Clustered Model Reduction of Interconnected Second-Order Systems
and Its Applications to Power Systems

Takayuki Ishizaki, Jun-ichi Imura (Tokyo Inst. of Tech.)Outline
Introduction: Why clustered model reduction?
Clustered Model Reduction Theory
Interconnected first-order systems
Extension to second-order networks

Application to power networks
Conclusion
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Control of Large-Scale Networks
Power Networks

Thermal power
Power line

Tokyo area: 20 million houses
Instability may be caused byrenewables such as PV, wind

homes
Micro Gridhomes

PV

Hydroelectricpower

Modelreduction is one prospective approach
TrafficNetworksCenter of Tokyo area: 5 million cars
Heavy traffic jam

How to manage?

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Standard Model Reduction Framework✓Main goal: Findsuch thatis small enough+ stability of error systems, error analysis, low computation costStandard methods:
Balanced truncation, Hankel norm approximation
error bound, stability preservation  high computational cost Krylov projection
lower computation cost  possibly unstable model, no error bound 4/21

Application of Standard Methods toNetwork Systems
Drawback: Network structure is lost through reduction
Network system

Reduced model

Dense matrix

Sparse Dense 
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Clustered Model Reduction
Network system

Aggregated model

Aggregated state

Cluster state: row vector

Sparse Sparse 

Preservation of network structure among clusters

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Why Clustered Model Reduction?
Gene Network [Mochizuki et al. , J. Theoretical Biology (2010)]

Clustered model reduction

Extract essential structure
to study mechanism of functions
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Outline
Introduction: Why clustered model reduction?
Clustered Model Reduction Theory
Interconnected first-order systems
Extension to second-order networks

Application to power networks
Conclusion
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System Description (First-Order Subsystems)[Definition] Bidirectional Networkwith
is said to be bidirectional network if

andis symmetric and stable.

Reaction-diffusion systems:

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Clustered Model Reduction Problem[Problem] Given
where, find a cluster set

such thatand

Bidirectional network
Cluster stateSparse Aggregated model

Aggregated stateSparse 

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How to Formulate Reducibility?
Bidirectional network[State trajectory under random]

7 clusters

50 trajectories

50 nodes, nonzero
israndomly chosen fromcan be aggregatedintoLocal
7-dim.uncontrollability!
variable[Definition] Reducible cluster
A clusteris said to be reducible if under any input signal
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Positive Tridiagonalization[Lemma] For every bidirectional network
such that
Bidirectional network, there exists a unitary
has the following structure.

Positive tridiagonal realization(not necessarily positive)

Metzler✓

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Reducibility Characterization
Bidirectional network

reducible

reducible－（graph Laplacian＋diagonal）

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Reducibility Characterization
Bidirectional network: positive tridiagonal realization: transformation matrix
Index matrix

reducible

reducible

Characterization in frequency domain

identical
identical

Equivalent characterization of cluster reducibility14/21

-Reducible Cluster Aggregation[Definition]-Reducibility of Clusters

A clusteris said to be -reducible if

Similar behavior[Theorem] If all clusters are -reducible, thenwhere: coarseness parameter1000 nodes

47 clusters

Extract essential
cluster structure!

aggregation

about 5% error
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Outline
Introduction: Why clustered model reduction?
Clustered Model Reduction Theory
Interconnected first-order systems
Extension to second-order networks

Application to power networks
Conclusion
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Formulation by Second-Order Systems
Second-order networks[Problem]

whereGiven

Aggregated model, find a cluster set

such thatand
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Extension to Second-Order Networks
First-order representation (2n-dim. system)✓whereIndex matrix[Definition]

w.r.t.
A cluster

position
velocityis said to be -reducible if[Theorem] If all clusters are -reducible, then
where
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Outline
Introduction: Why clustered model reduction?
Clustered Model Reduction Theory
Interconnected first-order systems
Extension to second-order networks

Application to power networks
Conclusion
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Numerical Example
Power network modeled by swing equation
Original network (300 nodes) Agg. model (42 clusters)

Zoom up

Position trajectory of a mass
smaller error

Relative error

when20/21

Concluding Remarks
Clustered model reduction
extract essential information on input-to-state mapping
application to power networks model by swing equation

Future works
application to more realistic power networks
extension to nonlinear systems
application to control system design[T. Ishizaki et al. IEEE TAC (2014)], [T. Ishizaki et al. NOLTA (2015)], My website, etc.

Thank you for your attention!
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