Kron Reduction of Graphs with Applications to ... - Semantic Scholar
Motivation: the current power grid is . . . Kron Reduction of Graphs with Applications to Electrical Networks Florian D¨orfler and Francesco Bullo Center for Control, Dynamical Systems & Computation University of California at Santa Barbara http://motion.me.ucsb.edu “. . . the greatest engineering achievement of the 20th century.” Center for Nonlinear Studies Los Alamos National Labs, New Mexico, June 8, 2011 Article available online at: http://arxiv.org/abs/1102.2950
[National Academy of Engineering ’10]
“. . . the largest and most complex machine engineered by humankind.” [P. Kundur ’94, V. Vittal ’03, . . . ]
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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Motivation: the envisioned power grid
Florian D¨ orfler (UCSB)
Kron Reduction
Energy is one of the top three national priorities
Expected developments in “smart grid”:
Expected developments in “smart grid”:
1
large number of distributed power sources
1
large number of distributed power sources
2
increasing adoption of renewables
2
increasing adoption of renewables
3
sophisticated cyber-coordination layer
3
sophisticated cyber-coordination layer
challenges: increasingly complex networks & stochastic disturbances opportunity: some smart grid keywords: control/sensing/optimization distributed/coordinated/decentralized
Today: “reducing the complexity by means of circuit and graph theory” Florian D¨ orfler (UCSB)
Kron Reduction
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Motivation: the envisioned power grid
Energy is one of the top three national priorities
/ ,
Center for Nonlinear Studies
Center for Nonlinear Studies
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/ ,
challenges: increasingly complex networks & stochastic disturbances opportunity: some smart grid keywords: control/sensing/optimization distributed/coordinated/decentralized
Today: “reducing the complexity by means of circuit and graph theory” Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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Kron reduction of a resistive circuit Nodal analysis by Kirchho↵’s and Ohm’s laws: I =Y ·V
Y = YT
n P
Yjk
Ik
Yk,shunt
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.. .
..
Yik + Yk,shunt . . . .. .
..
Kron Reduction
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T · Ybound-int
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Kron Reduction
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an undirected, connected, & weighted graph with boundary nodes ⇤ ⌅, interior nodes • , & possibly self-loops
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1.73
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“equivalent” reduced circuit
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Ired = Yred · Vboundary
I =Y ·V
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a symmetric and irreducible loopy Laplacian matrix Y with partition (⇤ ⌅ ,• ), & possibly diagonally dominance
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original circuit
1 1
⇤ ⌅ , interior nodes
1.92
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0.11
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a connected electrical network with conductance matrix Y , terminals • , & possibly shunt conductances
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Consider either of the following three equivalent setups:
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Florian D¨ orfler (UCSB)
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Boundary nodes ⇤ ⌅ arise as natural terminals in applications.
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• via Schur complement:
0.39
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Kron reduction of graphs
Ybound-int · Yinterior
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Center for Nonlinear Studies
Kron reduction: eliminate interior nodes
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Kron reduction of a resistive circuit
Yred = Y /Yinterior = Yboundary
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.. . 7 7 Yin 7 7 5 .. .
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= { weighted Laplacian matrix } + diag Yk,shunt = “loopy Laplacian”
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conductance matrix Y
G
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loopy Laplacian matrix Y
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k=1,k6=i
Florian D¨ orfler (UCSB)
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.. .. . 6 . 6 =6 6 Yi1 . . . 4 .. . .. .
k
Yik
I 2 Cn nodal current injections V 2 Cn nodal voltages/potentials Y 2 Cn⇥n nodal conductance matrix 2
Partition circuit equations via boundary nodes & interior nodes : Yboundary Ybound-int Vboundary Iboundary = T Iinterior Vinterior Ybound-int Yinterior
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Kron reduction of a resistive circuit
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G. Kron, “Tensor Analysis of Networks,” Wiley, 1939. Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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Kron reduction of graphs
Kron reduction of graphs
Kron reduction via Schur complement: 8
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Yred = Y /Yinterior G
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reduction 0.08
Kron
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Kron
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Florian D¨ orfler (UCSB)
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Relation of spectrum and algebraic properties of Q and Qred ?
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How about the graph topologies and the e↵ective resistances?
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What is the e↵ect of a perturbation in the original graph on the reduced graph, its spectrum, and its e↵ective resistance?
reduction
Gred
Finally, why is this graph reduction process of practical importance and in which application areas?
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Kron-reduced loopy Laplacian Yred
1.73
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reduction
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transfer conductance matrix Yred
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Center for Nonlinear Studies
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Florian D¨ orfler (UCSB)
Kron reduction of graphs: applications
Kron Reduction
Center for Nonlinear Studies
Electrical impedance tomography
Smart grid monitoring
transformation
Yred
[A. E. Kennelly 1899, A. Rosen 1924]
Ycut
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Kron reduction
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Kron reduction of graphs: applications
Purpose: construct low-dimensional equivalent circuits / graphs / models Simplest non-trivial case: star-
Yred = Y /Yinterior
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loopy Laplacian matrix Y
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Kron
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to reconstruct spatial conductivity
through cut-set variables
[E. Curtis and J. Morrow ’94 & ’00]
[I. Dobson ’11]
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Reduced power network modeling
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OFCIRCUITS INTEGRATED 29, NO. 1, JANUARY 2010 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED ANDCIRCUITS SYSTEMS,AND VOL.SYSTEMS, 29, NO. 1,VOL. JANUARY 2010 !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899:
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knot theory, Yang-Baxter equations and applications, high-energy physics, statistical mechanics, vortices in fluids, entanglement of polymers & DNA, . . . [F. D¨ orfler & F. Bullo ’11, J.H.H. Perk & H. Au-Yang ’06]
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Physics applications:
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sparse matrix algorithms, finite-element methods, sparse multi-grid solvers, Markov chain reduction, stochastic complementation, applied linear algebra & matrix analysis, Dirichlet-to-Neumann map, . . .
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Mathematics applications:
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Florian D¨ orfler (UCSB)
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smart grid monitoring, circuit theory, model reduction for power and water networks, power electronics, large-scale integration chips, electrical impedance tomography, data-mining, . . .
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Engineering applications:
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Representation of integration chips 8
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conductance matrix Y
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Kron reduction via Schur complement:
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Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16.
Fig. 5. conductance Two reducedmatrices conductance of the same network with 59 Fig. 5. Two reduced of thematrices same network with 59 matrix on the lefteliminating results when all internal nodes; terminals. The terminals. matrix on The the left results when all eliminating internal nodes; onbut the five right, all butnodes five internal nodes are on the right, all internal are eliminated. Theeliminated. matrix on The the matrix on the has fewer fivethan more than the first matrix, right has fewerright nonzeros, and nonzeros, only five and moreonly rows, therows, first matrix, hence the network corresponding network smaller: 721resistors. versus 1711 resistors. and hence the and corresponding is smaller: 721isversus 1711 By selecting (five toin be thispreserved case) to inbe preserved in By selecting specific internalspecific nodes internal (five in nodes this case) the reduced fill-in during elimination of the nodes other internal nodes is the reduced network, fill-in network, during elimination of the other internal is limitedThe considerably. difference bigger as limited considerably. difference The becomes even becomes bigger as even the number of the number of terminals increases. terminals increases.
for sparse computation
[J. Rommes and W. H. A. Schilders ’09]
T T Gk = G11 − G QQ k = G11 − QQ
0
2
4
6
8
10
TIME / s
for stability analysis and control
test system can be represented by ˙i = Hi ˙i = fs
i,
Di
10
i
+ Pmi
Gii Ei2
Ei E j ·
(11)
are provided to discuss whether the instability in Fig. 10 occurs in the corresponding real power system. First, the classical model with constant voltage behind impedance is used for first swing criterion of transient stability [1]. This is because second and multi swings may be affected by voltage fluctuations, damping effects, controllers such as AVR, PSS, and governor. Second, the fault durations, which we fixed at 20 cycles, are normally less than 10 cycles. Last, the load condition used above is different from the original one in [11]. We cannot hence argue that global instability occurs in the real system. Analysis, however, does show a possibility of global instability in real power systems.
[F. D¨ orfler and F. Bullo ’09] j=1,j=i
· {Gij cos(
i
j)
+ Bij sin(
i
j )},
where i = 2, . . . , 10. i is the rotor angle of generator i with respect to bus 1, and i the rotor speed deviation of generator i relative to system angular frequency (2 fs = 2 60 Hz). 1 is constant for the above assumption. The parameters fs , Hi , Pmi , Di , Ei , Gii , Gij , and Bij are in per unit system except for Hi and Di in second, and for fs in Helz. The mechanical input power Pmi to generator i and the magnitude Ei of internal voltage in generator i are assumed to be constant for transient stability studies [1], [2]. Hi is the inertia constant of generator i, Di its damping coefficient, and they are constant. Gii is the internal conductance, and Gij + jBij the transfer impedance between generators i and j; They are the parameters which change with network topology changes. Note that electrical loads in the test system are modeled as passive impedance [11].
IV. T OWARDS A C ONTROL FOR G LOBAL S WING I NSTABILITY Global instability is related to the undesirable phenomenon that should be avoided by control. We introduce a key mechanism for the control problem and discuss control strategies for preventing or avoiding the instability.
B. Numerical Experiment Coupled swing dynamics of 10 generators in the test system are simulated. Ei and the initial condition ( i (0), i (0) = 0) for generator i are fixed through power flow calculation. Hi is fixed at the original values in [11].
A. Internal Resonance as Another Mechanism Inspired by [12], we here describe the global instability with dynamical systems theory close to internal resonance [23], [24]. Consider collective dynamics in the system (5). For the system (5) with small parameters pm and b, the set
Florian D¨ orfler (UCSB) Kron Reduction reducedand network), andcan since this can bebefore computed before the reduced the network), since this be computed doing elimination the actual elimination we always include doing the actual we always include a check on a check on this. If ittoiseliminate decided to all internal nodes, the reduced this. If it is decided alleliminate internal nodes, the reduced matrix Gk can beefficiently computedusing efficiently conductanceconductance matrix Gk can be computed the using the Cholesky factorization Cholesky factorization G22 = LLT G22 = LLT
-5
Fig. 10. Coupled swing of phase angle i in New England test system. The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtained by numerical integration of eqs. (11).
Center for Nonlinear Studies
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Kron reduction of graphs: properties
Kron reduction of graphs: properties 2
Kron reduction of a graph with boundary ⇤ ⌅ , interior • , non-neg self-loops loopy Laplacian matrix Y Schur complement: Yred = Y /Yinterior
8
k+1 k / Yred = Yred
Iterative 1-dim Kron reduction:
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) topological evolution of the corresponding graph
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Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16.
11 10
n | |
0 Y = Yred
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iterative Kron reduction
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Kron reduction of
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Well-posedness: set of loopy Laplacian matrices is closed !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899:
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Properties of Kron reduction:
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Florian D¨ orfler (UCSB)
Kronswing Reduction Center for Nonlinear Studies Coupled of phase angle i in New England test system.
Fig. 10. The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtained by numerical integration of eqs. (11).
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Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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test system can be represented by are provided to discuss whether the instability in Fig. 10 occurs in the corresponding real power system. First, the classical model with constant voltage behind impedance is used for first swing criterion of transient stability [1]. This is because second and multi swings may be affected by voltage fluctuations, damping effects, controllers such as AVR, PSS, and governor. Second, the fault durations, which we fixed at 20 cycles, are normally less than 10 cycles. Last, the load condition used above is different from the original one in [11]. We cannot hence argue that global instability occurs in the real system. Analysis, however, does show a possibility of global instability in real power systems.
Kron reduction of graphs: properties
by edge to grounded node ⌥ ⌃
B. Numerical Experiment Coupled swing dynamics of 10 generators in the test system are simulated. Ei and the initial condition ( i (0), i (0) = 0) for generator i are fixed through power flow calculation. Hi is fixed at the original values in [11]. Pmi and constant power loads are assumed to be 50% at their ratings [22]. The damping Di is 0.005 s for all generators. Gii , Gij , and Bij are also based on the original line data in [11] and the power flow calculation. It is assumed that the test system is in a steady operating condition at t = 0 s, that a line-to-ground fault occurs at point F near bus 16 at t = 1 s 20/(60 Hz), and that line 16–17 trips at t = 1 s. The fault duration is 20 cycles of a 60-Hz sine wave. The fault is simulated by adding a small impedance (10 7 j) between bus 16 and ground. Fig. 10 shows coupled swings of rotor angle i in the test system. The figure indicates that all rotor angles start to grow coherently at about 8 s. The coherent growing is global instability. 27
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A. Internal Resonance as Another Mechanism Inspired by [12], we here describe the global instability with dynamical systems theory close to internal resonance [23], [24]. Consider collective dynamics in the system (5). For the system (5) with small parameters pm and b, the set {( , ) 2 S 1 R | = 0} of states in the phase plane is called resonant surface [23], and its neighborhood resonant band. The phase plane is decomposed into the two parts: resonant band and high-energy zone outside of it. Here the initial conditions of local and mode disturbances in Sec. II indeed exist inside the resonant band. The collective motion before the onset of coherent growing is trapped near the resonant band. On the other hand, after the coherent growing, it escapes from the resonant band as shown in Figs. 3(b), 4(b), 5, and 8(b) and (c). The trapped motion is almost integrable and is regarded as a captured state in resonance [23]. At a moment, the integrable motion may be interrupted by small kicks that happen during the resonant band. That is, the so-called release from resonance [23] happens, and the collective motion crosses the homoclinic orbit in Figs. 3(b), 4(b), 5, and 8(b) and (c), and hence it goes away from the resonant band. It is therefore said that global instability 27
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Kron reduction of 27
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Kron Reduction
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Kron reduction of
Yred Center for Nonlinear Studies
Topological properties: interior network connected ) reduced network complete at least one node in interior network features a self-loop ) all nodes in reduced network feature self-loops
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Yred
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(')$ Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply.
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) Equivalence: the following diagram commutes:
C. Remarks It was confirmed that the system (11) in the New England test system shows global instability. A few comments
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IV. T OWARDS A C ONTROL FOR G LOBAL S WING I NSTABILITY Global instability is related to the undesirable phenomenon that should be avoided by control. We introduce a key mechanism for the control problem and discuss control strategies for preventing or avoiding the instability.
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Augmentation: replace self-loops 8
Kron reduction of graphs: properties
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i
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3
i
where i = 2, . . . , 10. i is the rotor angle of generator i with respect to bus 1, and i the rotor speed deviation of generator i relative to system angular frequency (2 fs = 2 60 Hz). 1 is constant for the above assumption. The parameters fs , Hi , Pmi , Di , Ei , Gii , Gij , and Bij are in per unit system except for Hi and Di in second, and for fs in Helz. The mechanical input power Pmi to generator i and the magnitude Ei of internal voltage in generator i are assumed to be constant for transient stability studies [1], [2]. Hi is the inertia constant of generator i, Di its damping coefficient, and they are constant. Gii is the internal conductance, and Gij + jBij the transfer impedance between generators i and j; They are the parameters which change with network topology changes. Note that electrical loads in the test system are modeled as passive impedance [11].
8
(11)
j )},
30
Ei E j ·
j=1,j=i
8
Gii Ei2
j ) + Bij sin(
30
10
+ Pmi
i
8
Di
· {Gij cos(
30
i,
8
˙i = Hi ˙i = fs
Algebraic properties: self-loops in interior network decrease mutual coupling in reduced network increase self-loops in reduced network
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Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
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We start by considering “regular” electrical networks with scalar valued currents and p electrical network consists of an interconnection of purely resistive elements. Such generally described by graphs whose nodes connection points between resis Kron reduction of graphs: properties Kron reduction of represent graphs: properties correspond to the resistors themselves. The effective resistance between two nodes in Spectral properties: E↵ective resistance is defined as the potential drop between the Rtwo nodes, when a current source with ij : interlacing property: i (Y ) i (Yred ) i+n |⌅| (Y ) Ampere is connected across the two nodes (cf. Figure 1). For a general network, the com 6
7
) algebraic connectivity e↵ect of self-loops
2
is non-decreasing
u
on loop-less Laplacian matrices: + max{ }
2 (Lred )
2 (L)
+ min{ }
) self-loops weaken the algebraic connectivity
2
PSfrag replacements
Example: all mutual edges have unit weight 8
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Ampere o
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+
1 Amp
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without self-loops: 2 (L) = 0.39 0.69 = 2 (Lred ) with unit self-loops: 2 (L) = 0.39 0.29 = 2 (Lred )
Figure 1: A resistive electrical network for which a 1 Ampere current was is injected at at the reference node o. The resulting potential difference Vu Vo is the effective resis o. Kron reduction of graphs: properties Conclusions Florian D¨ orfler (UCSB)
Center for Nonlinear Studies
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Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
resistances relies on the usual Kirchoff’s and Ohm’s laws.
E↵ective resistance Rij :
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establish Equivalence and invariance of Rij among ⇤ ⌅To nodes:
1 1
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conductance matrix Y
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the connection between graph effective resistances defined in the prev loopy Laplacian trical networks we need to consider an abstract generalized electrical network in which matrix Y augment and resistors are k k matrices. For such networks,KronKirchoff’s current law can be defi Y Kron reduction reduction Kron reduction except that currents are added as matrices. Kirchoff’s voltage law can also be defined in G transfer conductance Rij potentials Kron reduction Y drops across edges are added as matrices.matrix Kirchoff’s voltage laws show the of i, j Kron-reduced valued node potential function. Ohm’s law takes loopy the following matrix form Laplacian Y 1
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red
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red
Yred
self-loops: Rij among ⇤ ⌅ &⌃ ⌥ uniform Kron Reduction
Ve = Re ie , Kron reduction is important in various applications
where k matrix flowing through edge e of the electric Analysis ofcurrent Kron reduction via algebraic graphthe theory 1 ⌅ ie is a generalized k R = 2 |Yred (i, j)| generalized resistance of that edge, and Ve is a &generalized k k matrix potential dr Open problem: directed complex-weighted graphs , R1 = ⌅2 |Yred (i, j)| + max{ } Generalized resistances are always symmetric positive definite matrices.
no self-loops: Rij among ⇤ ⌅ uniform ,
Florian D¨ orfler (UCSB)
Yred
augment
ij
ij
Center for Nonlinear Studies
17 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
18 / 18
The generalized electrical networks so defined share many of the properties of “re works. In particular, Kirchoff’s and Ohm’s laws uniquely define all edge currents an
[National Academy of Engineering ’10]
“. . . the largest and most complex machine engineered by humankind.” [P. Kundur ’94, V. Vittal ’03, . . . ]
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
1 / 18
Motivation: the envisioned power grid
Florian D¨ orfler (UCSB)
Kron Reduction
Energy is one of the top three national priorities
Expected developments in “smart grid”:
Expected developments in “smart grid”:
1
large number of distributed power sources
1
large number of distributed power sources
2
increasing adoption of renewables
2
increasing adoption of renewables
3
sophisticated cyber-coordination layer
3
sophisticated cyber-coordination layer
challenges: increasingly complex networks & stochastic disturbances opportunity: some smart grid keywords: control/sensing/optimization distributed/coordinated/decentralized
Today: “reducing the complexity by means of circuit and graph theory” Florian D¨ orfler (UCSB)
Kron Reduction
2 / 18
Motivation: the envisioned power grid
Energy is one of the top three national priorities
/ ,
Center for Nonlinear Studies
Center for Nonlinear Studies
3 / 18
/ ,
challenges: increasingly complex networks & stochastic disturbances opportunity: some smart grid keywords: control/sensing/optimization distributed/coordinated/decentralized
Today: “reducing the complexity by means of circuit and graph theory” Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
3 / 18
Kron reduction of a resistive circuit Nodal analysis by Kirchho↵’s and Ohm’s laws: I =Y ·V
Y = YT
n P
Yjk
Ik
Yk,shunt
1
.. .
..
Yik + Yk,shunt . . . .. .
..
Kron Reduction
.
1
8
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1 1
30
1
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30
1
1
1 1
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1 1
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1 1 1
1
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1
0.97
0.15
1
1
0.08
T · Ybound-int
-1
-0.66
1
0.21
0.06
-1
Kron Reduction
Center for Nonlinear Studies
5 / 18
an undirected, connected, & weighted graph with boundary nodes ⇤ ⌅, interior nodes • , & possibly self-loops
3
1.73
8
-1
0.72
30
30
8
“equivalent” reduced circuit
1
1 1
1
Ired = Yred · Vboundary
I =Y ·V
1
a symmetric and irreducible loopy Laplacian matrix Y with partition (⇤ ⌅ ,• ), & possibly diagonally dominance
2
8
original circuit
1 1
⇤ ⌅ , interior nodes
1.92
0.05
0.11
0.98
-1
30
a connected electrical network with conductance matrix Y , terminals • , & possibly shunt conductances
1
8
1
1
1
Consider either of the following three equivalent setups:
8
30
1
-1
Florian D¨ orfler (UCSB)
8
1 1
30
1 30
Boundary nodes ⇤ ⌅ arise as natural terminals in applications.
8
1
1 1
1
• via Schur complement:
0.39
30
30
1
1
1 1 1
Kron reduction of graphs
Ybound-int · Yinterior
-1
1 30
1 1
1
30
1 8
Center for Nonlinear Studies
Kron reduction: eliminate interior nodes
8
1
1
30
Kron reduction of a resistive circuit
Yred = Y /Yinterior = Yboundary
30
1
1
.. . 7 7 Yin 7 7 5 .. .
.
= { weighted Laplacian matrix } + diag Yk,shunt = “loopy Laplacian”
1
30
1
30
30
1
1 8
1 1
1 1
30
1
1 1 1
1 1
1 30
conductance matrix Y
G
1 1
1
loopy Laplacian matrix Y
1
27
8
1 1 1
30
1 8
3
30
1
3
k=1,k6=i
Florian D¨ orfler (UCSB)
-1
1
8
8
.. .. . 6 . 6 =6 6 Yi1 . . . 4 .. . .. .
k
Yik
I 2 Cn nodal current injections V 2 Cn nodal voltages/potentials Y 2 Cn⇥n nodal conductance matrix 2
Partition circuit equations via boundary nodes & interior nodes : Yboundary Ybound-int Vboundary Iboundary = T Iinterior Vinterior Ybound-int Yinterior
2
8
1
Kron reduction of a resistive circuit
1
1
27
1
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27
8
1
1
1 27
1
1 1
8
27
1
27
1 27
1
1 1
1 1
8
1
G. Kron, “Tensor Analysis of Networks,” Wiley, 1939. Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
6 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
7 / 18
Kron reduction of graphs
Kron reduction of graphs
Kron reduction via Schur complement: 8
30
Yred = Y /Yinterior G
30
8
1
1 1
1 1
30
1 1
30
1
1
1 30
1 1
1
0.39
1
reduction 0.08
Kron
8
0.15
8
Kron
0.06 8
Florian D¨ orfler (UCSB)
27
1
1
1
27
1
Relation of spectrum and algebraic properties of Q and Qred ?
1
How about the graph topologies and the e↵ective resistances?
1 1
1
8
What is the e↵ect of a perturbation in the original graph on the reduced graph, its spectrum, and its e↵ective resistance?
reduction
Gred
Finally, why is this graph reduction process of practical importance and in which application areas?
8
8
0.08
0.39
Kron-reduced loopy Laplacian Yred
1.73
1 27
1 27
reduction
0.21
0.05
0.11 8
27
1 1
transfer conductance matrix Yred
1.92
8
0.98
1
0.15 0.11
0.98
Kron Reduction
0.21 0.05 0.06
8
Center for Nonlinear Studies
1.92 1.73
8
8 / 18
Florian D¨ orfler (UCSB)
Kron reduction of graphs: applications
Kron Reduction
Center for Nonlinear Studies
Electrical impedance tomography
Smart grid monitoring
transformation
Yred
[A. E. Kennelly 1899, A. Rosen 1924]
Ycut
8
8
Kron reduction
1.0
1/3
1/3
30
1.0
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Kron reduction of graphs: applications
Purpose: construct low-dimensional equivalent circuits / graphs / models Simplest non-trivial case: star-
Yred = Y /Yinterior
1
8
1
1
1
27
loopy Laplacian matrix Y
8
Kron
1
30
1
1
1
27
to reconstruct spatial conductivity
through cut-set variables
[E. Curtis and J. Morrow ’94 & ’00]
[I. Dobson ’11]
1.0 1/3
8
8
8
32
32
Reduced power network modeling
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OFCIRCUITS INTEGRATED 29, NO. 1, JANUARY 2010 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED ANDCIRCUITS SYSTEMS,AND VOL.SYSTEMS, 29, NO. 1,VOL. JANUARY 2010 !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899:
!"#$%&' 8
8
15 37
δi / rad
10
32
5
7
8
8
8 8
8
8
8
8 8
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knot theory, Yang-Baxter equations and applications, high-energy physics, statistical mechanics, vortices in fluids, entanglement of polymers & DNA, . . . [F. D¨ orfler & F. Bullo ’11, J.H.H. Perk & H. Au-Yang ’06]
8
Physics applications:
23 16
11 10
32
19 20
33
33
3
3
34
34
4
4
5
5
8
sparse matrix algorithms, finite-element methods, sparse multi-grid solvers, Markov chain reduction, stochastic complementation, applied linear algebra & matrix analysis, Dirichlet-to-Neumann map, . . .
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06 07 08 09
19 20
14
6
36
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7
33
4
21
24 12
23 16
2
34
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10
15
31
10
0
Mathematics applications:
Center for Nonlinear Studies
8
24
7 14
13
2
36
11
31
2
32
Kron Reduction
9
21
6 12
31
20
13 7
18 4
8
45 6
11
8
28
7
15
23
29
27
3
39
δi / rad
39
17 8
2
26 2
35
0
22 19
38 25
30
22
4
9
-5
35
21 14 12
35
15 4
18
9
37
10
1 28
6
3
1
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27
3
6
1
0
16
39 5
25
2
5 6
F
9
Florian D¨ orfler (UCSB)
30
26
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smart grid monitoring, circuit theory, model reduction for power and water networks, power electronics, large-scale integration chips, electrical impedance tomography, data-mining, . . .
38
10
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28 27
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Engineering applications:
10 02 03 04 05
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10 30
7
8
Representation of integration chips 8
8
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1
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conductance matrix Y
1
1 1 1
27
8
30
1 1
1
Kron reduction via Schur complement:
8
Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16.
Fig. 5. conductance Two reducedmatrices conductance of the same network with 59 Fig. 5. Two reduced of thematrices same network with 59 matrix on the lefteliminating results when all internal nodes; terminals. The terminals. matrix on The the left results when all eliminating internal nodes; onbut the five right, all butnodes five internal nodes are on the right, all internal are eliminated. Theeliminated. matrix on The the matrix on the has fewer fivethan more than the first matrix, right has fewerright nonzeros, and nonzeros, only five and moreonly rows, therows, first matrix, hence the network corresponding network smaller: 721resistors. versus 1711 resistors. and hence the and corresponding is smaller: 721isversus 1711 By selecting (five toin be thispreserved case) to inbe preserved in By selecting specific internalspecific nodes internal (five in nodes this case) the reduced fill-in during elimination of the nodes other internal nodes is the reduced network, fill-in network, during elimination of the other internal is limitedThe considerably. difference bigger as limited considerably. difference The becomes even becomes bigger as even the number of the number of terminals increases. terminals increases.
for sparse computation
[J. Rommes and W. H. A. Schilders ’09]
T T Gk = G11 − G QQ k = G11 − QQ
0
2
4
6
8
10
TIME / s
for stability analysis and control
test system can be represented by ˙i = Hi ˙i = fs
i,
Di
10
i
+ Pmi
Gii Ei2
Ei E j ·
(11)
are provided to discuss whether the instability in Fig. 10 occurs in the corresponding real power system. First, the classical model with constant voltage behind impedance is used for first swing criterion of transient stability [1]. This is because second and multi swings may be affected by voltage fluctuations, damping effects, controllers such as AVR, PSS, and governor. Second, the fault durations, which we fixed at 20 cycles, are normally less than 10 cycles. Last, the load condition used above is different from the original one in [11]. We cannot hence argue that global instability occurs in the real system. Analysis, however, does show a possibility of global instability in real power systems.
[F. D¨ orfler and F. Bullo ’09] j=1,j=i
· {Gij cos(
i
j)
+ Bij sin(
i
j )},
where i = 2, . . . , 10. i is the rotor angle of generator i with respect to bus 1, and i the rotor speed deviation of generator i relative to system angular frequency (2 fs = 2 60 Hz). 1 is constant for the above assumption. The parameters fs , Hi , Pmi , Di , Ei , Gii , Gij , and Bij are in per unit system except for Hi and Di in second, and for fs in Helz. The mechanical input power Pmi to generator i and the magnitude Ei of internal voltage in generator i are assumed to be constant for transient stability studies [1], [2]. Hi is the inertia constant of generator i, Di its damping coefficient, and they are constant. Gii is the internal conductance, and Gij + jBij the transfer impedance between generators i and j; They are the parameters which change with network topology changes. Note that electrical loads in the test system are modeled as passive impedance [11].
IV. T OWARDS A C ONTROL FOR G LOBAL S WING I NSTABILITY Global instability is related to the undesirable phenomenon that should be avoided by control. We introduce a key mechanism for the control problem and discuss control strategies for preventing or avoiding the instability.
B. Numerical Experiment Coupled swing dynamics of 10 generators in the test system are simulated. Ei and the initial condition ( i (0), i (0) = 0) for generator i are fixed through power flow calculation. Hi is fixed at the original values in [11].
A. Internal Resonance as Another Mechanism Inspired by [12], we here describe the global instability with dynamical systems theory close to internal resonance [23], [24]. Consider collective dynamics in the system (5). For the system (5) with small parameters pm and b, the set
Florian D¨ orfler (UCSB) Kron Reduction reducedand network), andcan since this can bebefore computed before the reduced the network), since this be computed doing elimination the actual elimination we always include doing the actual we always include a check on a check on this. If ittoiseliminate decided to all internal nodes, the reduced this. If it is decided alleliminate internal nodes, the reduced matrix Gk can beefficiently computedusing efficiently conductanceconductance matrix Gk can be computed the using the Cholesky factorization Cholesky factorization G22 = LLT G22 = LLT
-5
Fig. 10. Coupled swing of phase angle i in New England test system. The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtained by numerical integration of eqs. (11).
Center for Nonlinear Studies
1
11 / 18
Kron reduction of graphs: properties
Kron reduction of graphs: properties 2
Kron reduction of a graph with boundary ⇤ ⌅ , interior • , non-neg self-loops loopy Laplacian matrix Y Schur complement: Yred = Y /Yinterior
8
k+1 k / Yred = Yred
Iterative 1-dim Kron reduction:
30
27
28
) topological evolution of the corresponding graph
25
1)
8
2)
8
•
8
8
28
30
30
30
30
30
30 8
8 27
27
25
27
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28
X
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30
8 30 30 8 8 30 8 8
39
1
2
45
6
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24 7
12
14
15
12
23
2
20
10
2
8
36
34
5
32 9
10
7
33
δi / rad
13 11
31
4
3
10
19
32
5
20
32
1 Yred
20 33
33
34
4
34
4
5
5
Yred = Yred
19
3
3
0
Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16.
11 10
n | |
0 Y = Yred
23 16
13
06 07 08 09
11
7
15
31
16
13
14
6
36
6
2
15
23
2 Yred
2
4
6
8
n | |+1
...
Yred
iterative Kron reduction
-5 0
30
21
5
10
24
7
31
19
17 8
36
0
22
14 12
Kron reduction of
22
4
9
21
7
-5
35
21 4
17 8
7
15
1 39
18
22
4
9
6
28
1
28 18
0
16
) Equivalence: the following diagram commutes:
27
3
35
F 3
1
27
3
29
39
1
26 2
35
17
5
9 38
25
30
6
9
24
29
2
37 10
6
δi / rad
38
27
6
25
8
30
26
1
5
10 02 03 04 05
9 38
10
8
18
30 8 8
8
2
...
25
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8
8
37 10
28
8
30
25
30
8
8
15 29 26
25
8
X
30
!"#$%&'
8 37
10
30 8
Well-posedness: set of loopy Laplacian matrices is closed !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899:
7)
8
30
27
30
8
8
X 1
30
30 8
8
3)
Properties of Kron reduction:
30
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30 8
10
TIME / s
Florian D¨ orfler (UCSB)
Kronswing Reduction Center for Nonlinear Studies Coupled of phase angle i in New England test system.
Fig. 10. The fault duration is 20 cycles of a 60-Hz sine wave. The result is obtained by numerical integration of eqs. (11).
12 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
13 / 18
test system can be represented by are provided to discuss whether the instability in Fig. 10 occurs in the corresponding real power system. First, the classical model with constant voltage behind impedance is used for first swing criterion of transient stability [1]. This is because second and multi swings may be affected by voltage fluctuations, damping effects, controllers such as AVR, PSS, and governor. Second, the fault durations, which we fixed at 20 cycles, are normally less than 10 cycles. Last, the load condition used above is different from the original one in [11]. We cannot hence argue that global instability occurs in the real system. Analysis, however, does show a possibility of global instability in real power systems.
Kron reduction of graphs: properties
by edge to grounded node ⌥ ⌃
B. Numerical Experiment Coupled swing dynamics of 10 generators in the test system are simulated. Ei and the initial condition ( i (0), i (0) = 0) for generator i are fixed through power flow calculation. Hi is fixed at the original values in [11]. Pmi and constant power loads are assumed to be 50% at their ratings [22]. The damping Di is 0.005 s for all generators. Gii , Gij , and Bij are also based on the original line data in [11] and the power flow calculation. It is assumed that the test system is in a steady operating condition at t = 0 s, that a line-to-ground fault occurs at point F near bus 16 at t = 1 s 20/(60 Hz), and that line 16–17 trips at t = 1 s. The fault duration is 20 cycles of a 60-Hz sine wave. The fault is simulated by adding a small impedance (10 7 j) between bus 16 and ground. Fig. 10 shows coupled swings of rotor angle i in the test system. The figure indicates that all rotor angles start to grow coherently at about 8 s. The coherent growing is global instability. 27
8
27
27
27
27
27
27
A. Internal Resonance as Another Mechanism Inspired by [12], we here describe the global instability with dynamical systems theory close to internal resonance [23], [24]. Consider collective dynamics in the system (5). For the system (5) with small parameters pm and b, the set {( , ) 2 S 1 R | = 0} of states in the phase plane is called resonant surface [23], and its neighborhood resonant band. The phase plane is decomposed into the two parts: resonant band and high-energy zone outside of it. Here the initial conditions of local and mode disturbances in Sec. II indeed exist inside the resonant band. The collective motion before the onset of coherent growing is trapped near the resonant band. On the other hand, after the coherent growing, it escapes from the resonant band as shown in Figs. 3(b), 4(b), 5, and 8(b) and (c). The trapped motion is almost integrable and is regarded as a captured state in resonance [23]. At a moment, the integrable motion may be interrupted by small kicks that happen during the resonant band. That is, the so-called release from resonance [23] happens, and the collective motion crosses the homoclinic orbit in Figs. 3(b), 4(b), 5, and 8(b) and (c), and hence it goes away from the resonant band. It is therefore said that global instability 27
8
8
27
27
27
27
Y
augment
Kron reduction of 27
3)
28
27
27
Florian D¨ orfler (UCSB)
Kron Reduction
25
27
25
30
8
X
8
7)
8
8
27
27
30
30 8
30
8
8
X 30
30
...
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25
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8
30
8
8
8
4
Kron reduction of
Yred Center for Nonlinear Studies
Topological properties: interior network connected ) reduced network complete at least one node in interior network features a self-loop ) all nodes in reduced network feature self-loops
Y
27
augment
X
8
28
5
Yred
25
28
X
30 8
(')$ Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June 10, 2009 at 14:48 from IEEE Xplore. Restrictions apply.
27
28
8
27
) Equivalence: the following diagram commutes:
C. Remarks It was confirmed that the system (11) in the New England test system shows global instability. A few comments
25
28
8
30
30
G
8
8
30
8
IV. T OWARDS A C ONTROL FOR G LOBAL S WING I NSTABILITY Global instability is related to the undesirable phenomenon that should be avoided by control. We introduce a key mechanism for the control problem and discuss control strategies for preventing or avoiding the instability.
8
2)
8
30
30
27
8
30
8
G
27
1)
8
Augmentation: replace self-loops 8
Kron reduction of graphs: properties
8
i
30
3
i
where i = 2, . . . , 10. i is the rotor angle of generator i with respect to bus 1, and i the rotor speed deviation of generator i relative to system angular frequency (2 fs = 2 60 Hz). 1 is constant for the above assumption. The parameters fs , Hi , Pmi , Di , Ei , Gii , Gij , and Bij are in per unit system except for Hi and Di in second, and for fs in Helz. The mechanical input power Pmi to generator i and the magnitude Ei of internal voltage in generator i are assumed to be constant for transient stability studies [1], [2]. Hi is the inertia constant of generator i, Di its damping coefficient, and they are constant. Gii is the internal conductance, and Gij + jBij the transfer impedance between generators i and j; They are the parameters which change with network topology changes. Note that electrical loads in the test system are modeled as passive impedance [11].
8
(11)
j )},
30
Ei E j ·
j=1,j=i
8
Gii Ei2
j ) + Bij sin(
30
10
+ Pmi
i
8
Di
· {Gij cos(
30
i,
8
˙i = Hi ˙i = fs
Algebraic properties: self-loops in interior network decrease mutual coupling in reduced network increase self-loops in reduced network
14 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
15 / 18
We start by considering “regular” electrical networks with scalar valued currents and p electrical network consists of an interconnection of purely resistive elements. Such generally described by graphs whose nodes connection points between resis Kron reduction of graphs: properties Kron reduction of represent graphs: properties correspond to the resistors themselves. The effective resistance between two nodes in Spectral properties: E↵ective resistance is defined as the potential drop between the Rtwo nodes, when a current source with ij : interlacing property: i (Y ) i (Yred ) i+n |⌅| (Y ) Ampere is connected across the two nodes (cf. Figure 1). For a general network, the com 6
7
) algebraic connectivity e↵ect of self-loops
2
is non-decreasing
u
on loop-less Laplacian matrices: + max{ }
2 (Lred )
2 (L)
+ min{ }
) self-loops weaken the algebraic connectivity
2
PSfrag replacements
Example: all mutual edges have unit weight 8
27
Ampere o
8
27
8
8
Kron reduction 27
27
27
27
27
27
8
+
1 Amp
8
8
8
without self-loops: 2 (L) = 0.39 0.69 = 2 (Lred ) with unit self-loops: 2 (L) = 0.39 0.29 = 2 (Lred )
Figure 1: A resistive electrical network for which a 1 Ampere current was is injected at at the reference node o. The resulting potential difference Vu Vo is the effective resis o. Kron reduction of graphs: properties Conclusions Florian D¨ orfler (UCSB)
Center for Nonlinear Studies
16 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
resistances relies on the usual Kirchoff’s and Ohm’s laws.
E↵ective resistance Rij :
8
30
G
30
8
1
establish Equivalence and invariance of Rij among ⇤ ⌅To nodes:
1 1
1
1
1
1
1 1 1
1
conductance matrix Y
8
1
30
30
30
1
1
8
1
27
1
8
1
1
1
30
1
1
30
1
1
1
30
1
27
1
1
27
1
0.39
0.08
27
1
1
1
27
1
27
1
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1
1 1
1
1
1
8
red
1.92
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Kron reduction of
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the connection between graph effective resistances defined in the prev loopy Laplacian trical networks we need to consider an abstract generalized electrical network in which matrix Y augment and resistors are k k matrices. For such networks,KronKirchoff’s current law can be defi Y Kron reduction reduction Kron reduction except that currents are added as matrices. Kirchoff’s voltage law can also be defined in G transfer conductance Rij potentials Kron reduction Y drops across edges are added as matrices.matrix Kirchoff’s voltage laws show the of i, j Kron-reduced valued node potential function. Ohm’s law takes loopy the following matrix form Laplacian Y 1
8
8
8
red
0.15
27
0.05
0.11
0.98
0.06
0.08
0.39
0.21
0.15 0.11
0.98
1.73
8
8
7
Kron Reduction
8
0.21 0.05
0.06
1.92
1.73
8
red
Yred
self-loops: Rij among ⇤ ⌅ &⌃ ⌥ uniform Kron Reduction
Ve = Re ie , Kron reduction is important in various applications
where k matrix flowing through edge e of the electric Analysis ofcurrent Kron reduction via algebraic graphthe theory 1 ⌅ ie is a generalized k R = 2 |Yred (i, j)| generalized resistance of that edge, and Ve is a &generalized k k matrix potential dr Open problem: directed complex-weighted graphs , R1 = ⌅2 |Yred (i, j)| + max{ } Generalized resistances are always symmetric positive definite matrices.
no self-loops: Rij among ⇤ ⌅ uniform ,
Florian D¨ orfler (UCSB)
Yred
augment
ij
ij
Center for Nonlinear Studies
17 / 18
Florian D¨ orfler (UCSB)
Kron Reduction
Center for Nonlinear Studies
18 / 18
The generalized electrical networks so defined share many of the properties of “re works. In particular, Kirchoff’s and Ohm’s laws uniquely define all edge currents an
