principles and practices of digital current regulation for
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AUPEC 2017Tutorial
Principles and Practices of Digital Current Regulation for AC Systems
Professor Grahame Holmes
Dr Brendan McGrath
RMIT Smart Energy Systems 2AUPEC 2017 Tutorial
Summary
• Power electronic AC conversion systems require precise current regulation with high bandwidth
• Conventional wisdom: Stationary Frame PI regulator gains can never be set high enough to
get acceptably small steady state error Synchronous Frame (P+Resonant) control is mandatory
• Unresolved question:What are the stationary frame PI gain limits? - simple theory says they can be arbitrarily large without regulator instability
• Answering this question allows best possible PI regulator performance to be determined for any particular application
RMIT Smart Energy Systems 3AUPEC 2017 Tutorial
Primary Objectives of a Current Regulation System
• Minimize steady state magnitude (and phase for an AC system) output variable error, preferably achieving zero steady state error.
• Accurately track the commanded reference variable during transient changes. This requires a regulation system to have as high a bandwidth as possible, to achieve the best possible dynamic response.
• Limit the peak value of the output variable, to avoid overload conditions.
• Minimize low order harmonics in the load variable, and compensate for DC-link voltage ripple, deadtime delays, semiconductor device voltage drops, load parameter variation and other practical second order effects associated with the operation of the electrical conversion system.
Introduction
RMIT Smart Energy Systems 4AUPEC 2017 Tutorial
Electrical Source Switched Power Converter
Introduction
General Arrangement of Current Regulation for an Electrical Motor Drive System
Load Motor
Current MeasurementTarget Current ReferenceController
RMIT Smart Energy Systems 5AUPEC 2017 Tutorial
Introduction
Electrical Schematic of AC Current Regulated Systems
Single Phase Current Regulated System
Three Phase Current Regulated System (L or LCL Filter)
RMIT Smart Energy Systems 6AUPEC 2017 Tutorial
Principle of Pulse Width Modulation
W.R. Bennett:1933 H.R. Black: 1953 S. Bowes: 1975
dcV
V0
1S
dcV
L
1D
2S2D
swv
oE+
-
R
0.1
0.1−0.1
0.1−
dcV
dcV−
( )tM oωcos
RMIT Smart Energy Systems 7AUPEC 2017 Tutorial
Analytical Solution is obtained by Double Fourier Analysis of the Switched Phase Leg Output Waveform:
Fundamental
Baseband Harmonics
Carrier
Carrier Sideband Harmonics
Frequency
Mag
nitu
de
The solution contains:• Fundamental component• Baseband Harmonics• Carrier Harmonics• Sideband Harmonics
Fundamental and Baseband Harmonics are the relevant harmonics for current regulation.
Pulse Width Modulation of Voltage Source Inverter
RMIT Smart Energy Systems 8AUPEC 2017 Tutorial
Sampled PWM of Inverter Phase Leg
NOTE: Sampled references are delayed wrt their sinusoidal sources.
RMIT Smart Energy Systems 9AUPEC 2017 Tutorial
Combining Phase Legs creates the well known single phase and three phase Voltage Source Inverters,
Three Phase VSI
Two Level Converter Topologies
dcV
V01S
dcV
L
1D
2S 2D
aswv oE3S 3D
4S 4D
bswv
R
dcV
V0 1S
dcV
L1D
4S 4D
aswvbE3S 3D
6S 6D
bswv5S 5D
2S 2D
cswv
L
L
aE
cE
R
R
R
Single Phase VSI
which cancel various harmonics between the phase legs depending on the modulation strategy, i.e.
RMIT Smart Energy Systems 10AUPEC 2017 Tutorial
PWM of Single Phase VSI
t
Fundamental for Phase Leg a
t
Switched l−lOutput Waveform
t
+Vdc
t
t
Triangular Carrier
Fundamental for Phase Leg b
Triangular Carrier
−Vdc
+Vdc
−Vdc
+2Vdc
−2Vdc
Switched Waveformfor Phase Leg b
Vactive
Vz VactiveVactiveVactive
Phase Leg b
Phase Leg a
l−l Output Voltage
ti−1
Vz Vz Vz
Vz Vz Vz Vz
Switched Waveformfor Phase Leg a
+Vdc
−Vdc
+Vdc
−Vdc+2Vdc
−2Vdc
ti ti+1 tj−1 tj tj+1
Single Phase Three level Continuous PWM - sampled
0 10 20 30 40 50 6010-4
10-3
10-2
10-1
100
Harmonic Number
0 10 20 30 40 50 6010-4
10-3
10-2
10-1
100
Harmonic Number
RMIT Smart Energy Systems 11AUPEC 2017 Tutorial
PWM of Three Phase VSI
t
t
Triangular CarrierFundamental for Phase Leg aFundamental for Phase Leg b
Switched Waveform for Phase Leg a
Switched Waveform for Phase Leg b
t
Fundamental for Phase Leg c
t
Switched Waveform for Phase Leg c
t
t
tSwitched l−l Output Waveform (vca)
+Vdc
−Vdc
+Vdc
−Vdc
+Vdc
−Vdc
+2Vdc
−2Vdc
+2Vdc
−2Vdc
+2Vdc
−2Vdc
Switched l−l Output Waveform (vbc)
Switched l−l Output Waveform(vab)
Phase Leg a
Phase Leg b
Phase Leg c
∆T/2
vab l−l Output Voltage
vbc l−l Output Voltage
vca l−l Output Voltage
+Vdc
−Vdc
+Vdc
−Vdc
+Vdc
−Vdc
+2Vdc
−2Vdc
+2Vdc
−2Vdc
+2Vdc
−2Vdc
∆T/2
VzVzVzVz
VzVzVzVz
VzVzVzVz
V1 V1
V2 V2
V1 + V2 V1 + V2
T1 − T3
T3 − T5
ti−1 ti
T5 − T1
ti+1
T3
T1
T5
RMIT Smart Energy Systems 12AUPEC 2017 Tutorial
Third harmonic injection and Space Vector modulation allow an increase in modulation index without reaching saturation limits.
0 50 100 150 200 250 300 350
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Third Harmonic Injection
Space Vector Modulation
0 50 100 150 200 250 300 350
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Third Harmonic Injection/Space Vector Modulation
RMIT Smart Energy Systems 13AUPEC 2017 Tutorial
Pulse Width Modulation
Common Myths about PWM:
• Must have odd carrier ratio to avoid even harmonics
• Must have triplen carrier ratio for three phase system
• Space Vector Modulation is superior to Sine-triangle Modulation
• Instantaneous Third Harmonic Injection is impossible
• Analog SV modulation is impossible
RMIT Smart Energy Systems 14AUPEC 2017 Tutorial
Analog Implementation of SV modulator
Pulse Width ModulationInstantaneous SV/3rd Harmonic Implementation
2),,min(),,max( cbacba
offvvvvvvv +
=
Digital Implementation of SV modulator
RMIT Smart Energy Systems 15AUPEC 2017 Tutorial
PWM of Three Phase Voltage Source Inverter
Phase Leg L-L Voltage
Phase Leg L-L Voltage
Sinusoidal Reference
Sine+ 3rd Harmonic
Experimental Confirmation of Theoretical PWM Harmonics
RMIT Smart Energy Systems 16AUPEC 2017 Tutorial
• Ideal “Average” Linear Model
• Transport and Sampling Delay Limitations
• Maximum Gain Determination – backemf compensation, Proportional-Resonant, d-q synchronous frame
• Three Phase Current Regulation
• Discrete Time Domain Models
• Current Regulation with LCL Filters
Linear Current Regulation
RMIT Smart Energy Systems 17AUPEC 2017 Tutorial
• Differential equation of system:
Ideal “Average” Model of Single Phase VSI
• Averaged model in Laplace form:
( ) ( ) ( ) ( )dt
tdILtRItEtV oooo +=−
( ) ( ) ( )sLR
sEsVsI ooo +
−=
( ) ( ) DCo VtMtV 2=
Vo(t) Eo(t)Io(t)2VDC
RMIT Smart Energy Systems 18AUPEC 2017 Tutorial
The ideal system reduces (in control terms) to the s-domain form of:
under the simplifications of
• The Modulator/VSI reduces to a linear gain (2VDC), ignoring switching harmonics (which cannot be controlled anyway)
• The input disturbance (Eo) represents the motor back-emf
• Plant block is defined by:
Ideal Linear AC Current Regulated System
( )psTRsLR
sG+
⋅=+
=1
111
RLTp =
H(s)∆I(s)Iref(s) G(s) Io(s)Vo(s)
2VDC
Eo(s)Controller Inverter
Gain Plant
RMIT Smart Energy Systems 19AUPEC 2017 Tutorial
Ideal Linear AC Current Regulated System
• Analysis of the system block diagram:
• Solving for Io (s) gives:
• This is the closed loop transfer function of the system
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )sGVsHsGsE
sGVsHsGVsHsIsI
DCo
DC
DCrefo 2121
2+
−+
=
( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] ( ) ( )sEsGsIsIVsHsG
sEsVsGsI
oorefDC
ooo−−=
−=2
( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )sEsGsIVsHsGVsHsGsI orefDCDCo −=+ 221
RMIT Smart Energy Systems 20AUPEC 2017 Tutorial
Ideal Linear AC Current Regulated System
• This transfer function includes two terms:–Reference tracking component
–Disturbance rejection:
• Primary performance objectives are:
( )( )
( )( ) ( )sGVsH
sGsEsI
DCo
o21+
−=
( )( )
( ) ( )( ) ( )sGVsH
sGVsHsI
sI
DC
DC
ref
o21
2+
=
( )( ) ε≤sEsI
o
o( )( ) 1≈sI
sI
ref
o Make H(s) as large as possible
RMIT Smart Energy Systems 21AUPEC 2017 Tutorial
Ideal Linear AC Current Regulated System
• Try a simple PI regulator:
• Open Loop Forward Path Gain:
• Classical control theory states system is stable provided:
• Simple PI forward path phase response asymptotes to -90o
at high frequencies. Hence system is unconditionally stable irrespective of PI gain Kp and integrator time constant τi.
THIS CLEARLY DOES NOT MATCH REALITY ????
( )
+=
+=
i
iP
iP s
sKs
KsHτ
ττ
111
( ) ( )( )p
i
i
PDCDC
sTs
sR
KVsGVsH+
+⋅=
1
12 ττ
( ) ( ) ( ) ( ) 121802 =°−>∠ ccDCccDC jGjHVwhenjGjHV ωωωω
RMIT Smart Energy Systems 22AUPEC 2017 Tutorial
Ideal Linear AC Current Regulated System
Simulated response of ideal average value model AC current regulator with
very high PI gains.
Circuit Parameter Value
Resistive load (R) (Ω) 2
Inductive load (L) (mH) 20
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 80
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
Test circuit parameters
RMIT Smart Energy Systems 23AUPEC 2017 Tutorial
Practical Linear AC Current Regulated System
PSIM Current Regulator Simulation with PWM and Controller Delays
RMIT Smart Energy Systems 24AUPEC 2017 Tutorial
Practical Linear AC Current Regulated System
Circuit Parameter Value
Resistive load (R) (Ω) 1.2
Inductive load (L) (mH) 10.0
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 220
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
Kp = 0.145Steady state error, damped transient
Kp = 0.218Reducing steady
state error, underdamped
transient
Kp = 0.247Less steady state
error, very underdamped
transient
• Clearly gain cannot be arbitrarily increased despite prediction of ideal theory.
• Need to identify the reason, so that PI gains can be set to maximum possible.
Ierr
Ierr
Ierr
RMIT Smart Energy Systems 25AUPEC 2017 Tutorial
• Average model must be extended to incorporate 2nd order effects which constrain practical controller gains.
• The two most likely candidates are:– Transport delay due to the PWM process– Sampling delay caused by digital controller
Practical Linear AC Current Regulated System
RMIT Smart Energy Systems 26AUPEC 2017 Tutorial
Sampling and Transport Delay
• Asymmetrical regular sampled PWM introduces a quarter carrier-period sampling delay (ZOH) into the control process.
• Computation delay and synchronous sampling cause a further half-carrier transport delay (z-1).
• Overall delay of 0.75 of the carrier period is introduced into the forward path of the control loop.
sd fTT /75.075.0 =∆=
Sampling and Transport Delay caused by PWM process and digital
controller sampling/computation.
RMIT Smart Energy Systems 27AUPEC 2017 Tutorial
Stationary frame PI Regulator with sampling and transport delay
Bode plot of open loop forward path gain for ideal and regulators with transport delay
The gain limitation is now clear – forward path phase goes below 1800 as frequency increases. Forward loop gain magnitude must fall less than 1.0 before this occurs.
Effect of Regulator Delay on Frequency Response
H(s) e-sTd
Sampling &Transport Delay
∆I(s)Iref(s) G(s) Io(s)Vo(s)
Eo(s)Controller Inverter
Gain Plant
2VDC
RMIT Smart Energy Systems 28AUPEC 2017 Tutorial
PI Regulator Response with Regulator Delay
Simulated stationary frame PI regulator with regulator delay(a) “ideal” high gains (b) gains reduced to maintain stability
WHICH RAISES THE QUESTION – WHAT ARE THE MAXIMUM POSSIBLE PRACTICAL GAINS?
(a)
(b)
RMIT Smart Energy Systems 29AUPEC 2017 Tutorial
Stationary Frame PI Regulator Maximum Gains
• Forward path transfer function with regulator delay Td
• Transfer function phase angle at cross over frequency s = jωc
Hence for a given regulator delay, ωc is the system bandwidth at φm = selected phase margin
( ) ( ) ( ) ( ) mpcdciccDCc TTjGVjH φπωωπτωωω +−=−−−=∠ −− 11 tan2/tan2
ωc =tan−1 ωcτ i( )−φm
Td
≈π 2−φm
Td
( ) ( )
+
+= −−
p
sT
i
iDCpDC
sTsT
es
sRVK
sGVesH dd1
1122
ττ
pc T
1>>ω ( ) 2tan 1 πω ≈−
pcT
RMIT Smart Energy Systems 30AUPEC 2017 Tutorial
Stationary Frame PI Regulator Maximum Gains
• Maximum possible Kp is for a given ωc is:
assuming
• Integral time constant τi can be determined by making
• These gains are independent of the plant time constant, and are determined only by φm Td VDC and L
For any given system with regulator delay Td , these are the maximum possible PI gains for a phase margin of φm
( )( )22
22
1
12
ic
pcicDC
pT
VRK
τω
ωτω
+
+=
( ) 2/tan 1 πτω ≈−ic
ci ω
τ 10≈
122 >>pc Tω
122 >>icτωDC
cp V
LK2ω
≈
RMIT Smart Energy Systems 31AUPEC 2017 Tutorial
Example Grid Inverter: Calculations & Results
Circuit parameters
krad/s81.55.1
698.056.124 =
−=
−= −eTd
mc
φπω
V/A145.0400
1081.52
33=
∗==
−eeV
LKDC
cp
ω
msec72.182.51010
3 ===ec
i ωτ
sec5.15/75.0 43 −== eeTd
rad698.040 == omφ
Circuit Parameter Value
Resistive load (R) (Ω) 1.2
Inductive load (L) (mH) 10.0
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 220
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
RMIT Smart Energy Systems 32AUPEC 2017 Tutorial
Practical Linear AC Current Regulated SystemCircuit Parameter Value
Resistive load (R) (Ω) 1.2
Inductive load (L) (mH) 10
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 220
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
Kp = 0.145φ = 40o
Steady state error, damped transient
Kp = 0.218φ = 15o
Reducing steady state error,
underdamped transient
Kp = 0.247φ = 5o
Less steady state error, very
underdamped transient
• Damping response of transient depends on designed gain margin.
• φ = 40o is a typical value that gives a good balanced response.
• But are these gains enough to get an acceptable AC regulator performance?
Ierr
Ierr
Ierr
RMIT Smart Energy Systems 33AUPEC 2017 Tutorial
Practical PI Regulator Performance Evaluation
• The current error ∆I(s) as a function of control inputs is :
• It has two elements: Closed Loop Tracking Error - the ability of regulation system to follow current reference:
• Closed Loop Disturbance Rejection Error – regulation system sensitivity to backemf disturbance:
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )sIsIsGVesH
sGsEsGVesH
sIsI DIDC
sToDC
sTref dd∆+∆=
++
+=∆ −− 2121
1
( ) ( )( ) ( ) ( )sGVesHsIsIsE
DCsT
ref
II d 21
1−+
=∆
=
( ) ( )( )
( )( ) ( )sGVesH
sGsEsIsE
DCsT
o
DD d 21 −+
=∆
=
RMIT Smart Energy Systems 34AUPEC 2017 Tutorial
Practical PI Regulator Performance Evaluation
• Both tracking error and disturbance rejection error increase with frequency (controller gain reduces)
• Disturbance error increases more slowly because of plant pole
• BUT, magnitude of disturbance injection (emf) is usually larger still creates more error
Irrespective of the contribution, the error at AC fundamental
frequency is often unacceptably large with PI controller even with
maximum possible gains.
RMIT Smart Energy Systems 35AUPEC 2017 Tutorial
• Maximised gains are often insufficient to achieve acceptable regulator performance, especially when error caused by back EMF is considered
• The following enhancements can be considered–Minimise regulator delay to achieve the maximum possible gains.
Use asymmetrical regulator sampled PWM and synchronous sampling as a matter of course.
–Reduce the back EMF disturbance effect with feed forward compensation
–Increase the regulator gain at the fundamental target frequency (Proportional-Resonant Controller)
• Lets see how these ideas work
Stationary Frame Regulator Enhancements
RMIT Smart Energy Systems 36AUPEC 2017 Tutorial
• Require to measure average load current (no switching harmonic components) without filter delay, to avoid increasing Td.
• Synchronously sample at top/bottom of (implied) carrier
• At this point VSI phase leg volt-second output equals commanded voltage minimum measured current ripple
Synchronously sampling the “average” current
Synchronous sampling of measured current
RMIT Smart Energy Systems 37AUPEC 2017 Tutorial
Synchronously sampling the “average” current
Calc.k
Calc.k+1
Calc.k+2
Calc.k+3
Sampled Current
Voltage Reference
Actual Current
i(k)i(k+1)
i(k+2) i(k+3)
m(k) m(k+1) m(k+2)m(k-1)
T
Carrier
m(k+3) m(k+4)
Calc.k+4
Calc.k+5
i(k+4) i(k+5)
• Ensures that the sampled currents correspond to the average currents
• Still has ½ carrier interval transport delay due to finite calculation time
RMIT Smart Energy Systems 38AUPEC 2017 Tutorial
Further Minimise Transport Delay
• At low switching frequencies, tracking error can still be too high
• Further reducing transport delay allows higher gains
• Achieved by pre sampling before each half carrier cycle and dynamically compensating for ripple current in sample
• Feasible using modern DSP with synchronous sampling
ADC1 ADC2 ADC3ADC4
RMIT Smart Energy Systems 39AUPEC 2017 Tutorial
• Back EMF is predictable, and so its effect can be reduced by feedforwarding the measured (or estimated) value
Feedforward Mitigation of Backemf Disturbance
Average model AC current regulation system with back EMF feedforward
H(s) e-sTd
Sampling &Transport Delay
∆I(s)Iref(s) G(s) Io(s)Vo(s)
F(s) e-sTd
EMF Feedforward
Eo(s)
Eo(s)
Controller InverterGain
Sampling &Transport Delay
EMFMismatch
Plant
2VDC
RMIT Smart Energy Systems 40AUPEC 2017 Tutorial
• The transfer function of system with feedforward becomes:
• With perfect feedforward (F(s)=1) influence of backemf is completely eliminated (ED(s)=0), and only ER(s) remain.
• The controller design process still maximises KP as before.
Feedforward Mitigation of Backemf Disturbance
( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )
( ) ( )sIsIsGVesH
sGesFsEsGVesH
sIsI DIDC
sT
sT
oDC
sTref d
d
d∆+∆=
+
−+
+=∆ −
−
− 211
211
( ) ( ) ( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )sGVesH
sGesFsEsGVesH
sGVesHsIsIDC
sT
sT
oDC
sTDC
sT
refo d
d
d
d
211
212
−
−
−
−
+
−−
+=
RMIT Smart Energy Systems 41AUPEC 2017 Tutorial
Feedforward Mitigation of Backemf Disturbance
Simulated and Experimental system response with 110Vrms backemf.
PI controller without backemf feed forward
PI controller with backemf feed forward
RMIT Smart Energy Systems 42AUPEC 2017 Tutorial
Accuracy of Back EMF Estimation
(a) Performance of PI regulator with backemf feedforward(b) Miss estimated back EMF estimation (80% magnitude, 40o phase lead)
RMIT Smart Energy Systems 43AUPEC 2017 Tutorial
• Key limitation of PI regulator is high ED(s) at fundamental reference frequency (ω0)
• An alternative approach is to increase controller gain at ω0without affecting high frequency gain determined by Kp
• This is the strategy of the P+Resonant (PR) controller, with a resonant gain peak at ω0 :
Increase Fundamental Gain with PR Regulator
( )
++= 22
11oi
pPRs
sKsHωτ
RMIT Smart Energy Systems 44AUPEC 2017 Tutorial
• Properties of the P+Resonant controller:–Low and high frequency behaviour approximates a proportional
controller–Controller provides near infinite gain at the target AC frequency–Reference tracking error and disturbance error are near zero at the
AC frequency–Hence zero steady magnitude and phase error
Note : PR regulator can be a little sensitive to mismatch between the resonant term and AC fundamental frequency.
Increase Fundamental Gain with PR Regulator
RMIT Smart Energy Systems 45AUPEC 2017 Tutorial
• Forward Path Gain:
• At the cross-over frequency:
• If , then:
• This is identical to the PI expression.
( ) ( )
+
++= −−
p
sT
oi
DCpDC
sTPR sT
es
sRVK
sGVesH dd1
1112
2 22 ωτ RLTp =
oc ωω >>
+
+=
+
+≈
−
−
pc
Tj
ic
icDCp
pc
Tj
ic
DCp
Tje
jj
RVK
Tje
jRVK
GAINPATHFORWARD
dc
dc
ωτωτω
ωτω
ω
ω
1112
1111
2
Maximum Gains with PR Regulator
cjs ω=
( ) ( )
+
−+= −−
pc
Tj
co
c
i
DCpcDC
TjcPR
Tjej
RVK
jGVejH dcdcωωω
ωτ
ωω ωω1
1112
2 22
( ) ( )
+
+= −−
p
sT
i
iDCpDC
sTsT
es
sRVK
sGVesH dd1
1122
ττ
RMIT Smart Energy Systems 46AUPEC 2017 Tutorial
• Therefore the maximum gains for the PR controller are the same as for the PI controller, and are defined by:
– Target Phase Margin φm
– DC link voltage 2VDC
– Load Inductance L– PWM/Current regulator transport delay Td
DC
cp V
LK2ω
=
Maximum Gains with PR Regulator
ci ω
τ 10=
d
mc T
φπω −=
2
RMIT Smart Energy Systems 47AUPEC 2017 Tutorial
Maximum Gains with PR Regulator
• Resonant gain peak improves backemf disturbance rejection at ω0
• Also reduce EI(s) at ω0, leading to better tracking
Bode plot of GH(s) for PI and PR regulators with transport delays
Closed Loop Disturbance Rejection Error ED(s) for PI and PR controllers
RMIT Smart Energy Systems 48AUPEC 2017 Tutorial
Practical Linear AC Current Regulated System
Circuit Parameter Value
Resistive load (R) (Ω) 1.2
Inductive load (L) (mH) 10
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 220
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
PI controller:Kp = 0.145
φ = 40o
Steady state error, damped transient
PR controller:Kp = 0.145
φ = 40o
Neligible steady state error,
damped transient
PI verses PR controller response with same gains.
Ierr
Ierr
RMIT Smart Energy Systems 49AUPEC 2017 Tutorial
Experimental Confirmation: PR Regulator
Simulated and Experimental system response with 110Vrms backemf
Same transient response for all three alternatives
PI controller without backemf feed forward
PI controller with backemf feed forward
PR controller
RMIT Smart Energy Systems 50AUPEC 2017 Tutorial
Three Phase AC Current Regulated VSI
Three phase Voltage Source Inverter (VSI) connected to a back EMF load through a series R-L impedance
RMIT Smart Energy Systems 51AUPEC 2017 Tutorial
Three Phase AC Current Regulated VSI
• KVL loops:( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sVsEsEsZsIsZsIsV bbabaa +−+−= 2
• For balanced system:
( ) ( ) ( ) 0=++ sVsVsV cba ( ) ( ) ( ) 0=++ sEsEsE cba
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sVsEsEsZsIsZsIsV ccbabb +−+−= 2
( ) ( ) ( )sVsVsV bac −−= ( ) ( ) ( )sEsEsE bac −−=
RMIT Smart Energy Systems 52AUPEC 2017 Tutorial
• For an isolated three phase system
• Substituting voltage and current constraints into KVL loops gives
Thus only two currents need to be controlled for a three phase system, each regulated only by their individual phase
voltages using previous single phase theory
• Modulate third phase to achieve balanced system using:
Three Phase AC Current Regulated VSI
( ) ( ) ( ) 0=++ sisisi cba
( ) ( ) ( )sLR
sEsVsi aaa +
−= ( ) ( ) ( )
sLRsEsVsi bb
b +−
= ( ) ( ) ( )sLR
sEsVsi ccc +
−=
( ) ( ) ( )sVsVsV bac −−=
RMIT Smart Energy Systems 53AUPEC 2017 Tutorial
Topology of a three phase AC current regulated voltage source inverter
Three Phase AC Current Regulated VSI
RMIT Smart Energy Systems 54AUPEC 2017 Tutorial
Three Phase AC Current Regulated VSI
Note - effective DC source voltage for Kp gain calculation changes from 2VDC for single phase system to VDC for three phase system
Single Phase Current Regulated System
Three Phase Current Regulated System (L or LCL Filter)
RMIT Smart Energy Systems 55AUPEC 2017 Tutorial
Example Grid Inverter: Calculations & Results
Circuit Parameter Value
Resistive load (R) (Ω) 1.2
Inductive load (L) (mH) 20
Switching Freq. (fs) (kHz) 5.0
DC Bus volt. (2VDC) (V) 400
Back EMF volt. (VEMF) (VRMS) 80
Back EMF freq. (Hz) 50
Sampling period (T) (sec) 10-4
Circuit parameters
krad/s81.55.1
698.056.124 =
−=
−= −eTd
mc
φπω
V/A582.0200
2081.5 33=
∗==
−eeV
LKDC
cp
ω
msec73.182.51010
3 ===ec
i ωτ
sec5.15/75.0 43 −== eeTd
rad698.040 == omφ
RMIT Smart Energy Systems 56AUPEC 2017 Tutorial
Example Grid Inverter: Calculations & Results
No back EMF, PI maximised gains (fs = 5kHz)
Back EMF, PI maximised gains (fs = 5kHz)
3 phase simulation 3 phase experimental
3 phase simulation 3 phase experimental
RMIT Smart Energy Systems 57AUPEC 2017 Tutorial
Example Grid Inverter: Calculations & Results
Back EMF, PI maximised gains plus feedforward EMF compensation (fs = 5kHz)
Back EMF, PR regulator maximised gains (fs = 5kHz)
3 phase simulation 3 phase experimental
3 phase simulation 3 phase experimental
RMIT Smart Energy Systems 58AUPEC 2017 Tutorial
Example Grid Inverter: Calculations & Results
Comparison of Current Error – all regulator alternatives
RMIT Smart Energy Systems 59AUPEC 2017 Tutorial
Back EMF Estimation by Direct Measurement
• Synchronously measure scaled values of back EMF at inverter output terminals, if it has internal filter
• Back EMF given by :
• Sensitive to noise, particularly PWM switching ripple.
( )( )
( ) ( )sZsZsZ
Egridfilter
filtermeasa +
=
Circuit arrangement to allow measurement of back EMF
RMIT Smart Energy Systems 60AUPEC 2017 Tutorial
Back EMF Estimation by Predictive Calculation
• Back EMF estimation, as used in predictive current regulator
• Load inductance estimation constraints
ea i[ ]= Va i − 2[ ]− 2Va i − 3[ ]− ia i[ ]R'+ L'∆T /2
−3ia i −1[ ]+ 5ia i − 2[ ]− 2ia i − 3[ ]( )
Response with back EMF, estimated feedforward EMF (fs = 5kHz)
0.8L ≤ L'≤1.23L
RMIT Smart Energy Systems 61AUPEC 2017 Tutorial
Further Minimise PWM Transport Delay
• At low switching frequencies, tracking error can still be too high
• Further reducing transport delay allows higher gains
• Achieved by pre sampling before each half carrier cycle and dynamically compensating for ripple current in sample
• Feasible using modern DSP with synchronous sampling
ADC1 ADC2 ADC3ADC4
RMIT Smart Energy Systems 62AUPEC 2017 Tutorial
Further Minimise PWM Transport Delay
• Ideally reduce overall regulator delay to 0.25 carrier period
• Theoretically should increase ωc and gains by factor of 3
• Experimentally found to be susceptible to noise
• Only approx 2 times increase in gain was achievable
without back EMF, PI controller, reduced sampling time by measurement compensation (fs = 2.5 kHz)
RMIT Smart Energy Systems 63AUPEC 2017 Tutorial
• ABC frame : Three non-orthogonal variables with two degrees of freedom
• αβ frame: Two orthogonal variables
• Clarke (alpha-beta) transform:
• Inverse Clarke transform:
−−−=
=
β
α
β
ααβ i
iii
Tiii
T
c
b
a
23212321
01
32
Orthogonal Reference Frames: αβ stationary frame
−−−
=
=
c
b
a
c
b
a
iii
iii
Tii
2323021211
32
αββ
α
RMIT Smart Energy Systems 64AUPEC 2017 Tutorial
• The (αβ) quantities are sinusoidal
• Hence gain maximisation approach is identical to the (abc) frame current regulator
• The regulator output is compatible with a direct space vector modulator – no need to indirectly calculate the phase C PWM command for SV implementation (but still need to decode SV output to phase leg switching commands)
• References can be defined in either the (abc) or the (αβ) frames as suits application
Orthogonal Reference Frames: αβ stationary frame
RMIT Smart Energy Systems 65AUPEC 2017 Tutorial
Orthogonal Reference Frames: αβ stationary frame
RMIT Smart Energy Systems 66AUPEC 2017 Tutorial
Orthogonal Reference Frames: αβ stationary frame
RMIT Smart Energy Systems 67AUPEC 2017 Tutorial
Orthogonal Reference Frames: dq rotating frame
• Synchronous frame – AC quantities are transformed to equivalent DC quantities
• Park transform – (αβ) to (dq)
• Inverse Park transform:
−
=
=
β
α
β
α
θθθθ
ii
ii
Tii
dqq
d
cossinsincos
−=
=
q
d
q
dTdq i
iii
Tii
θθθθ
β
α
cossinsincos
tωθ =
tωθ =
RMIT Smart Energy Systems 68AUPEC 2017 Tutorial
Orthogonal Reference Frames: dq rotating frame
RMIT Smart Energy Systems 69AUPEC 2017 Tutorial
Reference Frames: regulator transformation
Stationary frame αβ controller Rotating frame DQ controller
+
+
iP
iP
sK
sK
τ
τ110
011
++
+−
+
++
2222
2222
111
111
oiP
o
o
i
o
o
ioiP
ssK
s
sssK
ωτωω
τ
ωω
τωτ
+
+
iP
iP
sK
sK
τ
τ110
011
++
+−
+
++
2222
2222
111
111
oiP
o
o
i
o
o
ioiP
ssK
s
sssK
ωτωω
τ
ωω
τωτ
Stationary Frame PI regulator transforms to Synchronous Frame Resonant
Synchronous Frame PI regulator transforms to Stationary Frame Resonant
Gains are the same in both frames of reference and hence can be calculated as before
RMIT Smart Energy Systems 70AUPEC 2017 Tutorial
Simulated and Experimental PR Current Responses
RMIT Smart Energy Systems 71AUPEC 2017 Tutorial
Simulated and Experimental PR Current Responses
RMIT Smart Energy Systems 72AUPEC 2017 Tutorial
( ) ( ) ( )( )sss
ksi rrr
pqs ωωτ ω
ω −
+= ** 11 ( ) ( ) ( )
m
drrds L
sssi
** 1
λτ+=
Torque Controller Flux Controller
Example: AC Induction Motor Variable Speed Drive
RMIT Smart Energy Systems 73AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed Drive
VSI IM
1/Lm
PI
Slip Est.
PIABC
DQ PI ABC
DQ
prτ+1
rωrω
*drλ
*rω
*di
*qi
*qv
*dv
esθ e
sθ
−
−=
*
*
*
*
cossin
sincos
23
21
01
q
d
es
es
es
es
b
a
v
v
v
v
θθ
θθ
Synchronous Frame PI Regulator
−=
b
a
es
es
es
es
q
d
i
i
i
i
32
31
01
cossin
sincos
θθ
θθ
( )( )
( ) ( )( ) ( )
−−
+=
sisisisi
sK
svsv
dd
ip
q
d*
*
*
* 11τ
RMIT Smart Energy Systems 74AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed Drive
ABC
DQ PR
PR
VSI IM
1/Lm
PI
Slip Est.
rωrω
prτ+1*drλ
*rω
*di
*qi
*ai
*bi
*av
*bv
*cve
sθ
Stationary Frame PR Regulator
( )( )
( ) ( )( ) ( )
−−
++=
sisisisi
s
sKsvsv
bb
aa
eip
b
a*
*
22*
* 11ωτ
RMIT Smart Energy Systems 75AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed Drive
• Resonance was implemented using the Second Order Generalised Integrator:
• Provides the transfer function:
• Advantage of this form is that the integrators are unaffected by changes in resonance frequency, simplifying the implementation in a DSP.
Stationary Frame PR Regulator
1s
ωe2
1s
y(s)x(s) 1τi
( )( ) 22
1
ei s
ssxsy
ωτ +=
RMIT Smart Energy Systems 76AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed Drive
( )( )
( ) ( )( ) ( )
+
−−
+=
)(
)(*
*
*
* 11estb
esta
bb
aa
ip
b
asisisisi
sK
svsv
εε
τ
0)( 90+∠≈ e
snom
rnomesta V θ
ωωε
Stationary Frame PI Regulator with EMFest
RMIT Smart Energy Systems 77AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed Drive
ABC
DQ
VSI IM
1/Lm
PI
Slip Est.
rωrω
prτ+1*drλ
*rω
*di
*qi
*ai
*ci
*bi
esθ
hxx iii >−*DCx Vv −→
hxx iii −<−*DCx Vv +→
, (19a)
,
Implemented digitally for this example at 100 kHz update rate, average switching frequency of approx 5 kHz
Variable Band Hysteresis (digital implementation)
RMIT Smart Energy Systems 78AUPEC 2017 Tutorial
Gains matched for the three different strategies
Gains can be set to same maximum gains irrespective of current regulation strategy – sampling and transport delay
are the limiting factors
Example: AC Induction Motor Variable Speed Drive
RMIT Smart Energy Systems 79AUPEC 2017 Tutorial
Example: AC Induction Motor Variable Speed DriveParameter Value
Induction Motor (4 pole) 3.2 kW, 415 Vl-l, 6.3 A, YIM - rs 3.24 Ω
- rr 2.6 Ω- Lls = Ls – Lm 19 mH- Llr = Lr – Lm 21.1 mH- Lm 504.4 mH
Inertia 0.65 kgm2
System Load Dynamics 1.2 + 0.0034ωr+70e-6ωr2
Controller - kpω 0.95- τrω 0.034 sec- kpi 0.7- τri 0.00192 sec- Calc. Freq. 10 kHz
PWM Frequency 5 kHzVSI DC Link Voltage 520 V
• 15 kW 4 phase leg inverter ( 3 phase VSIplus DC bus chopper)
• DSP Controller:Texas Instruments TMS320F2810 fixed point DSP
• 3 phase 5kW Induction Motor• Quadrature pulse
encoder to measure speed and direction
• Coupled DC motor and drive to provide variable loading
Experimental System
RMIT Smart Energy Systems 80AUPEC 2017 Tutorial
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
200
400
600
Time (s)
Spe
ed (R
PM
)
Stationary Frame PI
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-5
0
5
AB
C C
urre
nt (A
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
Time (ms)
DQ
Cur
rent
(A)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-4
-2
0
2
Experimental Current Step Change: PIstationary
Example: AC Induction Motor Variable Speed Drive
RMIT Smart Energy Systems 81AUPEC 2017 Tutorial
Experimental Current Step Change: PRstationary
Example: AC Induction Motor Variable Speed Drive
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
200
400
600
Time (s)
Spe
ed (R
PM
)
Stationary Frame PR
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-5
0
5
AB
C C
urre
nt (A
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
Time (ms)
DQ
Cur
rent
(A)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-4
-2
0
2
RMIT Smart Energy Systems 82AUPEC 2017 Tutorial
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
200
400
600
Time (s)
Spe
ed (R
PM
)
Synchronous Frame PI
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-5
0
5
AB
C C
urre
nt (A
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
Time (ms)
DQ
Cur
rent
(A)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-4
-2
0
2
Experimental Current Step Change: PIsynchronous
Example: AC Induction Motor Variable Speed Drive
RMIT Smart Energy Systems 83AUPEC 2017 Tutorial
Experimental Current Step Change: Hysteresis
Example: AC Induction Motor Variable Speed Drive
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
200
400
600
Time (s)
Spee
d (R
PM)
Hysteresis
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-5
0
5
ABC
Curre
nt (A
)
Time (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
5
10
Time (s)
DQ C
urre
nt (A
)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
-4
-2
0
2
RMIT Smart Energy Systems 84AUPEC 2017 Tutorial
• LCL filter Third order line filter for grid connected inverters
• Benefits of LCL filters (over L filters)–Greater attenuation, less size, less cost
• Major design challenges–LC network resonance, low harmonic impedance
LCL Filters for AC Inverters
RMIT Smart Energy Systems 85AUPEC 2017 Tutorial
Comparable L and LCL Filters
• L filter impedance
• LCL filter impedance
• To meet IEEE519 impedance >100pu @ 10kHz
@ 50Hz @ 10kHzL = 55mH 0.5pu 100puL = 16.5mH 0.15pu 30pu
L1 = 3mH @ 50Hz @ 10kHzCf = 7.5uF 0.036pu 154puL2 = 1mH
RMIT Smart Energy Systems 86AUPEC 2017 Tutorial
L and LCL Filter Response
LCL filter with comparable switching frequency can never achieve comparable baseband filtering
LCL has greatest switching attenuation
L filters have greatest baseband attenuation
LCL dominated by series inductance in
baseband region
RMIT Smart Energy Systems 87AUPEC 2017 Tutorial
• LCL resonance damping:– Passively using series or shunt resistances– Actively using an additional state variable in the control law
• How to synthesise the current controller?– Analyse using discrete time models (i.e. state space or z-domain)– Sampling delay accounted for using a ZOH transformation of the
LCL filter s-domain transfer function(s)– Transport delay accounted for using a z-1 shift operator
• Active damping is proposed in multiple forms:– Capacitor current feedback is the most commonly accepted
• Damping may not be required under some circumstances
Control Loop Synthesis and Analysis
RMIT Smart Energy Systems 88AUPEC 2017 Tutorial
LCL Filter Controller Options
• Single loop –Difficult to manage LCL resonance
• Active damping (dual loop)–Commonly feedback of ic (or variant) for damping, through gain K
Which controller is suitable and when ? use Bode analysis
Plant (LCL)
Controller
Inverter
RMIT Smart Energy Systems 89AUPEC 2017 Tutorial
L and LCL Filter Equations
• L Filter
• LCL Filter
• NOTE : filter resistance neglected to model worst case undamped filter characteristic
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]sEsVsYsEsVsL
si oLo −=−=1
2
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )sEsYsVsY
sEs
sCLCLL
sVssCLL
si
Gov
res
f
fo
resf
+=+
+−
+=
22
21
2122212
1111
ωω
is the filter admittance seen by the grid and converter for an L filter( )sYL
are the filter admittances seen by the converter and grid for an LCL filter, where
( ) ( )sYsY Gv and( ) ( )2
1 2 1 2res fL L L L Cω = +
RMIT Smart Energy Systems 90AUPEC 2017 Tutorial
Discrete Time Model – Single Loop Controller
• Plant transfer function (grid current)
• where and
• using ZOH discretization (sampled system)
( ) ( )( ) ( )
22
2 2 21
1 LCi
o res
i sG s
V s sL s
γ
ω= =
+
( ) ( )( ) ( ) ( )
( )( ) ( )
22 2
1 2 1 2
sin 11 2 cos 1
resi
o res res
i z TT zG zV z L L z L L z z T
ωω ω
−= = − ×
+ − + − +
1 2
1 2res
f
L LL L C
ω +=
2
1LC
fL Cγ =
RMIT Smart Energy Systems 91AUPEC 2017 Tutorial
Discrete Time Model – Single Loop Controller
• PR fundamental controller
• Inverter model with PWM transport delay
• Open loop forward path equation (without active damping)( )( ) ( ) ( )2 1
22
DC c ie
i zz V G z G z
i z−=
( ) ( )( )( )
20
20 0
sin1 112 2 cos 1c p
r
T zG z KT z z T
ωω ω
− = + − +
( ) ( )1o DC oV z z V m z−=
( ) 2 20
11c pr
sG s KT s ω
= + +
RMIT Smart Energy Systems 92AUPEC 2017 Tutorial
Discrete Time Model – Dual Loop Controller
• Inner loop plant transfer function (capacitor current)– Active damping
• Using ZOH discretization (sampled system)
• Closed inner loop with gain K (including inverter model)
( ) ( )( )
( )( )2
1
sin 12 cos 1
c resic
o res res
i z T zG zV z L z z T
ωω ω
−= = ×
− +
( ) ( )( ) ( )
2
2 21
1cic
o res
i s sG sV s sL s ω
= =+
( )( )
( )( )
c DC ic
o DC ic
i z V G zm z z KV G z
=+
RMIT Smart Energy Systems 93AUPEC 2017 Tutorial
Discrete Time Model – Dual Loop Controller
• Equation relaying i2 to ic–Impulse invariant discretization (delay already in model)
• Overall open loop forward path equation with active damping
( )( )
( )( )
22 2
2i LC
c ic
i s G si s G s s
γ= =
( )( ) ( )
2 22
21LC
c
i z T zi z z
γ=
−Zimp
( )( ) ( ) ( )
( )( )( )
( ) ( ) ( ) ( )( )
2 2
2
2
cce
o c
DC ic cc
DC ic
i z i z i zG z
i z m z i z
V G z i z i zG z
z KV G z
= × ×
= ×+
RMIT Smart Energy Systems 94AUPEC 2017 Tutorial
LCL Filter Active Damping Regions
• Low LCL resonant frequency–-180º phase jump with
unbounded gain–Unstable without damping–Stabilised with active damping
• High LCL resonant frequency– -180º phase jump occurs after
phase crosses -180º due to delay roll-off
– Stable without damping
• Critical LCL frequency (ωcrit) forward path phase reaches -180º
RMIT Smart Energy Systems 95AUPEC 2017 Tutorial
Critical LCL Resonant Frequency
• To find ωcrit
• PR controller response well below crossover frequency–Phase contribution of controller is negligible:
• LCL resonance no phase contribution until the resonant frequency is reached
– Hence plant phase reduces to
• with substitution and manipulation
• Below ωcrit damping is required
• Above ωcrit single control loop is sufficient
( ) ( ) ( )22
2
j T j T j T j TDC c ie
i z e e V G e G ei
ω ω ω ω π−∠ = = ∠ = −
( ) 0j TcG e ω∠ ≈
( )21
1j T
i j TG ee
ωω∠ = ∠
−
( )2
2 2 2j T
e
i Tz e Ti
ω π ωω π∠ = = − − − = −3crit Tπω =
RMIT Smart Energy Systems 96AUPEC 2017 Tutorial
Exploring the Active Damping Regions
• Closed loop poles Required for root loci
• Using conventional 1+G(z)=0, from forward path G(z)–Single loop controller
–Inner loop of dual loop controller
–Overall dual loop controller
( )2 0DC p iz V K G z+ =
( ) 0DC icz V KG z+ =
( ) ( ) ( ) ( )2 0DC ic DC P ic cz V KG z V K G z i z i z+ + =
RMIT Smart Energy Systems 97AUPEC 2017 Tutorial
Single Loop (i2) Controller Root Loci
High frequencyCritical frequencyLow frequency
• No damping capability at critical or low frequencies
• Provides damping of poles at high frequency– No active damping required agree with Bode
• All systems have loci which track unstable through 0.5 3 2z j= ±
RMIT Smart Energy Systems 98AUPEC 2017 Tutorial
Single Loop Controller when Resonance Above ωcrit
• Single loop controller is suitable–Stable, active damping unnecessary
• Gain limitation Low frequency poles, not resonance–Same mechanism as L filters
Low frequency poles cause
instabilityResonant poles will be damped
RMIT Smart Energy Systems 99AUPEC 2017 Tutorial
Gain Selection when Resonance Above ωcrit
• Using the same method as for standard L filters [Holmes et. al. 2009]
–Resonance and controller minimal magnitude and phase contribution at crossover frequency
–Only low frequency model required series inductance plant
–Calculate crossover ωc frequency for a desired phase margin φm
( )( ) ( ) ( )
2 1
2 1 21DC pe
i z Tz V Ki z z L L
−=− +
( )( ) ( )
2
2 1 2
11
32 2 2 2
c
c c
DC pj Te j T j T
cc c
V K Ti z ei L L e e
T T T
ω
ω ω
ωπ πω ω
∠ = = ∠+ −
= − − − = − −
( )2
2
32 2
cj Tm ce
i z e Ti
ω πϕ π ω= + ∠ = = − 23 2
mc T
π ϕω −=
d
mc T
ϕπω −=
2
RMIT Smart Energy Systems 100AUPEC 2017 Tutorial
Gain Selection when Resonance Above ωcrit
• Proportional gain is set to achieve unity gain at desired crossover frequency ωc
• Design PR controller time constant (Tr)– Phase contribution minimal at ωc
( ) ( )1 2 1cj T
pDC
L L eK
V T
ω+ −=
( )1 2cp
DC
L LK
Vω +
≈
10r cT ω≈
High Frequency SystemL1 = 6mH L2 = 2mHCf = 1.5μF Leq = 1.5mH
2VDC = 650V fs = 5kHzωres = 21.1krads-1 Τ = 0.1msecωcrit=10.5krads-1 Τd = 0.15msec
φm = 45º ωc=5.24krads-1
Kp = 0.129A-1 Tr = 1.9msec
-135º
ωc
ωcrit
ωres
RMIT Smart Energy Systems 101AUPEC 2017 Tutorial
Inner Loop (ic) of Active Damping Root Loci
Low frequency
• Only low frequency region examined– High frequency not required– Critical frequency not effective (next slide)
• Provides damping of poles – Limited range of K
Critical frequency0.5 3 2z j= ±
RMIT Smart Energy Systems 102AUPEC 2017 Tutorial
Dual Loop Controller Root Loci
Critical frequencyLow frequency
• At low frequency – Poles are moved inside unit circle by active damping– Limitation to damping gain K before instability
• At critical frequency – never enter unit circle
Varying K(active damping)
Poles start unstable
RMIT Smart Energy Systems 103AUPEC 2017 Tutorial
Controller when Resonance Below ωcrit
• Dual loop controller (active damping) required–Capacitor current feedback selected
• Gain selection will ensure maximally damped poles
Poles originate outside unit
circle
Active damping draws poles into stability
RMIT Smart Energy Systems 104AUPEC 2017 Tutorial
Gain Selection when Resonance Below ωcrit
• Low frequency still dominated by series filter inductance
• Phase margin must consider phase contribution of resonance– Crossover frequency ratio of resonant frequency [Tang et. al. 2012]
• Proportional gain and PR time constant Same manner as earlier
• Best possible damping gain– Maximally damped poles
– Closed loop characteristic equation
– Bounded range for K
10r cT ω≈
0.3c resω ω≈
( )1 2cp
DC
L LK
Vω +
≈
Min K
Max K
( ) ( ) ( ) ( )2 0DC ic DC P ic cz V KG z V K G z i z i z+ + =
RMIT Smart Energy Systems 105AUPEC 2017 Tutorial
Bounded Range for K (Resonance Below ωcrit)
• Maximum value (Kmax)– Magnitude of characteristic equation adapted to equal 1
– Substitute in critical point
– Solve for Kmax
• Minimum value (Kmin)– Ratio of K to Kp identified using
– Routh’s Stability Criterion [Liu et. al. 2012]
– Rearranged as limitation
Kmin
Kmax
( ) ( )( ) ( )( )
2 2 2
21
1sin1
1 2 cos 1p LCDC res
res res
K z K T zV TL z z z z T
γωω ω
− +× =
− − +
( ) ( ) 221max cos21
sinTKT
TVLK LCpres
resDC
res γωω
ω+−=
0 0.5 3 2z j= +
1 2
1
pK L LK L
+≤ 1
min1 2
pK LK
L L=
+
RMIT Smart Energy Systems 106AUPEC 2017 Tutorial
Final Gain Selection (Resonance Below ωcrit)
• Bounded range Solid line– Kmin=0.044A-1
– Kmax=0.124A-1
• Maximum damping K=0.083A-1
Low Frequency SystemL1 = 6mH L2 = 2mHCf=15μF fs = 5kHz
ωres = 6.67krads-1 ωc=0.36ωres
ωcrit=10.5krads-1 ωc=2.4krads-1
Kp=0.059A-1 Tr=4.2msec K=0.083A-1
ωresωcrit
RMIT Smart Energy Systems 107AUPEC 2017 Tutorial
5kVA Experimental Setup
Other System ParametersL1=6mH L2=2mH 2VDC=650V E=415Vrms(l-l)
fsw=5kHz fsamp=10kHz T=100μs
IGBT Power Stage
ControllerTMS320F2810
LCL Filter
RMIT Smart Energy Systems 108AUPEC 2017 Tutorial
High Resonant Frequency System Results
• Simulation (PSIM) • Experimental
• No visible oscillation
• Rapid transient response maximised gains
• Similar to single L filter system
• High ripple poor filtering
RMIT Smart Energy Systems 109AUPEC 2017 Tutorial
Low Resonant Frequency System Results
• Simulation (PSIM) • Experimental
• Transient damped in 3 to 4 oscillations– Agreement with root locus design underdamped poles
• Minor dynamic performance reduction– Limited bandwidth despite maximised gains
RMIT Smart Energy Systems 110AUPEC 2017 Tutorial
Summary of Linear Regulation for LCL Systems
• Discrete time model of actively damped LCL system developed– Including sampling and PWM transport delay
• Two distinct regions of operation– High resonant frequency Single loop controller Simple gain selection based on L filter Dynamic performance and filtering capacity similar to L filter
– Low resonant frequency Actively damped controller (dual loop) Root locus gain selection for maximally damped poles Reduced dynamic performance
– Critical frequency dual loop active damping ineffective• Simulation and experimental results validate theoretical
models, controller selection and gain calculation
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