Stability and power sharing in microgrids

Stability and power sharing in microgrids J. Schiffer1 , R. Ortega2 , J. Raisch1,3 , T. Sezi4 Joint work with A. Astolfi and T. Seel 1 Control 2 Systems Group, TU Berlin 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, 24 2014 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Renewables change power system structure Generation Transmission and distribution SG .. . High voltage (HV) transmission system SG DG synchronous generator distributed generation unit SG Consumption Medium voltage (MV) distribution system Large loads Low voltage (LV) distribution system Small loads 3 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Renewables change power system structure Generation Transmission and distribution SG .. . High voltage (HV) transmission system SG DG synchronous generator distributed generation unit SG Consumption DG .. . Medium voltage (MV) distribution system Large loads Low voltage (LV) distribution system Small loads DG DG .. . DG 3 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook The microgrid concept Main electrical network PCC Transformer PV PV 1 Load 2 4 Load 8 Load PV 3 PV 11 Load Load 7 10 PV PV Storage Load FC Load 5 Storage Wind 9 PV FC Load SG 6 Load PV FC Load 5 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Basic functionality of DC-AC voltage inverters vAC,2 vAC,1 vDC t t t L vDC vAC,1 C1 C2 vAC,2 Power electronics ∼ = Inverter 8 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Inverter operated in grid-forming mode vDC 2 Rf 1 Lf vDC vDC 2 v Ia Rg Lg v Ga v Ib vG b vIc vGc Cf Rf 2 9 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Inverter operated in grid-forming mode vDC 2 Rf 1 Lf vDC Digital signal processor (DSP) vDC 2 v Ia Rg Lg v Ga v Ib vG b vIc vGc Cf Rf 2 Modulator Current controller iI iref Voltage controller vI vref 9 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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) vref Inverter with LC filter and inner control loops vIa vI b vIc Rg Lg vGa vG b vGc 10 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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) vref Inverter with LC filter and inner control loops vIa vI b vIc Rg Lg vGa vG b vGc 10 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Model of a single grid-forming inverter Inverter dynamics α̇i = uiδ , τP Ṗim = −Pim + Pi , i Vi = uiV , τP Q̇im = −Qim + Qi i Inverter output = b symmetric three-phase voltage  yi = vI abci uiδ uiV control inputs Pi Qi active power reactive power Pim measured active power measured reactive power Qim  sin(αi )   = Vi sin(αi − 2π 3 ) sin(αi + 2π 3 ) τP i time constant of meas. filter ⇒ MIMO (multiple-input-multiple-output) system 11 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook The dq0-transformation Let φ : R≥0 → T dq0-transformation Tdq0 : T → R3 ,  r Tdq0 (φ) := cos(φ) 2  sin(φ) √ 3 2 2 cos(φ − 32 π) sin(φ√− 23 π) 2 2  cos(φ + 32 π) sin(φ√+ 23 π)   2 2 dq0-transformation applied to symmetric three-phase signal     sin(α) xa     xabc = xb  = A sin(α − 2π 3 ) 2π xc sin(α + 3 ) xdq0 ⇓     r xd sin(α − φ) 3     = xq  = Tdq0 (φ)xabc = A cos(α − φ) 2 x0 0 ⇒ x0 = 0 for all t ≥ 0 12 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook The dq0-transformation Let φ : R≥0 → T dq0-transformation Tdq0 : T → R3 ,  r Tdq0 (φ) := cos(φ) 2  sin(φ) √ 3 2 2 cos(φ − 32 π) sin(φ√− 23 π) 2 2  cos(φ + 32 π) sin(φ√+ 23 π)   2 2 dq0-transformation applied to symmetric three-phase signal     sin(α) xa     xabc = xb  = A sin(α − 2π 3 ) 2π xc sin(α + 3 ) xdq0 ⇓     r xd sin(α − φ) 3     = xq  = Tdq0 (φ)xabc = A cos(α − φ) 2 x0 0 ⇒ x0 = 0 for all t ≥ 0 12 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Model of a grid-forming inverter in dq-coordinates Inverter dynamics in new coordinates δ̇i = ωi − ω com = uiδ − ω com , τP Ṗim = −Pim + Pi , i Vi = uiV , τP Q̇im = −Qim + Qi i Inverter output in new coordinates " ydq = Vi i t Z δi := αi − φ = α0i + Z φ= 0 t ω com dτ, 0 sin(δi ) cos(δi ) # α̇i − ω com dτ = α0i + Z 0 t ωi − ω com dτ ∈ T, ω com ∈ R (e.g. synchronous frequency) 13 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Power flow equations Active power flow at i -th node Pi (δ1 , . . . , δn , V1 , . . . , Vn ) = Gii Vi2 − X (Gik cos(δik ) − |Bik | sin(δik )) Vi Vk k ∼Ni Reactive power flow at i -th node Qi (δ1 , . . . , δn , V1 , . . . , Vn ) = |Bii |Vi2 − X (Gik sin(δik ) + |Bik | cos(δik )) Vi Vk k ∼Ni neighbors of node i conductance between nodes i and k ∈ Ni Ni ⊆ N ⊆ [0, n] ∩ N Gik ∈ R>0 P Gii = Ĝii + k∼N Gik susceptance between nodes i and k ∈ Ni Bik ∈ R<0 P |Bii | = |B̂ii | + k ∼N |Bik | i i 14 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Example network { δ2 , V2 } { δ5 , V5 } Σ2 P12 , Q12 { δ1 , V1 } P45 , Q45 P21 , Q21 Σ1 P32 , Q32 P23 , Q23 { δ3 , V3 } P71 , Q71 { δ4 , V4 } Σ4 Σ5 P54 , Q54 Σ3 P64 , Q64 P17 , Q17 P46 , Q46 P36 , Q36 { δ8 , V8 } Σ8 P47 , Q47 P74 , Q74 P78 , Q78 P87 , Q87 Σ7 P63 , Q63 Σ6 { δ6 , V6 } { δ7 , V7 } inverters in grid-forming mode represented by Σi , i = 1, . . . , 8 15 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 γi , γk , χi and χk Proportional active, respectively reactive, power sharing between units at nodes i ∈ N and k ∈ N , if Pis Ps = k, γi γk respectively Qis Qs = k χi χk Power sharing... allows to prespecify utilization of units avoids high circulating currents in network 16 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Power sharing is an agreement problem N⊆N U = diag(1/γi ), i ∈N D = diag(1/χi ), i ∈N Control objective lim UPN (δ, V ) = υ1|N| , t→∞ lim DQN (δ, V ) = β1|N| , t→∞ υ ∈ R>0 , β ∈ R>0 17 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 (δ̇ 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Power flows over a power line Assumptions Dominantly inductive power line with admittance Yik ∈ C between nodes i and k Small phase angle differences, i.e. δi − δk ≈ 0 ⇒ Approximations Yik = Gik + jBik ≈ jBik , sin(δik ) ≈ δik , cos(δik ) ≈ 1 ⇒ Active and reactive power flows simplify to Pik = −Bik Vi Vk δik , Qik = −Bik Vi2 + Bik Vi Vk = −Bik Vi (Vi − Vk ) 19 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Standard droop control for inverters kP ∈ R>0 i Frequency droop control uiδ = ω d − kP (Pim − Pid ) i Voltage droop control uiV = Vid − kQ (Qim − Qid ) ω d ∈ R>0 Pid ∈ R Pim ∈ R kQ ∈ R>0 i i Vid ∈ R>0 Further details: see, e.g., Chandorkar et al. (1993) Qid ∈ R Qim ∈ R frequency droop gain desired (nominal) frequency active power setpoint active power measurement voltage droop gain desired (nominal) voltage amplitude reactive power setpoint reactive power measurement 20 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Droop control - schematic representation − kP + − i Pid R δi + ω d Vi − kQ + − i + Qid Qim Pim Inverter with LC filter and inner control loops vIa Rg vI b vIc Lg v Ga vG b vGc Vid Low-pass filter Power calculation 21 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Closed-loop droop-controlled microgrid δ̇i = ω d − kP (Pim − Pid ) − ω com , i δ = col(δi ) ∈ Rn + Pi , ω = col(ωi ) ∈ Rn Vi = Vid − kQ (Qim − Qid ), V = col(Vi ) ∈ Rn τP Ṗim i = −Pim i τP Q̇im = −Qim + Qi i ⇓ change of variables + vector notation ⇓ δ̇ = ω − 1n ω com , T ω̇ = −ω + 1n ω d − KP (P − P d ), V d = col(Vid ) ∈ Rn T = diag(τP ) ∈ Rn×n i KP = diag(kP ) ∈ Rn×n i KQ = diag(kQ ) ∈ Rn×n i P = col(Pi ) ∈ Rn Q = col(Qi ) ∈ Rn P d = col(Pid ) ∈ Rn Q d = col(Qid ) ∈ Rn T V̇ = −V + V d − KQ (Q − Q d ) 22 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Closed-loop droop-controlled microgrid δ̇i = ω d − kP (Pim − Pid ) − ω com , i δ = col(δi ) ∈ Rn + Pi , ω = col(ωi ) ∈ Rn Vi = Vid − kQ (Qim − Qid ), V = col(Vi ) ∈ Rn τP Ṗim i = −Pim i τP Q̇im = −Qim + Qi i ⇓ change of variables + vector notation ⇓ δ̇ = ω − 1n ω com , T ω̇ = −ω + 1n ω d − KP (P − P d ), V d = col(Vid ) ∈ Rn T = diag(τP ) ∈ Rn×n i KP = diag(kP ) ∈ Rn×n i KQ = diag(kQ ) ∈ Rn×n i P = col(Pi ) ∈ Rn Q = col(Qi ) ∈ Rn P d = col(Pid ) ∈ Rn Q d = col(Qid ) ∈ Rn T V̇ = −V + V d − KQ (Q − Q d ) 22 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Port-Hamiltonian systems ẋ = (J(x) − R(x)) ∇H + g(x)u, y = g T (x)∇H, x ∈ X ⊆ Rn , u ∈ Rm , y ∈ Rm J(x) ∈ Rn×n , J(x) = −J(x)T (interconnection matrix) R(x) ≥ 0 ∈ Rn×n for all x ∈ X (damping matrix) H : X → R (Hamiltonian) T ∇H = ∂H ∂x Power balance equation Ḣ |{z} stored power = − ∇H T R(x)∇H | {z } dissipated power + uT y |{z} ≤ uT y supplied power 25 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Assumptions Lossless admittances, i.e. Gik = 0, i ∼ N , k ∼ N Power balance feasibility There exist constants δ s ∈ Θ, ω s ∈ R and V s ∈ Rn>0 , where n o s π | < , i ∼ N , k ∼ Ni , Θ := δ ∈ Tn |δik 2 such that 1n ω s − 1n ω d + KP [P(δ s , V s ) − P d ] = 0n , V s − V d + KQ [Q(δ s , V s ) − Q d ] = 0n 26 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Synchronized motion Synchronized motion of microgrid starting in (δ s , 1n ω s , V s ) ( ∗ δ (t) = mod2π s δ + 1n s !) t Z ω t− ω com (τ )dτ , 0 ω ∗ (t) = 1n ω s , V ∗ (t) = V s Aim: derive conditions, under which synchronized motion is attractive 27 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Power flows depend on angle differences δik Pi (δ1 , . . . , δn , V1 , . . . , Vn ) = X |Bik | sin(δik )Vi Vk , k∼Ni Qi (δ1 , . . . , δn , V1 , . . . , Vn ) = |Bii |Vi2 − X |Bik | cos(δik )Vi Vk k∼Ni ⇒ Flow of system can be described in reduced angle coordinates ⇒ Transform convergence problem into classical stability problem 28 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook A condition for local asymptotic stability Proposition Fix τPi , kPi , ω d and Pid If Vid , kQi and Qid are chosen such that D + T − W > L−1 W > 0 ⇒ x s is locally asymptotically stable L > 0, T > 0, D = diag Vid + kQ Qid i kQ (Vis )2 i ! >0 29 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Physical interpretation of stability condition D + T − W > L−1 W > 0 L= b network coupling strengths between θ and P T = b network coupling strengths between V and Q But P = P(δ, V ) 6= P(δ) and Q = Q(δ, V ) 6= Q(V ) ⇔ W= b cross-coupling strength Stability condition: couplings represented by L and T have to dominate over cross-couplings contained in W 30 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 kP γi = kP γk i k and kP Pid = kP Pkd , i k i ∼ N, k ∼N Stability condition + parameter selection criterion ⇒ limt→∞ UP(δ, V ) = υ1|N| , U = diag(1/γi ), i ∈ N, υ∈R 31 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Distributed voltage control (DVC) Alternative: consensus-based distributed voltage control (DVC) uiV = Vid − ki ei = t Z 0 ei (τ )dτ, X Qm Qm i − k χi χk k∼Ci More on reactive power sharing? ⇒ visit our talk ”A Consensus-Based Distributed Voltage Control for Reactive Power Sharing in Microgrids”, Thursday, June, 26, 10.00–10.20, Session Th A9 ”Power Systems” 33 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook CIGRE MV benchmark model Main electrical network 110/20 kV PCC 1 2 ∼ = 3 ∼ = 4 ∼ = 8 5a ∼ = 5b ∼ = 5c ∼ = 11 5 ∼ = 10a 10b 10 10c 9a ∼ = ∼ = 9b 9 ∼ = 9c ∼ = 7 ∼ = 6 ∼ = ∼ = ∼ = Photovoltaic plant (PV) Storage Fuel cell (FC) Wind power plant ∼ Inverter = Load 34 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook Simulation example ·10−2 V [pu] ∆f [Hz] 1.03 5 0 −5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1.02 1.01 1 0.99 0 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 0.1 Q [pu] P [pu] 0.3 0.2 0.1 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.4 Q/S N [-] P/S N [-] 1 0.8 0.6 0.4 0.2 00 0.3 0.2 0.1 0.5 1 1.5 2 2.5 t [s] 3 3.5 4 4.5 5 0 t [s] 35 / 38 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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 Motivation Microgrids Problem statement Stability and power sharing Simulation example Conclusions and outlook 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, 2013 38 / 38

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