Stability and power sharing in microgrids - TU Berlin · Stability and power sharing in microgrids J. Schiffer 1, R. Ortega2, J. Raisch;3, T. Sezi4 Joint work with A. Astolﬁ and

Stability and power sharing in microgrids

J. Schiffer1, R. Ortega2, J. Raisch1,3, T. Sezi4

Joint work with A. Astolfi and T. Seel

1Control Systems Group, TU Berlin2 Laboratoire des Signaux et Systemes, SUPELEC

3Systems and Control Theory Group,Max Planck Institute for Dynamics of Complex Technical Systems

4Siemens AG, Smart Grid Division, Energy Automation

HYCON2 Workshop, ECC 2014

Strasbourg, June, 24 2014

Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook

Outline

1 Motivation

2 Microgrids: concept and modeling

3 Problem statement

4 Asymptotic stability and power sharing in droop-controlledmicrogrids

5 Simulation example

6 Conclusions and outlook

2 / 38


Renewables change power system structure

Generation

Consumption

Transmission and distribution

High voltage (HV)transmission system

SG...

SG

SG synchronousgenerator

DG distributedgeneration unit

Medium voltage (MV)distribution system Large loads

Low voltage (LV)distribution system Small loads

3 / 38


Renewables change power system structure

Generation

Consumption

Transmission and distribution

High voltage (HV)transmission system

SG...

SG

SG synchronousgenerator

DG distributedgeneration unit

Medium voltage (MV)distribution system Large loads

DG...

DG

Low voltage (LV)distribution system Small loads

DG...

DG

3 / 38


Need change in power system operation

Increasing amount of renewable DG units highly affects in-feedstructure of existing power systemsMost renewable DG units interfaced to network via AC invertersPhysical characteristics of inverters largely differ fromcharacteristics of SGs

⇒ Different control and operation strategies are needed

Source: siemens.com

4 / 38


The microgrid concept

PCC

Transformer

Main electrical network

1

2

3

4

5 6

7

8

910

11

Load

PV Load

PV

Load

PV

Storage

FCLoad

PV

Load

Wind

Load

PV

Load

PV

FC

SGLoad

PV

Storage

FCLoad

PV

Load

5 / 38


Modeling of microgrids

Main network componentsDG units interfaced to network via inverters or SGsLoadsPower lines and transformers

Standard modeling assumptionsLoads can be modeled by impedancesLine dynamics can be neglected

⇒ Work with Kron-reduced network⇒ DG unit connected at each node in reduced network

Main focus: inverter-based microgrids

6 / 38


Main operation modes of inverters in microgrids

1 Grid-feeding modeInverter provides prespecified amount of active and reactivepower to grid

2 Grid-forming modeInverter is operated as AC voltage source with controllablefrequency and amplitude

Grid-forming units are essential components in power systemsTasks

To provide a synchronous frequencyTo provide a certain voltage level at all buses in the network

⇔ To provide a stable operating point

⇒ Focus on inverters in grid-forming mode

7 / 38


Basic functionality of DC-AC voltage inverters

Powerelectronics

L

C2C1

t

vAC,2

t

vDC

t

vAC,1

∼=

Inverter

vDC vAC,1 vAC,2

8 / 38


Inverter operated in grid-forming mode

vDC2

vDC2

vDC

Rf1 Lf

Cf

Rf2

vIaRg Lg vGa

vIb vGb

vIc vGc

9 / 38


Inverter operated in grid-forming mode

vDC2

vDC2

vDC

Rf1 Lf

Cf

Rf2

vIaRg Lg vGa

vIb vGb

vIc vGc

Modulator

Current controller

Voltage controller

iref

vref

Digital

sign

alprocessor

(DSP)

vI

iI

9 / 38


Grid-forming inverter as controllable voltage source

Model assumptions:Inverter is operated in grid-forming modeIts inner current and voltage controllers track references ideallyIf inverter connects intermittent DG unit (e.g., a PV plant) tonetwork, it is equipped with some sort of fast-reacting storage(e.g., a flywheel or a battery)

Inverterwith LCfilter and

innercontrolloops

Rg Lg vGa

vGb

vGc

vIa

vIb

vIc

vref

10 / 38


Grid-forming inverter as controllable voltage source

Model assumptions:Inverter is operated in grid-forming modeIts inner current and voltage controllers track references ideallyIf inverter connects intermittent DG unit (e.g., a PV plant) tonetwork, it is equipped with some sort of fast-reacting storage(e.g., a flywheel or a battery)


innercontrolloops

Rg Lg vGa

vGb

vGc

vIa

vIb

vIc

vref

10 / 38


Model of a single grid-forming inverter

Inverter dynamics

αi = uδi ,

τPiPm

i = −Pmi + Pi ,

Vi = uVi ,

τPiQm

i = −Qmi + Qi

Inverter output = symmetric three-phase voltage

yi = vIabci= Vi

sin(αi )

sin(αi − 2π3 )

sin(αi + 2π3 )

uδi control inputsuV

i

Pi active powerQi reactive power

Pmi measured

active powerQm

i measuredreactive power

τPitime constantof meas. filter

⇒ MIMO (multiple-input-multiple-output) system

11 / 38


The dq0-transformation

Let φ : R≥0 → Tdq0-transformation Tdq0 : T→ R3,

Tdq0(φ) :=

√23

cos(φ) cos(φ− 23π) cos(φ+ 2

3π)

sin(φ) sin(φ− 23π) sin(φ+ 2

3π)√2

2

√2

2

√2

2

dq0-transformation applied to symmetric three-phase signal

xabc =

xa

xbxc

= A

sin(α)

sin(α− 2π3 )

sin(α + 2π3 )

⇓

xdq0 =

xdxq

x0

= Tdq0(φ)xabc =

√32

A

sin(α− φ)

cos(α− φ)

0

⇒ x0 = 0 for all t ≥ 0

12 / 38


The dq0-transformation

Let φ : R≥0 → Tdq0-transformation Tdq0 : T→ R3,

Tdq0(φ) :=

√23

cos(φ) cos(φ− 23π) cos(φ+ 2

3π)

sin(φ) sin(φ− 23π) sin(φ+ 2

3π)√2

2

√2

2

√2

2

dq0-transformation applied to symmetric three-phase signal

xabc =

xa

xbxc

= A

sin(α)

sin(α− 2π3 )

sin(α + 2π3 )

⇓

xdq0 =

xdxq

x0

= Tdq0(φ)xabc =

√32

A

sin(α− φ)

cos(α− φ)

0

⇒ x0 = 0 for all t ≥ 0

12 / 38


Model of a grid-forming inverter in dq-coordinatesInverter dynamics in new coordinates

δi = ωi − ωcom = uδi − ω

com,

τPiPm

i = −Pmi + Pi ,

Vi = uVi ,

τPiQm

i = −Qmi + Qi

Inverter output in new coordinates

ydqi= Vi

[sin(δi )

cos(δi )

]

δi := αi − φ = α0i+

∫ t

0

(αi − ω

com) dτ = α0i+

∫ t

0

(ωi − ω

com) dτ ∈ T,

φ =

∫ t

0ωcomdτ, ωcom ∈ R (e.g. synchronous frequency)

13 / 38


Power flow equations

Active power flow at i-th node

Pi (δ1, . . . , δn,V1, . . . ,Vn) = GiiV2i −

∑k∼Ni

(Gik cos(δik )− |Bik | sin(δik )) ViVk

Reactive power flow at i-th node

Qi (δ1, . . . , δn,V1, . . . ,Vn) = |Bii |V2i −

∑k∼Ni

(Gik sin(δik ) + |Bik | cos(δik )) ViVk

neighbors of node i Ni ⊆ N ⊆ [0, n] ∩ Nconductance between nodes i and k ∈ Ni Gik ∈ R>0

Gii = Gii +∑

k∼NiGik

susceptance between nodes i and k ∈ Ni Bik ∈ R<0|Bii | = |Bii |+

∑k∼Ni

|Bik |

14 / 38


Example network

{δ1,V1}

{δ2,V2}

{δ3,V3}

{δ4,V4}

{δ5,V5}

{δ6,V6}

{δ7,V7}

{δ8,V8}

P12,Q12

P21,Q21

P17,Q17P71,Q71

P23,Q23

P32,Q32

P36,Q36

P63,Q63

P45,Q45

P54,Q54

P46,Q46P64,Q64

P47,Q47

P74,Q74P78,Q78

P87,Q87

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

inverters in grid-forming mode represented by Σi , i = 1, . . . , 815 / 38


Power sharing in microgrids

Definition (Power sharing)

Consider an AC electrical network, e.g. an AC microgrid

Denote its set of nodes by N = [1, n] ∩ N

Choose positive real constants γi , γk , χi and χk

Proportional active, respectively reactive, power sharing betweenunits at nodes i ∈ N and k ∈ N , if

Psiγi

=Ps

kγk, respectively

Qsiχi

=Qs

kχk

Power sharing...allows to prespecify utilization of unitsavoids high circulating currents in network

16 / 38


Power sharing is an agreement problem

N ⊆ N

U = diag(1/γi ), i ∈ N

D = diag(1/χi ), i ∈ N

Control objective

limt→∞

UPN (δ,V ) = υ1|N|,

limt→∞

DQN (δ,V ) = β1|N|, υ ∈ R>0, β ∈ R>0

17 / 38


Motivation for droop control of inverters

1 Droop control is widely used control solution in SG-based powersystems to address problems of frequency stability and activepower sharing

⇒ Adapt droop control to inverters⇒ Make inverters mimick behavior of SGs with respect to frequency

and active power

2 How to couple actuactor signals (δ and V ) with powers (P and Q)to achieve power sharing?

⇒ Pose MIMO control design problem as set of decoupled SISOcontrol design problems

⇒ Analyze couplings in power flow equations over a power line

18 / 38


Power flows over a power line

AssumptionsDominantly inductive power line with admittance Yik ∈ C betweennodes i and k

Small phase angle differences, i.e. δi − δk ≈ 0

⇒ Approximations

Yik = Gik + jBik ≈ jBik , sin(δik ) ≈ δik , cos(δik ) ≈ 1

⇒ Active and reactive power flows simplify to

Pik = −Bik ViVkδik ,

Qik = −Bik V 2i + Bik ViVk = −Bik Vi (Vi − Vk )

19 / 38


Standard droop control for inverters

Frequency droop control

uδi = ωd − kPi(Pm

i − Pdi )

Voltage droop control

uVi = V d

i − kQi(Qm

i −Qdi )

Further details: see, e.g., Chandorkar et al.(1993)

kPi∈ R>0 frequency droop

gainωd ∈ R>0 desired (nominal)

frequencyPd

i ∈ R active powersetpoint

Pmi ∈ R active power

measurementkQi∈ R>0 voltage droop

gainV d

i ∈ R>0 desired (nominal)voltage amplitude

Qdi ∈ R reactive power

setpointQm

i ∈ R reactive powermeasurement

20 / 38


Droop control - schematic representation


innercontrolloops

Rg Lg vGa

vGb

vGc

vIa

vIb

vIc

Powercalculation

Low-passfilter

Qmi

Pmi

−

+

Qdi

kQi−

+

V di

Vi

−

+

Pdi

kPi−

+

ωd

∫δi

21 / 38


Closed-loop droop-controlled microgrid

δi = ωd − kPi(Pm

i − Pdi )− ωcom,

τPiPm

i = −Pmi + Pi ,

Vi = V di − kQi

(Qmi −Qd

i ),

τPiQm

i = −Qmi + Qi

⇓change of variables

+vector notation

⇓

δ = ω − 1nωcom,

T ω = −ω + 1nωd − KP(P − Pd ),

T V = −V + V d − KQ(Q −Qd )

δ = col(δi ) ∈ Rn

ω = col(ωi ) ∈ Rn

V = col(Vi ) ∈ Rn

V d = col(V di ) ∈ Rn

T = diag(τPi) ∈ Rn×n

KP = diag(kPi) ∈ Rn×n

KQ = diag(kQi) ∈ Rn×n

P = col(Pi ) ∈ Rn

Q = col(Qi ) ∈ Rn

Pd = col(Pdi ) ∈ Rn

Qd = col(Qdi ) ∈ Rn

22 / 38


Closed-loop droop-controlled microgrid

δi = ωd − kPi(Pm

i − Pdi )− ωcom,

τPiPm

i = −Pmi + Pi ,

Vi = V di − kQi

(Qmi −Qd

i ),

τPiQm

i = −Qmi + Qi

⇓change of variables

+vector notation

⇓

δ = ω − 1nωcom,

T ω = −ω + 1nωd − KP(P − Pd ),

T V = −V + V d − KQ(Q −Qd )

δ = col(δi ) ∈ Rn

ω = col(ωi ) ∈ Rn

V = col(Vi ) ∈ Rn

V d = col(V di ) ∈ Rn

T = diag(τPi) ∈ Rn×n

KP = diag(kPi) ∈ Rn×n

KQ = diag(kQi) ∈ Rn×n

P = col(Pi ) ∈ Rn

Q = col(Qi ) ∈ Rn

Pd = col(Pdi ) ∈ Rn

Qd = col(Qdi ) ∈ Rn

22 / 38


Problem statement

1 Derive conditions for asymptotic stability of genericdroop-controlled microgrids

2 Investigate if droop control is suitable to achieve control objectiveof power sharing

23 / 38


Stability analysis

Coordinate transformation

Follow interconnection and damping assignment passivity-basedcontrol (IDA-PBC) approach (Ortega et al. (2002))

Represent microgrid dynamics in port-Hamiltonian form

Can easily identify energy function

24 / 38


Port-Hamiltonian systems

x = (J(x)− R(x))∇H + g(x)u, x ∈ X ⊆ Rn, u ∈ Rm,

y = gT (x)∇H, y ∈ Rm

J(x) ∈ Rn×n, J(x) = −J(x)T (interconnection matrix)R(x) ≥ 0 ∈ Rn×n for all x ∈ X (damping matrix)H : X→ R (Hamiltonian)

∇H =(∂H∂x

)T

Power balance equation

H︸︷︷︸stored power

= −∇HT R(x)∇H︸︷︷︸dissipated power

+ uT y︸︷︷︸supplied power

≤ uT y

25 / 38


Assumptions

Lossless admittances, i.e. Gik = 0, i ∼ N , k ∼ N

Power balance feasibility

There exist constants δs ∈ Θ, ωs ∈ R and V s ∈ Rn>0, where

Θ :={δ ∈ Tn ∣∣ |δs

ik | <π

2, i ∼ N , k ∼ Ni

},

such that

1nωs − 1nω

d + KP [P(δs,V s)− Pd ] = 0n,

V s − V d + KQ [Q(δs,V s)−Qd ] = 0n

26 / 38


Synchronized motion

Synchronized motion of microgrid starting in (δs, 1nωs,V s)

δ∗(t) = mod2π

{δs + 1n

(ωst −

∫ t

0ωcom(τ)dτ

)},

ω∗(t) = 1nωs,

V∗(t) = V s

Aim: derive conditions, under which synchronized motion isattractive

27 / 38


Power flows depend on angle differences δik

Pi (δ1, . . . , δn,V1, . . . ,Vn) =∑

k∼Ni

|Bik | sin(δik )ViVk ,

Qi (δ1, . . . , δn,V1, . . . ,Vn) = |Bii |V2i −

∑k∼Ni

|Bik | cos(δik )ViVk

⇒ Flow of system can be described in reduced angle coordinates

⇒ Transform convergence problem into classical stability problem

28 / 38


A condition for local asymptotic stability

Proposition

Fix τPi, kPi

, ωd and Pdi

If V di , kQi

and Qdi are chosen such that

D + T −W>L−1W > 0

⇒ xs is locally asymptotically stable

L > 0, T > 0, D = diag(

V di + kQi

Qdi

kQi(V s

i )2

)> 0

29 / 38


Physical interpretation of stability condition

D + T −W>L−1W > 0

L = network coupling strengths between θ and P

T = network coupling strengths between V and Q

But P = P(δ,V ) 6= P(δ) and Q = Q(δ,V ) 6= Q(V )

⇔ W = cross-coupling strength

Stability condition:couplings represented by L and T have to dominate overcross-couplings contained in W

30 / 38


A condition for active power sharing

Lemma (Proportional active power sharing)

Assume microgrid possesses synchronized motion

Then all generation units share active power proportionally insteady-state if

kPiγi = kPk

γk and kPiPd

i = kPkPd

k , i ∼ N , k ∼ N

Stability condition + parameter selection criterion

⇒ limt→∞ UP(δ,V ) = υ1|N|, U = diag(1/γi ), i ∈ N, υ ∈ R

31 / 38


Reactive power sharing

Voltage droop control does, in general, not guarantee desiredreactive power sharing

⇒ Several other or modified (heuristic) decentralized voltage controlstrategies proposed in literature(Zhong (2013), Li et al. (2009), Sao et al. (2005), Simpson-Porcoet al. (2014),...)

But: no general conditions or formal guarantees for reactivepower sharing are given

32 / 38


Distributed voltage control (DVC)

Alternative: consensus-based distributed voltage control (DVC)

uVi = V d

i − ki

∫ t

0ei (τ)dτ,

ei =∑

k∼Ci

(Qm

iχi−

Qmkχk

)

More on reactive power sharing?⇒ visit our talk ”A Consensus-Based Distributed Voltage Controlfor Reactive Power Sharing in Microgrids”,Thursday, June, 26, 10.00–10.20, Session Th A9 ”PowerSystems”

33 / 38


CIGRE MV benchmark model

PCC

110/20 kV

Main electrical network

1

2

3

4

5

6

7

8

910

11

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

∼=

5a

5b

5c

9a

9b

9c

10a

10b

10c

Storage

Wind power plant

Photovoltaic plant (PV)

Fuel cell (FC)

∼= Inverter

Load

34 / 38


Simulation example

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

P[pu]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

t [s]

P/S

N[-]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.1

0.2

0.3

0.4

t [s]

Q/S

N[-]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

Q[pu]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.99

1

1.01

1.02

1.03

V[pu]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−5

0

5

·10−2

∆f[Hz]

35 / 38


Conclusions and outlook

Microgrids are a promising concept in networks with largeamount of DG

Condition for local asymptotic stability in losslessdroop-controlled inverter-based microgrids

Selection criterion for parameters of frequency droop control thatensures desired active power sharing in steady-state

Proposed distributed voltage control (DVC), which solves openproblem of reactive power sharing

36 / 38


Outlook and related work

Analysis of lossy networks with variable frequencies and voltagesAnalysis of microgrids with frequency droop control anddistributed voltage control (DVC)Control schemes for highly ohmic networks

Secondary frequency control (Simpson-Porco et al. (2013),Burger et al. (2014), Andreasson et al. (2012), Bidram et al.(2013), Shafiee et al. (2014))Optimal operation control (Dorfler et al. (2014), Bolognani et al.(2013), Hans et al.(2014))Alternative inverter control schemes (Torres et al. (2014), Dhopleet al. (2014))

37 / 38


Publications

Schiffer, Johannes and Ortega, Romeo and Astolfi, Alessandro and Raisch, Jorg and Sezi,TevfikConditions for Stability of Droop-Controlled Inverter-Based Microgrids,Automatica, 2014

Schiffer, Johannes and Ortega, Romeo and Astolfi, Alessandro and Raisch, Jorg and Sezi,TevfikStability of Synchronized Motions of Inverter-Based Microgrids Under Droop Control,To appear at 19th IFAC World Congress, Cape Town, South Africa

Schiffer, Johannes and Seel, Thomas and Raisch, Jorg and Sezi, Tevfik,Voltage Stability and Reactive Power Sharing in Inverter-Based Microgrids withConsensus-Based Distributed Voltage ControlSubmitted to IEEE Transactions on Control Systems Technology, 2014

Schiffer, Johannes and Seel, Thomas and Raisch, Jorg and Sezi, Tevfik,A Consensus-Based Distributed Voltage Control for Reactive Power Sharing in Microgrids13th ECC, Strasbourg, France, 2014

Schiffer, Johannes and Goldin, Darina and Raisch, Jorg and Sezi, TevfikSynchronization of Droop-Controlled Microgrids with Distributed Rotational and ElectronicGeneration,IEEE 52nd CDC, Florence, Italy, 2013

38 / 38

Documents

Stability and power sharing in microgrids - TU Berlin · Stability and power sharing in microgrids J. Schiffer 1, R. Ortega2, J. Raisch;3, T. Sezi4 Joint work with A. Astolﬁ and