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Decentralized and optimal control of inter-area...

Florian Dörfler
October 16, 2024
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Decentralized and optimal control of inter-area oscillations in power networks

Berlin, 2015

Florian Dörfler

October 16, 2024
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  1. Decentralized and optimal control of inter-area oscillations in power networks

    Technische Universit¨ at Berlin 2015 Florian D¨ orfler
  2. Electro-mechanical oscillations in power networks Dramatic consequences: blackout of August

    10, 1996, resulted from instability of the 0.25 Hz mode in the Western interconnected system 10 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 South Arizona SoCal NoCal PacNW Canada North Montana Utah Source: http://certs.lbl.gov 0.25 Hz 2 / 50
  3. Less dramatic but quite common . . . usually well

    behaved 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 Source: G. Andersson (ETH) & W. Sattinger (Swissgrid) 3 / 50
  4. A closer look at some European incidents 0.5Hz Myles et

    al., 1988 2/3/1982 Ding et al., 2007 0.7Hz 8/5/2005 0.26Hz short circuit disconnection of Denish grid T P disconnection of Denish grid disconnection of Danish grid short circuit T P 5/29/2007 4/1/2007 AlAli et al., 2007 0.22Hz 0.15Hz !"# $%# $!# &#' (#' "#" )*+ ,"#"( ,"#"$ -'#'' -'#'& -'#', -'#'( ./**012/1 "'#"!#!""%34$$5!%467 8 19 /0:*;<=6 >*;</;<?/1 @07A1?/1 BC61?/1 T ! 4.3 s f ! 0.23 Hz Spain Germany Czech Romania !"# $%# $!# &#' (#' "#" )*+ ,"#"( ,"#"$ -'#'' -'#'& -'#', -'#'( ./**012/1 "'#"!#!""%34$$5!%467 8 19 /0:*;<=6 >*;</;<?/1 @07A1?/1 BC61?/1 T ! 4.3 s f ! 0.23 Hz Spain Germany Czech Romania ! !"#$%&'()' "#$%&'(&%)'*+,-..)$-/#! )0$%&! 1/2%&! 1.)#$! /3$)4%! -#! !!0!5678! $!5+8! 5/1/2005 ! ! 9/18/2010 AlAli et al., 2011 monitoring application was able to determine the post- disturbance damping of the East-west mode close to its true damping using only the ambient pre-disturbance data. This gives additional confidence in its effectiveness as an early- warning system against poorly damped oscillation that may arise also due to transient events. 49.94 49.96 49.98 50 50.02 50.04 50.06 50.08 50.1 09:22:00 09:22:30 09:23:00 09:23:30 09:24:00 09:24:30 09:25:00 09:25:30 09:26:00 09:26:30 09:27:00 09:27:30 09:28:00 f [Hz] Larrson et al., 2012 … 9/18/2010 arise also due to transient events. Frequency deviation (mHz) 21 78 24 74 10/25/2011 0.33Hz 0.48Hz Uhlen et al., 2008 0 20 40 60 80 100 120 140 160 180 -20 -10 0 10 20 30 40 50 Time (seconds) Angle (degrees) Relative Voltage phasor angles (Ref. Hasle) Nedre Røssåga Fardal Kristiansand 8/14/2007 15! Wilson et al., 2008 0.8Hz xx/xx/2007 5 / 50
  5. This is not a “solved problem” Europe: transmission network upgrades

    & expansion, renewable generation in remote locations, & deregulated markets, . . . United states: aging transmission infrastructure, sparse grid with load & generation hubs, & remote renewables, . . . Impact of Increasing Wind Power Generation on the North-South Inter-Area Oscillation Mode in the European ENTSO-E System Salaheddin AlAli *, Torsten Haase**, Ibrahim Nassar***, Harald Weber* *Institute of Electrical Power Engineering, University of Rostock **Dong Energy, Hamburg *** Department of Electrical Engineering, Al-Azhar University, Egypt Germany (Tel.: 0049-3814987125; e-mail: [email protected]) Abstract: After the enlargement of the European ENTSO-E power system towards Turkey at the end of 2010, the East-West Inter-Area Oscillation mode in the enlarged the European ENTSO-E power system has been identified in the frequency range of 0.15 Hz (TP = 7s) accompanied by insufficient damping. By the end of 2012, more than 107 GW of wind generation capacity had been installed across Europe, representing about 25% of the peak demand of ENTSO-E power system. In this paper, the impact of large scale wind power generation in the European ENTSO-E system on the North-South Inter-Area Oscillation mode using a detailed dynamic model of the European ENTSO-E system is investigated by gradually replacing the power generated by the synchronous generators in the system either Full Size Converter or Doubly Fed Induction Generator (DFIG) wind turbines. Because the whole system is extremely nonlinear, the analysis method in state space is senseless; therefore the damping behavior of Inter-Area-Oscillations of the whole system was analyzed in detail using the analysis method in time domain. The model was created using DIgSILENT software. Oscillation behaviour of the enlarged European power system under deregulated energy market conditions M. KurthÃ, E. Welfonder Department for Power Generation and Automatic Control (IVD), University of Stuttgart, Pfaffenwaldring 23, 70550 Stuttgart, Germany Received 11 February 2004; accepted 17 March 2005 Available online 26 May 2005 Abstract Aimed power system simulations are carried out to analyse the bad damping behaviour of slow inter-area oscillations sporadically occurring within the European power system. To obtain application-oriented results, the simulations are carried out by a detailed power system dynamic model and compared with corresponding oscillation measurements. Using analysis methods in the time and state space, it is shown that the damping behaviour can be improved by easily applicable countermeasures. Based on this, the foreseen enlargement of the European power system is investigated, when coupling both system ends step by step around the Mediterranean Sea to the so-called Mediterranean Ring. Also these predictive considerations lead to very interesting oscillation and damping results. r 2005 Elsevier Ltd. All rights reserved. Keywords: Inter-area oscillations; Damping; Power flow; Power plants; Voltage control; Speed control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mpact of Low Rotational Inertia on Power System Stability and Operation Andreas Ulbig, Theodor S. Borsche, Göran Andersson ETH Zurich, Power Systems Laboratory Physikstrasse 3, 8092 Zurich, Switzerland ulbig | borsche | andersson @ eeh.ee.ethz.ch Abstract: Large-scale deployment of Renewable Energy Sources (RES) has led to significant generation shares of variable RES in power systems worldwide. RES units, notably inverter- connected wind turbines and photovoltaics (PV) that as such do not provide rotational inertia, are effectively displacing conventional generators and their rotating machinery. The traditional assumption that grid inertia is sufficiently high with only small variations over time is thus not valid for power systems with high RES shares. This has implications for frequency dynamics and power system stability and operation. Frequency dynamics are faster in power systems with low rotational inertia, making frequency control and power system operation more challenging. This paper investigates the impact of low rotational inertia on power system stability and operation, contributes new analysis insights and offers mitigation options for low inertia impacts. Optimal coordinated control of multiple HVDC links for power oscillation damping based on model identification Robert Eriksson*,y and Lennart So ¨der Department of Electric Power Systems, Royal Institute of Technology (KTH), Stockholm, Sweden SUMMARY This paper deals with optimal coordinated control of several high voltage direct current (HVDC) links based on an estimated model of large power systems. The model of the power system is estimated by using subspace system identification techniques. An optimal controller is designed based on the estimated model with the aim to improve the damping in the system. The main contribution of this paper is the development of a new method which uses global Phasor measurement units (PMUs) signals for coordinated damping control of multiple HVDC links. The input signals are the controllable set-points of the HVDC links, the output signals are the speed signals of selected generators obtained from PMU. The PMU signals are used to estimate the current state of the model, i.e., the state of the system, an appropriate control action can then be applied to dampen the system. The benefit of the method is that the used output signals, i.e., the used PMU signals, are independent of the system equilibrium and therefore makes it possible to use state-feedback control, i.e., coordinated control. The method is applied to the Cigre ´ Nordic 32-bus system including two HVDC links. The consistent results show that the damping can be significantly increased. Copyright # 2010 John Wiley & Sons, Ltd. 6 / 50
  6. Remedies against electro-mechanical oscillations conventional control blue layer: interconnected generators

    fully decentralized control implemented locally effective against local oscillations ineffective against inter-area oscillations 7 / 50
  7. Remedies against electro-mechanical oscillations wide-area control (WAC) blue layer: interconnected

    generators fully decentralized control implemented locally distributed wide-area control using remote signals 8 / 50
  8. Outline Introduction Slow Coherency Modeling Conventional Wide-Area Analysis & Control

    Variance Amplification as Performance Metric Sparsity-Promoting Wide-Area Control Fully Decentralized & Optimal Control Large-Scale Case Study: NE-NY grid Conclusions
  9. Dominant electro-mechanical swing dynamics coarse-grained power network = coupled, forced,

    & heterogeneous pendula generator swing equations: Mi ¨ θi +Di ˙ θi = Pi − j Bij sin(θi −θj ) linearized at equilibrium (θ∗, ˙ θ∗, P∗): M ¨ θ + D ˙ θ + Lθ = P P3 P2 P1 where M, D are diagonal inertia and damping matrices & L is a Laplacian: L =      . . . ... . . . ... . . . −Bi1 cos(θ∗ i − θ∗ 1 ) · · · n j=1 Bij cos(θ∗ i − θ∗ j ) · · · −Bin cos(θ∗ i − θ∗ n ) . . . ... . . . ... . . .      9 / 50
  10. The main controllers to dampen oscillations Automatic Voltage Regulator (AVR)

    objective: voltage = const. ⇒ diminishing damping Power System Stabilizer (PSS) objective: net damping > 0 ⇒ damping of oscillations generator exciter grid AVR Σ PSS P ω E Eref EPSS HVDC (high voltage DC) & FACTS (flexible AC transmission systems): control by modulating lines 30 30 10 / 50
  11. Control-induced oscillations fact: multi-machine power systems have unstable zeros ⇒

    multiple local controllers interact in an adverse way ⇒ numerous tuning rules & heuristics for PSS design large interconnected power system consists of numerous generators connected through a high-voltage transmission network, supply- By Joe H. Chow, Juan J. Sanchez-Gasca, Haoxing Ren, and Shaopeng Wang uses both generator speed and electrical output power as in- put signals, although the main purpose is to synthesize a sig- nal less susceptible to torsional interactions [2]. We investigate the use of multiple input signals on the control de- signs for two realistic systems to demonstrate the applicabil- ity to two different control devices. The first system is a small equivalent Brazilian system in which the unstable open-loop system cannot be stabilized by a single conventional PSS. Here we show that the system can be stabilized with a single PSS using two input signals. The second system is a 24-gener- ator model of a real power system. We show that a thyris- tor-controlled series-compensation (TCSC) damping The use trol region trol schem signals are level contr of multiple of signals, ment, beca ipating reg The con and root-l controller There is no tions, whic PSS De In this sec propose th two input mote sign cus” analy delays, tw investigate PSS De The PSS d chine equ system fir plete syste The mo cates that quency of is due to t against th quency of South syst oscillating Itaipu gen cillations Areia and 12 10 8 6 4 2 0 –2 –2 –1.5 –1 –0.5 0 0.5 1 1.5 Root Locus Design Imaginary Axis Real Axis Figure 4. Root-locus plot of closed-loop system. Table 1. System damping ratios (%). Mode 1 Mode 2 Mode 3 Open loop −12.7 2.8 >50 Design R 11.5 11.4 10.6 Design RD 4.85 5.43 4.97 For a power system covering a large geographic area, communication systems incorporating multiple relay sta- tions, especially when the primary communication path is blocked and backup alternatives have to be used, will increase the time delay. Interarea oscillations involving machines spread over a wide geographical area tend to be of lower frequency, however, and thus can tolerate longer delays. TCSC Design for a 24-Generator System In the second example, we show the control design for a TCSC using two input signals. A TCSC consists of capacitor banks controlled by solid-state thyristor switches and hence has the capability of rapidly modulating the effective impedance on the transmission line where it is located, in response to interarea oscillations. This response capability makes TCSC a very effective damping device to allow for in- creased power transfers. The model used in this example represents a tightly interconnected system and includes 24 generators and their associated controllers. The bulk of the load is connected in the southern part of the system and consists of large industrial and urban centers. Fossil, nu- clear, steam, and hydro turbines are represented in the model. The backbone of the system is a 500-kV transmission southern p The sy interarea m north-sout creases, th relation i north-sout damping o TCSC M Fig. 10 sho takes a tot vides a com while takin its of the p compensa trolled. Th and is simp lay associa of the TCS of 15 ms. T signal u int put summ model can Measur The applic work, such tors, provi of interare control iss these devi selection o depends o the system trate the fe ing contro into a dam mote signa ler [7], [19 This se on the ap interarea m August 2000 IEEE Control Systems Magazin + + + + Xmodulation XTCSC Xorder Xmax Xmin Xref Xfixed Xdelive 1 1+sTTCSC Figure 10. TCSC model block diagram. 9 8 7 6 5 4 3 2 1 0 −1 −2 −1.5 −1 −0.5 0 0.5 Imaginary Axis Real Axis Figure 11. Root-locus plot with y = ω 2 as measured signal. 11 / 50
  12. Inter-area oscillations in power networks 220 309 310 120 103

    209 102 102 118 307 302 216 202 RTS 96 power network swing dynamics Groups of generators oscillate relative to each other due to . . . 1 heterogeneity in responses (inertia Mi and damping Di ) 2 topology: modular & clustered 3 power transfers between areas: aij = Bij cos(θ∗ i − θ∗ j ) 4 interaction of multiple local controllers 12 / 50
  13. Slow coherency and area aggregation aggregated RTS 96 model swing

    dynamics of aggregated model Aggregate model of lower dimension & with less complexity for 1 analysis and insights into inter-area dynamics [Chow & Kokotovic ’85] 2 measurement-based id of equivalenced models [Chakrabortty et.al.’10] 3 design of remedial actions [Xu et. al. ’11] & wide-area control (later) 13 / 50
  14. Setup in slow coherency 220 309 310 120 103 209

    102 102 118 307 302 216 202 original model aggregated model r given areas (spectral partition [Chow et al. ’85]) parameter capturing modularity/clustering: δ = maxα(Σ external connections in area α) minα(Σ internal connections in area α) inter-area dynamics by center of inertia: yα = i∈α Mi θi i∈α Mi , α ∈ {1, . . . , r} intra-area dynamics by area differences: zα i−1 = θi − θ1 , i ∈ α \ {1}, α ∈ {1, . . . , r} 14 / 50
  15. Linear transformation & time-scale separation swing equation T ⇐ =

    = ⇒ T−1 singular perturbation standard form M ¨ θ + D ˙ θ + Lθ = 0 T ⇐ = = ⇒ T−1        d dts     y ˙ y √ δ z √ δ ˙ z     =     ... . . . ... · · · A · · · ... . . . ...         y ˙ y z ˙ z     slow motion given by center of inertia: yα = i∈α Mi θi i∈α Mi fast motion given by intra-area differences: zα i−1 = θi − θ1 slow time scale: ts = δ · t · “max internal area degree” 15 / 50
  16. Area aggregation & approximation singular perturbation standard form: aggregated swing

    equations obtained by δ ↓ 0: d dts     y ˙ y √ δ z √ δ ˙ z    =     ... . . . ... · · · A · · · ... . . . ...         y ˙ y z ˙ z     Ma ¨ ϕ + Da ˙ ϕ + Lred ϕ = 0 Properties of aggregated model [D. Romeres, FD, & F. Bullo, ’13] 1 inertia & damping: Ma =   ... i∈α Mi ...   & Da =   ... i∈α Di ...   2 Laplacian: Lred = “inter-area” + “intra-area contributions” = positive semidefinite Laplacian with possibly negative weights 3 approximation: ∃ δ∗ such that for all δ ≤ δ∗: y(ts) = ϕ(ts) + O( √ δ) 16 / 50
  17. RTS 96 swing dynamics revisited 220 309 310 120 103

    209 102 102 118 307 302 216 202 17 / 50
  18. Canonical setup in wide-area control local actuators, remote measurements, &

    communication backbone power network dynamics generator transmission line wide-area measurements (e.g. PMUs) remote control loops + + + channel noise local control loops ... system noise FACTS PSS & AVR communication & processing wide-area controller ⇒ problem I: signal selection (sensors & actuators) ⇒ problem II: WAC design (subject to control signals) 18 / 50
  19. Spectral analysis reveals the critical modes & areas 1 recall

    solution of ˙ x = Ax: x(t) = i vi eλi t mode #i · wT i x0 contribution from x0 2 analyze eigenvectors & participation factors of weakly damped modes 3 spectral partitioning reveals coherent groups in eigenvectors polarities ℜ(λ) ℑ(λ) x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ℜ(v) ℑ(v) 220 309 310 120 103 209 102 102 118 307 302 216 202 19 / 50
  20. Modal signal selection metrics 572 IEEE TRANSACTIONS ON POWER SYSTEMS,

    VOL. 23, NO. 2, MAY 2008 Assessment of Two Methods to Select Wide-Area Signals for Power System Damping Control Annissa Heniche, Member, IEEE, and Innocent Kamwa, Fellow, IEEE Abstract—In this paper, two different approaches are applied to the Hydro-Québec network in order to select the most effective signals to damp inter-area oscillations. The damping is obtained by static var compensator (SVC) and synchronous condenser (SC) modulation. The robustness analysis, the simulations, and statis- tical results show, unambiguously, that in the case of wide-area sig- nals, the geometric approach is more reliable and useful than the residues approach. In fact, this study shows that the best robustness and performances are always obtained with the stabilizer configu- ration using the signals recommended by the geometric approach. In addition, the results confirm that wide-area control is more ef- fective than local control for damping inter-area oscillations. Index Terms—Compensator, control loop selection, geometric measures, inter-area oscillations, power system stabilizer, residues, wide-area control. I. INTRODUCTION INTER-AREA oscillations have been observed in electrical networks for many years [1]. Many power systems in the world are affected by these oscillations [2]–[4] whose frequency varies between 0.1 and 1 Hz. Currently, inter-area oscillation damping is done with devices that use local signals. The basic question we are asking here is: are these signals really the most efficient? In practice, the choice of measurement and control signals is a problem regularly faced by designers. In fact, to obtain the desired performances and robustness, we have to select signals that allow good observability and controllability of the system modes. To quantify the observability and controllability of the modes, measures have been defined in [5] and [6]. These mea- sures, which are deduced from the Popov Belevich Hautus test [7] and from residues, respectively, indicate how the th mode is observable from available measurements and how it is con- trollable from the system inputs. Thus, it is possible to select, for each mode, the most efficient control loop. By scientific curiosity, we wanted to know if the two methods the results concern only the Hydro-Québec network, it is impor- tant to notice that a statistical analysis was realized. This anal- ysis allowed the test of the two approaches using 1140 different configurations of the network. The aims of this paper are on one hand to show that the two measures do not provide the same conclusion in terms of con- trol loop selection and on the other hand to demonstrate the effi- ciency and reliability of one measure in comparison to the other. To do that, the two measures were applied in order to select the most effective control loops for damping the 0.6-Hz inter-area mode of Hydro-Québec network. Local and global angle shifts were considered. The inter-area damping is obtained by com- pensators modulation. The modulation signal is produced by a multi-band power system stabilizer (MBPSS) which uses only intermediate frequency band [8]. The description and the pa- rameters of this stabilizer are given in the Appendix. This paper is organized as follows. Section II is devoted to system modeling, while Section III presents a brief review of the controllability-observability measures used in this work. Sec- tion IV describes the application. Section V contains the re- sults. Sections VI is devoted to the discussion of the results, and Section VII is the conclusion. II. SYSTEM MODELING An electrical network is a nonlinear system which can be de- scribed by the following nonlinear state equation: (1) where , and are the state, input and output vectors, respectively. n is the dimension of the system, m is the number of inputs, and p is the number of outputs. 1 geometric criteria [H.M.A. Hamdan & A.M.A. Hamdan ’87]: modal controllability: effect of control input #j on mode #i modal observability: effect of control mode #i on sensor #j 2 frequency criteria [M. Tarokh ’92]: modal residues of transfer function ⇒ suboptimal procedures & many requirements: (i) identification of critical modes, (ii) sensor/actuator catalog, & (iii) combinatorial evaluation 20 / 50
  21. Decentralized WAC control design . . . subject to structural

    constraints is tough ⇒ . . . usually handled with suboptimal heuristics in MIMO case IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 4, NOVEMBER 2004 1951 Decentralized Power System Stabilizer Design Using Linear Parameter Varying Approach Wenzheng Qiu, Student Member, IEEE, Vijay Vittal, Fellow, IEEE, and Mustafa Khammash, Senior Member, IEEE Abstract—In this paper, the power system model is formulated as a finite dimensional linear system whose state-space entries depend continuously on a time varying parameter vector called the scheduling variables. This system is referred to as the linear parameter varying (LPV) system. Although the trajectory of the changing parameters such as load levels and tie line flows is not known in advance, in most situations, they can be measured in real time. The LPV technique is applied to the decentralized design of power system stabilizers (PSS) for large systems. In the approach developed, instead of considering the complete system model with all the interconnections, we develop a decentralized approach where each individual machine is considered separately with arbitrarily changing real and reactive power output in a defined range. These variables are chosen as the scheduling variables. The designed controller automatically adjusts its parameters depending on the scheduling variables to coordinate with change of operating conditions and the dynamics of the rest of the system. The resulting decentralized PSSs give good performance in a large operating range. Design procedures are presented and comparisons are made between the LPV decentralized PSSs and conventionally designed PSSs on the 50-generator IEEE test system. Index Terms—Decentralized control, gain scheduling, LPV, os- cillation damping, power system stabilizer. I. INTRODUCTION POWER system operating conditions vary with system con- figuration and load level in a complex manner. The system typically operates over a wide range of conditions. A variety of controllers are employed to ensure that the system operates in a stable manner within its operating range. In the past, many efforts have dealt with the application of robust control tech- niques to power systems, such as Kharitonov’s theorem [1], [2]–[6], [7], [8], and Structured Singular Value (SSV or ) techniques [9], [10]. These methods mainly use one Linear Time Invariant (LTI) controller to guarantee the robust stability and robust performance after describing the changes of oper- ating condition as uncertainties. With the advent of competi- tion and deregulation, systems are being operated closer than ever to their limits, which makes it hard to design a LTI con- troller that performs well at all operating conditions because Manuscript received December 15, 2003. This work was supported by the National Science Foundation under Grants ECS-0338624 and EEC-9908690 and by the Power System Engineering Research Center. Paper no. TPWRS- 00578-2003. W. Qiu and V. Vittal are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50010 USA. M. Khammash is with the Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106-5070 USA. Digital Object Identifier 10.1109/TPWRS.2004.836269 of the inherent system nonlinearity. Gain scheduling is a de- sign technique that has been successfully applied in many en- gineering applications including power systems [11]–[15]. In these attempts, a typical procedure for classical gain scheduling design was followed. This procedure consists of the following steps. Select several operating points which cover the range of the plant’s dynamics and obtain a LTI approximation to the plant at each operating point. For each linearized plant, design a LTI controller to meet the performance requirements; then, using some scheduling scheme, interpolate or schedule the local linear designs to yield an overall nonlinear controller that covers the entire operating range. Although these controllers work well in practice, stability and performance guarantees can not be pro- vided except for slow varying parameters [16], [17]. Further more, since these operating points are usually indexed by some combination of state or reference state trajectories, complex pa- rameter identification blocks are needed to perform scheduling and to deal with delicate stability questions in the switching zone. LPV theory [19], [20] has been developed in the past ten years. It is a natural extension of the conventional gain sched- uling approach. With real measurable scheduling variable(s), it can achieve larger system operating range while guaranteeing the stability and performance not only for slowly changing parameters but also for arbitrarily fast changing parameters. Compared with classical gain scheduling design, not only does it get rid of the strict limitations on the changing rates of scheduling variables, but also it has theoretical guarantees for stability and performance instead of the rule of thumb. LPV gain scheduling technique has been successfully applied in many engineering applications such as flight and process con- trol [21]–[24]. In the flight control problem, the LPV approach based on a single quadratic Lyapunov function is generally applied. Different variables such as altitude, attack angle, and Mach number, are taken as scheduling variables in different cases. The approach in [20] is employed in [24] to achieve improvement by introducing the variation rate bound of the scheduling variable and designing multiple LPV controllers over different operating ranges. These applications demonstrate the usefulness of LPV theory for real engineering problems. The promising results obtained and the actual implementation of this approach in safety critical systems like aircrafts and process control highlight the potential of this technique when applied to large power systems. We focus on PSS design in this paper. The PSS is often used to provide positive damping for power system oscillations. They are mostly single-loop local controllers, which use speed, power input signal, or frequency and synthesize a control signal based 0885-8950/04$20.00 © 2004 IEEE Published in IET Generation, Transmission & Distribution Received on 25th November 2009 Revised on 17th March 2010 doi:10.1049/iet-gtd.2009.0669 ISSN 1751-8687 Robust and coordinated tuning of power system stabiliser gains using sequential linear programming R.A. Jabr1 B.C. Pal2 N. Martins3 J.C.R. Ferraz4 1Department of Electrical & Computer Engineering, American University of Beirut, P.O. Box 11-0236, Riad El-Solh, Beirut 1107 2020, Lebanon 2Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2BT, UK 3CEPEL, Rio de Janeiro, RJ 21941-911, Brazil 4ANEEL, SGAN 603, Brasilia, DF 70830-030, Brazil E-mail: [email protected] Abstract: This study presents a linear programming (LP)-based multivariable root locus following technique for coordinating the gain settings of power system stabilisers (PSSs). The stabiliser robustness is accounted for in the design problem by simultaneously considering the state-space representations and multivariable root loci corresponding to different operating scenarios. The proposed technique computes a curve in the PSS gain parameter space such that when the PSS gains move along this curve to their optimal values, the branches of the corresponding multivariable root loci terminate at satisfactory points in the complex plane. The curve in the gain parameter space is computed via a linear program that successively minimises the Euclidean distance between the unsatisfactory and satisfactory eigenvalue locations. The design method is demonstrated on a 39-bus test system with 14 operating scenarios. A comparison is carried out between the coordination results of two PSS structures, one involving two phase-lead blocks and the other comprised of two phase-lead blocks and a phase-lag block. 1 Introduction The power system stabiliser (PSS) is designed to add damping to the generator rotor oscillations by proper modulation of its excitation voltage [1]. The PSS provides oscillation damping by producing an electrical torque component in phase with the rotor speed deviations. The basic structure of the PSS comprises a gain, phase compensation blocks, a washout filter and output limiters. With rotor speed employed as the PSS input signal, a torsional filter is also commonly used. The phase compensation blocks are used to provide a phase lead that compensates for the phase lag between the exciter input and the generator electrical torque. In practice, the phase-lead network should provide compensation over the entire frequency range of interest (0.1–2 Hz) and under different operating scenarios. It is generally desirable to have some under-compensation so that in addition to significantly increasing the damping torque, the PSS would promote a slight increase in the synchronising torque [1]. A PSS having two phase-lead blocks and a phase-lag block has been proposed as an alternative design to damp inter-area modes without compromising the effect of synchronising torques in the low-frequency spectrum [2]. PSSs of this type were manufactured and have been in continuous operation, for a decade, in three major hydro power stations of Northeastern Brazil. In related research, Kamwa et al. [3] presented a comparison between the main differences in behaviour of two modern digital-based PSSs: the PSS2B and the PSS4B. The modern PSS2B can be easily tuned as a speed-based PSS and has gained widespread use; the multi-channel PSS4B has also been used to achieve higher damping levels for ultra-low-frequency modes, but may require more elaborate tuning. The gain and phase compensation approach [4] has been the most effective and widely used method for designing IET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 8, pp. 893–904 893 doi: 10.1049/iet-gtd.2009.0669 & The Institution of Engineering and Technology 2010 www.ietdl.org IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013 1599 Robust and Low Order Power Oscillation Damper Design Through Polynomial Control Dumisani D. Simfukwe, Student Member, IEEE, and Bikash C. Pal, Senior Member, IEEE Abstract—The paper presents a method for designing low order robust controllers for stabilizing power system oscil- lations. The method uses polynomial control techniques. For single-input/single-output systems (SISO), the variability in operating conditions is captured using an interval polynomial. Kharitonov’s theorem is then used to characterize a fixed order robust controller guaranteeing specified damping. This gives bi-linear matrix inequality (BMI) stability conditions which are solved using the BMI solver PENBMI. The effectiveness of the method is demonstrated by designing power oscillation damping (POD) controllers for single-, four-, and 16-machine power system models. Index Terms—Bi-linear matrix inequality (BMI), controller de- sign, Kharitonov theorem, polynomial methods, power oscillation damping, power system stability. NOMENCLATURE Interval polynomials. Coefficient of polynomial for the term. Maximum and minimum limits on polynomial coefficient . th Kharitonov polynomial. Real and imaginary parts of coefficients of a th controller parameter. Even and odd parts of the polynomial . Hermite-Fujiwara matrix. th complex Kharitonov polynomial of the th polynomial. I. INTRODUCTION THE interconnected power systems inherently exhibit electromechanical oscillations when subjected to dis- turbance. The time scale of such oscillations ranges from tens of milliseconds to several minutes. One of the important oscillations in the range of seconds (0.2 to 1.0 Hz) involves many generators in the interconnected system—commonly known as inter-area oscillations [1]. Often the damping asso- ciated with these oscillations is poor and is dependent on the operating conditions: e.g., level of generation, demand, power flow through the interconnections and network topology and strength. Such variability in the system operation has motivated many researchers over the years to look for a robust oscillation damping strategies [1], [2]. Power system stabilizers (PSSs) as damping aid through generator excitation control and power oscillation dampers (PODs) through various types of flexible IEEE TUANSACTIOSS os POWER SYSTBMS, VOL. IS, so. I, FEBRUARY 2000 313 Robust Pole Placement Stabilizer Design Using Linear Matrix Inequalities P. Shrikant Rao and I. Sen Abstract-This paper presents the design of robust power system stabilizers which place the system poles in an acceptable region in the complex plane for a given set of operating and system conditions. It therefore, guarantees a well damped system response over the entire set of operating conditions. The proposed controller uses full state feedback. The feedback gain matrix is ohtained as the solution of a linear matrix inequality expressing the pole region constraints for polytopic plants. The techniqne is illustrated with applications to the design of stabilizers for a single machine and a 9 bus, 3 machine power system. Index Terms-Linear matrix inequalities, power system dy- namic stahility, robustness. Fig. I. 'The V cotmur, 1. INTRODUCTION system is modeled in terms of the hounds on the frequency response. A H , optimal controller is then synthesized which guarantees robust stability of the closed loop, Other perfor. P O W E R system stabilizers ( " ' ) are now commonly PSS [l] is quite popular with the industry due to its simplicity. However, the performance of these stabilizers can be consid- erably degraded With the changes in the Operatin& "Iition during normal operation. condition due to changes in the loads, generation and the transmission network resulting in accompanying changes in the system dynamics. A well designed stabilizer has to perform satisfactorily in the presence of such variations in the system. In other words, the stabilizer should be robust to changes in the system over its entire operating range. The nonlinear differential equations governing the behavior by utlhtles for dampin& the low frequency oscillations in power systems. The conventional lead com~ens:dion type of specifications such as disturbance attenuation criteria are also imposed on the system. However, it should he noted that the main objective of using a PSS is to provide a good transient behavior, Guaranteed robust stability of the closed loop, though necessary, is not adequate as a specification in pole-zero cancellations and choice of functions used in the design limit the of this techniqLle for pss design, H , design, being essentially a frequency domain approach, does not provide much control over thc transient behavior and closed loop pole location, It would be more desir. able to have a robust stabilizer which, in addition, guarantees an level of small signal transient performance, This can Power systems continually undergo changes in the operating this application, In addition, the problems of poorly damped of a power systeln can be linearized &out a particular operating be achieved by proper placelnellt of the closed loop poles ofthe 810 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 18, NO. 2, MAY 2003 Robust Power System Stabilizer Design Using Loop Shaping Approach Chuanjiang Zhu, Member, IEEE, Mustafa Khammash, Senior Member, IEEE, Vijay Vittal, Fellow, IEEE, and Wenzheng Qiu, Student Member, IEEE Abstract—A robust power system stabilizer (PSS) is designed using Glover-McFarlane’s loop shaping design procedure. Guidance for setting the feedback configuration for loop shaping and synthesis are presented. The resulting PSS ensures the stability of a set of perturbed plants with respect to the nominal system and has good oscillation damping ability. Comparisons are made between the resulting PSS, a conventionally designed PSS, and a controller designed based on the structured singular value theory. Index Terms—Gap metric, loop shaping, oscillation damping, power system stabilizer, structured singular value. I. INTRODUCTION POWER system stabilizers (PSS) have been used for many years to add damping to electromechanical oscillations. They were developed to extend stability limits by modulating the generator excitation to provide additional damping to the oscillations of synchronous machine rotors [1]. Many methods have been used in the design of PSS, such as root locus and sen- sitivity analysis [1], [2], pole placement [3], adaptive control [4], etc. Conventional design tunes the gain and time constants of the PSS, which are mostly lead-lag compensators, using modal frequency techniques. Such designs are specific for a given op- erating point; they do not guarantee robustness for a wide range of operating conditions. To include the model uncertainties at the controller design stage, modern robust control methodologies have been used in recent years to design PSS. The resulting PSS has the ability to controller design is relatively simpler than the synthesis in terms of the computational burden. This paper uses the Glover- McFarlane loop shaping design procedure to design the PSS. It combines the robust stabilization with the classical loop shaping technique. In contrast to the classical loop shaping approach, the loop shaping is done without explicit regard to the nominal plant phase information. The design is both simple and systematic. It does not require an iterative procedure for its solution. The design procedure guarantees the stabilization of a plant set within a ball of certain radius in terms of the gap metric. It is naturally tied to the concept of gap metric and is an elegant approach to synthesize controllers. For power system applications, the Glover-McFarlane loop shaping design has been used by Ambos [12], Pannett [13] et al. to design a controller for generator control. Graham [14] has designed robust controllers for FACTS devices to damp low fre- quency oscillations. In this work, we introduce this design procedure to PSS de- sign both on a four machine system and a 50-machine mod- erate sized system, and provide some basic guidelines for loop shaping weighting selection and controller design paradigm for- mulation. After obtaining the controller, nonlinear simulations are performed and comparisons of the performances are made with the conventional PSS and the controller. Finally, the structured singular value based analysis is performed to eval- uate the robustness of the controller. The rest of the paper is organized as follows: Section II briefly 294 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 Simultaneous Coordinated Tuning of PSS and FACTS Damping Controllers in Large Power Systems Li-Jun Cai and István Erlich, Member IEEE Abstract—This paper deals with the simultaneous coordinated tuning of the flexible ac transmission systems (FACTS) power oscillation damping controller and the conventional power system stabilizer (PSS) controllers in multi-machine power systems. Using the linearized system model and the parameter-constrained nonlinear optimization algorithm, interactions among FACTS controller and PSS controllers are considered. Furthermore, the parameters of the damping controllers are optimized simultane- ously. Simulation results of multi-machine power system validate the efficiency of this approach. The proposed method is effective for the tuning of multi-controllers in large power systems. Index Terms—Comprehensive damping index, coordination, damping control, FACTS, interaction, nonlinear optimization, power oscillation damping (POD), power system stabilizer (PSS), tuning. I. INTRODUCTION DAMPING of power system oscillations between inter- connected areas is very important for the system secure operation. Besides power system stabilizers (PSSs), flexible ac transmission systems (FACTS) devices are also applied to enhance system stability [1], [3], [8], [13], [18], [21]. Particu- larly, in multi-machine systems, using only conventional PSS may not provide sufficient damping for inter-area oscillations. In these cases, FACTS power oscillation damping (POD) con- trollers are effective solutions. Furthermore, in recent years, with the deregulation of the electricity market, the traditional concepts and practices of power systems have changed. Better utilization of the existing power system to increase capaci- ties by installing FACTS devices becomes imperative [25]. FACTS devices are playing an increasing and major role in the operation and control of competitive power systems. However, uncoordinated local control of FACTS devices and PSSs may cause destabilizing interactions. To improve overall system performance, many researches were made on the coor- dination between PSSs and FACTS POD controllers [12]–[16], [27]. Some of these methods are based on the complex non- linear simulation [12], [13], while the others are based on the linearized power system model. In this paper, an optimization-based tuning algorithm is pro- posed to coordinate among multiple controllers simultaneously. This algorithm optimizes the total system performance by means of sequential quadratic programming method. By min- imizing the objective function in which the influences of both Manuscript received March 3, 2004. Paper no. TPWRS-00016-2004. The authors are with the Department of Electrical Power Systems, Univer- sity of Duisburg-Essen, 47057, Germany (e-mail: [email protected]; er- [email protected]). Digital Object Identifier 10.1109/TPWRS.2004.841177 PSSs and FACTS POD controllers are considered, interactions among these controllers are improved. Therefore, the overall system performance is optimized. This paper is organized as follows. Following the introduc- tion, the test system comprising a series FACTS device and 16 generators is described. In Section III, the PSSs and FACTS POD controllers are introduced. In Section IV, simultaneous tuning method is discussed in detail. The simulation results are given in Section V. Finally, brief conclusions are deduced. II. MULTIMACHINE TEST SYSTEM The 16-machine 68-bus simplified New-England power system [6] modified with a series FACTS device, as shown in Fig. 1, is simulated in this study. Each generator is described by a sixth-order model and the series FACTS device is simulated using a power-injection model [4], [10], [12]. By means of the modal analysis, the test system can be di- vided into five areas [6]. The main inter-area oscillations are between area 1, 2, 3 and area 4 because of the relative weak in- terconnections between them. Series FACTS devices are the key devices of the FACTS family and they are recognized as effective and economical means to damp power system oscillation. Therefore, in this research, a series FACTS device, the thyristor-controlled series capacitor (TCSC) is employed for damping of the inter-area oscillations. The TCSC is located between bus A and bus A0 (on the tie line between areas 3 and 4). Its location is determined using the residue method for damping of inter-area oscillations [9], [21], [22]. This paper considers only the dynamic damping control of the FACTS devices. In practice, the placement of the FACTS devices will be based on their cost functions [26]. III. PSS AND FACTS POD CONTROLLER A. PSS PSS acts through the excitation system to introduce a compo- nent of additional damping torque proportional to speed change. It involves a transfer function consisting of an amplification block, a wash out block and two lead-lag blocks [6], [24], [27]. The lead-lag blocks provide the appropriate phase-lead charac- teristic to compensate the phase lag between the exciter input and the generator electrical torque. The lead-lag time constants are determined using the method given in [6], [24], [27]. The structure of the PSS controller is illustrated in Fig. 2. B. FACTS POD Controller In general, the structure of series FACTS POD controller, as shown in Fig. 3, is similar to the PSS controllers [8], [19], [27]. 0885-8950/$20.00 © 2005 IEEE 21 / 50
  22. Challenges in wide-area control 1 signal selection is combinatorial 2

    control design is suboptimal 3 identification of critical modes is somewhat ad hoc what information do you want to extract from the spectrum of a non-normal matrix ? Example: ˙ x = −1 102 0 −1 x x ’ = − x − 100 y y ’ = − y −40 −30 −20 −10 0 10 20 30 40 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y x1 x2 +2 2 +40 40 Today: ⇒ performance metric: variance amplification of stochastic system ⇒ simultaneously optimize performance & control architecture ⇒ fully decentralized & nearly optimal controller 22 / 50
  23. Input-output analysis in H2 - metric . . . complementing/improving

    modal analysis . . . same metric used later for control synthesis linear system with white noise input: ˙ x = Ax + B1 η energy of homogeneous network as performance output: z = Q1/2x power spectral density quantified by Hilbert-Schmidt norm G(j ω) 2 HS = trace (G(j ω) · G∗(j ω)) = i σ2 i G(j ω) steady-state variance of the output quantified by H2-norm G 2 H2 := lim t → ∞ E x(t)T Q x(t) = 1 2π ∞ −∞ G(j ω) 2 HS dω 23 / 50
  24. Slow coherency performance objectives recall sources for inter-area oscillations: 220

    309 310 120 103 209 102 102 118 307 302 216 202 linearized swing equation: M ¨ θ + D ˙ θ + Lθ=P mechanical energy: 1 2 ˙ θM ˙ θ + 1 2 θT Lθ heterogeneities in topology, power transfers, & machine responses (inertia & damp) ⇒ performance objectives = energy of homogeneous network: xT Q x = ˙ θT M ˙ θ + θT In − (1/n) · 1n×n θ other choices possible: center of inertia, inter-area differences, etc. 24 / 50
  25. Case study: IEEE 39 New England power grid model features

    (75 states): sub-transient generator models [Athay et. al. ’79] open loop is unstable exciters & tuned PSSs frequency & damping ratios of dominant inter-area modes 15 5 12 11 10 7 8 9 4 3 1 2 17 18 14 16 19 20 21 24 26 27 28 31 32 34 33 36 38 39 22 35 6 13 30 37 25 29 23 1 10 8 2 3 6 9 4 7 5 F Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16. test system can be represented by ˙ δi = ωi , Hi πfs ˙ ωi = −Di ωi + Pmi − Gii E2 i − 10 j=1,j=i Ei Ej · · {Gij cos(δi − δj ) + Bij sin(δi − δj )},        (11) where i = 2, . . . , 10. δi is the rotor angle of generator i with !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899: 1 10 1.2Hz @ 1.2% 1.1Hz @ 2.6% 1.0Hz @ 3.7% 1.1Hz @ 6.8% 0.7Hz @ 7.8% 25 / 50
  26. Power spectral density . . . reveals inter-area modes &

    local mode # 4 Variance amplification via diagonal elements of output covariance matrix . . . reveal #1 & #9 as crucial 26 / 50
  27. Optimal linear quadratic regulator (LQR) model: linearized ODE dynamics ˙

    x(t) = Ax(t) + B1 η(t) + B2u(t) control: memoryless linear state feedback u = −Kx(t) optimal centralized control with quadratic H2 - performance index: minimize J(K) lim t → ∞ E x(t)T Qx(t) + u(t)T Ru(t) subject to linear dynamics: ˙ x(t) = Ax(t) + B1 η(t) + B2u(t), linear control: u(t) = −Kx(t), stability: A − B2K Hurwitz. (no structural constraints on K) 2 4 6 27 / 50
  28. Sparsity-promoting optimal LQR [Lin, Fardad, & Jovanovi´ c ’13] simultaneously

    optimize performance & architecture minimize lim t → ∞ E x(t)T Qx(t) + u(t)T Ru(t) + γ card(K) subject to linear dynamics: ˙ x(t) = Ax(t) + B1 η(t) + B2u(t), linear control: u(t) = −Kx(t), stability: A − B2K Hurwitz. ⇒ for γ = 0: standard optimal control (typically not sparse) ⇒ for γ > 0: sparsity is promoted (problem is combinatorial) ⇒ card(K) convexified by weighted 1-norm i,j wij |Kij | 28 / 50
  29. Parameterized family of feedback gains K(γ) = arg min K

    J(K) + γ · i,j wij |Kij | 29 / 50
  30. Algorithmic approach to sparsity-promoting control 1 equivalent formulation via observability

    Gramian P: minimize Jγ(K) trace BT 1 PB1 + γ · i,j wij |Kij | subject to A − B2K)T P + P(A − B2K) = −(Q + KT RK) 2 warm-start at optimal centralized H2 - controller with γ = 0 3 homotopy path: continuously increase γ until the desired value γdes 4 ADMM: iterative solution for each value of γ ∈ [0, γdes] 5 update weights: update wij in each ADMM step: wij → 1 |Kij |+ε 6 polishing: structured optimization with desired sparsity pattern 30 / 50
  31. Some ADMM details 0 minimize f (K) + γ ·

    g(K) = H2 - performance + γ · sparsity 1 additional variable/constraint decoupling smooth & separable objectives: minimize f (K) + γ · g(L) subject to K − L = 0 2 introduce augmented Lagrangian Lρ(K, L, Λ) = f (K) + γ · g(L) + trace(Λ(K − L)) + ρ 2 K − L 2 F 3 alternating direction method of multipliers (ADMM): K+ argminK Lρ(K, L, Λ) (iteratively via smooth method) L+ argminL Lρ(K+, L, Λ) (analytically via soft-thresholding) Λ+ Λ + ρ · (K+ − L+) ⇒ guarantees: stabilizing gains (always) & convergent (if locally convex) 31 / 50
  32. Regularization of rotational symmetry rotational symmetry of power flow (absence

    of reference angle) Mi ¨ θi + Di ˙ θi = Pi − j Bij sin(θi − θj ) ⇒ [θ r] = [1n O] is eigenvector of linearized power system models ⇒ eigenvector is not detectable: [1n O]T Q [1n O] = 0 ⇒ no numeric LQR solution with standard Ricatti solvers regularization: xT Q x = ˙ θT M ˙ θ + θT (1 + ε)In − (1/n) · 1n×n θ ⇒ resulting feedback requires absolute angle: Kε [1n O] = ε · [ O] 32 / 50
  33. Performance vs. sparsity Q = energy of homogeneous network ,

    R = In , γ ∈ 10−4, 100 card(K⇤) card(K⇤ 0 ) [%] J⇤ J⇤ 0 J⇤ 0 [%] 10−4 10−3 10−2 10−1 100 0% 20% 40% 60% 80% 10−4 10−3 10−2 10−1 100 0% 0.4% 0.8% 1.2% 1.6% card(K⇤) card(K⇤ 0 ) [%] J⇤ J⇤ 0 J⇤ 0 [%] 10−4 10−3 10−2 10−1 100 0% 20% 40% 60% 80% 10−4 10−3 10−2 10−1 100 0% 0.4% 0.8% 1.2% 1.6% card(K⇤) / card(K⇤ 0 ) [%] (J⇤ J⇤ 0 ) / J⇤ 0 [%] for γ = 1 =⇒ 1.6 % relative performance loss 5.5 % non-zero elements in K 33 / 50
  34. Control architecture & signal exchange network 10 20 30 40

    50 60 70 2 4 6 8 ! = 0.0001604 , card(K ! *) = 460 10 20 30 40 50 60 70 2 4 6 8 ! = 0.0008377 , card(K ! *) = 321 10 20 30 40 50 60 70 2 4 6 8 ! = 0.004375 , card(K ! *) = 178 10 20 30 40 50 60 70 2 4 6 8 ! = 1 , card(K ! *) = 37 For γ = 1: local decentralized optimal control + K∗ 19 θ9(t) 34 / 50
  35. Sparse & nearly optimal wide-area control architecture single wide-area control

    link =⇒ nearly centralized performance 15 5 12 11 10 7 8 9 4 3 1 2 17 18 14 16 19 20 21 24 26 27 28 31 32 34 33 36 38 39 22 35 6 13 30 37 25 29 23 1 10 8 2 3 6 9 4 7 5 F Fig. 9. The New England test system [10], [11]. The system includes 10 synchronous generators and 39 buses. Most of the buses have constant active and reactive power loads. Coupled swing dynamics of 10 generators are studied in the case that a line-to-ground fault occurs at point F near bus 16. 0 -5 0 5 10 15 δ i / rad 0 -5 0 5 10 15 δ i / rad !"#$%&'''%()(*%(+,-.,*%/012-3*%)0-4%5677*%899: 1 10 caveat: ε-regularization results in feedback requiring the absolute angle θ9 . . . but there is no absolute angle ! 35 / 50
  36. too much plain vanilla . . . . . .

    need a closer look at rotational symmetry
  37. Taking the rotational symmetry into account structural constraint: there is

    no absolute angle open-loop: A 1 O = O O =⇒ closed-loop: (A − B2K) 1 O = O O ⇒ elimination of the average mode 1 x = θ r = U 0 0 I T ξ + 1 0 ¯ θ where U is orthonormal with columns ⊥ span (1) embedding in ADMM to promote sparsity in original coordinates minimize f (K) + γ · g(L) subject to KTT − L = 0 36 / 50
  38. Control architecture & signal exchange network under symmetry considerations leads

    to fully decentralized control γ = 0.0818, card (K) = 43 γ = 0.1548, card (K) = 38 γ = 0.2500, card (K) = 35 37 / 50
  39. Performance vs. sparsity Q = energy of homogeneous network ,

    R = In , γ ∈ 10−4, 0.25 (J − Jc ) /Jc card (K) /card (Kc ) 10−4 10−3 10−2 10−1 0 0.5 1 1.5 2 2.5 3 3.5 γ percent 10−4 10−3 10−2 10−1 0 20 40 60 80 100 γ percent γ = 0.25 ⇒ 3.0 % relative performance loss 5.2 % non-zero elements in K ⇒ fully decentralized control is nearly optimal ! 38 / 50
  40. Extensions to block sparsity minimize J(F) + γθ · gθ(Kθ)

    + γr · gr (Kr ) subject to F TT − Kθ Kr = 0 where gθ(Kθ) = i, j wij |Kθij | & off-diagonal block-regularizations are element-wise gr = i, j wij | (Is ◦ Kr )ij | block-wise gr = i = k βik wik || eT i (Is ◦ Kr ) ◦ vT k ||2 row-wise gr = i βi wi || eT i (Is ◦ Kr ) ||2 39 / 50
  41. Block-sparse control architecture under symmetry considerations & block sparsity leads

    to fully decentralized control γ = 0.0697, card (K) = 66 γ = 0.0818, card (K) = 64 γ = 0.2500, card (K) = 62 40 / 50
  42. Performance vs. sparsity Q = energy of homogeneous network ,

    R = In , γ ∈ 10−4, 0.25 (J − Jc ) /Jc card (K) /card (Kc ) 10−4 10−3 10−2 10−1 0 0.5 1 1.5 2 2.5 3 3.5 γ percent 10−4 10−3 10−2 10−1 0 20 40 60 80 100 γ percent γ = 0.25 ⇒ 2.3 % relative performance loss 9.2 % non-zero elements in K ⇒ fully decentralized control is nearly optimal ! 41 / 50
  43. Robustness: optimal control reduces sensitivity nominal controller applied to 10,

    000 operating points with ±20% randomized loading open-loop system centralized controller sparse controller block-sparse controller 43 / 50
  44. Case study: New England – New York test system model

    features (242 states): sub-transient generator models [Singh et. al. ’14] open loop is unstable exciters & tuned PSSs frequency & damping ratios of dominant inter-area modes Figure 1- Line Diagram of the 68-bus system 15 2 3 5 12 13 14 16 7 6 9 8 1 11 10 4 7 23 6 22 4 5 3 20 19 68 21 24 37 27 26 28 29 9 62 65 66 67 63 64 52 55 2 58 57 56 59 60 25 8 1 54 53 47 30 61 36 17 13 12 11 32 33 34 35 45 44 43 39 51 50 18 16 38 10 31 46 49 48 40 41 14 15 42 NETS NYPS AREA 3 AREA 4 AREA 5 1.1Hz @ 3.8% 1.3Hz @ 4.2% 1.1Hz @ 4.7% 1.3Hz @ 4.9% 45 / 50
  45. Block-sparse control architecture γ = 0.0429, card (K) = 115

    γ = 0.0655, card (K) = 109 γ = 0.1, card (K) = 107 46 / 50
  46. Performance vs. sparsity Q = energy of homogeneous network ,

    R = In , γ ∈ 10−4, 10−1 (J − Jc ) /Jc card (K) /card (Kc ) 10−4 10−3 10−2 10−1 0 0.5 1 1.5 2 2.5 3 γ percent 10−4 10−3 10−2 10−1 0 20 40 60 80 100 γ percent γ = 0.1 ⇒ 2.6 % relative performance loss 6.1 % non-zero elements in K ⇒ fully decentralized control is nearly optimal ! 47 / 50
  47. Summary & conclusions 1 analysis of inter-area dynamics via slow

    coherency theory 2 sparsity-promoting distributed optimal wide-area control ⇒ trade-off: sparse control architecture vs. performance 3 extensions to rotational symmetry & block sparsity ⇒ yields fully decentralized & nearly optimal controllers 4 illustrations with New England & New York power grid models Code available online sparsity-promoting wide-area control: http://www.ece.umn.edu/users/mihailo/software/lqrsp/wac.html extensions to rotational symmetry & block sparsity: www.umn.edu/∼mihailo/software/lqrsp/matlab-files/lqrsp wac.zip 49 / 50
  48. Acknowledgements & main references D. Romeres, F. D¨ orfler, and

    F. Bullo. Novel results on slow coherency in consensus and power networks. European Control Conference, July 2013. F. D¨ orfler, M. Jovanovi´ c, Michael Chertkov, and F. Bullo. Sparsity-Promoting Optimal Wide-Area Control of Power Networks. IEEE Transactions on Power Systems, July 2014. X. Wu, F. D¨ orfler, and M. Jovanovi´ c. Input-output analysis and decentralized optimal control of inter-area oscillations in power systems. Available at http://arxiv.org, February 2015. Xiaofan Wu Diego Romeres Mihailo Jovanovi´ c Michael Chertkov Francesco Bullo