Measurement-Based Methods For Model Reduction .

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Measurement-Based Methods for Model Reduction,Identification, and Distributed Optimization ofPower SystemsSeyed Behzad NabaviDepartment of Electrical and Computer EngineeringNorth Carolina State University24/4/20151/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Part I– Identification of DynamicReduced-Order Models of Power Systems2/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsIntroductionIntroductionMathematical modeling of dynamicequivalents of large-scale electric powersystems has seen some 40 years of longand rich research history.Chow and Kokotovic established therelationship between the slow coherencyand weak connections using singularperturbation theory.Slow coherency arises from the slowerinter-area modes. These interarea modes,Figure: R. Podemore: Coherency inif not properly damped, lead to systemPower Systemsseparation and extensive loss of load.3/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsIntroductionModel-based Dynamic EquivalencingReal-TimeMonitoringRecentevidences Wide-Areaof blackouts haveshown the discrepancy between theoffline models and the response of the system.W. Winter, K. Elkington, G. Bareux, and J. Kostevc, “Pushing the Limits: Europe's New Grid: Innovative Toolsto Combat Transmission Bottlenecks and Reduced Inertia," Power and Energy Magazine, 13(1), 20154/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsIntroductionModel-based Dynamic EquivalencingDynamic equivalencing has seen 40 years of active research:–––––linear modal decomposition [Undrill, 71]circuit-theoretic approaches [de Mello, 75]machine aggregation [Germond, 78]enumerative clustering algorithms [Zaborsky, 82]software programs such as DYNEQ and DYNRED [Price, 95]Model based methods:– need the exact knowledge of the entire power system model,– are computationally challenging,– are based on idealistic assumption about system structure and clustering.5/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsIntroductionMeasurement-based Dynamic EquivalencingPMUs provide high-resolution GPS-synchronized three-phasemeasurements of voltage, current, phasor, and frequency.System operators are, therefore, inclining more towards online modelsconstructed from PMU (Phasor Measurement Unit) measurements.We next propose two algorithms to identify these dynamic equivalentmodels using PMU measurements:* Identification of the equivalent linear models* Identification of the equivalent nonlinear DAE modelsG8G10PMU81026G4s292825G1s9273124G9G4sI ps4V ps4183917G132166PMU15143313420730PMUPMU3832G2V ps3I ps3G3s23191311V sI ps 1 p122353637G1sG6211245G3G4G7G2sG3sG56/1V ps2I sp2G2sModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionPower System Swing EquationNonlinear Electromechanical Model:δ̇i (t) ωs (ωi (t) 1),Mi ω̇i (t) Pmi Pei (t) Di (ωi (t) 1),Pe (t) kXl NkPkl (t) PL (t) 0, Qe (t) kkXl NkQkl (t) QL (t) 0kLinearized Kron-Reduced Model (around (δi0 , 1)):" δ̇(t) ω̇(t)#" 0n nωs In n#" δ(t)M 1 L M 1 D ω(t) {z}A T δ , δ1 · · · δn ,# Bd(t), T ω , ω1 · · · ωn , δ, ω RnM diag(Mi ) Rn n , D diag(Di ) Rn n , d(t) : unknown disturbance[L]i,j Ei Ej (Gij cos(δi0 δj0 ) Bij sin(δi0 δj0 )) i 6 j,[L]i,i nX[L]i,k ,k 17/1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionLinear Dynamic Equivalent nsmissionNetwork GraphG3sModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionIdentification of Linear Equivalent ModelsThe reduced-order model:" δ̇ s (t) ω̇ s (t)#" 0r r(M s ) 1 Ls ωs Ir r#" δ s (t)# (M s ) 1 D s ω s (t){z}As B s d(t).Assumptions:– The area partitioning for our system is known apriori.– There is at least one PMU at a generator bus in each area (S).– As and B s are a controllable pair.Objective:– Finding the equivalent linear model of a power system from yi (t) i S:yi (t) {Ṽi (t), Ĩi,j (t)}, i S, j Ni .Proposed Identification Steps:– Extract δks (t) for each area k from yi (t), i S.G1sG2s– Identification of As ork GraphG3sModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionExtraction of δks (t) for Each Area kStep 1: Extract δi (t) from yi (t):Ei (t) δi (t) jxd0 i Ii (t) φi (t)jxd iI iVi iEi i Vi (t) θi (t) δ̂i (t) (jxd0 i Ii (t) φi (t) Vi (t) θi (t)) δ̂i (t) δ̂i (t) δ̂i (t0 ).Step 2: Extract δ̂ks (t) from δ̂i,k (t), (generator i belonging to area k ):0 δi,k (t) δi,k (t) r 1Xρil e( σl jΩl )t ( σl jΩl )t ρil en 1Xρil e( σl jΩl )t ( σl jΩl )t, ρil el rl 1 {z δ s (t), inter-area or slow modesi,k} {z} δ f (t), intra-area or fast modesi,kUse a modal decomposition technique such as Prony to decompose δi,k (t)sForm δi,k(t) by retaining only the modes in [0.1,1] Hz.10 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionExtraction of δks (t) for Each Area nNetwork GraphG3ssWe truncate δi,k (t) to extract δi,k(t).From the coherency assumptionsss δ1,k(t) δ2,k(t) · · · δm(t)k ,ksWe set δks (t) δi,k(t).11 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionIdentification of AsThe reduced-order model:" δ̇ s (t)# ω̇ s (t)" ωs Ir r0r r(M s ) 1 Ls #" δ s (t)# (M s ) 1 D s ω s (t){z}As B s d(t).Solve the following NLS problem (assuming d(t) is a momentaryperturbation at t t0 ):ZminsAtm kt1 s δ s (t, As ) δ̂ (t) 2 k dt ω s (t, As ) ω̂ s (t) 2where, s δ s (t, As ) δ̂ (t1 )s exp(A (t t1 )), ω s (t, As ) ω̂ s (t1 ) ω̂ s (t) is calculated from the numerical differentiation of δ̂ s (t)normalized by ωs .12 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionIdentifiability Analysis of AsLemma: [Bellman and Astrom-70] Consider the systemẋ Ax Bu, y CxIf the matrix C is full column-rank and the system is controllable, then Aand B can be determined uniquely from input output data.In our identification problem, we assume (As , B s ) to be a controllable pair,and C I2n (full column-rank), thus As is identifiable.More results on identifiability analysis will be provided in Part III (jointwork with Dr. P. P. Khargonekar).13 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionA Case Study– NPCC 48 Machine ModelG34 G35979610G33921 G48103G36911131 102110112 948912383116 84G3861057190G24 68119G39 2214133G4416130G4334G27817577677 553738281342750 7G18457064 636713715G4112112 55492117G856 G1948G46G26 72G25G6861G114711G7251413152662 58533 G23106113 888587G523115952G16107111G15 G21G1795108G321405101 G98733367879 8082G29G3016G11714 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionA Case Study- NPCC 48 Machine Model3 δ1COI δ1s model-based δ8COI106 δ8s model-based40 1(deg)51(deg)(deg)20COI δ17smodel-based δ1720 2 5 2 4 10 305101505Time (sec)3 δ1COI10 6015 δ1s Case 2 δ8COI101015Time (sec)6COI δ17 δ8s Case 22s δ17Case 240 1(deg)51(deg)(deg)5Time (sec)020 2 5 2 4 10 305101505Time (sec)10Time (sec)15 6051015Time (sec)Defining the error:Ja (k ) 1tm t1Ztm δks ,reduced (t) δks ,actual (t) dt.t1PPk Ja (k ) 10.2232(deg) for the model-based method, andk Ja (k ) 4.6017(deg) for our measurement-based method.15 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionIdentification of the Equivalent DAE ModelsThe linear equivalent models are in the Kron’s form.This model is not a very suitable choice for:1 Identification of the individual equivalent parameters such as inertia (Mi )2 Shunt controller design purposes3 Describing the system behavior for large disturbances (transient stability)Area 1Area 4G4s I ps4V ps4G1sI ps 1V ps1V ps3I ps3G3sV ps2I sp2Area 2G2sArea 3δ̇is (t)s sMi ω̇i (t)δ̇i (t) ωs (ωi (t) 1),Mi ω̇i (t) Pmi Pei Di (ωi (t) 1),16 / 1 ωis (t) ωs ,s Pesi Dis (ωis (t) 1), PmiModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionIdentification of the Equivalent DAE ModelsAssumptions:– The area partitioning for our system is known apriori.– The boundary buses of all areas are equipped with PMUs (denoted by S).Objective:– Finding the equivalent DAE model of a power system from yi (t) i S:yi (t) {Ṽi (t), Ĩi,j (t)}, i S, j Ni .Proposed Identification Steps:G4s– Finding the equivalent pilot bus voltages and currents.– Estimating the equivalent area impedances.I ps4V ps4G1sV sI ps 1 p1V ps3I ps3G3s– Estimating the equivalent generator parameters.– Estimating the inter-area impedances.V ps2I sp2G2s17 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionEquivalent Pilot Bus Voltage and CurrentStep 1: Use yk to calculate Ṽpk (t) and Ĩpk (t)PĨpk (t) , Ipk (t) φpk (t) XĨi (t), Ṽpk (t) , Vpk (t) θpk (t) i Bki BkṼi (t)Ĩi (t)Ĩp k (t)PMUPMUCoherentArea kStep 1V pkCoherentArea kPMU18 / 1I pkStep 2V pskCoherentArea kI pskModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionEquivalent Pilot Bus Voltage and CurrentStep 2: Construction of Ṽpsk (t) and Ĩpsk (t)The modal decomposition of δis (t):δis (t) 2rXρjl eλl t l 12r X2rXρ0jkl e(λl λk )t Vpsk (t) 2rXαlk eλl t l 1k 1 l 12r X2rXα0ijk e(λi λj )t ,i 1 j 1Use Prony to decompose Vpk (t):Vpk (t) NXβlk eγl tl 1Retain only those modal components within the [0.1,1] Hz. The sum of theseselected modal components are classified as Vpsk (t).Apply the same procedure to extract θpsk (t), Ipsk (t), and φspk (t)PMUPMUCoherentArea kStep 1V pkCoherentArea kPMU18 / 1I pkStep 2V pskCoherentArea kI pskModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionEquivalent Area ImpedanceKVL in the equivalent circuit:Eks (t) δks (t) (rks jxd0sk )Ĩpsk (t) Ṽpsk (t).For any time instance:Φ0 , (rks jxd0sk )(Îpsk (t0 ) φ̂spk (t0 )) V̂psk (t0 ) θ̂psk (t0 ) ,.Φm , (rks jxd0sk )(Îpsk (tm ) φ̂spk (tm )) V̂psk (tm ) θ̂psk (tm ) .The estimation of rks and xd0sk can be posed as the following NLS problem: min var Φ0 , . . . , Φm ,xd0s , rkskrksEks ksI pskjx dskV psk V psk psk19 / 1the kth equivalentpilot busModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionEstimating the equivalent generator parametersSolve the following NLS problemZminss sMk ,Dk ,Pmktmt0 δ̂ks (t) δks (t, Mks , Dks , Pms k ) 2 dt,whereδ̇ks (t)s sMk ω̇k (t) ωks (t) ωs ,s Pm Pesk Dis (ωks (t) 1),kδks (t) δks (t0 ), ωks (t) ωks (t0 ), Pesk (t) Re Êks (t) δ̂ks (t) Ĩps (t)krksEks ksI pskjx dskV psk V psk psk20 / 1the kth equivalentpilot busModel Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionEstimating the inter-area impedancesKCL on equivalent pilot buses: Y s Ṽ s (t0 ) · · · Ṽ s (tm ) Ĩ s (t0 ) · · · Ĩ s (tm ) ,{z} {z} Ṽ sĨ ssEstimate Y by solving:minkY s Ṽ s Ĩ s k2F ,sYs.t. Y s (Y s )TG4sI ps4V ps4G1sV sI ps 1 p1V ps3I ps3G3sV ps2I sp2G2s21 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionA Case Study– IEEE 39 Bus Model.0074 j.02689D4s 1.6541M 4sArea 4293022V ps22320PMU42 jD3s 0 3 3x 10H 3s 267.2335G2sD2s 0.4417 M 2sG4 31.5H 2s 82.3485G7G5G3G2Area 275320.050119353738.0100 j.0267.0037 j.045113j 0.08152411159.0j0G1s63436-0.0301 -j 1.1996j0.00582112G3sV ps37G6151433G1V ps116971732.1j0391 D1s 0.3227 M 1s18PMU1V ps405Area 3PMU27H1s 510.6557j0.081726.0-031x 101x 1010.5(rad/sec)(rad/sec)10 0.5ω̂2s ω̂1sω2s ω1s 1 1.50.511.5Time (sec)22.5(rad/sec)Area 1PMU280.0076 25PMU178G1010H 4s 106.6757G4sG90.00G80 1 20.51.5Time (sec)22 / 120 0.5ω̂3s ω̂1sω3s ω1s10.52.5 10.5ω̂4s ω̂1sω4s ω1s11.522.5Time (sec)Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Identification of Dynamic Reduced-Order Models of Power SystemsPower System Model ReductionFuture WorkInvestigating the utility of the reduced order models for shunt controllerdesign purposes (such as Static Var Compensator (SVC)).V psk (t )GksI psk (t )ControlInversionCoherentArea kSVCSVC23 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringPart II– Distributed Optimization Algorithmsfor Wide-Area Oscillation Monitoring inPower Systems24 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringIntroductionIn Part I, we describe methods to identifythe equivalent models from PMU. In PartII, we use PMUs to identify the (inter-area)oscillation modes from PMUs in adistributed way.Majority of modal estimation algorithmsare centralized such as: EigenvalueRealization Algorithm (ERA)[Sanchez-Gasca-99], Prony analysis[Hauer-90], Robust Least Squares[Zhuo-08], and Hilbert-Huang transform[Messina-06].Figure: http://www.eia.gov/As the number of PMUs scales up into the thousands, the currentstate-of-the art centralized architectures will no longer be sustainable.25 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringWide-Area Oscillation MonitoringUsing PMU measurements to estimate the frequency, damping factor andresidue of the different electro-mechanical oscillation modesG14PMUG166 404148474231385149G1152G163664374468 43G13 53 6861154191005Time 1413562 MUG1046229G8 56G1560130 3326267PMU5305Time (sec)26 / 1G72057G50.240.220.20.180.1605Time (sec)Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringWide-Area Oscillation MonitoringWide-Area Oscillation MonitoringUsing PMUmeasurementsto estimatethe thefrequency,dampingUsingPMU measurementsto estimatefrequency,damping factor andresidue ionmodesof theelectro-mechanical oscillationmodesState-of-the-Art MonitoringArchitectureThe Proposed DistributedMonitoring Architecture26 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringProblem FormulationOscillation Monitoring011112212 20E1 1G1111122122 I L1V1 1E2 2G2En nGnVn nV2 2 I L2 I L ,n01Vi iGiVn 1 n 1011112222 12222 G n 1Ei i1111En 1 n 1 I Li I L , n 10111127 / 112222 Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringProblem FormulationCentralized Prony MethodStep 1. Find a1 through a2n θi (2n 1)··· θi (2n) θi (2n)··· θi (2n 1) . . θi (2n ) θi (2n 1) · · ·{z} {z ciHi θi (0) θi (1) . . . θi ( )} a1 a2 . . . a2n {z } aFinding the global a using all available measurements by solving: c1H1 . . . . a . . Hpcp {z}Solve this using Batch Least Squares - Centralized Prony MethodStep 2. Find the eigenvalues of A (i.e., σi jΩi ) by– Finding the roots of discrete-time transfer function (z1 through z2n )– Converting them from discrete-time to continuous-time28 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringProblem FormulationCentralized Prony Methodθi (Hi , ci ), i 1, . . . , p H1c1 . a . PMUG14414738673162516318G11354550393664374468 43G13762 G126121110G22717211413554 H1c11 a arg min k . a . k22a2HpcpG915455G31956G4cp 61262598333452G16286016G1046493229G8532130 342G15HpG14866 402423225859G6G72057G529 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringProblem FormulationDistributing the Prony MethodN Computational Areas: θj,i : PMU i in area j θj,i Hj,i , cj,iTTTĤj , [Hj,1Hj,2· · · Hj,N]T ,jG144866 4041473162386751G1135453952G16183664374443G1362 G122114135612115410G2Nj : is the total number of PMUs in Area 32130 342G15TTT· · · cj,N]Tcj,2ĉj , [cj,1jG155G31956G424Global Consensus Problem:23225859G6minimizeG720PNa1 ,.,aN ,z57G51i 1 2 kĤi ai ĉi k22subject to ai z 0, for i 1, . . . , NUse Alternating Direction Method of Multipliers(ADMM) to solve it30 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringProblem FormulationDistributing the Prony MethodThree Distributed Cyber-PhysicalArchitectures (Using ADMM):Standard ADMM– Asynchrnous ADMMG14G14866 2 G126121110G2Distributed ADMM271721141355461G91548333450G16282616526825130 332G104649602Hierarchical ADMM29G85355G31956G42423225859G6G72057G530 / 1Model Reduction, Identification, and Distributed Optimization of Power Systemsc 2015 by S. Nabavi

Distributed Oscillation MonitoringDis

Figure: R. Podemore: Coherency in Power Systems 3 / 1 Model Reduction, Identification, and Distributed Optimization of Power Systems . Identification of Dynamic Reduced-Order Models of Power Systems Power System Model Reduction A Case Study– NPCC 48 Machine Model G 34 G 48 G 46 G 45 43 G 7 G 6 G 4 G 5 G 10 36 97 92 33 91 96 95 93 94 111 .

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