- Email: [email protected]

ScienceDirect

IFAC PapersOnLine 52-4 (2019) 188–193

Impedance Specification and Stability Analysis for the AC Grid-Converter Impedance Specification and Stability Analysis for the AC Grid-Converter Impedance Stability Analysis Systemand in Modified Impedance Specification Specification Stability Sequence-Domain Analysis for for the the AC AC Grid-Converter Grid-Converter Systemand in Modified Sequence-Domain System in Modified Sequence-Domain System in Haitao Modified Sequence-Domain Yi Zhou* Hu* Pengyu Pan* Zhengyou He*

Yi Zhou* Haitao Hu* Pengyu Pan* Zhengyou He* Yi Zhou* Haitao Hu* Pengyu Pan* Yi Zhou* Haitao Hu* Pengyu Pan* Zhengyou Zhengyou He* He* * School of Electrical Engineering, Southwest Jiaotong University, Southwest Jiaotong University, * School of Electrical Engineering, * School ofChengdu, ElectricalChina. Engineering, Southwest Jiaotong University, (e-mail:[email protected]) * School ofChengdu, ElectricalChina. Engineering, Southwest Jiaotong University, (e-mail:[email protected]) Chengdu, China. (e-mail:[email protected]) Chengdu, China. (e-mail:[email protected]) Abstract: In the ac grid-converter system, the mismatching between the load-impedance and sourceAbstract: In thecaused ac grid-converter system, the mismatching between the load-impedance sourceimpedance have many instability issues, such as low-frequency oscillation, harmonicand instability Abstract: In sourcethe ac grid-converter system, the mismatching between the load-impedance and impedance have caused many instability issues, such as low-frequency oscillation, harmonic instability Abstract: In the ac grid-converter system, the mismatching between the load-impedance and sourceand sub-synchronous oscillation. Therefore, the individual impedance specification ofharmonic every subsystem is impedance have caused many instability issues, such as low-frequency oscillation, instability and sub-synchronous oscillation. Therefore, the individual impedance specification of every subsystem is impedance have caused many instability issues, such as low-frequency oscillation, harmonic instability very important to avoid impedance mismatching. In this paper, a typical impedance model and of the and sub-synchronous oscillation. Therefore, the individual specification of every subsystem is very important avoid impedance In this impedance paper, a typical impedance model and of the and sub-synchronous oscillation. the individual impedance specification of every subsystem is single-phase ac to grid-converter is Therefore, builtmismatching. in modified sequence-domain. And then, based on the forbiddenvery important to avoid impedance mismatching. In this paper, a typical impedance model and of the single-phase ac grid-converter is built in modified sequence-domain. And then, based on the forbiddenvery important to avoid impedance mismatching. In this paper, a typical impedance model and of the region-based criterion (FRBC),is abuilt novel impedance specification ofAnd the then, ac grid-converter system is single-phase ac grid-converter in modified sequence-domain. based on the forbiddenregion-based criterion (FRBC), novel impedance specification the ac the grid-converter system is single-phase ac grid-converter is abuilt in modified And then, based on the impedance forbiddenproposed, which provides a sufficient condition forsequence-domain. system stability of and guide individual region-based criterion (FRBC), a novel impedance specification of the ac grid-converter system is proposed, which provides a sufficient condition for system stability and guide the individual impedance region-based (FRBC), a novel impedance specification of the specification ac grid-converter system is design of which the criterion source and load. Compared withforthesystem existing impedance of the ac gridproposed, provides a sufficient condition stability and guide the individual impedance design of system, the source and aload. Compared withforthesystem existing impedance specification of the ac gridproposed, which provides sufficient condition stability andhas guide the individual impedance converter the proposed impedance specification is simpler and less conservatism. Simulation design of system, the source and load.impedance Comparedspecification with the existing impedance specification of the ac gridconverter the proposed is simpler and has less conservatism. Simulation design of the the source and load.and Compared with the existing impedance specification of ac gridresults verify effectiveness the correctness of proposed impedance specification for the the ac gridconverter system, the proposed impedance specification is simpler and has less conservatism. Simulation results verify the effectiveness and the correctness of proposed impedance specification for the ac gridconverter system, the proposed impedance specification is simpler and has less conservatism. Simulation converter system. results verify the effectiveness and the correctness of proposed impedance specification for the ac gridconverter system. results verify the effectiveness and the correctness of proposed impedance specification for the ac gridconverter system. © 2019, IFAC (International Federation ac of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Impedance specification, grid-converter system, stability analysis, modified sequenceconverter system. Keywords: Impedance specification, ac grid-converter system, stability analysis, modified sequencedomain, dq-domain. Keywords: Impedance specification, ac grid-converter system, stability analysis, modified sequencedomain, dq-domain. Keywords: Impedance specification, ac grid-converter system, stability analysis, modified sequencedomain, dq-domain. domain, dq-domain. grid-converter system cannot directly apply in the ac grid 1. INTRODUCTION grid-converter system cannot directly apply in the ac grid converter system. At present, some impedance specifications 1. INTRODUCTION grid-converter system cannot directly apply in the gridconverter system. At present, some impedance grid-converter system cannot directly apply inspecifications the ac ac grid1. INTRODUCTION of the ac grid-converter system have been proposed, such as Recent years, the instability issues have occurred in many dc of converter system. At present, some impedance specifications 1. INTRODUCTION the ac grid-converter systemsome haveimpedance been proposed, such as system. At present, specifications Recent years, the instability issues have occurred in many dc converter d-channel criterion (see Wen et al. (2017)), singular value and ac years, grid-converter systems the mismatching of the ac grid-converter been proposed, as Recent the instability issues due have to occurred in many dc d-channel criterion (see system Wen ethave al. (2017)), singularsuch value of the ac (see grid-converter system been proposed, such as and ac years, grid-converter systems to the mismatching Recent the instability issues have occurred in many criterion Chandrasekaran ethave al. (1999)), norm criterion between the source-impedance anddue load-impedance, such dc as criterion d-channel criterion (see Wen et al. (2017)), norm singular value and ac grid-converter systems due to the mismatching (see Chandrasekaran et al. (1999)), criterion criterion (see Wen (2017)), singular value between the source-impedance anddue load-impedance, such as d-channel and ac grid-converter systems to the mismatching (see Liu et al. (2013)). The d-channel criterion recommends low-frequency oscillation (see Hu et al. (2018)), harmonic criterion (see et al. (1999)), criterion between the source-impedance load-impedance, such as (see Liu et al. Chandrasekaran (2013)). The d-channel criterionnorm recommends (see Chandrasekaran etof al. norm criterion low-frequency oscillation (see and Hu et al. (2018)), harmonic between the(see source-impedance and load-impedance, such as criterion that the d-d channel impedance the(1999)), source modular should instability Wang et al. (2018)) and sub-synchronous (see Liu et al. (2013)). The d-channel criterion recommends low-frequency Hu et al. harmonic that the channel impedance of the source modular should (see Liud-d et al. (2013)). Thechannel d-channel criterion recommends instability (see oscillation Wang et (see al. (2018)) and(2018)), sub-synchronous low-frequency oscillation (see Hu et al. (2018)), harmonic be smaller than the d-d impedance of the load oscillation (see Shu et al. (2018)). Therefore, the impedance that the d-d channel impedance of the source modular instability (see (see Shu Wang et (2018)). al. (2018)) and sub-synchronous smaller than theimpedance d-d channel impedance of theshould load that the d-d channel of the source modular should oscillation et al. Therefore, the impedance be instability Wang et al. (2018)) and sub-synchronous modular. However, it neglects the impact of the off-diagonal specification plays an important role to avoid impedance be smaller than the d-d channel impedance the load However, it neglects the impact of the of off-diagonal oscillation (see Shu et al. (2018)). Therefore, the impedance modular. be smaller d-d channel impedance of the load specification plays an important role to avoid oscillation (see Shu et the al. (2018)).stable. Therefore, the impedance elements in than the the impedance ratio matrix on the stability mismatching and keep system modular. However, it neglects the impact of the off-diagonal specification plays an important role to avoid impedance elements in the impedance ratio matrix on the stability modular. However, it neglects the impact of the off-diagonal mismatching and keep the system stable. specification plays an important role to avoid impedance analysis, which may obtain the incorrect analysis result. The elements which in themay impedance matrix on the stability mismatching and keep the system stable. obtain theratio incorrect analysis result. The elements in the impedance ratio matrix onlimited the stability Both the dc and grid-converter systems can be equivalent analysis, mismatching andac keep the system stable. singular value criterion and norm criterion are by the analysis, value which may obtain the incorrect result. Both the dc and ac grid-converter systems can be equivalent singular criterion and norm criterionanalysis are limited byThe the which may s-plane, obtain thewhich incorrect analysis result. The to thethesource-load model shown systems in Fig.1. Therefore, the analysis, unit circle in criterion the makes the impedance Both dc and ac grid-converter can be equivalent singular value and norm criterion are limited by the to thethesource-load model shown systems in Fig.1. the unit circle in criterion the s-plane, which makesarethe impedance Both dc and ac grid-converter canTherefore, be equivalent singular value and norm criterion limited by the stability can be determined by the impedance ratio criterion specification more conservatism. In addition, the calculation to the source-load model shown in Fig.1. Therefore, the unit circle in the s-plane, which makes the impedance stability can be determined by the impedance ratio criterion In addition, the calculation to the source-load model shown in Fig.1. Therefore, the specification inmore the conservatism. s-plane, which makes the (see Middlebrook (1976)). of thecircle singular value and norm are complex. In impedance recently, a stability can be determined the impedance ratio criterion unit specification more conservatism. In addition, the (see Middlebrook (1976)). by the singular value and norm are complex. In calculation recently, a stability can be determined by the impedance ratio criterion of specification more conservatism. In addition, the calculation forbidden-region-based criterion (FRBC) in dq-domain (see (see Middlebrook (1976)). of the singular value and norm are complex. In recently, forbidden-region-based criterion (FRBC) in dq-domain (seeaa iout (see Middlebrook (1976)). i pcc of theetsingular value and norm are complex. In recently, iin Liao al. (2017)) proposes sufficient conditions of one ac iout forbidden-region-based criterion (FRBC) in dq-domain i pcc iin Liao et al. (2017)) proposes sufficient conditions of one(see ac forbidden-region-based criterion (FRBC) in dq-domain (see i i grid-converter system. However, the FRBC only provides out pcc iinin Source Load Liao et al. (2017)) proposes sufficient conditions one ac iuout u pcci pcc out uin iin pcc grid-converter system. However, the FRBC onlyof provides out Liao et al. conditions (2017)) proposes conditions ofelements one ac Source Load constraint forHowever, the sufficient real-part of diagonal u pcc uout uin Modular Modular grid-converter system. the FRBC only provides Source Load constraint conditions for the real-part of diagonal elements u grid-converter system. However, the FRBC only provides u uinin Modular Modular Source Load and magnitude of the off-diagonal elements in the impedance pcc out u pcc constraint conditions for the real-part of diagonal elements uout uin pcc out Modular and magnitude of the off-diagonal elements in the impedance constraint conditions for the give real-part of diagonal elements Z l ( s ) Modular Modular Z g ( s) Modular ratio matrix, which cannot the individual impedance and magnitude of the off-diagonal elements in the impedance impedance Z g (s) Z l (s) ratio matrix, which cannot give the individual and magnitude of the off-diagonal elements in the specification every subsystem. Z gg ( s ) Z ll ( s ) ratio matrix, ofwhich give the individual impedance Z g (s) Z l (s) specification every cannot subsystem. ratio matrix, ofwhich cannot give the individual impedance specification of every subsystem. Fig. 1. Source-load equivalent model. In this paper,ofthe FRBC is derived in modified sequencespecification every subsystem. Fig. 1. Source-load equivalent model. In this paper, is derivedofin the modified domain. Due the to FRBC the symmetry ideal sequencegrid, the Fig. 1. Source-load equivalent model. In this paper, the FRBC is derivedofinthe modified sequenceFig. 1. Source-load equivalent model. domain. Due to the symmetry ideal grid, the In this paper, the FRBC is derived in modified According to the impedance ratio criterion, a sufficient impedance matrix of the grid becomes one diagonalsequencematrix in domain. Due to of the ideal grid, According to the impedance ratio criterion, a sufficient impedance matrix the symmetry grid becomes one matrixthe in Due to ofthe the symmetry ofbrings thediagonal ideal grid, the condition oftoone stable dc grid-converter systema issufficient that the domain. modified sequence-domain, which convenience to According the impedance ratio criterion, impedancesequence-domain, matrix of the grid becomes one diagonal matrix to in condition dc grid-converter systema issufficient that the modified Accordingofof toone thestable impedance ratio iscriterion, which brings convenience impedance matrix of the grid becomes one diagonal matrix in magnitude the source-impedance always smaller than make deconstruction of the impedance ratio matrix for condition of one stable dc grid-converter system is that the modified sequence-domain, which brings convenience to magnitude of the source-impedance is always smaller than condition of one stable(see dc grid-converter system isHowever, that the make deconstruction of the impedance ratio matrix for modified sequence-domain, which brings convenience to that of load-impedance Wildrick et al. (1995)). the relationship between the source-impedance and magnitude of the source-impedance is al. always smaller than getting make deconstruction the impedance ratio matrix and for that of load-impedance (see Wildrick et (1995)). However, magnitude the source-impedance isa always than getting the relationshipof the source-impedance make deconstruction ofbetween the impedance ratio matrix for since the acofgrid-converter system iset kind of smaller theHowever, multiple load-impedance in the ac grid-converter system. that of load-impedance (see Wildrick al. (1995)). getting the relationship between the source-impedance and since the ac grid-converter system iseta al. kind of theHowever, multiple load-impedance that of load-impedance (see Wildrick (1995)). in the ac grid-converter system. between the source-impedance and input and multiple output (MIMO) cascade system and its getting the relationship since the grid-converter system aa kind the in the ac grid-converter system. input and ac multiple output (MIMO) system and its load-impedance since the ac grid-converter system is iscascade kind of of the multiple multiple Based on the FRBC sequence-domain, a novel load-impedance in thein acmodified grid-converter system. impedance ratio matrix is normally a non-diagonal matrix input and output system its on the FRBC in modified sequence-domain, a novel impedance ratio matrix is (MIMO) normally cascade a non-diagonal matrix input and etmultiple multiple output (MIMO) cascade systemofand and its Based impedance specification of ac sequence-domain, grid-converter system is (see Wen al. (2016)), the impedance specification the dc Based on the FRBC in modified aa novel impedance ratio matrix is normally a non-diagonal matrix of ac sequence-domain, grid-converter system is (see Wen et ratio al. (2016)), the dc impedance Based on thespecification FRBC in modified novel impedance matrix the is impedance normally aspecification non-diagonalof matrix specification of ac grid-converter system is (see Wen et al. (2016)), the impedance specification of the dc impedance (see Wen et al. (2016)), the impedance specification of the dc impedance specification of ac grid-converter system is

2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2019 IFAC 205 Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 205Control. 10.1016/j.ifacol.2019.08.177 Copyright © 2019 IFAC 205 Copyright © 2019 IFAC 205

2019 IFAC CSGRES Jeju, Korea, June 10-12, 2019

Yi Zhou et al. / IFAC PapersOnLine 52-4 (2019) 188–193

proposed. As the FRBC breaks the limitation of the unit circle in s-plane, the proposed impedance specification in modified sequence-domain has less conservatism compared with singular value criterion and norm criterion. Compared with d-channel criterion, as this specification considers the impact of the off-diagonal elements in the impedance ratio matrix, it is more accurate in stability analysis. Moreover, as the proposed impedance specification has no complex calculation, it provides a simpler sufficient condition for system stability and guide the individual impedance design of the source and load. The simulation results verify the effectiveness and the correctness of proposed impedance specification for the ac grid-converter system.

2.1 System Modeling in Modified Sequence-Domain The single-phase ac system studied in this paper is shown in Fig. 2. The impedance of the grid is simplified as a series circuit of resistor and inductor. In dq-frame, the impedance matrix of the grid in Fig. 2 can be expressed as: 0 Ls . (1) Rs sLs

The controller of the converter includes phase locked loop (PLL), current controller and dc-link voltage controller. The current controller is realized in dq-frame with the PI controller. The point of common coupling (PCC) voltage space vector and current space vector are us= [ud uq]T and is=[id iq]T, respectively. The modulation voltage space vector is uab= [uabd uabq]T. The main circuit parameters are listed in Table 1. Rs us

is

Ls

uab

Grid PLL

us

udc

Cd

dq

ud uq

K ipll

Voltage loop

H v ( s ) K pv

udc

T /4

id

dq ref q

i Current loop

Hc (s) K pc

udcref

is

K iv s

1

iq

Kic s

K Hc (s) K pc ic s

i 0 i G ( s) uds ids s s i 0 i G ( s) uq iq

ud

RMS of ac voltage

(4)

c s 0 uabdref uabdref 0 uabqref uds G pll ( s) c s s (5) 0 uabqref uabqref 0 uabdref G pll ( s) uq

dq

s

G dpll ( s )

1

where, the superscript ‘0’ means the steady state value.

Modulation

In the controller dq-frame, the reference signal of the modulation voltage in Fig.2 can be expressed as: c uabdref udc idref idc 0 idc 0 L c c H i ( s ) ref c c uabqref uq iq iq 0 L 0 iq

Table 1. Basic Parameters of the Single-Phase AC GridConverter System

us

(3)

G ipll ( s ) ref uab

uq

0 q pll 0 d pll

c d c q

PWM

0 L 0 L

Description

(2)

0 udc uds 1 c s 0 uq 0 1 ud G pll ( s) uq

RL

Fig. 2. Single-phase ac grid-converter system.

Symbol

uqs .

G upll ( s )

1

s

s u0 H pll ( s )

The relationships of the voltage space vector, the current space vector and the modulation voltage space vector between the system dq-frame and the controller dq-frame are expressed respectively,

0 H pll ( s ) K ppll

H pll ( s) G pll ( s )

Converter

T /4

=

L

R

Fundamental frequency 50Hz Input inductor of rectifier 10mH Dc-link capacitance 0.009F Load resistor 1000 0 Steady-state dc-link voltage 3600V u dc Steady-state d-axis voltage 2503V u d0 0 Steady-state d-axis current 10.4A id fs Switching frequency 250Hz Proportional parameters of the dc-link 0.3 Kpv voltage controller Integral parameter of the dc-link Kiv 1 voltage controller Proportional parameters of the current 1 Kpc controller Integral parameter of the current Kic 0.1 controller Ls Equivalent inductor of the grid 2.5mH Due to the PLL, the converter system has two dq-frames: one is the system dq-frame (superscript ‘s’), and another is the controller dq-frame (superscript ‘c’) (see Wen et al. (2016)). From Fig. 2, the transfer function of the PLL is derived as: f0 L Cd RL

2. SYSTEM MODELING AND STABILITY CRITERION

Rs sLs Z dq g (s) 0 Ls

189

G L

Simulation Parameters 1770V

(6) H c (s)

where, H i ( s ) 0 206

0

.

H c ( s )

2019 IFAC CSGRES 190 Jeju, Korea, June 10-12, 2019

Yi Zhou et al. / IFAC PapersOnLine 52-4 (2019) 188–193

Also, in the system dq-frame, the modulation voltage can be expressed as:

Y l (s) Ypnl (s) (14) Yl pn (s) AZ Yl pn (s) AZ1 ppl l Ynp (s) Ynn (s)

s uabd uds R sL 0 L ids (7) s s s . uabq uq 0 L R sL iq

where AZ

1 j 1 1 1 1 1 * 1 j , AZ AZ . 2 2 j j

G rl ( s )

Due to the PLL and dc-link voltage controller, the admittance matrix of the converter is asymmetrical in dq-domain. Therefore, in modified sequence-domain, the admittance matrix of the converter includes the positive sequence admittance Yppl , the negative sequence admittance Ynnl , the

Due to the delay of the digital control and pulse width modulator (PWM) (see Wen et al. (2016)), the relationship between the reference signal of the modulation voltage and the actual modulation voltage is expressed as: s s uabd uabdref s Gd ( s) s uabq uabqref

positive-negative sequence coupled admittance Ypnl and the

(8)

negative-positive sequence coupled admittance Ynpl . The

where Gd (s) eTd s , Td is the delay time of PWM. It can be then equivalent as a one-order inertial link: 1 Gd ( s) . (9) Td s 1

impedance ratio matrix in modified sequence-domain is expressed as:

In Fig.2, the reference signal of the d-axis current is expressed as:

where

idref = H v ( s )udc .

L ( s) Lpn (s) L pn (s) Z gpn ( s) Yl pn ( s) pp (15) Lnp (s) Lnn (s) g g Lpp ( s) Z pp (s)Yppl (s) , Lpn (s) Z pp (s)Ypnl (s ) . g l g l Lnp (s) Z nn (s)Ynp (s) , Lnn (s) Z nn (s )Ynn (s )

(10)

According to the principle of power conservation, and neglecting the power loss of the switches of the rectifier, the dc-link voltage disturbance can be transformed to ac side in the system dq-frame as:

i s u s udc G1 (s) G2 (s) ds G3 (s) G4 (s) ds iq uq G (s) G (s) i

2.2 FRBC Based on Impedance Ratio Matrix in Modified Sequence-Domain The FRBC is based on the eigenvalue estimation of the impedance ratio matrix in modified sequence-domain by using Gerschgorin Disc (GD). One eigenvalue must locate in corresponding GD, which satisfies:

(11)

v

where G1 =

0.5ud0 0 sCd udc 2udc0 / RL

0.5id0 G3 0 sCd udc 2udc0 / RL

G2 G4

0.5uq0 sCd udc0 2udc0 / RL 0.5iq0

1 ( s) Lpp ( s) Lpn ( s) and 2 ( s) Lnn ( s) Lnp ( s) . (16) In order to guarantee all of eigenvalues do not encircle the point (-1,0) in the s plane, a forbidden region is proposed in the s plane (see Liao et al. (2017)), which is shown in Fig. 3.

.

sCd udc0 2udc0 / RL

Im

Therefore, the admittance model of the converter in dqdomain can be expressed as:

Forbidden Region

Yldq G rl Gd H v Hi G i Gd (Hi G wL ) 1

1

L pp

L pn . (12) G dpll G upll H v Hi G v (Hi G wL )G ipll -1 0 Re Lnp In (1) and (12), the impedance matrix of the grid and the admittance matrix of the converter are non-diagonal matrixes. Lnn 2 It is very difficult to make deconstruction of the elements in their product matrix for getting the relationship between the source-impedance and load-impedance. Therefore, the impedance matrixes in dq-domain can be transformed to modified sequence-domain by a linear transformation as Fig. 3. Forbidden region and GDs. follows (see Rygg et al. (2016)): According to the geometric relation between the forbidden g Z pp (s) 0 region and GDs in Fig. 3, the FRBC in modified sequence1 pn pn Z g ( s) AZ Z g ( s) AZ (13) domain is expressed as: Z nng ( s) 0

I G

d

207

2019 IFAC CSGRES Jeju, Korea, June 10-12, 2019

Yi Zhou et al. / IFAC PapersOnLine 52-4 (2019) 188–193

191

Remark 2：if the grid and converter are symmetrical, (23) Re Lpp (s) Lpn (s) 1 . (17) can be rewritten as Yppl ( j ) Yppg ( j ) . It is similar with the Re Lnn (s) Lnp (s) 1 impedance specification of the single input and single output According to the mirror frequency symmetry (see Zhou et al. (SISO) system. (2018)), let s j , in positive and negative frequency, the Remark 3：if a gain margin M is considered, (23) can be FRBC in modified sequence-domain is rewritten as: expressed as: Re L pp ( j ) (18) L pn ( j ) 1 . Yppl ( j ) Ypnl ( j ) M Yppg ( j ) . (24) 3. PROPOSED IMPEDANCE SPECIFICATION 3.2 Conservatism Comparison 3.1 The Impedance Specification in Modified SequenceDomain

The conservatism of existing impedance specifications can be compared by the stability margin defined in Table 2 (see Liu et al. (2015)).

According to (18), the element Lpp can be written as: g L pp ( j ) Z pp ( j ) Yppl ( j )

Table 2. Definitions of Stability Margin for Impedance Specifications

(19)

g Z pp ( j ) Yppl ( j ) ( j ) ( j )

Impedance specifications

where, ( j ) and ( j ) are phases of source-impedance and load-admittance, respectively.

Stability margins

D-channel criterion Therefore, the real part of Lpp can be expressed as: g l Re Lpp ( j ) Z pp ( j ) Ypp ( j ) cos ( j ) ( j ) . (20)

g Z pp ( j ) Yppl ( j ) cos ( j ) ( j ) g Z pp ( j ) Ypnl ( j ) 1

.

(21)

Re Y ( j ) cos ( j ) 1

.

g l l Z pp ( j ) Y pp ( j ) Ypn ( j ) 1

For the single-phase ac grid-converter system shown in Fig.2, with increasing the inductor Ls, the stability margins of each impedance specification are shown in Fig.4.

(22)

Z ( j ) g pp

100 D-channel criterion

If the minimum value of the left side is larger than the right side in (22) under any frequency, the inequality (18) always holds in whole frequency range and the system is stable.

Singular-value criterion Infinity norm criterion Proposed impedance specification

Stability Margin/dB

80

Obviously, the minimum value of the left side in (22) is Yppl ( j ) . Therefore, the impedance specification in modified sequence-domain is expressed as: Yppl ( j ) Ypnl ( j ) Yppg ( j ) .

1 20 log g l max Z dd ( j ) Ydd ( j )

Proposed impedance 20 log specification max

l pp

Infinity-norm criterion

Hence, the relationship between the source-impedance and load-impedance can be expressed as:

l Im Yppl ( j ) sin ( j ) Ypn ( j )

1 20 log g l max Z dd ( j ) Ydd ( j )

Singular value criterion

Substituting (20) into (18), yields,

1 20 log g l ( j ) max Z dd ( j ) Ydd

60 40

20 0

(23) -20 0

Remark 1：in modified sequence-domain, if the magnitude of the sum of the positive sequence admittance and the positive-negative coupled admittance for the converter is smaller than the magnitude of the positive admittance for the grid in whole frequency range, the ac grid-converter system is always stable. However, (13) is a symmetrical matrix only when the grid is symmetrical. Hence, the precondition of this impedance specification is that the grid is symmetrical.

1

2

3

4

5

Ls/mH

Fig. 4. Variation of stability margins of different impedance specifications with the increasing inductor Ls. From Fig.4, the d-channel criterion has the least conservatism. However, the d-channel criterion neglects the impact of the non-diagonal elements in the impedance ratio matrix on 208

2019 IFAC CSGRES 192 Jeju, Korea, June 10-12, 2019

Yi Zhou et al. / IFAC PapersOnLine 52-4 (2019) 188–193

stability analysis, which may obtain the incorrect analysis result. Moreover, the stability margin of the proposed impedance specification is larger than that of the infinity norm criterion in all conditions. Compared with the singular value criterion, the stability margins of the proposed impedance specification are close to that of the singular value criterion. However, as the proposed impedance specification avoids the complex calculation of singular values, it is simpler. Therefore, the proposed impedance specification is simpler and has less conservatism.

Yppg (Ls=2.5mH) Magnitude/dB

Yppg (Ls=10mH) Yppl Ypnl

Fig. 5. Bode plots with increasing inductor Ls. 4. SIMULATION VERIFICATION In order to verify the correctness and effectiveness of the proposed impedance specification, the single-phase ac gridsystem shown in Fig.2 is built in the Matlab/Simulink. In addition, some cases are set to discuss the impact of parameters on the stability of the ac grid-converter system by using proposed impedance specification. Three different cases are listed in Table 3.

Ls=2.5mH

udc/kV

us/kV is/kA

Ls=10mH

us

is

udc

Table 3. Cases with Different Parameters Case No. 1 2 3

Time/s

Parameters Change Increasing Ls from 2.5 mH to 10mH Increasing Kpv from 0.3 to 1 Decreasing Kpc from 1 to 0.5

Fig. 6. Simulation waveform when Ls is increased from 2.5mH to 10mH.

Yppg (Ls=2.5mH) Yppl Ypnl (Kpv=0.3)

Magnitude/dB

The Bode plots of case 1 are shown in Fig. 5. The magnitude of Yppl Ypnl exceeds the magnitude of Y ppg when Ls is increased from 2.5mH to 10mH. The simulation waveform shown in Fig. 6 shows the oscillatory voltage and current under Ls=10mH.

Yppl Ypnl (Kpv=1)

In case 2, the Bode plots shown in Fig. 7 indicates that increasing Kpv will increase the magnitude of Yppl Ypnl .

Frequency/Hz

The magnitude of Yppl Ypnl exceeds the magnitude of Fig. 7. Bode plots with increasing Kpv.

Y ppg when Kpv is 1 and the system is determined as an

unstable case. However, corresponding simulation waveform shown in Fig. 8 is a stable case. This is because the proposed impedance specification is a sufficient condition for system stability.

Kpv=0.3

0.5, the magnitude of Y Y

g pp

Y

l pn

udc/kV

us/kV is/kA

In case 3, Fig. 9 shows that decreasing Kpc will increase the magnitude of Yppl Ypnl . When Kpc is decreased from 1 to l pp

Kpv=1

us

is

udc

exceeds the magnitude of Time/s

. In Fig. 10, Simulation waveform shows that the system

is unstable when Kpc is decreased from 1 to 0.5.

Fig. 8. Simulation waveform when Kpv is increased from 0.3 to 1.

All cases verify that the correctness and effectiveness of the proposed impedance specification. In addition, based on the proposed impedance specification, the analysis results also indicate that increasing Kpv and decreasing Kpc will increase the magnitude of the sum of the positive sequence admittance and positive-negative coupled sequence admittance for the rectifier, which increases the risk of system instability.

209

2019 IFAC CSGRES Jeju, Korea, June 10-12, 2019

Yi Zhou et al. / IFAC PapersOnLine 52-4 (2019) 188–193

Yppg (Ls=2.5mH) Magnitude/dB

Yppl Ypnl (Kpc=1) Yppl Ypnl (Kpc=0.5)

Frequency/Hz

Fig. 9. Bode plots with decreasing Kpc.

udc/kV

us/kV is/kA

Kpc=1

us

is

udc

Kpc=0.5

Time/s

Fig. 10. Simulation waveform when Kpc is decreased from 1 to 0.5. 5. CONCLUSION A novel impedance specification for ac grid-converter system is proposed in this paper. In modified sequence-domain, if the magnitude of the sum of the positive sequence admittance and the positive-negative sequence coupled admittance for the converter is smaller than the magnitude of the positive sequence admittance for the grid in whole frequency range, the ac grid-converter system is always stable. The proposed impedance specification provides a sufficient condition for stability of the ac grid-convert system, which guides the individual impedance design when load or source impedance is known. Moreover, compared with existing impedance specifications of the ac grid-converter system, proposed impedance specification is simpler and has less conservatism. REFERENCES Chandrasekaran, D., Borojevic, S. and Lindner, D.K. (1999). Input filter interaction in three phase AC–DC converters. In IEEE Power Electronics Specialists Conference, volume 2, 987–992. Hu, H., Tao, H., Blaabjerg, F., Wang, X., He, Z., and Gao, S. (2018). Train-Network Interactions and Stability Evaluation in High-Speed Railways--Part I: Phenomena and Modeling. IEEE Transactions on Power Electronics, 33(6), 4627–4642. Hu, H., Tao, H., Blaabjerg, F., Wang, X., He, Z., and Gao, S. (2018). Train-Network Interactions and Stability Evaluation in High-Speed Railways--Part II: Influential Factors and Verifications. IEEE Transactions on Power Electronics, 33(6), 4643–4659. 210

193

Liao, Y., Liu, Z., Zhang, G., and Xiang, C. (2017). Vehiclegrid system modeling and stability analysis with forbidden region-based criterion. IEEE Transactions on Power Electronics, 32(5), 3499-3512. Liu, F., Liu, J., Zhang, H., Xue, D. Hasan, S. U. and Zhou, L. (2013). Modified norm type stability criterion for cascade AC system. In IEEE Energy Conversion Congress and Exposition, 442–447. Liu, Z., Liu, J., Bao, W., and Zhao, Y. (2015). Infinity-norm of impedance-based stability criterion for three-phase ac distributed power systems with constant power loads. IEEE Transactions on Power Electronics, 30(6), 30303043. Middlebrook, R.D. (1976). Input filter considerations in design and application of switching regulators. In IEEE Industry Applications Society Conference, 94–107. Rygg, A., Molinas, M., Chen, Z., and Cai, X. (2016). A modified sequence domain impedance definition and its equivalence to the dq-domain impedance definition for the stability analysis of ac power electronic systems. IEEE Journal of Emerging & Selected Topics in Power Electronics, 4(4), 1383-1396. Shu, D., Xie, X., Rao, H., Gao, X., Jiang, Q., and Huang, Y. (2018). Sub- and super- synchronous interactions between statcoms and weak ac/dc transmissions with series compensations. IEEE Transactions on Power Electronics, 33(9), 7424-7437. Wang, X., and Blaabjerg, F. (2018). Harmonic stability in power electronic based power systems: concept, modeling, and analysis. IEEE Transactions on Smart Grid, to be published, DOI: 10.1109/TSG.2018.2812712. Wen, B., Boroyevich, D., Burgos, R., Mattavelli, P., and Shen, Z. (2015). Analysis of d-q small-signal impedance of grid-tied inverters. IEEE Transactions on Power Electronics, 31(1), 675-687. Wen, B., Burgos, R., Boroyevich, D., Mattavelli, P., and Shen, Z. (2017). Ac stability analysis and dq frame impedance specifications in power electronics based distributed power systems. IEEE Journal of Emerging & Selected Topics in Power Electronics, 5(4), 1455-1465. Wildrick, C. M., Lee, F. C., Cho, B. H., and Choi, B. (1995). A method of defining the load impedance specification for a stable distributed power system. IEEE Transactions on Power Electronics, 10(3), 280-285. Zhou, Y. Hu, H. Yang, J. and He, Z. (2018). A Novel Forbidden-Region-Based Stability Criterion in Modified Sequence-Domain for AC Grid-Converter System. IEEE Transactions on Power Electronics, to be published. DOI: 10.1109/TPEL.2018.2869422