Online Feedback Optimization with Applications to Power Systems Florian Dörfler ETH Zürich European Control Conference 2020
Acknowledgements Lukas Adrian Hauswirth Saverio Bolognani Ortmann ć Irina Subotić Gabriela Hug Miguel Picallo Verena Häberle 1 / 31 • • • • • • • • • • “Vuk Karadžić diploma“ “Life Activities Advancement Center” • • for patients’ electronic cards • • • • • The “Dositeja” reward for studying abroad in the year of • • The “Dositeja” reward for the extraordinary success in the year of 2009 and 2010 • •
feedforward feedback vs. optimization control w w d estimate y y r + System Controller Optimization System u u − complex specifications & decision simple feedback policies optimal, constrained, & multivariable suboptimal, unconstrained, & SISO strong requirements forgiving nature of feedback measurement driven, robust to precise model, full state, disturbance estimate, & computationally intensive uncertainty, fast & agile response → typically complementary methods are combined via time-scale separation y + r Optimization System Controller u − � � offline & feedforward real-time & feedback � 2 / 31
Example: power system balancing offline optimization : dispatch based 50 Hz 51 49 on forecasts of loads & renewables generation 200 marginal costs in €/MWh load Renewables 150 Nuclear energy Lignite 100 Hard coal Natural gas 50 Fuel oil control 0 [Milano, 2018] 0 10 20 30 40 50 60 70 80 90 100 Capacity in GW Re-scheduling costs online control based on frequency %!*! Germany [mio. €] %!() y 50 Hz + Frequency Power Control System u − %%(# %%"& frequency measurement re-schedule set-point to mitigate severe !&' !&( !!" #$% forecasting errors (redispatch, reserve, etc.) more uncertainty & fluctuations → infeasible !"## !"#! !"#$ !"#% !"#& !"#' !"#( !"#) & inefficient to separate optimization & control [Bundesnetzagentur, Monitoringbericht 2011-2019] 3 / 31
Synopsis & proposal for control architecture power grid : separate decision layers hit limits under increasing uncertainty similar observations in other large-scale & uncertain control systems : process control systems & queuing/routing/infrastructure networks proposal: open and online optimization algorithm as feedback control � �� � � �� � � �� � with inputs & outputs iterative & non-batch real-time interconnected operational disturbance w constraints u optimization dynamical actuation algorithm system u ∈ U e.g., x = f ( x , u , w ) ˙ y y = h ( x , u , w ) u = −∇ φ ( y , u ) ˙ measurement 4 / 31
Historical roots & conceptually related work process control : reducing the effect of uncertainty in sucessive optimization Optimizing Control [Garcia & Morari, 1981/84], Self-Optimizing Control [Skogestad, 2000], Modifier Adaptation [Marchetti et. al, 2009], Real-Time Optimization [Bonvin, ed., 2017] , ... extremum-seeking : derivative-free but hard for high dimensions & constraints [Leblanc, 1922], ...[Wittenmark & Urquhart, 1995], ...[Krstić & Wang, 2000], ..., [Feiling et al., 2018] MPC with anytime guarantees (though for dynamic optimization): real-time MPC [Zeilinger et al. 2009] , real-time iteration [Diel et al. 2005] , [Feller & Ebenbauer 2017] , etc. optimal routing, queuing, & congestion control in communication networks : e.g., TCP/IP [Kelly et al., 1998/2001], [Low, Paganini, & Doyle 2002], [Srikant 2012], [Low 2017], ... optimization algorithms as dynamic systems : much early work [Arrow et al., 1958], [Brockett, 1991], [Bloch et al., 1992], [Helmke & Moore, 1994], ... & recent revival [Holding & Lestas, 2014], [Cherukuri et al., 2017], [Lessard et al., 2016], [Wilson et al., 2016], [Wibisono et al, 2016], ... recent system theory approaches inspired by output regulation [Lawrence et al. 2018] & robust control methods [Nelson et al. 2017], [Colombino et al. 2018] 5 / 31
Theory literature inspired by power systems lots of recent theory development stimulated by power systems problems [Simpson-Porco et al., 2013], [Bolognani A Survey of Distributed Optimization and Control et al, 2015], [Dall’Anese & Simmonetto, Algorithms for Electric Power Systems 2016], [Hauswirth et al., 2016], [Gan & Daniel K. Molzahn, ∗ Member, IEEE , Florian D¨ orfler, † Member, IEEE , Henrik Sandberg, ‡ Member, IEEE , Steven H. Low, § Fellow, IEEE , Sambuddha Chakrabarti, ¶ Student Member, IEEE , Ross Baldick, ¶ Fellow, IEEE , and Javad Lavaei, ∗∗ Member, IEEE Low, 2016], [Tang & Low, 2017], ... early adoption : KKT control [Jokic et al, 2009] Steven Low Enrique Mallada Na Li Krishnamurthy Dvijotham literature kick-started ∼ 2013 by groups from Changhong Zhao Florian Dörfler Caltech, UCSB, UMN, Padova, KTH, & Groningen John Simpson-Porco Sandro Zampieri Yue Chen Jorge Cortez Claudio De Persis Andrey Bernstein changing focus : distributed & simple Saverio Bolognani Henrik Sandberg Emiliano Dall’Anese Andre Jokic Nima Monshizadeh → centralized & complex models/methods Andrea Simonetto Sergio Grammatico Arjan Van der Schaft Marcello Colombino Karl Johansson Sairaj Dhople implemented in microgrids (NREL, DTU, EPFL, ...) Ioannis Lestas & conceptually also in transactive control pilots (PNNL) 6 / 31
Overview algorithms & closed-loop stability analysis projected gradient flows on manifolds robust implementation aspects power system case studies throughout 7 / 31
ALGORITHMS & CLOSED-LOOP STABILITY ANALYSIS
Stylized optimization problem & algorithm simple optimization problem minimize φ ( y, u ) y,u subject to y = h ( u ) u ∈ U cont.-time projected gradient flow projected dynamical system � �� � u = Π g x ∈ Π g ˙ −∇ φ h ( u ) , u X [ f ]( x ) � arg min ˙ � v − f ( x ) � g ( x ) U v ∈ T x X � �� � ∂h � = Π g � − ∂u I ∇ φ ( y, u ) U � y = h ( u ) ◮ domain X ◮ vector field f Fact: a regular † solution u :[0 , ∞ ] →X ◮ metric g converges to critical points if φ has Lip- ◮ tangent cone T X schitz gradient & compact sublevel sets. all sufficiently regular † † regularity conditions made precise later 8 / 31
Algorithm in closed-loop with LTI dynamics optimization problem LTI dynamics x = Ax + Bu + Ew ˙ minimize φ ( y, u ) y,u y = Cx + Du + Fw subject to y = H io u + R io w const. disturbance w & steady-state maps u ∈ U x = − A − 1 B u − A − 1 E w → open & scaled projected gradient flow � �� � � �� � � � H is R ds � � H T u = Π U ˙ − ǫ io I ∇ φ ( y , u ) � � � � D − CA − 1 B F − CA − 1 E y = u + w � �� � � �� � H io R do + u x ǫ � � B U + + + w E A ∇ u φ D F − y + + − + + H T io ∇ y φ C 9 / 31
Stability, feasibility, & asymptotic optimality Theorem: Assume that regularity of cost function φ : compact sublevel sets & ℓ -Lipschitz gradient LTI system asymptotically stable : ∃ τ > 0 , ∃ P ≻ 0 : PA + A T P � − 2 τP sufficient time-scale separation (small gain): 0 < ǫ < ǫ ⋆ � 2 τ 1 cond ( P ) · ℓ � H io � Then the closed-loop system is stable and globally converges to the critical points of the optimization problem while remaining feasible at all times. Proof: LaSalle/Lyapunov analysis via singular perturbation [Saberi & Khalil ’84] e T P e � � Ψ δ ( u, e ) = δ · + (1 − δ ) · φ h ( u ) , u � �� � � �� � LTI Lyapunov function objective function with parameter δ ∈ (0 , 1) & steady-state error coordinate e = x − H is u − R ds w Ψ δ ( u, e ) is non-increasing if ǫ ≤ ǫ ⋆ and for optimal choice of δ → derivative ˙ 10 / 31
Example: optimal frequency control ◮ linearized swing dynamics dynamic LTI power system model power balancing objective ◮ 1st-order turbine-governor control generation set-points ◮ primary frequency droop unmeasured load disturbances ◮ DC power flow approximation measurements : frequency + constraint variables (injections & flows) optimization problem + 1 2 � max { 0 , y − y }� 2 Ξ + 1 2 � max { 0 , y − y }� 2 → objective : φ ( y, u ) = cost ( u ) Ξ � �� � � �� � economic generation operational limits (line flows, frequency, ...) → constraints : actuation u ∈ U & steady-state map y = H io u + R do w → control ˙ u = Π U ( . . . ∇ φ ) ≡ super-charged Automatic Generation Control 11 / 31
Test case: contingencies in IEEE 118 system events: generator outage at 100 s & double line tripping at 200 s Power Generation (Gen 37) [p.u.] 6 Setpoint Output 4 2 0 0 50 100 150 200 250 300 Time [s] 12 / 31
How conservative is ǫ < ǫ ⋆ ? still stable for ǫ = 2 ǫ ⋆ unstable for ǫ = 10 ǫ ⋆ · 10 − 2 Frequency Deviation from f 0 [Hz] Frequency Deviation from f 0 [Hz] 4 System Frequency System Frequency 5 2 0 0 − 2 − 5 Line Power Flow Magnitudes [p.u.] Line Power Flow Magnitudes [p.u.] 3 23 → 26 90 → 26 flow limit other lines 4 23 → 26 90 → 26 flow limit other lines 2 2 1 0 0 0 5 10 15 20 0 5 10 15 20 Time [s] Time [s] Note: conservativeness problem dependent & depends, e.g., on penalty scalings 13 / 31
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