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

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1 Using Effective Generator Impeance for Force Oscillation Source Location Samuel Chevalier, Stuent Member, IEEE, Petr Vorobev, Member, IEEE, Konstantin Turitsyn, Member, IEEE ariv:78.893v cs.sy 6 Aug 27 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 all 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 response to fluctuations of its terminal voltage an turbine an exciter inputs. 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 efficiency is confirme by simulations on the recommene testbes (eg. WECC 79-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, 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 vibration, or cyclical loas 2 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 effect is the 25 WECC isturbance where a reasonably small 2MW oscillation at the Nova Joffre cogeneration power 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 plant in Canaa resonate with one of the main inter-area moes resulting in a 2MW 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 6. 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 7. Similarly, 8 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, 9 proposes using the self-coherence spectrum of a PMU signal an its time shifte version to perform force oscillation etection. In, 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. One of the most promising methos, which has alreay shown its practical performance, is the Transient Energy Flow (TEF) metho, initially evelope in. One of the main avanteges 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 inepenant. 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 2, 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.

2 2 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 3. 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 track 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: ) A systematic moel is evelope to escribe a generator response to arbitrary perturbations of its terminal voltage along with turbine an exciter inputs. 2) A representation of a generator as an equivalent circuit with amittance matrix Y an current injection I in response to force oscillation is erive. 3) It is shown that the presence of current injection I is an inicator of the source of force oscillations, while a negative real part of amittance Y signifies an effective negative amping contribution from a generator. 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 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 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 79-bus test case of 6 in the presence of measurement noise an generator parameter uncertainty. Finally, conclusions an plans for future work are offere in Section V. II. INTERPRETING GENERATOR DYNAMICS IN THE PRESENCE OF FORCED OSCILLATIONS This section introuces the concept of a so-calle effective amittance matrix Y of a generator in respect to low frequency oscillations. If the generator is a source, then in aition to ~ Fig.. 2 n orer generator voltage E e jδ, terminal voltage V te jθ t, an swing bus voltage V se jθs with θ s. Fig. 2. 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). 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. Classical Generator Tie to a Network We start by consiering a 2 n orer generator with it internal EMF magnitue fixe. This generator is connecte to some terminal bus with positive sequence phasor voltage V t e jθt at frequency ω. Somewhere in this system, there is a swing bus with voltage V s e jθs. This configuration is shown by Fig.. B. Polar an Rectangular Coorinate Transformation We consier small perturbations of the voltage magnitue V t an phase θ t on the terminal bus Ṽt (these perturbations may be cause by the network or the generator itself). Ṽ t + Ṽt (V t + V t )e j(θt+ θt) () We linearize () to compute first orer eviation ( Ṽt) terms. Ṽt V t θ t (j cos(θ t ) sin(θ t )) (2) + V t (cos(θ t ) + j sin(θ t )) 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 ; Fig. 2 graphically portrays these relationships. Re( Ṽ t ) Im( Ṽt) cos(θt ) V t sin(θ t ) sin(θ t ) V t cos(θ t ) Vt T θ t Vt θ t (3)

3 3 C. State Space Formulation In orer to quantify the amittance matrix (Y) an current injection (I), a linearize state space formulation is use. ẋ A x + B u (4) y C x + D u (5) The state variable vector x associate with the 2 n orer generator contains the torque angle (δ) an frequency eviation ( ω) of the generator. The input vector u is efine as containing the mechanical torque variations, two orthogonal terminal bus voltages, an generator EMF. x δ ω (6) u τ m Re(Ṽt) Im(Ṽt) E (7) The swing equation for the 2 n orer generator is formulate with polar variables using a quasi-stationary power flow approximation. We neglect armature resistance R a since it is typically % of transient reactance. δ ω (8) M ω τ m V te sin(δ θ t ) D ω (9) In this expression, we have also assume P m ωτ m τ m since the spee eviations are small. We may linearize this expression an express it with state space formulation. u Vp is the input vector of polar voltage perturbations, u τ is the input torque perturbation, an u E is the input EMF variation. We efine the power angle as φ δ θ t. δ ω ẋ A x + B Vp u Vp + B τ u τ + B E u E () δ VtE M cos(φ) D + () M ω Vt E V M sin(φ) te + M cos(φ) θ t τm + E sin (φ) M V t M Since we wish to be able to relate the currents of the system generators in global rectangular coorinates (an use the components of 7 as inputs), we use the inverse transformation matrix T from (3) to transform the polar input variables into rectangular input variables. We efine u Vr as the input vector of rectangular voltage perturbations an corresponing state space matrix B Vr, where B Vr B Vp T. This is use to reformulate the system s state space representation. B Vr ẋ A x + B Vr u Vr + B τ u τ + B E u E (2) has the following analytical structure. B Vr E M sin(δ) cos (δ) (3) We now consier the moel output. We efine state space output y 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 (4) j I is linearize an split into real an imaginary currents. y C x + D Vr u Vr + D E u E (5) Re (I) E cos(δ ) δ Im (I) E sin(δ ) + (6) ω Re (Vt ) sin(δ) E Im (V t ) cos(δ) D. Frequency Domain Analysis + With the state space moel formulate, we consier a system in which some generator is experiencing a force oscillation (such as τ m (t) G sin(ω t)). In taking the Fourier transform of the stable state space system, ẋ jω x. jω x A x + B Vr ũ Vr + B τ ũ τ + B E ũ E (7) ỹ C x + D Vr ũ Vr + D E ũ E (8) The transfer functions, which irectly relate the inputs to the outputs, can be solve for, where Θ (jω A). x Θ(B Vr ũ Vr + B τ ũ τ + B E ũ E ) (9) ỹ CΘB Vr + D Vr ũ Vr + (2) CΘB τ ũ τ + CΘB E + D E ũ E In this formulation, we may make the following observations. The transfer function which relates terminal bus voltage ifferentials to the current flows acts as an amittance matrix. Similarly, the transfer function relating the torque phasor to the currents flows, in conjunction with the torque phasor, acts as one current source, an the transfer function relating the generator EMF phasor to the currents flows, in conjunction with the EMF phasor, acts as a secon current source. Y C(jΩ A) B Vr + D Vr (2) I τ C(jΩ A) B τ τm (22) I E C(jΩ A) B E + D E Ẽ (23) With this observation, we have the following intuitive moel formulation, where Y is a 2 2 matrix. Re (I) Re (V Im Y t ) (I) Im + I τ + I E (24) (V t ) The structure of Y may be written explicitly. sin δ cos δ cos Y Γ 2 δ sin 2 + δ sin δ cos δ Γ ( V te E 2 2 ) cos(φ) MΩ 2 + j (Ω D) (25) (26)

4 4 ~ ~ Fig. 3. Orientation of the irect () an quarature (q) axes. The negative current injections I τ an I E are similar. I τ Γ cos(δ) E τ sin(δ) m (27) ( I E Γ V ) t sin (φ) cos(δ) sin(δ) E + sin(δ) cos(δ) Ẽ (28) E. Transformation to a Local DQ Reference Frame When consiering the structures of (25), (27), an (28), it is clear that significant simplification may occur by passing to a q reference frame, i.e. rotating all in the irection of the rotor angle δ. We use the convention of q axes orientation from 4, so the rotational matrix which we efine as T 2 is: cos(δ) sin(δ) T 2 (29) sin(δ) cos(δ) We apply this transformation to the state space current flows Ĩ YṼ + I τ + I E of (24). We use the superscript q to enote if the vector of variables is given in the q reference frame, while no superscript enotes variables in the real an imaginary reference frame. For instance, r i is efine in the real an imaginary coorinate system while q q is efine in the q coorinate system. Ĩ q Y q Ṽ q + I q τ + I q E (3) where Y q T 2 YT 2 an q T 2 for any vector. 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. We consier the structure of the amittance matrix an the negative current injections once transforme. Y q Γ Iτ q Γ E I q E Vt sin(φ) Γ E (3) τ m (32) Ẽ (33) It is clear that a torque source will inject no current into the quarature axis circuit since the untransforme current injection is aligne with the axis. 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 Ṽq are all complex phasors. This is a eviation from the stanar power systems literature relate to generator analysis (such as 5) Fig. 4. Circuit iagram interpretation of equation (3), where Iτ Γ E τ m, IE Γ V t sin(φ) E Ẽ, an I q E Ẽ as taken from (32) an (33). At non-source buses, Iτ I E Iq E an all current flows are cause by terminal voltage eviations. 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 (3-33) an Fig. 4. F. 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 metho 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 6. The source generator moel may be approximate by the 6 th orer synchronous moel presente in 6, where the an q subscripts enote the Park reference frames. We choose to investigate this particular moel since it will be use to collect test results in Section IV. δ ω (34) M ω P m P e D ω (35) T ė q E f ( γ ) i e q (36) T qė ( q q γ q ) iq e (37) T ė q e q e q ( + γ i (38) T qė e e + ( q q ) + γ q iq (39) where γ x T x x ( x x) / (T x x), x {, q}. With stator resistance neglecte, the electrical power is P e e i + e q i q, an we write the terminal currents (i, i q ) 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 q e V t sin (δ θ t ) i q R e q V t cos (δ θ t ) (4) where R. We may efine the real an imaginary negative current injections by simply rotating i an i q in rectangular space 5 an negating. Equation (4) is a time omain transformation an shoul not be confuse with the phasor reference frame transformation of (29). sin (δ) cos (δ) i Re (I) Im (I) cos (δ) sin (δ) i q (4) The static AC voltage excitation system associate with the source generator in 6 is approximate by the following block iagram (limits exclue). The source of the force oscillation is given by G sin(ω t) with gain G an forcing frequency Ω. The exciter s associate ifferential equation follows.

5 5 TABLE I CURRENT AND VOLTAGE NOTATION. ALL VECTORS AND VECTOR ELEMENTS ARE COMPLE. Fig. 5. Voltage excitation system associate with the source bus. T A Ė f K A V i E f (42) Now that the full set of ifferential algebraic equations (DAEs) is specifie, the set of ifferential equations (34-39, 42) may be linearize to buil state matrix A an input matrices B ux, where x {P m, E f, etc...}. Similarly, the set of algebraic equations (4, 4) may be linearize to buil output matrix C an feethrough matrix D Vp. To be clear, D Vp is initially written in terms of polar variables V t, θ t since 4 is given in these terms. All matrices relating to V t an θ t (such as D Vp an B Vp ) must be transforme (into D Vr an B Vr ) with (3) as shown in (2) above. Finally, the effective generator amittance matrix Y may be compute using equation 2. Current injection I x may also be compute for any oscillating extraneous system input x. Equations (23) an (22) yiel two such examples for a 2 n orer system, but many more may be constructe base on where the source is locate. III. LEVERAGING Y FOR SOURCE DETECTION At all non-source generators, the negative injection currents Ĩ will be cause exclusively by terminal voltage oscillations since there will be no internal current source. If a generator s moel parameters are available, then the transforme amittance matrix of (2) can be constructe. PMU measurements from the generator terminal (if available) can then be use to calculate Ĩ an Ṽ. Finally, if Ĩ YṼ, then a source may be present in the generator. Of course, generators are complex machines which may have a variety of controllers (AVR, PSS, etc.), but this process may be generalize for an arbitrarily size state matrix A which may incorporate a variety of states. In Algorithm (), we formalize the steps for using generator terminal ata to etermine whether or not a generator is the source of a force oscillation, where I is the magnitue an ϑ is the phase of the current flowing into the generator. There are three primary sources of error in this process: nonlinearities, measurement noise an generator parameter uncertainty. The extent of unavoiable nonlinear system behavior epens on the size of the oscillation, but the associate error is typically small enough to be neglecte. IEEE Stanar C specifics that PMU magnitue error must be below.%, an timing error must be better than µs (or.2 ) 7. Since our metho epens on capturing the phase an magnitue of the force oscillation waves, spectral analysis may be use over many cycles to obtain accurate phase an magnitue ata. Generator parameters may have a much larger egree of uncertainty, with error ranging up to To quantify phasor magnitue M of a noisy, sinusoial time series vector x tn, the authors use M 2max( fft(x tn) ), where fft is the fast Fourier transform. To quantify the phase ϕ of x tn relative to reference signal x rn, the authors use ϕ ccf(hil(x tn), hil(x rn)), where ccf is the correlation coefficient function an hil is the Hilbert transform. Ṽ Ṽr Ṽi Measure Ĩ Ĩr Ĩi Measure (YṼ)r, (YṼ) i real (r) an imaginary (i) generator terminal voltage phasors real (r) an imaginary (i) negative generator current injection phasors Calculate real (r) an imaginary (i) negative generator current injection phasors several percent. Consequently, we propose to efine a Total Vector Error (TVE) threshol for interpreting the ifference between the measure current Ĩ an the compute currents YṼ. Stanar TVE is efine in 8; we formulate it slightly ifferently to quantify the aggregate error of both the real an imaginary current phasors. Equation (43) efines the TVE while Table I summarizes the notation. Ĩr (YṼ) r + Ĩi (YṼ) i TVE (43) Ĩr + Ĩi In (43), we sum the vector error magnitues of the measure an calculate generator currents. We then ivie this total error by the sum of the measure current magnitues to computing an effective average TVE. If this average is below over some threshol ɛ, then the measurements an calculations are sufficiently ifferent an the generator may be a source. START Use available generator moel ata (M,, etc.) to buil state space matrices A, B Vp, C an D Vp. 2 Convert B Vr B Vp T an D Vr D Vp T with 3 to move into rectangular phasor space. 3 Buil Y of 2 4 Collect generator terminal PMU ata 5 Define voltage, current magnitue an phase eviation phasors Ṽt, Ĩt an θ t, ϑ t (set t as phase) 6 Convert magnitue an phase eviation phasors into Ṽ r, Ṽ i an Ĩr, Ĩi appropriately with (3) 7 Determine the TVE with (43) if TVE < ɛ then Current source I not present: Generator not a source else Current source I is present: Generator is a source en Algorithm : Generator Source Detection Metho IV. TEST RESULTS In this section, we present two sets of test results. First, we consier a 3 bus system of two 2 n orer generators tie to an infinite bus. Secon, we test our metho on the WECC 79- bus system. In this system, the source bus generator is ientical to the 6 th orer generator + AVR presente in Section II. A. Raial Generators Tie to Infinite Bus It is well know in the literature 8, that relying on the location of the largest etecte oscillations is an unreliable

6 Voltage Mag: (pu) 6 ~ ~ TABLE III MAGNITUDE OF CURRENT AND VOLTAGE OSCILLATIONS (SCALED BY 4 ); Γ FOR BOTH GENERATORS Ṽ θ Ĩ ϑ Γ Gen e j7 Gen e j37 Fig Bus Diagram with Infinite Bus. All generators are 2 n orer, an a mechanically force oscillation τ m τ + τ is place on generator. TABLE II GENERATOR PARAMETERS M D E V t φ Gen Gen way for etermining the source of a force oscillation, an the excitation of local resonances is typically cite as the reason. While local resonances are certainly to blame, we may use our formulation to explain why non-resonant frequencies may still cause large current oscillations on non-source buses. To o so, we consier the structure of eq. (3): Ĩ Ṽ Ĩq Γ Ṽ q (44) It is clear that large current will flow, regarless of resonance, if Γ is large an the bus voltage oscillations are primarily in the irection of the q axis (see Fig. 3). We emonstrate this phenomenon on a simple eterministic system (no measurement noise or parameter uncertainty). We consier a 3 bus system of two raial generators tie to an infinite bus as given by Fig. 6. Particularly, we wish to show the situation where terminal voltage oscillations are largest on bus (the source), but generator current oscillations are largest on bus 2. The transmission lines have. an R., an other system parameters are summarize in Table II. When tie to an infinite bus, the classical generator moel has a natural V E M frequency of ω n cos (φ). When tie into a network though, we must consier the set of DAEs use to moel the system in orer to fin the natural frequency (local moe) associate with any single generator. ẋ f (x, y) (45) g (x, y) (46) We linearize this set of ifferential (f) an algebraic (g) equations, with state variable vector x an algebraic variable vector y, such that ẋ f x x + f y y an g x x + g y y. Next, we perform the necessary manipulations to buil the state matrix A s f x f y gy g x of the system. The imaginary parts of the complex eigenvalues of this matrix yiel the set of natural frequencies. The natural moes associate with generators an 2 are Ω.78 ra sec an Ω 2.95 ra sec, respectively. We choose to mechanically force the system at Ω 2. since this is between the natural frequency of generator 2 an the frequency which maximizes Γ in generator 2 s amittance matrix. After simulating this system, we measure 5-5 # -5 "V "V Time (s) Fig. 7. Terminal voltage magnitue oscillations for a mechanical oscillation on generator of Fig. 6. The FO commences at t. the magnitue of the voltage an current oscillations. These are summarize in Table III, where bol text enotes the largest magnitues between the generators. Voltage oscillations are largest at generator (the source), but current oscillations are largest at generator 2. Again, this is ue to the fact that Γ for generator 2 is much larger, as given in Table III, an because its voltage oscillations are primary in the irection of it own q axis. Magnitue of oscillation cannot consistently be use to locate the source of a force oscillation since magnitue has a complex relationship with both local resonance an amittance matrix structure. For visualization, Fig. 7 plots the voltage magnitue oscillations associate with generators an 2. In orer to use virtually collecte PMU ata to locate the source of the oscillation, we perform the steps presente in Algorithm () in orer to etermine Ṽ, Ĩ, an Y of both generators. Finally, we choose to rotate the measure (Ĩ) an the calculate (YṼ) currents into the q reference frame given by Fig. 3, since we are ealing with secon orer generators. In table IV, we compare these measure an calculate currents in the q reference frame. Since noise an uncertainty are neglecte, we o not bother computing the total vector errors. At generator 2, the an q axis currents measure from the PMU are equal to the currents compute by Y q Ṽ q. Therefore, it can be conclue that there is no source present on generator 2. The measure an calculate q axis currents on generator are equal, as expecte. There clearly is some sort of source present on the axis though, since the measurements an the calculations are not equal. TABLE IV MEASURED AND CALCULATED CURRENTS Ĩq (BOTH GENERATORS). MAGNITUDES ARE SCALED BY 4. Ĩ Ĩ q Ĩ 2 Ĩ q2 Meas. 4.4e j49.6e j83 8.7e j.e j28 Calc. 6.7e j57.6e j83 8.7e j.e j28

7 Prob: Density Prob: Density Prob: Density Voltage Phase (ra) TVE TVE Detren(3 4) Time (s) Fig. 8. The effects of manually etrening a rifting bus voltage angle are shown. This process is repeate for all reporte angular ata (a) Source Gen Non! Source Gens Generator Inex (b) Source Gen + I Non! Source Gens Generator Inex.4.2 (a) Fig.. TVE is plotte for all generators with at least / th the magnitue of the peak power flow oscillation. Panel (a) gives the TVE for the source bus when only the amittance matrix is use to calculate the current flows. Panel (b) gives the TVE for the source bus when I τ is calculate an ae Generator Parameter Error (%) (b) (c) PMU Magnitue Error (%) PMU Phase Error (Degs:) (a) : r :55 : 2 ~i r9 Meas: ~i r9 Est: ~i i9 Meas: ~i i9 Est: : (b) 2 ~i r36 Meas: ~i r36 Est: ~i i36 Meas: ~i i36 Est: : r :25 Fig. 9. PDFs relate to generator parameter uncertainty (panel (a)), PMU magnitue noise (panel (b)), an PMU phase noise (panel (c)) are plotte. To counteract the effects of the noise, a 3 r orer Savitzky-Golay filter is applie to all time series ata. No uncertainty is ae to generator inertia M since it is a static parameter which is typically known quite accurately. B. WECC-79 Bus System For further valiation, we apply these methos on ata collecte from the WECC-79 Bus System. As suggeste in 6, we ownloae the stanarize test case files an simulate the system using Power Systems Analysis Toolbox (PSAT) 6. We chose to investigate the performance of our methos on test case F. In this system, all loas are constant power while all non-source generators are moele as 2 n orer classical machines with D 4,.25, an various inertias aroun H 3 (machine base). The source generator is a high orer synchronous machine with an Automatic Voltage Regulator (AVR) moele by Fig. 5. Since this is a real system with no infinite bus to anchor the phase angles, there can be consierable angular rifting. This is a secon orer effect which generally will not have a large affect on the linearization analysis which we perform. For this reason, we manually etren the linear rifting by subtracting off a linear offset. Sample results are given in by Fig. 8. In test case F, a.86 Hz sinusoi is ae to the reference signal of the AVR attache to the source generator at bus 4 (see 6 for a full system map). We simulate this system an then ownsample the virtual PMU ata to 3Hz. We then ae white noise to the magnitue an phase time series ata accoring to IEEE Stanar C To account for generator parameter uncertainty, all generator parameters were altere by a percentage chosen from a normal istribution characterize by µ an σ. These istributions are visualize in Fig. 9; parameter uncertainty inclues AVR settings. Following the steps of Algorithm (), we quantifie the Fig.. Panel (a) compares the measure an estimate current phasors at bus 9 (TVE.69). Panel (b) compares the measure an estimate current phasors at bus 36 (TVE.32). We note that ĩ rx an ĩ ix enote the real an imaginary current phasors at bus x, respectively. generator voltage phasors (Ṽr, Ṽi) an negative current injection phasors (Ĩr, Ĩi). The phase of these phasors is quantifie relative to a reference signal of x r cos(ω t) (see footnote for remarks on phasor quantification). Using the uncertain generator parameter ata, we constructe the amittance matrix Y for each generator to compute current estimate Ĩ YṼ. Once these estimates were compute, we use equation (43) to quantify the average total vector error (TVE) for all current phasor calculations. TVE is plotte in Fig.. In Panel (b), we have ae the calculate internal current source which is riven by the oscillating AVR reference signal to the estimation of the source bus current. Of course, knowing the phase an amplitue of such a source is not realistic, but given this information, we use a formulation similar to (23) to quantify the exact current injection I τ. The TVE associate with the source bus in panel (a) is 57 times larger than that of any non source bus. It is clear that the calculate current is rastically ifferent than the measure current, so there must be some sore of internal current source. For further visualization of these results, Fig. presents two sets of measure an calculate phasor currents from generators 9 an 36. The first set of phasor currents (given by the thick line vectors) represent the current phasors irectly measure at the generator terminals. The secon set of phasor currents (given by the thin line vectors) represent the estimate phasor currents when calculate using Ĩ YṼ. Since at both generators Ĩ est Ĩ meas, current is a strict function of voltage, so neither generator has an internal current source an neither is the source of the oscillation. It is important to consier the maximum allowable level of

8 8 2.4 (a) TVE Prob: Density Source Gen Non! Source Gens (b) Generator Parameter Error % Generator Inex Fig. 2. Panel (a) gives the PDF for a large generator parameter uncertainty (σ ). Panel (b) gives the TVE for all generators with at least /th the magnitue of the peak power flow oscillation over, trials. generator parameter uncertainty for which this metho is still robust. Accoringly, we increase the stanar eviation of the generator parameter uncertainty to σ an quantifie the TVE over, instantiations of generator parameter selection. The uncertainty PDF, an the istributions of the results, are both given in Fig. 2. Even with a high egree of generator parameter uncertainty (up to 3%), the TVE at the source bus is still noticeably larger than the error at nonsource buses. In this example, system operators woul be able to ientify the source of the oscillation with little uncertainty. V. C ONCLUSION AND F UTURE W ORK We have evelope a metho for using generator terminal PMU ata to etermine the source of a force oscillation. This is accomplishe by representing any non-source generator as an effective amittance matrix in respect to low-frequency perturbations an any source generator as amittance plus current injection. Our erivations o not rely on any strong moeling assumptions, so there are no restrictions on their applications; generalization for more complex systems is rather straightforwar. Aitionally, through the realistic example provie in subsection IV-B, we have shown that the metho is robust to parameter uncertainty meaning that very etaile moel 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 2 an provie clarification for it s current inaequacies. R EFERENCES S. Achanta, M. Danielson, P. Evans, et al., Time synchronization in the electric power system, North American synchrophasor Initiative, Tech Report NASPI-27-TR-, March L. Vanfretti, S. Bengtsson, V. S. Peri, an J. O. Gjere, Effects of force oscillations in power system amping estimation, in 22 IEEE International Workshop on Applie Measurements for Power Systems (AMPS) Proceeings, pp. 6, Sept 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. 4 46, Aug W. uanyin, L. iaoxiao, 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 , 2. 5 S. A. N. Sarmai, V. Venkatasubramanian, an A. Salazar, Analysis of november 29, 25 western american oscillation event, IEEE Transactions on Power Systems, vol. 3, pp , Nov S. Maslennikov, B. Wang, et al., A test cases library for methos locating the sources of sustaine oscillations, in 26 IEEE Power an Energy Society General Meeting (PESGM), pp. 5, July 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. 5 59, Mar 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 , Jan 25. N. Al-Ashwal, D. Wilson, an M. Parashar, Ientifying sources of oscillations using wie area measurements, in Proceeings of the CIGRE US National Committee 24 gri of the future symposium, Houston, vol. 9, 24. L. Chen, Y. Min, an W. Hu, An energy-base metho for location of power system oscillation source, IEEE Transactions on Power Systems, vol. 28, pp , May 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 , L. Chen, F. u, 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. 9, pp. 2 28, 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 2..8., 2..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. 7, March K. Narenra, D. R. Gurusinghe, an A. D. Rajapakse, Dynamic performance evaluation an testing of phasor measurement unit (pmu) as per ieee c stanar, in Doble Client Committee Meetings Int. Protect. Testing Users Group, Chicago, IL, USA, 22. Samuel C. Chevalier (S 3) receive M.S. (26) an B.S. (25) 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 renewable energy penetration. Petr Vorobev (M 5) receive his Ph.D. egree in theoretical physics from Lanau Institute for Theoretical Physics, Moscow, in 2. 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, an multi-timescale approaches to power system moeling. 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 27. Currently, he is an Assistant 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.