Stability and power sharing in microgrids

Transcription

1 Stability and power sharing in microgrids J. Schiffer 1, R. Ortega 2, J. Raisch 1,3, T. Sezi 4 Joint work with A. Astolfi and T. Seel 1 Control Systems Group, TU Berlin 2 Laboratoire des Signaux et Systémes, SUPELEC 3 Systems and Control Theory Group, Max Planck Institute for Dynamics of Complex Technical Systems 4 Siemens AG, Smart Grid Division, Energy Automation HYCON2 Workshop, ECC 2014 Strasbourg, June,

2 Outline 1 Motivation 2 Microgrids: concept and modeling 3 Problem statement 4 Asymptotic stability and power sharing in droop-controlled microgrids 5 Simulation example 6 Conclusions and outlook 2 / 38

3 Renewables change power system structure Generation SG. SG Transmission and distribution High voltage (HV) transmission system SG DG synchronous generator distributed generation unit Consumption Medium voltage (MV) distribution system Large loads Low voltage (LV) distribution system Small loads 3 / 38

4 Renewables change power system structure Generation SG. SG DG. DG Transmission and distribution High voltage (HV) transmission system Medium voltage (MV) distribution system SG DG synchronous generator distributed generation unit Consumption Large loads DG. DG Low voltage (LV) distribution system Small loads 3 / 38

5 Need change in power system operation Increasing amount of renewable DG units highly affects in-feed structure of existing power systems Most renewable DG units interfaced to network via AC inverters Physical characteristics of inverters largely differ from characteristics of SGs Different control and operation strategies are needed Source: siemens.com 4 / 38

6 The microgrid concept Main electrical network PCC PV Transformer PV Load 4 PV Load Load 11 PV Load Load PV PV Storage 10 9 FC PV 7 Load Wind Storage Load Load 5 FC SG 6 Load FC Load PV 5 / 38

7 Modeling of microgrids Main network components DG units interfaced to network via inverters or SGs Loads Power lines and transformers Standard modeling assumptions Loads can be modeled by impedances Line dynamics can be neglected Work with Kron-reduced network DG unit connected at each node in reduced network Main focus: inverter-based microgrids 6 / 38

8 Main operation modes of inverters in microgrids 1 Grid-feeding mode Inverter provides prespecified amount of active and reactive power to grid 2 Grid-forming mode Inverter is operated as AC voltage source with controllable frequency and amplitude Grid-forming units are essential components in power systems Tasks To provide a synchronous frequency To provide a certain voltage level at all buses in the network To provide a stable operating point Focus on inverters in grid-forming mode 7 / 38

9 Basic functionality of DC-AC voltage inverters v DC v AC,1 v AC,2 t t t L v DC C 1 v AC,1 C 2 v AC,2 Power electronics = Inverter 8 / 38

10 Inverter operated in grid-forming mode v DC 2 R f1 L f v Ia R g L g vga v DC v Ib v Ic v Gb v Gc v DC 2 C f R f2 9 / 38

11 Inverter operated in grid-forming mode v DC 2 R f1 L f v Ia R g L g vga v DC v Ib v Ic v Gb v Gc v DC 2 C f R f2 Digital signal processor (DSP) Modulator Current controller i ref Voltage controller i I v I v ref 9 / 38

12 Grid-forming inverter as controllable voltage source Model assumptions: Inverter is operated in grid-forming mode Its inner current and voltage controllers track references ideally If inverter connects intermittent DG unit (e.g., a PV plant) to network, it is equipped with some sort of fast-reacting storage (e.g., a flywheel or a battery) v ref Inverter with LC filter and inner control loops v I a v Ib v I c R g L g vg a v Gb v G c 10 / 38

13 Grid-forming inverter as controllable voltage source Model assumptions: Inverter is operated in grid-forming mode Its inner current and voltage controllers track references ideally If inverter connects intermittent DG unit (e.g., a PV plant) to network, it is equipped with some sort of fast-reacting storage (e.g., a flywheel or a battery) v ref Inverter with LC filter and inner control loops v I a v Ib v I c R g L g vg a v Gb v G c 10 / 38

14 Model of a single grid-forming inverter Inverter dynamics α i = ui δ, τ Pi Ṗi m = Pi m + P i, V i = ui V, τ Pi Qm i = Qi m + Q i Inverter output = symmetric three-phase voltage sin(α i ) y i = v Iabci = V i sin(α i 2π 3 ) sin(α i + 2π 3 ) ui δ ui V P i Q i Pi m Qi m τ Pi control inputs active power reactive power measured active power measured reactive power time constant of meas. filter MIMO (multiple-input-multiple-output) system 11 / 38

15 The dq0-transformation Let φ : R 0 T dq0-transformation T dq0 : T R 3, T dq0 (φ) := cos(φ) cos(φ π) cos(φ π) sin(φ) sin(φ π) sin(φ π) dq0-transformation applied to symmetric three-phase signal x 0 = 0 for all t 0 x a sin(α) x abc = x b = A sin(α 2π 3 ) x c sin(α + 2π 3 ) x d sin(α φ) 3 x dq0 = x q = T dq0 (φ)x abc = 2 A cos(α φ) x / 38

16 The dq0-transformation Let φ : R 0 T dq0-transformation T dq0 : T R 3, T dq0 (φ) := cos(φ) cos(φ π) cos(φ π) sin(φ) sin(φ π) sin(φ π) dq0-transformation applied to symmetric three-phase signal x 0 = 0 for all t 0 x a sin(α) x abc = x b = A sin(α 2π 3 ) x c sin(α + 2π 3 ) x d sin(α φ) 3 x dq0 = x q = T dq0 (φ)x abc = 2 A cos(α φ) x / 38

17 Model of a grid-forming inverter in dq-coordinates Inverter dynamics in new coordinates δ i = ω i ω com = ui δ ω com, τ Pi Ṗi m = Pi m + P i, V i = ui V, τ Pi Qm i = Qi m + Q i Inverter output in new coordinates [ ] sin(δ y dqi = V i ) i cos(δ i ) t ( δ i := α i φ = α 0i + αi ω com) t ( dτ = α 0i + ωi ω com) dτ T, 0 0 t φ = ω com dτ, 0 ω com R (e.g. synchronous frequency) 13 / 38

18 Power flow equations Active power flow at i-th node P i (δ 1,..., δ n, V 1,..., V n) = G ii V 2 i k N i (G ik cos(δ ik ) B ik sin(δ ik )) V i V k Reactive power flow at i-th node Q i (δ 1,..., δ n, V 1,..., V n) = B ii V 2 i k N i (G ik sin(δ ik ) + B ik cos(δ ik )) V i V k neighbors of node i N i N [0, n] N conductance between nodes i and k N i G ik R >0 G ii = Ĝii + k N G i ik susceptance between nodes i and k N i B ik R <0 B ii = ˆB ii + k N B i ik 14 / 38

19 Example network {δ 2,V 2} {δ 1,V 1 } Σ 1 Σ 2 P 12,Q 12 P 21,Q 21 P 32,Q 32 P 23,Q 23 {δ 4,V 4 } Σ 4 P 45,Q 45 P 54,Q 54 {δ 5,V 5} Σ 5 {δ 3,V 3} Σ 3 P71,Q 71 P 17,Q 17 P 64,Q 64 P 46,Q 46 {δ 8,V 8} P 47,Q 47 P 36,Q 36 Σ 8 P78,Q 78 P 87,Q 87 Σ 7 P 74,Q 74 P 63,Q 63 Σ 6 {δ 6,V 6} {δ 7,V 7} inverters in grid-forming mode represented by Σ i, i = 1,..., 8 15 / 38

20 Power sharing in microgrids Definition (Power sharing) Consider an AC electrical network, e.g. an AC microgrid Denote its set of nodes by N = [1, n] N Choose positive real constants γ i, γ k, χ i and χ k Proportional active, respectively reactive, power sharing between units at nodes i N and k N, if P s i γ i = Ps k γ k, respectively Q s i χ i = Qs k χ k Power sharing... allows to prespecify utilization of units avoids high circulating currents in network 16 / 38

21 Power sharing is an agreement problem N N U = diag(1/γ i ), i N D = diag(1/χ i ), i N Control objective lim UP N(δ, V ) = υ1 N, t lim t DQ N(δ, V ) = β1 N, υ R >0, β R >0 17 / 38

22 Motivation for droop control of inverters 1 Droop control is widely used control solution in SG-based power systems to address problems of frequency stability and active power sharing Adapt droop control to inverters Make inverters mimick behavior of SGs with respect to frequency and active power 2 How to couple actuactor signals ( δ and V) with powers (P and Q) to achieve power sharing? Pose MIMO control design problem as set of decoupled SISO control design problems Analyze couplings in power flow equations over a power line 18 / 38

23 Power flows over a power line Assumptions Dominantly inductive power line with admittance Y ik C between nodes i and k Small phase angle differences, i.e. δ i δ k 0 Approximations Y ik = G ik + jb ik jb ik, sin(δ ik ) δ ik, cos(δ ik ) 1 Active and reactive power flows simplify to P ik = B ik V i V k δ ik, Q ik = B ik V 2 i + B ik V i V k = B ik V i (V i V k ) 19 / 38

24 Standard droop control for inverters Frequency droop control u δ i = ω d k Pi (P m i P d i ) Voltage droop control u V i = V d i k Qi (Q m i Q d i ) Further details: see, e.g., Chandorkar et al. (1993) k Pi R >0 frequency droop gain ω d R >0 desired (nominal) frequency Pi d R active power setpoint Pi m R active power k Qi R >0 measurement voltage droop gain V d i R >0 desired (nominal) voltage amplitude Q d i R reactive power setpoint Q m i R reactive power measurement 20 / 38

25 Droop control - schematic representation + P d i k Pi + Qi d + ω d k Qi + V d i δ i V i Inverter with LC filter and inner control loops v I a v Ib v I c R g L g vg a v Gb v G c Q m i P m i Low-pass filter Power calculation 21 / 38

26 Closed-loop droop-controlled microgrid δ i = ω d k Pi (P m i P d i ) ω com, τ Pi Ṗ m i = P m i + P i, V i = V d i k Qi (Q m i Q d i ), τ Pi Qm i = Q m i + Q i change of variables + vector notation δ = ω 1 n ω com, T ω = ω + 1 n ω d K P (P P d ), T V = V + V d K Q (Q Q d ) δ = col(δ i ) R n ω = col(ω i ) R n V = col(v i ) R n V d = col(v d i ) R n T = diag(τ Pi ) R n n K P = diag(k Pi ) R n n K Q = diag(k Qi ) R n n P = col(p i ) R n Q = col(q i ) R n P d = col(p d i ) R n Q d = col(q d i ) R n 22 / 38

27 Closed-loop droop-controlled microgrid δ i = ω d k Pi (P m i P d i ) ω com, τ Pi Ṗ m i = P m i + P i, V i = V d i k Qi (Q m i Q d i ), τ Pi Qm i = Q m i + Q i change of variables + vector notation δ = ω 1 n ω com, T ω = ω + 1 n ω d K P (P P d ), T V = V + V d K Q (Q Q d ) δ = col(δ i ) R n ω = col(ω i ) R n V = col(v i ) R n V d = col(v d i ) R n T = diag(τ Pi ) R n n K P = diag(k Pi ) R n n K Q = diag(k Qi ) R n n P = col(p i ) R n Q = col(q i ) R n P d = col(p d i ) R n Q d = col(q d i ) R n 22 / 38

28 Problem statement 1 Derive conditions for asymptotic stability of generic droop-controlled microgrids 2 Investigate if droop control is suitable to achieve control objective of power sharing 23 / 38

29 Stability analysis Coordinate transformation Follow interconnection and damping assignment passivity-based control (IDA-PBC) approach (Ortega et al. (2002)) Represent microgrid dynamics in port-hamiltonian form Can easily identify energy function 24 / 38

30 Port-Hamiltonian systems ẋ = (J(x) R(x)) H + g(x)u, x X R n, u R m, y = g T (x) H, y R m J(x) R n n, J(x) = J(x) T (interconnection matrix) R(x) 0 R n n for all x X (damping matrix) H : X R (Hamiltonian) ( ) T H = H x Power balance equation }{{} Ḣ = H T R(x) H }{{} stored power dissipated power + u T y }{{} u T y supplied power 25 / 38

31 Assumptions Lossless admittances, i.e. G ik = 0, i N, k N Power balance feasibility There exist constants δ s Θ, ω s R and V s R n >0, where such that Θ := { δ T n δik s < π } 2, i N, k N i, 1 n ω s 1 n ω d + K P [P(δ s, V s ) P d ] = 0 n, V s V d + K Q [Q(δ s, V s ) Q d ] = 0 n 26 / 38

32 Synchronized motion Synchronized motion of microgrid starting in (δ s, 1 n ω s, V s ) { )} t δ (t) = mod 2π δ s + 1 n (ω s t ω com (τ)dτ, 0 ω (t) = 1 n ω s, V (t) = V s Aim: derive conditions, under which synchronized motion is attractive 27 / 38

33 Power flows depend on angle differences δ ik P i (δ 1,..., δ n, V 1,..., V n) = Q i (δ 1,..., δ n, V 1,..., V n) = B ii V 2 i k N i B ik sin(δ ik )V i V k, k N i B ik cos(δ ik )V i V k Flow of system can be described in reduced angle coordinates Transform convergence problem into classical stability problem 28 / 38

34 A condition for local asymptotic stability Proposition Fix τ Pi, k Pi, ω d and P d i If V d i, k Qi and Q d i are chosen such that D + T W L 1 W > 0 x s is locally asymptotically stable L > 0, T > 0, D = diag ( V d i + k Qi Qi d k Qi (Vi s ) 2 ) > 0 29 / 38

35 Physical interpretation of stability condition D + T W L 1 W > 0 L = network coupling strengths between θ and P T = network coupling strengths between V and Q But P = P(δ, V ) P(δ) and Q = Q(δ, V ) Q(V ) W = cross-coupling strength Stability condition: couplings represented by L and T have to dominate over cross-couplings contained in W 30 / 38

36 A condition for active power sharing Lemma (Proportional active power sharing) Assume microgrid possesses synchronized motion Then all generation units share active power proportionally in steady-state if k Pi γ i = k Pk γ k and k Pi P d i = k Pk P d k, i N, k N Stability condition + parameter selection criterion lim t UP(δ, V ) = υ1 N, U = diag(1/γ i ), i N, υ R 31 / 38

37 Reactive power sharing Voltage droop control does, in general, not guarantee desired reactive power sharing Several other or modified (heuristic) decentralized voltage control strategies proposed in literature (Zhong (2013), Li et al. (2009), Sao et al. (2005), Simpson-Porco et al. (2014),...) But: no general conditions or formal guarantees for reactive power sharing are given 32 / 38

38 Distributed voltage control (DVC) Alternative: consensus-based distributed voltage control (DVC) u V i = V d i e i = k C i t k i e i (τ)dτ, 0 ( Q m i χ i ) Qm k χ k More on reactive power sharing? visit our talk A Consensus-Based Distributed Voltage Control for Reactive Power Sharing in Microgrids, Thursday, June, 26, , Session Th A9 Power Systems 33 / 38

39 CIGRE MV benchmark model Main electrical network 110/20 kv 1 PCC 3 2 = 5 4 5a 5b 5c = = = 11 = = 10 10a 10b = = 9 9a 9b = = = = = 10c = 9c = Photovoltaic plant (PV) Fuel cell (FC) Storage Wind power plant = Inverter Load 34 / 38

40 Simulation example f [Hz] V [pu] P [pu] Q [pu] P/S N [-] t [s] Q/S N [-] t [s] 35 / 38

41 Conclusions and outlook Microgrids are a promising concept in networks with large amount of DG Condition for local asymptotic stability in lossless droop-controlled inverter-based microgrids Selection criterion for parameters of frequency droop control that ensures desired active power sharing in steady-state Proposed distributed voltage control (DVC), which solves open problem of reactive power sharing 36 / 38

42 Outlook and related work Analysis of lossy networks with variable frequencies and voltages Analysis of microgrids with frequency droop control and distributed voltage control (DVC) Control schemes for highly ohmic networks Secondary frequency control (Simpson-Porco et al. (2013), Bürger et al. (2014), Andreasson et al. (2012), Bidram et al. (2013), Shafiee et al. (2014)) Optimal operation control (Dörfler et al. (2014), Bolognani et al. (2013), Hans et al.(2014)) Alternative inverter control schemes (Torres et al. (2014), Dhople et al. (2014)) 37 / 38

43 Publications Schiffer, Johannes and Ortega, Romeo and Astolfi, Alessandro and Raisch, Jörg and Sezi, Tevfik Conditions for Stability of Droop-Controlled Inverter-Based Microgrids, Automatica, 2014 Schiffer, Johannes and Ortega, Romeo and Astolfi, Alessandro and Raisch, Jörg and Sezi, Tevfik Stability of Synchronized Motions of Inverter-Based Microgrids Under Droop Control, To appear at 19th IFAC World Congress, Cape Town, South Africa Schiffer, Johannes and Seel, Thomas and Raisch, Jörg and Sezi, Tevfik, Voltage Stability and Reactive Power Sharing in Inverter-Based Microgrids with Consensus-Based Distributed Voltage Control Submitted to IEEE Transactions on Control Systems Technology, 2014 Schiffer, Johannes and Seel, Thomas and Raisch, Jörg and Sezi, Tevfik, A Consensus-Based Distributed Voltage Control for Reactive Power Sharing in Microgrids 13th ECC, Strasbourg, France, 2014 Schiffer, Johannes and Goldin, Darina and Raisch, Jörg and Sezi, Tevfik Synchronization of Droop-Controlled Microgrids with Distributed Rotational and Electronic Generation, IEEE 52nd CDC, Florence, Italy, / 38