Statistical Power System Line Outage Detection Under Transient Dynamics

Transcription

1 1 Statistica Power System Line Outage Detection Under Transient Dynamics Georgios Rovatsos, Student Member, IEEE, Xichen Jiang, Member, IEEE, Aejandro D Domínguez-García, Member, IEEE, and Venugopa V Veeravai, Feow, IEEE Abstract A quickest change detection (QCD) agorithm is proposed to address the probem o detecting and identiying ine outages in a power system The statistics o eectricity demand are assumed to be known and propagated through a inearized mode o the equations describing the power ow baance at each node o the network The proposed agorithm incorporates the transient dynamics o the power system oowing a ine outage and is appied to the measurements o votage phase anges, which are coected using phasor measurement units (PMUs) This adaptive agorithm is deveoped by treating the QCD probem as a dynamic composite hypothesis testing probem It is shown to have superior perormance compared to other ine outage detection agorithms previousy proposed in the iterature Case studies demonstrating this gain in perormance are iustrated through the IEEE 118-bus test system Index Terms Line outage detection and identiication, quickest change detection, CuSum test, power systems dynamics, ine aut I INTRODUCTION Timey detection o ine outages in a power system is crucia or maintaining operationa reiabiity In this regard, many onine decision-making toos rey on a system mode that is obtained oine, which can be inaccurate due to bad historica or teemetry data These inaccuracies have been a contributing actor in many recent backouts For exampe, in the 211 San Diego backout, operators were unabe to determine overoaded ines because the network mode was not up to date [1] This ack o situationa awareness imited the abiity o the operators to identiy and prevent the next critica contingency, and ed to a cascading aiure Simiary, during the 23 US Northeastern backout, operators aied to initiate the correct remedia schemes because they had an inaccurate mode o the power system and coud not identiy the oss o key transmission eements [2] These backouts highight the importance o deveoping onine measurement-based techniques to detect and identiy system topoogica changes that arise rom ine outages In this paper, we tacke such topoogy change detection probems by utiizing measurements provided by phasor measurement units (PMUs), which are samped at a much higher rate o 3 sampes every second, or rea-time monitoring o ine outages Rovatsos, Domínguez-García, and Veeravai are with the ECE Department o the University o Iinois at Urbana-Champaign Emai: rovatso2, aedan, vvv@illinoisedu Jiang is with the EE Department o Western Washington University Emai: xichenjiang@wwuedu The work o G Rovatsos and VV Veeravai was supported in part by the Nationa Science Foundation under grants CCF and EECS The work o X Jiang and AD Dominguez-Garcia was supported in part by PSERC, and the Nationa Science Foundation under grant EECS Our work extends the resuts o [3], [4], where the authors deveoped a method or ine outage detection and identiication based on the theory o quickest change detection (QCD) [5], [6] In this method, the incrementa changes in rea power injections are modeed as independent zero-mean Gaussian random variabes Then, the probabiity distribution o such incrementa changes is mapped to that o the incrementa changes in votage phase anges via a inear transormation obtained rom the power ow baance equations The PMUs provide a random sequence o votage phase ange measurements in rea-time; when a ine outage occurs, the probabiity distribution o the incrementa changes in the votage phase anges changes abrupty The objective is to detect a change in this probabiity distribution ater the occurrence o a ine outage as quicky as possibe whie maintaining a desired ase aarm rate In the previous work in [3], a Generaized Cumuative Sum (G-CuSum) based agorithm was proposed to sove this probem For this agorithm, a set o test statistics is computed in parae, one or each ine in the system An outage is decared the irst time any one o the statistics crosses a pre-speciied threshod In this paper, we improve on the method proposed in [4] by considering the power system transient response immediatey oowing the ine outage For exampe, ater an outage, the transient behavior o the system is dominated by the inertia response rom the generators This is oowed by the governor response and then the automatic generation contro (AGC) We incorporate these dynamics into the power system mode by reating incrementa changes in active power demand to active power generation We use this mode to deveop the Dynamic CuSum test (D-CuSum), which is used to capture the transient behavior in the non-composite QCD probem (see eg, [5], [6]) Then, the Generaized Dynamic CuSum test (G-D-CuSum) is derived by cacuating a D-CuSum statistic or each possibe ine outage scenario; an outage is decared the irst time any o the test statistics crosses a pre-speciied threshod The proposed test has better perormance because it considers the transient behavior in addition to the persistent change in the distribution that resuts rom the outage Power companies currenty use dierent approaches or ine outage detection depending on the type o system In distribution systems, which are typicay radia networks, when a reay responds to a aut in a transmission ine by actuating a circuit breaker, the oads that are served downstream rom the aut ocation wi experience outages Otentimes, these outages remain unnoticed unti reported by customers and then restorative actions are taken In transmission systems,

2 2 state estimation is perormed through the supervisory contro and data acquisition (SCADA) system However, the measurements acquired by this system are coected at a rate o one sampe every ive seconds, which is too sow or ine outage detection in rea-time The agorithms or topoogica change detection incude those based on state estimation [7], [8], and rues that mimic system operator decisions [9] More recent methods expoit the ast samping o votage magnitudes and phases provided by PMUs [1] [12] However, these schemes do not expoit the persistent nature o ine outages and do not incorporate transient behavior Ony the most recent PMU measurement is used to determine i an outage has occurred The authors o [13] proposed a method to detect ine outages using statistica cassiiers where a imum ikeihood estimation is perormed on the PMU data The authors aso considered the transient response o the system ater a ine outage by comparing synthesized data against actua data However, their method requires the exact time the ine outage occurs to be known beore appying the agorithm, whereas our proposed method does not have this restriction The remainder o this paper is organized as oows In Section II we describe the mode o the power system adopted in this work, and introduce the statistics describing the votage phase ange beore, during, and ater the occurrence o an outage In Section III, the proposed QCD-based ine outage detection agorithm that accounts or transient behavior ater a ine outage is introduced Other agorithms or ine outage detection in iterature are discussed in Section IV Section V discusses how to identiy ine outages using the proposed agorithm In Section VI, we iustrate the proposed ideas via numerica case studies on the IEEE 118-bus test systems, and compare the resuts o our agorithm with other ine outage detection schemes Finay, concuding remarks and directions or uture work are provided in Section VII II POWER SYSTEM MODEL Let L = 1,, L denote the set o ines in a system with N buses At time t, et V i (t) and θ i (t) denote the votage magnitude and phase ange at bus i, respectivey, and et P i (t) and Q i (t) denote the net active and reactive power injection at bus i, respectivey Then, the quasi steady-state behavior o the system can be described by the power ow equations (see eg, [14]), which or bus i can be compacty written as: P i (t) = p i (θ 1 (t),, θ N (t), V 1 (t),, V N (t)), Q i (t) = q i (θ 1 (t),, θ N (t), V 1 (t),, V N (t)), where the dependence on the system network parameters is impicity captured by p i ( ) and q i ( ) The outage o ine L at time t = t is assumed to be persistent (ie, the ine is not restored unti it is detected to be outaged), with (γ 1) t t < γ t, where t is the time between successive PMU sampes and γ is the index o the sampe immediatey ater the outage In addition, assume that the oss o ine does not cause isands to orm in the post-event system (ie, the underying graph representing the interna power system remains connected) (1) A Pre-outage Mode Let P i [k] := P i (k t) and Q i [k] := Q i (k t), t >, k =, 1, 2,, denote the k th measurement sampe o active and reactive power injections into bus i Simiary, et V i [k] and θ i [k], k =, 1, 2,, denote bus i s k th votage magnitude and ange measurement sampe Furthermore, deine variations in votage magnitudes and phase anges between consecutive samping times k t and (k 1) t as V i [k] := V i [k 1] V i [k], and θ i [k] := θ i [k 1] θ i [k], respectivey Simiary, variations in the active and reactive power injections at bus i between two consecutive samping times are deined as P i [k] = P i [k1] P i [k] and Q i [k] = Q i [k1] Q i [k] Proceeding in the same manner as in [4], we inearize (1) around (θ i [k], V i [k], P i [k], Q i [k]), i = 1,, N, and use the DC power ow assumptions (see eg, [14]), namey, i) at votage proie, ii) negigibe ine resistances, and iii) sma phase ange dierences, to decoupe the rea and reactive power ow equations Then, ater omitting the equation corresponding to the reerence bus, the reationship between votage phase anges and the variations in the rea power injection can be expressed as: P [k] H θ[k], (2) where P [k], θ[k] R (N 1) and H R (N 1) (N 1) is the imaginary part o the system admittance matrix with the row and coumn corresponding to the reerence bus removed In an actua power system, random uctuations in the oad drive the generator response Thereore, in this paper, we use the so-caed governor power ow mode (see eg, [15]), which is more reaistic than the conventiona power ow mode, where the sack bus picks up any changes in the oad power demand Suppose the power system has N d oad buses and N g generator buses In the governor power ow mode, at time instant k, the reation between changes in the oad demand vector, P d [k] R N d, and changes in the power generation vector, P g [k] R Ng, is described by P g [k] = B(t) P d [k], (3) where B(t) is a time dependent matrix o participation actors We approximate B(t) by quantizing it to take vaues B i, i =, 1,, T, where i denotes the time period o interest Let B(t) = B and M := H 1 during the pre-outage period Then, we can substitute (3) into (2) to obtain a pre-outage reation between the changes in the votage anges and the rea power demand at the oad buses as oows: θ[k] M P [k] = M [ P g [k] P d [k] where M = M (1) B M (2) = [M (1) M (2) ] [ B P d [k] P d [k] = (M (1) B M (2) ) P d [k] = M P d [k], ] ] (4)

3 3 B Instantaneous Change During Outage At the time o outage, t = t, there is an instantaneous change in the mean o the votage phase ange measurements that aects ony one incrementa sampe, namey, θ[γ ] = θ[γ 1] θ[γ ] The measurement θ[γ ] is taken immediatey prior to the outage, whereas θ[γ 1] is the measurement taken immediatey ater the outage Suppose the outaged ine connects buses m and n Then, the eect o an outage in ine can be modeed with a power injection o P [γ ] at bus m and P [γ ] at bus n, where P [γ ] is the pre-outage ine ow across ine rom m to n Foowing a simiar approach as the one in [4], the reation between the incrementa votage phase ange at the instant o outage, θ[γ ], and the variations in the rea power ow can be expressed as: θ[γ ] M P [γ ] P [γ 1]M r, (5) where r R N 1 is a vector with the (m 1) th entry equa to 1, the (n 1) th entry equa to 1, and a other entries equa to Furthermore, by using the governor power ow mode o (3), substituting into (5), and simpiying, we obtain: C Post-Outage θ[γ ] M P d [γ ] P [γ 1]M r (6) Foowing a ine outage, the power system undergoes a transient response governed by B i, i = 1, 2,, T 1 unti quasi steady-state is reached, in which B(t) settes to a constant B T For exampe, immediatey ater the outage occurs, the power system is dominated by the inertia response o the generators, which is then oowed by the governor response As a resut o the ine outage, the system topoogy changes, which maniests itse in the matrix H This change in the matrix H resuting rom the outage can be expressed as the sum o the pre-outage matrix and a perturbation matrix, H, ie, H = H H Then, by etting M := H 1 = [M (1) M (2) ], and proceeding in the same manner as the pre-outage mode o (4), we obtain the post-outage reation between the changes in the votage anges and the rea power demand as: θ[k] M,i P d [k], γ i 1 k < γ i, (7) where M,i = M (1) B i M (2), i = 1, 2,, T D Measurement Mode Since the votage phase anges, θ[k], are assumed to be measured by PMUs, we aow or the scenario where the anges are measured at ony a subset o the oad buses, and denote this reduced measurement set by ˆθ[k] Suppose that there are N d oad buses and we seect p N d ocations to depoy the PMUs Then, there are ( N d ) p possibe ocations to pace the PMUs In this paper, we assume that the PMU ocations are ixed; in genera, the probem o optima PMU pacement is NP-hard and its treatment is beyond the scope o this paper Let M = M, i 1 k < γ, M,T, i k γ T Then, the absence o a PMU at bus i corresponds to removing the i th row o M Thus, et ˆM R p N d be the matrix obtained by removing N p 1 rows rom M Thereore, we can reate ˆM to M in (8) as oows: (8) ˆM = C M, (9) where C R p (N 1) is a matrix o 1 s and s that appropriatey seects the rows o M Accordingy, the increments in the phase ange can be expressed as oows: ˆθ[k] ˆM P d [k] (1) The sma variations in the rea power injections at the oad buses, P d [k], can be attributed to random uctuations in eectricity consumption In this regard, we may mode the P d [k] s as independent and identicay distributed (iid) random vectors By the Centra Limit Theorem [16], it can be argued that each P d [k] is a Gaussian vector, ie, P d [k] N (, Λ), where Λ is the covariance matrix Note that the eements P d [k] are roughy independent Since ˆθ[k] depends on P d [k] through the inear reationship given in (1), we have that: := N (, ˆM Λ ˆM ), i 1 k < γ, () := N (µ, ˆM Λ ˆM ), i k = γ, (1) := N (, ˆM,1 Λ ˆM,1), i γ 1 k < γ 2, ˆθ[k] (11) (i) := N (, ˆM,i Λ ˆM,i), i γ i k < γ i1, (T ) := N (, ˆM,T Λ ˆM,T ), i γ T k, where µ := P [γ 1]CM r is the instantaneous meanshit and γ 1 = γ 1 It is important to note that or N (, ˆMΛ ˆM ) to be a nondegenerate probabiity density unction (pd), its covariance matrix, ˆMΛ ˆM, must be u rank We enorce this by ensuring that the number o PMUs aocated, p, is ess than or equa to the number o oad buses, N d, and that they are depoyed at nodes such that the measured votage phase anges are independent The matrices ˆM are known based on the system topoogy oowing a ine outage and Λ can be estimated rom historica records III LINE OUTAGE DETECTION USING QCD In the ine outage detection probem setting, the goa is to detect the outage in ine as quicky as possibe subject to ase aarm constraints The outage induces a change in the statistica characteristics o the observed sequence ˆθ[k] k 1 The aim is to design stopping rues that detect this change A stopping time τ, adapted to the observed sequence, is a random time during which a ine outage is decared The design o such stopping rues is a topic widey studied in the statistica signa processing iterature, under the broader category o quickest change detection (QCD) theory (see eg, [5], [6], [17])

4 4 A Probem Setup The goa in QCD is to design stopping rues to detect the change in the statistica behavior o the observed process as ast as possibe under ase aarm constraints The ase aarm constraint that we choose is based on the mean time to ase aarm (MTFA); thus, we woud ike E [τ] β, where β > is a pre-determined parameter, and E is the expectation under the probabiity measure when no outage has occurred In order to quantiy the detection deay or ine outages, we introduce the oowing deay metric: D (τ) = sup ess sup E γ, γ 1 [ (τ γ ) ˆθ[1],, ˆθ[γ 1] (12) According to (12), the deay o stopping time τ or detecting an outage in ine is measured by taking the expected vaue o (τ γ ) ater (i) assuming that the underying distribution is the one induced on the observations when an outage occurs in ine at time instant t, and (ii) conditioning on a set o preoutage observations ˆθ[1],, ˆθ[γ 1] In addition, the presence o the ess sup operation is equivaent to conditioning on the worst possibe set o observations beore the change, which is the set o observations that imize the deay or a given γ Finay, since the instant o outage is unknown, a supremum is taken over a possibe vaues o γ It shoud be noted that athough the deay metric o (12) is generay diicut to compute, or the agorithms studied in this paper, it can be easiy estimated by Monte Caro simuations (see eg, [6]) We now provide an overview o QCD adapted to the ine outage setting that incudes transient dynamics In the setting described in Section II, we assume that a sequence o observations ˆθ[k] k 1 is measured by PMUs and passed sequentiay to a decision maker According to the statistica mode in (11), beore an outage has occurred, ˆθ[k] At an unknown time instant t, an outage occurs in ine and the distribution o ˆθ[k] changes rom to () Then, the system undergoes a series o transient responses which corresponds to the distribution o ˆθ[k] evoving rom () to (T ) First, a meanshit takes pace during the instant o change t, where the pd is () Then, the statistica behavior o the process is characterized by a series o changes ony in the covariance matrix o the measurements In cassic QCD theory, the CuSum agorithm or the noncomposite setting (with known pre- and post-outage distribution) and Generaized CuSum (G-CuSum) agorithm or the composite setting (where pre-outage distribution is known and post-outage distribution beongs to a known set o distributions), are used to detect a persistent change in the distribution o a sequence These tests have optimaity properties with respect to popuar deay-mtfa ormuations (see eg, [18]- [2]) However, these agorithms are derived or statistica modes that do not consider the transient behavior o the system oowing a change event Here, we propose a stopping rue that expoits the transient phenomena oowing a ine outage which resuts in perormance gains over other methods For an intuitive interpretation o the detection agorithms discussed in this paper, we introduce the Kuback-Leiber (KL) divergence, which is an inormation theoretic measure ] o the discrepancy between two probabiity distributions, and g, deined as: D( g) := (x) og (x) dx (13) g(x) It is easy to show that D( g), with equaity i and ony i = g B Generaized CuSum Test The Generaized CuSum (G-CuSum) based test was proposed as a ine outage detection scheme in [3] with the understanding that the transition between pre- and post-outage periods is not characterized by any transient behavior other than the meanshit that occurs at the instant o outage The meanshit was captured by introducing an additiona ogikeihood ratio term between the distribution at the time o change and the distribution beore the change The ina test statistic takes the imum o this og-ikeihood ratio and the traditiona G-CuSum test recursion Athough the G-CuSum agorithm does not take any transient dynamics into consideration, it can sti perorm we when the transient distributions and the ina post-change distribution are simiar, ie, when the KL divergence between (i), i = 1, 2,, T 1, and (T ) is sma As a resut, it is useu to compare the perormance o the G-CuSum test with the perormance o the G-D-CuSum test that is proposed in this work Since the ine that is outaged is not known a priori, the G-CuSum test works by using the CuSum test statistics in a generaized manner As a resut, we compute L CuSum statistics in parae, one corresponding to each ine outage scenario, and decare a change when an outage to any ine is detected The CuSum recursion or ine is cacuated by accumuating og-ikeihood ratios between (T ) and In particuar, deine the G-CuSum statistic corresponding to ine outage recursivey as: W C [k] = W C [k 1] og (T ), og (),, (14) with W C [] = or a L The goa is to decare an outage as soon as any ine is outaged; thus, the agorithm decares a detection the irst time any o the ine statistics crosses its corresponding threshod Accordingy, the stopping time o the test is as oows: τ C = min ink 1 : W C [k] > A, (15) L with A > being the threshod corresponding to ine C Generaized Dynamic CuSum Test Since the statistica mode used in this paper incudes an arbitrary number o transient periods with inite duration, each one corresponding to a respective transient distribution induced on the observations, it is cear that the G-CuSum test

5 5 o [3] needs to be modiied to take this transient behavior into consideration Toward this end, we introduce the Generaized Dynamic CuSum (G-D-CuSum) test This test is derived by expoiting the so-caed Dynamic CuSum (D-CuSum) test, a test aso proposed in this work This test arises as a soution to the non-composite QCD probem under the presence o an arbitrary number o transient periods The D-CuSum test statistic is derived by ormuating the transient QCD probem as a dynamic composite hypothesis testing probem at each time instant The G-D-CuSum agorithm uses the test statistics o the D-CuSum test in a generaized manner, ie, cacuates a test statistic or each possibe ine outage in parae, and decares an outage when one o the ine statistics crosses a pre-determined positive threshod corresponding to the ine We reer the reader to the Appendix or an in depth anaysis o the D-CuSum test and the derivation o the test statistics based on the imum ikeihood ratio interpretation o the CuSum test By using the D-CuSum test statistic as a basis, we propose the G-D-CuSum test The statistic or ine is given as oows: W D [k] = Ω () [k],, Ω (T ) [k],, (16) where Ω (i) [k] = Ω(i) [k 1], Ω(i 1) [k 1] og (i), (17) or i 1,, T, Ω () [k] := og () and Ω (i) [] := or a L and a i The corresponding stopping rue is deined as τ D = min ink 1 : W D [k] > A (18) L Cacuating the test statistic or ine invoves cacuating the statistics Ω (),,Ω (T ) The ina test statistic is given by taking the imum o these terms Note that since the meanshit eect hods ony or one time instant, Ω () corresponds to a og-ikeihood ratio between the distribution at the outage and the pre-outage distribution To renew each Ω statistic, the vaue o the statistic in the previous time instant and the vaue o the statistic used to detect the previous distribution change is used The basis o this agorithm is that each statistic is used to capture one o the transient distributions As a resut, at each dierent period that the process goes through, one o the Ω statistics wi dominate the others, eading to the adaptive nature o the agorithm The test statistics are designed to use prior inormation rom other test statistics, expoiting the act that distribution changes occur in a sequentia manner It is aso important to note that the structure o the agorithm is not aected by the duration o any o the transient periods This is because the test statistic is cacuated by imizing a og-ikeihood ratio over a possibe changepoint aocations (see the Appendix) IV OTHER ALGORITHMS FOR CHANGE DETECTION In this section, we present two additiona change detection agorithms that are o ower compexity than the G-D-CuSum test First, we present the Meanshit test, a detection scheme that can be shown to be equivaent to that proposed in [1] or detecting ine outages Next, we present a modiied version o the Shewhart test (see eg, [6]) that accounts or transient dynamics but without using the history o the observations A Meanshit Test The Meanshit test presented here is a oneshot detection scheme, ie, ony the most recent observation is used to cacuate the test statistics For this agorithm, a singe ogikeihood ratio between the distribution o the observations at the changepoint and beore the changepoint is used to detect the outage Thus, when using the Meanshit test, the ine outage detection probem is treated as a probem o detecting the meanshit that occurs at the changepoint In particuar, deine the meanshit statistic corresponding to ine as oows: W M [k] = og () (19) Since an outage can occur at any ine, a generaized test structure is used, ie, the decision maker decares a change when one o the L statistics crosses the corresponding threshod, A Consequenty, the stopping time or the meanshit test is deined as τ M = min L ink 1 : W M [k] > A (2) It is expected that the Meanshit test wi perorm worse than the G-D-CuSum test or the oowing reasons First, the transient behavior is not incorporated in the deinition o the scheme Furthermore, the Meanshit test is designed without taking the persistency o the covariance shits into account and without expoiting past observations As a resut, the og-ikeihood ratios used in the test does not match the true distribution o the observations ater the time o change For exampe, during the irst transient period (γ < k γ 1 ), the expected vaue o the test statistic can be negative since [ E (1) og () ] = D( (1) ) D( (1) () ), where E (1) denotes the expectation under distribution (1) B Shewhart Test (21) A test widey used in QCD theory to detect a persistent change in the distribution o a sequence, mainy due to the ease o its impementation is the Shewhart test [6] In the cassic QCD setting, where no transient dynamics are present, a ogikeihood ratio between the persistent post-outage distribution and the pre-outage distribution is used By modiying the structure o the test to account or the meanshit and transient phenomenon ater an outage, we derive a test that is better in terms o perormance compared to the Meanshit test This is done by introducing an additiona og-ikeihood ratio term or each transient response period and one or the meanshit

6 6 Simiar to the Meanshit test, the Shewhart test is aso a oneshot detection scheme; thus, its perormance is inerior compared to our proposed G-D-CuSum test However, the introduction o the additiona og-ikeihood ratio terms aows the Shewhart test to have superior perormance compared to the Meanshit test Deine the Shewhart test statistic or ine outage as: W S [k] = i,1,,t og (i) (22) The Shewhart test incudes the meanshit og-ikeihood ratio o (19) aong with T additiona terms that are associated with dierent transient periods The Shewhart stopping time is deined as: τ S = min L ink 1 : W S [k] > A (23) From (22), it is easy to see that the Shewhart test uses a matching og-ikeihood ratio even ater the outage, meaning that the test statistic corresponding to the outaged ine is nonnegative on average post-change V LINE OUTAGE IDENTIFICATION The detection agorithm proposed in Section III, aong with those discussed in Section IV can aso be used to identiy the outaged ine One strategy woud be to decare the outaged ine as the one corresponding to the argest statistic, ie, the ine that is identiied as outaged is given by: ˆ = arg L W [τ] (24) A drawback to this method o ine identiication is that the statistics or other ines may aso increase oowing a ine outage Due to the structure o a power system, certain ine outages may cause mutipe ine statistics, in addition to the one corresponding to the true outaged ine, to increase Thereore, in order to reduce the probabiity o ase isoation, a set o ines can be identiied as potentiay outaged In this case, more than one ine shoud be checked by the system operator ater an outage is decared To this end, we generaize the idea behind (24) to account or the case o mutipe growing statistics In particuar, ater an outage is decared, we create a ranked ist containing r entries o ine indices, one or each one o the arge ine statistics at the stopping time, and the indices are ordered with respect to the vaues o these statistics The idea is simiar to ist decoding in digita communications (see eg [21]) Deine the ranked ist or an outage in ine as R = 1,, r, (25) where r is the cardinaity o the ranked ist The cardinaity r can be either ixed beorehand, or additiona constraints can be added to make the size variabe across dierent sampe paths (eg, by imposing an additiona constraint that a statistic not ony has to be among the argest, but aso has to be comparabe to the argest one or the corresponding ine to beong to the ranked ist) To quantiy the perormance o our agorithm with respect to its abiity to identiy the outaged ine accuratey, we deine the probabiity o ase isoation (PFI) For the case o ine outage, a ase isoation event occurs when is not incuded in the ranked ist R Deine the PFI when ine is outaged as: PFI (τ) = P R ine outage (26) The imum ength o the ranked ist shoud be chosen to optimize the tradeo between PFI and number o ines that need to be checked ater an outage detection has occurred In particuar, arger ranked ists ead to ower PFI, but to a arger set o possiby outaged ines to check VI CASE STUDIES In this section, the agorithm proposed in (16)-(18) is appied to the IEEE 118-bus test system (or the mode data, see [22]) In order to compute the transient dynamics oowing a ine outage, we use the simuation too Power System Toobox (PST) [23] For simpicity, we used the statistica mode in (11) with T = 2, ie, we assumed one transient period, with a duration o 1 sampes, ater the ine outage occurs Additiona transient periods coud easiy be incorporated into the simuations The power injection proies at the oad buses are assumed to be independent Gaussian random variabes with variance o 3 and the PMU samping rate is assumed to be 3 measurements per second In order to satisy the constraint that the number o PMUs, p, has to be ess than or equa to the number o oad buses, N d, we have chosen to pace a the PMUs at the N d oad buses However, it is important to note that simiar resuts were observed, but not presented in this paper, when PMUs were paced at other buses For our simuations, we ound that the error bounds or a the simuated vaues are within 5% o the means A Line Statistic Evoution First, we simuate two dierent ine outages occurring at k = 1, one in which the detection takes pace during the transient period and one in which the detection occurs ater the transient period; the resuts are shown in Fig 1 Figure 1(a) shows some typica progressions o W 18 [k]s or the various ine outage detection schemes discussed earier with a threshod o A = 12 From the igure, we concude that or this sampe run, a ine outage is decared ater 5 sampes when the G-D-CuSum stream crosses the threshod irst The other agorithms incur a much arger detection deay since they do not cross the threshod o A = 12 Fig 1(b) shows the typica progressions o W 32 [k]s or an outage in ine 32 For a threshod o A = 125, the G-D-CuSum detects a ine outage 156 sampes ater the outage occurs In this exampe, the detection occurs ater the transient dynamics have subsided From the pots, we concude that even though detection takes pace ater the transient dynamics subside (at k = 11), the G-D-CuSum agorithm sti has a smaer detection deay than the G-CuSum agorithm Note that the G-CuSum statistic does not grow during the transient period, which resuts in the G-CuSum having a arge deay Through cose inspection,

7 7 W[k] 12 G-D-CuSum G-CuSum 1 Shewhart Test Meanshit Test k (a) Line 18 outage with A =12 W[k] 12 G-D-CuSum G-CuSum 1 Shewhart Test Meanshit Test k (b) Line 32 outage with A =125 Fig 1: Sampe paths o dierent agorithms or IEEE 118-bus system W[k] 25 Line 18 Line 32 Line Line W[k] Line 36 Line 37 Line 38 Line k (a) Line 18 outage with A = k (b) Line 36 outage with A =13 Fig 2: Sampe paths o the G-D-CuSum agorithm or IEEE 118-bus system Detection Deay [s] Meanshit Test Shewhart Test G-CuSum G-D-CuSum Detection Deay [s] G-CuSum G-D-CuSum 1/24 1/4 1/ Mean Time to Fase Aarm [day] (a) Detection deay vs mean time to ase aarm or dierent agorithms 1/24 1/4 1/ Mean Time to Fase Aarm [day] (b) Detection deay vs mean time to ase aarm or G-D-CuSum and G-CuSum agorithms Fig 3: Monte Caro simuation resuts or an outage in ine 36 or IEEE 118-bus system we aso notice that the sopes o the G-D-CuSum and G- CuSum statistic are identica ater the transient period is over, something that can be veriied by the theory Next, we simuate two dierent ine outages occurring at time k = 1 and demonstrate the evoution o the G-D-CuSum statistic or dierent ines in Fig 2 For an outage in ine 18, it is seen in Fig 2(a) that W 18 [k] grows aster than other ine statistics An outage is decare ater 135 sampes, when W 18 [k] crosses a threshod o A = 25 In Fig 2(b), we show an exampe o a misdetection event In particuar, it can be noted that, or an outage in ine 36, other ine statistics can sometimes cross the test threshod beore W 36 [k] In particuar, in Fig 2(b) we see that W 37 [k] crosses a threshod o A = 13, thus a misdetection event occurs This can be seen as a justiication o the use o a ranked ist to identiy outaged ines B Deay Perormance We perormed Monte Caro simuations or outages in ines 36, 18, and 14, and show detection deay versus MTFA resuts or a the detection schemes presented in this work We compared the perormance o our proposed agorithm against the Meanshit test, Shewhart test, and the G-CuSum agorithm or an outage in ine 36, which is the ine outage case that

8 8 corresponds to the argest deay The resuts are shown in Fig 3 From Fig 3(a), we concude that the G-D-CuSum agorithm achieves the owest detection deays among a agorithms or a given MTFA The perormance o the Meanshit and Shewhart test is consideraby worse than the G-CuSum and G-D-CuSum test In Fig 3(b) we demonstrate the perormance gain that is achieved when using the G-D-CuSum test For an outage in ine 36, the G-D-CuSum test achieves more than an order o magnitude ess deay or given ase aarm rate Next, we evauate the perormance o the proposed agorithm or dierent ine outage cases Among a the ines o the system, detection deay or ine 14 is the owest or a ixed MTFA whie ine 36 has the worst detection deay Line 18 was chosen as a representative ine or intermediate deay vaues The resuts are shown in Fig 4 C Probabiity o Fase Isoation Finay, the PFI versus MTFA is obtained or outages in ines 36, 14, and 18; the resuts are recorded in Tabes I- III The PFI was cacuated by using the ranked ist method discussed in Section V or a ranked ist o ixed ength 1, 3, and 5 In Tabe I we demonstrate the PFI resuts or a ranked ists o ength 1 This is equa to identiication using (24), ie, identiying the ine with the highest statistic at the stopping time as outaged We note that, athough some outages can be handed eicienty with this simpe technique (eg outage in ine 18 and 14), some ine outages may ead to arge PFI vaues, eg an outage in ine 36 This happens due to the act that many ine statistics other than the one corresponding to the outaged ine grow post-outage, as was discussed in Section V In Tabe II we demonstrate the PFI vaues or a ranked ist o ength 3 The PFI is signiicanty reduced or ine 36 Finay, Tabe III shows the PFI resuts or a ranked ist o ength 5 In this case, the PFI or ine 36 is beow 5% Note that the PFI decreases as the MTFA increases This is because arger MTFA corresponds to arger threshods, which resut in smaer PFI vaues Remark 1: It shoud be noted that or this work we do not present any theoretica resuts reating the threshod choice and the MTFA or the proposed agorithm Consuting [6] one can ind threshod choices that provide guarantees or the MTFA vaue or the CuSum and G-CuSum tests Bounds reating the threshod choice and MTFA can be derived even or the agorithms presented in this paper, but they are generay Detection Deay [s] Line 36 Line 18 Line 14 1/24 1/4 1/ Mean Time to Fase Aarm [day] Fig 4: Detection deay vs mean time to ase aarm or dierent ine outages or IEEE 118-bus system TABLE I: Probabiity o ase isoation or IEEE 118-bus system simuated with a ranked ist o ength o 1 E [τ] [day] 1/24 1/4 1/ Line Line Line 14 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 TABLE II: Probabiity o ase isoation or IEEE 118-bus system simuated with a ranked ist o ength o 3 E [τ] [day] 1/24 1/4 1/ Line Line Line 14 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 TABLE III: Probabiity o ase isoation or IEEE 118-bus system simuated with a ranked ist o ength o 5 E [τ] [day] 1/24 1/4 1/ Line Line Line 14 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 < 1 6 conservative In practice, we expect the operator to choose the threshod vaues by running simuations and deriving Deay- MTFA curves such as those in our simuation resuts These curves can be derived by using dierent threshods that wi correspond to dierent MTFA vaues Additiona simuations can be perormed to reate threshod choice with the MTFA, two quantities that are connected by an exponentia reation VII CONCLUSION In this paper, we proposed an agorithm or detecting and identiying ine outages that expoits the statistica properties o votage phase ange measurements obtained rom PMUs in rea-time The proposed agorithm perorms better than previous methods o ine outage detection because it is adaptabe to the transient dynamics that occur in the system oowing a ine outage This agorithm eatures a set o statistics which are used to capture each distribution shit The agorithm is derived as a generaization o the generaized ikeihood ratio soution o the transient QCD probem The detection deay perormance o the proposed agorithm is compared against other ine outage detection agorithms or ine outages simuated on the IEEE 118-bus test system The same system is aso used to simuate the deay perormance o the test or dierent ine outage scenarios, as we as to iustrate the use o the agorithm or ine outage identiication To achieve the atter we used ranked ists, sets o possiby outaged ines that are highy ikey to contain the true outaged ine It is worth mentioning that the techniques proposed in this paper can be easiy extended to doube-ine outage detection (and in genera to mutipe ine outage detection), as in [4],

9 9 by correcty characterizing the distribution o the observations ater a doube-ine outage This can be done by empoying power ow equations Then, our proposed G-D-CuSum scheme can be used to detect such a change in the distribution In particuar, to hande doube-ine outages, a test statistic needs to be cacuated or every possibe pair o ines Future work in this area incudes deveoping schemes to optimay pace imited PMUs to imize agorithmic perormance in terms o detection deay and probabiity o ase isoation, as we as, impementing ow compexity soutions or detecting mutipe and cascading ine outages X k APPENDIX Assume a random process X k k=1 statistica behavior:, i 1 k < γ, (), i γ k < γ 1, (i), i γ i k < γ i1, (T ), i γ T k, with the oowing (27) where γ i N, i =,, T Note that the case o γ i1 = γ i 1 corresponds to a transient stage with a duration o one time instant Thus, the instantaneous meanshit distribution that is present in the power mode o this work can be captured by (27) The goa is to design a stopping rue to detect the change in the statistica behavior o the observed process that takes pace at time instant γ A heuristic test soution can be derived by considering this probem as a dynamic composite hypothesis testing probem Thus, at every time instant k, choose between the oowing two hypotheses: H k : k < γ, H k 1 : k γ The nomina hypothesis H k corresponds to the case that the time instant γ has not been reached yet, whie the aternative hypothesis H k 1 corresponds to the case that γ has been reached Each hypothesis induces a dierent set o distributions on the data X 1, X 2,, X k In particuar, H k is a singe hypothesis under which the data oow distribution iid and H k 1 is a composite hypothesis, ie, it induces one distribution beonging to a set o distributions The distribution that is induced depends on the vaues o the γ s and k To ind the test statistic we irst orm the ikeihood ratio o this hypothesis testing probem or an arbitrary choice o γ s: minγ 1 1,k j=γ () (X j ) k k j=minγ T 1,k1 j=γ (X j ) (T ) (X j ) This ikeihood ratio shoud be interpreted with the understanding that := 1 or i =,, T This is a k natura j=k1 (i) (X j) (X j) generaization o the imum ikeihood interpretation o the CuSum statistic [6] The test statistic is derived by taking the imum with respect to γ,, γ T An equivaent test statistic can be derived by imizing the ogarithm o the above quantity As a resut, we have that W [k] = minγ 1 1,k og () (X j ) γ < <γ T j=γ (X j ) (28) k og (T ) (X j ), (X j ) j=minγ T 1,k1 with the understanding that γ k hods This imization is the reason why the test is independent o the transient duration, as wi be seen ater The expression in (28) can be written in the oowing way: where W [k] = Ω () [k],, Ω (i) [k],, Ω (T ) [k], (29) Ω (i) [k] = γ1 1 og () (X j ) γ <γ 1< <γ i k j=γ (X j ) k og (i) (X j ), i =,, T, j=γ (X j ) i by using the act that k j=k1 og (i) (X k ) (X k ) (3) := We caim that (3) can be written in a recursive manner as oows: Ω (i) [k] = Ω (i) [k 1], Ω (i 1) [k 1] og (i) (X k ) (X k ), or i =,, T and Ω ( 1) [k] := or a k Z First, consider the case o i = : k Ω () [k] = γ k = γ k = γ k = og () (X j ) j=γ (X j ) k 1 og () (X j ) j=γ (X j ) og () (X k ) (X k ) k 1 og () (X j ) og () (X k ) j=γ (X j ) (X k ) γ k 1 k 1 j=k [ k 1 og () (X j ) j=γ (X j ) og () (X j ) (X j ) ], og () (X k ) (X k ) = Ω () [k 1], og () (X k ) (X k ) Since Ω ( 1) [k] :=, the argument we attempt to prove hods or the case o i = Now or the case o an arbitrary i:

10 1 Ω (i) [k] = γ1 1 og () (X j ) γ <γ 1< <γ i k j=γ (X j ) = γ <γ 1< <γ i k γ <γ 1< <γ i k k og (i) (X j ) j=γ (X j ) i γ1 1 og () (X j ) j=γ (X j ) k 1 og (i) (X j ) j=γ (X j ) i og (i) (X k ) (X k ) Consider the irst term o this expression We have that: γ1 1 k 1 og (i) (X j ) j=γ (X j ) i = k 1 og () (X j ) j=γ (X j ) γ <γ 1< <γ i k 1 og (i) (X j ) j=γ (X j ) i γ <γ 1< <γ i 1 k 1 k 1 ], og (i 1) (X j ) j=γ (X j ) i 1 [ γ1 1 og () (X j ) j=γ (X j ) [ γ1 1 og () (X j ) j=γ (X j ) ] = Ω (i) [k 1], Ω (i 1) [k 1] Since the test statistic wi be compared to a positive threshod, an equivaent test can be derived by not aowing the test statistic to take negative vaues Thus, the ina D-CuSum test statistic is deined as oows: W [k] = Ω () [k],, Ω (T ) [k],, (31) where Ω (i) [k] = Ω (i) [k 1], Ω (i 1) [k 1] og (i) (X k ) (X k ), (32) or i =,, T, Ω ( 1) [k] := or a k Z and Ω (i) [] := or a i The corresponding stopping time is given by comparing W [k] against a pre-determined positive threshod: τ = ink 1 : W [k] > A [3] G Rovatsos, X Jiang, A D Domínguez-García, and V V Veeravai, Comparison o statistica agorithms or power system ine outage detection, in Proc o the IEEE Internationa Conerence on Acoustics, Speech, and Signa Processing, Apr 216 [4] Y C Chen, T Banerjee, A D Domínguez-García, and V V Veeravai, Quickest ine outage detection and identiication, IEEE Trans Power Syst, vo 31, no 1, pp , Jan 216 [5] H V Poor and O Hadjiiadis, Quickest Detection Cambridge University Press, 29 [6] V V Veeravai and T Banerjee, Quickest Change Detection Esevier: E-reerence Signa Processing, 213 [7] K A Cements and P W Davis, Detection and identiication o topoogy errors in eectric power systems, IEEE Trans Power Syst, vo 3, no 4, pp , Nov 1988 [8] F F Wu and W E Liu, Detection o topoogy errors by state estimation, IEEE Trans Power Syst, vo 4, no 1, pp , Feb 1989 [9] N Singh and H Gavitsch, Detection and identiication o topoogica errors in onine power system anaysis, IEEE Trans Power Syst, vo 6, no 1, pp , Feb 1991 [1] J E Tate and T J Overbye, Line outage detection using phasor ange measurements, IEEE Trans Power Syst, vo 23, no 4, pp , Nov 28 [11] H Zhu and G B Giannakis, Sparse overcompete representations or eicient identiication o power ine outages, IEEE Trans Power Syst, vo 27, no 4, pp , Nov 212 [12] G Feng and A Abur, Identiication o auts using sparse optimization, in Proc o Communication, Contro, and Computing (Aerton Conerence), Sept 214, pp [13] M Garcia, T Catanach, S V Wie, R Bent, and E Lawrence, Line outage ocaization using phasor measurement data in transient state, IEEE Trans Power Syst, vo 31, no 4, pp , Ju 216 [14] A R Bergen and V Vitta, Power Systems Anaysis Prentice Ha, 2 [15] M Lotaian, R Schueter, D Idizior, P Rusche, S Tedeschi, L Shu, and A Yazdankhah, Inertia, governor, and AGC/economic dispatch oad ow simuations o oss o generation contingencies, IEEE Trans Power App Syst, vo PAS-14, no 11, pp , Nov 1985 [16] B Hajek, Random Processes or Engineers Cambridge university press, 215 [17] A G Tartakovsky, I V Nikiorov, and M Bassevie, Sequentia Anaysis: Hypothesis Testing and Change-Point Detection, ser Statistics CRC Press, 214 [18] M Poak, Optima detection o a change in distribution, Ann Statist, vo 13, no 1, pp , Mar 1985 [19] T L Lai, Inormation bounds and quick detection o parameter changes in stochastic systems, IEEE Trans In Theory, vo 44, no 7, pp , Nov 1998 [2] G V Moustakides, Optima stopping times or detecting changes in distributions, Ann Statist, vo 14, no 4, pp , Dec 1986 [Onine] Avaiabe: [21] P Eias, Error-correcting codes or ist decoding, IEEE Trans In Theory, vo 37, no 1, pp 5 12, Jan 1991 [22] Power system test case archive, Oct 212 [Onine] Avaiabe: [23] J Chow and K Cheung, A toobox or power system dynamics and contro engineering education and research, IEEE Trans Power Syst, vo 7, no 4, pp , Nov 1992 REFERENCES [1] FERC and NERC (212, Apr) Arizona-southern Caiornia outages on September 8, 211: Causes and recommendations [Onine] Avaiabe: [2] US-Canada Power System Outage Task Force (24, Apr) Fina report on the August 14th backout in the United States and Canada: causes and recommendations [Onine] Avaiabe: Georgios Rovatsos (S 15) received the MS degree in eectrica engineering rom the University o Iinois, Urbana-Champaign, in 216 and the BS degree in eectrica engineering rom the University o Patras, Rio, Greece in 214 He is currenty pursuing the PhD degree at the Coordinated Science Laboratory and the ECE Department at the University o Iinois at Urbana- Champaign His research interests are in detection and estimation theory, sequentia anaysis and machine earning

11 11 Xichen Jiang (S 12, M 16) is currenty an assistant proessor at Western Washington University with the department o eectrica engineering He received the BS, MS, and PhD degrees in eectrica engineering rom the University o Iinois, Urbana-Champaign, in 21, 212, and 216, respectivey His research interests incude contro theory, power system reiabiity, and cyber-physica systems Aejandro D Domínguez-García (S 2, M 7) received the degree o Eectrica Engineer rom the University o Oviedo (Spain) in 21 and the PhD degree in eectrica engineering and computer science rom the Massachusetts Institute o Technoogy, Cambridge, MA, in 27 He is an Associate Proessor with the Department o Eectrica and Computer Engineering (ECE), and a Research Associate Proessor with the Coordinated Science Laboratory and the Inormation Trust Institute, a at the University o Iinois at Urbana- Champaign He is aiiated with the ECE Power and Energy Systems area, and has been a Grainger Associate since August 211 His research interests are in the areas o system reiabiity theory and contro, and their appications to eectric power systems, power eectronics, and embedded eectronic systems or saety-critica/aut-toerant aircrat, aerospace, and automotive appications Dr Domínguez-García received the NSF CAREER Award in 21, and the Young Engineer Award rom the IEEE Power and Energy Society in 212 In 214, he was invited by the Nationa Academy o Engineering to attend the US Frontiers o Engineering Symposium, and was seected by the University o Iinois at Urbana-Champaign Provost to receive a Distinguished Promotion Award In 215, he received the U o I Coege o Engineering Dean s Award or Exceence in Research He is an editor o the IEEE Transactions on Power Systems and IEEE Power Engineering Letters Venugopa V Veeravai (M 92 SM 98 F 6) received the BTech degree (Siver Meda Honors) rom the Indian Institute o Technoogy, Bombay, in 1985, the MS degree rom Carnegie Meon University, Pittsburgh, PA, in 1987, and the PhD degree rom the University o Iinois at Urbana- Champaign, in 1992, a in eectrica engineering He joined the University o Iinois at Urbana- Champaign in 2, where he is currenty the Henry Magnuski Proessor in the Department o Eectrica and Computer Engineering, and where he is aso aiiated with the Department o Statistics, the Coordinated Science Laboratory, and the Inormation Trust Institute He served as a Program Director or communications research at the US Nationa Science Foundation in Arington, VA rom 23 to 25 He has previousy hed academic positions at Harvard University, Rice University, and Corne University, and has been on sabbatica at MIT, IISc Bangaore, and Quacomm, Inc His research interests incude statistica signa processing, machine earning, detection and estimation theory, inormation theory, and stochastic contro, with appications to sensor networks, cyberphysica systems, and wireess communications A recent emphasis o his research has been on signa processing and machine earning or data science appications Pro Veeravai was a Distinguished Lecturer or the IEEE Signa Processing Society during He has been on the Board o Governors o the IEEE Inormation Theory Society He has been an Associate Editor or Detection and Estimation or the IEEE Transactions on Inormation Theory and or the IEEE Transactions on Wireess Communications Among the awards he has received or research and teaching are the IEEE Browder J Thompson Best Paper Award, the Nationa Science Foundation CAREER Award, and the Presidentia Eary Career Award or Scientists and Engineers (PECASE), and the Wad Prize in Sequentia Anaysis