Stability of Power Electronic Systems Part 2 S.D. Sudhoff Fall 2010
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Stability of Power Electronic Systems
Part 2
S.D. Sudhoff
Fall 2010
DC Stability Toolbox
• Available at
https://engineering.purdue.edu/ECE/Research/Areas/PEDS
2
Example 4a
% DC Stability Toolbox
% S.D. Sudhoff
% example4a
% Step 1: Create the s-vector.
freqmin=1.0; % minimum frequ ency considered
freqmax=1000.0; % maximum frequ ency considered
nfreq=60; % number of fre quencies considered
s = svec1( freqmin,freqmax,nfreq ); % create the s vector for the Nyquists = svec1( freqmin,freqmax,nfreq ); % create the s vector for the Nyquist
% contour
3
Example 4a (Continued)
% Step 2: Parameters of the generalized sets.
NS=8; % number of sid es to generalized
% source impedance
NI=8; % number of int ermediate point to
% generalized source impedance
% Step 3: Define the stability criteria to use
gm=3; % gain margin i n dB
pm=30; % phase margin in degreespm=30; % phase margin in degrees
tr=60; % truncation ra dious in dB (how far out
% into lefthand plane we go
n1=20; % number of poi nts in horizontel
% portion of criteria (upper half)
n2=20; % number of poi nts to use in sloped
% portion of criteria (upper half)
4
Example 4a (Continued)
% Step 4: Generalized description of source
vpllmin=560.0*0.95; % minimum sourc e voltage (l-l,rms)
vpllmax=560.0*1.05; % maximum sourc e voltage (l-l,rms)
nv=3; % number of vol tages to consider
wemin=2*pi*60*0.95; % minimum sourc e frequency (rad/s)
wemax=2*pi*60*1.05; % maximum sourc e frequency (rad/s)
nwe=3; % number of sou rce frequencies to try
llpmin=4.16e-3; % minimum trans former leakage
% inductance (primary side)% inductance (primary side)
llpmax=2.08e-3; % maximum trans former leakage
% inductance (seconary side)
nl=3; % number of rea ctances to try
nps=1.513; % turns ratio
ldc=3.5e-3; % dc link induc tance
rdc=0.33; % dc link induc tor resistance
cdc=1.5e-3; % output capaci tance
rcdc=0.0; % output capaci tance ESR
kp=0.0; % controller pr oportional gain
ki=0.0; % controller in tegral gain
[PCM4_S] = rectsrce(vpllmin,vpllmax,nv,wemin,wemax, nwe, ...
llpmin,llpmax,nl,nps,ldc,rdc,cdc,rcdc,kp,ki,s,NS,NI );5
Example 4a (Continued)
spaceplt(1,4,s,PCM4_S,0.0);
title('Source Impedance');
% Step 5: Calculated a load admitance constraint
[L_const] = esacdspec(gm,pm,tr,n1,n2,PCM4_S,s);
% Step 6: Characterize load 1
c=25.0e-6; % input capacit ance
r=0.01; % input capacit ance series resistancer=0.01; % input capacit ance series resistance
nv=5; % number of vol tages to use
pmin=0.0; % minimum power
pmax=45000; % maximum power
np=10; % number of pow ers to use
% calculate the maximum dc voltage
vmax=(3.0*sqrt(2.0)/pi)*vpllmax/nps;
% calculate the minimum dc voltage
vmin0=(3.0*sqrt(2.0)/pi)*vpllmin/nps; % no load dc voltage
% with minimum input
reffmax=(3/pi)*wemax*(llpmax/nps^2)+rdc; % max imum impedance case
vmin=(vmin0+sqrt(vmin0^2-4.0*reffmax*pmax))*0.5;
[L1] = cpld(c,r,vmin,vmax,nv,pmin,pmax,np,s,NS,NI);6
Example 4a (Continued)
% Step 7: See if system 1 will be stable
spaceplt(2,4,s,L_const,0.0);
hold on;
spaceplt(2,4,s,L1,0.0);
hold off;
title('Generalized Load Admittance Constraint and L oad 1 Admittance');
% Step 8: Characterize load 2 and see if system 2 i s stable
nnsc =1.0; % order of nonlinear stabiliz ing controlnnsc =1.0; % order of nonlinear stabiliz ing control
taunsc=1.0e-2; % time constant of nonlin ear stabilizing control
[L2] = cpldnsc(c,r,nnsc,taunsc,vmin,vmax,nv,pmin,pm ax,np,s,NS,NI);
% Step 9: See if system 2 will be stable
spaceplt(3,4,s,L_const,0.0);
hold on;
spaceplt(3,4,s,L2,0.0);
hold off;
title('Generalized Load Admittance Constraint and L oad 2 Admittance');
7
Source Impedance
0
5
10
15
20
Source Impedance
mag
nitu
de,
dB
8
0
0.5
1
1.5
2
2.5
3
-100
-50
0
50
-20
-15
-10
-5
log of frequency, Hz Purdue DC Stability Toolbox, V. 3.0
phase, degrees
mag
nitu
de,
dB
Load 1
40
60
80
100
Generalized Load Admittance Constraint and Load 1 Admittance
9
0 0.5 1 1.5 2 2.5 3
50
100
150
200
250
300
350
-80
-60
-40
-20
0
20
40
phase, degrees
log of frequency, Hz Purdue DC Stability Toolbox, V. 3.0
mag
nitu
de,
dB
Load 2
20
40
60
80
100
Generalized Load Admittance Constraint and Load 2 Admittance
mag
nitu
de,
dB
10
0
0.5
1
1.5
2
2.5
3
50
100
150
200
250
300
350
-80
-60
-40
-20
0
log of frequency, Hz Purdue DC Stability Toolbox, V. 3.0
phase, degrees
mag
nitu
de,
dB
Treating Complex Systems
11
Single Port Converter
• S Converters
• L Converters
xx
x Si
v=
xS
xSx D
NS
,
,=
12
xx
x Lv
i =xL
xLx D
NL
,
,=
Multi Port Converters
• C Converter
• Z Converter
• Y Converter
• H Converter
13
• H’ Converter
C Converters
• Physically:
• Representation
x
xxxx C
vvii
)( ,2,1,2,1
−=−=
14
x
xC
xCx D
NC
,
,=
Z Converters
• Physically:
• Representation
=
x
x
xx
xx
x
x
i
i
ZZ
ZZ
v
v
,2
,1
,22,21
,12,11
,2
,1
15
xxxx iZZv ,2,22,21,2
xZ
xZxZ
xZxZ
xx
xx
D
NN
NN
ZZ
ZZ
,
,22,21
,12,11
,22,21
,12,11
=
Y Converters
• Physically:
• Representation
=
x
x
xx
xx
x
x
v
v
YY
YY
i
i
,2
,1
,22,21
,12,11
,2
,1
16
xxxx vYYi ,2,22,21,2
xY
xYxY
xYxY
xx
xx
D
NN
NN
YY
YY
,
,22,21
,12,11
,22,21
,12,11
=
H Converters
• Physically:
• Representation
=
x
x
xx
xx
x
x
i
v
HH
HH
v
i
,2
,1
,22,21
,12,11
,2
,1
17
xxxx iHHv ,2,22,21,2
xH
xHxH
xHxH
xx
xx
D
NN
NN
HH
HH
,
,22,21
,12,11
,22,21
,12,11
=
H’ Converters
• Physically:
• Representation
18
Mapping Functions
• SC to S• LC to L• HL to L• SH to S• Parallel L to L
19
• Parallel L to L• Parallel Y to Y• HLH’ to Y• YS to L• Y to L
SC to S
yxe CSS +=
20
LC to L
yx
xe CL
LL
+=
1
21
LC to L
22
LC to L
23
HL to L
yx
yxxxe LH
LHHHL
,22
,21,12,11 1+
−=
24
HL to L
25
HL to L
26
SH to S
yx
yyx
HS
HHSye HS
,11
,12,21
1,22 +−=
27
Parallel L to L
Ω+++= LLLLe Lβα
28
Ω+++= LLLLe Lβα
Parallel Y to Y
Ω+++= YYYYe Lβα
29
HLH’ to Y
( )βααβ
βααα
,22,22,22,22
,22,21,12,11,11
1
HHLHH
HLHHHY
x
xe ++
+−=
βααβ
βα
,22,22,22,22
,21,12,12 HHLHH
HHY
xe ++
=
30
βααβ ,22,22,22,22 HHLHH x++
βααβ
βα
,22,22,22,22
,12,21,21 HHLHH
HHY
xe ++
=
( )βααβ
αβββ
,22,22,22,22
,22,21,12,11,22
1
HHLHH
HLHHHY
x
xe ++
+−=
HLH’ to Y
31
HLH’ to Y
32
HLH’ to Y
33
HLH’ to Y
34
HLH’ to Y
35
HLH’ to Y
36
YS to L
yx
yxyxe SY
YYSYL
,22
,21,12,11 1+
−=
37
yx SY ,221+
Y to L
xxxxe YYYYL ,22,21,12,11 +++=
38
xxxxe YYYYL ,22,21,12,11 +++=
An Example: The NCS Testbed
39
Power Supply
pllv
eωpsn
cL
456-504 V, 358-396 rad/s, 1.30, 1.24 mH
40
cL
Converter Module
41
)1)(1()(
21
1
++=
ss
sKsH
sfsf
sfsfsf ττ
τ
Table 4.4-1. Converter Module Control Parameters msinvc 17.0=τ msinvout 17.0=τ msiniout 17.0=τ
1=d A/V 628.0=pvK AV 197=ivK AS/V
1.0=sfK 201 =sfτ ms 42 =sfτ ms
7.15=iiK kA/s 20* =∆ outmaxv V 20=limiti A
2=intlimi A 548, =mninv V 450, =mxinv V
Load
Table 4.4-2. Load Converter Parameters Component µF,xC
mΩ,xr
V,0 xv kW,xp
IM 590 127 400-420 0-5
42
IM 590 127 400-420 0-5 MC 580 253 400-420 0-5 CPL 479 189 400-420 0-5
System Analysis
43
Zone 1 Reductions
• The Plan
44
Hybrid Coordinates
−+
++
+
≤++
=+ if20log20
20if10
'' 22
22
2220/
Byxyx
jyx
yxjyx
jyx
B
45
≥+ 2022 yx
Zone 1 Check 1
46
Figure 4.5-2. Admittance Space Plot of CM to CM Interface (Testing 1,22
−cmH of CM-P1 as load admittance with admittance
constraint based on cmH ,22 of CM-S1 as a source).
Zone 1 Check 2
47
Figure 4.5-3. Admittance Space Plot of Dual CM to IM Interface (Testing
imL as load admitance with admittance constraint based
on cmH ,22 of CM-P1 in parallel with cmH ,22 of CM-S1 as a
source).
Zone Aggregation
48
Zone Aggregation
49
Next Step: YS to L
50
Figure 4.5-6. Admittance Space Plot of Three Zones / PS-2 Interface (Testing
zY 3,22 as a load admittance with admittance constraint based on
psS as a source).
At This Point
51
Results
52
Figure 4.5-8. Admittance Space Plot of PS-1 / Rest of System Interface. (Testing zpsL3 as a load admittance with admittance constraint
based on psS as a source).
Just For Fun
• Remove Power Supply Output Capacitance
• Reduce CM Input Capacitance to 100 µF
• Set Ksf to zero
53
Results
54
Figure 4.4-9. Admittance Space Plot of Three Zones / PS-2 Interface for Modified System (Testing zpsL3 as a load admittance with
admittance constraint based on psS as a source).
Methods to Obtain Immittance
• Average Value Model
• Hardware Measurement
55
• Switched Model
Reading: S. Sudhoff, B. Loop, J. Byoun, A. Cramer, “A New Procedure for Calculating Immittance Characteristics Using Detailed Computer Simulation,” Power Electronic Specialist Conference, Orlanda, FL, June 17-21, 2007
General Approach
56
( )∑=
=N
nipipp vv
1,, cosθ ipipip t ,,, φωθ +=
An Approach for Detailed Simulations
• Perturbation
( )∑=
=N
nipipp vv
1,, cosθ
ipipip t ,,, φωθ +=
57
• Properties of Perturbation
Waveform Representation
[ ])()()()( 21 tttt Nθθθ L=θ
nnn tt 0)( θωθ +=
scdcapr ttftf fθSfθC ))(())(()( ++=
58
scdcapr ttftf fθSfθC ))(())(()( ++=
)cos()( ,, jiji ααC =
( )jiji ,, sin)( ααS =
Waveform Estimation
• Lets sample the waveform at M times where M>2N+1
• Let tm denote time at sample m of M samples
• We will look for representation such that
59
)()( maprm tftf =
Waveform Estimation
• At the m’th time
• Thus, we arrive at system
( ) ( )( ) ( )( )[ ]
=
s
c
dc
mmm
f
tttf
f
fθSθC1
60
bAx =
( )[ ])(1 ΘSΘC1A ×= M
( ) ( ) ( )[ ]TMtftftf L21=b
[ ]TTs
Tcdcf ffx =
)(, mnnm tθ=Θ
Solution
61
• Thus ( ) bAAAx TT 1−=
Need for Filtering
• Why
• Filtered waveform
62
∫∞−
−=t
dthxtx λλλ )()()(
Filtering
• For Our Purposes
( ) scdcdcapr ttfHtf fθSfθC ))(())(( ++=
∫∞−
−=t
lpfij dtht λλλα )( ))(cos()( ijC
63
• Note
∞−
∫∞−
−=t
lpfij dtht λλλα )( ))(sin()( ijS
Filtering
• Thus, our solution strategy becomes the same, but
( ) ( )[ ]ΘSΘC1A dcM H1×=
[ ]TTs
TcdcdcHf ffx =
( ) ( ) ( )[ ]T
64
• If a bandpass filter is used
( ) ( ) ( )[ ]TMsmsmsm tftftf L21=b
( )[ ])(ΘSΘCA =
[ ]TTs
Tc ffx =
Impedance Measurement
• For our case
[ ]pr θθθ =
[ ]TTT fff =
ispicpi jfff ,,~ −=
65
[ ]Tcp
Tcr
Tc fff =
[ ]Tsp
Tsr
Ts fff = i
ii
i
vz ~
~=
Example – InV Admittance
66
Example – InV Admittance
67
Example – InV Admittance
68
Alternate Approach - DFT
Example – InV Admittance
69
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