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Virtual Inertia Emulation and Placement in Powe...

Florian Dörfler
October 16, 2024
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Virtual Inertia Emulation and Placement in Power Grids

Suppelec, 2016

Florian Dörfler

October 16, 2024
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  1. Virtual Inertia Emulation and Placement in Power Grids S´ eminaire

    d’Automatique du Plateau de Saclay Laboratoire de Signaux et Syst` emes du Supelec Florian D¨ orfler
  2. At the beginning of power systems was . . .

    At the beginning was the synchronous machine: M d dt ω(t) = Pgeneration(t) − Pdemand(t) change of kinetic energy = instantaneous power balance Fact: the AC grid & all of power system operation has been designed around synchronous machines. Pgeneration Pdemand ω 2 / 35
  3. Operation centered around bulk synchronous generation 49.88 49.89 49.90 49.91

    49.92 49.93 49.94 49.95 49.96 49.97 49.98 49.99 50.00 50.01 50.02 16:45:00 16:50:00 16:55:00 17:00:00 17:05:00 17:10:00 17:15:00 8. Dezember 2004 f [Hz] 49.88 49.89 49.90 49.91 49.92 49.93 49.94 49.95 49.96 49.97 49.98 49.99 50.00 50.01 50.02 16:45:00 16:50:00 16:55:00 17:00:00 17:05:00 17:10:00 17:15:00 8. Dezember 2004 f [Hz] Frequency Athens f - Setpoint Frequency Mettlen, Switzerland PP - Outage PS Oscillation Source: W. Sattinger, Swissgrid Primary Control Secondary Control Tertiary Control Oscillation/Control M echanical Inertia 3 / 35
  4. A few (of many) game changers . . . synchronous

    generator new workhorse scaling location & distributed implementation Almost all operational problems can principally be resolved . . . but one (?) 5 / 35
  5. Fundamental challenge: operation of low-inertia systems We slowly loose our

    giant electromechanical low-pass filter: M d dt ω(t) = Pgeneration(t) − Pdemand(t) change of kinetic energy = instantaneous power balance Pgeneration Pdemand ω 6 / 35
  6. Low-inertia stability: # 1 problem of distributed generation # frequency

    violations in Nordic grid (source: ENTSO-E) Number * 10 0 5000 10000 15000 20000 25000 30000 Duration [s] Events [-] Months of the year 75 mHz Criterion Summary - Short View - Year 2001-2011 Number * 10 Duration 2001 2002 2003 2004 2006 2005 2007 2008 2009 2010 Fig. 3.2: Frequency quality behaviour in Continental Europe during the last ten years. Source: Swissgrid It can clearly be observed how the accumulated time continuously increases with higher frequency deviations as well as the number of corresponding events. 3.1.2. CAUSES The unbundling process has separated power generation from TSO, imposing new commercial rules in the system operating process. Generation units are considered as simple balance responsible parties without taking dynamic behaviour into account: slow or fast units. Following the principle of equality, the market has created unique rules for settlement: energy supplied in a time frame versus energy calculated from schedule in the same time frame. Energy is traded as constant power in time frame. The market, being orientated on energy, has not developed rules for real time operation as power. In consequence we are faced with the following unit behaviour (Figure 3.3): Power basepoint scheduled A: Fast units response B: Slow unit response Load evolution which must be covered Energy to be compensated - real cause of frequency deterministic deviations same in Switzerland (source: Swissgrid) inertia is shrinking, time-varying, & localized, . . . & increasing disturbances Solutions in sight: none really . . . other than emulating virtual inertia through fly-wheels, batteries, super caps, HVDC, demand-response, . . . 7 / 35
  7. Virtual inertia emulation devices commercially available, required by grid-codes or

    incentivized through markets !""" #$%&'%(#!)&' )& *)+"$ ','#"-'. /)01 23. &)1 2. -%, 2456 5676 !89:;8;<=><? />@=AB: !<;@=>B >< CD!EFGBH;I +><I *JK;@ E;<;@B=>J< -JLB88BI@;MB DBNLB@> -J?LBIIB8 %@B<>! "#$%&'# (&)*&+! ,---. B<I "LBO D1 ":F'BBIB<P! "&'./+ (&)*&+! ,--- !"#$%&'"'() %* +$,(-.'() /'-#%(-' .( 0.1$%2$.3- 4-.(2 5.$)6,7 !('$)., 8.".-9 :%(.! "#$%&'# (&)*&+! ,---; :6$<,(,$,<,(, =%%77,! (&)*&+! ,---; ,(3 06>67 ?@ ?9,(3%$>,$! (&)*&+! ,--- Virtual synchronous generators: A survey and new perspectives Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b a Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran b Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan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d dt ω(t) = Pgeneration(t)−Pdemand(t) . . . essentially D-control ⇒ plug-&-play (decentralized & passive), grid-friendly, user-friendly, . . . ⇒ today: where to do it? how to do it properly? 8 / 35
  8. Classification & choice of actuators (source: Stephan Masselis) each of

    these (& far more) have been proposed for virtual inertia emulation 9 / 35
  9. Inertia emulation & virtual synchronous machines 1 naive D-control on

    ω(t): M d dt ω(t) = Pgeneration(t) − Pdemand(t) 2 more sophisticated emulation of virtual synchronous machine Virtual synchronous generators: A survey and new perspectives Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b a Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran b Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan a r t i c l e i n f o Article history: Received 31 December 2012 Received in revised form 12 June 2013 Accepted 13 July 2013 Keywords: Virtual inertia Renewable energy VSG Frequency control Voltage control Microgrid a b s t r a c t In comparison of the conventional bulk power plants, in which the synchronous machines dominate, the distributed generator (DG) units have either very small or no rotating mass and damping property. With growing the penetration level of DGs, the impact of low inertia and damping effect on the grid stability and dynamic performance increases. A solution towards stability improvement of such a grid is to pro- vide virtual inertia by virtual synchronous generators (VSGs) that can be established by using short term energy storage together with a power inverter and a proper control mechanism. The present paper reviews the fundamentals and main concept of VSGs, and their role to support the power grid control. Then, a VSG-based frequency control scheme is addressed, and the paper is focused on the poetical role of VSGs in the grid frequency regulation task. The most important VSG topologies with a survey on the recent works/achievements are presented. Finally, the relevant key issues, main technical challenges, further research needs and new perspectives are emphasized. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The capacity of installed inverter-based distributed generators A solution towards stabilizing such a grid is to provide addi- tional inertia, virtually. A virtual inertia can be established for DGs/RESs by using short term energy storage together with a Electrical Power and Energy Systems 54 (2014) 244–254 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes 3 everything in between . . . and much more . . . ⇒ by measuring AC current/voltage/power/frequency ⇒ software model of virtual machine provides converter setpoints ⇒ actuation via modulation (switching) or DC injection (batteries etc.) 10 / 35
  10. Challenges in real-world converter implementations Virtual synchronous generators: A survey

    and new perspectives Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b a Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran b Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan a r t i c l e i n f o Article history: Received 31 December 2012 Received in revised form 12 June 2013 Accepted 13 July 2013 Keywords: Virtual inertia Renewable energy VSG Frequency control Voltage control Microgrid a b s t r a c t In comparison of the conventional bulk power plants, in which the synchronous machines dominate, the distributed generator (DG) units have either very small or no rotating mass and damping property. With growing the penetration level of DGs, the impact of low inertia and damping effect on the grid stability and dynamic performance increases. A solution towards stability improvement of such a grid is to pro- vide virtual inertia by virtual synchronous generators (VSGs) that can be established by using short term energy storage together with a power inverter and a proper control mechanism. The present paper reviews the fundamentals and main concept of VSGs, and their role to support the power grid control. Then, a VSG-based frequency control scheme is addressed, and the paper is focused on the poetical role of VSGs in the grid frequency regulation task. The most important VSG topologies with a survey on the recent works/achievements are presented. Finally, the relevant key issues, main technical challenges, further research needs and new perspectives are emphasized. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The capacity of installed inverter-based distributed generators (DGs) in power system is growing rapidly; and a high penetration level is targeted for the next two decades. For example only in Ja- pan, 14.3 GW photovoltaic (PV) electric energy is planned to be connected to the grid by 2020, and it will be increased to 53 GW by 2030. In European countries, USA, China, and India significant targets are also considered for using the DGs and renewable energy sources (RESs) in their power systems up to next two decades. Compared to the conventional bulk power plants, in which the synchronous machine dominate, the DG/RES units have either very small or no rotating mass (which is the main source of inertia) and damping property. The intrinsic kinetic energy (rotor inertia) and damping property (due to mechanical friction and electrical losses in stator, field and damper windings) of the bulk synchronous gen- erators play a significant role in the grid stability. With growing the penetration level of DGs/RESs, the impact of low inertia and damping effect on the grid dynamic performance and stability increases. Voltage rise due to reverse power from PV generations [1], excessive supply of electricity in the grid due to full generation by the DGs/RESs, power fluctuations due to var- iable nature of RESs, and degradation of frequency regulation (especially in the islanded microgrids [2], can be considered as some negative results of mentioned issue. A solution towards stabilizing such a grid is to provide addi- tional inertia, virtually. A virtual inertia can be established for DGs/RESs by using short term energy storage together with a power electronics inverter/converter and a proper control mecha- nism. This concept is known as virtual synchronous generator (VSG) [3] or virtual synchronous machine (VISMA) [4]. The units will then operate like a synchronous generator, exhibiting amount of inertia and damping properties of conventional synchronous ma- chines for short time intervals (in this work, the notation of ‘‘VSG’’ is used for the mentioned concept). As a result, the virtual inertia concept may provide a basis for maintaining a large share of DGs/RESs in future grids without compromising system stability. The present paper contains the following topics: first the funda- mentals and main concepts are introduced. Then, the role of VSGs in microgrids control is explained. In continuation, the most important VSG topologies with a review on the previous works and achievements are presented. The application areas for the VSGs, particularly in the grid frequency control, are mentioned. A frequency control scheme is addressed, and finally, the main tech- nical challenges and further research needs are addressed and the paper is concluded. 2. Fundamentals and concepts The idea of the VSG is initially based on reproducing the dynamic properties of a real synchronous generator (SG) for the power electronics-based DG/RES units, in order to inherit the advantages of a SG in stability enhancement. The principle of the VSG can be applied either to a single DG, or to a group of DGs. The first 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.07.009 ⇑ Corresponding author at: Dept. of Electrical and Computer Eng., University of Kurdistan, Sanandaj, PO Box 416, Iran. Tel.: +98 8716660073. E-mail address: [email protected] (H. Bevrani). Electrical Power and Energy Systems 54 (2014) 244–254 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes 1 Abstract- The method to investigate the interaction between a Virtual Synchronous Generator (VSG) and a power system is presented here. A VSG is a power-electronics based device that emulates the rotational inertia of synchronous generators. The development of such a device started in a pure simulation environment and extends to the practical realization of a VSG. Investigating the interaction between a VSG and a power system is a problem, as a power system cannot be manipulated without disturbing customers. By replacing the power system with a real time simulated one, this problem can be solved. The VSG then interacts with the simulated power system through a power interface. The advantages of such a laboratory test-setup are numerous and should prove beneficial to the further development of the VSG concept. I. INTRODUCTION Short term frequency stability in power systems is secured mainly by the large rotational inertia of synchronous machines which, due to its counteracting nature, smoothes out the various disturbances. The increasing growth of dispersed generation will cause the so-called inertia constant of the power system to decrease. This may result in the power system becoming instable [1]-[3]. A promising solution to such a development is the Virtual Synchronous Generator (VSG) [4]-[8], which replaces the lost inertia with virtual inertia. The VSG consists of three distinctive components, namely a power processor, an energy storage device and the appropriate control algorithm [4] as shown in Fig. 1. This system has been tested in a full Matlab/Simulink [21] simulation environment with promising results. Fig. 1. The VSG Concept. This work is a part of the VSYNC project funded by the European Commission under the FP6 framework with contract No:FP6 – 038584 (www.vsync.eu). To better study and witness the effects of virtual inertia, the hardware of a real VSG should be tested within a power system. Investigating the interaction between a real VSG and a power system is not easy as a power system cannot be manipulated without disturbing customers. Building a real power system for testing purposes would be too costly. By replacing the power system with a real time simulated one, this problem can be solved. In this paper the testing of a real hardware VSG in combination with a simulated power system is described. The power processor from Fig.1 is built from a Triphase® [9], [10] inverter system. The Matlab/simulink VSG algorithm is directly implemented on the inverter system through a dedicated FPGA interface developed by Triphase®. In order to test the hardware implemented VSG and to study its effects within a power system, it is connected with a real time digital simulator from RTDS® [17] through a power interface (Fig 2). Fig. 2. RTDS and Power Interface and VSG in a closed loop. The RTDS® simulates power systems in real time and is often used in closed loop testing with real external hardware. Keeping in mind that the ADCs and DACs, which are the inputs and outputs of the RTDS, have a dynamic range of ±10V max rated at 5mA max and the Triphase® inverter system is rated at 16kVA, it is clear that a power interface has to come in between to make this union possible as it is shown in Fig. 2. The main function of the power interface is to replicate the voltage waveform of a bus in a network model to a level of 400VLL at terminal 1 in Fig. 2. This terminal is loaded by the VSG and the current flowing from/to the VSG is fed back to the RTDS, to load the bus in the network model with that current. The simulated power system is a transfer from the Matlab/Simulink environment, in which the system was developed initially, to RSCAD [18] format. In section II the requirements for testing a VSG and the principle of a VSG are discussed and in section III the test set Real Time Simulation of a Power System with VSG Hardware in the Loop Vasileios Karapanos, Sjoerd de Haan, Member, IEEE, Kasper Zwetsloot Faculty of Electrical Engineering, Mathematics and Computer Science Delft University of Technology Delft, the Netherlands E-mails: [email protected], [email protected], [email protected] k,(((  1 delays in measurement acquisition, signal processing, & actuation 2 accuracy in AC measurements (averaged over ≈ 5 cycles) 3 constraints on currents, voltages, power, etc. 4 guarantees on stability and robustness today: use DC measurement, exploit analog storage, & passive control 11 / 35
  11. Averaged inverter model iload + − vx iαβ R L

    ic C − + vαβ idc Gdc Cdc ix DC cap & AC filter equations: Cdc ˙ vdc = −Gdc vdc + idc − 1 2 m iαβ C ˙ vαβ = −iload + iαβ L ˙ iαβ = −Riαβ + 1 2 mvdc − vαβ modulation: ix = 1 2 m iαβ , vx = 1 2 mvdc passive: (idc , iload )→(vdc , vαβ) model of a synchronous generator ˙ θ = ω M ˙ ω = −Dω + τm + iαβ Lm if − sin(θ) cos(θ) C ˙ vαβ = −Gload vαβ + iαβ Ls ˙ iαβ = −Riαβ − vαβ − ωLm if − sin(θ) cos(θ) if ✓ 12 / 35
  12. standard power electronics control would continue by 1 constructing voltage/current/power

    references (e..g, droop, synchronous machine emulation, etc.) 2 tracking these references at the converter terminals typically by means of cascaded PI controllers let’s do something different (smarter?) today . . . 13 / 35
  13. See the similarities & the differences ? iload + −

    vx iαβ R L ic C − + vαβ idc Gdc Cdc ix DC cap & AC filter equations: Cdc ˙ vdc = −Gdc vdc + idc − 1 2 m iαβ C ˙ vαβ = −iload + iαβ L ˙ iαβ = −Riαβ + 1 2 mvdc − vαβ modulation: ix = 1 2 m iαβ , vx = 1 2 mvdc passive: (idc , iload )→(vdc , vαβ) model of a synchronous generator ˙ θ = ω M ˙ ω = −Dω + τm + iαβ Lm if − sin(θ) cos(θ) C ˙ vαβ = −Gload vαβ + iαβ Ls ˙ iαβ = −Riαβ − vαβ − ωLm if − sin(θ) cos(θ) if ✓ 14 / 35
  14. Model matching (= emulation) as inner control loop iload +

    − vx iαβ R L ic C − + vαβ idc Gdc Cdc ix DC cap & AC filter equations: Cdc ˙ vdc = −Gdc vdc + idc − 1 2 m iαβ C ˙ vαβ = −iload + iαβ L ˙ iαβ = −Riαβ + 1 2 mvdc − vαβ matching control: ˙ θ = η · vdc , m = µ · − sin(θ) cos(θ) with η, µ > 0 ⇒ pros: is balanced, uses natural storage, & based on DC measurement ⇒ virtual machine with M = Cdc η2 , D = Gdc η2 , τm = idc η , if = µ ηLm ⇒ base for outer controls via idc & µ, e.g., virtual torque, PSS, & inertia 15 / 35
  15. Some properties & different viewpoints 1 quadratic curves for stationary

    P vs. (|V |, ω) ⇒ P ≤ Pmax = i2 dc /4Gdc ⇒ reactive power not directly affected ⇒ (P, ω)-droop ≈ 1/η ⇒ (P, |V |)-droop ≈ 1/µ 2 reformulation as virtual & adaptive oscillator 3 remains passive: (idc , iload )→(vdc , vαβ) 0 0.5 1 1.5 2 Active power P ×104 0 50 100 150 200 Amplitude (V) 0 0.5 1 1.5 2 2.5 Active power P ×104 0 10 20 30 40 50 Frequency (Hz) Cdc ˙ vdc = −Gdc vdc + i∗ dc − 1 2 m⊤iαβ C ˙ vαβ = −iload + iαβ L ˙ iαβ = −Riαβ + 1 2 mvdc − vαβ ˙ ξ = vdc η · 0 1 −1 0 ξ η µ _ m vdc (idc , iload ) (vdc , vαβ ) inverter modulation 16 / 35
  16. Eye candy: response to a load step 0 0.1 0.2

    0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(s) 0 0.05 0.1 0.15 0.2 g load (Ω-1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time(s) -200 -100 0 100 200 V x Gload iload + − vx iαβ R L ic C − + vαβ idc Gdc Cdc ix 17 / 35
  17. Linearized & Kron-reduced swing equation model mi ¨ θi +

    di ˙ θi = pin,i − pe,i generator swing equations pe,i ≈ j∈N bij (θi − θj ) linearized power flows likelihood of disturbance at #i: ti ≥ 0 Pgeneration Pdemand ω Pgeneration + η Pdemand state space representation: ˙ θ ˙ ω = 0 I −M−1L −M−1D A θ ω + 0 M−1 T1/2 B η where M = diag(mi ), D = diag(di ), T = diag(ti ), & L = LT (Laplacian) 18 / 35
  18. Performance metric for emulation of rotational inertia f f restoration

    time nominal frequency nominal frequency max deviation effort ROCOF System norm: amplification of disturbances: impulse (fault), step (loss of unit), white noise (renewables) to performance outputs: integral, peak, ROCOF, restoration time, . . . 19 / 35
  19. Coherency performance metric & H2 norm Energy expended by the

    system to return to synchronous operation: ∞ 0 {i, j}∈ E aij (θi (t) − θj (t))2 + n i=1 si ω2 i (t) dt H2 norm interpretation: 1 associated performance output: y = Q1/2 1 0 0 Q1/2 2 θ ω 2 impulses (faults) −→ output energy ∞ 0 y(t)T y(t) dt 3 white noise (renewables) −→ output variance lim t→∞ E y(t)T y(t) 20 / 35
  20. Algebraic characterization of the H2 norm Lemma: via observability Gramian

    G 2 2 = Trace(BTPB) where P is the observability Gramian P = ∞ 0 eATtCTCeAt dt P solves a Lyapunov equation: P A + ATP + Q = 0 A has a zero eigenvalue → restricts choice of Q y = Q1/2 1 0 0 Q1/2 2 θ ω Q1/2 1 1 = 0 P is unique for P [1 0] = [0 0] 21 / 35
  21. Problem formulation minimize P , mi G 2 2 =

    Trace(BTPB) → performance metric subject to n i=1 mi ≤ mbdg → budget constraint mi ≤ mi ≤ mi , i ∈ {1, . . . , n} → capacity constraint P A + ATP + Q = 0 → observability Gramian P [1 0] = [0 0] → uniqueness Insights 1 m appears as m−1 in system matrices A , B 2 product of B(m) & P in the objective 3 product of A(m) & P in the constraint        ⇒ large-scale & non-convex 22 / 35
  22. Building the intuition: results for two-area networks Fundamental learnings 1

    explicit closed-form solution is rational function 2 sufficiently uniform (t/d)i → strongly convex & fairly flat cost 3 non trivial in the presence of capacity constraints m 1 0 2 4 6 8 10 f(m 1 ) 0 1 2 3 4 5 6 dissimilar t/d identical t/d Dissimilar and Identical t/d ratios performance metric t 1 =1-t 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Optimal inertia allocation 0 5 10 15 20 25 m1 ∗ m2 ∗ mbdg m1 ∗ + m2 ∗ Budget, Sum, Inertia1, Inertia2 optimal inertia allocation 23 / 35
  23. Closed-form results for cost of primary control P/ ˙ θ

    primary droop control (ωi − ω∗) ∝ (Pi ∗ − Pi (θ)) Di ˙ θi = Pi ∗ − Pi (θ) P2 P1 P 𝜔 𝜔* 𝜔sync Primary control effort → accounted for by integral quadratic cost ∞ 0 ˙ θ(t)TD ˙ θ(t) dt which is the H2 performance for the penalties Q1/2 1 = 0 and Q1/2 2 = D 24 / 35
  24. Primary Control . . . cont’d Theorem: the primary control

    effort optimization reads equivalently as minimize mi n i=1 ti mi subject to n i=1 mi ≤ mbdg mi ≤ mi ≤ mi , i ∈ {1, . . . , n} Key take-aways: optimal solution independent of network topology allocation ∝ √ ti or mi = min{mbdg , mi } Location & strength of disturbance are crucial solution ingredients 25 / 35
  25. Taylor & power series expansions Key idea: expand the performance

    metric as a power series in m G 2 2 = Trace(B(m)TP(m)B(m)) Motivation: scalar series expansion at mi in direction µi : 1 (mi + δµi ) = 1 mi − δµi m2 i + O(δ2) Expand system matrices as Taylor series in direction µ: A(m + δµ) = A(0) (m,µ) + A(1) (m,µ) δ + O(δ2) B(m + δµ) = B(0) (m,µ) + B(1) (m,µ) δ + O(δ2) Expand the observability Gramian as a power series in direction µ: P(m + δµ) = P(0) (m,µ) + P(1) (m,µ) δ + O(δ2) 26 / 35
  26. Explicit gradient computation Expansion of system matrices & Gramian ⇒

    match coefficients . . . Formula for gradient at m in direction µ 1 nominal Lyapunov equation for O(δ0): P(0) = Lyap(A(0) , Q) 2 perturbed Lyapunov equation for O(δ1) terms: P(1) = Lyap(A(0) , P(0)A(1) + A(1)T P(0)) 3 expand objective in direction µ: G 2 2 = Trace(B(m)TP(m)B(m)) = Trace((. . .) + δ(. . .)) + O(δ2) 4 gradient: Trace(2 ∗ B(1)T P(0)B(0) + B(0)T P(1)B(0)) ⇒ use favorite method for reduced optimization problem 27 / 35
  27. Modified Kundur case study: 3 regions & 12 buses transformer

    reactance 0.15 p.u., line impedance (0.0001+0.001i) p.u./km 10 9 5 1 11 12 7 6 3 4 2 8 28 / 35
  28. Heuristics outperformed by H2 - optimal allocation Scenario: disturbance at

    #4 locally optimal solution outperforms heuristic max/uniform allocation optimal allocation ≈ matches disturbance inertia emulation at all undisturbed nodes is actually detrimental ⇒ location of disturbance & inertia emulation matters node 1 2 4 5 6 8 9 10 12 inertia 0 40 80 120 160 m m∗ m trace 0 0.05 0.1 0.15 0.2 0.25 Cost Original, Optimal, and Capacity allocations allocation subject to capacity constraints node 1 2 4 5 6 8 9 10 12 inertia 0 40 80 120 160 m m∗ muni trace 0 0.05 0.1 0.15 0.2 0.25 Cost Original, Optimal, and Uniform allocations allocation subject to the budget constraint 29 / 35
  29. Eye candy: time-domain plots of post fault behavior Time(s) 0

    50 100 150 ∆θ 1 -∆θ 4 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 (a) Time(s) 0 50 100 150 ∆ω4 -0.1 -0.05 0 0.05 0.1 0.15 (b) Time(s) 0 50 100 150 ∆ω5 ×10-3 -2 -1 0 1 2 3 4 5 (c) Time(s) 0 50 100 150 Control effort -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (d) m muni m∗ Control Effort Angle Diff. Freq #4 Freq #5 Original, Optimal, and Uniform allocations Take-home messages: best oscillation performance smallest peak frequency at #4 undisturbed sites are irrelevant minimal control effort mi · ¨ θi 30 / 35
  30. Conclusions on virtual inertia emulation Where to do it? 1

    H2-optimal (non-convex) allocation 2 closed-form results for cost of primary control 3 numerical approach via gradient computation How to do it? 1 down-sides of naive inertia emulation 2 novel machine matching control What else to do? Inertia emulation is . . . decentralized, plug-and-play (passive), grid-friendly, user-friendly, . . . suboptimal, wasteful in control effort, & need for new actuators 31 / 35
  31. Recall: operation centered around (virtual) sync generators 49.88 49.89 49.90

    49.91 49.92 49.93 49.94 49.95 49.96 49.97 49.98 49.99 50.00 50.01 50.02 16:45:00 16:50:00 16:55:00 17:00:00 17:05:00 17:10:00 17:15:00 8. Dezember 2004 f [Hz] 49.88 49.89 49.90 49.91 49.92 49.93 49.94 49.95 49.96 49.97 49.98 49.99 50.00 50.01 50.02 16:45:00 16:50:00 16:55:00 17:00:00 17:05:00 17:10:00 17:15:00 8. Dezember 2004 f [Hz] Frequency Athens f - Setpoint Frequency Mettlen, Switzerland PP - Outage PS Oscillation Source: W. Sattinger, Swissgrid Primary Control Secondary Control Tertiary Control Oscillation/Control Inertia & Em ulation 32 / 35
  32. A control perspective of power system operation Conventional strategy: emulate

    generator physics & control M ˙ ω(t) (virtual) inertia = Pmech tertiary control − Dω(t) primary control − t 0 ω(τ) d τ secondary control − Pelec Essentially all PID + setpoint control (simple, robust, & scalable) M ˙ ω(t) D = P set-point − Dω(t) P − t 0 ω(τ) d τ I − Pelec Control engineers should be able to do better . . . 33 / 35
  33. This “ what else? ” has been broadly recognized by

    TSOs, device manufacturers, academia, etc. Massive InteGRATion of power Electronic devices “The question that has to be examined is: how much power electronics can the grid cope with?” (European Commission) current controls what else? all options are on the table and keep us busy . . . 34 / 35
  34. Spectral perspective on different inertia allocations Real Axis -0.18 -0.16

    -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 Imaginary Axis -3 -2 -1 0 1 2 3 cone m muni m∗ Cone, Original, Optimal, and Uniform allocations • m = m → best damping asymptote & best damping ratio • Spectrum holds only partial information !!
  35. The planning problem sparse allocation of limited resources 1-regularized inertia

    allocation (promoting a sparse solution): minimize P , mi Jγ(m, P) = G 2 2 + γ m − m 1 subject to n i=1 mi ≤ mbdg mi ≤ mi ≤ mi i ∈ {1, . . . , n} P A + ATP + Q = 0 P[1 0] = [0 0] where γ ≥ 0 trades off sparsity penalty and the original objective Highlights: 1 regularization term is linear & differentiable 2 gradient computation algorithm can be used with some tweaking
  36. Relative performance loss (%) as a function of γ 0%

    → optimal allocation, 100% → no additional allocation γ ×10-4 0 0.5 1 1.5 2 2.5 3 Cardinality 0 2 4 6 8 10 Cardinality-Loc Cardinality-Uni Performance-Loc Performance-Uni Relative Performance Loss (%) 0 20 40 60 80 100 0 20 40 60 80 100 Localized and Uniform disturbances Card Perf. % 1 uniform disturbance ⇒ ∃ γ 1.3% loss ≡ (9 → 7) 2 localized disturbance ⇒ (2 → 1) without affecting performance
  37. Uniform disturbance to damping ratio power sharing → d ∝

    P∗, assuming t ∝ source rating P∗ Theorem: for ti /di = tj /dj the allocation problem reads equivalently as minimize mi n i=1 si mi subject to n i=1 mi ≤ mbdg mi ≤ mi ≤ mi , i ∈ {1, . . . , n} Key takeaways: optimal solution independent of network topology allocation ∝ √ si or mi = min{mbdg , mi } What if freq. penalty ∝ inertia? → norm independent of inertia
  38. Taylor & power series expansions Key idea: expand the performance

    metric as a power series in m G 2 2 = Trace(B(m)TP(m)B(m)) Motivation: scalar series expansion at mi in direction µi : 1 (mi + δµi ) = 1 mi − δµi m2 i + O(δ2) Expand system matrices in direction µ, where Φ = diag(µ): A(0) (m,µ) = 0 I −M−1L −M−1D , A(1) (m,µ) = 0 0 ΦM−2L ΦM−2D B(0) (m,µ) = 0 M−1T1/2 , B(1) (m,µ) = 0 −ΦM−2T1/2
  39. Taylor & power series expansions cont’d Expand the observability Gramian

    as a power series in direction µ P(m) = P(m + δµ) = P(0) (m,µ) + P(1) (m,µ) δ + O(δ2) Formula for gradient in direction µ 1 nominal Lyapunov equation for O(δ0): P(0) = Lyap(A(0) , Q) 2 perturbed Lyapunov equation for O(δ1) terms: P(1) = Lyap(A(0) , P(0)A(1) + A(1)T P(0)) 3 expand objective in direction µ: G 2 2 = Trace(B(m)TP(m)B(m)) = Trace((. . .) + δ(. . .)) + O(δ2) 4 gradient: Trace(2 ∗ B(1)T P(0)B(0) + B(0)T P(1)B(0))
  40. Gradient computation Algorithm: Gradient computation & perturbation analysis Input →

    current values of the decision variables mi Output → numerically evaluated gradient ∇f of the cost function 1 Evaluate the system matrices A(0) , B(0) based on current inertia 2 Solve for P(0)=Lyap(A(0) , Q) using a Lyapunov routine 3 For each node- obtain the perturbed system matrices A(1) , B(1) 4 Compute P(1)=Lyap(A(0) , P(0)A(1) + A(1)T P(0)) 5 Gradient ⇒ Trace(2 ∗ B(1)T P(0)B(0) + B(0)T P(1)B(0))
  41. Heuristics outperformed also for uniform disturbance Cost Original, Optimal, and

    Capacity allocations node 1 2 4 5 6 8 9 10 12 inertia 0 50 100 150 m m∗ m trace 0 0.05 0.1 0.15 allocation subject to capacity constraints node 1 2 4 5 6 8 9 10 12 inertia 0 30 60 90 trace 0 0.05 0.1 0.15 m m∗ muni Cost Original, Optimal, and Uniform allocations allocation subject to the budget constraint Scenario: uniform disturbance Heuristics for placement: 1 max allocation in case of capacity constraints 2 uniform allocation in case of budget constraint Results & insights: 1 locally optimal solution outperforms heuristics 2 optimal solution = max inertia at each bus