Project Samples I. Outline. Power Grids as Prototypical Complex Networks. Project Samples II. Cyber-physical security

Transcription

1 Dynamics and Control in Power Grids and Complex Oscillator Networks Electric Energy & Power Networks Florian Dörfler Center for Nonlinear Studies Los Alamos National Laboratories Department of Energy Center for Control, Dynamical Systems & Computation University of California at Santa Barbara Electric energy is critical for our technological civilization Purpose of electric power grid: generate/transmit/distribute Op challenges: multiple scales, nonlinear, & complex / / Trends, Advances, & Tomorrow s Power Grid increasing renewables & deregulation The Envisioned Power Grid complex, cyber-physical, & smart growing demand & operation at capacity Rapid technological and scientific advances: increasing volatility & complexity, decreasing robustness margins smart grid keywords interdisciplinary: power, control, comm, optim, comp, physics,... industry, & society physics & dynamics multi-scale complex nonlinear control smart grid monitoring optimization distributed comp & comm operation & control re-instrumentation: PMUs & FACTS complex & cyber-physical systems research themes: understanding & taming complexity decentralized smart & cyber-physical cyber-coordination layer for smart grid / /

2 Project Samples I Introduction and motivation Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments Cyber-physical security Coarse-graining of networks (with F. Pasqualetti & F. Bullo) (with D. Romeres, I. Dobson, & F. Bullo)!"#$%&'''%()(*%(+,-.,*%/-*%)-%*%: δi / rad / Fig.. The New England test system [], []. The system includes synchronous generators and buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of generators are studied in the case that a line-to-ground fault occurs at point F near bus. - F Conclusions δi / rad!"#$%&' - TIME / s / Fig.. Coupled swing of phase angle δi in New England test system. The fault duration is cycles of a -Hz sine wave. The result is obtained by numerical integration of eqs. (). test system can be represented by δ i = ωi,! Hi ω i = Di ωi + Pmi Gii Ei Ei E j πfs Project Samples II j=,j!=i {Gij cos(δi δj ) + Bij sin(δi δj )}, single wide-are comm link local decentralized control F " [rad]!'&!'&!!!!'&!!'&!"!"'&! " # $ % &!"'& [s]! " # $ % & transportation networks B. Numerical Experiment Coupled swing dynamics of generators in the test system are simulated. Ei and the initial condition (δi (), ωi () = ) for generator i are fixed through power flow calculation. Hi is fixed at the original values in []. Pmi and constant power loads are assumed to be % at their ratings []. The damping Di is. s for all generators. Gii, Gij, and Bij are also based on the original line data in [] and the power flow calculation. It is assumed that the test system is in a steady operating condition at t = s, that a line-to-ground fault occurs at point F near bus at t = s /( Hz), and that line trips at t = s. The fault duration is cycles of a -Hz sine wave. The fault is simulated by adding a small impedance ( j) between bus and ground. Fig. shows coupled swings of rotor angle δi in the test system. The figure indicates that all rotor angles start to grow coherently at about s. The coherent growing is global instability. A. Internal Resonance as Another Mechanism Inspired by [], we here describe the global instability with dynamical systems theory close to internal resonance [], []. Consider collective dynamics in the system (). For the system () with small parameters pm and b, the set {(δ, ω) S R ω = } of states in the phase plane is called resonant surface [], and its neighborhood resonant band. The phase plane is decomposed into the two parts: resonant band and high-energy zone outside of it. Here the initial conditions of local and mode disturbances in Sec. II indeed exist inside the resonant band. The collective motion before the onset of coherent growing is trapped near the resonant band. On the other hand, after the coherent growing, it escapes from the resonant band as shown in Figs. (b), (b),, and (b) and (c). The trapped motion is almost integrable and is regarded as a captured state in resonance []. At a moment, the integrable motion may be interrupted by small kicks that happen during the resonant band. That is, the so-called release from resonance [] happens, and the collective motion crosses the homoclinic orbit in Figs. (b), (b),, and (b) and (c), and hence it goes away from the resonant band. It is therefore said that global instability social networks & epidemics!" [s] IV. T OWARDS A C ONTROL FOR G LOBAL S WING I NSTABILITY Global instability is related to the undesirable phenomenon that should be avoided by control. We introduce a key mechanism for the control problem and discuss control strategies for preventing or avoiding the instability. biochemical reaction networks are provided to discuss whether the instability in Fig. occurs in the corresponding real power system. First, the classical model with constant voltage behind impedance is used for first swing criterion of transient stability []. This is because second and multi swings may be affected by voltage fluctuations, damping effects, controllers such as AVR, PSS, and governor. Second, the fault durations, which we fixed at cycles, are normally less than cycles. Last, the load condition used above is different from the original one in []. We cannot hence argue that global instability occurs in the real system. Analysis, however, does show a possibility of global instability in real power systems. Similar challenges & tools in wide-area control " [rad] where i =,...,. δi is the rotor angle of generator i with respect to bus, and ωi the rotor speed deviation of generator i relative to system angular frequency (πfs = π Hz). δ is constant for the above assumption. The parameters fs, Hi, Pmi, Di, Ei, Gii, Gij, and Bij are in per unit system except for Hi and Di in second, and for fs in Helz. The mechanical input power Pmi to generator i and the magnitude Ei of internal voltage in generator i are assumed to be constant for transient stability studies [], []. Hi is the inertia constant of generator i, Di its damping coefficient, and they are constant. Gii is the internal conductance, and Gij + jbij the transfer impedance between generators i and j; They are the parameters which change with network topology changes. Note that electrical loads in the test system are modeled as passive impedance []. Distributed wide-area control (with M. Jovanovic, M. Chertkov, & F. Bullo) () Power Grids as Prototypical Complex Networks rotor speeds.% of centralized H control performance robotic coordination & sensor ntkws Inverters in microgrids (with J. Simpson-Porco, J.M. Guerrero, & F. Bullo)... C. Remarks It was confirmed that the system () in the New England test system shows global instability. A few comments (')$ Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June, at : from IEEE Xplore. Restrictions apply. Plenty of synergies and cross-fertilization / /

3 Mathematical Model of Power Transmission Network Introduction and motivation Synchronization in power networks & coupled oscillators V i e i i Y ij V j e i j P m,i V i e i i Y ij Y ik P l,i V i e i i Y ij D i Synchronization analysis & conditions Applications & experiments Conclusions active power flow on line i j: V i V j Y ij sin ( ) θ i θ j }{{} a ij =max power transfer power balance at node i: P }{{} i = a ij sin(θ i θ j ) j power injection (DAE) power network dynamics [A. Bergen & D. Hill ]: : swing eq with P i > M i θ i + D i θ i = P i j a ij sin(θ i θ j ) : P i < and D i D i θ i = P i j a ij sin(θ i θ j ) / / Models of DC Sources with Inverters & Load Models DC source with droop-controlled DC/AC power converter [M.C. Chandorkar et. al. ]: D (droop) i θ i = P (setpoint) i j a ij sin(θ i θ j ) Synchronization in Power Networks Sync is crucial for the functionality and operation of the AC power grid. Generators have to swing in sync despite fluctuations/faults/contingencies. Def: θ i = θ j & θ i θ j bounded branches {i, j} constant current and admittance loads in Kron-reduced network [F. Dörfler et al. ]: M i θ i +D i θ i = P (red) i j a(red) ij sin(θ i θ j ) Y ik Y i,shunt V i e iθi Y ij I i = sync d frequencies & constrained active power flows Given: network parameters & topology and load & generation profile Q: an optimal, stable, and robust synchronous operating point? Security analysis [Araposthatis et al., Wu et al. &, Ilić,... ] constant motor loads [P. Kundur ]: Load flow feasibility [Chiang et al., Dobson, Lesieutre et al.,... ] Optimal generation dispatch [Lavaei et al., Bose et al.,... ] M i θi + D i θi = P (load) i j a ij sin(θ i θ j ) P (load) i V i e iθi Y ij Transient stability [Sastry et al., Bergen et al., Hill et al.,... ] Inverters in microgrids [Chandorkar et. al., Guerrero et al., Zhong,... ] / /

4 Synchronization in Complex Oscillator Networks Pendulum clocks & an odd kind of sympathy [C. Huygens, Horologium Oscillatorium, ] Today s canonical coupled oscillator model [A. Winfree, Y. Kuramoto ] Coupled oscillator model: θ i = ω i n a ij sin(θ i θ j ) j= n oscillators with phase θ i S non-identical natural frequencies ω i R elastic coupling with strength a ij = a ji undirected & connected graph G(V, E, A)!!! a a a Synchronization in Complex Oscillator Networks applications Coupled oscillator model: θ i = ω i n j= a ij sin(θ i θ j ) A few related applications: Sync in Josephson junctions [S. Watanabe et. al, K. Wiesenfeld et al. ] Sync in a population of fireflies [G.B. Ermentrout, Y. Zhou et al. ] Canonical model of coupled limit cycle oscillators [F.C. Hoppensteadt et al., E. Brown et al. ] Countless sync phenomena in sciences/bio/tech. [S. Strogatz, J. Acebrón et al., F. Dörfler et al. ] / / Synchronization in Complex Oscillator Networks phenomenology and challenges Synchronization is a trade-off: coupling vs. heterogeneity i (t) coupling small & ω i ω j large incoherence θ i = ω i n a ij sin(θ i θ j ) j= i (t) coupling large & ω i ω j small frequency sync Introduction and motivation Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments A central question: quantify coupling vs. heterogeneity [S. Strogatz, A. Arenas et al., S. Boccaletti et al. ] / Conclusions /

5 main result () Power network model: Family of dynamical system Hλ : d θ = ( λ) () + λ (), d t θ X Mi θ i + Di θ i = Pi aij sin(θi θj ) j X Di θ i = Pi aij sin(θi θj ) j (.) Variation of coupled oscillator model: θ i = Pi X j Theorem: Properties of the Hλ family aij sin(θi θj ) (.) Add decoupled frequency dynamics: θ i = θ i λ [, ] [F. Do rfler & F. Bullo ] Invariance of equilibria: For all λ [, ] the equilibria are P θ, θ : θ i =, Pi = j aij sin(θi θj ). Invariance of local stability: For all equilibria and λ [, ], the Jacobian has constant number of stable/unstable/zero eigenvalues. Homotopy: construct continuous interpolation between () and () / / topological equivalence interpretation near the equilibrium manifolds F. Do rfler and F. Bullo Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments. θ (t) θ (t) [rad/s] θ (t) θ (t) [rad/s] Introduction and motivation () synchronizes () synchronizes.... θ (t) θ(t) [rad]... θ(t) θ(t) [rad] Fig... Phase space plot of a network of n = second-order Kuramoto oscillators (.) with n = m (left plot) and the corresponding first-order scaled Kuramoto oscillators (.) together with the scaled frequency dynamics (.) (right plot). The natural frequencies ωi, damping terms Di, and coupling strength K are such that ωsync = and K/Kcritical =.. From the same initial configuration θ() (denoted by ) both first and second-order oscillators converge exponentially to the same nearby phase-locked equilibria (denoted by ) as predicted by Theorems. and.. main message: w.l.o.g. focus on coupled oscillator model / Conclusions /

6 Synchronization in a Complete & Homogeneous Graph Classic Kuramoto model: [Y. Kuramoto ] θ i = ω i K n n j= sin(θ i θ j ) Theorem: Explicit sync condition [F. Dörfler & F. Bullo ] The following statements are equivalent: Coupling dominates heterogeneity, i.e., K > K critical ω max ω min. Synchronization in a Complete & Homogeneous Graph main proof ideas Arc invariance: θ(t) in γ arc arc-length V (θ(t)) is non-increasing V ( (t)) { V (θ(t)) = maxi,j {,...,n} θ i (t) θ j (t) D + V (θ(t)) true if K sin(γ) K critical Kuramoto models with {ω,..., ω n } [ω min, ω max ] synchronize. Binary synchronization condition: K > K critical Strictly improves existing cond s [F. de Smet et al., N. Chopra et al., G. Schmidt et al., A. Jadbabaie et al., S.J. Chung et al., J.L. van Hemmen et al., A. Franci et al., S.Y. Ha et al., G.B. Ermentrout, A. Acebron et al. ] / Bounds on transient dynamics: K critical /K = sin(γ min ) = sin(γ max ) region of attraction includes angles θ(t = ) in γ max arc, & asymptotic cohesiveness of angles θ(t ) in γ min arc / Synchronization in a Complete & Homogeneous Graph main proof ideas Arc invariance: θ(t) in γ arc arc-length V (θ(t)) is non-increasing V ( (t)) { V (θ(t)) = maxi,j {,...,n} θ i (t) θ j (t) D + V (θ(t)) true if K sin(γ) K critical Frequency synchronization linear time-varying system (consensus) d dt θ i = n j= a ij(t) ( θ i θ j ), where a ij (t) = K n cos(θ i(t) θ j (t)) becomes positive in finite time Introduction and motivation Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments Conclusions / /

7 Primer on Algebraic Graph Theory Laplacian matrix L = degree matrix adjacency matrix L = L T = n a i j= a ij a in Notions of connectivity spectral: nd smallest eigenvalue of L is algebraic connectivity λ (L) topological: degree n j= a ij or degree distribution Notions of heterogeneity ω E, = max {i,j} E ω i ω j, ω E, = ( {i,j} E ω i ω j ) / Synchronization in Sparse Graphs a brief overview θ i = ω i n j= a ij sin(θ i θ j ) necessary sync condition: [C. Tavora and O.J.M. Smith ] n j= a ij ω i sync sufficient sync condition: λ (L) > ω E, sync [F. Dörfler and F. Bullo ] similar conditions with diff. metrics on coupling & heterogeneity Problem: sharpest general conditions are conservative / / A Nearly Exact Synchronization Condition main result Theorem: Sharp sync condition [F. Dörfler, M. Chertkov, & F. Bullo ] Under one of following assumptions: If ) extremal topologies: trees, homogeneous graphs, or {, } rings ) extremal parameters: L ω is bipolar, small, or symmetric (for rings) ) arbitrary one-connected combinations of ) and ) L ω E, < a unique & locally exponentially stable synchronous solution θ T n satisfying θ i θ j arcsin( L ω E, ) for all {i, j} E... and result is statistically correct. A Nearly Exact Synchronization Condition statistical accuracy for power networks Randomized power network test cases with % randomized loads and % randomized generation Randomized test case Numerical worst-case Analytic prediction of Accuracy of condition: ( instances) angle differences: angle differences: arcsin( L ω E, ) max θi θj arcsin( L ω E, ) max θi θj {i,j} E {i,j} E bus system. rad. rad. rad IEEE bus system. rad. rad. rad IEEE RTS. rad. rad. rad IEEE bus system. rad. rad. rad New England. rad. rad. rad IEEE bus system. rad. rad. rad IEEE RTS. rad. rad. rad IEEE bus system. rad. rad. rad IEEE bus system. rad. rad. rad Polish bus system. rad. rad. rad (winter peak /) similar results have been reproduced by / /

8 A Nearly Exact Synchronization Condition comments Monte Carlo studies: for range of random topologies & parameters with high prob & accuracy: sync for almost all G(V, E, A) & ω Possibly thin sets of degenerate counter-examples for large rings Intuition: the condition L ω E, < [ ] λ... (L) eigenvectors of L λ n (L) is equivalent to [ ] T eigenvectors of L ω includes previous conditions on λ (L) and degree ( λ n (L)) E, < Introduction and motivation Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments Conclusions / / Power Flow Approximation AC power flow: P i = n j= a ij sin(θ i θ j ) DC power flow: P i = n j= a ij (δ i δ j ) Conventional DC approximation: θ i θ j δ i δ j Our modified DC approximation: θ i θ j arcsin(δ i δ j ) Power Flow Approximation Security-Constrained Power Flow AC power flow with security constraints P i = n j= a ij sin(θ i θ j ), θ i θ j < γ ij {i, j} E DC power flow with security constraints P i = n j= a ij(δ i δ j ), δ i δ j < γ ij {i, j} E DC approximation ed C modified DC approximation edc..... x Error histograms for samples of randomized IEEE system apps: convexify OPF, planning, contingency screening, etc. / Novel test P i = n ij(δ i δ j ), j= δ i δ j < sin(γ ij ) {i, j} E Proof of equivalence for a tree: θi θj = arcsin(δi δj ) /

9 Contingency Analysis Contingency Analysis two contingencies {, } {, } #: increase generation & increase loads IEEE Reliability Test System at nominal operating point #: generator is tripped / / Distributed Averaging PI Droop Control in Microgrids predicting transition to instability design based on coupled oscillator insights Distributed & Averaging PI droop-controller (DAPI) ] (t) [rad s ] Microgrid modeled as network of loads and inverters (t) [rad s i (t) j (t) [rad] Contingency Analysis t t t t [s] t > t (t) [rad] (t) [rad] Continuously increase loads: condition arcsin( L ω E, ) < γ predicts that thermal limit γ of Decentralized primary control sync: θ i (t) ωsync line {, } is violated at. % of additional loading line {, } is tripped at.% of additional loading / Distributed secondary control frequency regulation: ωsync /

10 Distributed Averaging PI Droop Control in Microgrids theoretic guarantees Distributed Averaging PI Droop Control in Microgrids Practical implementation at Aalborg University, Denmark Theorem (Properties DAPI control) [J. Simpson-Porco, F. Dörfler, & F. Bullo, ] unique & exponentially stable closed-loop sync manifold; frequency regulation & optimal power sharing; robustness to voltage variations, losses, & uncertainties; plug n play & arbitrary tuning. Distributed & Averaging PI droop-controller (DAPI) Implementation (together with Q. Shafiee & J.M. Guerrero) DC Power Supply V DC Power Supply V Inverter PC-Simulink RTW&dSPACE Control Desk Inverter i L i L LCLFilter v v LCLFilter i o i o Load / Experimental results are remarkable: off-the-shelf, robust, small transients / Distributed Averaging PI Droop Control in Microgrids Practical implementation at Aalborg University, Denmark Implementation (together with Q. Shafiee & J.M. Guerrero) DC Power Supply V DC Power Supply V Inverter PC-Simulink RTW&dSPACE Control Desk Inverter i L i L LCLFilter v v LCLFilter i o i o Load Experimental results are remarkable: off-the-shelf, robust, small transients / Introduction and motivation Synchronization in power networks & coupled oscillators Synchronization analysis & conditions Applications & experiments Conclusions /

11 Summary Related Publications F. Do rfler and F. Bullo.Synchronization in Complex Oscillator Networks: A Survey. In Automatica, April, Note: submitted. Lessons learned today: power networks are coupled oscillators F. Do rfler, M. Chertkov, and F. Bullo. Synchronization in Complex Oscillator Networks and Smart Grids. In Proceedings of the National Academy of Sciences, February. sync if coupling > heterogeneity F. Do rfler, F. Pasqualetti and F. Bullo. Continuous-Time Distributed Observers with Discrete Communication. In IEEE Journal of Selected Topics in Signal Processing, March. necessary, sufficient, & sharp sync cond s J.W. Simpson-Porco, F. Do rfler, and F. Bullo. Synchronization and Power-Sharing for Droop-Controlled Inverters in Islanded Microgrids. In Automatica, Februay, Note: provisionally accepted. theory is useful, robust & applicable F. Do rfler and F. Bullo. Kron Reduction of Graphs with Applications to Electrical Networks. In IEEE Transactions on Circuits and Systems I., January. Further results & applications (not shown) F. Pasqualetti, F. Do rfler, and F. Bullo. Attack Detection and Identification in Cyber-Physical Systems. In IEEE Transactions on Automatic Control, December, Note: to appear. Related ongoing and future work: F. Do rfler and F. Bullo. Synchronization and Transient Stability in Power Networks and Non-uniform Kuramoto Oscillators. In SIAM Journal on Control and Optimization, June. more complete theory & more detailed models F. Do rfler and F. Bullo. On the Critical Coupling for Kuramoto Oscillators. In SIAM Journal on Applied Dynamical Systems, September. from analysis to control synthesis: cont. control design, hybrid remedial action schemes, computation & optimization Research supported by / Acknowledgements Francesco Bullo Michael Chertkov Frank Allgo wer M. Jovanovic Diego Romeres Sandro Zampieri Ian Dobson Josep Guerrero Bruce Francis Fabio Pasqualetti J. Simpson-Porco Hedi Bouattour Ullrich Mu nz Scott Backhaus Qobad Shafiee