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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
A. Network representation
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
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
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
Transmission line 1 - 3 Transmission line 3 - 2Generator 1
1 1531
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
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
A. Network representation
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
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
(deviation in power consumption) ~ (voltage phase frequency)
Structure-preserving (SP) model
1 5 23 4
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
Coupled oscillator representation
A. Network representation
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
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
Structure-preserving (SP) model
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
�̇2 = ASP2 �KSP
24 sin(�2 � �4),
D3
!R
�̇3 = ASP3 �KSP
31 sin(�3 � �1)�KSP35 sin(�3 � �5),
D4
!R
�̇4 = ASP4 �KSP
42 sin(�4 � �2)�KSP45 sin(�4 � �5),
D5
!R
�̇5 = ASP5 �KSP
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
�̇2 = ASM2 �KSM
21 sin(�2 � �1)�KSM23 sin(�2 � �3),
2H3
!R
�̈3 +D3
!R
�̇3 = ASM3 �KSM
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
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
(#oscillators) = (#loads) + 2 x (#generators)
Synchronous moter (SM) model
A. Network representation
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
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
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
Model as a “generator” with negative power generation
Kron reduction
Synchronous moter (SM) modelA. Network representation
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
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
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
1
3
2
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
Coupled oscillator representation
Synchronous moter (SM) model
1
3
2
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
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
�̇2 = ASP2 �KSP
24 sin(�2 � �4),
D3
!R
�̇3 = ASP3 �KSP
31 sin(�3 � �1)�KSP35 sin(�3 � �5),
D4
!R
�̇4 = ASP4 �KSP
42 sin(�4 � �2)�KSP45 sin(�4 � �5),
D5
!R
�̇5 = ASP5 �KSP
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
�̇2 = ASM2 �KSM
21 sin(�2 � �1)�KSM23 sin(�2 � �3),
2H3
!R
�̈3 +D3
!R
�̇3 = ASM3 �KSM
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
(#oscillators) = (#loads) + (#generators)
The effective network (EN) model
A. Network representation
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
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
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
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
Model as constant impedance
Kron reduction
The effective network (EN) modelA. Network representation
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
1 2
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
Coupled oscillator representation
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
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
The effective network (EN) model
1 2
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
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
�̇2 = ASP2 �KSP
24 sin(�2 � �4),
D3
!R
�̇3 = ASP3 �KSP
31 sin(�3 � �1)�KSP35 sin(�3 � �5),
D4
!R
�̇4 = ASP4 �KSP
42 sin(�4 � �2)�KSP45 sin(�4 � �5),
D5
!R
�̇5 = ASP5 �KSP
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
�̇2 = ASM2 �KSM
21 sin(�2 � �1)�KSM23 sin(�2 � �3),
2H3
!R
�̈3 +D3
!R
�̇3 = ASM3 �KSM
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
(#oscillators) = (#generators)
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)
Transmission line 1 - 3 Transmission line 3 - 2Generator 1
1 1 31
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
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
A. Network representation
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
1 2
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
Coupled oscillator representation
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
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
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
A. Network representation
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