Modeling power-grid synchronization dynamics

Modeling power-grid synchronization dynamicsTakashi NishikawaDepartment of Physics & AstronomyNorthwestern University

Funding: NSF, LANL, Argonne, ISEN

Reference: T.N. & A. E. Motter, New J. Phys. 17, 015012 (2015)Reinventing the Grid: Designing Resilient, Adaptive and Creative Power Systems - Santa Fe Institute - April 14, 2015

Modeling power-grid synchronization as complex networks

- G. Filatrella, A. H. Nielsen, & N. F. Pedersen, Eur. Phys. J. B 61, 485 (2008)- L.Buzna, S.Lozano, & A. Díaz-Guilera, Phys. Rev. E 80, 066120 (2009)- M. Rohden, A. Sorge, M. Timme, & D. Witthaut, PRL 109, 064101 (2012)- F. Dörfler & F. Bullo, SIAM J. Control. Optim. 50,1616 (2012)- S. Lozano, L. Buzna, A. Díaz-Guilera, Eur. Phys. J. B 85, 1 (2012)- D. Witthaut & M. Timme, New J. Phys. 14, 083036 (2012)- F. Dörfler, M. Chertkov, & F. Bullo, PNAS 110, 2005 (2013)- A. E. Motter, S. A. Myers, M. Anghel, & T. N., Nat. Phys. 9, 191 (2013) - P. J. Menck, J. Heitzig, J. Kurths, & H. J. Schellnhuber, Nat. Commun. 5, 3969 (2014)- L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, & R. Roy, Nat. Commun. 5,

4079 (2014)- A. Gajduk, M. Todorovski, & L. Kocarev, Eur. Phys. J. 223, 2387 (2014)- K. Schmietendorf, J. Peinke, R. Friedrich, & O. Kamps, Eur. Phys. J. 223, 2577 (2014)- P. H. J. Nardelli, N. Rubido, C. Wang, M. S. Baptista, C. Pomalaza-Raez, P. Cardieri, & M.

Latva-aho, Eur. Phys. J. 223, 2423 (2014)- T.N. & A. E. Motter, New J. Phys. 17, 015012 (2015)

Modeling power-grid networks

A. Network representation

1 3 2

In a steady state:- Power flows are balanced.- Generator phase angles are frequency-synchronized.

generator generator

load

Modeling power-grid networks


1 3 2

Load

Transmission line 1 - 3 Transmission line 3 - 2 Generator 2Generator 1

1 1 2 2 25 431

B. Electric circuit representation

3

FIG. 1. Modeling of power-grid network dynamics. (A) Simple network representation with nodes

representing generators or loads, and links representing transmission lines or transformers. Here

we used the example case consisting of two generators (nodes 1 and 2) and one load node (node 3),

which is discussed in Section VIA. (B) Representation of the electrical properties of the components

in the same network. There are three possible ways to represent the load node, which are used in

three di↵erent models. (C) Representation of the system as a network of coupled oscillators for

each choice of load representation in (B). For the system parameter values given in Section VIA,

the network dynamics obeys Eq. (2) [or equivalently, Eqs. (28), (29), and (30) for the EN, SP, and

SM model, respectively] with the indicated values of Ai

, Kij

, and �ij

. Each of the three dynamical

models has its own definition of nodes that are di↵erent from that used in (A). In (B), these nodes

are shown as black dots and indicated by orange, blue, and green indices for the EN, SP, and SM

models, respectively. The same coloring scheme is used for the node indices in (C). Note that in the

SP model the generator terminals are treated as load nodes with zero power consumption (nodes

3 and 4), separately from the generator internal nodes (nodes 1 and 2), leading to the 5-node

representation.

3

Load

Transmission line 1 - 3 Transmission line 3 - 2Generator 1

1 1531


3










, Kij

, and �ij







representation.

3

Transient reactance

Voltage source with constant magnitude

Generator dynamics — The swing equation

stability condition we derive below is necessary and sufficient for heterogeneously coupled powergrids, with the assumption that a certain function of generator parameters is homogeneous. Thiscondition helps us address the question of how to strengthen synchronization.

Power grids deliver a growing share of the energy consumed in the world and will soon un-dergo substantial changes owing to the increased harnessing of intermittent energy sources, thecommercialization of plug-in electric automobiles, and the development of real-time pricing andtwo-way energy exchange technologies. These advances will further increase the economical andsocietal importance of power grids, but they will also lead to new disturbances associated withfluctuations in production and demand, which may trigger desynchronization of power generators.Although power grids can, and often do, rely on active control devices to maintain synchronism,the question of whether proper design would allow the same to be achieved while reducing de-pendence on existing controllers is extremely relevant. In the U.S., for example, data on reportedpower outages show that over 3/4 of all large events involve equipment misoperation or humanerrors among other factors28. This illustrates why stability drawn from the network itself would bedesirable.

The dynamics

We represent the power grid as a network of power generators and substations (nodes) connectedby power transmission lines (links). The nodes may thus consume, produce, and distribute power,while the links transport power and may include passive elements with resistance, capacitance, andinductance. The state of the system is determined using power flow calculations given the powerdemand and other properties of the system (see Methods), which is a procedure we implementfor the systems in Table 1. In possession of all variables that describe the steady-state alternatingcurrent flow, we seek to identify the conditions under which the generators can remain stablysynchronized.

The starting point of our analysis is the equation of motion

2Hi

!R

d2�idt2

= Pmi � Pei, (2)

the so-called swing equation (ref. 29 and Methods), which, along with the power flow equations,describes the dynamics of generator i. The parameter Hi is the inertia constant of the generator, !R

is the reference frequency of the system, Pmi is the mechanical power provided by the generator,and Pei is the power demanded of the generator by the network (including the power lost to damp-

3

generator’s phase angle deviation

inertia constant for the generatorreference frequency

const. mechanical power input to the generatorpower demanded of the generator by the network (includes damping effects and nonlinear coupling terms)

= =

= =

=

Anderson & Fouad, Power system control and stability (2003)

damping constant of the generator=

Generator dynamics — The swing equation

stability condition we derive below is necessary and sufficient for heterogeneously coupled powergrids, with the assumption that a certain function of generator parameters is homogeneous. Thiscondition helps us address the question of how to strengthen synchronization.

Power grids deliver a growing share of the energy consumed in the world and will soon un-dergo substantial changes owing to the increased harnessing of intermittent energy sources, thecommercialization of plug-in electric automobiles, and the development of real-time pricing andtwo-way energy exchange technologies. These advances will further increase the economical andsocietal importance of power grids, but they will also lead to new disturbances associated withfluctuations in production and demand, which may trigger desynchronization of power generators.Although power grids can, and often do, rely on active control devices to maintain synchronism,the question of whether proper design would allow the same to be achieved while reducing de-pendence on existing controllers is extremely relevant. In the U.S., for example, data on reportedpower outages show that over 3/4 of all large events involve equipment misoperation or humanerrors among other factors28. This illustrates why stability drawn from the network itself would bedesirable.

The dynamics

We represent the power grid as a network of power generators and substations (nodes) connectedby power transmission lines (links). The nodes may thus consume, produce, and distribute power,while the links transport power and may include passive elements with resistance, capacitance, andinductance. The state of the system is determined using power flow calculations given the powerdemand and other properties of the system (see Methods), which is a procedure we implementfor the systems in Table 1. In possession of all variables that describe the steady-state alternatingcurrent flow, we seek to identify the conditions under which the generators can remain stablysynchronized.

The starting point of our analysis is the equation of motion

2Hi

!R

d2�idt2

= Pmi � Pei, (2)

the so-called swing equation (ref. 29 and Methods), which, along with the power flow equations,describes the dynamics of generator i. The parameter Hi is the inertia constant of the generator, !R

is the reference frequency of the system, Pmi is the mechanical power provided by the generator,and Pei is the power demanded of the generator by the network (including the power lost to damp-

3

generator’s phase angle deviation

inertia constant for the generatorreference frequency

const. mechanical power input to the generatorpower demanded of the generator by the network (includes damping effects and nonlinear coupling terms)

= =

= =

=

Anderson & Fouad, Power system control and stability (2003)

damping constant of the generator=

Since generators are frequency-synchronized in a steady state,

Load


1 1531


3










, Kij

, and �ij







representation.

3

3 different ways to model loads

Structure-preserving (SP) model(deviation in power consumption) ~ (voltage phase frequency)A. R. Bergen and D. J. Hill, IEEE Trans. Power Appar. Syst. PAS-100, 25–35 (1981)

Synchronous motor (SM) modelModel as synchronous motors (equivalent to the generator model)G. Filatrella, A. H. Nielsen, & N. F. Pedersen, Eur. Phys. J. B 61, 485 (2008)

Effective network (EN) modelModel as constant impedancesAnderson & Fouad, Power system control and stability (2003) A. E. Motter, S. A. Myers, M. Anghel, & T. N., Nat. Phys. 9, 191 (2013)

Forcing: (part of )

Coupling to other generators

Modeling as coupled oscillator network

Structure-preserving (SP) model


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

(deviation in power consumption) ~ (voltage phase frequency)


1 5 23 4










, Kij

, and �ij







representation.

3

Coupled oscillator representation


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3


EN model:

2H1

!R

�̈1 +D1

!R

�̇1 = AEN1 �KEN sin(�1 � �2 � �EN),

2H2

!R

�̈2 +D2

!R

�̇2 = AEN2 �KEN sin(�2 � �1 � �EN),

(28)

where

AEN1 = P ⇤

g,1 � |E⇤1 |2GEN

11 , AEN2 = P ⇤

g,2 � |E⇤2 |2GEN

22

KEN = |E⇤1E

⇤2Y12|, �EN = ↵EN

12 � ⇡

2, Y EN

12 = |Y EN12 | exp(j↵EN

12 ).

SP model:

2H1

!R

�̈1 +D1

!R

�̇1 = ASP1 �KSP

13 sin(�1 � �3),

2H2

!R

�̈2 +D2

!R


24 sin(�2 � �4),

D3

!R


31 sin(�3 � �1)�KSP35 sin(�3 � �5),

D4

!R


42 sin(�4 � �2)�KSP45 sin(�4 � �5),

D5

!R


53 sin(�5 � �3)�KSP54 sin(�5 � �4),

(29)

where

ASP1 = P ⇤

g,1, ASP2 = P ⇤

g,2,

ASP3 := �|V ⇤

1 |2G0033, ASP

4 := �|V ⇤2 |2G0

044, ASP5 := �P ⇤

`,3 � |V ⇤3 |2G0

055,

KSP13 = KSP

31 = |E1V⇤1 /x

0d,1|, KSP

24 = KSP42 = |E2V

⇤2 /x

0d,2|,

K35 = KSP53 = |V ⇤

1 V⇤3 G

0035|, KSP

45 = KSP54 = |V ⇤

2 V⇤3 G

0045|.

SM model:

2H1

!R

�̈1 +D1

!R

�̇1 = ASM1 �KSM

12 sin(�1 � �2)�KSM13 sin(�1 � �3),

2H2

!R

�̈2 +D2

!R


21 sin(�2 � �1)�KSM23 sin(�2 � �3),

2H3

!R

�̈3 +D3

!R


31 sin(�3 � �1)�KSM32 sin(�3 � �2),

(30)

where

ASM1 = P ⇤

g,1 � |E⇤1 |2GSM

11 , ASM2 = P ⇤

g,2 � |E⇤2 |2GSM

22 , ASM3 = �P ⇤

`,3 � |E⇤3 |2GSM

33 ,

KSMij

:= |E⇤i

E⇤j

Y SMij

|, i 6= j, i, j = 1, 2, 3.

24

1 5 23 4










, Kij

, and �ij







representation.

3

(#oscillators) = (#loads) + 2 x (#generators)

Synchronous moter (SM) model


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

3










, Kij

, and �ij







representation.

3

Model as a “generator” with negative power generation

Kron reduction

Synchronous moter (SM) modelA. Network representation

1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

3










, Kij

, and �ij







representation.

3

1

3

2










, Kij

, and �ij







representation.

3


The effective network (EN) model


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

Model as constant impedance

Kron reduction

The effective network (EN) modelA. Network representation

1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3

1 2










, Kij

, and �ij







representation.

3











, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

Effective network of generator interactions

nodes

Northern Italy network

Admittance matrix Y0 of size N x NPhysical network of transmission lines

Kirchhoff’s law:

Effective network of generator interactions

Admittance matrix Y0 of size N x NPhysical network of transmission lines

Effective network of generator interactionsEffective admittance matrix YEN of size ng x ng

nodes

nodes

Northern Italy network

Kirchhoff’s law:

Kirchhoff’s law:

Kron reduction

3 different network dynamics models

Structure-preserving (SP) model• Preserves structure of physical network of transmission lines.• (#oscillators) = (#loads) + 2 × (#generators)

Synchronous motor (SM) model• Generators and loads have the same type of dynamics.• (#oscillators) = (#loads) + (#generators)

Effective network (EN) model• Allows study of effective network of interactions between generators.• (#oscillators) = (#generators)

Benefits of modeling power grids as coupled oscillator networks

- Generalized Kuramoto models- Network symmetry and

synchronization- Master stability function

framework

Enhancing synchronization stability by adjusting generator parameters

Master stability function for stability analysis

Stability enhancement

A. E. Motter, S. A. Myers, M. Anghel, & T. N., Nat. Phys. 9, 191 (2013)


1 1 31










, Kij

, and �ij







representation.

3

Non-identical oscillator model for identical generators

the generators. Also note that a similar e↵ective network can be derived in the case of constant

current injections at load nodes and generator terminal nodes [nonzero current vectors It and I`

replacing the zero entries on the left hand side of Eq. (13)].

Accounting for the transient reactances x0d,i

is important, since they are typically not small19

for real generators producing small power and vary widely from generator to generator. This is

because the values need to be expressed in p.u. with respect to the system base (a common set of

reference units for the system) when writing the admittance matrices Yd

and Y00. If instead the

values were expressed in p.u. with respect to the rated power PR

of the generator, they tend to

lie in a relatively narrow range19. A consequence of having x0d,i

> 0 for all i is that the e↵ective

network represented by YEN, which is obtained after eliminating all nodes except for the generator

internal nodes, has the topology of a complete graph (Yij

6= 0 for all i and j). This follows from

a general property of Kron reduction that two nodes are connected in the reduced network if and

only if the two nodes are connected in the original network by a path in which all intermediate

nodes are eliminated by the reduction process28. If one neglects the transient reactances and sets

x0d,i

= 0, we see from Eq. (14) that we would have YEN = Y0. The matrix Y0 can be shown to

equal the admittance matrix between the generator terminal nodes obtained by eliminating the

load nodes through Kron reduction, which may or may not have the topology of a complete graph

depending on the location of the load nodes.

Using the e↵ective admittance matrix YEN, the single sinusoidal coupling term in Eq. (9) can

be replaced by an expression for Pe

that comes from a power balance equation equivalent to Eq. (3)

with |Vi

| replaced by |E⇤i

|, Y0 by YEN, and �i

by �i

:

Pe,i

=

ngX

j=1

|E⇤i

E⇤j

Y ENij

| cos(�j

� �i

+ ↵ENij

), (15)

where Y ENij

= |Y ENij

| exp(j↵ENij

) and all quantities must be expressed in p.u. with respect to the

system base. This can be used to show that writing the swing equation (8) for each generator leads

to an equation of the same form as Eq. (2), and the EN model is thus given by

2Hi

!R

�̈i

+D

i

!R

�̇i

= AENi

�ngX

j=1,j 6=i

KENij

sin(�i

� �j

� �ENij

), i = 1, . . . , ng

,

AENi

:= P ⇤g,i

� |E⇤i

|2GENii

, KENij

:= |E⇤i

E⇤j

Y ENij

|, �ENij

:= ↵ENij

� ⇡

2,

(16)

where GENii

is the real part of the complex admittance Y ENii

. Notice that the number of phase

oscillators in Eq. (2) in this case is ng

, since generators are the only dynamical elements over the

14

the generators. Also note that a similar e↵ective network can be derived in the case of constant

current injections at load nodes and generator terminal nodes [nonzero current vectors It and I`

replacing the zero entries on the left hand side of Eq. (13)].

Accounting for the transient reactances x0d,i

is important, since they are typically not small19

for real generators producing small power and vary widely from generator to generator. This is

because the values need to be expressed in p.u. with respect to the system base (a common set of

reference units for the system) when writing the admittance matrices Yd

and Y00. If instead the

values were expressed in p.u. with respect to the rated power PR

of the generator, they tend to

lie in a relatively narrow range19. A consequence of having x0d,i

> 0 for all i is that the e↵ective

network represented by YEN, which is obtained after eliminating all nodes except for the generator

internal nodes, has the topology of a complete graph (Yij

6= 0 for all i and j). This follows from

a general property of Kron reduction that two nodes are connected in the reduced network if and

only if the two nodes are connected in the original network by a path in which all intermediate

nodes are eliminated by the reduction process28. If one neglects the transient reactances and sets

x0d,i

= 0, we see from Eq. (14) that we would have YEN = Y0. The matrix Y0 can be shown to

equal the admittance matrix between the generator terminal nodes obtained by eliminating the

load nodes through Kron reduction, which may or may not have the topology of a complete graph

depending on the location of the load nodes.

Using the e↵ective admittance matrix YEN, the single sinusoidal coupling term in Eq. (9) can

be replaced by an expression for Pe

that comes from a power balance equation equivalent to Eq. (3)

with |Vi

| replaced by |E⇤i

|, Y0 by YEN, and �i

by �i

:

Pe,i

=

ngX

j=1

|E⇤i

E⇤j

Y ENij

| cos(�j

� �i

+ ↵ENij

), (15)

where Y ENij

= |Y ENij

| exp(j↵ENij

) and all quantities must be expressed in p.u. with respect to the

system base. This can be used to show that writing the swing equation (8) for each generator leads

to an equation of the same form as Eq. (2), and the EN model is thus given by

2Hi

!R

�̈i

+D

i

!R

�̇i

= AENi

�ngX

j=1,j 6=i

KENij

sin(�i

� �j

� �ENij

), i = 1, . . . , ng

,

AENi

:= P ⇤g,i

� |E⇤i

|2GENii

, KENij

:= |E⇤i

E⇤j

Y ENij

|, �ENij

:= ↵ENij

� ⇡

2,

(16)

where GENii

is the real part of the complex admittance Y ENii

. Notice that the number of phase

oscillators in Eq. (2) in this case is ng

, since generators are the only dynamical elements over the

14

Suppose Hi, Di, x’d,i, Pg,i* are the same for all i.

Non-identical oscillator model for identical generators


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3

1 2










, Kij

, and �ij







representation.

3











, Kij

, and �ij







representation.

3










, Kij

, and �ij







representation.

3

d 10-1

10-3

10-5

10-7

Figure 3: Enhancement of the stability of synchronous states. a–d, Response of the originalnetwork (a), the network with adjusted x0

d,i (b), the network with �i =¯� (c), and the network with

�i = �opt (d) to a perturbation in the Northern Italy power grid. The perturbation was applied tothe phase of each generator in the synchronous state at t = 0, and was drawn from the Gaussiandistribution with mean zero and standard deviation 0.01 rad. In each case, the network layout is thesame as in Fig. 1b and the bottom panel shows the time evolution of �0i1 = �i1��⇤i1, where �i1 is thephase of generator i relative to generator 1 and �⇤i1 is the corresponding phase in the synchronousstate. Generator 1, shown as a white node in the network, is used as a reference to discount phasedrifts common to all generators. The other nodes and their time-evolution curves are color-codedby the maximum value of |�0i1| for 2 t 3. By adjusting the transient reactance of the generators,the divergence from the unstable steady state is converted to exponential convergence (a and b).This stability is improved upon adjusting the generator parameters to ensure a common value for�i, but is further improved when this common value is tuned to �opt = 2

p↵2

(c and d).

21

Summary and remarks

References: A. E. Motter, S. A. Myers, M. Anghel, & T. N., Nat. Phys. 9, 191 (2013)T. N. & A. E. Motter, New J. Phys. 17, 015012 (2015)

Summary:

- 3 load models leading to 3 ways of modeling the grid dynamics: structural-preserving (SP), synchronous motor (SM), and effective network (EN) models

- Effective network of interactionsRemarks:

- Which load model?

- Fluctuations in demand and generation

- Distributed/renewable energy sources


1 3 2

Load


1 1 2 2 25 431


3










, Kij

, and �ij







representation.

3

Modeling power-grid synchronization dynamics

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