Robust Decentralized Control In Power Systems

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Robust decentralized control in power systemsClaudio De PersisInstitute of Engineering and TechnologyJ.C. Willems Center for Systems and ControlPowerWeb Lunch LectureTU Delft, December 13, 2018Joint work with Weitenberg (RUG), Jiang-Mallada (JohnsHopkins), Zhao (NREL), Dörfler (ETH)

AC power systems Power system networkof generation, loads,transmission lines Power system control maintain system securityat minimal cost Basic security requirement keeping frequencyaround nominal value1 / 22

Frequency control: BasicsFrequency control Any instantaneous load-generation imbalance results in afrequency deviation from the nominal one (50-60 Hz) Small load changes on a fast time scale are dealt with theAutomatic Generation Control (AGC)2 / 22

The three control layersfound that a20% of annbarriers,” hoptions sucoutput contis potentialvery high wwind turbinseveral EurIn countrrelatively lagrid frequencrucial. A 2Fig. 6. Inertial, primary frequency controls, and AGC (secondary) responsethat replaci[figure EPRI](figure courtesy Pouyan Pourbeik of EPRI).percentage plementedby individual power plants adjusting their power levels in can potentiviaresponselocal droopcontrol fromof turbinegovernorsfrequency. Tto requeststhe systemsoperator. Secondary AGC is centralized and uses integral control to restorelevels of intB. Motivation for Active Power Control in Wind TurbinesGreek islesfrequency22Active power control (APC) is the purposeful control ofThe3 /nece

Conventional operational strategyCentral control authority AGC is implemented with a central regulator Frequency deviation ω is measured at low-voltage network andintegrated to generate the regulator output signal pT ṗ ω The incremental contribution of the individual generatingunits to the total generation is obtained via participationsfactors Ki 1ui Ki 1 p,i 1, 2, . . . , n4 / 22

Conventional operational strategy The conventional strategy is developed for conventionalgenerators which have high inertia, hence abrupt changes arebetter absorbed by the system, thus easing the task offrequency restoration Renewable generation leads to significant reduction of inertia,hence to a more volatile network, which challenges existingcontrol schemes5 / 22

Distributed controlAn answer to this challenge has leveraged the use of localcontrollers that cooperate over a communication networkPHYSICALNETWORKP1eθ̇1 ω1M1 ω̇1 A1 ω1 u1 P1l P1ey1 ω1ω1u1.Pneθ̇n ωnMn ω̇n An ωn un Pnl Pneyn ωn! 1θ̇1 j N1 (θj θ1 ) q1 ω1u1 q1 1 ω1θ.ωnun! 1θ̇n j Nn (θj θn ) qn ωnun qn 1 ωnθCOMMUNICATION NETWORKICT for Network Systems Semi-centralizedPT ṗ i ωi ,ui Ki 1 p Distributed averaging integralP 1Ti ṗi j Nic (pj pi ) Ki ωiui Ki 1 pi15 / 32Yet, due to security, robustness and economic concerns, it isdesirable to regulate the frequency without relying oncommunication6 / 22

Power networkLossless, network-reduced power system with n generating unitsgener. unit 1POWER NETWORKgener. unit 2gener. unit 4gener. unit 6generating unit ifrequency deviation ωicontroll. power inj. uiuncontroll. power inj. Pi?gener. unit 3gener. unit 5gener. unit 7powertransmissiongener. unit 8generating unit jfrequency deviation ωjcontroll. power inj. ujuncontroll. power inj. Pj?[figure Stegink]7 / 22

Frequency dynamicsθ̇i ωi ,γijX z } {Vi Vj Bij sin(θi θj ) ui Pi?Mi ω̇i Ai ωi j Niu4Local measurements: ωiω4 Swing equations ωi frequency deviationi6 jγ12M6u61XMi ωi22 iω6u2M2γ36 mechanical equivalent Bij Vi Vj cos(θi θj ) γ23ω1M1γ56 purely inductive linesBij Bji12ω2M5 voltages Vi constantH M3γ34γ45ω5 θi phase angle deviationEnergy function:XM4u5u3ω3u1γ81γ67M8M7ω7ω8u8u78 / 22[figure Stegink]

Synchronization frequency controlθ̇i ωiPMi ω̇i Di ωi ui Pi? j Bij Ei Ej sin(θi θj ) Synchronous solutionωi ωsync Synchronous frequencyωsync PCase n 2M1 ω̇1 M2 ω̇2 iPPi? i uiPi Di D1 ω1 u1 P1? B12 E1 E2 sin(θ1 θ2 ) D2 ω2 u2 P2? B21 E2 E1 sin(θ2 θ1 )If ω1 ω2 ωsync const, summing up0 (D1 D2 )ωsync u1 u2 P1? P2? Zero frequency deviation0 PiPi? Piui9 / 22

Optimal frequency restorationManifold choices of ui? to achieveXX0 Pi? ui?iiOptimal dispatch problemPminimizeu Rn Pi ai ui2 P subject toi ui 0i Pi SolutionP ?Pui? ai 1 Pi ii aiOptimal frequency restorationGiven unknown Pi? , designui (ωi )that stabilizes the powersystem model to(θi? , ωi? 0, ui? )Fair proportional sharingai ui? aj uj? i, j10 / 22

Fully decentralized frequency controlTi ṗi ωiui pi No communicationrequired Frequency regulationω 0Frequency (Hz)IEEE 39-node ‘New England’ benchmark network60 Frequency at G159.9 Noisy measurementsωi ηi59.859.7 Non-zero mean noise ηno noisenoisy59.6020406080 Noise bound η 0.01Hz11 / 22

Fully decentralized frequency control No optimalityu u(p(0), θ(0)) 6 u ?Active power output of all generators (no noise)100500-50150Power (MW)Power (MW)150020406080Time (s)10050 Unstable behavior0-50 Steady state0204060Time (s)Active power output of all generators (noise)800 Ti ṗi ωi ηi 6 0 Tj ṗj ωj ηj12 / 22

Ti ṗi ωi Ki piui piFrequency (Hz)Leaky integral control6059.959.859.70 No communicationrequiredi 1KiP iPi?ε6080for the othersPiDi Pi?PiKi 1 Banded frequencyregulationX40Leaky integral control Ti 0.05s, Ki 0.005 for G3, G5, G6, G9, G10, Ki 0.01X Di ωsync εiFrequency (Hz)i20Time (s) Synchronousfrequencyωsync Pno noisenoisy59.66059.959.859.7no noisenoisy59.6020406080Time (s)Decentralized pure integral control13 / 22

150Ti ṗi ωi Ki piui piPower (MW)Leaky integral control100500-50020406080Time (s) Steady-state Power sharingKi ui Kj uj Approx steady-state optimalityminimizeu Rnsubject toP2Pi Ki ui Pi Pi i (1 Di Ki )ui 0Leaky integral control Ti 0.05s, Ki 0.005 for G3, G5, G6,G9, G10, Ki 0.01 for the others150Power (MW)ui? Ki 1 ωsync100500-50020406080Time (s)Decentralized pure integral control14 / 22

150Ti ṗi ωi ηi Ki piui pi Noisy measurements ωi ηi Non-zero mean noise ηPower (MW)Leaky integral control100500-50020406080Time (s) Noise bound η 0.01HzLeaky integral control Ti 0.05s, Ki 0.005 for G3, G5, G6,G9, G10, Ki 0.01 for the othersRobust frequency regulation (ISS)kx(t)k2 λe α̂t kx(0)k2 γ( sup kη(t)k)2t R 0where λ, α̂, γ are positive constants andx col(δ δ ? , ω ω ? , p p ? )measures the deviation from the synchronous solution15 / 22

Ki k for G3 G5 G6 G9 G10Ki 2k for othersTi τ 0.05s0.20.10Frequency RMSE (Hz) Convergence time (s)Tuning of the gains KiFrequency error (Hz)Impact of control 60.0080.010.0040.0060.0080.015040As k % Noise-free steady-state frequencyerror % α̂ % implies convergence time & γ & implies RMSE &3020107 10-3654300.002Inverse DC gain kIncreasing gains Ki leads to reduced accuracy in frequency regulation faster response increased robustness to noise16 / 22

Impact of control parametersKi 0.005 for G3 G5 G6 G9 G10Ki 0.01 for the othersTi τAs τ % α̂ & implies convergence time % γ & implies RMSE &Frequency RMSE (Hz) Convergence time (s)Tuning of the time constants Ti3020100100.020.040.060.080.10.040.060.080.1 10-386400.02Time constant τ (s)Increasing time constants Ti leads to slower response increased robustness to noise17 / 22

0.20.100Frequency RMSE (Hz) Convergence time (s)Frequency error (Hz)Tuning recommendations0.0020.0040.0060.0080.01 Fix ratios between Ki 1 fromgenerating units 0.0040.0060.0080.01 10-3654300.002Frequency RMSE (Hz) Convergence time (s)Inverse DC gain kii302010010ui? Ki 1 ωsyncP Fix i Ki 1 for bandedfrequency regulationP ? XXP 1Ki i i Diε0.020.040.060.080.10.040.060.080.1 10-38 Fix Ti to strike a trade-offbetween frequency rejectionrate and noise rejection6400.02Time constant τ (s)Ti % α̂ & and γ &18 / 22

A further comparison6059.959.859.7no noisenoisy59.60204060Decentralized leaky6059.959.859.7no noisenoisy59.680020Time (s)02040600020P6020406080j Nic (pj pi ) Ki 1 ωi8060806080002040Time (s)150100500-506050-5080Power (MW)Power (MW)Power (MW)0Time (s)DAI Ti ṗi 401505040100Time (s)10002015050-5080150-50no noisenoisy59.6Time (s)100Time (s)Powerwith noise59.70Power (MW)Power (MW)Power (MW)50059.880150100-50606059.9Time (s)150Powerno noise40DAIFrequency (Hz)Frequencyinte-Frequency (Hz)Frequency (Hz)Decentralizedgral0204060Time (s)ui Ki 1 pi80100500-5002040Time (s)19 / 22

Robust stabilityProof is Lyapunov-based, using a strict Lyapunov functionW U(δ) U(δ ) U(δ ) (δ δ )11 (ω ω ) M(ω ω ) (p p ) T (p p )22 ( U(δ) U(δ )) M(ω ω ). U(δ) 12Pi6 jBij Vi Vj cos(δi δj ) potential energy W with 0 is the “shifted” energy function H 21 p Tp For sufficiently small , W is strictly decreasing along thesolutions This allows for quantification of robustness to noise20 / 22

ConclusionsA fully decentralized stabilizing integral control for achieving robustnoise-rejection, satisfactory steady-state regulation, desirable transientperformance These objectives are not aligned and trade-offs must be foundTuning guidelines are providedResulting time constants Ti /Ki compatible with actuator response timeLow-pass filter compares favourably wrt droop (noise rejection)Future work Lead compensators could improve transient performance Extension more accurate physical models Impact of topology on the diffusion of noise and scalabilityReferenceWeitenberg, Jiang, Zhao, Mallada, De Persis, Dörfler (2018). Robust decentralized secondary frequency control inpower systems: merits and trade-offs. IEEE Transactions on Automatic Control, in press, available asarXiv:1711.07332 Weitenberg, De Persis, Monshizadeh (2018). Exponential convergence under distributed averaging integralfrequency control. Automatica, 98, 103-113.?Weitenberg, De Persis (2018). Robustness to noise of distributed averaging integral controllers. Systems &Control Letters, 119, 1-7.21 / 22

Thank you!22 / 22

Robust decentralized control in power systems Claudio De Persis Institute of Engineering and Technology J.C. Willems Center for Systems and Control . Power system control maintain system security at minimal cost Basic security requirement keeping frequency around nominal value 1/22.

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