 2 = 1.5 (𝑢+

𝑔𝑑+𝑖−

𝑔𝑞− − 𝑢+

𝑔𝑞+𝑖−

𝑔𝑑− − 𝑢−

𝑔𝑑−𝑖+

𝑔𝑞+

+ 𝑢−

𝑔𝑞−𝑖+

𝑔𝑑+) ,

𝑞𝑔0 = 1.5 (𝑢+

𝑔𝑞+𝑖+

𝑔𝑑+ − 𝑢+

𝑔𝑑+𝑖+

𝑔𝑞+ + 𝑢−

𝑔𝑞−𝑖−

𝑔𝑑−

− 𝑢−

𝑔𝑑−𝑖−

𝑔𝑞−) ,

𝑞𝑔 cos 2 = 1.5 (𝑢−

𝑔𝑞−𝑖+

𝑔𝑑+ − 𝑢−

𝑔𝑑−𝑖+

𝑔𝑞+ + 𝑢+

𝑔𝑞+𝑖−

𝑔𝑑−

− 𝑢+

𝑔𝑑+𝑖−

𝑔𝑞−) ,

𝑞𝑔 sin 2 = 1.5 (𝑢+

𝑔𝑑+𝑖−

𝑔𝑑− − 𝑢+

𝑔𝑞+𝑖−

𝑔𝑞− − 𝑢−

𝑔𝑑−𝑖+

𝑔𝑑+

− 𝑢−

𝑔𝑞−𝑖+

𝑔𝑞+) .

(28)

In the above formula, 𝑝𝑔0 and 𝑞𝑔0 denote average com-ponents of active and reactive transient power, respectively,𝑝𝑔 cos 2 and 𝑝𝑔 sin 2 denote doubled frequency component ofactive transient power, and 𝑞𝑔 cos 2 and 𝑞𝑔 sin 2 denote doubledfrequency component of reactive transient power.

If we take positive sequence axis 𝑑 grid voltage vectororiented control strategy, that is, 𝑢+𝑔𝑞+ = 0, given commandvalues of positive and negative sequence current componentscan be obtained, respectively, in order to eliminate doubledfrequency fluctuation of active power:

𝑖+

𝑔𝑑+

∗=

𝑝𝑔0∗

[1.5𝑢+𝑔𝑑+ (1 − 𝑘2𝑑𝑑 − 𝑘

2𝑞𝑑)]

,

𝑖+

𝑔𝑞+

∗=

−𝑞𝑔0∗

[1.5𝑢+𝑔𝑑+ (1 + 𝑘2𝑑𝑑 + 𝑘

2𝑞𝑑)]

,

𝑖−

𝑔𝑑−

∗= −𝑘𝑑𝑑 ⋅ 𝑖

+

𝑔𝑑+

∗− 𝑘𝑑𝑞 ⋅ 𝑖

+

𝑔𝑞+

∗,

𝑖−

𝑔𝑞−

∗= 𝑘𝑑𝑑 ⋅ 𝑖

+

𝑔𝑞+

∗− 𝑘𝑑𝑞 ⋅ 𝑖

+

𝑔𝑑+

∗.

(29)


PWM generator

S1

S1

S3

S3

S5

S5

C1

C2

L2L1Ia

Ib

Ic

Usa

Usb

Usc

AD AD AD ADS4

S4

S6

S6

S2

S2

+−

PI

PI

PLL

PI

SVPWM

Feedforward

MPPTSa , Sb , Sc

U𝛼 , U𝛽

Urd , Urq

Usd , Usq

Ud

Uq

+

+

−

−

𝜃

𝜃

𝜃

Iq

Id

Usa , Usb , Usc

I∗d

I∗q

Idc

Udc

Udc

U∗dc

Ia , Ib , Ic

I∗d

dq/𝛼𝛽

abc/dq

abc/dq

Figure 10: Single axis 𝑑𝑞 PI control method.

In the above formula, 𝑘𝑑𝑑 and 𝑘𝑞𝑑 meet the followingrelations:

𝑘𝑑𝑑 =𝑢−𝑔𝑑−

𝑢+𝑔𝑑+

,

𝑘𝑞𝑑 =𝑢−𝑔𝑞−

𝑢+𝑔𝑑+

.

(30)

3. Existing Control Strategies

By deducing grid-side current model and active/reactivepower model under unbalanced grid, during LVRT, invertercontrol algorithm will solve three major problems.

Question 1. How to eliminate doubled frequency fluctuationof active power during grid voltage dip?

Question 2. How to eliminate doubled frequency fluctuationof reactive power during grid voltage dip?

Question 3. How to eliminate negative sequence injectioncurrent during grid voltage dip?

The academic circle has put forward some control strate-gies and mainly studies on these three points. For com-parison, single axis 𝑑𝑞 PI control method will be added tothe following control algorithms under symmetrical grid asfollows.

3.1. Control Strategy Based on Current Loop under Single Axis𝑑𝑞 Rotating Coordinate System. Thebasic idea of using single𝑑𝑞 PI controller method is to conduct 𝑎𝑏𝑐/𝑑𝑞 conversionof IGBT inverter-side current feedback signal, obtain 𝑖𝑑 and

𝑖𝑞, and send to PI controller. This control method does notconduct positive and negative sequence decomposition ofcurrent, and the control block diagram is shown in Figure 10.

This control method will lead to deterioration of controleffects during grid unbalance or drop, as it considers feedbackcurrent as a whole and does not restrain doubled frequencyfluctuation of negative sequence current and active/reactivepower. Figure 11 is about waveform of active/reactive powerunder single-phase drop (drop) of grid. Grid voltage hassingle-phase drop at 0.1 s, and active power and reactivepower have single-phase drop 0.1 s later and lead to doubledfrequency fluctuation due to absence of responsive reactivepower control method.

3.2. Control Strategy Based on Current Loop under Dual 𝑑𝑞Rotating Coordinate System. Current loop control methodunder dual 𝑑𝑞 rotating coordinate system is used to controlpositive and negative sequence components of current underdifferent rotating coordinate systems, respectively.Therefore,each rotating coordinate system realizes control of DC vari-ables, and thus PI controller could achieve very good stablestate and dynamic performance [14, 15]. Under unbalancedthree-phase grid voltage, three-phase voltage and positiveand negative sequence components of current are in grid-side mathematical models under positive and negative 𝑑𝑞rotating coordinate systems, respectively. It can be seen thatthere aremutual coupling phenomena of components on axes𝑑 and 𝑞 under two 𝑑𝑞 rotating coordinate systems, whichmakes current control more difficult. Thus, current loopsunder these two coordinate systems are of decoupling control,respectively, to realize independent regulation of axes 𝑑 and𝑞.

With the example of current regulation under positive 𝑑𝑞rotating coordinate system, when grid voltage is unbalanced,


0 0.05 0.1 0.15 0.2

Time (s)

6

4

2

0

−2

−4

×105

Pow

er ((

kW),

(kva

r))

Grid voltage drop point Active power

Reactive power

Figure 11: Active/reactive power output waveform under single-phase grid drop.

positive sequence components of grid-side inverter outputvoltage under positive 𝑑𝑞 rotating coordinate system areshown in the following formula:

V+𝑑(𝑃) = 𝐿 ⋅𝑑𝑖

+

𝑑(𝑃)

𝑑𝑡+ 𝑒

+

𝑑(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑞(𝑃),

V+𝑞(𝑃) = 𝐿 ⋅𝑑𝑖

+𝑞(𝑃)

𝑑𝑡+ 𝑒

+

𝑞(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑑(𝑃).

(31)

Make approximation, and

ΔV+𝑑(𝑃) = 𝐿 ⋅𝑑𝑖

+

𝑑(𝑃)

𝑑𝑡,

ΔV+𝑞(𝑃) = 𝐿 ⋅𝑑𝑖

+

𝑞(𝑃)

𝑑𝑡.

(32)

Formula (31) can be expressed as

V+𝑑(𝑃) = ΔV+

𝑑(𝑃) + 𝑒+

𝑑(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑞(𝑃),

V+𝑞(𝑃) = ΔV+

𝑞(𝑃) + 𝑒+

𝑞(𝑃) − 𝜔𝐿 ⋅ 𝑖+

𝑑(𝑃).

(33)

To eliminate error under stable state, current loop takesPI regulation, and controller can be expressed as

V+𝑑(𝑃) = 𝑘𝑝 ⋅ (𝑖+

𝑑(𝑃)

∗− 𝑖

+

𝑑(𝑃)) + 𝑘𝑖

⋅ ∫ (𝑖+

𝑑(𝑃)

∗− 𝑖

+

𝑑(𝑃)) 𝑑𝑡,

V+𝑞(𝑃) = 𝑘𝑝 ⋅ (𝑖+

𝑞(𝑃)

∗− 𝑖

+

𝑞(𝑃)) + 𝑘𝑖

⋅ ∫ (𝑖+

𝑞(𝑃)

∗− 𝑖

+

𝑞(𝑃)) 𝑑𝑡.

(34)

In the above formula, 𝑖+𝑑(𝑃)∗ and 𝑖

+𝑞(𝑃)

∗ are positivesequence current reference components on axis 𝑑𝑞 and 𝑘𝑝and 𝑘𝑖 are proportion and integral coefficient of PI controller,respectively. Cross decoupling control block diagram ofpositive sequence components of grid-side current on axis 𝑑𝑞under positive synchronous rotating coordinate system can

−𝜔L

−𝜔L

+

+

+

+

+

+

+

+

−

−

i+q(P)∗

i+d(P)∗

i+q(P)

i+d(P)

i+d(P)

e+q(P)

e+d(P) + je+q(P)

PI

PI

Figure 12: Axis 𝑑𝑞 decoupling control block diagram.

be obtained according to formula (33) and (34), as is shownin Figure 12.

Similarly, use PI controller to control negative sequencecomponents of current and controller:

V−𝑑(𝑁) = 𝑘𝑝 ⋅ (𝑖−

𝑑(𝑁)

∗− 𝑖

−

𝑑(𝑁)) + 𝑘𝑖

⋅ ∫ (𝑖−

𝑑(𝑁)

∗− 𝑖

−

𝑑(𝑁)) 𝑑𝑡,

V−𝑞(𝑁) = 𝑘𝑝 ⋅ (𝑖−

𝑞(𝑁)

∗− 𝑖

−

𝑞(𝑁)) + 𝑘𝑖

⋅ ∫ (𝑖−

𝑞(𝑁)

∗− 𝑖

−

𝑞(𝑁)) 𝑑𝑡.

(35)

Assume that there are positive and negative 𝑑𝑞 rotatingcoordinate systems that are symmetrical about axis 𝛼 withrotating angular frequency of 𝜔 and initial angle of ±Ψ𝑑.Rotating vector 𝑉 corresponding to output voltage of grid-side inverter can be seen as the vector sum of positive andnegative sequence components; that is

𝑉 = 𝑉+

𝑑𝑞(𝑃) ⋅ 𝑒𝑗𝜃𝑑 + 𝑉

+

𝑑𝑞(𝑁) ⋅ 𝑒−𝑗𝜃𝑑 , (36)

where

𝜃𝑑 = 𝜔𝑡 + Ψ𝑑,

𝑉+

𝑑𝑞(𝑃) = V+𝑑(𝑃) + 𝑗V+

𝑞(𝑃),

𝑉+

𝑑𝑞(𝑁) = V+𝑑(𝑁) + 𝑗V+

𝑞(𝑁).

(37)

Combine formula (34) with (35), and obtain doublecurrent loop control block diagram of inverter under positiveand negative 𝑑𝑞 coordinate systems, as is shown in Figure 13.

According to Figure 14, positive sequence component𝑉+𝑑𝑞 that inverter outputs is current loop output correspond-

ing to positive sequence current, while negative sequencecomponent 𝑉−

𝑑𝑞 that inverter outputs is current loop outputcorresponding to negative sequence current. Through 𝑑𝑞/𝛼𝛽conversion,𝑉𝛼𝛽(𝑃) and𝑉𝛼𝛽(𝑁) are obtained, andoutput voltagevector 𝑉𝛼𝛽 of inverter is obtained through vector synthesis.Vector 𝑉𝛼𝛽 would be input of space vector pulse widthmodulation (SVPWM) to control turning on/off of switchingdevices like IGBT and so forth and finally achieve grid-connected current control objective.


PI

PI

−𝜔L

−𝜔L

PI

PI

−𝜔L

−𝜔L

+

+

++

+

++

+

++

+

++

+

−

+

−

−

+−

i−q(P)∗

i+q(P)∗

i+d(P)∗

i+q(P)

i+d(P)

i+d(P)

e+q(P)

e−q(P)

e+d(P) + je+q(P)

V𝛼𝛽 SVPWM Inverter

i−d(P)∗

e−d(P) + je−q(P)

i−q(P)

i−d(P)

i−d(P)

Figure 13: Block diagram of three-phase grid-connected inverter system control.

Gi(s)

Gi(s)

2r/3s

2r/3s

−

−

−

−+ +

+

++

+ + +

+

kc

Ginv

ic

iinv

uinv e

ig

iNinv_dq

iPinv_dq

1

sLinv

1

sLg

1

sCf

iP∗

inv_dq

iN∗

inv_dq

Figure 14: Block diagram of LCL active damping control under unbalanced control strategy.

3.3. Active Damping Compensation Feedback Control Strat-egy. In practical application, in order to conduct real-timedetection of inverter-side current to facilitate safety protec-tion, place current sensors at the output side of inverter.Meanwhile, to obtain grid synchronous signals, place voltagesensors at the grid side. As inverter-side output voltage ispulse width modulation (PWM) wave and is inconvenientto measure, in order to use the known measurement ofgrid voltage and obtain accurate command current, convertformula (31) and obtain the relationship between inverter-side voltage and grid voltage under fundamental waves asfollows:

𝑢𝑃

inv(𝑑) = 𝑒𝑃

𝑑 − 𝜔𝐿 inv𝑖𝑃

inv(𝑞) − 𝜔𝐿𝑔𝑖𝑃

𝑔𝑞,

𝑢𝑃

inv(𝑞) = 𝑒𝑃

𝑞 − 𝜔𝐿 inv𝑖𝑃

inv(𝑑) − 𝜔𝐿𝑔𝑖𝑃

𝑔𝑑,

𝑢𝑁

inv(𝑑) = 𝑒𝑃

𝑑 − 𝜔𝐿 inv𝑖𝑁

inv(𝑞) − 𝜔𝐿𝑔𝑖𝑁

𝑔𝑞,

𝑢𝑁

inv(𝑞) = 𝑒𝑃

𝑞 − 𝜔𝐿 inv𝑖𝑁

inv(𝑑) − 𝜔𝐿𝑔𝑖𝑁

𝑔𝑑.

(38)

Substitute formula (38) with formula (28) and obtain [16,17]:


[[[[[

[

𝑃0

𝑄0

𝑃cos 2

𝑃sin 2

]]]]]

]

=3

2

[[[[[[

[

𝑒𝑃

𝑑 𝑒𝑃

𝑞 𝑒𝑁

𝑑 𝑒𝑁

𝑞

𝑒𝑃

𝑞 −𝑒𝑃

𝑑 𝑒𝑁

𝑞 −𝑒𝑁

𝑑

𝑒𝑁

𝑑 𝑒𝑁

𝑞 𝑒𝑃

𝑑 𝑒𝑃

𝑞

𝑒𝑁

𝑞 −𝑒𝑁

𝑑 −𝑒𝑃

𝑞 𝑒𝑃

𝑑

]]]]]]

]

[[[[[[

[

𝑖𝑃

inv(𝑑)

𝑖𝑃

inv(𝑞)

𝑖𝑁

inv(𝑑)

𝑖𝑁

inv(𝑞)

]]]]]]

]

+3

2

[[[[[[[[[

[

−𝜔𝐿 (𝑖𝑃inv(𝑞) + 𝑖

𝑃𝑔𝑞) 𝜔𝐿 (𝑖

𝑃inv(𝑑) + 𝑖

𝑃𝑔𝑑) 𝜔𝐿 (𝑖

𝑁inv(𝑞) + 𝑖

𝑁𝑔𝑞) −𝜔𝐿 (𝑖

𝑁inv(𝑑) + 𝑖

𝑁𝑔𝑑)

𝜔𝐿 (𝑖𝑃

inv(𝑑) + 𝑖𝑃

𝑔𝑑) 𝜔𝐿 (𝑖𝑃

inv(𝑞) + 𝑖𝑃

𝑔𝑞) −𝜔𝐿 (𝑖𝑁

inv(𝑑) + 𝑖𝑁

𝑔𝑑) −𝜔𝐿 (𝑖𝑁

inv(𝑞) + 𝑖𝑁

𝑔𝑞)

𝜔𝐿 (𝑖𝑁


𝑔𝑞) −𝜔𝐿 (𝑖𝑁


𝑔𝑑) −𝜔𝐿 (𝑖𝑃


𝑔𝑞) 𝜔𝐿 (𝑖𝑃


𝑔𝑑)

−𝜔𝐿 (𝑖𝑁


𝑔𝑑) −𝜔𝐿 (𝑖𝑁


𝑔𝑞) −𝜔𝐿 (𝑖𝑃


𝑔𝑑) −𝜔𝐿 (𝑖𝑃


𝑔𝑞)

]]]]]]]]]

]

[[[[[[[[[

[

𝑖𝑃

inv(𝑑)

𝑖𝑃inv(𝑞)

𝑖𝑁

inv(𝑑)

𝑖𝑁inv(𝑞)

]]]]]]]]]

]

.

(39)

Formula (39) is the relationship between knowndetectionand inverter output power, to eliminate negative sequencecurrent (𝑖𝑁inv(𝑑) = 𝑖

𝑁

inv(𝑞) = 0). Considering full poweroperation, to simplify computation under designed operatingparameters, simplify the above formula and given value ofcurrent reference is

𝑖𝑃∗

inv(𝑑) =2𝑒

𝑃

𝑑𝑃0 + 𝑒𝑃

𝑞𝑄0

3 [(𝑒𝑃𝑑)2+ (𝑒𝑃𝑞)

2

]

,

𝑖𝑃∗

inv(𝑞) =2𝑒

𝑃

𝑞𝑃0 − 𝑒𝑃

𝑑𝑄0

3 [(𝑒𝑃𝑑)2+ (𝑒𝑃𝑞)

2

]

,

𝑖𝑁

inv(𝑑) = 0,

𝑖𝑁

inv(𝑞) = 0,

(40)

Δ𝑄 = 𝜔𝐶 [(𝑒𝑃

𝑐𝑑)2

+ (𝑒𝑃

𝑐𝑞)2

+ (𝑒𝑁

𝑐𝑞)2

+ (𝑒𝑁

𝑐𝑑)2

] . (41)

Under stable operating state, set Δ𝑄 = 𝑄0 and substitutewith formula (41), and obtain given value of inner loopcurrent under unit power factor grid-connected operation.When conducting process of LVRT, carry out real-timemodification of the value of 𝑄0 to regulate reactive poweroutput according to detected grid voltage drop depth. Toensure that grid-connected current does not exceed outputprotection value, active and reactive power output will meet:

𝑖𝑃∗

inv(𝑑) ≤ √(𝑖max)2− (𝑖𝑃

∗

inv(𝑞))2

. (42)

In the formula, 𝑖max denotes the maximum amplitudelimit of output current. Stable control and reactive power

support after voltage drop are realized by dynamic changeof current command value on axis 𝑑𝑞, and select appro-priate controller parameters to ensure system stability. Aspositive and negative sequence dual 𝑑𝑞 current loop controlmethod would separate voltage and current into positiveand negative sequence components under unbalanced sit-uation while filter capacitive current 𝑖𝐶 is mainly high-frequency component, commonly used positive and negativesequence separation method cannot obtain positive andnegative sequence components under high frequency. Thus,active damping control method is taken under balancedsituation but cannot realize damping control under unbal-anced situation and even affect system stability. Add activedamping method of filter capacitive current feedforward tomodulation signal, and current loop control block diagramis shown in Figure 14. Dashed block on the left side is thecontrol block diagram of unbalanced control algorithm, anddashed block on the right side is the equivalent mathematicalmodel of LCL. Feedforward method shown in Figure 14avoids positive and negative sequence separation feedfor-ward variable 𝑖𝐶 effectively, and regulation of coefficient 𝑘𝐶does not affect transfer function of active damping algo-rithm in Figure 5 and can restrain resonance of LCL filtereffectively.

4. Control Methods Proposed in This Paper

4.1. Phase Lock Control Strategy. Under unbalanced or dis-torted grid, axis 𝑞 has secondary fluctuation, and thus phaselock method under unbalanced grid is not applicable. Whenthree-phase grid is unbalanced or distorted, the influence ofnegative sequence components and harmonic components


will be considered. For asymmetrical grid three-phase volt-age,

[[

[

𝑒𝑎

𝑒𝑏

𝑒𝑐

]]

]

= 𝑉𝑝

[[[[[

[

cos (𝜃𝑢)

cos(𝜃𝑢 −2𝜋

3)

cos(𝜃𝑢 +2𝜋

3)

]]]]]

]

+

∞

∑

𝑛=−∞𝑛 =−1

𝑉𝑛

[[[[[

[

cos (𝑛𝜃𝑢 + 𝜙𝑛)

cos(𝑛𝜃𝑢 + 𝜙𝑛 −2𝜋

3)

cos(𝑛𝜃𝑢 + 𝜙𝑛 +2𝜋

3)

]]]]]

]

+

∞

∑

𝑛=1

𝑉𝑛

[[[

[

cos (𝑛𝜃𝑢 + 𝜙0𝑛)cos (𝑛𝜃𝑢 + 𝜙0𝑛)cos (𝑛𝜃𝑢 + 𝜙0𝑛)

]]]

]

.

(43)

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 sequencevoltage (𝑛 = −1) and harmonic voltage (𝑛 = ±1), and thethird term denotes zero sequence voltage.

For voltage expression of three-phase balanced system,

[[

[

𝑒𝑎

𝑒𝑏

𝑒𝑐

]]

]

= 𝑉𝑝

[[[[[[

[

cos (𝜃𝑢)

cos(𝜃𝑢 −2𝜋

3)

cos(𝜃𝑢 +2𝜋

3)

]]]]]]

]

, (44)

where 𝑒𝑎, 𝑒𝑏, and 𝑒𝑐 denote A, B, and C three-phase voltage,𝑉𝑝 is voltage amplitude, and 𝜃𝑢 is A-phase angle.

Replace formula (44) with formula (43), and computenew phase error:

𝑒𝑞 (𝑘) = 𝑉𝑝 sin (𝜃ref (𝑘) − 𝜃𝑢 (𝑘))

+

∞

∑

𝑛=−∞𝑛 =−1

𝑉𝑛 sin (𝜃ref (𝑘) − 𝑛𝜃𝑢 (𝑘) − 𝜙𝑛) .(45)

When locking phase of positive sequence fundamentalcomponents of grid voltage, that is, 𝜃𝑢 = 𝜃ref, phase error canbe written in the following form:

𝑒𝑞 (𝑘) =

∞

∑

𝑛=−∞𝑛 =−1

𝑉𝑛 sin ((1 − 𝑛) 𝜃𝑢 (𝑘) − 𝜙𝑛) . (46)

From formula (46), negative sequence voltage in three-phase grid would have doubled frequency oscillation signalon phase error 𝑒𝑞; 𝑛 times of harmonic voltagewould generate𝑛 − 1 or 𝑛 + 1 times of frequency oscillation signal, and thus𝑒𝑞 cannot express accurately phase error between referencephase 𝜃ref and actual positive sequence fundamental voltagephase 𝜃𝑢 and thus could not conduct accurate phase lock. Tosum up, to accurately realize phase lock of grid-side positive

sequence voltage, all oscillation signals in 𝑒𝑞 will be filtered.This paper uses sliding Goertzel filter to filter oscillationsignals in phase error, and its implementation principle is asfollows: compute value of the 𝑛th harmonic through digitalFourier Transform. Sliding Goertzel filter used in this paperis to compute with assistance of a sliding window with lengthof𝑁SG based on Goertzel filter.

According to digital Fourier Transform, for continuoustime function 𝑥(𝑡) with period of 𝑇1, value of the 𝑛thharmonic is

𝑌(𝑛𝑓1)=

𝑁SG−1

∑

𝑘=0

𝑥 (𝑘𝑇𝑠) ⋅ 𝑒−𝑗2𝜋𝑓1 ⋅𝑘𝑇𝑠 ⋅𝑛, (47)

where 𝑇𝑠 is sampling period,𝑁SG is the number of samplingpoints in a function period, and

𝑓𝑠 =1

𝑇𝑠

= 𝑓1 ⋅ 𝑁SG. (48)

Substitute formula (48) with formula (47), and

𝑌(𝑛𝑓1)=

𝑁SG−1

∑

𝑘=0

𝑥 (𝑘 + 𝑞 − 𝑁SG + 1) ⋅ 𝑒−𝑗2𝜋𝑘𝑛/𝑁SG . (49)

It is quite complex to compute each harmonic valuethrough (49) directly, and each computation of harmonicvalue takes a function period. In filter algorithm based onsliding DFT used in this paper, filtering value of previousmoment is known, and filtering value of later moment canbe obtained through simple recursive operation.This filteringmethod is simple to compute and has important practicalsignificance. For periodic function 𝑥(𝑡), formula (50) iscorresponding to discrete Fourier expressions of this periodicfunction at times 𝑞 and 𝑞 + 1 and the 𝑛th harmonic values ofthis periodic function at times 𝑞 and 𝑞 + 1:

𝑌(𝑞,𝑛) =

𝑁SG−1

∑

𝑘=0

𝑥 (𝑘 + 𝑞 − 𝑁SG + 1) ⋅ 𝑒−𝑗2𝜋𝑘𝑛/𝑁SG ,

𝑌(𝑞+1,𝑛) =

𝑁SG−1

∑

𝑘=0

𝑥 (𝑘 + 𝑞 − 𝑁SG + 2) ⋅ 𝑒−𝑗2𝜋𝑘𝑛/𝑁SG .

(50)

Thus, the relationship between both 𝑛th harmonics ofperiodic function 𝑥(𝑡) at times 𝑞 and 𝑞 + 1 is

𝑌(𝑞+1,𝑛) = [𝑌(𝑞,𝑛) + 𝑥 (𝑞 + 1) − 𝑥 (𝑞 − 𝑁SG + 1)]

⋅ 𝑒𝑗2𝜋𝑘𝑛/𝑁SG .

(51)

According to formula (51), filter with a sliding Goertzelfilter, and derive harmonic value of later moment accordingto harmonic value of previous moment, which is relativelyeasy and conducive to digitalization. This paper uses slidingGoertzel filter to filter all oscillation signals in phase error andonly keepDCvariables. Take 𝑛 = 0, simplify formula (51), andobtain

𝑌(𝑘) = 𝑌(𝑘−1) + 𝑥(𝑘) − 𝑥(𝑘−𝑁SG). (52)


ControllerGoertzel

Signalsampling

Referencegenerator

ADe𝛼

e𝛽

eqea/eb/ecabc/𝛼𝛽

𝜃ref

2𝜋

NPLL

𝛼𝛽/dq

Ts

Figure 15: Principle block diagram of phase-locked loop.

2𝜋

NPLL 1

z − 1

𝜃ref

𝜃u

+−

+−

Vp sineq Ts

e∗q = 0

𝜔0

z − 1Gc(z) KGinvTsGSG(z)

Figure 16: Nonlinear mathematical model of phase-locked loop in domain 𝑍.

Transfer function of sliding Goertzel filter in domain𝑍 isshown in the following formula:

𝐺SG =1 − 𝑧

−𝑁SG

1 − 𝑧−1. (53)

Principle block diagram of phase-locked loop applied tothree-phase asymmetrical grid is shown in Figure 15.

This three-phase phase-locked loop uses sliding Goertzelfilter to filter disturbance generated by negative sequencecomponents and harmonic components in three-phase grid,and the system can lock phase of positive sequence funda-mental voltage very well. Expression of A-phase angle 𝜃𝑢 andreference phase angle 𝜃ref in domain 𝑍 is

𝜃𝑢 (𝑧) =𝜔

𝑧 − 1,

𝜃ref (𝑧) =2𝜋/𝑁pll

𝑧 − 1.

(54)

Combine formula (53) and (54) with control principlediagram of phase-locked loop, and mathematical model ofphase-locked loop in domain 𝑍 can be obtained as shown inFigure 16.

In Figure 16, 𝐺SG(𝑧) denotes 𝑍-domain transfer functioncorresponding to Goertzel filter, 𝐺𝐶(𝑧) is 𝑍-domain transferfunction corresponding to controller, 𝐾𝐺𝐶-𝑇𝑠(𝑧) is a propor-tion function from controller output to sampling period 𝑇𝑠,and 𝜔 is angular frequency of grid. Considering that 𝑒∗𝑞 = 0and when phase error is very small and formula (46) is met,Figure 16 can be simplified into Figure 17, and we finallyobtain a linear mathematical model in domain 𝑍 that is easyto achieve.

This phase-locked loop can eliminate the influence ofunbalanced grid voltage and has good dynamic trackingfeatures. Simulation results indicate that this phase-lockedloop can lock positive sequence fundamental components

in three-phase grid very well under unbalanced three-phase grid voltage, voltage distortion, and sudden frequencychanges and has advantages such as good dynamic perfor-mance, short dynamic response time, and high precision ofstable state.

4.2. Take PR Controller for Grid Current Loop. Under pos-itive and reverse synchronous rotating coordinate systems,although dual 𝑑𝑞 and PI current controller can meet controldemand of grid-connected inverter system under small faultof asymmetrical stable grid voltage, it has poor response todynamic transient process under large fault of asymmetricaltransient grid voltage. According to Figure 14, dual 𝑑𝑞 andPI controller need to conduct positive and negative sequencedecomposition of feedback current in current control loopunder positive and reverse synchronous rotating coordinatesystems. Therefore, wave trap link is introduced to innercurrent loop, while delay brought by wave trap will affectdynamic performance of system and thus decrease FRTability of grid-connected inverter. Therefore, new controlalgorithmwill be used to eliminate delay of inner current loopbrought by positive and negative sequence decomposition.Then, this paper puts forward an active damping controlalgorithm based on PR current controller combined withcapacitive current feedback.

Mathematical model of grid-connected inverter in 𝛼𝛽static coordinate system is

𝑉𝛼𝛽 = 𝑈𝛼𝛽 + 𝐿𝑑𝐼𝛼𝛽

𝑑𝑡. (55)

Here, introduce PR current controller, that is, propor-tional resonant controller [4–8], and set

𝑈∗

𝑔𝛼𝛽 =𝑑𝐼𝑔𝛼𝛽

𝑑𝑡= 𝐹PR (𝑠) (𝐼

∗

𝑔𝛼𝛽 − 𝐼𝑔𝛼𝛽) ,(56)


2𝜋

NPLL 1

z − 1

𝜃ref

𝜃u

+−

Vp

eq Ts 𝜔0

z − 1Gc(z) KGinvTsGSG(z)

Figure 17: Linear mathematical model of phase-locked loop in domain 𝑍.

14

12

10

8

6

4

2

0

−20 0.05 0.1 0.15 0.2

Grid voltage drop point

Time (s)

One-phase grid voltage drop

Uq

_PLL

(a) The wave of𝑈𝑞 under one-phase grid voltage drop

14

12

10

8

6

4

2

0

−2

Grid voltage drop point

0 0.05 0.1 0.15 0.2

Time (s)

Three-phase grid voltage drop

Uq

_PLL

(b) The wave of𝑈𝑞 under three-phase grid voltage drop

Figure 18: The wave of 𝑈𝑞.

where 𝐹PR(𝑠) denotes transfer function of PR current con-troller, and the expression is

𝐹PR (𝑠) = 𝐾𝑝 + 𝑅 (𝑠) = 𝐾𝑝 + 𝐾𝑟

𝑠

𝑠2 + 𝜔12. (57)

In formula (57), 𝐾𝑝 and 𝐾𝑟 denote proportional coeffi-cient and harmonic coefficient of PR controller, respectivelyand 𝐾𝑝 plays the same role with traditional PI controllerand is used to regulate dynamic performance of system. 𝑅(𝑠)is the transfer function of harmonic controller, harmonicangular frequency of which is ±𝜔1. In the situation of gridvoltage asymmetrical fault under 𝛼𝛽 static coordinate system,positive and negative sequence components of grid voltageand current rotate at synchronous angular speed of 𝜔1 and−𝜔1, respectively. Thus, positive and negative sequence com-ponents of output current can be controlled synchronouslyby a harmonic current controller. Transfer function of PRcontroller shown in formula (57) is an 𝑠-domain functionand will be discrete to domain 𝑍 to conduct digital controlof grid-connected inverter output current by PR controller.Use Tustin conversion method to make transfer function ofPR controller discrete, and then

𝐹PR (𝑧) = 𝐾𝑝 + 𝐾𝑟𝑅 (𝑧) , (58)

where 𝑅(𝑧) can be expressed as

𝑅 (𝑧) =𝑈 (𝑧)

𝐸 (𝑧)=𝑏0 + 𝑏1𝑧

−1+ 𝑏2𝑧

−2

1 + 𝑎1𝑧−1 + 𝑎2𝑧

−2, (59)

where

𝑏0 =2𝑇𝑠

𝜔12𝑇𝑠

2,

𝑏1 = 0,

𝑏2 =2𝑇𝑠

4 + 𝜔12𝑇𝑠

2,

𝑎1 =2𝜔1

2𝑇𝑠

2− 8

4 + 𝜔12𝑇𝑠

2,

𝑎2 = 1.

(60)

In the above formula, 𝑇𝑠 is the sampling period of controlsystem.

According to formula (58)∼formula (60), discrete differ-ence equation of PR controller can be obtained as follows:

𝑢 (𝑘) = 𝐾𝑝𝑒 (𝑘) − 𝑎1𝑢 (𝑘 − 1) − 𝑢 (𝑘 − 2) + 𝑏0𝐾𝑟𝑒 (𝑘)

+ 𝑏2𝐾𝑟𝑒 (𝑘 − 2) ,

(61)

where 𝑢(𝑘) is output value at the 𝑘th sampling time, 𝑒(𝑘) isoutput error at the 𝑘th sampling time, 𝑢(𝑘−1) is output valueat 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 controlthat uses PR current controller is shown in Figure 19. UsePR current controller, set according to control objective andpower, decompose positive and negative sequence compo-nents of three-phase grid voltage, and calculate and obtain


ugabc

abc/𝛼𝛽

abc/𝛼𝛽

ug𝛼𝛽

PLL

SVM

𝜔1, 𝜃1

ej𝜃1

e−j𝜃1

ej𝜃1

e−j𝜃1

u+gdq

u−gdq

u+gdq

u−gdq2nd wave trap Current reference

generator

+

+

i+gdq+

i−gdq−

Q∗g P∗

g

igabc ig𝛼𝛽 i∗g𝛼𝛽

PR controller

S1,2,3,4,5,6

u𝛼 u𝛽

Figure 19: Current control scheme for the GCIs based on PR regulator under grid voltage unbalance.

given command values of positive and negative sequencecomponents of output current. In control block diagram, cur-rent control loop has not any delay or error brought by suchdecomposition. Therefore, dynamic regulation performanceof control system under grid voltage asymmetrical fault isimproved, and thus FRT ability of grid-connected inverter isimproved.

4.3. Grid Voltage Feedforward. At the moment of voltagedrop, current shock during switching between stable grid-connected operation and LVRT operating state will becontrolled to ensure safe access of photovoltaic inverter tooperating interval after drop. For transient process, literature[18] puts forward a feedforward control method with respectto single-phase grid-connected inverter with L-shaped filter;literature [19] uses full voltage feedforward based on LCL fil-ter to restrain the influence of grid voltage harmonic on grid-connected current but could not restrain transient currentshock when grid voltage drops. Therefore, this paper, basedon the selected inverter grid-connected system, puts forwarda simplified grid voltage feedforward control algorithm thatretrains transient current shock in terms of transient stateof voltage drop. In Figure 14, system input variables are𝑖𝑃∗

inv(𝑑𝑞) and 𝑖𝑁∗

inv(𝑑𝑞) and output variable is 𝑖𝑔. To analyzethe influence of disturbance variable 𝑒 on system stability,simplify Figure 13 into control structure shown in Figure 20,where 𝑘𝑒 is grid voltage feedforward coefficient.

In Figure 20, closed-loop transfer function of the systemis

𝐼𝑔 =𝐺𝑖 (𝑠) 𝐺1 (𝑠) 𝐺2 (𝑠)

1 + 𝐺𝑖 (𝑠) 𝐺1 (𝑠) 𝐺2 (𝑠)⋅ 𝑖

∗

inv

−𝐺2 (𝑠) (𝑘𝑒𝐺1 (𝑠) − 1)

1 + 𝐺𝑖 (𝑠) 𝐺1 (𝑠) 𝐺2 (𝑠)⋅ 𝐸.

(62)

i∗ inv

+

++

+

−

−

Gi(s) G1(z) G2(z)ig

ke

e

Figure 20: Simplified block diagram of system control.

In the above formula, 𝐸 denotes grid voltage, where

𝐺1 (𝑠) =𝐺inv

𝐿 inv𝐶𝑓𝑠2 + (𝐺inv𝐺𝑖 (𝑠) − 𝑘𝑐𝐺inv) 𝐶𝑓𝑠 + 1

,

𝐺1 (𝑠)

=𝐿 inv𝐶𝑓𝑠

2+ (𝐺inv𝐺𝑖 (𝑠) − 𝑘𝑐𝐺inv) 𝐶𝑓𝑠 + 1

𝐿 inv𝐿𝑔𝑠3 + (𝐺inv𝐺𝑖 (𝑠) − 𝑘𝑐𝐺inv) 𝐿𝑔𝐶𝑓𝑠

2 + (𝐿 + 𝐿𝑔) 𝑠.

(63)

According to formula (63), control output variable 𝐼𝑔and input variable 𝑖∗inv and grid voltage 𝐸 are all related.Regard 𝐸 as disturbance term, and grid current will increaseproportionally at the moment of voltage drop. Therefore,to eliminate the influence of disturbance variable on grid-connected current, introduce grid voltage feedforward, as isshown in dashed part of Figure 24. If disturbance term iscompletely eliminated, it can be obtained that

𝑘𝑒 =1

𝐺1 (𝑠)

=1

𝐺inv[𝐿 inv𝐶𝑓𝑠

2+ (𝐺inv𝐺𝑖 (𝑠) − 𝑘𝑐𝐺inv) 𝐶𝑓𝑠 + 1] .

(64)

Disturbance variable 𝑒 can be completely restrained byusing feedforward coefficient 𝑘𝑒 of formula (64); that is,


Table 1: Parameters of photovoltaic inverter system.

Category Name Parameter

System parameters

Maximum DC voltage 1000Vdc

Maximum DC power 560 kWpDC voltage range with MPPT 480∼880Vdc

Maximum current output 1167AStartup DC voltage 520Vdc

Rated power output 500 kWRated grid voltage 315Vac

Total harmonic distortion of current <2% (rated power output)DC component of current <0.5% (rated current output)

Filter parameters

Inductance value of 𝐿1 100 uHInductance value of 𝐿2 50 uHCapacitance values for 𝐶2 200 uF (Δ-connected)The resonant frequency of LCL 1125Hz

Other parametersCapacitance values for 𝐶1 25.12mFSwitch frequency of IGBT 3000HzSample frequency 10000Hz

distortion of 𝑒 will not affect input of 𝐼𝑔. Differential termand 2nd-order differential term in formula (64) will restrainhigh-frequency components, and proportional term affectslow-frequency components. For grid voltage drop especiallytransient one, the existence of differential variable will makefeedforward signal go to infinity and result in unstablesystem. At the moment of grid voltage drop, to restrain grid-connected current transient shock, protect safe operationof grid-connected device, and meet requirement of LVRT,simplify formula (64), ignore differential term and 2nd-orderdifferential term in feedforward coefficient, and then obtain

𝑘𝑒 =1

𝐺inv. (65)

The introduction of feedforward variables would play agood retraining role in transient current shock caused bygrid voltage drop. Simplify feedforward coefficient into aproportional link, and add a low-pass filter to collectionchannel, in order to eliminate the influence of high-frequencyvariables in feedforward signal 𝑒 on grid-connected outputcurrent waveform.

5. Simulation and Test Waveforms

In order to verify the proposed control strategy, this paperapplies the simulations and experimental tests to compareresponse performances by using various methods. The studyutilizes Matlab/Simulink for simulation analysis of controlstrategy and tests on experimental platform. Regarding setupof tests, the situational part is carried out under standard testcondition (STC), while the experimental part is carried outunder nominal operating cell temperature (NOCT). Table 2shows the differences of these two conditions. The exper-imental platform includes the PV power model, inverter,grid simulator, and switch gears. The PV model consists ofcombination of PV arrays.The rated power of inverter in thispaper is 500 kW. Table 1 shows the parameters of inverter.

2000

1500

1000

500

0

−500

−1000

−1500

−2000Grid current (A)

Grid

curr

ent (

A)

Grid voltage (V)

Grid

vol

tage

(V)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Grid voltage recovery pointGrid voltage dip point

Time (s)

Figure 21: Diagrams of grid current and voltage when grid drops to30%.

The function of grid simulator is to produce the various gridvoltage drops. Regarding the realization solutions of the gridvoltage drops, there are generally two methods: the first isreactor combination simulator and the second is convertersimulator. In contrast, the advantage of the first one is that thetest condition is more close to the real circumstance of gridvoltage drops. However, the cost of the first one is muchmoreexpensive than the second method. There is no need to usethe first solution if not to do certification of the inverter. Thispaper utilizes the second solution and takes the convertersimulator as the source of voltage drops.

Table 2 shows electric parameters of single-block solar cellmodules, and a photovoltaic array consists of 20 modules inseries.

To verify control effects of this algorithm under balancedgrid voltage drop and unbalanced grid voltage drop, set twocompared groups, respectively, and standard during LVRTwill follow Figure 1(b). In order to obverse the simulationwaveform conveniently, this paper sets 𝑡1 = 0.1 s as gridvoltage drop time point and sets 𝑡2 = 0.2 s as grid voltagerecover time point. Figure 21 shows three-phase voltage and


Table 2: Parameters of photovoltaic cells.

Parameters Description Unit Value

Standard test condition (STC): 1000W/m2 inirradiance, battery temperature of 25∘C, and airquality of AM1.5, according to EN 60904-3

𝑝max Peak power Watt 260Δ𝑃max Power tolerance Watt 0/5𝜂𝑚 Component efficiency Percentage 15.9𝑉𝑚𝑝𝑝 Peak power voltage Volt 30.9𝐼𝑚𝑝𝑝 Peak power current Ampere 8.41𝑉oc Open circuit voltage Volt 38.9𝐼sc Short circuit current Ampere 8.98

Nominal operating cell temperature (NOCT):800W/m2 in irradiance, ambient temperature of20∘C, and temperature in the open state of thecomponents under the conditions of 1m/s windspeed.

𝑃max Peak power Watt 188.3𝑉𝑚𝑝𝑝 Peak power voltage Volt 28.1𝐼𝑚𝑝𝑝 Peak power current Ampere 6.7𝑉oc Open circuit voltage Volt 35.9𝐼sc Short circuit current Ampere 7.27

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

500

1000

1500

2000

−500

−1000

1

2

Time (s)

Activ

e and

reac

tive c

urre

nt (A

)

(a) Single axis 𝑑𝑞 PI control method

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

500

1000

1500

2000

−500

1

2

Time (s)

Activ

e and

reac

tive c

urre

nt (A

)

(b) Dual 𝑑𝑞 rotating coordinate controller

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1

2

Time (s)

0

500

1000

1500

2000

−500

Activ

e and

reac

tive c

urre

nt (A

)

(c) Active damping compensation feedback control strategy

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1

2

Time (s)

0

500

1000

1500

2000

−500

Activ

e and

reac

tive c

urre

nt (A

)

(d) PR controller

Figure 22: Active current and reactive current waveform when grid voltage dropped at a single phase. Note: 1: reactive current/A; 2: activecurrent/A.

current when voltage drops to 5%Un under balanced voltagedrop condition. Before the voltage drops, grid current keepshigh sinusoidal to stabilize grid-connected operation. At thetime point of 𝑡1, there is no current shock which benefitsfrom 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 iseasy to find that the reactive current entered the stable statein a time cycle. As required in grid code, the value of reactivecurrent should be equal to the reactive current in 100%𝑃n innormal grid voltage state. After the grid voltage drop, negativesequence current after regulation is rapidly restrained down.


1

2

Activ

e pow

er (k

W)

Reac

tive p

ower

(kva

r)×105

8

6

4

2

0

−2

−40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (s)

1

2

(a) Single axis 𝑑𝑞 PI control method

1

2Activ

e pow

er (k

W)

Reac

tive p

ower

(kva

r)

×105

6

4

2

0

−2

−40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (s)

1

2

(b) Dual 𝑑𝑞 rotating coordinate controller

Activ

e pow

er (k

W)

Reac

tive p

ower

(kva

r)

×105

6

4

2

0

−2

−40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (s)

2

1

(c) Active damping compensation feedback control strategy

Activ

e pow

er (k

W)

Reac

tive p

ower

(kva

r)

×105

6

4

3

5

2

1

0

−2

−1

−30 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (s)

2

1

(d) PR controller

Figure 23: Active power and reactive power when grid dropped at a single phase. Note: 1: reactive power/kvar; 2: active power/kW.

The inverter recovers balanced three-phase current outputand certain active power output. The recovery time is lessthan 20ms.

As required in [6], when grid voltage drops to 5%𝑈n,photovoltaic inverter will output reactive current at 1.05times of rated current. In order to observe active andreactive current fluctuation during the interval of voltagedrop, the paper gives active and reactive current waveformsafter 𝑎𝑏𝑐/𝑑𝑞 conversion. Figure 22 shows the comparedwaveforms of active/reactive power, respectively, in the fourdifferent control strategies conditions. It is clear to findthat active current and reactive current have not occurredduring asymmetrical voltage drop. Furthermore, there is nodouble frequency fluctuation during the voltage fault-modeinterval. Besides, in order to decrease the shock of transientcurrent, this study adds the soft start solution in all comparedcontrol strategies. Regarding the response time of current,Figure 22(a) takes up nearly 55ms (2.75 time cycles) atthe beginning of voltage drop and 10ms (0.5 time cycles)at the recovery process. However, this method brings themost serious current shock when voltage begins to recoverat 𝑡2 time point. Figures 22(b) and 22(d) show a goodresponse performance at the two time points 𝑡1 and 𝑡2. Bothresponse times of current are less than 10ms (0.5 time cycles).

Figure 22(c) has a rapid response at the recovery time pointbut fails to regulate the value of reactive current of whichdemerit is similar to Figure 22(a).

As discussed above, if asymmetrical grid drop conditionoccurs (as shown in Figure 18, where only two phases B andC drop), positive and negative sequence decomposition ofgrid voltage conduct phase lock of grid voltage very wellwith rapid response. Figure 23 gives waves the active andreactive power under four various control strategies. It canbe concluded that the first three strategies result in powerfluctuation inevitably. Figure 22(d) shows clearly that bothactive current and reactive current during LVRT intervalhave not doubled frequency fluctuation. So, the active powerand reactive power in Figure 23(d) are quite stable. As it isan algorithm simulation, the requirement that grid voltageresponse speed will be within 30ms in literature [6] cannotbe truly reflected in simulation model.

Figure 24 shows experimental waveforms under 500 kWfull power. Figure 24(a) shows triphase current under sym-metrical voltage drop circumstances. When the triphasevoltage decreases to 5%𝑈n, the triphase output current is reg-ulated into a new value rapidly to response the grid changes.When the triphase voltage recovers to the normal degree, theoutput triphase current is regulated into the formal value.


Grid voltage (V)

Grid voltage (V)

Grid current (A)

Grid current (A)

Grid voltage dropped point Grid voltage recovered point

(a) Current output when grid dropped at 𝑠 single-phase

Grid voltage (V)

Grid voltage (V)

Grid current (A)

Grid current (A)

Grid voltage dropped point Grid voltage recovered point

(b) Current output when grid dropped at three-phase

Figure 24: Grid current and grid voltage.


It can be judged that the inverter has ridden through thelow-voltage state successfully. Figure 24(b) shows triphasecurrent under asymmetrical voltage drop circumstances.Thevoltage of A phase drops to 30%𝑈n, while the two otherphases are stable. We can find that the total Thd of thetriphase current is less than 15% and recover time is lessthan 30ms, which proves the effectiveness of the proposedstrategy.

6. Conclusion

This paper proposes and analyzes the novel mathematicalmodels of photovoltaic inverter under balanced and unbal-anced grid.We focus on photovoltaic inverter control strategyto address the complex issues under unbalanced condition. Asystematic framework based on PR controller is proposed toenable the grid code control of photovoltaic inverter underLVRT condition. With the PR based control strategy, thesystem can eliminate negative sequence components of gridcurrent under unbalanced grid voltage and improve doubledsequence fluctuation of output current and power. The gen-eral performance of the proposed control method is superiorto the existing representative control approaches. In phaselock link of grid voltage, the sliding Goertzel filter is adoptedto purify the disturbance generated by negative sequence andharmonic components in three-phase grid. The system hasachieved very good performance in lock phase of positivesequence fundamental voltage. In order to eliminate shakingof grid voltage and asymmetrical influence, grid-side voltagefeedback link is added to the front end of output modulationwave. It is found that the introduction of feedforward variablewould play a very good restraining role in transient currentshock caused by grid voltage drop. In the implemented testplatform, a simplified feedforward coefficient is embeddedinto a proportional link, and a low-pass filter is inserted intothe collection channel in order to eliminate effectively theinfluence of high-frequency variables in voltage feedforwardsignal on grid-connected output current waveform. Theproposed mathematical models and principles are verifiedusing intensive simulations and experiments.The results haveindicated that grid-side current could fully track the positivesequence components of grid voltage under unbalanced gridvoltage. Moreover, the system could sustain a stable andbalanced output of inverter under both restraining negativesequence current and reactive power fluctuation on gridside.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors would like to thank the University of Macau forits funding support under Grants MYRG079(Y1-L2)-FST13-YZX and MYRG2015-00077-FST.

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