Sparsity-promoting wide-area control of power systems. F. Dörfler, Mihailo Jovanović, M. Chertkov, and F. Bullo American Control Conference

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1 Sparsity-promoting wide-area control of power systems F. Dörfler, Mihailo Jovanović, M. Chertkov, and F. Bullo 203 American Control Conference

2 Electro-mechanical oscillations in power systems Local oscillations single generators swing relative to the rest of the grid typically damped by Power System Stabilizers (PSSs) Inter-area oscillations groups of generators oscillate relative to each other associated with dynamics of power transfers

3 Blackout of Aug. 0, 996 Inter-area oscillations resulted from instability of the 0.25 Hz mode western interconnected system: California-Oregon power transfer: 2

4 Slow coherency theory WHERE ARE THE INTER-AREA MODES COMING FROM? slow coherency theory Chow, Kokotović, et al. 78, 82 RTS 96 power system: generator angles linearized swing equation: 3 time time

5 Conventional control Blue layer: generators with transmission lines 4 Fully decentralized controller effective against local oscillations ineffective against inter-area oscillations

6 Wide-area control Blue layer: generators with transmission lines wide-area controller 5 KEY CHALLENGE: identification of a signal exchange network performance vs sparsity

7 + raft u wac (t) remote control signals local control loops PSS & AVR u loc (t) generator wide-area controller wide-area measurements (e.g. PMUs) + channel and measurement noise u loc (t) transmission FACTS line power network dynamics (t) system noise

8 ❶ SPARSITY-PROMOTING WIDE-AREA CONTROL Safeguard against inter-area oscillations Performance vs sparsity ❷ CASE STUDY IEEE New England power grid model Outline 7 ❸ SUMMARY AND OUTLOOK

9 MODEL FEATURES raft detailed sub-transient generator models exciters carefully tuned PSS data F Case study: IEEE New England Power Grid

10 single wide-area comm link F Preview of a key result single long range interaction nearly centralized performance

11 linearized dynamics: raft ẋ = A x + B d + B 2 u objective function: J = lim t E ( x T (t) Q x(t) + u T (t) R u(t) ) memoryless controller: no structural constraints u = K x Optimal wide-area control 0 globally optimal controller: A T P + P A P B 2 R B T 2 P + Q = 0 K c = R B T 2 P

12 minimize J(K) + γ i, j variance amplification W ij K ij sparsity-promoting penalty function Sparsity-promoting optimal control γ > 0 performance vs sparsity tradeoff W ij 0 weights (for additional flexibility) Lin, Fardad, Jovanović, IEEE TAC 3 (in press; arxiv:.688)

13 K(γ) := arg min (J(K) + γ g(k)) K Parameterized family of feedback gains 2 ALGORITHM: alternating direction method of multipliers

14 Energy of power network without inter-area modes inspired by slow coherency theory ( J := lim E θ T (t) Q θ θ(t) + θ T (t) θ(t) ) + u T (t) u(t) t Q θ := ɛ I + (I N ) T Performance index 3 other choices possible

15 , 5, 6, 7, 9 Mode 3 raft Mode ,3 all others Mode Mode 2 90, 2, 3, 8, ,5 6,7 4, 5, 6, Mode 5 others 0.5,8 others Open-loop dynamics Dominant inter-area modes with local PSSs

16 percent raft (J J c ) /J c card (K) /card (K c ) γ γ Performance vs sparsity 5 γ = relative to K c {.6 % performance loss 5.5 % non-zero elements in K

17 γ = , card (K) = 90 Signal exchange network 6 γ =, card (K) = 37

18 (t) [Hz] 0(t) i(t) [rad] raft Mode all others local PSS control t [s] t [s] (t) [Hz] 0(t) i(t) [rad] Mode others 330 local PSS control & wide-area control generator 0 generator t [s] t [s] 7

19 + - raft wide-area control (t) K u wac (t) system noise multiplicative uncertainty m dynamics with local control local control loops... gain uncertainty power network dynamics x(t) 8 Robustness? + g

20 raft multivariable gain reduction margin γ multivariable gain amplification margin multivariable phase margin γ γ

21 SPARSITY-PROMOTING OPTIMAL CONTROL Performance vs sparsity tradeoff Software Lin, Fardad, Jovanović, IEEE TAC 3 (in press; arxiv:.688) WIDE-AREA CONTROL OF POWER NETWORKS Remedy against inter-area oscillations mihailo/software/lqrsp/ Summary and outlook 20 IEEE New England power grid model OPEN QUESTIONS Extension to structure-preserving descriptor models Theoretic analysis of robustness degradation Exploit the rotational symmetry of the models