Feedback Optimization on the Power Flow Manifold Institut für Automation und angewandte Informatik (IAI) Karlsruhe Institute of Technology (KIT)

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1 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, 28

2 Acknowledgements Adrian Hauswirth Saverio Bolognani Gabriela Hug 2

3 Power system operation: supply chain without storage Traditional Power Generation principle: deliver power from generators to loads physical constraints: Kirchhoff s and Ohm s laws operational constraints: thermal and voltage limits transmission grid performance objectives: running costs, reliability, quality of service fit-and-forget design: distribution grid historically designed according to worst-case possible demand 3

4 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 flexible but spatio-temporal patterns inverter-interfaced storage/generation extremely fast actuation modular & flexible control information & comm technology inexpensive reliable communication increasingly ubiquitous sensing power Germany 7 August 24 biomass hydro time of day 4GW 75% single residential load profile solar wind power 4KW Domestic consumer time of day Electric Vehicle Fast charging single PV plant 2KW Tesla supercharger 4

5 Recall: feedforward vs. feedback or optimization vs. control feedforward optimization feedback control p r Controller u System y r + Controller u System y highly model based computationally intensive optimal decision operational constraints model-free (robust) design fast response suboptimal operation unconstrained operation typically complementary methods are combined via time-scale separation r + Optimization Controller System u y offline & feedforward real-time & feedback 5

6 Example: power systems load / generation balancing optimization stage SC-OPF, market schedule real-time operation automated/manual services/re-dispatch generation setpoints state estimation low-level automatic controllers droop, AGC u x power system prediction (load, generation, downtimes) disturbance δt optimization stage economic dispatch based on load/renewable prediction real-time interface manual re-dispatch, area balancing services marginal costs in /MWh Renewables Nuclear energy Lignite Hard coal Natural gas Fuel oil Capacity in GW [Cludius et al., 24] low-level automatic control frequency regulation at the individual generators 5Hz + Frequency Control u Power System frequency measurement y 6

7 The price for time-scale separation: sky-rocketing re-dispatch re-dispatch to deal with unforeseen load, congestion, & renewables ever more uncertainty & fluctuations on all time scales operation architecture becomes infeasible & inefficient Cost of ancillary services of German TSOs in mio. Euros primary frequency control reserves secondary frequency control reserves tertiary frequency control reserves Redispatch actions in the German transmission grid in hours reactive power national & internat. redispatch [Bundesnetzagentur, Monitoringbericht 26] There must be a better way of operation. [Bundesnetzagentur, Monitoringbericht 26] 7

8 Synopsis... for essentially all ancillary services real-time balancing frequency control economic re-dispatch voltage regulation voltage collapse prevention line congestion relief reactive power compensation losses minimization recall new challenges: increased variability poor short-term prediction correlated uncertainties recall new opportunities: fast actuation ubiquitous sensing reliable communication 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

9 Control-theoretic core of the problem time-scale separation of complementary feedback/feedforward architectures r + Optimization Controller System u y ideal approach: optimal feedback policies (from HJB, Pontryagin, etc.) u(x) argmin T s.t. dynamics ẋ = h(x, u) l(x, u) dt + φ(x(t ), u(t )) s.t. constraints x X and u U u System disturbance δ 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

10 Today we will follow a different approach u(x) argmin T l(x, u) dt + φ(x(t ), u(t )) s.t. dynamics ẋ = h(x, u) s.t. constraints x X and u U u System disturbance δ x drop exact argmin drop integral/stage costs let physics solve equality constraints (dynamics) Instead we apply online optimization in closed loop with fast/stationary physics: operational constraints disturbance δ robust (feedback) fast response operational constraints steady-state optimal feedback control: online optimization algorithm, e.g., u + = Proj (... ) u actuation x real-time state measurements physical plant: steady-state power system h(x, u, δ) =

11 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. 998/2, Low, Paganini, and Doyle 22, Srikant 22,... ] lots of recent theory development in power systems & other infrastructures lots of related work: [Bolognani et. al, 25], [Dall Anese and Simmonetto, 26/27], [Gan and Low, 26], [Tang and Low, 27],... A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems Daniel K. Molzahn, Member, IEEE, Florian Dörfler, 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 Abstract Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems. Index Terms Distributed optimization, online optimization, electric power systems independent literature in process control [Bonvin et al. 29/2] or extremum I. INTRODUCTION ENTRALIZED computation has been the primary way seeking [Krstic and Wang 2],... andc probably that optimization and control much algorithms have more been applied to electric power systems. Notably, independent system operators (ISOs) seek a minimum cost generation dispatch for large-scale transmission systems by solving an optimal power flow (OPF) problem. (See [] [8] for related literature reviews.) Other control objectives, such as maintaining scheduled power interchanges, are achieved via an Automatic Generation Control (AGC) signal that is sent to the generators that provide regulation services. These optimization and control problems are formulated using network parameters, such as line impedances, system topology, and flow limits; generator parameters, such as cost providing demand response resources, etc.), the centralized paradigm most prevalent in current power systems will potentially be augmented with distributed optimization algorithms. Rather than collecting all problem parameters and performing a central calculation, distributed algorithms are computed by many agents that obtain certain problem parameters via communication with a limited set of neighbors. Depending on the specifics of the distributed algorithm and the application of interest, these agents may represent individual buses or large portions of a power system. Distributed algorithms have several potential advantages over centralized approaches. The computing agents only have to share limited amounts of information with a subset of the other agents. This can improve cybersecurity and reduce the expense of the necessary communication infrastructure. Distributed algorithms also have advantages in robustness with respect to failure of individual agents. Further, with the ability to perform parallel computations, distributed algorithms have the potential to be computationally superior to centralized algorithms, both in terms of solution speed and the maximum problem size that can be addressed. Finally, distributed algorithms also have the potential to respect privacy of data, measurements, cost functions, and constraints, which becomes increasingly important in a distributed generation scenario. This paper surveys the literature of distributed algorithms with applications to power system optimization and control. This paper first considers distributed optimization algorithms for solving OPF problems in offline applications. Many dis- MPC version of dropping argmin : real-time iteration [Diel et al. 25], real-time MPC [Zeilinger et al. 29],... and related papers with anytime guarantees plenty of interesting recent system theory coming out [Nelson and Mallada 27]

12 OVERVIEW. 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 2

13 AC power flow model, constraints, and objectives quasi-stationary (for now) dynamics 2 3 Ohm s Law Current Law AC power AC power flow equations nodal voltage current injection power injections line impedance line current 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 3

14 What makes power flow optimization interesting? graphical illustration of AC power flow Ohm s Law Current Law AC power AC power flow equations (all variables and parameters are -valued) [Hiskens, 2] imagine constraints slicing this set nonlinear, non-convex, disconnected additionally the parameters are ±2% uncertain... this is only the steady state! [Molzahn, 26] 4

15 Ancillary services as a real-time optimal power flow Offline optimal power flow (OPF) minimize φ(x, u) e.g., losses, generation subject to h(x, u, δ) = AC power flow (x, u) X U operational constraints exogenous variables u controllable generation δ exogenous disturbances (e.g., loads & renewables) x endogenous variables (voltages) Idea for an online algorithm goal: closed-loop gradient flow [ẋ ] = Proj u U X {linearization of h} φ(x, u) implement control u (as above) consistency of x ensured by non-singular physics h(x, u, δ) = discrete-time implementation feedback control: online optimization algorithm, e.g., u + = Proj (... ) operational constraints u actuation x real-time state measurements disturbance δ physical plant: steady-state power system h(x, u, δ) = 5

16 Pretty hand-waivy... I know. I will make it more precise later. Let s see if it works! 6

17 Preview: simple algorithm solves many problems G Generator C Synchronous Condensor S Solar W Wind W..5 Bus voltages [p.u.].95 G C 3 C2 S.9 Branch current magnitudes [p.u.] G 2 C,5, 5 Feedback OPF Optimal cost Generation cost Time [hrs] controller: gradient + saturation ( generation + voltage violation ) Active power injection [MW] Time [hrs] Gen Gen2 Solar Wind time-variant disturbances/constraints robustness to noise & uncertainty dynamics of physical system crude discretization/linearization 7

18 Preview cont d: robustness to model mismatch G Generator C Synchronous Condensor S Solar W Wind G C 3 G 2 C W C2 S feedback control: online optimization algorithm, e.g., u + = Proj (... ) operational constraints u actuation x real-time state measurements disturbance δ physical plant: steady-state power system h(x, u, δ) = gradient controller: saturation of generation constraints soft penalty for operational constraints no automatic re-dispatch feedback optimization model uncertainty feasible? f f v v feasible? f f v v loads ±4% no yes.. line params ±2% yes.9. yes..3 2 line failures no yes.9.7 conclusion: simple algorithm performs extremely well & robust closer look! 8

19 TECHNICAL INGREDIENT I: THE POWER FLOW MANIFOLD 9

20 Key insights about our physical equality constraint AC power flow is complex but it defines a smooth manifold local tangent plane approximations & h(x, u, δ) = locally solvable for x Bolognani & Dörfler (25) Fast power system analysis via implicit linearization of the power flow manifold v dc i dc τ ii R T L T i T v C G v m R I L I C I G I i I θ, ω m v τ e r s L θ M r f v f C G q i s v r s i f AC power flow is attractive steady state for ambient physical dynamics physics enforce feasibility even for non-exact (e.g., discretized) updates Gross, Arghir, & Dörfler (28) On the steady-state behavior of a nonlinear power system model 2

21 Geometric perspective: the power flow manifold node v =, θ = p, q y =.4.8j node 2 v 2, θ 2 p 2, q 2 v variables: all of x = ( V, θ, P, Q) power flow manifold: M = {x : h(x) = }.5 q p 2.5 submanifold in R 2n or R 6n (3-phase) tangent space h(x) (x x x x ) = best linear approximant at x accuracy depends on curvature 2 h(x) x 2 constant in rectangular coordinates v p q 2.5 2

22 Accuracy illustrated with unbalanced three-phase IEEE3 exact solution linear approximant dirty secret: power flow manifold is very flat (linear) near usual operating points Matlab/Octave 22

23 Coordinate-dependent linearizations reveal old friends flat-voltage/-injection point: x = ( V, θ, P, Q ) = (,,, ) [ ] [ ] [ ] R(Y ) I(Y ) V P tangent space parameterization = I(Y ) R(Y ) θ Q is linear coupled power flow and R(Y ) gives DC power flow approximation nonlinear change to quadratic coordinates V V 2 linearization is (non-radial) LinDistFlow [M.E. Baran and F.F. Wu, 88] more exact in V.5.4 power flow manifold.3.2 power flow manifold p v ! 2 linear coupled power flow DC power flow approximation (neglects PV coupling) 2 v p q 2 linear approximation linear approximation in quadratic coordinates

24 TECHNICAL INGREDIENT II: MANIFOLD OPTIMIZATION 24

25 Unconstrained manifold optimization: the smooth case geometric objects: manifold M = {x : h(x) = } tangent space T xm = ker h(x) x objective Riemann metric (degree of freedom) φ : M R g : T xm T xm R target state: local minimizer on the manifold x arg min x M φ(x) always feasible trajectory/sequence x(t) remains on manifold M continuous-time gradient descent on M: linear approximant Gradient of cost function. grad φ(x): gradient of cost function in ambient space 2. Π M (x, gradφ(x)): projection of gradient on tangent space T xm x(t) ẋ Projected gradient 3. flow on manifold: ẋ = Π M (x, gradφ(x)) manifold 25

26 Constrained manifold optimization: the wild west dealing with operational constraints g(x). penalty in cost function φ barrier: not practical for online implementation soft penalty: practical but no real-time feasibility 2. dualization and gradient flow on Lagrangian poor performance & no real-time feasibility theory: close to none available on manifolds Hauswirth, Bolognani, Hug, & Dörfler (28) Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow 3. projection of gradient flow trajectory x(t) on feasible set K = M {g(x) } ẋ = Π K (x, gradφ(x)) arg min grad φ(x) v g v T x > K where T > x K T xm is inward tangent cone 26

27 Projected gradient descent on manifolds Theorem (simplified) Let x : [, ) K be a Carathéodory solution of the initial value problem ẋ = Π K (x, gradφ(x)), x() = x. If φ has compact level sets on K, then x(t) will converge to a critical point x of φ on K. K = { x : x 2 2 =, x 2 } Hauswirth, Bolognani, Hug, & Dörfler (26) Projected gradient descent on Riemanniann manifolds with applications to online power system optimization Hidden assumption: existence of a Carathéodory solution x(t) K when does it exist, is forward complete, unique, and sufficiently regular? (in absence of convexity, Euclidean space, and other regularity properties) 27

28 Analysis via projected systems hit mathematical bedrock power flow manifold disconnected regions cusps & corners (convex and/or inward) constraint set gradient field metric manifold existence (Krasovski) loc. compact loc. bounded - C Krasovski = Carathéodory Clarke regular C C C uniqueness of solutions prox regular C, C, C, also forward-lipschitz continuity of time-varying constraints continuity with respect to initial conditions and parameters Hauswirth, Bolognani, Hug, & Dörfler (28) Projected Dynamical Systems on Irregular, Non-Euclidean Domain for Nonlinear Optimization Hauswirth, Subotic, Bolognani, Hug, & Dörfler (28) Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems 28

29 Implementation issue: how to induce the gradient flow? Open-loop system ẋ = u controlled generation = h(x, x 2, δ) AC power flow manifold relating x & other variables Desired closed-loop system ẋ = f (x, x 2 ) desired projected ẋ 2 = f 2 (x, x 2 ) gradient descent where f(x) = Π K (x, gradφ(x)) Solution use non-singularity of the physics: = h(x, x 2, δ) can be solved for x 2 Feedback equivalence The trajectories of the desired closed loop coincide with those of the open loop under the feedback u = f (x, x 2 ). feedback optimizer ΠK (x, gradφ(x)) actuate u x measure open-loop system ẋ = u = h(x, x 2, δ) closed-loop trajectory remains feasible at all times and converges to optimality no need to numerically solve the optimization problem or any power flow equation 29

30 Implementation issue: discrete-time manifold optimization always feasible trajectory/sequence x(t) remains on manifold M discrete-time gradient descent on M:. grad φ(x): gradient of cost function linear approximant Gradient of cost function Projected gradient 2. Π M (x, gradφ(x)): projection of gradient x(t) 3. Euler integration of gradient flow: x(t+) = x(t) ε Π M (x, gradφ(x)) 4. retraction step: x(t + ) = R x(t) ( x(t + ) ) manifold x(t + ) Retraction Discrete-time control implementation: manifold is attractive steady state for ambient dynamics retraction is taken care of by the physics: nature enforces feasibility can be made rigorous using singular perturbation theory (Tikhonov) 3

31 CASE STUDIES: TRACKING, FEASIBILITY, & DYNAMICS 3

32 Simple illustrative case study Objective Value [$] feedback optimizer Π K (x, gradφ(x)) actuate u x measure open-loop system ẋ = u = h(x, x 2, δ) 5 real time cost global minimum Bus voltages [p.u.] Active power generation [MW] 2 Slack bus Gen A Gen B iteration 32

33 The tracking problem power system affected by exogeneous time-varying inputs δ t under disturbances state could leave feasible region K (ill-defined) δ t feedback optimizer Π K (x, gradφ(x)) u U x open-loop system ẋ = u = h(x, x 2, δ t ) constraints satisfaction for non-controllable variables: K accounts only for hard constraints on controllable variables u (e.g., generation limits) gradient projection becomes input saturation (saturated proportional feedback control) soft constraints included via penalty functions in φ (e.g., thermal and voltage limits) 33

34 Tracking performance G Generator C Synchronous Condensor S Solar..5 Bus voltages [p.u.] W Wind W.95 G C 3 C2 S.9 Branch current magnitudes [p.u.] G 2 C.5 Aggregate Load &Available Renewable Power [MW] 4 Load Solar Wind Time [hrs] controller: penalty + saturation Active power injection [MW] Time [hrs] Gen Gen2 Solar Wind 34

35 Tracking performance G Generator C Synchronous Condensor S Solar Comparison W Wind W closed-loop feedback trajectory G C 3 C2 S benchmark: feedforward OPF (ground-truth solution of an ideal OPF with access to exact disturbance and without computation delay) G 2 C Generation cost practically exact tracking + trajectory feasibility,5, 5 Feedback OPF Optimal cost + robustness to model mismatch Time [hrs] 35

36 Trajectory feasibility The feasible region K = M X often has disconnected components. x K x M feedforward (OPF) optimizer x = arg min x K φ(x) can be in different disconnected component no feasible trajectory exists: x x must violate constraints feedback (gradient descent) continuous closed-loop trajectory x(t) guaranteed to be feasible convergence of x(t) to a local minimum is guaranteed 36

37 '('-)+'('* '(',)+'(! Illustration of continuous trajectories & reachability 5-bus example known to have two disconnected feasible regions: " 2 Objective Value [$] Feedback Feed-forward &! '('%)+'(!' '('$)'('* # '(%%)+'(*' '(%%)+'(*' $ % & [Molzahn, 26] [s,2s]: separate feasible regions [2s,3s]: loosen limits on reactive power Q 2 regions merge [4s,5s]: tighten limits on Q 2 vanishing feasible region Voltage Levels [p.u.] Active Power Generation P [MW] Gen Gen Gen Gen2 Q5min Reactive Power Generation Q [MVAR]

38 Feedback optimization with frequency Power [p.u.] frequency ω as global variable primary control: P = P G Kω secondary frequency control incorporated via dual multiplier 2% step increase in load Active Power Generation Cost [$] Power [p.u.] Time [s] Generation Cost Aggregated Reactive Power Generation Reference OPF deviation [Hz] Time [s] Frequency Time [s] Voltage [p.u.] Time [s] Bus voltages Time [s]

39 Same feedback optimization with grid dynamics Active Power Generation Frequency 2.2 Power [p.u.].5.5 deviation [Hz] Time [s] Time [s] Active Power Generation (zoomed) Frequency (zoomed).5.6 Power [MW].4.2 deviation [Hz] Time [s] Time [s] dynamic grid model: swing equation & simple turbine governor work in progress based on singular perturbation methods dynamic and quasi-stationary dynamics are close and converge to the same optimal solutions under sufficient time-scale separation 39

40 Feedback optimization in dynamic IEEE 3-bus system Active power injection [MW] G Generator C Synchronous Condensor.6 S Solar.4 W Wind W.2 G C 3 C2 S Time [hrs] Gen Gen 2 Gen 3 Solar Wind. Frequency deviation [Hz] G 2 C 5 2 events: 5 2 generator outage at 4: PV generation drops at : and 4:5 feedback optimization can provide all ancillary services + optimal + constraints + robust + scalable Time [hrs] Generation cost [$/hr] reference AC OPF Feedback OPF Time [hrs] 4

41 Conclusions Summary: necessity of real-time power system operation our starting point: online optimization as feedback control technical approach: manifold optimization & projected dyn. systems unified framework accommodating various constraints & objectives Ongoing and future work: fun: questions on existence of trajectories, reachability,... quantitative guarantees for robustness, tracking, etc. interaction of optimization algorithm with low level grid dynamics efficient implementation, discretization, experiments, RTE collaboration extensions: transient optimality à la MPC & model-free à la extremum seeking 4

42 Thanks! Florian Dörfler 42