Architectures and Algorithms for Distributed Generation Control of Inertia-Less AC Microgrids

Transcription

1 Architectures and Algorithms for Distributed Generation Control of Inertia-Less AC Microgrids Alejandro D. Domínguez-García Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign PSERC Webinar November 15, 2016

2 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 1 of 35

3 Microgrid Notion A group of loads and distributed energy resources (DERs) interconnected via an electrical network with a small physical footprint with the possibility of operating: M1. as part of a large power system [Grid-connected mode] M2. as an autonomous power system [Islanded mode] Examples of Distributed Energy Resources (DERs) PV systems Electric Vehicles Fuel Cells Residential Storage A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 2 of 35

4 G tie line G bulk grid microgrid Grid-Connected: May be viewed as a single entity with the idea of controlling the DERs within its boundaries to provide services to the bulk grid A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 3 of 35

5 G tie line G bulk grid microgrid Grid-Connected: May be viewed as a single entity with the idea of controlling the DERs within its boundaries to provide services to the bulk grid Islanded: Enables consumers to maintain electricity supply by appropriately controlling the DERs within its boundaries A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 3 of 35

6 G tie line G bulk grid microgrid Grid-Connected: May be viewed as a single entity with the idea of controlling the DERs within its boundaries to provide services to the bulk grid Islanded: Enables consumers to maintain electricity supply by appropriately controlling the DERs within its boundaries Different control objectives for each operational mode and each type of DER A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 3 of 35

7 G tie line G bulk grid microgrid Grid-Connected: May be viewed as a single entity with the idea of controlling the DERs within its boundaries to provide services to the bulk grid Islanded: Enables consumers to maintain electricity supply by appropriately controlling the DERs within its boundaries Different control objectives for each operational mode and each type of DER Focus of the Talk Control of inverter-interfaced DERs in islanded inertia-less AC microgrids A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 3 of 35

8 Architectural Solutions Centralized: Requires communication between a central processor and the various generation resources (and possibly loads) Requires up-to-date knowledge of generation resource availability Subject to failures at the decision maker (single-point-of-failure) Distributed: Inherent ability to handle incomplete global knowledge Potential resiliency to faults and/or unpredictable behavior n Controller 3 2 i n 5 4 i Centralized Distributed A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Introduction 4 of 35

9 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Preliminaries 5 of 35

10 Modeling Assumptions A1. Short lossless lines A2. Balanced three-phase sinusoidal regime A3. Generators and loads interfaced through voltage source inverters A4. Inverter frequency and voltage magnitude droop-based controls A5. All quantities in p.u. A6. Base voltages = voltage control reference values A7. Network, inverter filter, and voltage controller dynamics are the fastest A8. Voltage droop control much faster than frequency droop control A9. Voltage and current inner controller much faster than droop controls A10. Inverter reactive power capability sufficient to support voltage control Assumptions A1, A5 A10 = bus voltages approximately equal to 1 p.u. A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Preliminaries 6 of 35

11 Generator Buses For each generator bus i V (g) p D i dθ i (t) dt = u i (t) u i u i (t) u i, := {1, 2,..., m}: j N p(i) B ij sin(θ i (t) θ j (t)), where N p (i) is the set of buses to which bus i is electrically connected θ i(t) u i(t) u i, u i D i B ij Phase angle in reference frequency coordinate frame Active power set point Set point lower and upper limits Frequency control droop coefficient Absolute value of susceptance of line connecting nodes i and j A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Preliminaries 7 of 35

12 Load Buses For each load bus i V (l) p D i dθ i (t) dt := {m + 1, m + 2,..., n}: = (l 0 i + l i (t)) j N p(i) B ij sin(θ i (t) θ j (t)), where N p (i) is the set of buses to which bus i is electrically connected θ i(t) l 0 i l i(t) D i B ij Phase angle in reference frequency coordinate frame Nominal active power demand Active power demand perturbation Frequency control droop coefficient Absolute value of susceptance of line connecting nodes i and j A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Preliminaries 8 of 35

13 Generation Control Objectives O1. For l 0 i, i V(l) p P1. i V (g) p, find constant generator set points, u i, i V(g) p, so that u i = i V (l) p l 0 i, u i u i u i, i V (g) p, and there is a corresponding equilibrium point, θ i, i V(g) p V p (l), satisfying P2. θ i θ j φ for all i, j electrically connected, and φ [0, π/2) O2. For sufficiently small changes in l i (t), i V p (l), regulate the value of u i (t), i V p (g) around u i, i V(g) p so that P3. dθ i (t) dt 0 as t A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Preliminaries 9 of 35

14 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 10 of 35

15 Overview of Frequency Regulation Architecture For given l 0 i s, u i s are computed to meet the properties in Objective O1 After a load perturbation occurs, set points are iteratively adjusted around the u i s via closed-loop control These adjustments serve to eliminate the frequency error that results from the load perturbation In general, load perturbations and subsequent set point adjustments move the system away from the stable equilibrium point corresponding to the u i s The u i s are recomputed based upon an estimate for the amount by which the system has deviated from the aforementioned stable equilibrium point A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 11 of 35

16 Choosing Set Points For l 0 i, i V(l) p, find constant generator set points, u i, i V(g) p, so that P1. i V (g) p u i = i V (l) p l 0 i, u i u i u i, i V (g) p and there is a corresponding equilibrium point, θ i, i V(g) p V p (l), satisfying P2. θ i θ j φ, for all i, j, electrically connected, and φ [0, π/2) The u i s can be obtained by solving a feasibility problem involving U1. Node power balance equations U2. Set point lower and upper limits A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 12 of 35

17 Feasibility Problem Formulation Let G p = (V p, E p ) be an undirected graph, where Vp = V p (g) V p (l), Ep V p V p, with {i, j} E p if nodes i and j are electrically connected Then, the u i s can be obtained as a solution of find subject to u, θ [ ] u l 0 = MBsin(M T θ ) M T θ φ u u u, u R m Vector of active power set points θ R n Vector of phase angles l 0 R n m Vector of nominal active power demands u, u R m Vector of set point lower and upper limits M R n Ep Graph Incidence matrix for arbitrary edge orientation B R Ep Ep Diagonal matrix of line susceptance absolute values sin(m T θ ) R Ep Vector of angle across line sine s A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 13 of 35

18 Flow Space Formulation The feasibility problem is equivalent to F0: find u, θ [ ] u subject to l 0 = MBsin(M T θ ) M T θ φ u u u, F1: find u, f [ ] u subject to l 0 = Mf Narcsin(B 1 f ) = 0 sin(φ)b f sin(φ)b u u u f R Ep b R Ep N R ( Ep n+1) Ep arcsin(b 1 f ) R Ep Vector of line flows Vector of line susceptance absolute values Graph fundamental loop matrix Vector of normalized line flows arcsin s A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 14 of 35

19 Convexifying the Problem [Zholbaryssov, D-G, 16] The constraint Narcsin(B 1 f ) = 0 makes feasibility problem F1 non convex Trees: Nothing to do as N = 0 Networks with non-overlapping loops: Remove Narcsin(B 1 f ) = 0, and Replace sin(φ)b f sin(φ)b by f f f, with f = f = sin(φ) ( b N T β ), where β = [β 1,, β Ep n+1], and β i = 1 2 ( ( ) ) φ b i sin b C i 1 i b i Maximum of line susceptance absolute values along loop i b i Minimum of of line susceptance absolute values along loop i C i Number of lines in loop i A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 15 of 35

20 Set Point Computation Feasibility Problem F0: Feasibility Problem F2: find subject to u, θ [ ] u l 0 = MBsin(M T θ ) find subject to u, f [ ] u l 0 = Mf M T θ φ f f f u u u u u u Lemma For every solution (u, f) of F2, there exists a θ so that (u, θ) is a solution of F0 F2 is a special case of the class of problems studied in commodity networks F2 can be used to find set points that satisfy desired properties P1 and P2 A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 16 of 35

21 Example u1 1 f12 u2 2 f24 C1 4 u4 1 f 12 f 13 f 24 f M = f13 3 u3 f34 [ ] N = b ij = 1, {i, j} E p In this case, β 1 = 0.25 for φ = π 2 ; thus, f ij = 0.75, f ij = 0.75 Assume u 4 = 1, and u 1, u 2, and u 3 are constrained to lie within [0, 1] u = [1, 0, 0, 1] T and f = [0.6, 0.4, 0.6, 0.4] T form a solution for F2 From (u, f), we can form (u, f ) that solves F1 by adding to f a term laying in the null space of M: f = f 0.1N T = [0.5, 0.5, 0.5, 0.5] T ; clearly sin 1 (f 12) sin 1 (f 13) + sin 1 (f 24) sin 1 (f 34) = 0 Not every solution to F1 is a solution to F2 A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 17 of 35

22 Average Frequency Error Recall microgrid dynamic model: D i dθ i (t) dt D i dθ i (t) dt = u i (t) j N p(i) = (l 0 i + l i (t)) B ij sin(θ i (t) θ j (t)), j N p(i) i V (g) p B ij sin(θ i (t) θ j (t)), V (l) p Define the average frequency error at t 0 as follows ω(t) := n i=1 D i dθi(t) dt n i=1 D i Then, by adding the microgrid dynamics equations 1 ω(t) = n i=1 D u i (t) (l 0 i + l i (t)) i i V (g) p i V (l) p A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 18 of 35

23 Control Scheme [Cady, Zholbaryssov, D-G, Hadjicostis, 16] The intervals of control scheme execution are referred to as rounds Rounds are indexed by r = 0, 1, 2,... The duration of each round is T0 units of time tr := rt 0 denotes the time at the beginning of round r Assume a large number of rounds elapsed between load perturbations, i.e., (l) If the power demanded by load i V p is perturbed by l i at t = t 0, then l i(t) = l 0 i + l i for t 0 < t < r 0T 0 and r 0 sufficiently large Let u i [r] := u i (t), t r t < t r+1, then ω[r] = D u i [r] (l 0 i + l i ), where D := 1 n i=1 Di Control Objective i V (g) p i V (l) p Iteratively adjust set-points so as to drive the average frequency error to zero A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 19 of 35

24 Control Scheme [Cady, Zholbaryssov, D-G, Hadjicostis, 16 For a given u i, generator i adjusts its set point as follows: e i [r + 1] = e i [r] + κ i ω[r], u i [r] = u i + α i e i [r], where e i [0] = 0, and α i and κ i are appropriately chosen gains Let e[r] = [e 1 [r], e 2 [r],, e m [r]] T, then: [ ] [ ] ω[r + 1] β Dα T = e[r + 1] κ where β := D i V (g) p I m } {{ } :=Φ [ ] ω[r] + e[r] [ ] D l i, 0 m i V (l) p α i κ i, α := [α 1,..., α m ] T, and κ := [κ 1,..., κ m ] T A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 20 of 35

25 Gain Choice Proposition If α i and κ i for i V (g) p then are chosen such that 2 < D i V (g) p α i κ i < 0, The system is marginally stable and the average frequency error asymptotically approaches zero, i.e., ρ(φ) 1 and ω[r] 0 as r The value of the average frequency error at any round r is given by ω[r] = (1 + β) r 1 (β 1)D l i i V (l) p A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Architecture 21 of 35

26 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 22 of 35

27 Control Dependence on Global Information Recall the set-point adjustment scheme:: e i [r + 1] = e i [r] + κ i ω[r], u i [r] = u i + α i e i [r], where e i [0] = 0, and α i and κ i chosen so that 2 < D i V (g) p α i κ i < 0 In order to implement and execute this control, each generator requires some global information: Q1. Set point u i Q2. Gains α i and κ i Q3. Average frequency error ω[r] Objective Design distributed algorithms that enables the computation of Q1-Q3 A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 23 of 35

28 Cyber Layer for Distributed Computation and Control Bus local processors can perform simple computations (,, +, ) Exchange of information between local processors is described by a connected undirected graph G c = {V c, E c }: Vc := {1, 2,..., n} is the set of nodes Ec V V is the set of edges, where {i, j} E c if nodes i and j can exchange information Each element in Vc corresponds to a bus of the physical network, i.e., there is a one-to-one mapping between V c and V p Each element in Ec corresponds to a line of the physical network, i.e., there is a one-to-one mapping between E c and E p Nodes that exchange information which node i are said to be its neighbors and are represented by the set N c (i) = {i V : {i, j} E c } A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 24 of 35

29 Interaction Between Physical and Cyber Layers Cyber Layer, G c Communication Network Generator Buses, Vg Physical Layer, G p Electrical Network Transceiver 1 1 Local Processor 1 + u1(t) D1 Generator 1 = P1 1 V1 1 θ1... Transceiver m m Local Processor m + um(t) Dm Generator m = Pm m Vm 1 θm Load Buses, Vl Transceiver m + 1 m + 1 Local Processor m lm+1(t), Dm+1 Load m + 1 l m+1.. Pm+1 m + 1 Vm+1 1 θm+1. Transceiver n n Local Processor n + ln(t), Dn Load n l n Pn n Vn 1 θn A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Layer Interaction 25 of 35

30 Set Point Computation [Cady, Hadjicostis, D-G, 15] Recall feasibility problem F2: find subject to u, f [ ] u l 0 = Mf f f f u u u Distributed algorithm over graph G c for solving F2: Each node i Vp maintains and updates estimates of flows with its neighbors: [k], j Np(i), k = 0, 1, 2, f (i) ji (g) Each generator node i V p maintains an estimate for its set point: u i[k], k = 0, 1, 2, At each iteration k, each node i Vp computes its power flow balance: Algorithm Progress d i[k] := { j N f (i) p(i) ij j N p(i) f (i) ij [k] + ui[k], i [k] li, i V(g) p V(l) p Each node iteratively updates its estimates so as to drive its power flow balance to zero A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 26 of 35

31 Iteration Three-Step Process [Step 1] Estimates are adjusted to interim values using the flow balance: f (i) (i) ij [k + 1] = f ij [k] + d i [k] N p (i) + 1 d i [k] û i [k + 1] = u i [k] 2 ( N p (i) + 1) [Step 2] Pairs of neighbors average interim estimates of flow between them: ˆf (i) ij [k + 1] = 1 2 ( ) (i) (j) f ij [k + 1] f ji [k + 1] [Step 3] New estimate values are obtained by enforcing limits: f (i) ij [k + 1] = [ ˆf (i) ij [k + 1] ] f ij u i [k + 1] = [û i [k + 1]] ui u i f ij A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 27 of 35

32 Feasible Flow Algorithm Convergence Since f (i) (j) ij [k] = f ji [k] =: f ij[k], Steps 1-3 can be collapsed: f ij [k + 1] = u i [k + 1] = [ f ij [k] + 1 ( )] f d i [k] 2 N p (i) + 1 d j [k] ij N p (j) + 1 f ij (1) [ ] ui d i [k] u i [k] (2) 2( N p (i) + 1) u i Proposition Suppose there exists a solution, (u, f ), to feasibility problem F2. Then, as k, u i u i, f ij[k] f ij, d i[k] 0 Convergence proof relies on the observation that the iterations in (1) - (2) corresponds to those of a gradient descent algorithm for a quadratic program A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Distributed Implementation 28 of 35

33 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Simulation Results 29 of 35

34 Six-Node Network Case Study Generator Bus Load Bus Generator bus parameters i u i u i D i Load bus parameter i l 0 i D i Line parameters Line Index {1, 4} {1, 5} {2, 4} {2, 6} {3, 5} {3, 6} Susceptance A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Simulation Results 30 of 35

35 Initial Set-Point Computation Flow balances converge to within ±0.001 in the first 50 iterations, yielding: u 1 = 0.85 u 2 = 1.5 pu u 3 = 0.95 pu d 1 [k] d 2 [k] d 3 [k] d 4 [k] d 5 [k] d 6 [k] iterations, k A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Simulation Results 31 of 35

36 System Response to Load Perturbations l 4, l 6, and l 5 perturbed at t = 2 s, t = 4 s, and t = 6 s, respectively: l4 = pu l5 = pu l6 = pu u 1, u 2, u 3 recomputed at t = 2 s θ 1(t) θ 2(t) θ 3(t) θ 4(t) θ 5(t) θ 6(t) time, t [s] ω(t) [rad/s] time, t [s] 1.25 (a) Bus voltage angles (b) Average frequency error (c) Set-points u 1(t) u 2(t) u 3(t) u 1 u 2 u time, t [s] A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Simulation Results 32 of 35

37 Outline 1 Introduction 2 Preliminaries 3 Generation Control Architecture 4 Distributed Implementation 5 Simulation Results 6 Concluding Remarks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Concluding Remarks 33 of 35

38 Summary Things discussed today: Architecture for frequency regulation in islanded inertia-less AC microgrids Computation of set points guaranteed to yield a stable equilibrium point Distributed algorithm for set-point computation Things not discussed today [Cady, Zholbaryssov, D-G, Hadjicostis, 16]: Distributed computation of controller gains Distributed computingutation of average frequency error Criteria andd distributed protocol for triggering for set-point recomputation Future work: Voltage control architecture Extensions to lossy networks A. D. Domínguez-García aledan@illinois.edu Distributed Generation Control of Microgrids Concluding Remarks 34 of 35

39 Questions? Alejandro D. Domínguez-García A. D. Domínguez-García Distributed Generation Control of Microgrids Concluding Remarks 35 of 35