Networked Feedback Control for a Smart Power Distribution Grid

Transcription

1 Networked Feedback Control for a Smart Power Distribution Grid Saverio Bolognani 6 March Workshop 1

2 Future power distribution grids Traditional Power Generation transmission grid It delivers power from the transmission grid to the consumers. Very little sensing, monitoring, actuation. The easy part of the grid: conventionally fit-and-forget design. distribution grid 2

3 New challenges Distributed microgenerators (conventional and renewable sources) Electric mobility (large flexible demand, spatio-temporal patterns). Germany 17 August 2014 wind Installed renewable generation Germany 2013 solar 24 GW biomass hydro 41GW 75% solar wind hydro + biomass Transmission grid 6 GW Distribution grid 15 GW Energy consumption by sector (2010) Buildings 40.9% Industry 31.3% Transportation 27.8% 25.9% Electricity consumption 73.9% 4KW Domestic consumer Electric Vehicle Fast charging 120KW Tesla supercharger Physical grid limits grid congestion Fit-and-forget unsustainable grid reinforcement 3

4 Virtual grid reinforcement Transmission Grid Distribution Grid control control Curtailment x x Reduced hosting capacity x x control control CONTROL LAYER Higher renewable generation Larger hosting capacity Transmission Grid Distribution Grid distribution grid uncontrolled overvoltage controlled renewable generation controlled undervoltage uncontrolled power demand Virtual grid reinforcement same infrastructure more sensors controlled grid = stronger grid distributed ancillary services accomodate active power flows transparently 4

5 OVERVIEW 1. A feedback control perspective on power system operation 2. A tractable power grid model for feedback control design 3. Control design example: voltage regulation Distributed model-free control Centralized chance-constrained decision 5

6 A FEEDBACK CONTROL PERSPECTIVE ON POWER SYSTEM OPERATION 6

7 Power distribution grid model 0 supply point voltage magnitude vh voltage angle θh h active power ph reactive power qh microgenerator load Grid model Nonlinear complex valued power flow equations diag(u)yu = s Actuation Tap changer v 0 Reactive power compensators q h Active power management p h where u h = v h e jθ h s h = p h + jq h complex bus voltages complex bus powers Sensing Voltage meters v h (sometimes θ h ) Line currents, transformer loading,... Underdetermined: few sensors 7

8 A control perspective on distributed grid operation power generation power demands grid actuation plant Power distribution network state x grid sensing Ancillary services: voltage regulation / reactive power compensation / economic re-dispatch / loss minimization / line congestion control / energy balancing /... Control objective Drive the system to a target state x = [ v θ p q ] subject to soft constraints x = argmin x J(x) hard constraints x X chance constraints P [x X ] < ǫ 8

9 Feedforward control OPF power generation power demands grid actuation plant Power distribution network state x grid sensing Conventional approach Core tool: Optimal Power Flow Fast OPF solvers in radial networks Many variants, including distributed implementations However: Requires full state measurement - full communication Heavily model based 9

10 Feedback control disturbance power generation power demands grid actuation input plant Power distribution network state x grid sensing output FEED BACK Control theory answer Disturbance rejection grid state regulated despite demand/generation Model-free design Robustness against uncertainty Output feedback 10

11 A TRACTABLE POWER GRID MODEL FOR FEEDBACK CONTROL DESIGN 11

12 Power flow manifold Set of all states that satisfy the grid equations diag(u)yu = s power flow manifold M := {x F(x) = 0} 1.4 Best linear approximant v p q Tangent plane at a nominal power flow solution x M A x (x x ) = 0 A x := F(x) x x=x Implicit No input/outputs (not a disadvantage) Sparse The matrix A x has the sparsity pattern of the grid graph Structure preserving Elements of A x depend on local parameters Bolognani & Dörfler (2015) Fast power system analysis via implicit linearization of the power flow manifold 12

13 CONTROL DESIGN EXAMPLE: VOLTAGE REGULATION 13

14 Case 1: hard constraints disturbance power generation power demands microgenerator reactive power q h input q h q h q h plant Power distribution network state x FEED BACK microgenerator voltage v h v v h v output Inputs: reactive power q h of microgenerators Outputs: voltage measurement v h at the microgenerators Control objective: Soft constraints minimize v T Lv Hard constraints (voltage drops on the lines) V v h V at all sensors q q h q h h at all actuators 14

15 Case 1: hard constraints linear approximant 1. Modeling assumption power flow manifold Modeling assumption on the parameters: constant R/X ratio ρ. on the structure: Kron reduction to controllable nodes [ ρl L A x (x x ) = 0 L ρl I 0 ] v θ 0 I p = 0 q 15

16 Case 1: hard constraints linear approximant 1. Modeling assumption 2. Equilibrium power flow manifold Equilibrium: Saddle point of the Lagrangian L(x, λ, η) = v T Lv + λ T (v v) + η T (q q) +... Stable for the discrete-time trajectories in which we alternate exact minimization in the primal variable x projected gradient ascent in the dual variables λ, η 15

17 Case 1: hard constraints linear approximant search direction x 1. Modeling assumption 2. Equilibrium 3. Trajectory power flow manifold Search directions: By projecting each possible direction δq on the linear manifold ker A x, we obtain feasible search directions in the state space. 1 L δq 1+ρ 2 δx = ρ L δq 1+ρ 2 0 δq 15

18 Case 1: hard constraints linear approximant L x δx 1. Modeling assumption search direction x 2. Equilibrium 3. Trajectory 4. Feedback law power flow manifold Primal minimization step: we determine the step δx such that 2Lv + λ 1 L δq 1+ρ L x = and δx = ρ L δq 1+ρ 2 0 satisfy L x (x + δx, λ, η)t δx = 0 η δq 15

19 Case 1: hard constraints 1. Modeling assumption 2. Equilibrium 3. Trajectory 4. Feedback law Output feedback control law q q + (1 + ρ 2 ) (Lv + λ) + (1 + ρ 2 ) 2 Lη primal minimization } λ h [λ h + α(v h v)] 0 η h [ η h + β(q h q h ) ] dual ascent (integral action) 0 Lv, Lη Diffusion terms that requires nearest-neighbor communication. 15

20 Case 1: hard constraints performance same feasible equilibrium as ORPF networked control ORPF Output feedback control law convergence to OPF solution short-range communication no demand or generation measurement scalable plug-and-play limited model knowledge purely local control no power flow solver interleaved sensing and actuation communication complexity Proof of mean square convergence (with randomized async updates) S. Bolognani, R. Carli, G. Cavraro, & S. Zampieri (2015) Distributed reactive power feedback control for voltage regulation and loss minimization Communication is necessary: No local strategy can guarantee convergence to a feasible voltage profile. G. Cavraro, S. Bolognani, R. Carli, & S. Zampieri (2016) The value of communication in the voltage regulation problem 16

21 Case 2: chance constraints disturbance w power demands microgenerator active power input plant Power distribution network state x FEED BACK total power demand output y = hp h P[x X] < ǫ Inputs: active power p h of microgenerators Outputs: total grid demand y = p h h Control objective: Soft constraints maximize (minimize curtailment) generators h Chance constraint V v h V p h for all buses, with high probability 17

22 Case 2: chance constraints Scenario approach Convert stochastic constraint into large set of determistic ones P [x X(w)] < ǫ x X(w (i) ), i = 1,..., N 8 Two sources of information on the unknown w Historical samples w (i) of the prior distribution Online measurements y = Hw from the system Scenario approach based on conditional distribution High computational demand Large memory footprint Not suited for real-time feedback control 18

23 Case 2: chance constraints Preprocessing Two-phase algorithm Disturbance samples {w (i) } Offline algorithm Augmented polytope ˆP Real-time feedback Measurement y Online algorithm Decision (input) Express posterior distribution as a projection: ŵ y = w + K(y Hw) Construct a feasible region parametrized in y offline Compute the conditional feasible polytope online 15 p 38 [MW] 10 5 p 30 [MW] y = 5 [MW] y = 3 [MW] 10 no measurement y = 0 [MW] Offline Online Computation time Compute Σ and K Construct augmented polytope ˆP Compute minimal representation of ˆP Total offline computation time Slice ˆP at y = y meas to obtain ˆPy Solve LP defined on ˆPy Total online computation time Memory footprint 55 min 1.8 ms Offline Augmented polytope ˆP constraints Online Minimal representation of ˆP 12 constraints 19

24 CONCLUSIONS 20

25 Conclusions A tractable linear model structure preserving computationally efficient Ancillary services via feedback control model-free and robust limited measurement need for communication Next step Feedback on the power flow manifold linear approximant Gradient of cost function Bus voltages [p.u] 1.1 Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus Time [hrs] Active power injection [MW] 400 Gen 1 Gen 2 Gen 3 p Time [hrs] Reactive power injection [MVAR] Projected gradient 50 Gen 1 Gen 2 Gen 3 x(t) x(t +1) Retraction 0 power flow manifold Time [hrs] A. Hauswirth, A. Zanardi, S. Bolognani, F. Dörfler, & G. Hug (2017) Online Optimization in Closed Loop on the Power Flow Manifold 21

26 Saverio Bolognani 22

27 THE VALUE OF COMMUNICATION IN VOLTAGE REGULATION 23

28 Simulations and comparison Voltage [p.u.] v v 2 q 1 q Reactive power [p.u.] Modified IEEE 123 Distribution Test Feeder Light load + 2 microgenerators overvoltage 2 sets of constraints: { voltage limits v h v power converter limits q h q h github 24

29 Simulations and comparison Voltage [p.u.] steady state error saturation Reactive power [p.u.] q i Fully decentralized, proportional controller. q max i q h (t) = f(v h (t)) q max i v min v max v i Latest grid code draft Turitsyn (2011) Low (2012) Aliprantis (2013) Hiskens (2013) Kekatos (2015) 24

30 Simulations and comparison Voltage [p.u.] 1.1 steady state error saturation Reactive power [p.u.] δq i Fully decentralized, integral controller. v min v max v i Li (2014) Farivar (2015) q h (t + 1) = q h (t) f(v h (t)) 24

31 Simulations and comparison Voltage [p.u.] 1.1 no steady state error reactive power sharing Reactive power [p.u.] Networked feedback control (neighbor-to-neighbor communication) λ h [λ h + α(v h v)] 0 η h [ η h + β(q h q h ) ] 0 q q γ J(q) λ Lη 24

32 FEEDBACK OPTIMIZATION ON THE POWER FLOW MANIFOLD 25

33 Gradient descent on the power flow manifold Target state linear approximant Gradient of cost function Projected gradient x = arg min x M J(x) local minimizer on the power flow manifold x(t) ẋ power flow manifold Continuous time trajectory on the power flow manifold 1. J(x): gradient of the cost function (soft constraints) in ambient space 2. Π x J(x): projection of the gradient on the linear approximant in x 3. Flow on the manifold: ẋ = γπ x J(x) 26

34 Gradient descent on the power flow manifold ] x = [ xexo x endo Exogenous variables Inputs/disturbances Reactive power injection q i linear approximant x(t) Gradient of cost function Projected gradient Endogenous variables Determined by the physics of the grid. Voltage v i power flow manifold x(t +1) Retraction From gradient descent flow to discrete-time feedback control: 1. Compute Π x J(x) 2. Actuate system based on δx = γπ J (exogeneous variables / inputs) 3. Retraction step x(t + 1) = R x(t) (δx) x(t + 1) M. 27

35 Hard constraints: need for a new theory Feasible input region Not a smooth manifold Projected gradient descent Retraction preserves feasibility min J(x) x M K A. Hauswirth, S. Bolognani, G. Hug, & F. Dörfler (2016) Projected Gradient Descent on Riemannian Manifolds with Applications to Online Power System Optimization Output constraints No barrier function (backtracking not allowed) No time-varying penalty (persistent feedback control) Dualization: saddle / primal-dual trajectories on manifolds 28

36 Feedback optimization on the power flow manifold 200 Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Load profiles [MW] 1,500 Feedback OPF Optimal cost Generation cost 100 1, Time [hrs] Time [hrs] 400 Active power injection [MW] Bus voltages [p.u] 300 Gen 1 Gen 2 Gen 3 p Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus Time [hrs] Time [hrs] Reactive power injection [MVAR] Gen 1 Gen 2 Gen 3 Optimal tracking Time [hrs] Feasible trajectory Joint economic dispatch / volt-var optimization A. Hauswirth, A. Zanardi, S. Bolognani, F. Dörfler, & G. Hug (2017) Online Optimization in Closed Loop on the Power Flow Manifold 29