incremental energy functions for microgrid control - … · incremental energy functions for...

Incremental energy functions for microgrid control

C. De Persis

Institute of Engineering and TechnologyJ.C. Willems Center for Systems and Control

University of Groningen

Joint with N. Monshizadeh (Cambridge), J. Schiffer (Leeds), F. Dorfler (ETH)

Future Electric Power Systemsand the Energy TransitionChampery, Switzerland

5-9 February 2017

Outline

Aim

Control of DAE models of microgrids with coupled frequencyand voltage dynamics

Energy-inspired Lyapunov functionsPassivity-based frameworkSecondary frequency optimal controllers

De Persis-Monshizadeh. Bregman storage functions for microgrid control, IEEE-TAC and ECC’16Schiffer-Dorfler. On stability of DAPI frequency and active power controlled DA power system model, ECC’16De Persis-Monshizadeh-Schiffer-Dorfler. A Lyapunov approach to control of microgrids with a network-preserveddifferential-algebraic model. CDC’16

Outline

Microgrid modelsOptimal synchronous motionEnergy functions and passivityControllers

De Persis et al. Incremental functions for DAE microgrids Future Power Systems 1 / 23

Models

Microgrid topology

Microgrid topology G(V, E), with V = VI ∪ VL and E ⊆ V × VGenerators VI inverter-interfaced devices that measure powerand deliver an appropriate voltage waveformLoads VL consume generated powerPower lines E deliver power from generators to loads


Inverters

Phasor domain: idealsinusoidal voltagesrepresented by amplitude Viand phase θi

Inverters: AC (sinusoidal)voltage sources withcontrollable amplitude Vi andfrequency ωi = θi

θi = ωi

ωi = uωiVi = uV

i

-2⇡ -⇡ 0 ⇡ 2⇡

�1

0

1

�

xabc

(a) Symmetric three-phase ACsignal with constant amplitude

-2⇡ -⇡ 0 ⇡ 2⇡

�1

0

1

�

xabc

(b) Symmetric three-phase ACsignal with time-varying ampli-tude

-2⇡ -⇡ 0 ⇡ 2⇡

�1

0

1

�

xabc

(c) Asymmetric three-phase ACsignal with phases not shiftedequally by 2⇡

3

-2⇡ -⇡ 0 ⇡ 2⇡

�1

0

1

�

xabc

(d) Asymmetric three-phase ACsignal resulting of an asymmet-ric superposition of a symmetricsignal with signals oscillating athigher frequencies

Figure 1: Symmetric and asymmetric AC three-phase signals. Thelines correspond to xa ’—’, xb ’- -’, xc ’· · · ’.

2.2. The dq0-transformation

An important coordinate transformation known as dq0-transformation in the literature [71, 70, 5, 63, 92, 6, 103]is introduced.

Definition 2.9. [5, Chapter 4], [63, Chapter 11] Let x :R�0 ! R3 and % : R�0 ! S. Consider the mapping Tdq0 :S ! R3⇥3,

Tdq0(%) :=

r2

3

24

cos(%) cos(%� 23⇡) cos(% + 2

3⇡)sin(%) sin(%(t) � 2

3⇡) sin(% + 23⇡)p

22

p2

2

p2

2

35 .

(2.1)Then, fdq0 : R3 ⇥ S ! R3,

fdq0(x(t), %(t)) = Tdq0(%(t))x(t) (2.2)

is called dq0-transformation.

Note that the mapping (2.1) is unitary, i.e., T>dq0 = T�1

dq0.From a geometrical point of view, the dq0-transformationis a concatenation of two rotational transformations, see[70] for further details. The variables in the transformedcoordinates are often denoted by dq0-variables.

The dq0-transformation o↵ers various advantages whenanalyzing and working with power systems and is thereforewidely used in applications [70, 5, 4, 92, 103]. For exam-ple, the dq0-transformation permits, through appropriatechoice of %, to map three-phase AC signals to constant sig-nals, i.e., to transform periodic orbits into constant equi-libria. This simplifies the control design and analysis inpower systems, which is the main reason why the trans-formation (2.2) is introduced in the present case. In ad-dition, the transformation (2.2) exploits the fact that, in

a power system operated under symmetric conditions, athree-phase signal can be represented by two quantities.To see this, let xabc : R�0 ! R3 be a symmetric three-phase signal with amplitude A : R�0 ! R�0 and phaseangle ✓ : R�0 ! S, as in Definition 2.3. Applying themapping (2.1) with some angle % : R�0 ! S to xabc yields

xdq0 =

24

xd

xq

x0

35 = Tdq0(%)xabc =

r3

2A

24

sin(✓ � %)cos(✓ � %)

0

35 .

Hence, x0 = 0 for all t � 0. Therefore and as in this workonly symmetric three-phase signals are considered, it isconvenient to introduce the mapping Tdq : S ! R2⇥3,

Tdq(%) :=

r2

3

cos(%) cos(%� 2

3⇡) cos(% + 23⇡)

sin(%) sin(%� 23⇡) sin(% + 2

3⇡)

�,

(2.3)which, when applied to the symmetric three-phase signalxabc defined above, yields

xdq =

xd

xq

�= Tdq(%)xabc =

r3

2A

sin(✓ � %)cos(✓ � %)

�.

In the following, xdq are referred to as the dq-coordinatesof xabc. Note that xabc = Tdq(%)

>xdq.

Remark 2.10. There are several variants of the mapping(2.1) available in the literature. They may di↵er from themapping (2.1) in the order of the rows and the sign of theentries in the second row of the matrix given in (2.1), see[5, 6, 103]. However, all representations are equivalent inthe sense that they can all be represented by Tdq0 as givenin (2.1) by choosing an appropriate angle % and, possibly,rearranging the row order of the matrix Tdq0. The sameapplies to the mapping Tdq given in (2.3). Since—with aslightly di↵erent scaling factor—this transformation wasfirst introduced by Robert H. Park in 1929 [71] it is alsooften called Park transformation [92, Appendix A].

2.3. Instantaneous power

Power is one of the most important quantities in con-trol, monitoring and operation of electrical networks. Thefirst theoretical contributions to the definition of the powerflows in an AC network date back to the early 20th cen-tury. However, these first definitions are restricted to sinu-soidal steady-state conditions and based on the root-mean-square values of currents and voltages. As a consequence,these definitions of electric power are not well-suited forthe purposes of network control under time-varying oper-ating conditions [4].

The extension of the definition of electrical power totime-varying operating conditions is called “instantaneouspower theory” in the power system and power electronicscommunity [4, 92]. The development of this theory alreadybegun in the 1930s with the study of active and non-activecomponents of currents and voltages [32]. Among others,relevant contributions are [16, 21, 3, 100, 22, 72, 53].

4

Schiffer et al. Modeling of microgrids – from fundamental physics to

phasors and voltage sources. Automatica, arXiv 1505.00136


Inverter controllers

Frequency and voltage droop controllers (Chandorkar et al. ’03,Schiffer et al. ’14)

θi = ωiωi = −(ωi − ω∗)− KPi(Pi − P∗i ) + uPi

Vi = −(Vi − V ∗)− KQi(Qi −Q∗i ) + uQi i ∈ VI

Pi active power at inverter iP∗i active power set point at inverter iQi reactive power at inverter iQ∗i reactive power set point at inverter i


Microgrid circuitry

Lossless case lines are purely inductive

Pi =∑

j∈Ni

ViVj |Bij | sin(θij)

Qi = |Bii |V 2i −

∑

j∈Ni

ViVj |Bij | cos(θij)

whereBij = − 1

ω∗Lijsusceptance of line {i , j}

Bii = Bii +∑

j∈NiBij

Bii ≤ 0 shunt-conductance

No decoupling assumption – poorly motivated if working near anoperating point without a flat voltage profile


Loads

Constant power loads

0 = Pi − P∗i0 = Qi −Q∗i i ∈ VL

P∗i and Q∗i nonzero constant setpoints for active and reactive powerdemand


Control goals

θ = ωωI = −(ωI − 1ω∗)− KP(PI − P∗I ) + uP

VI = −(VI − V ∗I )− KQ(QI −Q∗I )+uQ0 = PL − P∗L0 = QL −Q∗L

Ideal scenario

Frequency regulation ωi ≈ ω∗

Voltage regulation Vi ≈ V ∗

Active power sharingPi

Pj≈ KPj

KPi

Reactive power sharingQi

Qj≈ KQj

KQi


Optimal synchronous motion


Synchronous motion

θi(t) = θi = θ0i + ω∗t i ∈ V

ωi(t) = ωi = ω∗ i ∈ VIVi(t) = V i i ∈ V

The synchronous motion must satisfy

θi = ω∗

0 = −KPi(P i − P∗i ) + uPi

0 = −(V i − V ∗i ) −KQi(Qi −Q∗i ) + uQi i ∈ VI

given uP ,uQ.



Optimal resource allocation

minuP

12

∑

i

K−1Pi u2

Pi

s.t. 0 =∑

i∈VI

uPi

KPi+∑

i∈VI

P∗i +∑

i∈VL

P∗i

which returns

u∗Pi = −∑

i∈V P∗i∑i∈VI

K−1Pi

Replacing in 0 = −(ω − 1ω∗)− KPi(P i − P∗i )+u∗Pi and solving for P i

KPiP∗i = KPjP∗j ⇒ P i

P j=

KPj

KPi

Active powersharing


Energy functions and passivity

Choice of coordinates

Center-of-inertia coordinates maps the synchronous motion to anequilibrium

δI = θI − 1θref = ΠθI ,

δL = θL − 1θref = θL −1nI11T θI ,

whereθref = 1

nI1T θI

Π = (I − 1nI11T )

The system becomes

δI = ΠωI ,ωI = −(ωI − 1ωI)− KP(PI − P∗I ) + uP ,

VI = −(VI − V ∗I )− KQ(QI −Q∗I ),0 = PL − P∗L ,0 = QL −Q∗L


Energy function

“Energy” inspired Lyapunov function

U(δ, ω,V ) =

kinetic︷︸︸︷12ωT

I K−1P ωI +

reactive power︷︸︸︷1T Q

=∑

i∈VI

12

K−1Pi ω

2i +

∑

i∈VQi

Tsolas-Arapostathis-Varaiya (1985), Chiang (2011)

The Lyapunov function is further shaped via the term

H(V ) = −(Q∗I + K−1Q V ∗I + K−1

Q uQ)T ln(VI)− (Q∗L)T ln(VL)

to absorb constant power injections and loads.


Incremental energy function

Incremental Lyapunov function (Bregman distance) of

S(δ, ωI ,V ) := U(δ, ωI ,V ) + H(V )

writes as

S(δ, ωI ,V ) = S(δ, ωI ,V )− S(δ, ωI ,V )− ∂S∂δ

∣∣∣∣T

−(δ − δ)

− ∂S∂ωI

∣∣∣∣T

−(ωI − ωI)−

∂S∂V

∣∣∣∣T

−(V − V )

1 S(δ, ωI ,V ) = 02 S(δ, ωI ,V ) strictly convex⇒ S(δ, ω,V ) > 0, (δ, ω,V ) 6= (δ, ω,V )

Keenan (1951), Bregman (1967), Alonso-Ydstie (2001)


A port-Hamiltonian model

Using the incremental Lyapunov function the system can be written as

δIωI

VI00

=

0 ΠKP 0 0 0

−KP −KP 0 0 0

0 0 −KQ[VI ] 0 0

0 0 0 −I 0

0 0 0 0 −I

∇δIS∇ωIS∇VIS∇δLS∇VLS

+

0 0I 00 I0 00 0

ñuP − uPuQ − uQ

ô

ñyP − yPyQ − yQ

ô=

ñ0 I 0 0 00 0 I 0 0

ô∇δIS∇ωIS∇VIS∇δLS∇VLS



A “skew-symmetric” interconnection structure

δIωI

VI00

=

0 ΠKP 0 0 0

−KP −KP 0 0 0

0 0 −KQ[VI ] 0 0

0 0 0 −I 0

0 0 0 0 −I


+

0 0I 00 I0 00 0


ô


ô=

ñ0 I 0 0 00 0 I 0 0




A dissipative structure

δIωI

VI00

=

0 ΠKP 0 0 0

−KP −KP 0 0 0

0 0 −KQ[VI ] 0 0

0 0 0 −I 0

0 0 0 0 −I


+

0 0I 00 I0 00 0


ô


ô=

ñ0 I 0 0 00 0 I 0 0



Regularity and passivityAlgebraic equations®

0 = PL − P∗L ,

0 = [VL]−1(QL −Q∗L )

⇔

0 =∂1T Q∂δL

− P∗L ,

0 =∂1T Q∂VL

− [VL]−1Q∗L ,

⇔ 0 = g(x , z)

with x = (δI , ωI ,VI) and z = (δL,VL).

Strict convexity and regularity of the AE If

Γ(V )[cos(DT δ)] [sin(DT δ)]Γ(V )|DL|T [VL]−1

[VL]−1|DL|Γ(V )[sin(DT δ)] ALL(cos(DT δ)) + [VL]−2[Q∗L ]

−

> 0

where (i) D the incidence matrix of G; (ii) Γ(V ) = diag(|Bij |ViVj){i,j}∈E ,then regularity of the AE holds.

Dvijotham-Low-Chertkov. Convexity of energy-like functions. arXiv 1501.04052DP–Monshizadeh. Bregman storage functions for microgrid control. IEEE-TAC arXiv1404.0576


Regularity and passivity

Strict convexity and regularity of the AE⇒ Incremental passivity

S(δ, ωI ,V ) = − ∂S∂ωI

TKP

∂S∂ωI− ∂S∂VI

T[VI ]KQ

∂S∂VI

+∂S∂ωI

T(uP − uP) +

∂S∂VI

T(uQ − uQ)

whereyP − yP =

∂S∂ωI

yQ − yQ =∂S∂VI


Variations on the theme

Other inverter controllers

Droop controller

VI = −VI − KQQI + uQ

Quadratic droop controller (Simpson-Porco et al ‘15)

VI = −[VI ]VI − KQQI + [VI ]uQ

Averaging reactive power controller (Schiffer et al ‘15,DP-Monshizadeh ‘15)

VI = −[VI ]KQLQKQQI + [VI ]uQ


A dissipative representation

S = U +H, H(V ) = H(V )− H(V )− ∂H∂V

∣∣∣∣T

−(V − V )

H(V ) X (VI) Y (VI)

Droop 1T VI − cT ln(V ) [VI ] IQuadratic droop 1

2 V TI VI − cT ln(VL) [VI ] [VI ]

Reactive consensus −cT lnV [VI ]LQ[VI ] [VI ]

δI

ωI

VI

00

=

0 ΠKP 0 0 0−KP −KP 0 0 0

0 0 −X (VI) 0 00 0 0 −I 00 0 0 0 −I

∇S +

0 0I 00 Y (VI)0 00 0

ïuP − uP

uQ − uQ

òï

yP − yPyQ − yQ

ò=

ï0 I 0 0 00 0 Y (VI) 0 0

ò∇S


Controllers

Regularity and passivity

Strict convexity and regularity of the AE⇒ Incremental passivity

S(δ, ωI ,V ) = − ∂S∂ωI

TKP

∂S∂ωI− ∂S∂VI

T[VI ]KQ

∂S∂VI

+∂S∂ωI

T(uP − uP) +

∂S∂VI

T(uQ − uQ)

whereyP − yP =

∂S∂ωI

yQ − yQ =∂S∂VI


Distributed internal model controller

Internal model A controller to generate uP = −1 1T P∗

1T K−1P 1

ξ = −Lcξ − K−1P (ωI − 1ω∗)

uP = ξ

where Lc is the Laplacian of a connected communication graphGc(VI , Ec)

ωI − 1ω∗ = 0⇒ ξ = −1 1T P∗

1T K−1P 1

is a solution of the controller

providing the optimal control uP

Incrementally passive system with storage functionC(ξ) = 1

2(ξ − ξ)>(ξ − ξ)

ξ = −LC∇C(ξ)− K−1P (ωI − 1ω∗)

uP − uP = ∇C(ξ)


Closed-loop system

δI

ωI

VI

00

=

0 ΠKP 0 0 0

−KP −KP 0 0 0

0 0 −K−1Q [VI ]

−1 0 0

0 0 0 −I 0

0 0 0 0 −I

∇δIU∇ωIU∇VIU∇δLU∇VLU

+

0I000

[uP − uP

]

yP − yP =[0 I 0 0 0

]

∇δIU∇ωIU∇VIU∇δLU∇VLU

I −I

ξ = −LC∇C(ξ) + uC − uCyC − yC = ∇C(ξ)

yP − yP

uC − uC = −(ωI − ω∗)yC − yC

uP − uP


Convergence

Using the Lyapunov function for the closed-loop system

W(δ, ωI ,V , ξ) = U(δ, ωI ,V ) + C(ξ) +12

(δI − δI)11T (δI − δI)

yields

W = −∂W∂ωI

>KP

∂W∂ωI− ∂W∂VI

>KQ[VI ]

∂W∂VI− ∂W

∂ξ

>LC∂W∂ξ

and the convergence

δ → δ, ωI → ω = 1ω∗, V → V , ξ → ξ, uP → u∗P

provided thatW(δ, ωI ,V , ξ) is strictly convex around (δ, ωI ,V , ξ)


Convergence

Using the Lyapunov function for the closed-loop system

W(δ, ωI ,V , ξ) = U(δ, ωI ,V ) + C(ξ) +12

(δI − δI)11T (δI − δI)

yields

W = −∂W∂ωI

>KP

∂W∂ωI− ∂W∂VI

>KQ[VI ]

∂W∂VI− ∂W

∂ξ

>LC∂W∂ξ

and the convergence

KPiP i

KPjP j= 1,

KQiQi + V i

KQjQj + V j=

uQi

uQj

provided thatW(δ, ωI ,V , ξ) is strictly convex around (δ, ωI ,V , ξ)


Strict convexity

Strict convexity ofW(δ, ωI ,V , ξ) around (δ, ωI ,V , ξ)

∂2W∂(δ, ωI ,V , ξ)

∣∣∣∣(δ,ωI ,V ,ξ)

> 0

m

Γ(V )[cos(DT δ)] [sin(DT δ)]Γ(V )|D|T [V ]−1

[V ]−1|D|Γ(V )[sin(DT δ)] A(cos(DT δ)) + h(V )

> 0

where h(V ) = diag([VI ]−2[Q∗I + K−1

Q V ∗I ], [VL]−2[Q∗L]).

This condition also implies the regularity of the AE


Conclusions

Conclusions

Secondary frequency controllers for joint frequencyregulation and power sharingNonlinear model and coupled voltage-frequency equations

Passivity-based framework for microgrid controlEnergy-inspired Lyapunov (Bregman) storage functionsDroop-controlled inverters

Future work

Analysis of other voltage regulation algorithms (quadraticdroop, reactive power consensus)Incremental passivity for dynamic voltage controlLyapunov-based robustness analysis (delays, informationlosses)


QUESTIONS?

Olimpia Zagnoli


References used for the presentationC. De Persis and N. Monshizadeh. Bregman storage functions for microgrid control. IEEE Transactions on AutomaticControl, arXiv preprint arXiv:1510.05811 and European Control Conference, 2016.

C. De Persis, N. Monshizadeh, J. Schiffer, F. Dorfler. A Lyapunov approach to control of microgrids with anetwork-preserved differential-algebraic model. In Proceedings of the 55th IEEE Conference on Decision and Control,Las Vegas, NV, USA, 2016, 2595-2600.

J. Schiffer and F. Dorfler. On stability of a distributed averaging PI frequency and active power controlleddifferential-algebraic power system model. European Control Conference, 2016.

J. Schiffer, T. Seel, J. Raisch, and T. Sezi. Voltage stability and reactive power sharing in inverter-based microgrids withconsensus-based distributed voltage control. IEEE Transactions on Control Systems Technology, 24(1):96–109, 2016.

J. W. Simpson-Porco, F. Dorfler, and F. Bullo. Voltage stabilization in microgrids via quadratic droop control. IEEETransactions on Automatic Control, 2016.

J. Schiffer, R. Ortega, A. Astolfi, J. Raisch, and T. Sezi. Conditions for stability of droop-controlled inverter-basedmicrogrids. Automatica, 50(10):2457–2469, 2014.

F. Dorfler, J. W. Simpson-Porco, and F. Bullo. Breaking the Hierarchy: distributed control & economic optimality inMicrogrids. IEEE Transactions on Control of Network Systems, 2016.

S. Trip, M. Burger, and C. De Persis. An internal model approach to (optimal) frequency regulation in power grids withtime-varying voltages. Automatica, 64:240–253, 2016.

J. Schiffer, D. Zonetti, R. Ortega, A. Stankovic, T. Sezi, and J. Raisch. Modeling of microgrids-from fundamental physicsto phasors and voltage sources. Automatica, arXiv preprint arXiv:1505.00136, 2015.

N. Monshizadeh, C. De Persis, A.J. van der Schaft, and J.M.A. Scherpen. A networked reduced model for electricalnetworks with constant power loads. IEEE Transactions on Automatic Control, arXiv preprint arXiv:1512.08250 and 2016American Control Conference.N.A. Tsolas, A. Arapostathis, P.V. Varaiya. A structure preserving energy function for power system transient stabilityanalysis. IEEE Transactions on Circuits and Systems, 32, 10, 1041–1049, 1985.

M. Burger, C. De Persis. Dynamic coupling design for nonlinear output agreement and time-varying flow control.Automatica, 51 (1), 210–222, 2015.

H.-D. Chiang. Direct Methods for Stability Analysis of Electric Power Systems. John Wiley & Sons, 2011.


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