Feedback Optimization on the Power Flow Manifold Institut für Automation und angewandte Informatik (IAI) Karlsruhe Institute of Technology (KIT) Florian Dörfler Automatic Control Laboratory, ETH Zürich March 26 - 29, 2018 1
Acknowledgements Adrian Hauswirth Saverio Bolognani Gabriela Hug 2
Power system operation: supply chain without storage principle: deliver power Traditional from generators to loads Power Generation physical constraints: Kirchhoff’s and Ohm’s laws operational constraints: thermal and voltage limits performance objectives: transmission grid running costs, reliability, quality of service fit-and-forget design: historically designed according distribution grid to worst-case possible demand 3
New challenges and opportunities variable renewable energy sources – poor short-range prediction & correlations – fluctuations on all time scales (low inertia) distributed microgeneration – conventional and renewable sources – congestion and under-/over-voltage electric mobility – large peak (power) & total (energy) demand single residential single PV plant load profile – flexible but spatio-temporal patterns power power inverter-interfaced storage/generation time of day time of day – extremely fast actuation – modular & flexible control Germany Electric Vehicle 17 August 2014 Fast charging wind information & comm technology 41GW 75% – inexpensive reliable communication 4KW 120KW biomass Domestic Tesla consumer supercharger – increasingly ubiquitous sensing hydro solar 4
Recall: feedforward vs. feedback or optimization vs. control feedforward optimization feedback control p y r + y r System System Controller Controller u u − highly model based model-free (robust) design computationally intensive fast response optimal decision suboptimal operation unconstrained operation operational constraints ⇒ typically complementary methods are combined via time-scale separation y + r Optimization System Controller u − � offline & feedforward real-time & feedback � � 5
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) 200 marginal costs in €/MWh optimization stage Renewables 150 Nuclear energy economic dispatch based Lignite 100 Hard coal on load/renewable prediction Natural gas 50 Fuel oil real-time interface 0 manual re-dispatch, 0 10 20 30 40 50 60 70 80 90 100 Capacity in GW [Cludius et al., 2014] area balancing services y low-level automatic control 50 Hz + Frequency Power Control System u − frequency regulation at the individual generators frequency measurement 6
The price for time-scale separation: sky-rocketing re-dispatch 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 ⇒ operation architecture becomes 227.6 154.8 infeasible & inefficient 104.2 67.4 tertiary frequency 156.1 control reserves 106.0 50.2 27.0 68.3 Redispatch actions in the German reactive power 15 811 33.0 transmission grid 26.7 32.6 in hours 41.6 164.8 national & internat. 113.3 redispatch 185.4 8 453 411.9 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] 7
Synopsis ...for essentially all ancillary services • real-time balancing recall new challenges: • frequency control increased variability • economic re-dispatch poor short-term prediction correlated uncertainties • voltage regulation • voltage collapse prevention recall new opportunities: • line congestion relief fast actuation • reactive power compensation ubiquitous sensing reliable communication • losses minimization Today: these services are partially automated, implemented independently, online or offline, based on forecasts (or not), and operating on different time/spatial scales. One central paradigm of “smart(er) grids” : real-time operation Future power systems will require faster operation, based on online control and monitoring, in order to meet the operational specifications in real time. 8
Control-theoretic core of the problem time-scale separation of complementary feedback/feedforward architectures y + r Optimization Controller System u − ideal approach: optimal feedback policies (from HJB, Pontryagin, etc.) � T disturbance δ u ( x ) ∈ argmin 0 ℓ ( x, u ) dt + φ ( x ( T ) , u ( T )) System x = h ( x, u ) ˙ s.t. dynamics u s.t. constraints x ∈ X and u ∈ U x → explicit ( T = ∞ ) feedback policies are not tractable analytically or computationally → usually a decent trade-off: receding horizon model predictive control MPC ⇒ not suited for power systems (due to dimension, robustness, uncertainty, etc.) 9
Today we will follow a different approach drop exact argmin � T disturbance δ u ( x ) ∈ argmin 0 ℓ ( x, u ) dt + φ ( x ( T ) , u ( T )) drop integral/stage costs System s .t. dynamics x = h ( x, u ) ˙ u let physics solve equality s.t. constraints x ∈ X and u ∈ U constraints (dynamics) x Instead we apply online optimization in closed loop with fast/stationary physics: disturbance δ operational constraints robust (feedback) feedback control: physical plant: u fast response actuation online optimization steady-state algorithm, e.g., power system operational constraints u + = Proj ∇ ( . . . ) h ( x , u , δ ) = 0 steady-state optimal x real-time state measurements 10
Very brief review on related online optimization in closed loop • 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], ... • 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 • plenty of interesting recent system theory coming out [Nelson and Mallada 2017] 11
OVERVIEW 1. Problem setup & preview of a solution 2. Technical ingredient I: the power flow manifold 3. Technical ingredient II: manifold optimization 4. Case studies: tracking, feasibility, & dynamics 12
AC power flow model, constraints, and objectives quasi-stationary (for now) dynamics Ohm’s Law Current Law 12 13 11 8 7 10 9 AC power 6 3 4 AC power flow equations 2 5 line impedance nodal voltage line current current injection power injections power flow (all variables and parameters are -valued) objective: economic dispatch, minimize losses, distance to collapse, etc. operational constraints: generation capacity, voltage bands, congestion control: state measurements and actuation via generation set-points 13
What makes power flow optimization interesting? Ohm’s Law Current Law graphical illustration of AC power flow AC power AC power flow equations (all variables and parameters are -valued) [Hiskens, 2001] imagine constraints slicing this set ⇒ nonlinear, non-convex, disconnected additionally the parameters are ± 20% uncertain ...this is only the steady state! [Molzahn, 2016] 14
Ancillary services as a real-time optimal power flow Offline optimal power flow (OPF) exogenous variables → u controllable generation minimize φ ( x , u ) e.g., losses, generation → δ exogenous disturbances subject to h ( x , u , δ ) = 0 AC power flow (e.g., loads & renewables) ( x , u ) ∈ X × U operational constraints x endogenous variables (voltages) Idea for an online algorithm goal: closed-loop gradient flow disturbance δ operational constraints � � x ˙ = − Proj U∩X∩{ linearization of h } ∇ φ ( x , u ) u ˙ feedback control: physical plant: u actuation online optimization steady-state algorithm, e.g., power system implement control ˙ u (as above) u + = Proj ∇ ( . . . ) h ( x , u , δ ) = 0 x consistency of x ensured by real-time state measurements non-singular physics h ( x , u , δ ) = 0 discrete-time implementation 15
Pretty hand-waivy ...I know. I will make it more precise later. Let’s see if it works! 16
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