Real-Time Feedback Optimization of Power Systems Florian Dörfler Automatic Control Laboratory, ETH Zürich 1
Acknowledgements Adrian Hauswirth Saverio Bolognani Gabriela Hug 2
feedforward optimization vs. feedback control d estimate d d y y r + Optimization System System Controller u u − complex optimal decision robust to model uncertainty operational constraints fast response MIMO (multi-input/output) measurement driven highly model-based suboptimal operation computationally intensive unconstrained operation → typically complementary methods are combined via time-scale separation y r + Optimization Controller System u − � � offline & feedforward real-time & feedback � 3
Example: power systems load / generation balancing generation setpoints u real-time low-level optimization stage schedule operation automatic power system controllers SC-OPF , market automated/manual services/re-dispatch droop, AGC x state estimation disturbance δ t prediction (load, generation, downtimes) marginal costs in €/MWh 200 optimization stage Renewables 150 Nuclear energy economic dispatch based Lignite 100 on predictions/markets Hard coal Natural gas 50 Fuel oil real-time operations 0 0 10 20 30 40 50 60 70 80 90 100 unforeseen deviations from Capacity in GW [Cludius et al., 2014] schedule (e.g. congestion) low-level automatic control 50 Hz + y Frequency Power Control u System frequency regulation at − the individual generators frequency measurement 4
Price for time-scale separation re-dispatch to deal with unforeseen Cost of ancillary services of German TSOs load, congestion, & renewables in mio. Euros 111.8 82.3 primary frequency 85.2 ⇒ ever more uncertainty & control reserves 103.4 110.9 fluctuations on all time scales 371.9 267.1 secondary frequency 352.9 control reserves 227.6 ⇒ operation architecture becomes 154.8 104.2 infeasible & inefficient 67.4 tertiary frequency 156.1 control reserves 106.0 50.2 27.0 68.3 reactive power Redispatch actions in the German 33.0 15 811 26.7 transmission grid 32.6 in hours 41.6 164.8 national & internat. 113.3 redispatch 185.4 411.9 8 453 7 965 7 160 2011 2012 2013 2014 2015 5 030 [Bundesnetzagentur, Monitoringbericht 2016] 1 588 There must be a better way of operation. 2010 2011 2012 2013 2014 2015 [Bundesnetzagentur, Monitoringbericht 2016] 5
Ancillary services: synopsis and proposal Today: partially automated, provided by separate mechanisms, hitting limits • reactive power • real time balancing • economic re-dispatch compensation • voltage regulation • collapse prevention • frequency control • loss minimization • line congestion relief Central paradigm of future “smart” grids: automation for real-time operation Proposal: online optimization disturbance w operational constraints algorithms as feedback control u → robust (feedback) feedback power actuation optimizer system → fast response u ∈ U → operational constraints e.g., x = f ( x , u , w ) ˙ y y = h ( x , u , w ) → steady-state optimal u = −∇ φ ( y , u ) ˙ real-time state → MIMO decision making measurements 6
Brief review on related literature • historical roots : optimal routing and queuing in communication networks, e.g., in the internet (TCP/IP) [Kelly et al. 1998/2001, Low, Paganini, and Doyle 2002, Srikant 2012, ...] • lots of recent theory development in power systems & other infrastructures lots of related work: [Bolognani et al, A Survey of Distributed Optimization and Control 2015], [Dall’Anese and Simmonetto, Algorithms for Electric Power Systems 2016/2017], [Gan and Low, 2016], 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 [Tang and Low, 2017], ... early adoptions: KKT control [Jokic et al, 2009] and Commelec [Bernstein et al, 2015] • MPC version of “dropping argmin”: real-time iteration [Diel et al. 2005] , real-time MPC [Zeilinger et al. 2009] , ...and related papers with anytime guarantees • independent literature in process control [Bonvin et al. 2009/2010] or extremum seeking [Krstic and Wang 2000] , ...and probably much more • recent system theory [Nelson et al. 2017], [Colombino et al. 2018], [Lawrence et al. 2018] • algorithms as dynamic control systems [Lessard et al., 2014], [Wilson et al., 2018] 7
OVERVIEW 1. Interconnected dynamics and stability analysis 2. Projected gradient flow on the power flow manifold 3. Numerical experiments 8
INTERCONNECTED DYNAMICS AND STABILITY ANALYSIS 9
Stylized problem description Optimization Problem LTI Dynamics x = Ax + Bu + Qw ˙ minimize φ ( y, u ) y,u y = Cx + Du subject to y = ( CH + D ) u + CRw with A Hurwitz & steady-state maps u ∈ U x = − A − 1 B u − A − 1 Q w → gradient control of steady state � �� � � �� � H R � � CH + D I � T ∇ φ − ǫ � y = ( CH + D ) u + CRw u = Π U ˙ ( u ) + u x � � ǫ B U + ∇ u φ D A + + + y + � T ∇ y φ � − CH + D I C 10
Stability, feasibility, & asymptotic optimality of closed loop Theorem: Assume that LTI system asymptotically stable : ∃ γ > 0 , ∃ P ≻ 0 : PA + A T P � − γP regularity of cost function φ : compact level sets and ℓ -Lipschitz gradient sufficient time-scale separation (small gain): 0 ≤ ǫ ≤ ǫ ⋆ � γ 2 ℓ � H � 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 inspired from singular perturbation theory e T Pe Ψ δ ( u, e ) = δ · + (1 − δ ) · φ ( e, u ) � �� � � �� � LTI Lyapunov function objective function with steady-state error coordinate e = x − Hu − Rw & coefficient δ ∈ (0 , 1) Ψ δ ( u, e ) is non-increasing if ǫ ≤ ǫ ⋆ and for optimal choice of δ → derivative ˙ 11
Example: optimal constrained frequency control Dynamic model: linearized swing dynamics primary frequency control 1st-order turbine-governor DC power flow approximation ˙ θ = ω x = Ax + Bu + Qw ˙ where = − M − 1 � Dω + B θ − p + p L ( t ) � ω ˙ � θ � , u = p C , w = p L ( t ) x = ω = − K � R − 1 ω + p − p C � ˙ p p Measurements: 1 0 0 frequency at node 1 0 . . . 0 θ = B ℓ y = 0 0 ω selected line flows I p active power injections 0 0 12
Example: optimal constrained frequency control optimization problem minimize φ ( y ) y,u y = CHu + CRw subject to u ∈ U where y = CHu + CRw is the steady-state input-output map economic cost and operational limits are encoded in 2 � max { 0 , y − y }� 2 1 Ξ + 1 2 � max { 0 , y − y }� 2 φ ( y ) = cost ( y ) + Ξ � �� � � �� � DC OPF operational limits (line flows, frequency, ...) U describes the saturation constraints on the actuation → control ˙ u = Π U ( . . . ∇ φ ) ≡ optimal Automatic Generation Control (AGC) 13
Response to contingencies Generator outage & double line tripping in IEEE 118-bus test system · 10 5 Generation cost [$/hr] 1.5 1 0.5 Feedback Opt of fl ine DC OPF Frequency Deviation from f 0 [Hz] 0.2 0.1 0 − 0.1 − 0.2 Line Power Flow Magnitudes [p.u.] Power Generation (Gen 37) [p.u.] 4 6 23 → 26 90 → 26 fl ow limit other lines Setpoint Output 3 4 2 2 1 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 Time [s] Time [s] 14
How conservative is ǫ ≤ ǫ ⋆ ? Simulation on IEEE 118-bus test case still stable for ǫ = 2 ǫ ⋆ unstable for ǫ = 10 ǫ ⋆ · 10 − 2 Frequency Deviation from f 0 [Hz] Frequency Deviation from f 0 [Hz] 4 System Frequency System Frequency 5 2 0 0 − 2 − 5 · 10 5 Generation cost [$/hr] · 10 7 Generation cost [$/hr] 8 Feedback Opt offline DC OPF Feedback Opt offline DC OPF 1.4 6 4 1.2 2 0 1 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 ranges from 1.2 to 1000, depending on penalty scalings. 15
Highlights and comparison of our contributions Weak assumptions on plant Parsimonious but powerful setup internal stability potentially conservative bound, but → no observability/controllability needed → minimal assumptions on optimization problem & plant reduced model dependency → constraints assured by general → need only steady-state map H plant dynamics (no primal/dual) [Jokic et al. 2009], [Zhao et al. 2013] Weak assumptions on cost → directly useful for design (no LMIs) Lipschitz gradient + properness [Nelson et al. 2017], [Colombino et al. 2018] → no (strict/strong) convexity required proof can be extended to other convexity ⇒ global convergence algorithms & nonlinear dynamics → Menta, Hauswirth, Bolognani, Hug & Dörfler (2018) take-home msg: online optimization “Stability of Dynamic Feedback Optimization algorithms can be applied to dynamics with Applications to Power Systems” 16
PROJECTED GRADIENT FLOW ON THE POWER FLOW MANIFOLD 17
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