A Statistical Framework for Real-time Event Detection in Power Systems

Transcription

1 1 A Statistica Framework for Rea-time Event Detection in Power Systems Noan Uhrich, Tim Christman, Phiip Swisher, and Xichen Jiang Abstract A quickest change detection (QCD) agorithm is appied to the probem of detecting and identifying ine outages in a power system. The statistics of eectricity demand are assumed to be known and propagated through a inearized mode of the equations describing the power fow baance. The votage phase ange measurements are coected using phasor measurement units (PMUs). It is shown that the proposed agorithm is appicabe to situations where the transient dynamics of the power system foowing a ine outage are considered. Case studies demonstrating the performance of the QCD agorithm to fauts with transient dynamics are iustrated through the IEEE 14-bus test system. I. INTRODUCTION Timey detection of ine outages in a power system is crucia for maintaining operationa reiabiity. Toward this end, many onine decision-making toos rey on a system mode that is obtained offine, which can be inaccurate due to bad historica or teemetry data. Such inaccuracies have been a contributing factor in many recent backouts. For exampe, in the 2011 San Diego backout, operators were unabe to determine overoaded ines because the network mode was not up to date [1]. This ack of situationa awareness imited the abiity of the operators to identify and prevent the next critica contingency, and ed to a cascading faiure. Simiary, during the 2003 US Northeastern backout, operators faied to initiate the correct remedia schemes because they had an inaccurate mode of the power system and coud not identify the oss of key transmission eements [2]. These backouts highight the importance of deveoping onine measurement-based techniques to detect and identify system topoogica changes that arise from ine outages. In this paper, we tacke such topoogy change detection probems by utiizing measurements provided by phasor measurement units (PMUs). The authors of [3] deveoped a method for ine outage detection and identification based on the theory of quickest change detection (QCD) [4], [5]. In this method, the incrementa changes in rea power injections are modeed as independent zero-mean Gaussian random variabes. Then, the probabiity distribution of such incrementa changes is mapped to that of the incrementa changes in votage phase anges via a inear transformation obtained from the power fow baance equations. The PMUs provide a random sequence of votage phase ange measurements in rea-time; when a ine outage occurs, the probabiity distribution of the incrementa changes in the votage phase anges changes abrupty. The objective is to detect a change in this probabiity distribution after the occurrence of a ine outage as quicky as possibe whie maintaining a desired fase aarm rate. This paper extends the method proposed in [3] by considering the power system transient response immediatey foowing the ine outage. For exampe, after an outage, the transient behavior of the system is dominated by the inertia response from the generators. This is foowed 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. Then, the Generaized Dynamic CuSum test (G-D-CuSum) is derived by cacuating a D-CuSum statistic for each possibe ine outage scenario; an outage is decared the first time any of the test statistics crosses a pre-specified threshod. The proposed test has better performance because it takes the transient behavior into account in addition to the persistent change in the distribution that resuts from the outage.

2 2 The remainder of this paper is organized as foows. In Section II, we describe the mode of the power system adopted in this work, and introduce the statistics describing the votage phase ange before, during, and after the occurrence of an outage. In Section III, the proposed QCD-based ine outage identification agorithm that accounts for transient behavior after a ine outage is introduced. In Section IV, we iustrate the proposed ideas via numerica case studies on the IEEE 14-bus test system. Finay, concuding remarks and directions for future work are provided in Section V. II. POWER SYSTEM MODEL Let L = {1,..., L} denote the set of ines in the system. 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 of the system can be described by the power fow equations, which for 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 of ine L at time t = t f is assumed to be persistent (i.e., the ine is not restored unti it is detected to be outaged), with γ 0 t t f < (γ 0 + 1) t, where t is the time between successive PMU sampes. In addition, assume that the oss of ine does not cause isands to form in the post-event system (i.e., the underying graph representing the interna power system remains connected). A. Pre-outage Mode Let P i [k] := P i (k t) and Q i [k] := Q i (k t), t > 0, k = 0, 1, 2,..., denote the k th measurement sampe of active and reactive power injections into bus i. Simiary, et V i [k] and θ i [k], k = 0, 1, 2,..., denote bus i s k th votage magnitude and ange measurement sampe. Furthermore, define 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 defined as P i [k] = P i [k+1] P i [k] and Q i [k] = Q i [k + 1] Q i [k]. Proceeding in the same manner as in [3], we inearize (1) around (θ i [k], V i [k], P i [k], Q i [k]), i = 1,..., N, and use the DC power fow assumptions to decoupe the rea and reactive power fow equations. Then, after omitting the equation corresponding to the reference bus, the reationship between votage phase anges and the variations in the rea power injection can be expressed as: P [k] H 0 θ[k], (2) where P [k], θ[k] R (N 1) and H 0 R (N 1) (N 1) is the imaginary part of the system admittance matrix with the row and coumn corresponding to the reference bus removed. In an actua power system, random fuctuations in the oad drive the generator response. Therefore, in this paper, we use the so-caed governor power fow mode (see e.g., [6]), which is more reaistic than the conventiona power fow mode, where the sack bus picks up any changes in the oad power demand. In the governor power fow 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 of participation factors. We approximate B(t) by quantizing it to take vaues B i, i = 0, 1,..., T, where i denotes the time period of interest. Let B(t) = B 0 and M 0 := H 1 0 (1)

3 3 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 foows: θ[k] M 0 P [k] [ ] P = M g [k] 0 P d [k] [ = [M0 1 M0 2 B0 P ] d [k] P d [k] = (M 1 0 B 0 + M 2 0 ) P d [k] = M 0 P d [k], ] (4) where M 0 = M 1 0 B 0 + M 2 0. B. Instantaneous Change During Outage At the time of outage, t = t f, there is an instantaneous change in the mean of the votage phase ange measurements that affects ony one incrementa sampe, namey, θ[γ 0 ] = θ[γ 0 + 1] θ[γ 0 ]. The measurement θ[γ 0 ] is taken immediatey prior to the outage, whereas θ[γ 0 + 1] is the measurement taken immediatey after the outage. Suppose the outaged ine connects buses m and n. Then, the effect of an outage in ine can be modeed with a power injection of P [γ 0 ] at bus m and P [γ 0 ] at bus n, where P [γ 0 ] is the pre-outage ine fow across ine from m to n. Foowing a simiar approach as the one in [3], the reation between the incrementa votage phase ange at the instant of outage, θ[γ 0 ], and the variations in the rea power fow can be expressed as: θ[γ 0 ] M 0 P [γ 0 ] P [γ 0 + 1]M 0 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 0. Furthermore, by using the governor power fow mode of (3) and substituting into (5), and simpifying, we obtain: θ[γ 0 ] M 0 P d [γ 0 ] P [γ 0 + 1]M 0 r. (6) C. Post-Outage Foowing a ine outage, the power system undergoes a transient response governed by B i, i = 1, 2,..., T 1 unti quasi-steadystate is reached, in which B(t) settes to a constant B T. For exampe, immediatey after the outage occurs, the power system is dominated by the inertia response of the generators, which is then foowed by the governor response. As a resut of the ine outage, the system topoogy changes, which manifests itsef in the matrix H 0. This change in the matrix H 0 resuting from the outage can be expressed as the sum of the pre-outage matrix and a perturbation matrix, H, i.e., ], and deriving in the same manner as the pre-outage mode of (4), we obtain the post-outage reation between the changes in the votage anges and the rea power demand as: H = H 0 + H. Then, by etting M := H 1 = [M 1 M 2 where M,i = M 1B i + M 2, i = 1, 2,..., T. θ[k] M,i P d [k], γ i 1 k < γ i, (7)

4 4 D. Measurement Mode Since the votage phase anges, θ[k], are assumed to be measured by PMUs, we aow for the scenario where the anges are measured at ony a subset of 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 ) possibe ocations to pace the PMUs. In this paper, we assume that the PMU ocations p are fixed; in genera, the probem of optima PMU pacement is NP-hard and its treatment is beyond the scope of this paper. Let M = M 0, if 1 k < γ 0,. M,K, if k γ T 1. Then, the absence of a PMU at bus i corresponds to removing the i th row of M. Thus, et ˆM R p N d be the matrix obtained by removing N p 1 rows from M. Therefore, we can reate ˆM to M in (8) as foows: ˆM = C M, (9) where C R p (N 1) is a matrix of 1 s and 0 s that appropriatey seects the rows of M. Accordingy, the increments in the phase ange can be expressed as foows: ˆθ[k] ˆM P d [k]. (10) The sma variations in the rea power injections at the oad buses, P d [k], can be attributed to random fuctuations in eectricity consumption. In this regard, we may mode the P d [k] s as independent and identicay distributed (i.i.d.) random vectors. By the Centra Limit Theorem [7], it can be shown that each P d [k] is a Gaussian vector, i.e., P d [k] N (0, Λ), where Λ is the covariance matrix. Note that the eements P d [k] need not be independent. Since ˆθ[k] depends on P d [k] through the inear reationship given in (10), we have that: ˆθ[k] f 0 := N (0, ˆM 0 Λ ˆM 0 T ), if 1 k < γ 0, f (0) := N ( P [γ + 1]CM 0 r, ˆM 0 Λ ˆM 0 T ), if k = γ 0,. f (T ) := N (0, ˆM,K Λ ˆM,K T ), if k γ T 1, It is important to note that for N (0, ˆMΛ ˆM ) T to be a nondegenerate p.d.f., its covariance matrix, ˆMΛ ˆM T, must be fu rank. We enforce this by ensuring that the number of PMUs aocated, p, is ess than or equa to the number of oad buses, N d. 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 fase aarm constraints. The outage induces a change in the statistica characteristics of 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. (8) (11)

5 5 A. Probem Setup The goa in QCD is to design stopping rues that wi detect the change in the statistica behavior of the observed process as fast as possibe under fase aarm constraints. The fase aarm constraint that we choose is based on the mean time to fase aarm; thus, we woud ike E [τ] β, where β > 0 is a pre-determined parameter, and E is the expectation under the probabiity measure where no outage has occurred. In order to quantify the detection deay for ine outages, we introduce the foowing deay metric: [ ] D (τ) = sup ess sup E γ0, (τ γ 0 ) + ˆθ[1],..., ˆθ[γ 0 1]. (12) γ 0 1 According to (12), the deay of stopping time τ for detecting an outage in ine is measured by taking the expected vaue of (τ γ 0 ) + under the assumption that (i) the underying distribution is the one induced on the observations when an outage occurs in ine at time instant γ 0, and (ii) conditioning on a set of pre-outage observations { ˆθ[1],..., ˆθ[γ 0 1]}. In the setting described in Section II, we assume that a sequence of observations { ˆθ[k]} k 1 is measured by PMUs and passed sequentiay to a decision maker. According to the statistica mode in (11), before an outage has occurred, ˆθ[k] f 0. At an unknown time instant γ 0, an outage occurs in ine and the distribution of ˆθ[k] changes from f 0 to f (0). Then, the system undergoes a series of transient responses which corresponds to the distribution of ˆθ[k] evoving from f (0) to f (T ). First, a meanshift takes pace during the instant of change γ 0, where the pdf is f (0). Then, the statistica behavior of the process is characterized by a series of changes ony in the covariance matrix of the measurements. B. Generaized CuSum Test The Generaized CuSum (G-CuSum) based test was proposed as a ine outage detection scheme in [8] with the understanding that the transition between pre- and post-outage periods is not characterized by any transient behavior other than the meanshift that occurs at the instant of outage. The meanshift was captured by introducing an additiona og-ikeihood ratio term between the distribution at the time of change and the distribution before the change. The fina test statistic takes the maximum of 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 perform we when the transient distributions and the fina post- change distribution are simiar, i.e., when the KL divergence between f (i), i = 1, 2,..., T 1, and f (T ) is sma. As a resut, it is insightfu to compare the performance of the G-CuSum test with the performance of the G-D-CuSum test that is proposed in this work. Since the ine that is outaged is not known, 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 for ine is cacuated by accumuating og-ikeihood ratios between f (T ) and f 0. In particuar, define the G-CuSum statistic corresponding to ine outage recursivey as: W C [k] = max {W C [k 1] + og f (T ) ( ˆθ[k]) f 0 ( ˆθ[k]), with W C [0] = 0 for a L. og f (0) ( ˆθ[k]) f 0 ( ˆθ[k]), 0 }, (13)

6 6 The goa is to decare an outage as soon as any ine is outaged; thus, the agorithm decares a detection the first time any of the ine statistics crosses its corresponding threshod. Accordingy, the stopping time of the test is as foows: { } τ C = inf inf{k 1 : W C [k] > A }, (14) E with A > 0 being the threshod corresponding to ine. C. Generaized Dynamic CuSum Test Since the statistica mode used in this paper incudes an arbitrary number of transient periods with finite duration, each one corresponding to a respective transient distribution induced on the observations, it is cear that the G-CuSum test of [8] needs to be modified 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), a test aso proposed in this work. This test arises as a soution to the non-composite QCD probem under the presence of an arbitrary number of transient periods. The D-CuSum test statistic is derived by formuating the transient QCD probem as a dynamic composite hypothesis testing probem at each time instant. The G-D-CuSum agorithm uses the test statistics of the D-CuSum test in a generaized manner, i.e., cacuates a test statistic for each possibe ine outage in parae, and decares an outage when one of the ine statistics crosses a pre-determined positive threshod corresponding to the ine. By using the D-CuSum test statistic as a basis, we propose the G-D-CuSum test. The statistic for ine is given as foows: where { W D [k] = max Ω (1) [k],..., Ω (T ) [k], og f (0) ( ˆθ[k]) f 0 ( ˆθ[k]) {[ Ω (i) [k] = max max{ω (i) [k 1], Ω(i 1) [k 1]}+ og f (i) ] } ( ˆθ[k]), 0, f 0 ( ˆθ[k]) }, (15) for i {1,..., T }, Ω (0) [k] := 0 for a k Z and Ω (i) [0] := 0 for a L and a i. The corresponding stopping rue is defined as { } τ D = inf inf{k 1 : W D [k] > A }. (17) L (16) Cacuating the test statistic for ine invoves cacuating the statistics Ω (1),...,Ω (T ). The fina test statistic is given by taking the maximum of these terms together with the og-ikeihood ratio between the distribution at the outage and the pre-outage distribution. Note that to renew each Ω statistic, the vaue of the statistic in the previous time instant and the vaue of the statistic used to detect the previous distribution change is used. IV. CASE STUDIES In this section, the agorithm proposed in (15)-(17) is appied to the IEEE 14-bus test system (see [9]), the one-ine diagram of which is shown in Fig. 1. In order to compute the transient dynamics foowing a ine outage, we use the simuation too Power System Toobox [10]. For simpicity, we used the statistica mode in (11) with T = 2, i.e., we assumed one transient period after the ine outage occurs. Additiona transient periods coud easiy be incorporated into the simuations. The power injection profie at the oad buses are assumed to be independent Gaussian random variabes with variance of 0.03 and the PMU samping rate is assumed to be 30 measurements per second.

7 7 Bus Ange [deg] Fig. 1: IEEE 14-bus system one-ine diagram. θ 4 θ 5 θ 6 θ 7 θ 8 θ 9 θ Time [s] Fig. 2: Votage phase anges of IEEE 14-bus system foowing an outage in ine 4. A. 14-bus System For the IEEE 14-bus system, we simuated an outage in ine 4 at t = 0.1s. The dynamic responses of the votage phase anges are shown in Fig. 2. From the pots, we concude that the transient period foowing a ine outage asts approximatey 3 seconds, which is assumed in the mode for our proposed detection agorithm. We then sampe the votage anges during the transient period and appy the detection agorithm of (15) to the coected data. The threshod A is seected to be 400. Starting from k = 100, which is when the outage occurs, the W4 D stream starts to grow whie a the other streams W D for 4 stay near 0. From the figure, we concude that at k = 275, an outage is decared, which is when W4 D > 400. The detection

8 Fauted ine 4 A other ines Fig. 3: Votage phase anges of IEEE 14-bus system foowing an outage in ine 4. deay is 175 sampes. V. CONCLUSION In this paper, we addressed the probem of detecting and identifying ine outages by expoiting the statistica properties of votage phase ange measurements obtained from PMUs in rea-time. A reation between the incrementa active power demand and active power generation to the votage phase anges is estabished through the inearized power baance equations. The system transient response foowing a ine outage is modeed using time dependent participation factors, which more reaisticay captures the actua system dynamics and resuts in better detection performance. Simuations are performed on the IEEE 14-bus test systems to iustrate the effectiveness of the proposed ine outage detection agorithm. REFERENCES [1] FERC and NERC. (2012, Apr.) Arizona-southern Caifornia outages on September 8, 2011: Causes and recommendations. [Onine]. Avaiabe: [2] U.S.-Canada Power System Outage Task Force. (2004, Apr.) Fina report on the August 14th backout in the United States and Canada: causes and recommendations. [Onine]. Avaiabe: [3] Y. C. Chen, T. Banerjee, A. D. Domínguez-García, and V. V. Veeravai, Quickest ine outage detection and identification, IEEE Transactions on Power Systems, vo. 31, no. 1, pp , [4] H. V. Poor and O. Hadjiiadis, Quickest Detection. Cambridge University Press, [5] V. V. Veeravai and T. Banerjee, Quickest Change Detection. Esevier: E-reference Signa Processing, [6] M. Lotfaian, R. Schueter, D. Idizior, P. Rusche, S. Tedeschi, L. Shu, and A. Yazdankhah, Inertia, governor, and AGC/economic dispatch oad fow simuations of oss of generation contingencies, IEEE Transactions on Power Apparatus and Systems, vo. PAS-104, no. 11, pp , nov [7] B. Hajek, Random Processes for Engineers. Cambridge university press, [8] G. Rovatsos, X. Jiang, A. D. Domínguez-García, and V. V. Veeravai, Comparison of statistica agorithms for power system ine outage detection, in Proc. of the IEEE Internationa Conference on Acoustics, Speech, and Signa Processing, Apr [9] Power system test case archive, Oct [Onine]. Avaiabe: [10] J. Chow and K. Cheung, A toobox for power system dynamics and contro engineering education and research, IEEE Transactions on Power Systems, vo. 7, no. 4, pp , Nov