Slow Coherency & Sparsity-Promoting Optimal Wide-Area Control

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1 Slow Coherency & Sparsity-Promoting Optimal Wide-Area Control Trends, Advances, & Tomorrow s Power Grid increasing renewables & deregulation growing demand & operation at capacity Florian Dörfler, Mihailo Jovanović, Diego Romers, Michael Chertkov, and Francesco Bullo increasing volatility & complexity, decreasing robustness margins Rapid technological and scientific advances: re-instrumentation: PMUs & FACTS complex & cyber-physical systems CNLS Smart Grid Seminar Series July, cyber-coordination layer for smart grid / Electro-Mechanical Oscillations in Power Networks Dramatic consequences: blackout of August,, resulted from instability of. Hz mode in Western interconnected system Taxonomy of Electro-Mechanical Oscillations Synchronous generator = electromech. oscillator local oscillations: Canada PacNW North 7 Montana = Single generators oscillate relative to rest of grid typically damped by local feedback via Power System Stabilizers PSSs). Hz NoCal Utah Power system = complex oscillator network inter-area oscillations: SoCal South Arizona = groups of generators oscillate relative to each or inter-area oscillations are only poorly controllable by local feedback Consequences of recent developments: increasing power transfers outpace capacity of transmission system ever more lightly damped electromechanical inter-area oscillations Source: / /

2 7 Outline Introduction and Motivation Modeling & Analysis: Slow Coherency in Power Networks Coordinated Control: Sparsity-Promoting Wide Area Control New England Power Grid: Coordinated Wide-Area Control Conclusions Power Network Swing Dynamics Coarse-grained power network dynamics = generator swing dynamics: M i θi + D i θi = P i n j= E ie j Y ij sinθ i θ j ) Swing equations linearized around an equilibrium θ, ): M & D R n n M θ + D θ + Lθ = diagonal inertia and damping matrices L R n n Laplacian matrix with coupling a ij = E i E j Y ij cosθi θj ) L = n a i j= a ij a in / sparsely coupled harmonic oscillators with heterogeneous frequencies / Causes for Inter-Area Dynamics in Power Networks Slow Coherency and Area Aggregation RTS power network swing dynamics of original model aggregated RTS model swing dynamics of aggregated model Inter-area oscillations are caused by heterogeneity: fast & slow responses inertia M i and damping D i ) topology: internally strongly and externally sparsely connected areas power transfers between areas: a ij = E i E j Y ij cosθi θj ) / Aggregate model of lower dimension & with less complexity for analysis and insights into inter-area dynamics [Chow and Kokotovic ] measurement-based id of equivalent models [Chakrabortty et.al. ] remedial action schemes [Xu et. al. ] & wide-area control [later today] 7 /

3 7 Setup in Slow Coherency Linear Transformation & Time-Scale Separation original model r given areas from spectral partition [Chow et al. & ]) small sparsity parameter: δ = max ασ external connections in area α) min α Σ internal connections in area α) inter-area dynamics by center of inertia: y α = i α M iθ i, α {,..., r} i α M i Swing equation = Singular perturbation standard form M θ + D θ + Lθ = = y d ẏ dt s δ z y A ẏ δ ż.... z..... ż Slow motion given by center of inertia: y α = i α M iθ i, α {,..., r} i α M i Fast motion given by intra-area differences: intra-area dynamics by area differences: z α i = θ i θ, i α \ {}, α {,..., r} aggregated model z α i = θ i θ, i α \ {}, α {,..., r} / Slow time scale: t s = δ t max internal area degree / Area Aggregation Singular Perturbation Approximation Singular perturbation standard form: d dt s y ẏ δ z =... y A ẏ δ ż.... z..... ż Singular perturbation standard form: d dt s y.... ẏ δ z =... y A ẏ δ ż.... z..... ż Aggregated swing equations obtained by δ : M a ϕ + D a ϕ + L red ϕ = Aggregated swing equations obtained by δ : M a ϕ + D a ϕ + L red ϕ = Properties of aggregated model [D. Romeres, F. Dörfler, & F. Bullo, ] M a = i α M i and D a = i α D i L red = inter-area Laplacian + intra-area contributions = positive semidefinite Laplacian with possibly negative weights / Singular perturbation approximation [D. Romeres, F. Dörfler, & F. Bullo, ] There exist δ sufficiently small such that for δ δ and for all t > : [ ] [ ] yts ) ϕts ) = + O [ ] [ ] zts ) ϕts ) δ), = Ã + O δ). ẏt s ) ϕt s ) żt s ) ϕt s ) Variations of result were known previously under restrictive assumptions on topology and damping [Chow & Kokotovic et al. 7 &, Biyik & Arcak 7]. /

4 7 RTS Swing Dynamics Revisited Outline Introduction and Motivation Modeling & Analysis: Slow Coherency in Power Networks Coordinated Control: Sparsity-Promoting Wide Area Control New England Power Grid: Coordinated Wide-Area Control Conclusions / / Remedies Against Electro-Mechanical Oscillations conventional control Blue layer: interconnected generators Remedies Against Electro-Mechanical Oscillations wide-area control Blue layer: interconnected generators Fully decentralized control implemented via PSS, AVR, or FACTS: effective against local oscillations ineffective against inter-area oscillations / Fully decentralized control Distributed wide-area control requires identification of sparse control architecture: actuators, measurements, & communication channels /

5 Challenges in Wide-Area Control Objectives: wide-area control should achieve optimal closed-loop performance low communication complexity Setup in Wide-Area Control remote control signals & remote measurements e.g., PMUs) excitation PSS & AVR) and power electronics FACTS) actuators communication backbone network Problem: objectives are conflicting design optimal) centralized control identify control architecture complete state info & measurements high communication complexity u wac t) remote control signals local control loops u loc t) wide-area controller wide-area measurements e.g. PMUs) + channel and measurement noise identify measurements & control architecture design control decentralized optimal) control is hard combinatorial criteria for control channels Today: simultaneously optimize closed-loop performance & identify sparse control architecture / + + PSS & AVR FACTS... u loc t) t) generator transmission line system noise power network dynamics / Optimal Wide-Area Damping Control Model: linearized ODE dynamics ẋt) = Axt) + B ηt) + B ut) Control: memoryless linear state feedback u = Kxt) Optimal centralized control with quadratic performance index: { } minimize JK) lim E xt) T Qxt) + ut) T Rut) t subject to linear dynamics: linear control: stability: ẋt) = Axt) + B ηt) + B ut), ut) = Kxt), A B K ) Hurwitz. no structural constraints on K) 7 / Sparsity-Promoting Optimal Wide-Area Damping Control Sparsity-promoting optimal control [Lin, Fardad, & Jovanović ]: simultaneously optimize control performance & control architecture minimize subject to linear dynamics: linear control: stability: { } lim E xt) T Qxt) + ut) T Rut) + γ ) t ẋt) = Axt) + B ηt) + B ut), ut) = Kxt), A B K ) Hurwitz. for γ = : standard optimal control typically not sparse) for γ > : sparsity is promoted problem is combinatorial) ) approximated by weighted l -norm i,j w ij K ij /

6 7 Parameterized Family of Feedback Gains Kγ) = arg min JK) + γ ) w ij K ij K i,j Slow Coherency Performance Objectives Sources for inter-area oscillations: linearized swing equation: M θ + D θ + Lθ = mechanical energy: θm θ + θt Lθ heterogeneities in topology, power transfers, & machine responses inertia & damp) Performance objectives = energy of homogeneous network: x T Q x = θ T M uniform }{{} I n θ + θt L } uniform {{} I n /n) n n θ / Or choices possible: center of inertia, inter-area differences, etc. / Algorithmic Approach to Sparsity-Promoting Control Outline Equivalent formulation via observability Gramian P: minimize J γ K) trace B T ) PB + γ i,j w ij K ij subject to A B K) T P + PA B K) = Q + K T RK) ; Introduction and Motivation Modeling & Analysis: Slow Coherency in Power Networks Warm-start at optimal centralized H controller with γ = Coordinated Control: Sparsity-Promoting Wide Area Control Homotopy path: continuously increase γ until desired value γ des ADMM: iterative solution for each value of γ [, γ des ] Update weights: update w ij in each ADMM step: w ij K ij +ε New England Power Grid: Coordinated Wide-Area Control Conclusions Polishing: structured optimization with desired sparsity pattern / /

7 Case Study: IEEE New England Power Grid "#$%&'''%)*%+,-.,*%/-*%)-%77*%: sub-transient generator models [Athay et. al. 7].... ors 7 7,, δ i = ωi, Hi ω i = Di ωi + Pmi Gii Ei Ei E j πf s ,,,, TIME / s,7 ) [%] % % % % % % % % %.% 7 J.%.% J ) / J [%].%.%.%.%.% % system. Fig.. Coupled swing of phase angle δi in New England test The fault duration is cycles of a -Hz sine The result is obtained % wave. by numerical integration of eqs. ). For γ = : local decentralized optimal control + θ t) K / Sparse & Nearly Optimal Wide-Area Control Architecture Control Architecture & Signal Exchange Network ors test system represented by,,,7 can be,,,7 7 % are providedto discuss wher instability in Fig.. % relative performance loss occurs in corresponding for real power γ = system. = First, 7 j=,j=i classical model with constant voltage behind impedance. is % non-zero elements in K {Gij cosδi δj ) + Bij sinδi δj )}, used for first swing criterion of transient stability []. This is / because second and multi swings may be affected by voltage where i =,...,. δi is rotor angle of generator i with to bus, and ωi rotor respect speed deviation of generator fluctuations, damping effects, controllers such as AVR, PSS, i relative to system angular frequency πfs = π Hz). and governor. Second, fault durations, which we fixed at for δ is constant above assumption. The parameters cycles, are normally less than cycles. Last, load fs, Hi, Pmi, Di, Ei, Gii, Gij, and Bij are in per unit condition used above is different from original one in system except for Hi and Di in second, and for fs in Helz. []. We cannot hence argue that global instability occurs in The mechanical input power Pmi to generator i and real system. Analysis, however, does show a possibility magnitude Eiofinternal voltage in generator i are assumed of global instability in real power systems. to be constant for transient stability studies [], []. Hi is "#$%&'''%)*%+,-.,*%/-*%)-%77*%: IV. T OWARDS A C ONTROL FOR G LOBAL S WING 7inertia constant of generator i, Di its damping coefficient, I NSTABILITY and y are constant. Gii is internal conductance, and Global instability is related to undesirable phenomenon Gij + jbij transfer impedance between generators i a key and j; They are parameters which change with network that should be avoided by control. We introduce mechanism for control problem and discuss control topology changes. Note that electrical7loads in test system 7 * strategies for preventing or avoiding instability. =. are, modeled as ) = passive impedance []. A. Internal Resonance as Anor Mechanism B. Numerical Experiment 7 Inspired by [], we here describe global instability Coupled swing dynamics of generators in test system are simulated. Ei and initial condition with dynamical systems ory close to internal resonance 7 δi ), ωi ) = ) for generator i are fixed through power [], []. Consider collective dynamics in system ). F For system ) with small parameters p and b, set flow calculation. H is fixed at original values in []. m i 7 * {δ, ω) S R ω = } of states in phase plane is constant power loads are assumed to be % at ir =.77 Pmi, and ) = its neighborhood resonant ratings []. The damping Di is. s for all generators. called resonant surface [], and Gii, Gij, and Bij are also based on original line data band. The phase plane is decomposed into two parts: zone outside of it. Here in [] and power flow calculation. It is assumed that resonant band and high-energy initial conditions of local and mode disturbances in Sec. II test system is in a steady operating condition at t = s, motion that a line-to-ground fault occurs at point F near bus at indeed exist inside resonant band. The collective t = s /Hz), and that line 7 trips at t = s. The before onset of coherent growing is trapped near 7 duration*)is =7 cycles of a -Hz sine wave. The fault resonant band. On or hand, after coherent growing, =.7 fault, 7 in Figs. b), is simulated by adding a small impedance 7 j) between it escapes from resonant band as shown 7 bus and ground. Fig. shows coupled swings of rotor b),, and b) and c). The trapped motion is almost angle δi in test system. The figure indicates that all rotor integrable and is regarded as a captured state in resonance may be interrupted angles start to grow coherently at about s. The coherent []. At a moment, integrable motion by small kicks that happen during resonant band. That is, growing is global instability. so-called release from resonance [] happens, and 7 C. Remarks collective motion crosses in Figs. b), test system [], []. The system includes =, * ) = 7 Fig.homoclinic. Theorbit New England It was confirmed that system ) in New Eng- b),, and b) and c), and hence it goes away from synchronous generators and buses. Most of buses have constant is refore said that global instability land test system shows global instability. A few comments resonant band. Itsingle wide-area control link = nearly centralized performance Fig.. The New England7test system [], []. The system includes synchronous generators and buses. Most of buses have constant active and reactive power loads. Coupled swing dynamics of generators are studied in case that amode line-to-ground fault occurs Mode at point Mode : : : F near bus. all ors Mode : - 7 Mode : dominant inter-area modes of New England grid with PSSs 7 F exciters & carefully tuned PSS data [Jabr et. al. ] ) / ) 7 7 ')$ / active and reactive power loads. Coupled swing dynamics of generators are studied in case that a line-to-ground fault occurs at point F near bus. Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on June, at : from IEEE Xplore. Restrictions apply. δi / rad - δi / rad ) ) [%] [%] ) ) Model features: Q = energy of homogeneous"#$%&' network, R = In, γ, δi / rad. Performance vs Sparsity J J J J [%] [%] J J δi / rad - / Fig.. Co The fault dura

8 Closed-Loop Performance for γ = modes #,,... are strongly damped and mode # is distorted local PSS control Mode all ors local PSS control & wide-area control Mode. 7. ors Robustness Achieved by Sparsity-Promoting Control tuning of PSSs & operating cond gain margins time delays, multiple SCADA rates, & comm uncertainties phase margins & input uncertainties )" & &' multivariable phase margin " # " $ " % " " " "&% "& multivariable gain reduction margin γ θt) [Hz] θt) θit) [rad] " '& '& " " generator generator "'& "'& " # $ % & " # $ % & t [s] t [s] ' '% '# '# '% " # $ % & t [s] θt) [Hz] θt) θit) [rad] " '& '& ' '% '# '# '% " # $ % & t [s] / wide-area control u wac t) + K - system noise t) multiplicative uncertainty + m dynamics with local control local control loops... gain uncertainty g power network dynamics xt) "&' " # " $ " % " " " &' &'# &'% multivariable gain amplification margin " # " $ " % " " " Additionally: sparsity pattern is not sensitive to operating point 7 / γ γ Sparsity Identification & Control by Alternative Means identified WAC channel: θ t) needs to be communicated to AVR # Outline proportional feedback u t) = K θ t) θ t) ) applied to nonlinear DAE system without local optimal decentralized control local PSS control local PSS control & WAC Introduction and Motivation Modeling & Analysis: Slow Coherency in Power Networks θt) [Hz] "& "% "$ "$ "% "& "& "# $ $"# % %"# & &"# ' '"# # "# $ $"# % %"# & &"# ' '"# # θt) [Hz] "& "% "$ "$ "% generator t [s] generator t [s] Coordinated Control: Sparsity-Promoting Wide Area Control New England Power Grid: Coordinated Wide-Area Control power output [pu] $' $% $ ) ' % % generator t [s] generator t [s] % % "# $ $"# % %"# & &"# ' '"# # "# $ $"# % %"# & &"# ' '"# # power output [pu] $' $% $ ) ' / Conclusions /

9 Conclusions Summary: analysis of inter-area dynamics via slow coherency ory damping inter-area modes via distributed wide-area control sparsity-promoting optimal control with slow coherency objectives trade-off: sparse control architecture vs. performance References & People D. Romeres, F. Dörfler, and F. Bullo. Novel results on slow coherency in consensus and power networks. In European Control Conference, July. F. Dörfler, M. Jovanović, Michael Chertkov, and F. Bullo. Sparse and optimal wide-area damping control in power networks. In American Control Conference, June. F. Dörfler, M. Jovanović, Michael Chertkov, and F. Bullo. Sparsity-Promoting Optimal Wide-Area Control of Power Networks. Submitted to IEEE Transactions on Power Systems, July. Available at F. Lin, M. Fardad, and M. R. Jovanović. Design of optimal sparse feedback gains via alternating direction method of multipliers. To appear in IEEE Transactions on Automatic Control,. Available at New England Power grid example shows: excellent performance, low communication complexity, & favorable robustness margins Open directions: extension to structure-preserving DAE models oretic analysis of robustness degradation exploit rotational symmetry of models / Diego Romeres Mihailo Jovanović Michael Chertkov Francesco Bullo /