FundamentalTheoryforScalablePowerSystemsControl

TakayukiIshizaki(TokyoInstituteofTechnology)

JST-NSF-RCNWSonDistributedEMS

Collaborative Works Overview

Wind-IntegratedPowerSystem

Prof.Chakrabortty

Prof.Imura

RetConIntervalOptimization

Prof.Ramdani

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Sadamotovisit

Multiperiod ElectricityMarketDesign

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Prof.Ilić

intelligentBAs

RetConNMR

Sasahara

visit

Profs.Johansson&Sandberg

NetworkControlTheory

RetConNMRIEEE TRANSACTION ON AUTOMATIC CONTROL, VOL. X, NO. X, ... 20XX 6

In addition to the preexistent controller, we suppose that aset of additional decentralized controllers is designed such thateach of them stabilizes the corresponding disjoint subsystem,namely !

˙xi = Aixi +Biu′i

y′i = Cixi.

In particular, we suppose that an additional dynamical mapKi(·) is designed such that

˙xi = Aixi +BiKi(Cixi) (27)

is stable. In the following, we denote the index set of sub-systems to which the additional decentralized controllers areimplemented by L. Unless otherwise stated, we denote the ithcomponent of a stacked symbol by that with the subscript ofi, whereas the stacked version of a symbol is denoted by thatwithout the subscript of i, e.g., xi and x.

On the premise of the definition above, we consider theinput signal in the form of

u = w + w, (28)

for which w and w are to be constructed by cooperative use ofK(·) and Ki(·) for i ∈ L. To realize a practical control system,it is desirable that the local subsystem control is individuallymanaged by each of Ki(·), while the entire system stability isto be ensured by K(·). The simplest way to use both K(·) andKi(·) for i ∈ L would be implementing them as

w = K(y), wi = Ki(yi),

where wi is assumed to be zero for i ∈ L. However, this simpleimplementation does not necessarily guarantee the stability ofthe feedback system; thereby possibly inducing the instabilityof the entire feedback system.

To prevent the induction of instability, let us considergiving a compensation signal to each additional decentralizedcontroller. This is performed in a manner such that u in (28)is constructed by

K : w = K(y) (29)

in conjunction with the combination of

Ki : wi = Ki(yi − yi),

Σi : yi = Fi

"wi, {γj}j∈Ni

# (30)

where wi is assumed to be zero for i ∈ L, and Fi(·)denotes the dynamical map of a compensator Σi. Note thateach compensator measures the interconnection output signalsfrom the neighborhood subsystems and the input signal fromthe preexistent controller, i.e., {γj}j∈Ni and wi. Thus, thecombination of Ki and Σi in (30) can be regarded as adistributed controller using the output signals of neighborhoodsubsystems. In this formulation, we address the followingcompensator design problem.

Problem 4.1: Consider an interconnected system Σ in (23),each of whose subsystem is given by Σi in (22). Let a feedbackcontroller K in (29) be given such that (26) is stable, and letKi in (30) be given such that (27) is stable for every i ∈ L.Then, design a set of compensators Σi for i ∈ L in the formof (30) satisfying the following specifications.

Preexistent controller

Additional controllers

Compensators

connection

Fig. 4. Signal-flow diagram of hierarchical distributed control systems.

– The entire closed-loop system is stable under the feed-back control of (28).

– The design scheme of each compensator Σi is reliant onlyon the corresponding subsystem model of Σi, and not onthe controller models of K and Ki.

These specifications are satisfied when the stability of theentire feedback system is guaranteed for any combination of Kand Ki for i ∈ L such that (26) and (27) are stable. This typeof distributed control systems, referred to as a hierarchicaldistributed control system, is reasonable in the sense thateach retrofit controller, which corresponds to the combinationof Ki and Σi in (30), can be designed and implementedindependently of the other controllers. The entire signal-flowdiagram is depicted as in Fig. 4.

B. SolutionOn the basis of hierarchical state-space expansion, we give

a solution to Problem 4.1. The key to solving the probleminvolves selecting the parameters in Lemma 3.2 as

A = diag(Ai), P = I,

which lead to Γ in (18) whose (i, j)-block is given by

Γi,j =

!αi,jLi, j ∈ Ni

0, otherwise.(31)

Then, we have the following result.Theorem 4.1: With the notation in Section IV, consider the

set of compensators each of whose dynamics is given by

Σi :

!˙xi = Aixi + Li

$j∈Ni

αi,jγj +Biwi

yi = Cixi(32)

for i ∈ L. Then, the closed-loop system under the feedbackcontrol of (28) is stable for any choice of an index set L andany combination of feedback controllers K and Ki such that(26) and (27) are stable.

Proof: See the proof of Theorem 4.2, which includesTheorem 4.1 as a special case.

In Theorem 4.1, the dynamics of Σi can be regarded as anobserver for the interconnection output signals {γj}j∈Ni , towhich the input signal wi from the preexistent controller Kis injected. The compensator Σi plays the role cancelling outthe effect of interferences with neighborhood subsystems and

RetrofitControlNetworkModelReductionConvexAnalysis

Me

Koike

Large-Scale Power Systems ControlIEEJEAST30Model (stablesystem composedof30generators)

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Controllerdesignbylocalmodel?Stability?Betterperformance?

parameters?

parameters?

parameters?

parameters?

parameters?

Completeglobalmodelingisunrealistic!!

retrofitAreaofinterest

localdisturbance

4/13

Problem Formulation: Retrofit Control

Subsystemofinterest (modelavailable)

Othersubsystem(s) (modelunavailable)

possiblylargescale

"Problem# Findaretrofitcontroller suchthat(a)thewholesystemiskeptstable and(b)ismade small forany

Assumption: (i)aremeasurable(ii)thepreexistingsystemwithoutisstable

Hierarchical State-Space ExpansionCoupledstateequationofanda

stable (assumption)

state-spaceexpansiontocascaderealization

Hierarchicalrealization -dim

stable stabilizedby

decoupled!

"Lemma#

and forany

If andthen

6/13

Localized Controller Design

7/13

Hierarchicalrealization modelavailable!

Howtoimplement??

"Lemma# Designacontrollersuchthat

isstableand

Thentheclosed-loopsystemisstableand

Generalizationtodynamicalcontrollerdesignisstraightforward

constant

Controller ImplementationHowtoimplement??

"Theorem#

Localizingcompensator

Theclosed-loopsystemwiththeretrofitcontroller

isinternallystableanditsatisfies

with

with

Demonstration by Swing Equation Model

Localperf.

Glob

alperf.

High

Time

:designbasedoninfinitebusmodel:retrofitcontrol

unstable!

Freq

. dev

iatio

n

highgainLQR

lowgainLQR

retrofit

Scalabledevelopmentoflarge-scalestablenetworksystemsbasedondistributeddesignandimplementation ofmultipleretrofitcontrollers

6 MVA

w/ controllerw/o controller

Enhanced Damping of Wind Power Systems

20 MVA 40 MVA

Time (s)

rela

tive

gen.

angl

es (d

eg)

retrofitretrofit

Retrofitcontrolofwindpowerplantcanenhancedampingperformance

Poorlydampedaswindpenetrationlevelincreases

Time (s)rela

tive

gen.

angl

es (d

eg)

Freq

. diff

. (H

z)

byT.Sadamoto,A.Chakrabortty

RetrofitcontrolLocalizationofcontrollerdesignandimplementationStabilityguaranteeandcontrolperformanceimprovement

Concluding Remarks

Hierarchicalstate-spaceexpansionRedundantrealizationwithcascadestructureSystematicanalysisforstabilityandcontrolperformance

Thankyouforyourattention!

GeneralTheory: T.Ishizaki,T.Sadamoto,J.Imura,H.Sandberg,K.H.Johansson:RetrofitControl:LocalizationofControllerDesignandImplementation.arXiv

PowerSystemsApplication: T.Sadamoto,A.Chakrabortty,T.Ishizaki,J.Imura:ARetrofitting-BasedSupplementaryControllerDesignforEnhancingDampingPerformanceofWindPowerSystems.ACC2017