# compositional transient stability analysis

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Compositional Transient Stability Analysis OF POWERSYSTEMTRANSCRIPT

Compositional Transient Stability Analysisof Multimachine Power Networks

Sina Yamac Caliskan and Paulo Tabuada

AbstractDuring the normal operation of a power system, all thevoltages and currents are sinusoids with a frequency of 60 Hz inAmerica and parts of Asia or of 50 Hz in the rest of the world.Forcing all the currents and voltages to be sinusoids with the rightfrequency is one of the most important problems in power systems.This problem is known as the transient stability problem in thepower systems literature. The classical models used to study tran-sient stability are based on several implicit assumptions that areviolated when transients occur. One such assumption is the use ofphasors to study transients.While phasors require sinusoidal wave-forms to be well dened, there is no guarantee that waveforms willremain sinusoidal during transients. In this paper, we use energy-based models derived from rst principles that are not subjectto hard-to-justify classical assumptions. In addition to eliminateassumptions that are known not to hold during transient stages, wederive intuitive conditions ensuring the transient stability of powersystems with lossy transmission lines. Furthermore, the conditionsfor transient stability are compositional in the sense that one inferstransient stability of a large power system by checking simpleconditions for individual generators.

Index TermsElectromechanical systems, nonlinear controlsystems, power system dynamics, power system stability.

I. INTRODUCTION

P OWER system is the name given to a collection of devicesthat generate, transmit, and distribute energy to consumingunits such as residential buildings, factories, and street lighting.Abusing language, we use the terms power and energy inter-changeably, as typically done in the power systems literature.Excluding a small portion of generating units, such as solar cellsand fuel cells, we can think of power generators in a powersystem as electromechanical systems [3], [29]. Natural sources,such as the chemical energy trapped in fossil fuels, are used togenerate mechanical energy, which is then converted into elec-trical energy.When power systems are working in normal operating con-

ditions, i.e., in steady state, the generators satisfy two mainconditions: 1) their rotors rotate with the same velocity, whichis also known as synchronous velocity and 2) the generatedvoltages are sinusoidal waveforms with the same frequency.Keeping the velocity of the generators at the synchronousvelocity and the terminal voltages at the desired levels is called

frequency stability and voltage stability, respectively [22].Whenall the generators are rotating with the same velocity, they aresynchronized and the relative differences between the rotorangles remain constant. The ability of a power system to recoverand maintain this synchronism is called rotor angle stability.Transient stability, as dened in [22], is the maintenance of rotorangle stability when the power system is subject to large dis-turbances. These large disturbances are caused by faults on thepower system such as the tripping of a transmission line.In industry, the most common way of checking transient

stability of a power system is to run extensive timedomainsimulations for important fault scenarios [26]. This way ofdeveloping action plans for the maintenance of transient stabilityis easy and practical if we know all the important scenariosthatwe need to consider.Unfortunately, power systems are large-scale systems and the number of possible scenarios is quite large.As an exhaustive search of all of these scenarios is impossible,power engineers need to guess the important cases that theyneed to analyze. These guesses, as made by humans, are proneto errors. Moreover, timedomain simulations do not provideinsight for developing control laws that guarantee transientstability [27]. Because of these reasons, additional methods arerequired for transient stability analysis. Currently, the methodsthat do not rely on timedomain simulations can be collected intwo different groups: direct methods and automatic learningapproaches. The automatic learning approaches [38] are based onmachine learning techniques. In this work, we do not considerautomatic learning approaches and focus on direct methods.

A. Direct Methods and Their Limitations

Directmethods are based on obtainingLyapunov functions forsimplemodelsofpower systems [16], [26], [37].To thebestofourknowledge, theoriginof the idea canbe found in the1947paperofMagnusson [23], which uses the concept of transient energy,which is the sum of kinetic and potential energies to study thestability of power systems. In 1958, Aylett, assuming that a two-machine system can be represented by the dynamical equation

showed that thereexists aseparatrixdividing the two-dimensionalplane of and into two regions [2]. One of the regions is aninvariant set with respect to the two-machine system dynamics,i.e., if the initial condition is in this set, trajectories stay inside thisset for all future time. Aylett concluded that in order to check thestability of the system, we only need to check whether the state isin the invariant set or not. Aylett also characterized the separatrixthat denes the invariant set and extended the results from thetwo-machine case to the three-machine case in the same

Manuscript received September 20, 2013; revised September 23, 2013;accepted January 6, 2014. Date of publication February 5, 2014; date ofcurrent version April 9, 2014. This work was supported in part by theNational Science Foundation (NSF) award 1035916. Recommended byAssociate Editor C. Canudas-de-Wit.

The authors are with the Department of Electrical Engineering, University ofCalifornia, Los Angeles, CA 90095-1594 USA (e-mail: caliskan@ee.ucla.edu;tabuada@ee.ucla.edu).

Color versions of one ormore of the gures in this paper are available online athttp://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TCNS.2014.2304868

4 TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 1, NO. 1, MARCH 2014

2325-5870 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

monograph. Although the term Lyapunov function was notstated explicitly in hiswork,Aylettswork usedLyapunov-basedideas. Some of the other pioneering works on direct methodsinclude Szendy [34], Gless [18], El-Abiad and Nagaphan [15],and Willems [39]. The work based on direct methods mainlyfocused on nding better Lyapunov functions that work for moredetailed models and provide less conservative results. TheseLyapunov functions are used to estimate the region of attractionof the stable equilibrium points that correspond to desiredoperating conditions. The stability of a power system after theclearance of a fault can then be tested by determiningwhether thepostfault state belongs to the desired region of attraction. Forfurther information, we refer the reader to [16], [26][28], and[37]. There are several problematic issues with direct methods.The rst problem is the set of assumptions used to construct

these models. The models used for transient stability analysisimplicitly assume that the angular velocities of the generators arevery close to the synchronous velocity. In other words, it isassumed that the system is very close to desired equilibrium andthe models developed based on this assumption are used toanalyze the stability of the same equilibrium. The standardanswer given to this objection is the following: the models thatare used in transient stability studies are used only for the rstswing transients and for these transients, the angular velocitiesof the generators are very close to the synchronous velocity.Unfortunately, in real-world scenarios, large swings need tobe considered. Citing the postmortem report [35, p. 25] ofAugust 14, 2003 blackout in Canada and the Northeast of theUnited States, the large frequency swings that were inducedbecame a principal means by which the blackout spread across awide area. Using models based on rst swing assumptions toanalyze cases such as the August 14, 2003 blackout does notseem reasonable.The second problem is that the models used for transient

stability analysis, again implicitly, pose certain assumptions onthe grid. The transmission lines are modeled as impedances andthe loads are modeled either as impedances or as constant currentsources. These modeling assumptions are used to eliminate theinternal nodes of the network via a procedure called Kronreduction [3], [13]. The resulting network after Kron reductionis a strongly connected network. Every generator is connected toevery other generator via transmission lines modeled as a seriesconnection of an inductor and a resistor. After this reductionprocess, the resistances in the reduced grid are neglected. Thefundamental reason behind the neglect of the resistances lies inthe strong belief, in the power systems community, about thenonexistence of Lyapunov functions when these resistances arenot neglected. This belief stems from the paper [9], which assertsthe nonexistence of global Lyapunov functions for power sys-tems with losses in the reduced power grid model. It is furthersupported by the fact that the Lyapunov functions that the powersystems community has developed contain path-dependentterms unless these resistances are neglected. The reader shouldnote that the resistors here represent both the losses on thetransmission lines and the loads. Hence, this assumption impliesthat there is no load in the grid (other than the loads modeled ascurrent injections), which is not a reasonable assumption. Inaddition to these problems that have their origin in neglecting the

resistances on the grid, the process of constructing these reducedmodels, i.

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