Dispatchable Virtual Oscillator Control: Theory and Experiments

Transcription

1 Dispatchable Virtual Oscillator Control: Theory and Experiments D. Groß Automatic Control Laboratory, ETH Zurich

2 Acknowledgements Control Design & Stability analysis: Marcello Colombino (NREL) Irina Subotic (ETH) Florian Dörfler (ETH) Jean Sebastien Brouillon Experimental validation: Gab-Su Seo (NREL) Marcello Colombino (NREL) Irina Subotic (ETH) Brian B. Johnson (University of Washington) High-fidelity numerical case study: Thibault Prevost (RTE) Guillaume Denis (RTE) Uros Markovic (ETH) Taoufik Qoria (LE2P) 1

3 What do we see here? Hz BEWAG UCTE *10 sec 2

4 Frequency of West Berlin re-connecting to Europe Hz December 7, 1994 BEWAG UCTE *10 sec before re-connection: islanded operation based on batteries & single boiler afterwards connected to European grid based on synchronous generation 3

5 The foundation of today s power system 4

6 The foundation of today s power system Synchronous machines with rotational inertia M d dt ω P generation P demand 4

7 The foundation of today s power system Synchronous machines with rotational inertia M d dt ω P generation P demand Today s grid operation heavily relies on 1. kinetic energy 1 2 Mω2 as safeguard against disturbances 2. self-synchronization of machines through the grid 3. robust stabilization of frequency and voltage by generator controls 4

8 The foundation of today s power system Synchronous machines with rotational inertia M d dt ω P generation P demand Today s grid operation heavily relies on 1. kinetic energy 1 2 Mω2 as safeguard against disturbances 2. self-synchronization of machines through the grid 3. robust stabilization of frequency and voltage by generator controls We are replacing this solid foundation with... 4

9 We are replacing the foundation of today s system synchronous machines renewables & power electronics 5

10 We are replacing the foundation of today s system synchronous machines + large rotational inertia renewables & power electronics no rotational inertia 5

11 We are replacing the foundation of today s system synchronous machines + large rotational inertia + kinetic energy 1 2 Mω2 as buffer renewables & power electronics no rotational inertia almost no energy storage 5

12 We are replacing the foundation of today s system synchronous machines + large rotational inertia + kinetic energy 1 2 Mω2 as buffer + self-synchronize through grid renewables & power electronics no rotational inertia almost no energy storage no inherent self-synchronization 5

13 We are replacing the foundation of today s system synchronous machines + large rotational inertia + kinetic energy 1 2 Mω2 as buffer + self-synchronize through grid + robust control of voltage & freq. renewables & power electronics no rotational inertia almost no energy storage no inherent self-synchronization fragile control of voltage & freq. 5

14 We are replacing the foundation of today s system synchronous machines + large rotational inertia + kinetic energy 1 2 Mω2 as buffer + self-synchronize through grid + robust control of voltage & freq. slow primary control renewables & power electronics no rotational inertia almost no energy storage no inherent self-synchronization fragile control of voltage & freq. + fast actuation & control 5

15 We are replacing the foundation of today s system synchronous machines + large rotational inertia + kinetic energy 1 2 Mω2 as buffer + self-synchronize through grid + robust control of voltage & freq. slow primary control renewables & power electronics no rotational inertia almost no energy storage no inherent self-synchronization fragile control of voltage & freq. + fast actuation & control what could possibly go wrong? 5

16 The concerns are not hypothetical UPDATE REPORT! BLACK SYSTEM EVENT IN SOUTH AUSTRALIA ON 28 SEPTEMBER 2016 AN UPDATE TO THE PRELIMINARY OPERATING INCIDENT REPORT FOR THE NATIONAL ELECTRICITY MARKET. DATA ANALYSIS AS AT 5.00 PM TUESDAY 11 OCTOBER lack of robust control: Nine of the 13 wind farms online did not ride through the six voltage disturbances experienced during the event. 6

17 The concerns are not hypothetical UPDATE REPORT! BLACK SYSTEM EVENT IN SOUTH AUSTRALIA ON 28 SEPTEMBER 2016 AN UPDATE TO THE PRELIMINARY OPERATING INCIDENT REPORT FOR THE NATIONAL ELECTRICITY MARKET. DATA ANALYSIS AS AT 5.00 PM TUESDAY 11 OCTOBER lack of robust control: Nine of the 13 wind farms online did not ride through the six voltage disturbances experienced during the event. between the lines: conventional system would have been more resilient (?) 6

18 The concerns are not hypothetical issues broadly recognized by TSOs, device manufacturers, academia, agencies, etc. UPDATE REPORT! BLACK SYSTEM EVENT IN SOUTH AUSTRALIA ON 28 SEPTEMBER 2016 MIGRATE project: Massive InteGRATion of power Electronic devices!!"#$%% "&'()*%")+,-.)'%/),-)0% 1"2%/).3**)456(-34'% AN UPDATE TO THE PRELIMINARY OPERATING INCIDENT REPORT FOR THE NATIONAL ELECTRICITY MARKET. DATA ANALYSIS AS AT 5.00 PM TUESDAY 11 OCTOBER 2016.!"#$%&!&$!&'"!()*!+$,,-&&""! Impact of Low Rotational Inertia on Power System Stability and Operation lack of robust control: Nine of the 13 wind farms online did not ride through the six voltage disturbances experienced during the event. Andreas Ulbig, Theodor S. Borsche, Göran Andersson ETH Zurich, Power Systems Laboratory Physikstrasse 3, 8092 Zurich, Switzerland ulbig borsche eeh.ee.ethz.ch ERCOT CONCEPT PAPER PUBLIC Future Ancillary Services in ERCOT Frequency Stability Evaluation Criteria for the Synchronous Zone of Continental Europe ERCOT is recommending the transition to the following five AS products plus one additional AS that would be used during some transition period: Synchronous Inertial Response Service (SIR), Fast Frequency Response Service (FFR), Primary Frequency Response Service (PFR), Renewable and Sustainable Energy (RR), Reviews 55 (2016) Up and Down Regulating Reserve Service and Contingency Reserve Service (CR). Contents lists available at ScienceDirect Supplemental Reserve Service (SR) (during transition period) Requirements and impacting factors RG-CE System Protection & Dynamics Sub Group Renewable and Sustainable Energy Reviews journal homepage: between the lines: The relevance of inertia in power systems n Pieter Tielens, Dirk Van Hertem conventional system would have been more resilient (?) ELECTA, Department of Electrical Engineering (ESAT), University of Leuven (KU Leuven), Leuven, Belgium and EnergyVille, Genk, Belgium However, as these sources are fully controllable, a regulation can be added to the inverter to provide synthetic inertia. This can also be seen as a short term frequency support. On the other hand, these sources might be quite restricted with respect to the available capacity and possible activation time. The inverters have a very low overload capability compared to synchronous machines. 6

19 Cartoon summary of today s approach Conceptually, inverters are oscillators that have to synchronize 7

20 Cartoon summary of today s approach Hypothetically, they could sync by communication (not feasible) 7

21 Cartoon summary of today s approach Colorful idea: inverters sync through physics & clever local control 7

22 Cartoon summary of today s approach Colorful idea: inverters sync through physics & clever local control Theory: sync of coupled oscillators & nonlinear decentralized control Experiments: hardware show superior performance Simulations: realistic power systems test cases in collaboration with RTE 7

23 Outline Problem Setup: Modeling and Specifications Dispatchable Virtual Oscillator Control Experimental Validation High-fidelity power systems simulations Conclusions

24 Modeling: the network L = Y k1 n j=1 Y kj Y kn

25 Modeling: the network L = Y k1 n j=1 Y kj Y kn Network: undirected connected graph 8

26 Modeling: the network L = Y k1 n j=1 Y kj Y kn Network: undirected connected graph Edges: lines with admittance Y kj = 1 r kj 2 +ω2 0 l2 kj 8

27 Modeling: the network L = Y k1 n j=1 Y kj Y kn Network: undirected connected graph Edges: lines with admittance Y kj = 1 r kj 2 +ω2 0 l2 kj Laplacian: L = B diag( Y l )B and extended Laplacian L := L I 2 8

28 Modeling: the network L = Y k1 n j=1 Y kj Y kn Network: undirected connected graph Edges: lines with admittance Y kj = 1 r kj 2 +ω2 0 l2 kj Laplacian: L = B diag( Y l )B and extended Laplacian L := L I 2 Local measurement of current Information on relative voltages R(κ)i a o,k = j k Y kj (v k v j), κ := tan 1 ( l kj r kj ω 0) 8

29 Modeling: the power converter v dc i dc i L R 1 G C v 2 v dc u network i o DC port modulation control (3-phase) LC output filter AC port to power grid 9

30 Modeling: the power converter v dc i dc i L R 1 G C v 2 v dc u network i o DC port modulation control (3-phase) LC output filter AC port to power grid passive DC port port (i dc, v dc ) for energy balance control details neglected today: assume v dc to be stiffly regulated modulation lossless signal transformer (averaged) controlled switching voltage 1 v 2 dcu with u [ 1, 1] LC filter to smoothen harmonics with R, G modeling filter/switching losses 9

31 Control: standard power electronics approach actuation of DC source/boost DC voltage control DC voltage cascaded voltage/current tracking control converter modulation PWM reference synthesis (e.g., droop or virtual inertia) (P, Q, kv k,!) measurement processing (e.g., via PLL) AC current & voltage 1. acquiring & processing of AC measurements 2. synthesis of references (voltage/current) 3. cascaded PI controllers track reference 4. actuation via modulation and DC-side supply DC/AC power inverter 10

32 Control: standard power electronics approach actuation of DC source/boost DC voltage control DC voltage cascaded voltage/current tracking control converter modulation PWM reference synthesis (e.g., droop or virtual inertia) (P, Q, kv k,!) measurement processing (e.g., via PLL) AC current & voltage 1. acquiring & processing of AC measurements 2. synthesis of references (voltage/current) 3. cascaded PI controllers track reference 4. actuation via modulation and DC-side supply DC/AC power inverter well actuated, modular, & fast control system controllable voltage source 10

33 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} 11

34 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system 11

35 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 11

36 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 11

37 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 We aim to stabilize the steady-state behavior via decentralized control 1. nominal synchronous frequency ω 0 : v k (t) = R(ω 0t)v k (0), k N (R(θ) : 2D rotation matrix) 11

38 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 We aim to stabilize the steady-state behavior via decentralized control 1. nominal synchronous frequency ω 0 : v k (t) = R(ω 0t)v k (0), k N (R(θ) : 2D rotation matrix) 2. voltage amplitude: v k = v k, k N 11

39 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 We aim to stabilize the steady-state behavior via decentralized control 1. nominal synchronous frequency ω 0 : v k (t) = R(ω 0t)v k (0), k N (R(θ) : 2D rotation matrix) 2. voltage amplitude: v k = v, k N (for ease of presentation) 11

40 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 We aim to stabilize the steady-state behavior via decentralized control 1. nominal synchronous frequency ω 0 : v k (t) = R(ω 0t)v k (0), k N (R(θ) : 2D rotation matrix) 2. voltage amplitude: v k = v, k N (for ease of presentation) 3. active & reactive power injection p k, q k : v k i o,k = p k, v k R( π 2 )i o,k = q k, k N 11

41 Control objectives in the stationary frame Notation: N nodes N := {1,..., N} balanced three-phase system node voltage v k R 2 output current i o,k R 2 We aim to stabilize the steady-state behavior via decentralized control 1. nominal synchronous frequency ω 0 : v k (t) = R(ω 0t)v k (0), k N (R(θ) : 2D rotation matrix) 2. voltage amplitude: v k = v, k N (for ease of presentation) 3. relative voltage angles θ jk : v k (t) = R(θ k1) v 1(t), k N \ {1} 11

42 Main control challenges v 2 ω 0 ω 0 v 2 θ 12 v 1 0 v 1 0 v 12

43 Main control challenges v 2 ω 0 ω 0 v 2 θ 12 v 1 0 v 1 0 v nonlinear objectives (v k, p k, q k) & stabilization of a limit cycle 12

44 Main control challenges v 2 ω 0 ω 0 v 2 θ 12 v 1 0 v 1 0 v nonlinear objectives (v k, p k, q k) & stabilization of a limit cycle decentralized control: only local measurements (v k, i o,k ) available 12

45 Main control challenges v 2 ω 0 ω 0 v 2 θ 12 v 1 0 v 1 0 v nonlinear objectives (v k, p k, q k) & stabilization of a limit cycle decentralized control: only local measurements (v k, i o,k ) available time-scale separation between slow sources & fast network may not hold 12

46 Main control challenges v 2 ω 0 ω 0 v 2 θ 12 v 1 0 v 1 0 v nonlinear objectives (v k, p k, q k) & stabilization of a limit cycle decentralized control: only local measurements (v k, i o,k ) available time-scale separation between slow sources & fast network may not hold + fully controllable voltage sources & stable linear network dynamics 12

47 Comparison of grid-forming control strategies ω idc θ, ω ω*0! m v!" ω!sync sync C!" G!" is τm τe rs RI LI i I v Lθ P2 P1 p(t) p rf if droop synchronous machine emulation control + v M vf rs good performance near steady state certificates for stability & attraction basin + backward compatible in nominal case poor performance & needs hacks to work + v - R g(v) L C PWM dc,k virtual oscillator control (VOC) + robust & almost globally stable sync cannot meet amplitude/power specifications 13

48 Comparison of grid-forming control strategies ω idc θ, ω ω*0! m v!" ω!sync sync C!" G!" is τm τe rs RI LI i I v Lθ P2 P1 p(t) p rf if droop synchronous machine emulation control + v M vf rs good performance near steady state certificates for stability & attraction basin + backward compatible in nominal case poor performance & needs hacks to work ching of synchronous machines by DC storage + v R g(v) L C PWM lorian Dörfler dc,k xαβ ch, Switzerland virtual oscillator control (VOC) Abstract + & almostconverters globally in stable sync power trategyrobust for grid-forming low-inertia by identifying the structural similarities specifications between the cannot meet amplitude/power today:summary foundational control approach [Colombino, Groß, Brouillon, & Dorfler, 17, 18] Structural similarities allowcolombino, model matching [Seo, Subotic, Johnson, Groß, & Dorfler, 18]13 by adding one integrator

49 Outline Problem Setup: Modeling and Specifications Dispatchable Virtual Oscillator Control Experimental Validation High-fidelity power systems simulations Conclusions

50 Recall problem setup 1. simplifying assumptions (will be removed later) i o,k to network converter controllable voltage source d dt v k(t) = u k (v k, i o,k ) grid quasi-static: l d dt i + ri ( j lω 0 + r ) i lines homogeneous κ = tan(l kj /r kj ) k, j 14

51 Recall problem setup 1. simplifying assumptions (will be removed later) i o,k to network converter controllable voltage source d dt v k(t) = u k (v k, i o,k ) grid quasi-static: l d dt i + ri ( j lω 0 + r ) i lines homogeneous κ = tan(l kj /r kj ) k, j 2. fully decentralized control of converter terminal voltage & current set-points for relative angles {θ jk} nonlocal measurements v j grid & load parameters local measurements (v k, i o,k ) local set-points (v k, p k, q k) 14

52 Recall problem setup 1. simplifying assumptions (will be removed later) i o,k to network converter controllable voltage source d dt v k(t) = u k (v k, i o,k ) grid quasi-static: l d dt i + ri ( j lω 0 + r ) i lines homogeneous κ = tan(l kj /r kj ) k, j 2. fully decentralized control of converter terminal voltage & current set-points for relative angles {θ jk} nonlocal measurements v j grid & load parameters local measurements (v k, i o,k ) local set-points (v k, p k, q k) ω 0 3. control objective stabilize desired quasi steady state ω 0 v 2 θ 12 v 1 (synchronous, 3-phase-balanced, and meet set-points in nominal case) 0 v 14

53 Colorful idea for closed-loop target dynamics objectives: frequency, phase, and voltage stability d dt v k = [ 0 ω0 ω 0 0 } {{ } rotation at ω 0 ] v k + c 1 e θ,k (v) }{{} synchronization + c 2 e v,k (v k ) }{{} magnitude regulation ω 0 e v,2 (v 2 ) v 2 e θ,2 (v) ω e 0 θ,1 (v) synchronization: e θ,k (v) = n j=1 Y jk ( v j R(θ jk)v k ) e v,1 (v 1 ) 0 v 1 amplitude regulation: e v,k (v k ) = ( v 2 v k 2) v k 15

54 Colorful idea for closed-loop target dynamics objectives: frequency, phase, and voltage stability d dt v k = [ 0 ω0 ω 0 0 } {{ } rotation at ω 0 ] v k + c 1 e θ,k (v) }{{} synchronization + c 2 e v,k (v k ) }{{} magnitude regulation e ω v,2 (v 2 ) 0 e v,1 (v 1 ) ω 0 v 2 v 1 synchronization: e θ,k (v) = n j=1 Y jk ( v j R(θ jk)v k ) 0 amplitude regulation: e v,k (v k ) = ( v 2 v k 2) v k 15

55 Colorful idea for closed-loop target dynamics objectives: frequency, phase, and voltage stability d dt v k = [ 0 ω0 ω 0 0 } {{ } rotation at ω 0 ] v k + c 1 e θ,k (v) }{{} synchronization + c 2 e v,k (v k ) }{{} magnitude regulation ω 0 ω 0 v 2 v 1 synchronization: e θ,k (v) = n j=1 Y jk ( v j R(θ jk)v k ) 0 amplitude regulation: e v,k (v k ) = ( v 2 v k 2) v k 15

56 Decentralized implementation of target dynamics j Y jk (v j R(θ jk)v k ) }{{} need to know Y jk, v j, v k and θ jk 16

57 Decentralized implementation of target dynamics Y jk (v j R(θjk)v k ) j }{{} need to know Y jk, v j, v k and θ jk = j Y jk (v j v k ) }{{} Laplacian feedback + j Y jk (I R(θ jk))v k }{{} local feedback: K k (θ )v k 16

58 Decentralized implementation of target dynamics Y jk (v j R(θjk)v k ) j }{{} need to know Y jk, v j, v k and θ jk = j Y jk (v j v k ) }{{} Laplacian feedback + j Y jk (I R(θ jk))v k }{{} local feedback: K k (θ )v k insight I: non-local measurements from communication through physics R(κ) i o,k }{{} local feedback (κ = l/r uniform) = j Y jk (v j v k ) }{{} distributed Laplacian feedback 16

59 Decentralized implementation of target dynamics Y jk (v j R(θjk)v k ) j }{{} need to know Y jk, v j, v k and θ jk = j Y jk (v j v k ) }{{} Laplacian feedback + j Y jk (I R(θ jk))v k }{{} local feedback: K k (θ )v k insight I: non-local measurements from communication through physics R(κ) i o,k }{{} local feedback (κ = l/r uniform) = j Y jk (v j v k ) }{{} distributed Laplacian feedback insight II: angle set-points & line-parameters from power flow equations p k = v 2 j qk = v 2 j r jk (1 cos(θ jk )) ω 0l jk sin(θ jk ) r 2 jk +ω2 0 l2 jk ω 0 l jk (1 cos(θ jk ))+r jk sin(θ jk ) r 2 jk +ω2 0 l2 jk K k (θ ) = 1 [ ] q R(κ) k p k v 2 p k qk }{{}}{{} global parameters local parameters 16

60 Main results 1. desired target dynamics can be realized via fully decentralized control : d dt v k = [ 0 ω 0 ω 0 0 ] vk }{{} rotation at ω 0 ( + η R(κ) ( 1 [ q k p v 2 k p k q k }{{} synchronization through physics ] ) ) v k i o,k + α (v 2 v k 2 ) v k }{{} local amplitude regulation 17

61 Main results 1. desired target dynamics can be realized via fully decentralized control : d dt v k = [ 0 ω 0 ω 0 0 ] vk }{{} rotation at ω 0 ( + η R(κ) ( 1 [ q k p v 2 k p k q k }{{} synchronization through physics 3. certifiable, sharp, and intuitive stability conditions : ] ) ) v k i o,k + α (v 2 v k 2 ) v k consistent v, p k, and q k satisfy AC power flow equations }{{} local amplitude regulation magnitude control slower than synchronization control (α small ) power transfer small enough compared to network connectivity synchronization slower than line dynamics (η small enough ) 17

62 Main results 1. desired target dynamics can be realized via fully decentralized control : d dt v k = [ 0 ω 0 ω 0 0 ] vk }{{} rotation at ω 0 ( + η R(κ) ( 1 [ q k p v 2 k p k q k }{{} synchronization through physics 3. certifiable, sharp, and intuitive stability conditions : ] ) ) v k i o,k + α (v 2 v k 2 ) v k consistent v, p k, and q k satisfy AC power flow equations }{{} local amplitude regulation magnitude control slower than synchronization control (α small ) power transfer small enough compared to network connectivity synchronization slower than line dynamics (η small enough ) 2. almost global stability result :... the system is almost globally asymptotically stable with respect to a limit cycle corresponding to a pre-specified solution of the AC power-flow equations at a synchronous frequency ω 0. 17

63 Proof sketch for algebraic grid: Lyapunov & center manifold ( Lyapunov function: V (v) = 1 2 v 2 S + c 2 v k v 2 v 2 k 2) 2 sync set S amplitude set A T 0 2N target set T 18

64 Proof sketch for algebraic grid: Lyapunov & center manifold ( Lyapunov function: V (v) = 1 2 v 2 S + c 2 v k v 2 v 2 k 2) 2 sync set S T 0 2N is globally attractive lim t v(t) T 0 = 0 2N amplitude set A T 0 2N target set T 18

65 Proof sketch for algebraic grid: Lyapunov & center manifold ( Lyapunov function: V (v) = 1 2 v 2 S + c 2 v k v 2 v 2 k 2) 2 sync set S T 0 2N is globally attractive lim t v(t) T 0 = 0 2N amplitude set A T T is stable v(t) T χ 2( v 0 T ) 0 2N target set T 18

66 Proof sketch for algebraic grid: Lyapunov & center manifold ( Lyapunov function: V (v) = 1 2 v 2 S + c 2 v k v 2 v 2 k 2) 2 Z {02N } 0-stable manifold sync set S T 0 2N is globally attractive lim t v(t) T 0 = 0 2N amplitude set A T T is stable v(t) T χ 2( v 0 T ) target set T 0 2N T is almost globally attractive 0 2N exponentially unstable = Z {02N } has measure zero v 0 / Z {02N } : lim t v(t) T = 0 18

67 Proof sketch for algebraic grid: Lyapunov & center manifold ( Lyapunov function: V (v) = 1 2 v 2 S + c 2 v k v 2 v 2 k 2) 2 Z {02N } 0-stable manifold sync set S T 0 2N is globally attractive lim t v(t) T 0 = 0 2N amplitude set A T T is stable v(t) T χ 2( v 0 T ) target set T 0 2N T is almost globally attractive 0 2N exponentially unstable = Z {02N } has measure zero v 0 / Z {02N } : lim t v(t) T = 0 stability & almost global attractivity = almost global asymptotic stability 18

68 Connection to virtual oscillator control (VOC) dynamics in rotating reference frame at ω 0 ( d [ dt v 1 q ] ) k = η R(κ) k p v 2 k p v k i o,k + ηα (v 2 v k 2 ) v k k q k }{{}}{{} synchronization through physics local amplitude regulation 19

69 Connection to virtual oscillator control (VOC) dynamics in rotating reference frame at ω 0 ( d [ dt v 1 q ] ) k = η R(κ) k p v 2 k p v k i o,k + ηα (v 2 v k 2 ) v k k q k }{{}}{{} synchronization through physics local amplitude regulation polar coordinates & resistive grid (microgrid, l jk = 0) ( d q θ d t k = η k v q ) k 2 v k 2 ( d p v d t k = η k v p ) k v 2 v k 2 k + ηα ( v k 1 v v 2 k 3) (phase) (magnitude) 19

70 Connection to virtual oscillator control (VOC) dynamics in rotating reference frame at ω 0 ( d [ dt v 1 q ] ) k = η R(κ) k p v 2 k p v k i o,k + ηα (v 2 v k 2 ) v k k q k }{{}}{{} synchronization through physics local amplitude regulation polar coordinates & resistive grid (microgrid, l jk = 0) ( d q θ d t k = η k v q ) k 2 v k 2 ( d p v d t k = η k v p ) k v 2 v k 2 k + ηα ( v k 1 v v 2 k 3) averaged virtual oscillator control d d q k θ d t k = η v k 2 p k v d t k = η v k v k + α ( v 2 k β v k 3 ) (phase) (magnitude) (phase) (magnitude) no set-points for power ( p k = 0, q k = 0) and voltage set-point v obscured 19

71 Connection to droop control dynamics in rotating reference frame at ω 0 ( d [ dt v 1 q ] ) k = η R(κ) k p v 2 k p v k i o,k + ηα (v 2 v k 2 ) v k k q k }{{}}{{} synchronization through physics local amplitude regulation polar coordinates & inductive grid (transmission network, r jk = 0) ( d p θ d t k = η k v p ) k (phase) 2 v k 2 ( d q v d t k = η k v q ) k v 2 v k 2 k + ηα (1 1 v v 2 k 2 ) v k (magnitude) 20

72 Connection to droop control dynamics in rotating reference frame at ω 0 ( d [ dt v 1 q ] ) k = η R(κ) k p v 2 k p v k i o,k + ηα (v 2 v k 2 ) v k k q k }{{}}{{} synchronization through physics local amplitude regulation polar coordinates & inductive grid (transmission network, r jk = 0) ( d p θ d t k = η k v p ) k (phase) 2 v k 2 ( d q v d t k = η k v q ) k v 2 v k 2 k + ηα (1 1 v v 2 k 2 ) v k (magnitude) active power droop & quadratic voltage droop d θ d t k = η (p k p k ) d v d t k = η (qk q k ) + α(v v k ) v k (p ω droop) (q v droop) feedback of power differences near nominal voltage v k v 1 20

73 Case study: IEEE 9 Bus system v 2 7 v v 3 21

74 Case study: IEEE 9 Bus system v 2 7 v v 3 t = 0 s: black start of three inverters initial state: v k (0) 10 3 convergence to set-point 21

75 Case study: IEEE 9 Bus system v 2 7 v v 3 t = 0 s: black start of three inverters initial state: v k (0) 10 3 convergence to set-point t = 5 s: load step-up 20% load increase at bus 5 consistent power sharing 21

76 Case study: IEEE 9 Bus system v v 3 t = 0 s: black start of three inverters initial state: v k (0) 10 3 convergence to set-point t = 5 s: load step-up 20% load increase at bus 5 consistent power sharing t = 10 s: loss of inverter 1 the remaining inverters synchronize they supply the load sharing power 21

77 Simulation of IEEE 9 Bus system pk [p.u.] vk [p.u.] ω [p.u.] io,k [p.u.] time [s] time [s] 22

78 Dropping assumptions: dynamic lines control gains

79 Dropping assumptions: dynamic lines control gains control gains

80 Dropping assumptions: dynamic lines control gains control gains observations inverter control interferes with the line dynamics controller needs to be artificially slowed down recognized problem [Vorobev, Huang, Hosaini, & Turitsyn, 17] 23

81 Dropping assumptions: dynamic lines control gains control gains observations inverter control interferes with the line dynamics controller needs to be artificially slowed down recognized problem [Vorobev, Huang, Hosaini, & Turitsyn, 17] networked control reason communication through currents to infer voltages very inductive lines delay the information transfer the controller must be slow in very inductive networks 23

82 Dropping assumptions: dynamic lines control gains control gains re-do the math leading to updated condition: magnitude control slower than sync control slower than line dynamics observations inverter control interferes with the line dynamics controller needs to be artificially slowed down recognized problem [Vorobev, Huang, Hosaini, & Turitsyn, 17] networked control reason communication through currents to infer voltages very inductive lines delay the information transfer the controller must be slow in very inductive networks 23

83 Main result: almost GAS with transmission lines dynamics Dynamics of the inverter based AC power system ( d 1 [ v d t k = η v R(κ) q ] ) k p k 2 p v k + R(κ) i o,k + α(1 1 v k q k v 2 k 2 ) v k L T d d t i = ZT i + B v 24

84 Main result: almost GAS with transmission lines dynamics Dynamics of the inverter based AC power system ( d 1 [ v d t k = η v R(κ) q ] ) k p k 2 p v k + R(κ) i o,k + α(1 1 v k q k v 2 k 2 ) v k L T d d t i = ZT i + B v stability condition for η > 0, α > 0, v > 0, θ jk θ π 2, and c > 0 N max Y jk 1 cos(θ jk ) 1 + α < k N j=1 2 η < η = ( 1 + cos( θ ) ) λ 2(L) c c τ Y (c + 5 K L ) 24

85 Main result: almost GAS with transmission lines dynamics Dynamics of the inverter based AC power system ( d 1 [ v d t k = η v R(κ) q ] ) k p k 2 p v k + R(κ) i o,k + α(1 1 v k q k v 2 k 2 ) v k L T d d t i = ZT i + B v stability condition for η > 0, α > 0, v > 0, θ jk θ π 2, and c > 0 N max Y jk 1 cos(θ jk ) 1 + α < k N j=1 2 η < η = ( 1 + cos( θ ) ) λ 2(L) c c τ Y (c + 5 K L ) Theorem Consider gains η R >0, α R >0, power and voltage set-points q k R, p k R, v R >0, and steady-state angles {θ k1} such that the power flow equations and stability condition holds. The system is almost globally asymptotically stable with respect to the steady-state given by the power set-points q k, p k, the voltage set-point v, and the nominal frequency ω 0. 24

86 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. 25

87 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading 25

88 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer 25

89 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer margin c for electromagnetic transients 25

90 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer margin c for electromagnetic transients bound on η shrinks due to increase in 25

91 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer margin c for electromagnetic transients bound on η shrinks due to increase in maximum network gain Ȳ m & transmission line time-constant τ 25

92 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer margin c for electromagnetic transients bound on η shrinks due to increase in maximum network gain Ȳ m & transmission line time-constant τ maximum apparent power s & maximum node degree d 25

93 Power-system interpretation of stability condition Sufficient condition for η > 0, α > 0, v > 0, θ jk π 2, and c > 0 max k N j:(j,k) E cos(κ) v 2 p jk + sin(κ) v 2 η < qjk + α 1 λ2(l) c, 2 c 2τȲ m(c + 5 v 2 s + 10 d ), with maximum line admittance Ȳ, the maximum number of transmission lines m connected to a single node, and d := max k j:(j,k) E Y jk. amplitude gain α small enough & not too heavy loading higher voltage allows for larger power transfer margin c for electromagnetic transients bound on η shrinks due to increase in maximum network gain Ȳ m & transmission line time-constant τ maximum apparent power s & maximum node degree d upgrading or adding lines can destabilize the system 25

94 Proof sketch for dynamic grid: perturbation-inspired Lyapunov d v = fv(v, i) d t v i d d t i = f i(v, i) 26

95 Proof sketch for dynamic grid: perturbation-inspired Lyapunov d v = fv(v, i) d t v i a = h(v) y = i h(v) h(v) d d t i = f i(v, i) 26

96 Proof sketch for dynamic grid: perturbation-inspired Lyapunov d v = fv(v, i) d t i a = h(v) v Individual Lyapunov functions slow system: V (v) for d v = fv(v, h(v)) d t fast system: W (y) for d y = fi(v, y + h(v)) d t where d v = 0 & coordinate y = i h(v) d t y = i h(v) h(v) d d t i = f i(v, i) 26

97 Proof sketch for dynamic grid: perturbation-inspired Lyapunov y = i h(v) d v = fv(v, i) d t i a = h(v) h(v) v Individual Lyapunov functions slow system: V (v) for d v = fv(v, h(v)) d t fast system: W (y) for d y = fi(v, y + h(v)) d t where d v = 0 & coordinate y = i h(v) d t Lyapunov function for the full system ν(x) = dw (i h(v)) + (1 d)v (v) where d [0, 1] is free convex coefficient d ν(x) is decaying under stability condition d t d d t i = f i(v, i) 26

98 Proof sketch for dynamic grid: perturbation-inspired Lyapunov y = i h(v) d v = fv(v, i) d t i a = h(v) h(v) v Individual Lyapunov functions slow system: V (v) for d v = fv(v, h(v)) d t fast system: W (y) for d y = fi(v, y + h(v)) d t where d v = 0 & coordinate y = i h(v) d t Lyapunov function for the full system ν(x) = dw (i h(v)) + (1 d)v (v) where d [0, 1] is free convex coefficient d ν(x) is decaying under stability condition d t d d t i = f i(v, i) Almost global asymptotic stability T {0 n} globally attractive & T stable Z {0n} has measure zero 26

99 Stability regions (three-bus transmission system) synchronization gain [p.u.] linear instability certified stability region damping ratios constraints violated amplitude gain [p.u.] vk [p.u.] increase of control gains by factor 10 oscillations, overshoots, & instability conditions are highly accurate 27

100 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o 28

101 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o idea: invert LC filter so that v 1 2 v dcu 28

102 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o idea: invert LC filter so that v 1 2 v dcu control: perform robust inversion of LC filter via cascaded PI 28

103 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o idea: invert LC filter so that v 1 2 v dcu control: perform robust inversion of LC filter via cascaded PI analysis: repeat proof via singular perturbation Lyapunov functions 28

104 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o idea: invert LC filter so that v 1 v 2 dcu control: perform robust inversion of LC filter via cascaded PI analysis: repeat proof via singular perturbation Lyapunov functions almost global stability for sufficient time scale separation (quantifiable) VOC model < line dynamics < voltage PI < current PI [Subotic, ETH Zürich Master thesis 18] 28

105 Dropping assumptions: detailed converter model voltage source model: i o d v(t) = u(v, io) dt detailed converter model with LC filter: v dc i 1 2 vdc u L R G C v i o idea: invert LC filter so that v 1 v 2 dcu control: perform robust inversion of LC filter via cascaded PI analysis: repeat proof via singular perturbation Lyapunov functions almost global stability for sufficient time scale separation (quantifiable) VOC model < line dynamics < voltage PI < current PI... similar steps for control of v dc in a more detailed model [Subotic, ETH Zürich Master thesis 18] 28

106 Experimental NREL 29

107 Experimental results [Seo, Subotic, Johnson, Colombino, Groß, & Dörfler, APEC 18] black start of inverter #1 under 500 W load (making use of almost global stability) connecting inverter #2 while inverter #1 is regulating the grid under 500 W load 250 W to 750 W load transient with two inverters active change of setpoint: p of inverter #2 updated from 250 W to 500 W 30

108 Outline Problem Setup: Modeling and Specifications Dispatchable Virtual Oscillator Control Experimental Validation High-fidelity power systems simulations Conclusions

109 Dropping assumptions: current limitation converter current limits not accounted for in the analysis typically addressed in inner control loops may comprise stability of outer loops (i.e., dvoc) actuation of DC source/boost DC voltage control DC voltage cascaded voltage/current tracking control converter modulation PWM reference synthesis (e.g., droop or virtual inertia) (P, Q, kv k,!) measurement processing (e.g., via PLL) AC current & voltage System-level controllers dispatchable virtual oscillator control droop control (P ω, Q V ) machine matching 1 DC/AC power inverter 1 Curi, Groß, Dörfler: Control for Low Inertia Power Grids: A Model Reduction Approach, CDC 17 2 Paquette, Divan: Virtual impedance current limiting for inverters in microgrids with synchronous generators, IEEE IA 15 31

110 Dropping assumptions: current limitation converter current limits not accounted for in the analysis typically addressed in inner control loops may comprise stability of outer loops (i.e., dvoc) actuation of DC source/boost DC voltage control DC voltage cascaded voltage/current tracking control converter modulation PWM reference synthesis (e.g., droop or virtual inertia) (P, Q, kv k,!) measurement processing (e.g., via PLL) AC current & voltage DC/AC power inverter System-level controllers dispatchable virtual oscillator control droop control (P ω, Q V ) machine matching 1 Device-level control current limitation via virtual impedance 2 cascaded PI control (current & voltage) DC control: PI (droop & dvoc), P (matching) 1 Curi, Groß, Dörfler: Control for Low Inertia Power Grids: A Model Reduction Approach, CDC 17 2 Paquette, Divan: Virtual impedance current limiting for inverters in microgrids with synchronous generators, IEEE IA 15 31

111 Three-converter transmission system high-fidelity simulation of a three-converter transmission system comparability of different system-level controls robustness to faults: tripping of a line & short circuit fault current-limitation during short circuit fault Inverter Control 0 α a1, α b1, α c1 1 I DC1 L f1 PCC 1 C I s1 R f1 I g1 DC part V DC1 Inverter 1 C f1 E g1 Z load K line R line K sc R line X line X line I g2 Rf2 L f2 Inverter Control 0 α a2, α b2, α c2 1 I DC2 I s2 R line3 R K sc line Inverter 2 V DC2 DC part X E line3 g2 C f2 PCC 2 Inverter Control K 3 X line2 0 α a3, α b3, α c3 1 I DC3 I s3 L f3 R f3 I g3 PCC 3 R line2 DC part V DC3 Inverter 3 C f3 E g3 Collaboration with RTE, L2EP, and the Power Systems Laboratory (PSL) at ETH within the project MIGRATE 32

112 Three-converter system: line tripping & short circuit fault Frequency [Hz] Terminal voltage [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Events t = 0.5 s: line from PCC1 to PCC3 disconnected t = 5 s: short circuit near PCC1 t = 1.65 s: fault cleared Output current [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] DC voltage [p.u.] 1.01 Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] 33

113 Three-converter system: line tripping & short circuit fault Frequency [Hz] Terminal voltage [p.u.] Output current [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) Events t = 0.5 s: line from PCC1 to PCC3 disconnected t = 5 s: short circuit near PCC1 t = 1.65 s: fault cleared Observations dvoc is compatible with droop and machine matching t [s] DC voltage [p.u.] 1.01 Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] 33

114 Three-converter system: line tripping & short circuit fault Frequency [Hz] Terminal voltage [p.u.] Output current [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Events t = 0.5 s: line from PCC1 to PCC3 disconnected t = 5 s: short circuit near PCC1 t = 1.65 s: fault cleared Observations dvoc is compatible with droop and machine matching system remains stable during & after faults DC voltage [p.u.] 1.01 Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] 33

115 Three-converter system: line tripping & short circuit fault Frequency [Hz] Terminal voltage [p.u.] Output current [p.u.] DC voltage [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Events t = 0.5 s: line from PCC1 to PCC3 disconnected t = 5 s: short circuit near PCC1 t = 1.65 s: fault cleared Observations dvoc is compatible with droop and machine matching system remains stable during & after faults virtual impedance limits current during transients 33

116 Three-converter system: line tripping & short circuit fault Frequency [Hz] Terminal voltage [p.u.] Output current [p.u.] DC voltage [p.u.] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Droop (250 MW) dvoc (500 MW) Matching (1 GW) t [s] Events t = 0.5 s: line from PCC1 to PCC3 disconnected t = 5 s: short circuit near PCC1 t = 1.65 s: fault cleared Observations dvoc is compatible with droop and machine matching system remains stable during & after faults virtual impedance limits current during transients dvoc is compatible with virtual impedance 33

117 Conclusions Summary grid-forming control strategy: dispatchable virtual oscillator control theoretic analysis: sharp and intuitive stability conditions experimental validation & high-fidelity simulations Ongoing & future work theoretical questions: robustness practical issue: compatibility with legacy system experimental ETH, NREL Main references D. Groß, M. Colombino, J.S. Brouillon, & F. Dörfler. The effect of transmission-line dynamics on grid-forming dispatchable virtual oscillator control. G. Seo, I. Subotic, B. Johnson, M. Colombino, D. Groß, & F. Dörfler. Dispatchable Virtual Oscillator Control for Decentralized Inverter-Dominant Power Systems. M. Colombino, D. Groß, J.S. Brouillon, & F. Dörfler. Global phase and magnitude synchronization of coupled oscillators with application to the control of grid-forming power inverters. 34

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