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

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1 Clustered Model Reduction of Interconnected Second-Order Systems and Its Applications to Power Systems Takayuki Ishizaki, Jun-ichi Imura (Tokyo Inst. of Tech.)

2 Outline Introduction: Why clustered model reduction? Clustered Model Reduction Theory Interconnected first-order systems Extension to second-order networks Application to power networks Conclusion 2/21

3 3/21 Control of Large-Scale Networks Power Networks Tokyo area: 20 million houses Instability may be caused by renewables such as PV, wind Micro Grid Power line homes PV Thermal power Model reduction is one prospective approach Traffic Networks Center of Tokyo area: 5 million cars Heavy traffic jam homes Hydroelectric power How to manage?

4 Standard Model Reduction Framework Main goal: Find such that is small enough + stability of error systems, error analysis, low computation cost Standard 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

5 Application of Standard Methods to Network Systems Drawback: Network structure is lost through reduction Network system Reduced model Dense matrix Sparse Dense 5/21

6 Clustered Model Reduction Network system Aggregated model Cluster state Aggregated state : row vector Sparse Sparse Preservation of network structure among clusters 6/21

7 7/21 Why Clustered Model Reduction? Gene Network [Mochizuki et al., J. Theoretical Biology (2010)] Clustered model reduction Extract essential structure to study mechanism of functions

8 Outline Introduction: Why clustered model reduction? Clustered Model Reduction Theory Interconnected first-order systems Extension to second-order networks Application to power networks Conclusion 8/21

9 9/21 System Description (First-Order Subsystems) [Definition] Bidirectional Network with is said to be bidirectional network if and is symmetric and stable. Reaction-diffusion systems:

10 Clustered Model Reduction Problem [Problem] Given, find a cluster set such that where and Bidirectional network Aggregated model Cluster state Aggregated state Sparse Sparse 10/21

11 11/21 How to Formulate Reducibility? Bidirectional network [State trajectory under random ] 7 clusters 50 trajectories 50 nodes, nonzero is randomly chosen from [Definition] Reducible cluster A cluster can be aggregated into Local 7-dim. uncontrollability! variable is said to be reducible if under any input signal

12 Positive Tridiagonalization [Lemma] For every bidirectional network such that, there exists a unitary has the following structure. Bidirectional network Positive tridiagonal realization (not necessarily positive) Metzler 12/21

13 13/21 Reducibility Characterization Bidirectional network reducible reducible -(graph Laplacian+diagonal)

14 Reducibility Characterization Bidirectional network : positive tridiagonal realization : transformation matrix Index matrix reducible reducible Characterization in frequency domain identical identical Equivalent characterization of cluster reducibility 14/21

15 -Reducible Cluster Aggregation [Definition] -Reducibility of Clusters A cluster is said to be -reducible if Similar behavior [Theorem] If all clusters are -reducible, then where : coarseness parameter 1000 nodes 47 clusters Extract essential aggregation cluster structure! about 5% error 15/21

16 Outline Introduction: Why clustered model reduction? Clustered Model Reduction Theory Interconnected first-order systems Extension to second-order networks Application to power networks Conclusion 16/21

17 17/21 Formulation by Second-Order Systems Second-order networks Aggregated model [Problem] Given, find a cluster set such that where and

18 Extension to Second-Order Networks First-order representation (2n-dim. system) where Index matrix w.r.t. position velocity [Definition] A cluster is said to be -reducible if [Theorem] If all clusters are -reducible, then where 18/21

19 Outline Introduction: Why clustered model reduction? Clustered Model Reduction Theory Interconnected first-order systems Extension to second-order networks Application to power networks Conclusion 19/21

20 20/21 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 when

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! 21/21