1

Adaptive Control of Grid-Connected Inverters Based on Online Grid

Impedance Measurements

2013 CFES Annual ConferenceTroy, New York

Mauricio Cespedes and Jian SunRensselaer Polytechnic Institute

2

Outline of the Presentation• Effect of Grid Impedance on Stability of

Grid-Connected Inverters• Inverter Impedance Models• Derivation of Adaptive Rules• Online Grid Impedance Identification• Experimental Demonstration• Conclusions

3

Inverter and Grid Impedances

L

PLL

PWM PLL

ia

ib

icdcV

va

vb

vc

Current Control System

Util

ity

Filte

r

Zg

Zi

Ac GridGrid-Connected Inverter

Characteristic Polynomial

p(s) = 1+Zg(s) / Zi(s)

Applies to Both Positive-Sequence and Negative-Sequence

4

-3 -2 -1 0 1 2 3 4

0

1

2

3

100 Hz 1 kHz 10 kHz-200-150-100-50

050

100150

100 Hz 1 kHz 10 kHz10

20

30

40

50

Imag

inar

y Pa

rtAnalysis of the Impedance Ratio

Nyquist Plot of Impedance Ratio

← =170°

←Hz

Sequence Gain Margin Phase Margin

Positive 2.7 dB @ 420 Hz 10° @ 440 Hz

Negative 7.5 dB @ 350 Hz 23° @ 440 Hz

Phas

e (D

EG)

Mag

nitu

de (d

B

)f ZpZn

Zg

5

Harmonic Resonanceib (4 A/div.)ia (4 A/div.) ic (4 A/div.)

6

0.00 0.01 0.02 0.03 0.04 0.05

-100

1020

0.00 0.01 0.02 0.03 0.04 0.05

-2.00

2.04.0

Sequence ComponentsC

urre

nt (A

)

icia ib

|I h|/|

I 1| (

%)

ipin

Cur

rent

(A)

ss ssss

thththrd stth th

Negative Seq.

Positive Seq.*Based on Last Cycle

7

Inverter Impedance Models

where TPLL(s) V1HPLL(s)/[1+V1HPLL(s)]

Symbol Value 2·(60 Hz)Vdc 550 VL 1.8 mH

Kp 0.098Ki 485Kp 3.14Ki 2365

V 120 √2 VI1 7.5 Ai1 0 rad

Td 25 s

Zn(s) =

Zp(s) =

Gv(s) ~

Applying the HarmonicLinearization Method:

8

Impedance Model Verification

10 Hz 100 Hz 1 kHz 10 kHz

1020304050

10 Hz 100 Hz 1 kHz 10 kHz

-150-100-50

050

100150

Phas

e (d

eg.)

Mag

nitu

de (d

B

)

nZ

pZ

nZ

pZ

• Around 60 Hz– Fundamental Components

Are Measurement Noise

• At Harmonic Frequencies– Capacitive Converter

Impedance Explains Resonance with Inductive Grids

– Wide PLL Bandwidth Introduces Negative Damping Around 100 Hz

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Adaptive Control Rule Derivation• Apply the Routh-Hurwitz Stability Criterion to

the Characteristic Polynomial– Positive Sequence– Negative Sequence

• Simplify the Grid Impedance Model– Assume Inductive Grid– Valid at Low Frequencies

• Simplify the Inverter Impedance Model

10

Inverter Impedance Simplification• Simplification of Inverter Impedance Models

1.

2.

3.

• Simplified Characteristic Polynomial

11

Adaptation Rule of PLL Bandwidth• Stability Conditions

1.

2.

3.

• System is Stable if and Only If

,

i: Current Loop Bandwidth; LB is base inductance

12

Roots of Characteristic Polynomial

(0.03 pu)

Introduce Adaptive Rule in the Full-Order System Model, No Simplifications:

13

Summary of PLL Adaptation• Adaptation Rule Derived Assuming f >> f1

– Hence Stability Cannot Be Guaranteed if Lg > 0.33 pu

• Adaptive Rule Does Not Guarantee Minimum Damping Ratio– Rotation Mapping Introduces Complex-Valued

Coefficients to the Characteristic Polynomial

z-planes-plane

Routh-Hurwitz Fails

14

CurrentControl

Online Zg(s) Identification

Ac

Grid

Use the Inverter DSP to Identify Zg(s)

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Impulse Response Methodia (3 A/div.) ic (3 A/div.)ib (3 A/div.)

Cur

rent

(p.u

.)

Cur

rent

(A)

DSP Sample Number (k)

←As Seen on the DSP

• Pros– Update Estimate Every Two Fundamental Cycles– Insensitive to the Power System Non-Stationarity

• Cons– Excitation of Nonlinear Dynamics

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Grid Impedance Identification

Cf

Rd

Element Value PerspectiveLg 3.3 mH 0.06 puRd 1.87 3 watts / phaseCf 20 F 0.12 pu

Utility Grid 164 H 0.03 pu

Lg

Util

ity

Zn

Zp

←Sweep Results

←DSP

←Sweep Results

←DSP

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Adaptation of Control System• Assume Grid Impedance

is Inductive– Extract it from the Non-

Parametric Estimation• Adjust PLL Bandwidth

Based on Grid Inductance– Gain Scheduling Control– No Dynamics Associated to

the Adjustment

18

Experimental DemonstrationL

PLL

PWM

Hi(s)

ibr

Hi(s)

PLL

ia

ib

icdcV

va

vb

vc Util

ity

Hi(s)

dq

abc iqr

idr

Cf

Rd Lg

Enable

Grid Parameter ValueLx 10 mH (0.34 pu)Lg 5 mH (0.17 pu)Rd 1.2 W

Cf 10 F (0.04 pu)

Lx

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Identification of Strong Grid

←Pulse for Identification

ib (2.5 A/div.)ia (2.5 A/div.) ic (2.5 A/div.)

PLL ~ 2·(120 Hz)Lg>

20

-3 -2 -1 0 1 2 3 4

-1

0

1

2

100 Hz 1 kHz 10 kHz-200-150-100-50

050

100150

100 Hz 1 kHz 10 kHz10

20

30

40

50

Impedance RatioIm

agin

ary

Part

Nyquist Plot of Impedance Ratio

Phas

e (D

EG)

Mag

nitu

de (d

B

)f

Zp

Zn

Zg

f

Sequence Gain Margin Phase Margin

Positive > 20 dB > 30o

Negative > 20 dB > 30o

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Weak Grid Transition

ib (2.5 A/div.)ia (2.5 A/div.) ic (2.5 A/div.)

PLL ~ 2·(120 Hz)Lg

>

Enable Changes State

22100 Hz 1 kHz 10 kHz

-200-150-100-50

050

100150

100 Hz 1 kHz 10 kHz10

20

30

40

50

-3 -2 -1 0 1 2 3 4

-1

0

1

2

Impedance RatioIm

agin

ary

Part

Nyquist Plot of Impedance Ratio

Phas

e (D

EG)

Mag

nitu

de (d

B

)

f

Zp

Zn

Zg

f

Sequence Gain Margin Phase Margin

Positive 2.5 dB @ 200 Hz 13.6o @ 300 Hz

Negative 12 dB @ 30 Hz 19.0o @ 300 Hz

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Resonance During Weak Grid

Inductance Estimate Has Not Been Updated

ib (2.5 A/div.)ia (2.5 A/div.) ic (2.5 A/div.)

PLL ~ 2·(120 Hz)Lg

>

Estimate Has Not Been Updated

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0.00 0.01 0.02 0.03 0.04 0.05

-5.00

5.010

0.00 s 0.01 s 0.02 s 0.03 s 0.04 s 0.05 s

-0.50

0.51.0

Sequence ComponentsC

urre

nt (A

)

icia ib

|I h|/|

I 1| (

%)

ipin

Cur

rent

(A)

thththrd stth th

Negative Seq.

Positive Seq.

*Based on Last Cycle

25

Adaptation to Weak Grid

ib (2.5 A/div.)ia (2.5 A/div.) ic (2.5 A/div.)

Lg

>

Estimate Has Been Updated PLL ~ 2·(40 Hz)

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Summary of Adaptation Transient

Weak Grid Transition

Harmonic Resonance

Adaptation

Estimate Is Updated Every 200 ms (12 Fundamental Cycles)

in: Negative Sequenceip: Positive Sequence

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Conclusions• Proposed and Demonstrated Adaptation of

PLL Bandwidth and Grid Voltage Feedforward Gains to Guarantee System Stability Based on Online Grid Impedance ID– Used the Conventional Routh Hurwitz Criterion

• Developing Conditions for Constant Damping Ratio Involve Complex-Valued Coefficients in the Characteristic Polynomial– Used the Generalized Routh-Hurwitz Criterion