Modeling of Power Converters Connected to the Electrical Grid

Chair of Power ElectronicsChristian-Albrechts-Universität zu KielKaiserstraße 224143 Kiel

Modeling of Power Converters Connected to the Electrical Grid

Prof. Marco Liserre, PhD, IEEE fellowHead of the Chair of Power [email protected]

Marco Liserre, Zhixiang Zou, Roberto Rosso, Giovanni De Carne

Chair of Power Electronics | Prof. Marco Liserre | [email protected] 1

Contents

Model Deepness of Grid-connected Power Converter

Modeling of PWM Converters

Current Control of PWM Converters

Grid Synchronization for Three-phase Power Converter

Power and dc Control of Grid-connected Power Converter


Model Deepness of Grid-connected Power Converter


DC-link capacitor

Semiconductors or voltage source?

AC filter

Inner control loop

Outer control loop

Only electrical or thermal too?

PLL

Key Components of Modeling


Stability issues:

AC small signal stability

Sub-Synchronous signal stability

DC voltage stability

AC transient stability

Frequency stability

High PE integration

Interaction among controllersOff-shore wind connection

HVDC + governor interactionHybrid AC/DC grids

Multi-terminal DC gridsLow Voltage Ride Through

Angle/Voltage stability with PEHigh PE penetration impact on system

Virtual Inertia strategies

Grid Integration Studies


A Mapping between the Forms of Harmonic Instability and Components

Source: X. Wang and F. Blaabjerg, "Harmonic Stability in Power Electronic Based Power Systems: Concept, Modeling, and Analysis," IEEE Transactions on Smart Grid, in press.


Linearized Modeling Methods for Power Converters

Source: X. Wang and F. Blaabjerg, "Harmonic Stability in Power Electronic Based Power Systems: Concept, Modeling, and Analysis," IEEE Transactions on Smart Grid, in press.


Goal:Study the harmonic behavior of power electronics-based converter.

Issue:Converter’s inner controller may resonate with the passive elements of the grid (e.g., filter).Focus on modeling:The PLL has to be included in the model to consider low order harmonic resonances. Difference of converter self-admittance with

or without PLL modeled

AC Small-signal Stability


0 1 2 3 4 5 6 7 8

Time (s)

-4

-2

0

2

4

Mag

nitu

de (p

u)

LPB to Gen torque

0 1 2 3 4 5 6 7 8

Time (s)

-4

-2

0

2

4

Mag

nitu

de (p

u)

Phase a current

0 1 2 3 4 5 6 7 8

Time (s)

-2

-1

0

1

2

Mag

nitu

de (p

u)

Electromagnetic torque

Goal:Study the low order (<50Hz) phenomena.

Issue:HVDC controller interacts with turbine-generation shaft, oscillating at low frequency.Focus on modeling:The outer control loop modeling impacts on the oscillation observed. Difference between PQ control mode (blue

line) and Vdc control mode (red line) of HVDC systems connected to a wind park

AC sub-synchronous signal stability


Difference of DC-link modeling during a power variation

Goal:Study the voltage stability and its dynamic behavior in DC grids.

Issue:Fast power variations have an impact on the grid stability and on the component stress

Focus on modeling:The DC-link passive components modeling shall be considered to evaluate the stability.

DC Voltage Stability


Goal:Study the grid behavior during large disturbances (e.g., faults)

Issue:HVDC DC-link voltage are affected by large AC voltage variations and voltage control loop dynamics.Focus on modeling:The inner control loop (voltage/current) shall be considered for stability issues. (a) Phasorial, and (b) average model

including inner control loop of HVDC converters

Transient stability


Goal:Estimate the frequency stability in micro-/ ST-based grids under fast power variations.

Issue:PLL and DC-link dynamics may interact, leading to large frequency oscillations.

Focus on modeling:PLL and DC-link models must be considered to avoid under-estimation of frequency issue. Difference of DC-link modeling during a

power variation

Frequency stability


Com

plex

ityof

the

mod

elModeling deepness scale


Two possibilities: Class D

• Ideal DC-link (piloted voltage source)• ideal PLL (infinite bandwidth)

Class C• Capacitor model included• Simplified PLL model (only dynamic)


CLASS C is the minimum requirement!

Modeling Scale - Example 1:Frequency Stability



Three possibilities: Class D

• Ideal DC-link (voltage source)• Equivalent control model (dynamic)

Class C• Capacitor model included• Equivalent control model

Class B• Capacitor model included• Detailed control modelCLASS C is sufficient for stability studies!

Modeling Scale - Example 2:dc Transient Voltage Stability

Source: Report of M.Sc. Freiber Rojas, Karlsruhe Institute of Technology


Modeling of PWM Converters


Switching Ripple due to PWM

Source: R.W. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Springer US, 2001.


Output Voltage Spectrum with Sinusoidal Modulation

Source: R.W. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Springer US, 2001.


Averaging to Remove Switching Ripple


Perturbation and Linearization


The Small-signal Assumption


Perturbation of Inductor Equation


Perturbation of Inductor Equation


Neglect of Second-order Terms


Linearized Inductor Equation


TopologyConsiderations:

• The switches are ideal

• The output filter is an inductor and a resistor

• Ts is the switching period

Equations of the circuit: The average voltage over one switching period is:

Modeling of the Single-Phase Inverter


Modulation and Switching Function • Switching function is the commutated signal

• Duty-cycle is the average value of the switching function:

- is the switching function of leg “a“.- is the switching function of leg “b“.

The voltage va and vb can be defined as:

The voltage va and vb can write in term of its swithing function:

The one period average voltages va and vb are:

Switching Function


(1)

(2)

(3)

(4)

Previous equations: Replace (3) and (4) in (1):

For the modulation strategy:

Then, equation (5) becomes:

(6)

(5)

The average equation of the ac side and the equivalent average circuit are shown below:

Equivalent average circuit of the ac side

Modeling of the ac Side


(6)

Thus, the average equivalent circuit of the dc side is:

Equivalent average circuit of the dc side

Equation of the circuit:

The leg currents are related with the switching function by:

Replacing (7) in (6):

(7)

The average dc current over one switching period is:

Modeling of the dc Side


Combining the equivalent circuits of the ac side and dc side, it is obtained the equivalent average circuit of the single-phase converter.

Average Model of the Inverter


Modeling of the Three-Phase Inverter

Topology with L filter:

The switching function:

where:

The ac voltage equation:

Equations of the circuit:


The ac voltage equation: Equations of the circuit:

From αβ reference frame to dq reference frame



The ac voltage equation in αβ frame : Equations of the circuit:


The ac voltage equation in dq frame :




The ac voltage equation in αβ frame :


The ac voltage equation in dq frame :

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

1

1

dq d d d dc

qd q q q dc

di ti t Ri t e t p t V

dt Ldi t

i t Ri t e t p t Vdt L

ω

ω

− = − − +

+ = − − +

Average circuit of the ac side in dq-frame:


Equivalent average circuit of the dc side:

The current of dc side can be described in terms of the switching function and output current, as:

Pertubing and linearizing the previous equation, the main dc side current equation is obtained:



Equivalent Circuit

Equivalent average circuit of the three-phase inverter in dq-frame:


Equivalent average circuit of the dc side:

Equivalent average circuit of the three-phase inverter in dq-frame:



Current Control of PWM Converters


PWM current control methods

ON/OFF controllers with pulse width modulator

hysteresis predictive optimized

linear Non - linear

fuzzy passivity

PI predictive Dead - beat

resonant repetitive

38

Current Control Overview

Source: R. Teodorescu, M. Liserre, and P. Rodriguez, Grid Converters for Photovoltaic and Wind Power Systems. John Wiley & Sons, 2011.


• Typically PI controllers are used for the current loop in grid inverters• Technical optimum design (damping 0.707 overshoot 5%)

gv

v∗ i( )dG s ( )fG s( )PIG si∗

e

+− ++ +−

( ) IPI P

kG s ks

= +

1( )1 1.5d

s

G sT s

=+

( ) 1( )( )f

i sG sv s R Ls

= =+

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

10-1 100 101 102 103 104-20

-15

-10

-5

0

Mag

nitu

de (D

b)

100 101 102 103 104-400

-300

-200

-100

0

Frequency (Hz)

Pha

se (D

egre

e)

PI Current Control



• When the current controlled inverter is connected to the grid, the phaseerror results in a power factor decrement and the limited disturbancerejection capability leads to the need of grid feed-forward compensation

• However the imperfect compensation action of the feed-forward controldue to the background distortion results in high harmonic distortion of thecurrent and consequently non-compliance with international power qualitystandards

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-0.2

0

0.2

0.4

0.6

0.8

1

error (scaled)

referenceactual

time [s]0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-1.5

-1

-0.5

0

0.5

1

error

actualreference

0.019 0.0192 0.0194 0.0196 0.0198 0.02 0.0202 0.0204 0.0206 0.0208 0.021

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.023 0.0235 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.02750.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

time [s]

steady-state magnitude and phase errorlimited disturbance rejection capability

Shortcomings of PI controller



• Resonant control is based on the use of Generalized Integrator (GI)• A double integrator achieves infinite gain at a certain frequency, called resonance

frequency, and almost no attenuation outside this frequency

• The GI will lead to zero stationary error and improved and selective disturbancerejection as compared with PI controller

101

102

103

-180

-90

0

90

180

Phas

e (de

g)

Bode Diagram

Frequency (Hz)

-200

-100

0

100

200

Magn

itude

(dB)

time [s]

inpu

t and

out

put o

f the

reso

nant

con

trolle

r

2 2

ss ω+

2 2

ss ω+

GI

Resonant Control


Grid Synchronization for Three-phase Power Converter


Grid Synchronization by Using a PLL


Grid Synchronization by Using a PLL


Synchronous Reference Frame PLL (SRF-PLL)


SRF-PLL under Unbalanced and Distorted Grid



SRF-PLL under Unbalanced and Distorted Grid



During the last decade control structures emulating the behaviour of a synchronous machine (SM)have been proposed. Such control structures have been categorized as virtual synchronousmachines (VSMs).

Since they can reproduce the power-synchronization mechanism of a SM, they do not need adedicated unit for synchronation to the grid. The generalized structure of a power-synchronization based control is reported below:

Source: R. Rosso, S. Engelken, and M. Liserre, ” A generalized formulation of active power synchronization based control algorithms for grid connected converters,” in Proc. 44th IEEE IECON, Washington DC.

PLL-less Converters:General structure


The synchronverter is among the most common control structuresproposed in the literature based on the power-synchronization mechanismof a SM.

Source: Q.-C. Zhong, P.-L. Nguyen, Z. Ma, and W. Sheng, ”Self-synchronized synchronverters: inverters without a dedicated synchronization unit,” IEEE Trans. Power Electron., vol. 29, no. 2, pp. 617-630, Feb. 2014.

Active power loop

Reactive power loop

PLL-less Converters: Synchronverter


The Active power loop is described by the swing equationof a SM:

The reactive power loop reproduces a Q-V droop behaviour according to thecoefficient 𝐷𝐷𝑞𝑞:

𝐽𝐽ω̇ = 𝑇𝑇𝑚𝑚 − 𝑇𝑇𝑒𝑒 − 𝐷𝐷𝑝𝑝ω

Mechanical inertia

Rotor speed Mechanical torque

Electrical torque

Mechanicalfriction

Δ𝑄𝑄 = −𝐷𝐷𝑞𝑞Δ𝑉𝑉

Voltage deviation at the PCC

Reactive power setpoint variation

The synchronverterControl structure


A linearized small-signal model of the synchronverter connected to the grid through an LCL filterhas been obtained for investigation purposes and a design procedure has been proposed.

The synchronverterSmall-signal model

Source: R. Rosso, J. Cassoli, S. Engelken, G. Buticchi, and M. Liserre, ”Analysis and Design of LCL Filter Based Synchronverter”, 2017 IEEE Energy conversion conference and exposition (ECCE), Cincinnati, 2017.R. Rosso, J. Cassoli, G. Buticchi, S. Engelken, and M. Liserre, ”Robust stability analysis of LCL filter based synchronverter under different grid conditions,” IEEE Trans. Power Electron., 2018, DOI 10.1109/TPEL.2018.2867040.

C G


The synchronverterParallel operation

In order to study the interactions among synchronverters operating in parallel, thecomponent connection method (CCM) has been adopted in the reference.

It allows obtaining the state-space representation of a complex system connecting toeach other separated subsystem, whose state-space representations are known.

Source: R. Rosso, S. Engelken, and M. Liserre, ”Analysis of the behavior of synchronverters operating in parallel by means of component connection method,” in Proc. 2018 IEEE Energy Conversion Congress and Exposition (ECCE), Portland, OR, 2018.


Once the state-space representation of the system is obtained, eigenvalue analysiscan be performed in order to assess system stability.

critical eigenvalues

Source: R. Rosso, S. Engelken, and M. Liserre, ”Analysis of the behavior of synchronverters operating in parallel by means of component connection method,” in Proc. 2018 IEEE Energy Conversion Congress and Exposition (ECCE), Portland, OR, 2018.

The synchronverterParallel operation


Park transformation:

Small phase perturbation (<7deg):

Large phase perturbation (<38deg):

Modeling of PLL-based Converter

Source: Z. Zou and M. Liserre, Study of Phase-locked-loop-based Synchronization of Grid Inverter During Large Phase Jump. in ECCE, 2018.


Average model:

Updating model with second-order terms:




The proposed model gives a general way to analyze PLL-based converter under different conditions– Stability analysis without PLL– Stability analysis with PLL during small phase jump– Stability analysis with PLL during large phase jump

Complete model of PLL-based grid converter:

Static analysis

Small-signal analysis with PLL

For large phase perturbation




Stability Analysis with Different Models

Conventional small-signal

model

Higher accurate model

Stable

Unstable

An unstable case of PLL-based converter due to phase jump


Interactions among PLL-basedconverters operating in parallel

A simplified converter model considering only the PLL in the inner control loop hasbeen adopted in order to investigate the interactions among PLL-based converters.

Impedance-based stability analysis has been performed for investigation purposes.



The problem formulation allows the calculation of a non-conservative stability marginbased on the small-gain theorem.

This not only allows to assess whether the system is stable or not, but also providesinformation about the robustness with respect to parameter variations.



Through a Monte Carlo analysis the following aspects have been highlighted:

The stability margin of a converter connected to the grid through a PLL decreases increasing thebandwidth of the PLL and the impedance of the grid.

The stability margin further decreases if there is another converter operating nearby.

The interactions among PLL based converters become stronger increasing the grid impedance.


Power and dc Control of Grid-connected Power Converter


Power Control in Synchronous Frame: PQ Open Loop Control

• Active and reactivepower feed-forwardcontrol

• Vdc control acts on thepower reference

d

q

qi di

i→

timetimeqi

ω

IP

kks

+

dcv

di∗

qi∗

dcV controller

PLLgv gv α

gv β

f gV

θ

je θ−

gdv

gqv

*vje θ−

iα

iβ

di

qi

di∗

di

gdv

qi∗

gqv

θ θ

je θ

vα∗

vβ∗

IP

kks

+

IP

kks

+

Lω−

Lω

i

2 2

1 gd gq

gq gdgd gq

v vv vv v

− +

gdv gqv

+-

+-

Currentcontroller

Currentcontroller

Σ

Σ

+- X

+-

dcV ∗

P∗

Q∗

αβ

abc

abc

αβabc

αβ



Power Control in Stationary Frame: PQ Open Loop Control

• PLL may be avoided but it is used for making the control freq. adaptive and compute the first harmonic

• Active and reactive powerfeed-forward control

• Vdc control acts on thepower reference

β

α

iβ

iα

i→

time

time

iβ2 2

IP

k sks ω

++

iα∗

iβ∗

iiα

iβ

iα∗

iβ∗

vα∗

vβ∗

PLLgvgv α

gv β

θ,f ω

*v

dcV controller

IP

kks

+

2 2

1 g g

g gg g

v vv vv v

α β

β αα β

− +

gv α gv β

X

dcv

+-

+-

Currentcontroller

Currentcontroller

+-

+-

P∗

Q∗

dcV ∗

αβ

abc

abc

αβ

2 2I

Pk sk

s ω+

+



+

iVSI

-v

-dcvC

+ +

-

TOTL

e

Produced power Delivered power

Capacitor power

Voltage decreases

+

iVSI

-v

-dcvC

+ +

-

TOTL

e

Produced power Delivered power

Capacitor power

Voltage increases

+

iVSI

-v

-dcvC

+ +

-

TOTL

e

Absorbed power Absorbed power

Capacitor power

Voltage increases

+

iVSI

-v

-dcvC

+ +

-

TOTL

e

Absorbed power Absorbed power

Capacitor power

Voltage decreases

DC Voltage Control



• The control of the dc voltage through the ac current can result in theidentification of two loops, an outer dc voltage loop and an internalcurrent loop

• The internal loop is designed to achieve short settling times• On the other hand, the outer loop main goals are optimum regulation and

stability thus the voltage loop could be designed to be some what slower• Therefore, the internal and the external loops can be considered

decoupled

CC VC

synch

+

-

+- +-

dcv

dcv∗i

i∗Modulator

+

-

i TOTL

e

e *

(3 )2

sdc

dc

P n Tv

C v∆ +

∆ =⋅ ⋅

DC Voltage Control: Cascaded Control


+

-

+

-

+-

oi dci iTOTL

dcv v e

+-

11pI

ksT

+

dcv∗ 11 xsT+

dcv

dc-voltage loop plantCurrent loopdc-voltage PI

=sampling time

DC Voltage Control: Small Signal Analysis


It must be nearby ωcr!!!

DC Voltage Control


DC Loads

4 5 6 7 8 9

200

400

600

800

1000

-10 -9 -8 -7 -6 -5 -4

200

400

600

800

1000

0 1 2 3 4 5 6 7 80

200

400

600

800

1000

V/I characteristics

Step response (Load/Source step change)

Constant power load

Constant power source

Resistive load

Resistive load

Constant power load

Constant power source

Constant current load

Constant current source

CCL RL CPL

Overshoot 47 V 44 V 50 V

Settling time 0.153 s 0.153 s 0.155 s


Conclusion

Model Deepness of Grid-connected Power Converter is related to the problem to be studied

Modeling of PWM Converters -> averaging and linearization

Current Control of PWM Converters -> generalized integrator

Grid Synchronization for Three-phase Power Converter -> PLL and power synchronization-based converters

Power and dc Control of Grid-connected Power Converter -> constant power case

Documents

Modeling of Power Converters Connected to the Electrical Grid