WITH the recent widescale deployment of Phasor Measurement. Using Effective Generator Impedance for Forced Oscillation Source Location

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1 1 Using Effective Generator Impeance for Force Oscillation Source Location Samuel Chevalier, Stuent Member, IEEE, Petr Vorobev, Member, IEEE, Konstantin Turitsyn, Member, IEEE arxiv: v cs.sy 4 May 18 Abstract Locating the sources of force low-frequency oscillations in power systems is an important problem. A number of propose methos emonstrate their practical usefulness, but many of them rely on strong moeling assumptions an provie poor performance in certain cases for reasons still not well unerstoo. This paper proposes a systematic metho for locating the source of a force oscillation by consiering a generator s response to fluctuations of its terminal voltages an currents. It is shown that a generator can be represente as an effective amittance matrix with respect to low-frequency oscillations, an an explicit form for this matrix, for various generator moels, is erive. Furthermore, it is shown that a source generator, in aition to its effective amittance, is characterize by the presence of an effective current source thus giving a natural qualitative istinction between source an nonsource generators. Detaile escriptions are given of a source etection proceure base on this evelope representation, an the metho s effectiveness is confirme by simulations on the recommene testbes (eg. WECC 179-bus system). This metho is free of strong moeling assumptions an is also shown to be robust in the presence of measurement noise an generator parameter uncertainty. Inex Terms Low frequency oscillations of power systems, force oscillations, phasor measurement unit (PMU), power system ynamics I. INTRODUCTION WITH the recent wiescale eployment of Phasor Measurement Units (PMUs) across the US transmission gri 1, system operators are becoming keenly aware of the pervasive presence of low frequency oscillations. Generally, low frequency oscillations are either natural moes, attribute to poorly tune control settings an large power flows across weak tie lines, or force oscillations, which are cause by extraneous isturbances. Such extraneous inputs may be relate to faulty controllers, turbine vibrations, or cyclical loas 4. The appearance of force oscillations reuces the quality of electric power an has potential etrimental effects on various equipment. More importantly, whenever a isturbance occurs at the frequencies close to one of the natural system moes, a resonance conition may lea to significant amplification of amplitue, where a relatively small perturbation on one bus can cause rather large power swings in ifferent locations aroun the system. An example of this This work was supporte in part by the MIT/Skoltech initiative, the Skoltech-MIT Next Generation grant, an the MIT Energy Initiative See Fun Program. S. Chevalier an K. Turitsyn are with Department of Mechanical Engineering, Massachusetts Institute of Technology. schev, turitsyn@mit.eu P. Vorobev is with Department of Mechanical Engineering, Massachusetts Institute of Technology an also with Skolkovo Institute of Science an Technology. petrvoro@mit.eu effect is the 5 WECC isturbance where a reasonably small MW oscillation at the Nova Joffre cogeneration power plant in Canaa resonate with one of the main inter-area moes resulting in a MW power oscillation on the Oregon- California intertie 5. Accoringly, there is a nee in the power systems community for the evelopment of methos which are capable of using on-line PMU ata to trace the source of a force oscillation. It is accepte that esigning control methos for amping of force oscillations is impractical 6; instea, isconnection of the ientifie source with subsequent investigation of the causes of the isturbance is the main solution. A variety of source ientification techniques have been evelope with varying levels of success; many are outline in a recent literature survey 7 where the main requirements for such methos are also state. A set of test cases for valiating ifferent source location methos is presente in 8. These cases were evelope in coorination with IEEE Task Force on Force Oscillations, an they will allow for a stanarize examination of all source etection algorithms. Before applying any source location proceure, the type of observe isturbance has to be ientifie. To ifferentiate between force oscillations an other types of isturbances, a metho base on statistical signatures of ifferent types of oscillations was propose in 9. Similarly, 1 uses spectral analysis of PMU ata to trigger a force oscillation warning. The authors then suggest using statistical tools (pattern mining an maximal variance ratios) from on-line generator SCADA ata to etermine the oscillation source. If oscillation magnitues are low an signal noise is high, 11 proposes using the self-coherence spectrum of a PMU signal an its time shifte version to perform force oscillation etection. In 1, phase coherency is use to ientify groups of generators which swing together. The source is ientifie as the generator in the source group which is proviing the smallest contribution to the overall amping. This will correspon to the generator whose rotor oscillation phase is leaing all other source group rotor oscillation phases. An important class of source location methos, which are terme the hybri methos in 7, leverage both a known system moel an measure PMU ata. Demonstrate in 13 an 14, these methos use measure PMU signals as inputs for a power system moel. After simulating this moel, the time omain moel outputs are compare with their corresponing measure PMU signals. Significant eviation between the moel preictions an the PMU measurements may inicate the presence of a force oscillation. These types of methos are also use for moel valiation.

2 One of the most promising methos, which has alreay shown its practical performance, is the Transient Energy Flow (TEF) metho, initially evelope in 15. One of the main avantages of this metho is that it tracks the flow of effective transient energy in all lines where PMU ata is available, thus being naturally moel inepenent. The authors show that the issipate energy is equivalent to amping torque. Aaption of the metho for use with actual PMU ata, as outline in 6, has been name the Dissipating Energy Flow (DEF). The metho was able to successfully locate the source of a force oscillation in a variety of simulate test cases an in over 5 actual events from both ISONE an WECC. While having the avantage of being moel free, this metho has certain shortcomings, the most important being its inability to istinguish between a true source bus an a bus having an effective negative amping contribution, since both such buses are seen as sources of Transient Energy. A number of rather strong assumptions are also crucial for the metho, namely, constant PQ loas an a lossless network. Accoringly, the metho performs poorly when constant impeance loas are present. In this situation, the metho triggers a false alarm 7 by ientifying such a loa as the isturbance source. This particular shortcoming raises a natural question about the proper efinition of oscillation energy. A full iscussion of the open questions concerning the DEF/TEF methos can be foun in the conclusion of 16. It is clear that a more systematic approach is neee to stuy force oscillations, especially in the evelopment of methos which o not heavily rely on strong moel assumptions. In this paper we have evelope a systematic proceure to locate the sources of force oscillations. We start by eriving a relation between generator terminal voltage an current fluctuations in the presence of persistent oscillation. We then show with minimal moeling assumptions that, base on this relation, it is possible to effectively istinguish between source an nonsource generators. We also apply our results to perturbations with frequencies close to a natural system moe so that the maximum amplitue is observe on a non-source generator. The specific contributions of this paper are as follows. 1) A systematic metho for calculating a generator s frequency response function, with respect to terminal voltage an current perturbations, is given. ) An equivalent circuit interpretation is introuce which treats a generator s frequency response function as an effective amittance matrix Y an any internal force oscillations as current injections I. 3) An explicit force oscillation source location algorithm, which compares preicte an measure current spectrums while making unique measurement noise consierations, is presente. The rest of the paper is structure as follows. In Section II we introuce an effective generator amittance matrix Y with respect to terminal voltage an current perturbations an show that a force oscillation source may be transforme into an effective current injection I. We show the explicit steps for builing the amittance matrix an current injections associate with a classical generator, an we then exten the ~ Fig. 1. n orer generator tie to a network. Internal generator voltage E e jδ, terminal voltage V te jθ t, an swing bus voltage V se jθs with θ s = are all shown. methos to a 6 th orer generator with voltage control. In Section III, we present an algorithm for using Y to etermine if a generator is the source of an oscillation. Section IV presents test results from a 3-bus system an from the stanarize 179-bus test cases of 8 in the presence of measurement noise an generator parameter uncertainty. Also, we inclue a comparison between our algorithm an the DEF metho in the context of a system with a resistive loa. Finally, conclusions an plans for future work are offere in Section V. II. REPRESENTING GENERATORS AS FREQUENCY RESPONSE FUNCTIONS This section introuces the concept of a generator s effective amittance matrix Y which characterizes its frequency response. If the generator is an oscillatory source, then in aition to matrix Y, we show that an effective current source I will appear in parallel with amittance Y. We analytically erive these expressions for a classical generator moel an then show how the methos exten to higher orer moels. A. State Space Formulation for a Classical Generator In this section, the amittance matrix which relates a classical generator s rectangular voltage an rectangular current perturbations is erive. Effective current sources relating to torque an EMF oscillations are also erive. We start by consiering a n orer generator with its internal EMF magnitue fixe. This generator is connecte to some terminal bus with positive sequence phasor voltage V t e jθt at frequency ω. This configuration is shown by Fig. 1. In orer to quantify the amittance matrix (Y) an current injection (I) associate with this generator, a linearize state space formulation is use. ẋ = A x + B u (1) y = C x + D u, () where the state variable vector x contains the torque angle (δ) an spee eviation ( ω) of the generator, an the input vector u contains the mechanical torque variations, two orthogonal terminal bus voltages, an the generator EMF. These are expresse as x = δ ω (3) u = τ m Re(Ṽt) Im(Ṽt) E. (4) The swing equation for the n orer generator is formulate with polar variables using a quasi-stationary power flow ap-

3 3 Fig.. Panel (a) shows the steay state phasor V te jθ t an phasor eviation Ṽt. Panel (b) expans eviation Ṽt from panel (a) an ecomposes the relationship between the rectangular eviations (Re( Ṽt), Im( Ṽt)) an the corresponing polar eviations ( V t, V t θ t). proximation. We neglect armature resistance R a since it is typically 1% of transient reactance X. δ = ω (5) M ω = τ m V te sin(δ θ t ) D ω, (6) X where in this expression, we have also assume P m = ωτ m τ m since the spee eviations are small. This expression may be linearize an expresse in state space formulation. u Vp is the input vector of polar voltage perturbations, u τ is the input torque perturbation, u E is the input EMF variation, an power angle is efine as ϕ = δ θ t : δ ω ẋ =A x + B Vp u Vp + B τ u τ + B E u E (7) 1 δ = VtE MX cos(ϕ) D + (8) M ω Vt E V MX sin(ϕ) te + MX cos(ϕ) θ t 1 τm + E. sin (ϕ) M V t MX In eriving this moel, we wish to relate terminal voltage an current perturbations in rectangular coorinates. To o so, small perturbations of the voltage magnitue V t an phase θ t on the terminal bus voltage Ṽt are consiere, such that Ṽ t + Ṽt =(V t + V t )e j(θt+ θt). (9) After linearizing, the Ṽt components may be separate into their real an imaginary parts, an the polar rectangular relationships may be expresse by employing transformation matrix T 1. Fig. graphically portrays the following relationships: Re( Ṽ t ) Im( Ṽt) = cos(θt ) V t sin(θ t ) sin(θ t ) V t cos(θ t ) u Vr = T 1 u Vp. Vt θ t (1) Accoringly, the inverse transformation matrix T 1 1 from (1) is employe to transform the vector of polar voltage perturbation variables ( u Vp ) into the vector of rectangular voltage perturbation variables ( u Vr ). The corresponing state space matrix is B Vr, where B Vr = B Vp T 1 1. This is use to reformulate the system s state space representation: ẋ =A x + B Vr u Vr + B τ u τ + B E u E. (11) B Vr has the following analytical structure: B Vr = E MX sin(δ) cos (δ). (1) The state space moel s output y is efine as the orthogonal real an imaginary current flows into the generator (we call these the negative current injections): Re(Ṽt) + jim(ṽt) E e jδ I =. (13) jx I is linearize an split into real an imaginary currents. y =C x + D Vr u Vr + D E u E (14) Re (I) E cos(δ ) X = δ Im (I) E sin(δ ) + (15) X ω 1 X Re (Vt ) sin(δ) E Im (V t ) cos(δ) 1 X + 1 X B. Frequency Response Function Construction With the state space moel formulate, the Fourier transform of the system may be taken, such that ẋ = jω x. In this analysis, we note that ũ E = Ẽ an ũ τ = τ are the Fourier transforms of oscillatory steay state eviations, where the respective steay state values are given by E an τ. jω x = A x + B Vr ũ Vr + B τ ũ τ + B E ũ E (16) ỹ = C x + D Vr ũ Vr + D E ũ E. (17) The Frequency Response Functions (FRFs), which irectly relate the inputs to the outputs, can be solve for, where Θ = (jω 1 A) 1 : x =Θ(B Vr ũ Vr + B τ ũ τ + B E ũ E ) (18) ỹ = CΘB Vr + D Vr ũ Vr + (19) CΘB τ ũ τ + CΘB E + D E ũ E. In this formulation, the following observations may be mae. The FRF which relates terminal bus voltage ifferentials to the current flows acts as an amittance matrix. Similarly, the FRF relating the torque phasor to the currents flows, in conjunction with the torque phasor, acts as one potential current source, an the FRF relating the generator EMF phasor to the currents flows, in conjunction with the EMF phasor, acts as a secon potential current source: Y = C(jΩ 1 A) 1 B Vr + D Vr () I τ = C(jΩ 1 A) 1 B τ τm (1) I E = C(jΩ 1 A) 1 B E + D E Ẽ. () With this observation, the following intuitive moel formulation may be observe: ṼR = Y + I τ + I E, (3) ĨRĨI ṼI

4 4 where Y is a matrix an the real (or imaginary) part of the voltage (or current), which is itself a phasor, is given by Ṽ R. The structure of Y may be written explicitly as sin δ cos δ cos Y = Γ δ sin δ sin δ cos δ Γ = ( V te X E X + ) cos(ϕ) MΩ + j (Ω D) 1 X 1 X (4) (5) an the negative current injections I τ an I E are given as I τ = Γ X cos(δ) E τ sin(δ) m (6) ( I E = Γ V ) t sin (ϕ) cos(δ) sin(δ) X E + sin(δ) cos(δ) Ẽ. (7) X When a generator is the source of negative amping, the angle associate with the complex amittance matrix parameter Γ will point into quarants I or II of the complex plane. Accoringly, the FRF of a generator provies a natural interpretation of negative amping with regars to the phase shift relationships between the input an output signals. Future work shall investigate how this property may be exploite to fin locations of negative amping in the system. C. Transformation to a Local q Reference Frame When consiering the structures of (4), (6), an (7), it is clear that significant simplification may occur by passing to a q reference frame, i.e. rotating each expression in the irection of the rotor angle δ. We use the convention of q axes orientation from 17, so the rotational matrix efine as T is cos(δ) sin(δ) T =. (8) sin(δ) cos(δ) This transformation is applie to the state space current injection equation Ĩ = YṼ + I τ + I E of (3). The superscript q enotes variables given in the q reference frame, while no superscript enotes variables in the real an imaginary reference frame. For instance, X = X r X i is efine in the real an imaginary coorinate system while X q = X X q is efine in the q coorinate system. Ĩ q = Y q Ṽ q + I q τ + I q E, (9) where Y q = T YT 1 an X q = T X for any vector X. Fig. 3 provies a visualization of these transformations. In the new coorinate system, the irect () axis is in line with δ, an the quarature (q) axis is perpenicular to the irect axis. Once transforme, the the amittance matrix an the negative current injections are given by Y q = 1 X Γ 1 X Iτ q X = Γ E I q E = Vt sin(ϕ) Γ E 1 X (3) τ m (31) Ẽ. (3) Fig. 3. Orientation of the irect () an quarature (q) axes. ~ ~ Fig. 4. Circuit iagram interpretation of equation (9), where Iτ = Γ X E τ m, IE = Γ V t sin(ϕ) E Ẽ, an I q E = 1 X Ẽ as taken from (31) an (3). At non-source buses, Iτ = IE = Iq E = an all current flows are cause by terminal voltage eviations. A conventional orthogonal circuit iagram interpretation of this result is given by Fig. 4. It is important to remember that Ṽ, Ṽ q, Ĩ, an Ĩq are all complex phasors. This is a eviation from the stanar power systems literature relate to generator analysis (such as 18) which uses orthogonal q ecomposition in orer to treat V q an V as real value signals. We note that the purpose of performing this q rotation is to buil the intuition provie by equations (3-3) an Fig. 4. In general, transforming voltages an current into a q reference frame is not necessary. D. Extension to a 6 th Orer Generator Moel with AVR Although the propose methos for quantifying the effective amittance an current injections of a generator are evelope for a low orer moel, the same techniques may be employe for an arbitrarily complex moel. We choose to consier the source bus generator moel presente in the set of stanarize test cases in 8. The source generator moel may be approximate by the 6 th orer synchronous moel presente in 19, where the an q subscripts enote the Park reference frames. This particular moel is chosen since it will be use to collect test results in Section IV: δ = ω (33) M ω = P m P e D ω (34) T ė q = E f (X X γ ) i e q (35) T qė = ( X q X q γ q ) iq e (36) T ė q = e q e q (X X + γ i (37) T qė = e e + ( X q X q ) + γ q iq, (38) where γ x = T xx x (X x X x) / (T xx x), x {, q}. With stator resistance neglecte, the electrical power is P e = e i + e q i q, an the terminal currents (i, i q ) can be written

5 5 Fig. 5. Voltage excitation system associate with source bus #1 in subsection IV-B. The force oscillation source is given by G sin(ω t). in terms of the terminal voltages (e = V t sin (δ θ t ), e q = V t cos (δ θ t )) an the subtransient voltages (e, e q ): i R X 1 q e = V t sin (δ θ t ) i q R e, q V t cos (δ θ t ) X (39) where R = when neglecte. The real an imaginary negative current injections are compute by simply rotating i an i q in rectangular space 18 an negating. Equation (4) is a time omain transformation an shoul not be confuse with the phasor reference frame transformation of (8): IR sin (δ) cos (δ) i I I = cos (δ) sin (δ) i q. (4) Finally, since PMUs measure the magnitue an phase of voltage an current signals, it is numerically convenient to have the generator s FRF relate voltage magnitue an phase perturbations with current magnitue an phase perturbations. Therefore, the generator moel nees some nonlinear function relating its state an algebraic variables to the current magnitue (I) an current phase (φ): I = I R + I I (41) ( ) φ = tan 1 II. (4) I R Controllers may also be inclue in the generator moel. The static voltage excitation system associate with the source generator of test case F1 in 8 is approximate by the block iagram in Fig. 5 (limits exclue). The source of the force oscillation is given by G sin(ω t) with gain G an forcing frequency Ω. This forcing function is not inclue in the system moel; it is only shown for illustration. The exciter s associate ifferential equation follows: T A Ė f = K A V i E f. (43) Now that the generator s full set of nonlinear Differential Algebraic Equations (DAEs), f an g respectively, have been specifie, they can be written as follows, with state variable vector x, algebraic variable vector y = V θ, an output vector I = I φ : ẋ = f (x, y) (44) I = g (x, y). (45) These DAEs are linearize such that ẋ = f x x + f y y an I = g x x + g y y. Finally, the generator s FRF Y can be built: FRF Y = g x (jω1 f x ) 1 f y + g y. (46) This FRF relates the Fourier transform of the inputs an the outputs across the full spectrum frequencies. In efining the Fourier transform of the time omain signal x(t) as x(ω) = + x(t)e jωt t, we see that the the FRF relates the Fourier transforms of the time omain voltages an currents in the following way: Ĩ(Ω) Ṽ(Ω) = Y(Ω), Ω ). (47) φ(ω) θ(ω) Of course, generators are complex machines which may have a variety of controllers (AVR, PSS, etc.) an a multitue of states, but this process may be generalize for arbitrarily complex DAE systems f an g so long as terminal current can be written as a function of terminal voltage. III. LEVERAGING Y FOR SOURCE DETECTION In a eterministic power system where generator moel parameters are fully known, measurement noise is negligible an perturbations are small, the FRF Y can fully preict the measure spectrum of the generator output Ĩ for a given measure spectrum input Ṽ at all non-source generators. In this ieal system, the following simple test may be naively applie at each generator across the full spectrum of frequencies. Ĩ = YṼ Ĩ YṼ Non-source generator (48) Source generator (49) In other wors, if the measure current spectrum Ĩ an the preicte current spectrum YṼ match, then the generator has no internal oscillation source. If, though, Ĩ YṼ at some particular frequency, then a current source (force oscillation) may be present in the generator at sai frequency. Of course, to implement this test on any given generator, there must be a PMU present which is capable of measuring the generator s terminal voltages an currents so that their respective spectrums may be compute. The realities of power system operation can prevent the naive tests of (48) an (49) from being irectly implemente. There are three primary sources of potential error in this process. First, nonlinearities may prevent the amittance matrix, which is built on a linearize system moel, from exactly preicting the generator ynamics. The extent of nonlinear system behavior epens on the size of the oscillation, but the associate error is typically small enough to be neglecte. Seconly, in builing the FRF, generator parameters (amping, time constants, etc.) may have a large egree of uncertainty. Accoringly, the results presente from tests on the 179-bus system in section IV consier this uncertainty. An thirly, espite the fact that IEEE Stanar C37.4 specifics that PMU magnitue error must be below.1%, an timing error must be better than 1 µs (or. ), aitive error from current an voltage transformer equipment may present aitional error. Since measure voltage an current spectral comparisons can breakown severely when this nontrivial PMU measurement noise is present, the next section introuces a framework for ealing with the problem of aitive measurement noise. A. Bouning Error Associate with PMU Measurement Noise We efine V(t), θ(t), I(t) an φ(t) to be the true voltage magnitue, voltage phase, current magnitue, an current

6 6 phase time series vectors, respectively, at some generator bus. We further assume these vectors are perturbations from their respective steay state operating points. We now efine the measure time series vectors to be ˆX(t), where the true signals are corrupte by Aitive White Gaussian Noise (AWGN) from ɛ X (t): ˆX(t) = X(t) + ɛ X (t), X V, θ, I, φ. (5) In measuring the spectrum of ˆX(t), we invoke the linearity property of the Fourier transform, such that Ĩ Ṽ Y Σ TABLE I DEFINITION OF LSD TERMS FROM (64) Measure 1 vector of complex value current magnitue an phase variables Ĩ(Ω) an φ(ω) Measure 1 vector of complex value voltage magnitue an phase variables Ṽ(Ω) an θ(ω) Moele frequency epenent complex amittance matrix, as given by (46) Estimate upper boun (frequency epenent) on measurement error effects, as given by the maximum l norm of the vector in (56) F{ ˆX(t)} = ˆX(Ω) (51) := X(Ω) + ɛ X (Ω). (5) The Fourier transform of AWGN will ieally have a flat magnitue spectrum (equal to λ ɛx ) an a uniformly istribute phase spectrum characterize by U(, π): ɛ X (Ω) = λ ɛx e ju(,π), Ω ). (53) In applying the amittance matrix transformation of (47) to calculate the ifference in the measure (Ĩ) an the preicte (YṼ) currents at some non-source bus, the following error may be approximate: ( Ĩ + ɛ Ĩ YṼ = I) Y 11(Ṽ + ɛv) Y1( θ + ɛ θ ) ( φ + ɛ φ ) Y 1(Ṽ + ɛv) Y( θ (54) + ɛ θ ) ɛi Y 11 ɛ V Y 1 ɛ θ (55) ɛ φ Y 1 ɛ V Y ɛ θ ɛm :=, (56) ɛp where the simplification in (55) is ue to the fact that, theoretically, Ĩ Y 11Ṽ Y θ 1 = an φ Y 1 Ṽ Y θ = for all frequencies. In (56), the variables ɛ m an ɛ p have been efine which represent the aggregate measurement error spectrums associate with Ĩ YṼ. We seek to quantify this error, at each frequency Ω, with the l norm such that Ĩ YṼ = ɛ m + ɛ p. (57) As can been seen from (55), this error norm will be maximize when the complex entries meet the following phase conitions: ɛ I = (Y 11 ɛ V ) = (Y 1 ɛ θ ) (58) ɛ φ = (Y 1 ɛ V ) = (Y ɛ θ ). (59) Since the measurement error spectrums have uniformly istribute phase angles U(, π), this is a plausible scenario an it provies us with a theoretical upper boun on the measurement error for a generator with known moel parameters an no force oscillation: Σ := max { ɛ m } + max { ɛ p }, (6) where we give the following efinitions for max { ɛ m } an max { ɛ p }: max { ɛ m } = ɛ I + Y 11 ɛ V + Y 1 ɛ θ (61) max { ɛ p } = ɛ φ + Y 1 ɛ V + Y ɛ θ. (6) In (6), Σ is the maximum upper boun on the aggregate measurement error, an it is uniquely efine for all frequencies since both ɛ m an ɛ p are irect functions of frequency. If Ĩ YṼ is significantly larger than Σ at some frequency, then PMU measurement error may not be the source of the error, an an internal force oscillation may be to blame. In calculating (61) an (6), the operator must have a sense of the PMU measurement noise strength. Ieally, this noise strength is constant in the frequency omain, but realistically, it fluctuates for a time limite signal ɛ X (t). Therefore, in estimating the measurement noise strength in any PMU signal, a system operator shoul be conservative in choosing values for λ ɛx from (53). One such conservative choice, which has been foun via experimentation, is to set λ ɛx equal to twice the expecte value of the magnitue of the fast Fourier transform (fft) of its associate time omain signal ɛ X (t), where ɛ X (t) is constructe by sampling length(t) times from N (, σpmu ). Therefore, λ ɛx E fft {ɛ X (t)}. (63) B. Defining a Practical Source Location Technique In computing the error between the measure an preicte currents at a given bus, (6) efines a useful approximate upper boun on the associate measurement error. As long as the strength of the measurement noise is known (or can be estimate, such as in 1), this upper boun can be compute for all frequencies. Assuming an accurate FRF, significant eviations from this upper boun at any given frequency may inicate the presence of an internal current source (force oscillation). To quantify the size of the spectral eviation at each frequency, we introuce a metric terme the Local Spectral Deviation (LSD). Its form is given as follows: LSD = Ĩ YṼ Σ. (64) Table I summarizes the terms in (64) which is compute at all generators for which terminal PMU ata ata is available. Formally, the LSD calculates the ifference in the preiction error an the maximum boun on the effects of measurement noise error. To apply the LSD, the operator shoul first etermine the central forcing frequency Ω of the system (there may be multiple forcing frequencies if the system is experiencing multiple force oscillations). In Algorithm (1), the steps for using generator terminal ata to etermine whether or not a generator is the source of a force oscillation are formalize. In this algorithm, the operator specifie threshol ι is use to etermine if the LSD is large enough for a generator to be eeme a source.

7 Real Power Deviation (pu) 7 We note that this algorithm shoul be applie in situations where an operator has a high egree of certainty that the etecte oscillations are in fact force oscillations (references such as an 3 can be useful to this en); the metho we have evelope will not locate the source of negative amping in a system, an therefore it will be unhelpful in locating the source of a natural oscillation. ~ ~ START 1 Use available generator moel ata to construct DEA sets (44) an (45). Buil the FRF Y of (46) which relates polar voltage an polar current eviations 3 Acquire PMU time series vectors V(t), θ(t), I(t), an φ(t) from the generator terminals 4 Subtract estimate steay state operating points from these time series vectors 5 Take the fft of these perturbation vectors to buil Ĩ(Ω) an Ṽ(Ω) 6 Ientify forcing frequency (or frequencies) Ω 7 Compute the LSD of (64) at Ω if LSD < then Preiction error is less than Σ : Generator is not a source else if < LSD < ι then Preiction error is larger than Σ but less that ι: Generator probably not a source else Preiction error is larger than threshol: Generator is a source en Algorithm 1: Generator Source Detection Metho IV. TEST RESULTS In this section, we present five sets of test results. First, we consier a 3-bus system of two n orer generators tie to an infinite bus. Secon, we test our metho on the moifie WECC 179-bus system in the presence of a force oscillation. Thir, we test our metho on the moifie WECC 179-bus system in the presence of a natural oscillation. Fourth, we apply a rectangular force oscillation in the WECC 179-bus system when a poorly ampe moe is present. An fifth, we contrast the effectiveness of the DEF metho an the FRF source location metho in the context of a three-bus system with a constant impeance loa. A. Raial Generators Tie to Infinite Bus It is well known in the literature 1, 1 that relying on the location of the largest etecte oscillations is an unreliable way for etermining the source of a force oscillation. Because of the excitation of local resonances, large power oscillations can occur at non-source generators. We emonstrate the effectiveness of our source location technique in the presence of resonance amplification occurring on a nonsource generator by simulating the simple 3-bus system of two raial generators tie to an infinite bus as given by Fig. 6. In this system, the lines have X =.1 an R =.1, an Fig Bus Diagram with Infinite Bus. Both generators are n orer, an a mechanically force oscillation τ m = τ + τ is place on generator 1. White noise is applie to the phase an magnitue of the infinite bus voltage "P 1 "P Time (s) Fig. 7. Active power injection eviations for generators 1 an. other system parameters are summarize in Table II. A force oscillation is applie to the mechanical torque of generator 1 via τ m = τ + α sin(ω t). Aitionally, ambient white noise is applie to the magnitue an the phase of the infinite bus voltage to mimic system fluctuations. The riving frequency of the force oscillation Ω is chosen by consiering the eigenvalues of the system. To fin these eigenvalues, the system DAEs of ẋ = f (x, y) an = g (x, y) were linearize such that ẋ = f x x + f y y an = g x x + g y y. The imaginary parts of the complex eigenvalues of the state matrix A s = f x f y gy 1 g x yiel the set of natural frequencies. The natural moes associate with generators 1 an are Ω 1 =.78 ra sec an Ω = ra sec, respectively. We therefore choose to mechanically force the system at f = since this is close to, but not irectly on top of, the natural moe of generator. Fig. 7 shows a time omain simulation plot of the power injection eviations at each generator. The stanar eviation of power injections at generator is almost twice as larger as that of generator 1, an the forcing frequency of f = π Hz can be seen unerneath the system noise. After collecting the time omain voltage an current ata from the simulation, the preicte (YṼ) an measure (Ĩ) current spectrums were compare. For illustrative purposes, measurement noise is not applie an generator moel parameter uncertainty is neglecte such that Y is known exactly for both generators. For a small frequency range, the magnitue spectrum comparisons are given by Fig. 8. There are two important observations concerning these comparisons. First, the spectral peaks of generator (the non-source generator) at the forcing frequency of f =.3 are much larger than the spectral peaks of generator 1 (the source generator) ue to resonance. Secon, the preicte an measure spectrums at the forcing frequency of the source generator (seen in panels (a) & (c)) misalign significantly. From irect visual inspection

8 j ~?1j j ~?j j ~ I1j j ~ Ij Voltage Magnitue (pu) (a) Generator 1 (Source) "! Preicte Measure Generator (Non! Source) (b) Preicte Measure V 7 V 7 + PMU Noise (c) "! Preicte Measure () Preicte Measure Fig. 8. Spectral magnitue of current magnitue (panels (a) & (b)) an current phase (panels (c) & ()) perturbations are given for each generator. The forcing frequency is locate at f =.3. The symbol in panels (a) an (c) highlight the locations of significant eviations between the preicte an expecte spectrums. TABLE II GENERATOR PARAMETERS M D X E V t ϕ Gen Gen of Fig. 8, it is clear that a moest internal oscillation is present on generator 1 which is causing eviations between the measure an the preicte spectrums (the LSD is not compute since measurement noise is not applie in this test). B. WECC 179-Bus System (Force Oscillation) For further valiation, we apply these methos on ata collecte from the WECC 179-bus system in the presence of multiple force oscillations. As suggeste in 8, the stanarize test case files were ownloae an simulate using Power Systems Analysis Toolbox (PSAT) 19. We chose to investigate the performance of our methos on a moifie version of test case F1. In F1, a scale.86 Hz sinusoi is ae to the reference signal of the AVR attache to the source generator at bus 4 (see 8 for a full system map). In the system, all loas are constant power while all non-source generators are moele as n orer classical machines with parameters D = 4, X =.5, an various inertias aroun H = 3 (machine base). The source generator is a sixth orer synchronous machine with an Automatic Voltage Regulator (AVR) moele by Fig. 5. To engener a realistic testing scenario, we moify this test case in three major ways. 1) Loa fluctuations are ae to all PQ loas. The ynamics of these fluctuations are moele by the Ornstein- Uhlenbeck process 4 of u(t) = Eu(t) + 1ξ, (65) where 1 is the nxn ientity matrix for n PQ loas an E is a iagonal matrix of inverse time correlations. ξ is a vector of zero-mean inepenent Gaussian ranom Time (s) Fig. 9. Actual an measure voltage magnitue at bus 7 (generator). variables (stanar eviation σ =.5e 3). The noise vector u(t) is ae to the PQ loas such that S(t) = S (1 + u(t)) (66) where S(t) = P(t) + jq(t). ) Two aitional force oscillations are ae to the system (along with the AVR oscillation at generator bus 4). Each new oscillation is ae to the mechanical torque of a n orer system generator accoring to τ m = τ (1 + α i sin(ω i t)). (67) These force oscillations are arbitrarily ae to generator buses 13 an 65, an in each case α i =.5. One of these oscillations is applie at f =.5 Hz an the secon is applie at f =. Hz. 3) PMU measurement noise is ae to the simulation ata. AWGN with a stanar eviation of σ =.3 (% pu) is applie to all PMU times series vectors. This value of σ was chosen since the associate istribution tails realistically exten up to ±1% pu. For a visualization of the effect of PMU measurement noise in the presence of system ynamics, Fig. 9 shows the bus voltage magnitue of a generator bus (bus 7). The applie noise greatly corrupts the fft calculations. After simulating the system for 1s, the PMU ata from each generator were collecte an analyze accoring to Algorithm (1). In builing the FRF of (46) for each generator, it was assume generator moel parameters were initially known precisely (the en of this subsection will consier parameter uncertainty). Fig. 1 shows a sample of the simulation results associate with generator bus 9 (a non-source generator). These results show three spectral lines in each panel: (i) a measure spectrum magnitue, (ii) a preicte spectrum magnitue, an (iii) a maximum boun on the associate PMU measurement error Σ. (6) was use to compute Σ along with the approximation given by (63). We further assume that σ PMU is roughly known for each PMU. Fig. 1 shows that the measure an preicte current (phase an magnitue) spectrums begin to eviate sharply for frequencies higher than 1 Hz. This is ue to the fact that the amittance matrix amplifies the mi an high frequency measurement noise, which begins the greatly ominate the voltage signal. Fig. 11 shows that the preiction error, though, is always lower than the measurement error boun. This implies that the generator at bus 9 is not an oscillation source.

9 Square Spectral Magnitue Square Spectral Magnitue Square Spectral Magnitue Square Spectral Magnitue Spectral Magnitue Spectral Magnitue Spectral Magnitue Spectral Magnitue 9 1 (a) - ~ I- - -Y 11 ~ V + Y1 ~ jmax f~mgj 1 (a) - ~ I- - -Y 11 ~ V + Y1 ~ jmax f~mgj (b) -? ~ - - -Y 1 ~ V + Y ~ jmax f~pgj 1 (b) -? ~ - - -Y 1 ~ V + Y ~ jmax f~pgj Fig. 1. The spectral magnitue of the measure current magnitue (panel (a)) an the measure current phase (panel (b)) perturbations at generator bus 9 are given by the blue traces. The associate preicte spectral magnitues are given by the black traces. Finally, the orange traces give the estimate maximum PMU measurement noise errors Fig. 1. The spectral magnitue of the measure current magnitue (panel (a)) an the measure current phase (panel (b)) perturbations at generator bus 4 are given by the blue traces. T The associate preicte spectral magnitues are given by the black traces. Finally, the orange traces give the estimate maximum PMU measurement noise errors. 1. ~ I! Y ~V. ' 1. ~ I! Y ~V. ' Fig. 11. The preiction error Ĩ YṼ an the maximum measurement noise error Σ associate with the non-source generator at bus 9 are plotte. Since there is no internal force oscillation, preiction error is mostly cause by measurement error. Accoringly, the preiction error is boune by the conservative maximum measurement noise error estimate Σ. The results of Figs. 1 an 11, which are associate with a non-source bus, can be contraste to Figs. 1 an 13, which are associate with source bus 4. At this generator, the AVR reference is oscillate at f =.86 Hz. This causes large observable ifferences in the measure an preicte magnitue spectrums. In Figs. 14 an 15, the preiction error an measurement noise error bouns are also contraste at generators 13 an 65 (both source generators). As can be seen, there is significant spectral error at the forcing frequencies which the measurement noise cannot account for. This implies that both of these generators are sources of force oscillations. After analyzing the generator spectrums, the LSD can be quantifie at each forcing frequency across all 9 system generators. These results are given in Fig 16. In plotting the LSD inices for each generator at each forcing frequency, the largest spectral eviations are easily foun at the correct source generators. We o not formally efine a threshol parameter ι, which is require in the final steps of Algorithm (1), since it woul have to be foun empirically, by a system operator, via PMU ata collecte over time. We currently o not have access to such ata. Although system generators may be moele reasonably accurately, the generator moel parameters may be known Fig. 13. The preiction error Ĩ YṼ an the maximum measurement noise error Σ associate with the source generator at bus 4 are plotte. Since there is an internal force oscillation at f =.86 Hz, the preiction error greatly excees the measurement noise error boun at this frequency ~ I! Y ~V. ' Fig. 14. The preiction error Ĩ YṼ an the maximum measurement noise error Σ associate with the source generator at bus 13 are plotte. Since there is an internal force oscillation at f =.5 Hz, the preiction error greatly excees the measurement noise error boun at this frequency ~ I! Y ~V. ' 5 1 Fig. 15. The preiction error Ĩ YṼ an the maximum measurement noise error Σ associate with the source generator at bus 65 are plotte. Since there is an internal force oscillation at f =. Hz, the preiction error greatly excees the measurement noise error boun at this frequency.

10 1 LSD for f = :5 Hz.4. (a) Source Gen Non! Source Gens LSD for f = :86 Hz LSD for f = : Hz Voltage Magnitue (pu).4. 1 (b) (c) Source Gen Non! Source Gens Source Gen Non! Source Gens Generator Inex Fig. 16. The LSD is compute at each generator for f =.5 Hz (panel (a)), f =.86 Hz (panel (b)), an f =. Hz (panel (c)). At each frequency, the correct generator is locate, espite strong PMU measurement noise. Generator inex 1 correspons to the generator at bus 4, generator inex 5 correspons to the generator at bus 13, an generator inex 15 correspons to the generator at bus 65. to a lesser egree of accuracy. To consier the effects of generator parameter uncertainty, the LSD is re-quantifie for each generator, but in builing the FRF of (46), generator parameter uncertainty is introuce over 1 trials. Parameter uncertainty inclues all amping, reactance, time constant, an AVR variables. Inertia uncertainty is not consiere since this is a static an typically very well efine parameter. All parameters are altere by a percentage chosen from a normal istribution characterize by µ = an σ = 1%, meaning uncertainty can range up to ±3%. This was the largest stanar eviation for which parameters uncertainty was tolerable. The results, given by Fig. 17, show that the LSD metric is fairly robust to moel parameter uncertainty, although future work will refine this metho for enhance accuracy. In general, this parameter uncertainty analysis inicates that a reasonably accurate generator moel is necessary to employ these frequency response methos at any particular generator. C. WECC-179 Bus System (Natural Oscillation) As a thir test, the amittance matrix source location technique was applie in the presence of a natural oscillation (no force oscillation sources). We use test case ND1 from 8, where a natural oscillation is excite in the WECC 179-bus system. In ND1, all generators are moele as n orer, an most are assigne a amping parameter of D = 4. The generators at buses 45 an 159, though, are assigne D = 1.5 an D = 1, respectively, such that there exists a poorly ampe moe with amping ratio ζ =.1. To excite the system s unerampe moe, a fault is applie at bus 159 for.5s. This system was simulate for 1 secons with the same loa ynamics an PMU measurement noise assumptions as were use in simulating test case F1. The bus voltage magnitue from generator buses 45 an 159 (oscillations are strongest at these generators) are given before, Fig. 17. The LSD is compute at each generator for f =.5 Hz (panel (a)), f =.86 Hz (panel (b)), an f =. Hz (panel (c)) over 1 trials to consier the impact of generator parameter uncertainty A :5s fault V 45 V Time (s) Fig. 18. The voltage magnitue at buses 45 an 159 are plotte before, while, an briefly after the system experiences a fault. Measurement noise is not shown. uring, an after the fault by Fig. 18. As can be inferre from this plot, the excite unerampe natural moe of this system has frequency f n = 1.41 Hz. Since the persisting oscillations are cause by the excitation of a poorly ampe moe, we say the system is experiencing a natural oscillation rather than a force oscillation. Therefore, the source location technique shoul inicate that no generator contains an internal forcing function. To test this theory, the preiction error aggregate Ĩ YṼ an the noise error boun Σ were calculate via Algorithm (1) an plotte for generator buses 45 an 159 (see Figs. 19 an, respectively). In each of these cases, the preiction error slightly excees the noise error at f n = 1.41 Hz. This eviation is very small, though, relative to the strength of the oscillation, an is likely ue to slight nonlinearity of the generator responses (generator current angular perturbations are very large). To further analyze the system, the LSDs were calculate at each generator (we again assume PMU ata were available). Since the LSD is a function of frequency, an there is no forcing frequency, we compute the LSDs at all generators in the range of f = 1.38 to f = 1.4 Hz. We then plot the maximum LSD in this frequency ban for each generator. This result is shown in Fig. 1. In this plot, the maximum LSDs at generators 13 (bus 45) an 8 (bus 159) are seen to cross the zero threshol. Given the strength of the oscillation, as seen in Fig. 18, an the very small eviation between

11 Max LSD for f = 1:38 to 1:4 Hz Square Spectral Magnitue Square Spectral Magnitue Voltage Magnitue (pu) ~ I! Y ~V. ' 1.1 V 65 V Fig. 19. The preiction error Ĩ YṼ an the noise error boun Σ associate with generator 45 are plotte for test case ND1. The preiction error slightly excees the noise error boun at f = 1.41 Hz ~ I! Y ~V. ' 5 1 Fig.. The preiction error Ĩ YṼ an the noise error boun Σ associate with generator 159 are plotte for test case ND1. The preiction error slightly excees the noise error boun at f = 1.41 Hz. the preiction an the measurement, none of the sample generators coul be force oscillation source caniates. More formally, all calculate LSD values are smaller than any realistically chosen ι parameter which woul represent the threshol for etermining if a generator is the source of a force oscillation. We may thus conclue that either the system is being forcibly oscillate by some non-generator piece of equipment or loa, or that a natural oscillation is riving the system s perioic ynamics. D. WECC-179 Bus System (Force + Natural Oscillation) As a fourth test case, we use the natural oscillation test case ND an we ae the force oscillation escribe in test case F63 (both are escribe in 8). Specifically, we set the amping parameters of the generators at buses 35 (D 35 =.5) an 65 (D 65 = 1) such that there exists a poorly ampe moe (ζ =.%) at.37 Hz. Aitionally, we forcibly oscillate generator 79 s AVR reference voltage with a aitive square wave of frequency.4 Hz. In this particular Time (s) Fig.. The voltage magnitue at buses 65 an 79 are plotte over 35 secons. The natural moe frequency of.37 Hz, the forcing frequency of.4 Hz, an the resulting beat frequency can all be seen clearly. Measurement noise is not shown. situation, the presence of a negative amping at generator 65 can cause the generator to be viewe as a source of the so calle transient energy in the DEF metho. Accoringly, the DEF metho will locate this generator as the source of the negative amping. Our FRF metho, though, may be use in a complimentary fashion to fin the force oscillation source. The voltage magnitues at buses 65 an 79 are shown in Fig. over 35 secons. Generator 79 s response to the aitive square wave on the AVR reference is evient. In this test, the force oscillation frequency is only slightly larger than the natural frequency of the poorly ampe moe. This elicits a strong response from the generator at bus 65. Accoringly, we compare the preiction error an measurement noise boun at both generators across the full spectrum of frequencies. In Fig. 3, the the preiction error is seen to be totally containe by the measurement noise error boun at generator 65. This is true for all other generators (asie from generator 79) in the system as well. The resulting negative LSDs at all of these generators, across all frequencies, along with the massively positive LSD at generator 79, inicates there is only one force oscillation source. This is shown by Fig. 5. Further evience that generator 79 is the source of the oscillation can be seen by the Fig. 4. There are a series of preiction error spikes which violate the measurement noise error boun. The statistical signatures of these spikes further inicate that the forcing function is a square wave. To unerstan why, equation (68) gives the Fourier series of a pure square wave g s (t) with funamental frequency f. This series contains frequencies f, 3f, 5f, an so on, just as spectral eviations in Fig. 4 occur at f =.4, 1.,. an.8 Hz: g s (t) = 4 π n=1,3,5... sin (πnft). (68) n Non! Source Gens Generator Inex Fig. 1. The maximum LSD, from f = 1.38 to f = 1.4, is plotte for each generator. The maximum LSDs at generators 13 (bus 45) an 8 (bus 159) are slightly positive, but are still sufficiently small. E. 3-Bus System with Constant Impeance As inicate in 16, network resistances embee in system transfer conuctances (shunt an series) an constant impeance loas may act as the source of transient energy from the viewpoint of the DEF metho. The simplest system known to exhibit this phenomena 16 can be moele as a two generator system with some constant impeance loa (or shunt), as given by Fig. 6. In this system, we apply light Ornstein-Uhlenbeck noise of (65) to the resistive loa in orer

12 Max LSD across all frequencies Square Spectral Magnitue Generator SSM Generator 1 SSM Square Spectral Magnitue W D (p:u:) 1 1. ~ I! Y ~V. '..1 W D 1 W D 13 W D Fig. 3. The preiction error Ĩ YṼ an the noise error boun Σ associate with generator 65 are plotte for the test case where an unerampe natural moe is excite by a force oscillation. Despite a strong oscillatory response from generator 65 at.37 Hz, the preiction error is entirely containe by the measurement noise error boun for all frequencies Time (sec) Fig. 7. The DEF is compute for lines {1}, {13}, an {3}. 1 - (a). ~ I! Y ~V. ' 1. ~ I! Y ~V. ' (b). ~ I! Y ~V. ' Fig. 4. The preiction error Ĩ YṼ an the noise error boun Σ associate with generator 79, the source bus, are plotte for the test case where an unerampe natural moe is excite by a force oscillation. The preiction error violates the measurement noise error at f =.4, 1.,. an.8 Hz Fig. 8. The preiction error Ĩ YṼ (given as the Square Spectral Magnitue (SSM)) an the noise error boun Σ associate with generator 1 (panel (a)) an generator (panel (b)) are given. to mimic system fluctuations, an we apply a force oscillation of Ω = ra sec to the torque on generator 1. After simulating this system an aing white PMU measurement noise with σ =.1 (% pu), the flow of issipating energy was compute accoring to 6, eq. (3). The results are given by Fig. 7. Accoring to the notation introuce in 6, Source Gen Non! Source Gens Generator Inex Fig. 5. The LSD is compute at each generator across the full range of measure frequencies for the natural + force oscillation test case. Only the largest LSD for each generator is plotte here, though. Generator inex 18, which correspons to the generator at bus 79, is correctly ientifie as the source generator. ~ ~ Fig bus system with two n orer generators an a resistive loa. Resistive Ornstein-Uhlenbeck noise is ae to mimic system fluctuations. eq. (5), we foun that DE 1 =.61, DE 13 =.48, an DE 3 =.94. These results inicate that energy is flowing from the resistive loa at bus 3 to the two generator buses. Energy is also flowing from the generator 1 (the source bus) to generator (the system sink). These results o not accurately locate the source of the oscillation ue to the resistive loa. The reasons why are explaine in 16 an shall not be investigate here. We then applie the FRF metho to both generators. In builing the FRF of Y, reactance an amping parameters were perturbe by a percentage pulle from a normal istribution with stanar eviation σ =.5%. The FO is clearly locate at generator 1 ue to the significantly positive LSD at.3hz in panel (a) of Fig. 8. Conversely, the LSD at.3 Hz on generator is effectively. Since the FRF metho presente in this paper is invariant to network ynamics, it is not constraine by loa moeling assumptions. V. CONCLUSION AND FUTURE WORK We have evelope a metho for using generator terminal PMU ata to etermine the source of a force oscillation. This is accomplishe by builing the Frequency Response Function (FRF) for a given generator an comparing its measure an preicte current spectrums. The FRF can be erive from any arbitrary generator moel without simplification, so it is thus unconstraine by moel orer reuction necessities. Unique measurement noise consierations are taken into account to etermine if measurement an preiction eviations are ue to noise or an internal forcing function. Similar to the hybri methos of 13 an 14, our metho assumes prior

13 13 knowlege of generator moels. Unlike the hybri moels though, our metho is simulation free an algebraically simple to implement. Also, PMU noise consierations are more straightforwar to hanle an results may be interprete more intuitively. Through the examples provie in Section IV, we have shown that the metho is robust to moel parameter uncertainty, meaning that very accurate system parameter knowlege is not a bining requirement. In subsequent work, we hope to leverage this technique an the properties of the erive amittance matrices to further characterize how oscillations propagate through the transmission gri. This will len aitional unerstaning into the mechanisms behin the successful Dissipating Energy Flow metho of 6 an provie a framework for improvement investigations. REFERENCES 1 S. Achanta, M. Danielson, P. Evans, et al., Time synchronization in the electric power system, North American synchrophasor Initiative, Tech Report NASPI-17-TR-1, March 17. L. Vanfretti, S. Bengtsson, V. S. Peri, an J. O. Gjere, Effects of force oscillations in power system amping estimation, in 1 IEEE International Workshop on Applie Measurements for Power Systems (AMPS) Proceeings, pp. 1 6, Sept 1. 3 N. Rostamkolai, R. J. Piwko, an A. S. Matusik, Evaluation of the impact of a large cyclic loa on the lilco power system using time simulation an frequency omain techniques, IEEE Transactions on Power Systems, vol. 9, pp , Aug W. Xuanyin, L. Xiaoxiao, an L. Fushang, Analysis on oscillation in electro-hyraulic regulating system of steam turbine an fault iagnosis base on psobp, Expert Systems with Applications, vol. 37, no. 5, pp , 1. 5 S. A. N. Sarmai, V. Venkatasubramanian, an A. Salazar, Analysis of november 9, 5 western american oscillation event, IEEE Transactions on Power Systems, vol. 31, pp , Nov S. Maslennikov, B. Wang, an E. Litvinov, Dissipating energy flow metho for locating the source of sustaine oscillations, International Journal of Electrical Power an Energy Systems, pp. 55 6, B. Wang an K. SUN, Location methos of oscillation sources in power systems: a survey, Journal of Moern Power Systems an Clean Energy, vol. 5, pp , Mar S. Maslennikov, B. Wang, et al., A test cases library for methos locating the sources of sustaine oscillations, in 16 IEEE Power an Energy Society General Meeting (PESGM), pp. 1 5, July X. Wang an K. Turitsyn, Data-riven iagnostics of mechanism an source of sustaine oscillations, IEEE Transactions on Power Systems, vol. 31, pp , Sept J. O Brien, T. Wu, et al., Source location of force oscillations using synchrophasor an scaa ata, in HICSS, N. Zhou an J. Dagle, Initial results in using a self-coherence metho for etecting sustaine oscillations, IEEE Transactions on Power Systems, vol. 3, pp. 5 53, Jan N. Al-Ashwal, D. Wilson, an M. Parashar, Ientifying sources of oscillations using wie area measurements, in Proceeings of the CIGRE US National Committee 14 gri of the future symposium, Houston, vol. 19, H. Wu, Q. Duan, an J. Ma, Disturbance source self-iagnosis of the smart gri, in 1 Spring Congress on Engineering an Technology, pp. 1 4, May J. Ma, P. Zhang, H. j. Fu, B. Bo, an Z. y. Dong, Application of phasor measurement unit on locating isturbance source for low-frequency oscillation, IEEE Transactions on Smart Gri, vol. 1, pp , Dec L. Chen, Y. Min, an W. Hu, An energy-base metho for location of power system oscillation source, IEEE Transactions on Power Systems, vol. 8, pp , May L. Chen, F. Xu, Y. Min, M. Wang, an W. Hu, Transient energy issipation of resistances an its effect on power system amping, International Journal of Electrical Power an Energy Systems, vol. 91, pp. 1 8, P. Kunur et al., Power system stability an control, vol. 7. McGraw-Hill New York, P. Sauer an M. Pai, Power System Dynamics an Stability. Stipes Publishing L.L.C., F. Milano, Power System Analysis Toolbox Reference Manual for PSAT version.1.8.,.1.8 e., Ieee guie for synchronization, calibration, testing, an installation of phasor measurement units (pmus) for power system protection an control, IEEE St C , pp. 1 17, March M. Brown, M. Biswal, S. Brahma, S. J. Ranae, an H. Cao, Characterizing an quantifying noise in pmu ata, in 16 IEEE Power an Energy Society General Meeting (PESGM), pp. 1 5, July 16. R. Xie an D. Trunowski, Distinguishing features of natural an force oscillations, in 15 IEEE Power Energy Society General Meeting, pp. 1 5, July H. Ye, Y. Liu, P. Zhang, an Z. Du, Analysis an etection of force oscillation in power system, IEEE Transactions on Power Systems, vol. 3, pp , March G. Ghanavati, P. D. H. Hines, an T. I. Lakoba, Ientifying useful statistical inicators of proximity to instability in stochastic power systems, IEEE Transactions on Power Systems, vol. 31, pp , March 16. Samuel C. Chevalier (S 13) receive M.S. (16) an B.S. (15) egrees in Electrical Engineering from the University of Vermont, an he is currently pursuing the Ph.D. in Mechanical Engineering from the Massachusetts Institute of Technology (MIT). His research interests inclue power system stability an PMU applications. Petr Vorobev (M 15) receive his PhD egree in theoretical physics from Lanau Institute for Theoretical Physics, Moscow, in 1. Currently, he is a Postoctoral Associate at the Mechanical Engineering Department of Massachusetts Institute of Technology (MIT), Cambrige. His research interests inclue a broa range of topics relate to power system ynamics an control. This covers low frequency oscillations in power systems, ynamics of power system components, multi-timescale approaches to power system moelling, evelopment of plug-anplay control architectures for microgris. Konstantin Turitsyn (M 9) receive the M.Sc. egree in physics from Moscow Institute of Physics an Technology an the Ph.D. egree in physics from Lanau Institute for Theoretical Physics, Moscow, in 7. Currently, he is an Associate Professor at the Mechanical Engineering Department of Massachusetts Institute of Technology (MIT), Cambrige. Before joining MIT, he hel the position of Oppenheimer fellow at Los Alamos National Laboratory, an Kaanoff-Rice Postoctoral Scholar at University of Chicago. His research interests encompass a broa range of problems involving nonlinear an stochastic ynamics of complex systems. Specific interests in energy relate fiels inclue stability an security assessment, integration of istribute an renewable generation.