KTH ROYAL INSTITUTE OF TECHNOLOGY Structured Model .

KTH ROYAL INSTITUTEOF TECHNOLOGYStructured Model Reduction ofNetworks of Passive SystemsHenrik SandbergDepartment of Automatic ControlKTH, Stockholm, Sweden

Joint Work With Bart BesselinkUniv. of GroningenKarl Henrik JohanssonChristopher SturkKTH Automatic ControlLuigi VanfrettiYuwa ChompoobutrgoolKTH Power Systems2

Outline Introduction Part I: Clustering-based model reduction of networkedpassive systems Part II: Coherency-independent structured modelreduction of power systems Summary3

Motivation: Networked SystemsChallenges Dynamics dependent on subsystems and interconnection Large-scale interconnection complicates analysis,simulation, and synthesisGoal. Model reduction of large-scale networked systems4

Related WorkGeneral methods Balanced truncation (Moore, Glover, ) Hankel-norm approximation (Glover, ) Moment matching/Krylov-subspace methods (Antoulas,Astolfi, Benner, )5

Related WorkReduction of subsystems, i.e., structured reduction Controller reduction/closed-loop model reduction(Anderson, Zhou, De Moor, ) Structured balanced truncation (Beck, Van Dooren,Sandberg, ) Example in Part II6

Related WorkClustering-based model reduction Time-scale separation (Chow, Kokotovic, ) Graph-based clustering (Ishizaki, Monshizadeh,Trentelman ) Structured balanced truncation (Besselink, ) Example in Part I and II7

Part I: Clustering-based model reduction ofnetworked passive systemsProblem and results Subsystems with identical higher-order dynamics Controllability/observability-based cluster selection A priori 𝐻 -error bound and preserved synchronization (cf.balanced truncation)Reference. Besselink, Sandberg, Johansson: "Clustering-BasedModel Reduction of Networked Passive Systems". IEEE Trans. onAutomatic Control, 61:10, pp. 2958--2973, October 2016.8

Modeling1.Identical subsystem dynamics2. Interconnection topology with 𝑤𝑖𝑗 03. External outputs9

AssumptionsA1. The subsystems Σ𝑖 are passive with storage function1𝑉𝑖 (𝑥𝑖 ) 𝑥𝑖𝑇 𝑄𝑥𝑖 (supply𝑖 𝑣𝑖𝑇 𝑧𝑖 )2A2. The graph 𝒢 (𝒱, ℰ) with graph Laplacian 𝐿 is such thata) The underlying undirected graph is a treeb) 𝒢 contains a directed rooted spanning tree10

Network Synchronization11

Problem and ApproachGoal. Approximate the input-output behavior of Σ by aclustering-based reduced-order system Σ12

Problem and ApproachWish list for approximation method1. Preserve synchronization and passivity2. Identify suitable clusters3. Provide a priori bound on 𝑦 𝑦4. Be scalable in system size (#nodes 𝑛, state dim. Σ 𝑛 𝑛)13

Problem and ApproachIdea. Find neighboring subsystems Σ𝑖 that are hard to steer individually from the inputs hard to distinguish from the outputs14

Edge Laplacian 𝑳𝐞15

Edge Dynamics and Controllability16

Edge Dynamics and ControllabilityProperties Gramian can be defined as Σe is asymptotically stable Π𝑐 ℝ 𝑛 1 𝑛 1 only dependent on interconnection properties Measure of controllability for each individual edge17

Edge Singular valuesGeneralized edge controllability GramianGeneralized edge observability GramianGeneralized squared edge singular valuesNote. Minimize trace of Π𝑐 and Π𝑜 to obtain unique Gramiansand small singular values18

One-step ClusteringReduced-order systemPetrov-Galerkin projection of graph Laplacian19

One-step ClusteringOpens up for repeated one-step clustering!20

Performance GuaranteesGeneralized edge singular values21

Summary So FarWish list for approximation method1. Preserve synchronization and passivity OK2. Identify suitable clusters Use generalized edge singular values3. Provide a priori bound on 𝑦 𝑦 Generalized edge singular values provide bounds4. Be scalable in system size (#nodes 𝑛, state dim. Σ 𝑛 𝑛) Solve two LMIs of size 𝑛 (independent of subsystemsize 𝑛) [and possibly one Riccati equation of size 𝑛 toverify passivity]22

Example: Thermal Model of a Corridor ofSix Rooms23

Example: Thermal Model of a Corridor ofSix Rooms24

Example: Thermal Model of a Corridor ofSix Rooms25

Summary Part I Clustering-based reduction procedure Edge controllability and observability properties Preservation of synchronization and error boundPossible extensions Arbitrary network topology Non-identical subsystems Nonlinear networked systems Lower boundsReference. Besselink, Sandberg, Johansson: "Clustering-BasedModel Reduction of Networked Passive Systems". IEEE Trans. onAutomatic Control, 61:10, pp. 2958--2973, October 201626

Part II: Coherency-independent structuredmodel reduction of power systemsProblem and results Model reduction of nonlinear large-scale power system Clustering, linearization, and reduction of external area Application of structured balanced truncationReference. Sturk, Vanfretti, Chompoobutrgool, Sandberg: "CoherencyIndependent Structured Model Reduction of Power Systems". IEEETrans. on Power Systems, 29:5, pp. 2418--2426, September 2014.27

Background Increasingly interconnected powersystems New challenges for dynamicsimulation, operation, and control oflarge-scale power systems Coherency-based power systemmodel reduction not always suitable28

ApproachDivide system into a study area and an external areaObjective: Reduce the external area so that the effect of theapproximation error in the study area is as small as possible29

ApproachDivide system into a study area and an external area Study area 𝑁 often set by utility ownership or market area.Nonlinear model will be retained here External area 𝐺 denotes other utilities. Will be linearized andreduced here Insight from structured/closed-loop model reduction: Reductionof 𝐺 should depend on 𝑁!30

Four-Step Procedure1. Define the model (DAE)2. Linearizing31

Four-Step Procedure3. Structured/closed-loop model reduction of external areamodel, 𝐺 𝐺 (details next)4. Nonlinear complete reduced modelReduced linearexternal areaUnreduced nonlinearstudy area32

Structured Model Reduction of 𝑮(Following Schelfhout/De Moor, Vandendorpe/Van Dooren,Sandberg/Murray): 𝑁, 𝐺 Σ(𝐴, 𝐵, 𝐶, 𝐷)Local balancing of 𝐺 only:Structured (Hankel) singular values of 𝐺:Truncation or singular perturbation of 𝐺 yields 𝐺Note 1. 𝐺 depends on study area 𝑁Note 2. Error bound and stability guarantee requiregeneralized Gramians (LMIs) [Sandberg/Murray]33

Model Reduction of Non-Coherent Areas:KTH-Nordic32 SystemModel info: Study area:SouthernSweden.Keepdetailed model52 buses52 lines28 transformers20 generators(12 hydro gen.)External area:Simplify as muchas possible34

Model Reduction of Non-Coherent Areas:KTH-Nordic32 System External area 𝐺 has 246 dynamic states. Reduced external area 𝐺 has 17 dynamic states35

Model Reduction of Non-Coherent Areas:KTH-Nordic32 System External area 𝐺 has 246 dynamic states. Reduced external area 𝐺 has 17 dynamic states36

What If Open-Loop Reduction Used toSimplify External Area 𝑮?[Sturk et al.:“Structured Model Reductionof Power Systems”, ACC 2012]37

Summary Part II Clustering, linearization, and reduction of external powersystem area Application of structured balanced truncation: Closed-loopbehavior matters! Verification on a model of the Nordic gridPossible extensions Nonlinear model reduction with error bounds and stabilityguaranteesReference. Sturk, Vanfretti, Chompoobutrgool, Sandberg: "CoherencyIndependent Structured Model Reduction of Power Systems". IEEETrans. on Power Systems, 29:5, pp. 2418--2426, September 2014.38

Concluding Remarks Model reduction of networked systems. Dynamics dependenton subsystems and interconnection. Many applications! Model reduction methods could reduce topology and/ordynamicsChallenge. Many heuristics possible. We want rigorous scalablemethods with performance guarantees. Balanced truncation and Hankel-norm approximation do notpreserve network structures very well LMIs are very expensive to solve [ 𝒪(𝑛5.5 )]39

Thank You!SponsorsContactHenrik SandbergKTH Department of Automatic Controlhsan@kth.sepeople.kth.se/ hsan/40

model reduction of power systems Problem and results Model reduction of nonlinear large-scale power system Clustering, linearization, and reduction of external area Application of structured balanced truncation Reference. Sturk, Vanfretti, Chompoobutrgool, Sandberg: "Coherency-Independent Structured Model Reduction of Power Systems".