Synchronverters: Inverters that mimicsynchronous generators

George WEISSTel Aviv University, Israel

QingChang ZhongLoughborough University, UK

New-ACE workshop, Loughborough, September 2010

Outline

We shall have a technological part (“how to do it”) followed by atheoretical part (“how to prove it”). Here are the items that weshall discuss in the first part:

I Motivation and relevant works;I Modelling of synchronous generators;I Implementation of a synchronverter;I Operation of a synchronverter;I Simulation results;I Experimental setup and results;I Potential applications.

Motivation

I Increasing share of renewable energy. In the UK, 20% target by2020.

I Regulation of system frequency and voltage: Currently mostinverters feed currents to the grid and do not take part in systemregulation and there is a need of voltage controlled inverters toconnect with weak grids.

I Threat to power system stability: Inverters have differentdynamics from conventional synchronous generators.

I The need for a smooth transition.

Our solution

I Synchronverters are inverters that mimic synchronousgenerators when viewed from the grid.

I The energy flow between the DC bus and the AC buschanges direction automatically according to the gridfrequency.

I Such converters can take part in the power systemregulation of frequency and voltage, in the same way assynchronous generators.

I Dynamically such converters behave like synchronousgenerators, thus, they can use the same controlalgorithms, developed over more than 100 years.

Relevant works

I Virtual synchronous machine (VISMA) by H. Beck and R.Hesse (2007)

I The voltages at the point of common coupling with the grid are measured to calculate the phase

currents of the VISMA in real time.

I These currents are used as reference currents for a current-controlled inverter. If the current

tracking error is small, then the inverter behaves like a synchronous machine, justifying the term

VISMA. However, a synchronous generator is a voltage source.

I The grid integration using control algorithms for SG was left as future work.

I Virtual synchronous generator (VSG) by the VSYNCproject (EU) since 2007.

I Add a short-term energy storage system to provide virtual inertia.

I The inverter does not have the dynamics of a synchronous generator.

I It can have frequency/voltage drooping.

I Project manager: Klaas Visscher (NL), also K. De Brabandere, B. Bolsens, J. Van den Keybus, A.

Woyte, J. Driesen, R. Belmans.

I Related work by C. Seo and P. Lehn.

I The inverter does not have the dynamics of a synchronous generator.

Some basics about inverters+

-

Rs, Ls va

vb

vc

ia

ib

ic

ea

eb

ec

VDC

C

Modelling of synchronous generators

I Motivation and relevant worksI Modelling of synchronous generators

I Electrical partI Mechanical part

I Implementation of a synchronverterI Operation of a synchronverterI Simulation resultsI Experimental setup and resultsI Potential applications

Synchronous generator - electrical partConsider a round rotor machine (without damper windings), with one pair of poles per phase (and one pair of poles

on the rotor) and with no saturation effects in the iron core. The stator windings can be regarded as concentrated

coils having self-inductance L and mutual inductance−M.

M

M M

Rs , L Rs , L

Rs , L

Rotor field axis

( 0=θ )

Field voltage

Rotation

N

Notation

Define

Φ =

ΦaΦbΦc

, i =

iaibic

and

cos θ =

cos θ

cos(θ − 2π3 )

cos(θ − 4π3 )

, sin θ =

sin θ

sin(θ − 2π3 )

sin(θ − 4π3 )

.

Flux linkages

The field (or rotor) winding can be regarded as a concentratedcoil having self-inductance Lf . The mutual inductance betweenthe field coil and each of the three stator coils is Mf cos θ.Assume that the neutral line is not connected, thenia + ib + ic = 0. The stator flux linkages are

Φ = Ls i + Mf if cos θ, (1)

where Ls = L + M, and the field flux linkage is

Φf = Lf if + Mf⟨i , cos θ

⟩, (2)

where 〈·, ·〉 denotes the conventional inner product. Thesecond term Mf

⟨i , cosθ

⟩is constant if the three phase currents

are sinusoidal (as functions of θ) and balanced.

Voltage

The phase terminal voltages v =[

va vb vc]T are

v = −Rs i − dΦ

dt= −Rsi − Ls

didt

+ e, (3)

where Rs is the resistance of the stator windings ande =

[ea eb ec

]T is the back emf, given by

e = Mf if θsin θ −Mfdifdt

cos θ. (4)

The field terminal voltage from (2) is

vf = −Rf if −dΦf

dt, (5)

where Rf is the resistance of the rotor winding.

Synchronous generator - mechanical part

The mechanical part of the machine is governed by

J θ = Tm − Te − Dpθ, (6)

where J is the moment of inertia of all parts rotating with therotor, Tm is the mechanical torque, Te is the electromagnetictoque and Dp is a damping factor. Te can be found from theenergy E stored in the machine magnetic field, i.e.,

E =12〈i , Φ〉+

12

if Φf

=12〈i , Lsi〉+ Mf if

⟨i , cos θ

⟩+

12

Lf i2f .

The electromagnetic torque Te

Te =∂E∂θm

∣∣∣∣Φ, Φf constant

= − ∂E∂θm

∣∣∣∣i, if constant

.

Since the mechanical rotor angle θm satisfies θ = pθm,

Te = pMf if⟨

i , sin θ⟩

. (7)

Note that if i = i0sin ϕ then

Te = pMf if i0⟨

sin ϕ, sin θ⟩

=32

pMf if i0cos(θ − ϕ).

Note also that if if is constant then (7) with (4) yield

Teθm = 〈i , e〉 .

Provision of a neutral line

The above analysis is based on the assumption that there is noneutral line. If a neutral line is connected, then

ia + ib + ic = iN ,

where iN is the current flowing through the neutral line. Then, theformula for the stator flux linkages (1) becomes

Φ = Ls i + Mf if cos θ −[

111

]MiN

and the phase terminal voltages (3) become

v = −Rs i − Lsdidt

+[

111

]M

diNdt

+ e,

where e is given by (4). The other formulas are not affected.

Real and reactive power

Define the generated real power P and reactive power Q as

P = 〈i , e〉 and Q = 〈i , eq〉 ,

where eq has the same amplitude as e but with a phase delayed byπ2 , i.e.,

eq = θMf if sin(θ − π

2) = −θMf if cos θ.

Then, the real power and reactive power are, respectively,

P = θMf if⟨

i , sin θ⟩

,

Q = −θMf if⟨i , cos θ

⟩. (8)

Note that if i = i0sin ϕ (as would be the case in the sinusoidalsteady state), then

P = θMf if⟨

i , sin θ⟩

=32θMf if i0cos(θ − ϕ),

Q = −θMf if⟨i , cos θ

⟩=

32θMf if i0sin(θ − ϕ).

These coincide with the conventional definitions for real powerand reactive power.

Implementation of a synchronverterI. The electronic part (without control)

It is advantageous to assume that the field (rotor) winding of thesynchronverter is fed by an adjustable DC current source ifinstead of a voltage source vf . In this case, the terminal voltagevf varies, but this is irrelevant. As long as if is constant, we have

e = Mf if θsin θ −Mfdifdt

cos θ.

= θMf if sin θ. (9)

The effect of the neutral current iN can be ignored if M ischosen as 0, because

v = −Rsi − Lsdidt

+[

111

]M

diNdt

+ e.

Virtual mechanical part:

θ =1J

(Tm − Te − Dpθ), Te = pMf if⟨

i , sin θ⟩

.

Back emf and reactive power:

e = θMf if sin θ, Q = −θMf if⟨i , cos θ

⟩.

Te Eqn. (7) Eqn. (8) Eqn. (9)

s

1

Dp

Tm

-

θ θ&

i

e

Mf if

Q

Js

1

-

II. The power partThis part consists of three phase legs and a three-phase LCfilter, which is used to suppress the switching noise. If theinverter is to be connected to the grid, then three moreinductors Lg (with series resistance Rg) and a circuit breakerare needed to interface with the grid.

+

-

Ls , Rs va

vb

vc

ia

ib

ic

ea

eb

ec

VDC

C

vga

vgb

vgc

Circuit Breaker

Lg , Rg

v = −Rsi − Lsdidt

+ e.

Interaction between the two parts

I The switches in the inverter are operated so that theaverage values of ea, eb and ec over a switching periodshould be equal to e given in (9), which can be achieved bythe usual PWM techniques.

I The phase currents are fed back to the electronic part.

Te Eqn. (7) Eqn. (8) Eqn. (9)

s

1

Dp

Tm

-

θ θ&

i

e

Mf if

Q

Js

1

-

+

-

Ls , Rs va

vb

vc

ia

ib

ic

ea

eb

ec

VDC

C

vga

vgb

vgc

Circuit Breaker

Lg , Rg

Operation of a synchronverterA. Operation objectives

I The frequency should be maintained, e.g., at 50Hz

I The output voltage should be maintained, e.g. at 230V.

I The generated/consumed real power should be regulated.

I The reactive power should be regulated, if connected tothe grid.

B. Frequency drooping

The speed regulation system of the prime mover for a conventionalsynchronous generator can be implemented in a synchronverter bycomparing the virtual angular speed θ with the angular frequencyreference θr before feeding it into the damping block Dp. As a result,the damping factor Dp actually behaves as the frequency droopingcoefficient, which is defined as the ratio of the required change oftorque ∆T to the change of speed (frequency) ∆θ:

Dp =∆T∆θ

=∆TTmn

θn

∆θ

Tmn

θn,

where Tmn is the nominal mechanical torque. Because of the built-infrequency drooping mechanism, a synchronverter automaticallyshares the load with other inverters of the same type and with SGsconnected on the same bus.

C. Voltage drooping

The regulation of reactive power Q flowing out of the synchronvertercan be realised similarly. Define the voltage drooping coefficient Dqas the ratio of the required change of reactive power ∆Q to thechange of voltage ∆v :

Dq =∆Q∆v

=∆QQn

vn

∆vQn

vn,

where Qn is the nominal reactive power and vn is the nominalamplitude of terminal voltage v . The difference between thereference voltage vr and the amplitude of the feedback voltage vfb isamplified with the voltage drooping coefficient Dq before adding to thedifference between the set point Qset and the reactive power Q. Theresulting signal is then fed into an integrator with a gain 1

K to generateMf if .

D. Complete electronic part

Js

1

Te Eqn. (7) Eqn. (8) Eqn. (9)

s

1

Dp

Tm

-

θ θ&

i

e

rθ&-

Dq

rv

Qset -

-

Mf if

Ks

1

Q

n

p

θ& Pset

PWM generation

From

\to th

e po

wer

par

t

fbv

Reset gθ

Amplitude detection

cθ

mv

The synchronverter - simulation and experiments

Parameters Values Parameters ValuesLs 0.45 mH Lg 0.45 mHRs 0.135 Ω Rg 0.135 Ω

C 22 µF Frequency 50 HzR 1000 Ω Line voltage 20.78 Vrms

Rated power 100 W DC voltage 42VDp 0.2026 Dq 117.88

Simulation results

I t = 0: Simulation started to allow the PLL

and synchronverter to start up;

I t = 1s: Circuit breaker on;

I t = 2s: Pset = 80W;

I t = 3s: Qset = 60Var;

I t = 4s: drooping mechanism enabled;

I t = 5s: grid voltage decreased by 5%.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 649.8

49.9

50

50.1

50.2Frequency (Hz)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2 Amplitude of v-vg (V)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.95

0.975

1

1.025

1.05Normalised vc

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-20

020406080

100120140 P (W)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-20

020406080

Time (Second)

Q (Var)

50Hz49.95Hz

Experimental setup

The synchronverter is connected to the grid, three-phase 400V 50Hz,via a circuit breaker and a step-up transformer.

Experimental results

The experiments were carried out according to the followingsequence of actions:

1. start the system, but keeping all the IGBTs off;2. start operating the IGBTs, roughly at 2s;3. turn the circuit breaker on, roughly at 6s;4. apply instruction Pset = 70W, roughly at 11s;5. apply instruction Qset = 30 Var, roughly at 16s;6. enable the drooping mechanism, roughly at 22s;7. stop data recording, roughly at 27s.

Case 1: Grid frequency > 50Hz

Time (Second)Fr

eque

ncy

(Hz)

(a) synchronverter frequency

Time (Second)

v−

v g(V

)

(b) voltage difference v − vg

Time (Second)

van

dv g

(am

plitu

de,V

)

v@I

vg@

@I

(c) amplitude of v and vg

Time (Second)

P(W

)and

Q(V

ar) P

XXyQ

¡ª

(d) P and Q

Case 2: Grid frequency < 50Hz

Time (Second)Fr

eque

ncy

(Hz)

(a) synchronverter frequency

Time (Second)

v−

v g(V

)

(b) voltage difference v − vg

Time (Second)

van

dv g

(am

plitu

de,V

)

v@I

vg@

@I

(c) amplitude of v and vg

Time (Second)

P(W

)and

Q(V

ar)

P@@I

Q¡¡ª

(d) P and Q

Potential applications

I Distributed generation and renewable energy, allowing thesesources to take part in the regulation of power system frequency,voltage and overall stability.

I Uninterrupted power supplies (UPS), in particular, the paralleloperation of multiple UPSs.

I Isolated/distributed power supplies, e.g., to replace rotaryfrequency converters.

I Static synchronous compensator (STATCOM) to improve thepower factor.

I HVDC transmission (at the receiving end).

I Induction heating.

Current status of the technology

I Patent application filed, entered into the PCT stageI Funding received for building proper prototypes & commercialisationI Conference paper appeared in ieeexploreI Journal paper to appear in IEEE Industrial ElectronicsI Applied to AC drives — AC Ward Leonard drive systems (PEMD2010,

SPEEDAM 2010)

Summary of the “how to do it” part

I An approach is proposed to operate inverters to mimic synchronous generatorsafter establishing the mathematical model of synchronous generators. Suchinverters are called synchronverters.

I Synchronverters can be operated in island mode or grid-connected mode. Whenit is connected to the grid, it can take part in the regulation of power systemfrequency and voltage, via frequency and voltage drooping.

I The energy flow between the DC bus and the AC bus changes directionautomatically according to the grid frequency.

I It can disconnect from the grid and can automatically re-synchronise andre-connect with the grid.

I Potential applications include grid connection of renewable energy sources,parallel operation of UPS, HVDC transmission, STATCOM, isolated/distributedpower supplies etc.

Modeling the synchronous generator as aport-Hamiltonian system

Rewriting the equations in the form

x = F (x , v), where x =

iaibifωθ

is very complicated. The situation becomes much better afterthe Park transformation (R.H. Park, 1929), which is unitary:

U(θ) =

√23

cos θ cos(θ − 2π3 ) cos(θ + 2π

3 )

− sin θ − sin(θ − 2π3 ) − sin(θ + 2π

3 )1√2

1√2

1√2

,

so that

idiqi0

= U(θ)

iaibic

,

vdvqv0

= U(θ)

vavbvc

.

Modeling ... - continued

Physical interpretation: id creates the flux parallel to the rotor, iqcreates the flux perpendicular to the rotor and i0 = 0. Thevoltage v0 is irrelevant (floating terminals). We introduce

x =

idiqifω

, v =

−vd−vq−vfTm

,

we denote, as usual, Ls = L + M, m =√

32Mf and

z =

ΦdΦqΦfJω

=

Ls 0 m 00 Ls 0 0m 0 Lf 00 0 0 J

·

idiqifω

.

Modeling ... - continuedWe denote the 4× 4 matrix on the last slide by L, so thatz = Lx . Assuming that Lf Ls > m2 we have L > 0. The totalenergy in the synchronous machine is

E =12〈Lx , x〉 = 〈L−1z, z〉.

The equations of the synchronous machine are very simple:

z = A(z)L−1z + v ,

y = L−1z (= x),

where

A(z) =

−Rs ωLs 0 0−ωLs −Rs 0 −mif

0 0 −Rf 00 mif 0 −Dp

.

Modeling ... - continuedThis is a port-Hamiltonian system (in the sense of van derSchaft and Maschke) since

L−1z =∂E∂z

(= x).

The general form of port-Hamiltonian systems is

z = A(z)∂E∂z

+ B(z)v ,

y = B∗(z)∂E∂z

,

where v is the input, z is the state and y is the output, E is apositive function and A(z) + A∗(z) ≤ 0. For such systems wehave E ≤ 〈u, y〉. In our case, B(z) = I and A(z) has a neatdecomposition into a skew-symmetric part (written in red) and astrictly negative part (written in black).

Properties of the synchronous generator system

From its special structure we can see easily that it is globallyasymptotically stable. Moreover, for every constant inputv0 ∈ R4 the system has a unique equilibrium state z0 suchthat

A(z)L−1z0 + v0 = 0.

This equilibrium state is locally asymptotically stable, but wedo not know if it is globally asymptotically stable.The synchronous machine is strictly output passive:

E ≤ 〈y , v〉 − µ‖y‖2, µ > 0.

From the facts that the system is passive and globallyasymptotically stable, it follows that the system is iISS stable,with iISS gain equal to the identity.

Properties ... -continued

This means the following: there exists a K∞ function α (growsfrom 0 to ∞) and a KL function β (grows from zero in the firstcoordinate, decays to zero in the second coordinate) such that

α(‖z(t , z0, v)‖) ≤ β(‖z0‖, t) +

∫ t

0‖v(σ)‖dσ.

This follows from a recent result of Chen Wang and GW(IEEE-TAC, 2008). It is based on earlier results by• D. Angeli, E. Sontag, Y. Wang (2000),• B. Jayawardhana, A. Teel, E. Ryan (2007).We would like to have an iISS type result around eachequilibrium point. We can only prove passivity around eachequilibrium point.Aim for later: stability of joint equilibrium points of severalsynchronous machines connected in parallel.