c 2016 Georgios Rovatsos

c 2016 Georgios Rovatsos

QUICKEST CHANGE DETECTION WITH APPLICATIONS TO LINE OUTAGE DETECTION BY GEORGIOS ROVATSOS THESIS Submitted in partia fufiment of the requirements for the degree of Master of Science in Eectrica and Computer Engineering in the Graduate Coege of the University of Iinois at Urbana-Champaign, 2016 Urbana, Iinois Adviser: Professor Venugopa V. Veeravai

ABSTRACT In this work, we focus on appications of quickest change detection (QCD) theory in the probem of ine outage detection and identification. We start by discussing fundamenta resuts of sequentia hypothesis testing and QCD, and by proposing an agorithm for the QCD setting under transient dynamics. Foowing, we appy these resuts in the ine outage detection probem. QCD agorithms are appied on measurements of votage phase anges, which are coected using phasor measurement units (PMUs), samping units that sampe at an approximate rate of 30 sampes per second and that are paced in the buses of the system. The goa is to detect a ine outage as fast as possibe, under fase aarm constraints. First, we study the ine outage setting where no transient dynamics are present. Then, we propose a QCD agorithm for the case where transient dynamics are present. Line outage identification schemes are aso discussed. ii

To my famiy iii

ACKNOWLEDGMENTS I woud ike to express my deepest gratitude to Professor Venugopa V. Veeravai for his support throughout this research process. Without his counse and guidance this work woud not have been possibe. Lasty, I woud ike to express the thanks I owe to my famiy for their patience and support throughout my ife. iv

TABLE OF CONTENTS LIST OF TABLES............................. vii LIST OF FIGURES............................ viii CHAPTER 1 INTRODUCTION.................... 1 1.1 Background............................ 1 1.2 Probem Statement........................ 2 1.3 Reated Work........................... 2 1.4 Contribution of Thesis...................... 3 CHAPTER 2 BINARY SEQUENTIAL HYPOTHESIS TESTING.. 5 2.1 Probem Statement........................ 5 2.2 Bayesian and non-bayesian Formuation............ 6 2.3 The Sequentia Probabiity Ratio Test (SPRT)......... 7 CHAPTER 3 QUICKEST CHANGE DETECTION.......... 10 3.1 Probem Statement........................ 10 3.2 QCD Agorithms......................... 13 3.3 Optimaity Properties of CuSum and Shiryaev-Roberts Agorithms.............................. 15 CHAPTER 4 QUICKEST CHANGE DETECTION UNDER TRAN- SIENT DYNAMICS.......................... 17 4.1 The Dynamic CuSum Agorithm................. 17 CHAPTER 5 QCD ALGORITHMS FOR NON-TRANSIENT POWER SYSTEM LINE OUTAGE DETECTION............... 22 5.1 Power System Mode....................... 22 5.2 Line Outage Detection Using QCD............... 27 5.3 QCD-based Line Outage Detection Agorithms......... 28 5.4 Other Line Outage Detection Agorithms............ 30 CHAPTER 6 LINE OUTAGE DETECTION AND IDENTIFI- CATION UNDER TRANSIENT DYNAMICS............ 32 6.1 QCD Based Line Outage Detection Agorithms Under Transient Dynamics.......................... 32 v

6.2 Other Agorithms for Line Outage Detection Under Transient Dynamics.......................... 35 6.3 Line Outage Identification.................... 37 CHAPTER 7 CASE STUDIES..................... 39 7.1 Simuation Resuts and Discussion for Chapter 5........ 39 7.2 Simuation Resuts and Discussion for Chapter 6........ 41 CHAPTER 8 CONCLUSIONS..................... 46 REFERENCES............................... 47 vi

LIST OF TABLES 7.1 Probabiity of fase isoation for IEEE 118-bus system simuated with a ranked ist of ength of 1.............. 45 7.2 Probabiity of fase isoation for IEEE 118-bus system simuated with a ranked ist of ength of 3.............. 45 7.3 Probabiity of fase isoation for IEEE 118-bus system simuated with a ranked ist of ength of 5.............. 45 vii

LIST OF FIGURES 2.1 Typica evoution of the SPRT statistic with f 0 = N (0, 1) and f 1 = N (0, 2)......................... 8 3.1 Exampe of sampe path of a process under the QCD setting. At γ = 20 the distribution changes from N (0, 1) to N (0, 2)............................... 11 3.2 Shewhart test run for data of Figure 3.1............. 13 3.3 CuSum agorithm run for data of Figure 3.1........... 15 3.4 Shiryaev-Roberts agorithm run for data of Figure 3.1..... 15 4.1 A typica reaization of a sequence of i.i.d. Gaussian variabes. The statistics of the process are characterized by (4.6)................................. 21 4.2 D-CuSum test run for data of Figure 4.1............. 21 7.1 Exampe of a run of the G-CuSum for the 14-bus system.... 40 7.2 Detection deay vs. mean time to fase aarm.......... 41 7.3 Sampe paths of different agorithms for IEEE 118-bus system. 42 7.4 Sampe paths of the G-D-CuSum agorithm for IEEE 118- bus system............................. 42 7.5 IEEE 118-bus Monte Caro simuation resuts for an outage in ine 36.............................. 43 7.6 118-bus: Detection deay vs. mean time to fase aarm for different ine outages........................ 44 viii

CHAPTER 1 INTRODUCTION The motivation behind this work stems from the increasing presence of phasor measurement units (PMUs) across the power grid. The introduction of this new measurement unit has ed to significant advances in the stabiity monitoring and state estimation capabiities for the power system. As a resut, the integration of the PMUs has paved the way for the use of rea-time agorithms that can be expoited to detect ine outages in an efficient and robust manner. Furthermore, the statistica behavior of the observed measurement process is another motivation. In particuar, a ine outage event eads to a change in the statistica behavior of the observed sequence of observations. The detection of changes of this nature is a probem that is studied thoroughy in the QCD iterature. 1.1 Background Timey detection of ine outages in a power system is crucia for maintaining operationa reiabiity. In this regard, 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 1

detect and identify system topoogica changes that arise from ine outages. In this work, we tacke such topoogy change detection probems by utiizing measurements provided by PMUs. 1.2 Probem Statement Our work extends the resuts of [3], where the authors deveoped a method for ine outage detection and identification based on the theory of quickest change detection (QCD) [4], [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 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. In this work, we focus on the probem of ine outage detection and identification in two settings. First, we assume that after the outage the system changes state instantaneousy, and ater we study a more reaistic setting, where the system mode is characterized by transient behavior. 1.3 Reated Work Eary approaches for topoogica change detection incude agorithms based on state estimation [7], [8], and rue-based agorithms that mimic system operator decisions [9]. More recent methods expoit the fast samping of votage magnitudes and phases provided by PMUs [10] [12]. However, these schemes do not expoit the persistent nature of ine outages and do not incorporate transient behavior. Ony the most recent PMU measurement is used to determine if an outage has occurred. The authors of [13] proposed a method to detect ine outages using statistica cassifiers where a maximum ikeihood estimation is performed on the PMU data. The authors aso considered the 2

transient response of the system after a ine outage by comparing synthesized data against actua data. However, their method requires knowedge of the exact time of the ine outage before appying the agorithm, whereas our proposed methods do not have this restriction. 1.4 Contribution of Thesis We first study the ine outage probem when no transient dynamics are present, i.e., the shift from pre- to post- change distribution happens amost instanty. As in [3], the agorithm that is proposed as a soution is based on adapting the Generaized Cumuative Sum (G-CuSum) test from the QCD iterature (see, e.g., [4], [6]) to the ine outage detection probem. Our agorithm not ony takes the persistent covariance change into consideration, but it aso expoits past observations to detect the occurrence of an outage. In [3], the statistics for each individua ine are compared to a common predetermined threshod, and an outage is decared if one of these statistics crosses the threshod. In this work, we present a method for setting a different threshod for each ine outage statistic by taking the dissimiarity between the pre- and post-change distribution into consideration. This difference between pre- and post-change distributions is described by the Kuback-Leiber (KL) divergence, a metric that quantifies the distance between two distributions. In addition, we compare the performance of our test to that of the Shewhart test, the meanshift test, and the agorithm of [3], and observe notabe improvements in terms of performance. Next, we study the ine outage probem under the presence of an arbitrary number of transient periods. We improve on 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 (see e.g., [4], [6]). Then, the Generaized Dynamic CuSum (G-D-CuSum) test 3

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 considers the transient behavior in addition to the persistent change in the distribution that resuts from the outage. Furthermore, we discuss ine outage identification techniques that can be empoyed easiy in practice. The remainder of this thesis is organized as foows. In Chapter 2, we review the probem of binary sequentia hypothesis testing. In Chapter 3, we study the probem of quickest change detection (QCD) and provide theoretica resuts that are going to be used throughout this work. In Chapter 4, we formuate the QCD probem under transient dynamics and propose the D-CuSum test. In Chapter, 5 we study the probem of ine outage detection and identification when no transient dynamics are present, and propose a G- CuSum based agorithm that uses varying threshods to achieve performance gains. In Chapter, 6 we introduce the G-D-CuSum test as a proposed agorithm for detecting outages under transient behavior. We aso demonstrate methods for identifying outaged ines. In Chapter, 7 we provide simuation resuts on the IEEE 14-bus and 118-bus systems. Finay, concuding remarks are made in Chapter 8. 4

CHAPTER 2 BINARY SEQUENTIAL HYPOTHESIS TESTING In this chapter, we provide a brief review of the probem of binary sequentia hypothesis testing, introduced in [14] (aso see [15]). In this setting, measurements foowing one of two candidate distributions are fed to a decision maker sequentiay. The goa is to design stopping procedures that dictate when to stop samping, and use the data avaiabe up to the time of stop to decide in favor of one of the two hypotheses. Note that, in contrast to the traditiona detection theory techniques where the sampe size is fixed beforehand, in sequentia hypothesis testing the sampe size is determined onine, by managing a tradeoff between the number of sampes and the desired eve of accuracy. 2.1 Probem Statement In the present setting, the measurements come from a stream of observations that is characterized by one of two potentia statistica behaviors (in this work we wi focus on the binary testing probem). As a resut, we have two hypotheses on the samped data: H 0 : X k f 0 i.i.d. H 1 : X k f 1 i.i.d. The goa here is to design a sequentia test (τ, δ), which is essentiay a stopping time τ accompanied by a decision rue δ. The notion of stopping time is defined formay as foows: Definition 1. A stopping time τ adapted on a random process {X k } k=1 is a random variabe with the property that {τ = k} σ(x 1,..., X k ), where σ(x 1,..., X k ) the σ-agebra generated by X 1,..., X k. Intuitivey, this means 5

that no observations after time instant k are necessary to decide whether to stop samping at k. The stopping time τ shoud be designed so that the observations {X k } τ k=1 are sufficient to make a correct decision with respect to given accuracy. Note that the present probem is characterized by a pair of conficting goas: on one hand we need to make an accurate decision regarding the statistica behavior of the process, which may require a arge number of sampes, on the other hand, a arger sampe size is more costy. Thus, we have to manage a tradeoff which wi depend on our budget and the desired eve of accuracy. 2.2 Bayesian and non-bayesian Formuation In [14], two formuations are proposed to capture the tradeoff between sampe size and detection accuracy. For the first one, we foow a Bayesian approach, meaning we assume known priors π 0, π 1, on the hypotheses. The second is based on a Neyman-Pearson (NP) approach. The soution to both instances is the Sequentia Probabiity Ratio Test (SPRT) proposed in [14]. We start by defining the Bayes risk for the first formuation: Definition 2. For a sequentia test (τ, δ), the Bayes risk is given by r(τ, δ) = c[π 0 E 0 [τ] + π 1 E 1 [τ]] + π 0 P F (τ, δ) + π 1 P M (τ, δ), (2.1) where E 0 is the expectation under distribution f 0, E 1 is the expectation under distribution f 1, c is the cost that we suffer for each additiona sampe measured, P F is the probabiity of fase aarm, and P M is the probabiity of misdetection. A Bayesian sequentia test is the sequentia test (τ, δ) B that minimizes the Bayes risk, i.e. (τ, δ) B = arg min r(τ, δ). (2.2) (τ,δ) When no priors on the hypotheses are avaiabe, we can approach the sequentia hypothesis testing probem through an NP approach. In particuar, 6

such a formuation is given as foows: min subject to E 0 [τ] and E 1 [τ] P F (τ, δ) α P M (τ, δ) β, (2.3) where α, β (0, 1). In practica terms, the goa in this formuation is to find the sequentia test that minimizes E 0 [τ] and E 1 [τ] among a tests that have sufficienty sma P F and P M. 2.3 The Sequentia Probabiity Ratio Test (SPRT) Here, we present the Sequentia Probabiity Ratio Test (SPRT), proposed in [14], which is the soution to the two tradeoff formuations presented in Sec. 2.2. To define the SPRT, we use the og-ikeihood ratio of X 1,..., X k as a test statistic. Define the test statistic at time k as: S k = k j=1 og f 1(X j ) f 0 (X j ). (2.4) The corresponding stopping and decision rue for the SPRT is given by: τ SP RT = inf{k 1 : S k (a, b)} (2.5) and 1 if S δ SP RT τ SP RT > b = 0 if S τ SP RT < a, (2.6) where a < 0 < b are the test threshods. The SPRT test invoves cacuating a test statistic S k at each time instant, with S 0 := 0. The SPRT statistic is then compared to threshods a and b. If S k > b, we stop and decide in favor of H 1. If S k < a, we stop and decide in favor of H 0. Otherwise, we continue samping unti we have accumuated enough data for an accurate decision. In [14] it was shown that the SPRT is optima for both the Bayesian and the non-bayesian tradeoff formuations that were presented in Sec. 2.2. There, it is shown that the optima test with respect to (2.2) is derived by seecting a 7

0 2 4 6 8 10 12 0 S k -2-4 -6-8 -10 a=-10-12 (a) When f 0 is the true distribution. 12 S k 10 b=10 8 6 S k 4 2 0-2 0 5 10 15 20 25 30 Time (b) When f 1 is the true distribution. Figure 2.1: Typica evoution of the SPRT statistic with f 0 = N (0, 1) and f 1 = N (0, 2). and b, so that the Bayes risk is minimized. For (2.3) it is shown that seecting threshods that satisfy the inequaity constraints with equaity resuts in an optima scheme. In Fig. 2.1 we show the evoution of the SPRT statistic for two different hypotheses on the data, namey, H 0 corresponding to f 0 = N (0, 1) and H 1 corresponding to f 1 = N (0, 2). In particuar, in Fig. 2.1(a) we see a sampe path for the case that the data is generated by the distribution f 0 = N (0, 1). Note how the test statistic decreases unti threshod a = 10 is crossed and decision in favor of H 0 is taken. Simiary, when the data are generated by f 1, the test statistic grows unti it crosses threshod b = 10 and we take a decision in favor of H 1. For an intuitive interpretation of the detection agorithms discussed in this work, we wi be using the Kuback-Leiber (KL) divergence, which is 8

an information theoretic measure of the discrepancy between two probabiity distributions. Definition 3. The KL divergence between two probabiity density functions, f and g, is defined as: D(f g) := f(x) og f(x) [ g(x) dx := E f og f(x) ]. (2.7) g(x) It is easy to show that D(f g) 0, with equaity if and ony if f = g. For a detaied study of the KL divergence see, e.g., [16]. A simpe justification as to why the SPRT is a suitabe agorithm for the binary sequentia hypothesis testing probem can be given by examining the expected vaue of the test statistic under both regimes. In particuar, for hypotheses H 0 and H 1, respectivey, we have that E f0 [S k ] = E f0 [ k j=1 and E f1 [S k ] = E f1 [ k j=1 og f ] 1(X j ) = f 0 (X j ) og f ] 1(X j ) = f 0 (X j ) k j=1 k j=1 [ E f0 og f ] 0(X j ) = kd(f 0 f 1 ) < 0 f 1 (X j ) [ E f1 og f ] 1(X j ) = kd(f 1 f 0 ) > 0. f 0 (X j ) As a resut, when H 0 is the true hypothesis, the test statistic wi decrease with an expected drift of D(f 0 f 1 ), eventuay passing a negative threshod of a. Simiary, when H 1 is the true hypothesis, the test statistic wi grow with an expected drift of D(f 1 f 0 ), eventuay crossing a positive threshod of b. Note that since the average drifts are fixed for a specific pair of distributions, making b arger or a smaer wi resut in a arger number of samped measurements on average, thus improving the decision accuracy. For a thorough anaysis of the SPRT performance evauation see [15]. 9

CHAPTER 3 QUICKEST CHANGE DETECTION In this chapter, we study the probem of quickest change detection (QCD). In QCD, the goa is to detect an abrupt change in the statistica behavior of a sequentiay observed process. Detection techniques aim to minimize the deay under fase aarm constraints, i.e., detect a change in the statistica behavior of the process fast enough whie maintaining a sufficienty ow occurrence rate of fase aarm events. The theoretica resuts of this chapter are fundamenta for the rest of the thesis. For a deeper anaysis of QCD theory we refer the reader to [4]; aso see [5] and [6]. 3.1 Probem Statement In Chapter 2, we studied the probem of binary sequentia hypothesis testing, where a sequence of measurements is characterized by one of two statistica behaviors, and we aim to detect the true behavior by processing sampes in a sequentia manner. In QCD, we observe a random process that initiay foows a distribution f 0 i.i.d. At some unknown time instant γ, the process switches to a distribution f 1 i.i.d. In summary, the statistica behavior of the process is as foows: X k f 0, for k < γ X k f 1, for k γ. This is aso caed the i.i.d. setting. For the traditiona, non-composite instance of the probem, both f 0 and f 1 are known beforehand. In Fig. 3.1 we generate a sequence of sampes that are characterized by a distribution shift of this form. In this work, we wi focus on the minimax setting of the probem, where it is assumed that γ 1 is unknown but not random. The goa in QCD is to create a procedure that wi be used to detect the 10

10 8 6 4 X k 2 0-2 -4-6 -8 0 10 20 30 40 50 60 70 80 90 100 Time Figure 3.1: Exampe of sampe path of a process under the QCD setting. At γ = 20 the distribution changes from N (0, 1) to N (0, 2). abrupt statistica behavior change that occurs at γ. This procedure wi have the structure of a stopping time τ adapted on {X k }, with the understanding that instead of stopping samping, we decare that a change has occurred. 3.1.1 Minimax QCD Tradeoff Formuations The QCD probem is characterized by an underying tradeoff: on one hand, we want to detect distribution changes as fast as possibe, whie on the other hand we want to avoid frequent fase aarm events, i.e., avoid decaring a change has occurred, when it has not occurred yet. To this end, we present two popuar deay mean time to fase aarm formuations that are used in the QCD iterature, due to Lorden [17] and Poak [18] respectivey. Before presenting these two tradeoff formuations we need to define two deay metrics that we wi be using. The first deay metric, which was proposed by Lorden, is based on the expected vaue of (τ γ) + conditioned on the worst possibe measurements before change. In particuar, for a stopping time τ, define the first deay metric as: [ WADD(τ) = sup ess sup E γ (τ γ) + X 1,..., X γ 1 ], (3.1) γ 1 11

where E γ denotes the expected vaue when the underying distribution is the one induced on the sequence of observations when an outage occurs at γ, and (x) + := max{x, 0}. Note that (3.1) invoves taking the expected vaue of (τ γ) + after conditioning on a set of observations {X 1,..., X γ 1 }. Using the ess sup can be seen as choosing the worst possibe set of observations to condition on, i.e., the set of observations that maximize the expected deay for given γ. Finay, since the time instant during which the distribution change happens is unknown, we have to take the sup over a possibe changepoints. The second deay metric was proposed by Poak, and is defined as foows: CADD(τ) = sup γ 1 E γ [τ γ ]. τ γ (3.2) It is easy to show that for any stopping rue τ we have that CADD(τ) WADD(τ), i.e., the WADD metric is a more pessimistic way of measuring the deay. It shoud be noted that athough the deay metrics of (3.1) and (3.2) are generay difficut to compute, for the agorithms studied in this thesis, they can be easiy computed by Monte Caro simuations (see [6] for more detais). With the two defined deays in mind we move on to present the reated tradeoff formuations. We start with Lorden s formuation: Formuation 1. min τ WADD(τ) subject to E [τ] β. (3.3) Lorden s formuation invoves searching among the set of stopping times that satisfy E [τ] β, where E the expectation under the measure that no distribution change occurs, to find the one that minimizes WADD. The inequaity constraint is imposed to guarantee that fase aarm events wi be sufficienty rare. Simiary, for Poak s deay we have a respective tradeoff formuation: Formuation 2. min τ CADD(τ) subject to E [τ] β. In the next section we present agorithms that are used in QCD theory. 12 (3.4)

45 40 A=40 35 30 25 W k 20 15 10 5 0-5 0 10 20 30 40 50 60 70 Time Figure 3.2: Shewhart test run for data of Figure 3.1. 3.2 QCD Agorithms A simpe QCD agorithm was proposed by Shewhart in [19]. In defining the Shewhart test, ony the current observation is used. In particuar, define the Shewhart test statistic by W SH k = og f 1(X k ) f 0 (X k ). (3.5) The Shewhart test is based on the fact that the expected vaue of the ogikeihood ratio after change is given by D(f 1 f 0 ), which is a positive quantity; thus, detection can be achieved by using a positive threshod. In particuar, the Shewhart stopping time is defined as τ SH = inf{k 1 : W SH k > A}, (3.6) where A > 0 the threshod. In Fig. 3.2, we show the evoution of the Shewhart statistic for the sampes shown in Fig. 3.1. A detection scheme that enjoys optimaity properties with respect to Poak s and Lorden s formuation is the so-caed Cumuative Sum (CuSum) agorithm, proposed by Page in [20]. The CuSum agorithm invoves accumuating og-ikeihood ratios between the pre- and post-change distribution (hence the name of the agorithm) to form the test statistic, and decaring a distribution change has occurred when said test statistic crosses a positive threshod. In particuar, the CuSum statistic at time k is given by the 13

foowing recursion: W C k = ( Wk 1 C + og f ) + 1(X k ), (3.7) f 0 (X k ) with W C 0 = 0. The corresponding CuSum stopping time is defined as τ C = inf{k 1 : W C k > A}. (3.8) In Fig. 3.3, we show the evoution of the CuSum statistic for the sampes shown in Fig. 3.1. The notion of KL divergence can aso be used here to provide an intuitive interpretation of the CuSum test. In particuar, understanding the idea behind the agorithm bois down to understanding the behavior of the ogarithmic term of (3.7) before and after the distribution change. Before the change occurs, we have that for the expected vaue of the ogarithmic term of (3.7): [ E f0 og f ] 1(X k ) = D(f 0 f 1 ) < 0, f 0 (X k ) which wi cause the test statistic to take non-negative vaues around zero. After the change occurs we have that [ E f1 og f ] 1(X k ) = D(f 1 f 0 ) > 0. f 0 (X k ) Thus, the CuSum statistic wi grow with a positive average drift of D(f 0 f 1 ), eventuay crossing a positive threshod and decaring a distribution change has occurred. An agorithm that has a very strong connection to the CuSum agorithm is the so caed Shiryaev-Roberts (SR) agorithm. The SR test statistic is given by the foowing recursion: W SR k = (1 + W SR k 1) og f 1(X k ) f 0 (X k ), (3.9) with W SR 0 = 0. The corresponding SR stopping time is defined as τ SR = inf{k 1 : W C k > A}. (3.10) 14

120 100 A=100 80 W k 60 40 20 0 0 5 10 15 20 25 30 35 40 Time Figure 3.3: CuSum agorithm run for data of Figure 3.1. 200 A=180 150 100 W k 50 0-50 0 5 10 15 20 25 Time Figure 3.4: Shiryaev-Roberts agorithm run for data of Figure 3.1. In Fig. 3.4 we show the evoution of the SR statistic for the sampes shown in Fig. 3.1. 3.3 Optimaity Properties of CuSum and Shiryaev-Roberts Agorithms In Sec. 3.2, we presented agorithms that can be used to detect an abrupt change in the statistica behavior of a sequence of measurements under the usua i.i.d. setting. In the current section, we provide theoretica justification for the use of the CuSum and SR agorithms by reviewing properties regard- 15

ing the optimaity of the agorithms for Lorden s and Poak s formuations of the QCD probem. For the i.i.d. setting, Lorden showed in [17] that the CuSum agorithm is asymptoticay optima with respect to Formuation 1 as γ. In detai we have the foowing theorem: Theorem 1. With a threshod choice of A = og γ, the CuSum stopping rue (3.8) satisfies and E [τ C ] γ inf WADD(τ) WADD(τ C ) τ:e (τ) γ og γ D(f 1 f 0 ), as γ, where the notation is used to denote that the ratio of the quantities on the two sides of the approaches 1 in the imit as γ. A stronger resut was proved in [21] and ater in [22], where it was shown that the CuSum agorithm is exacty optima with respect to Formuation 1. For the same QCD setting, Poak showed in [18] that the SR agorithm is asymptoticay optima with respect to Formuation 2 as γ. In detai, we have the foowing theorem: Theorem 2. With a threshod choice of A = γ, the SR stopping rue (3.10) satisfies and E [τ SR ] γ inf CADD(τ) CADD(τ SR ) τ:e (τ) γ og γ D(f 1 f 0 ), as γ, where the notation is used to denote that the difference of the quantities on the two sides of the approaches 0 in the imit as γ. Since the performance of both the CuSum and the SR agorithms are asymptoticay equa, we have that Theorem 1 and Theorem 2 impy that both the agorithms are asymptoticay optima with respect to Formuation 1 and Formuation 2. 16

CHAPTER 4 QUICKEST CHANGE DETECTION UNDER TRANSIENT DYNAMICS Up to now, we have studied the traditiona QCD probem, where at some time instant γ the distribution of the observed process changes from an initia distribution to a fina distribution. In this section, we further generaize the QCD probem by incorporating transient dynamics. As a resut, the shift from the initia to the fina distribution does not happen instantaneousy, but after a series of cascading transient stages of finite duration, each one corresponding to a different probabiity distribution. We study the non-composite version of the transient QCD probem, where a distributions are known beforehand. We introduce the Dynamic CuSum (D-CuSum) agorithm as the proposed agorithm. 4.1 The Dynamic CuSum Agorithm Assume a random process {X k } k=1 with the foowing statistica behavior: f 0, if 1 k < γ 0, f (0), if γ 0 k < γ 1,. X k (4.1) f (i), if γ i k < γ i+1,. f (T ), if γ T k, where γ i N, i = 0,..., T. Note that the case of γ i+1 = γ i + 1 corresponds to a transient stage with a duration of one time instant. The goa is to design a stopping rue that wi detect the change in the statistica behavior of the observed process that takes pace at time instant γ 0. A heuristic test soution can be derived by considering this probem as a dynamic composite hypothesis testing probem. Thus, at every time instant 17

k, choose between the foowing two hypotheses: H k 0 : k < γ 0, H k 1 : k γ 0. The nomina hypothesis H k 0 corresponds to the case that the time instant γ 0 has not been reached yet, whie the aternative hypothesis H k 1 corresponds to the case that γ 0 has been reached. Each hypothesis induces a different set of distributions on the data X 1, X 2,..., X k. In particuar, H k 0 is a singe hypothesis under which the data foow distribution f 0 i.i.d. and H1 k is a composite hypothesis, i.e., it induces one distribution beonging to a set of distributions. The distribution that is induced depends on the vaues of the γ s and k. To find the test statistic we first form the ikeihood ratio of this hypothesis testing probem for an arbitrary choice of γ s: min{γ 1 1,k} j=γ 0 f (0) (X j ) k k j=min{γ T 1,k}+1 j=γ 0 f 0 (X j ) f (T ) (X j ) Note that this ikeihood ratio shoud be interpreted with the understanding k that := 1 for i = 0,..., T. This is a natura generaization of j=k+1 f (i) (X j ) f 0 (X j ) the maximum ikeihood interpretation of the CuSum statistic [6]. The test statistic is derived by taking the maximum with respect to γ 0,..., γ T. An equivaent test statistic can be derived by maximizing the ogarithm of the above quantity. As a resut, we have that. W k = { min{γ 1 1,k} max og f (0) (X j ) γ 0 < <γ T f j=γ 0 (X j ) + + 0 k j=min{γ T 1,k}+1 og f } (T ) (X j ), f 0 (X j ) with the understanding that γ 0 k hods. This maximization is the reason the test is independent of the transient duration, as wi be seen ater. W k can be written in the foowing way: W k = max{ω (0) k,..., Ω(i) k,..., Ω(T ) k }, (4.2) 18

where { γ1 1 Ω (i) k = max og f (0) (X j ) γ 0 <γ 1 < <γ i k f j=γ 0 (X j ) +... 0 k + og f } (i) (X j ), i = 0,..., T, f j=γ 0 (X j ) i (4.3) by using the fact that k j=k+1 og f (i) (X k ) f 0 (X k ) written in a recursive manner as foows: = 1. We caim that the (4.3) can be Ω (i) k = max{ω (i) [k 1], Ω (i 1) [k 1]} + og f (i) (X k ) f 0 (X k ), for i = 0,..., T and Ω (0) k := 0 for a k Z. First, consider the case i = 0: Ω (0) k { k = max γ 0 k = max γ 0 k = max γ 0 k { = max og f (0) (X j ) f j=γ 0 (X j ) 0 { k 1 } og f (0) (X j ) f j=γ 0 (X j ) + og f (0) (X k ) f 0 (X k ) 0 } { k 1 og f (0) (X j ) f j=γ 0 (X j ) 0 max γ 0 k 1 k 1 j=k [ k 1 og f (0) (X j ) f j=γ 0 (X j ) 0 } og f (0) (X j ) f 0 (X j ) } + og f (0) (X k ) f 0 (X k ) ], + og f (0) (X k ) f 0 (X k ) = max{ω (0) [k 1], 0} + og f (0) (X k ) f 0 (X k ). Since Ω ( 1) k := 0, the argument we attempt to prove hods for the case of i = 0. Now for the case of an arbitrary i: 19

{ γ1 1 Ω (i) k = max γ 0 <γ 1 < <γ i k j=γ 0 og f (0) (X j ) f 0 (X j ) + + { γ1 1 max og f (0) (X j ) γ 0 <γ 1 < <γ i k j=γ 0 og f (i) (X k ) f 0 (X k ). k 1 f 0 (X j ) + + Consider the first term of this expression. We have that: { γ1 1 max γ 0 <γ 1 < <γ i k { = max max γ 0 <γ 1 < <γ i k 1 og f (0) (X j ) f j=γ 0 (X j ) + + k 1 0 [ γ1 1 og f (0) (X j ) f j=γ 0 (X j ) + k 1 0 [ γ1 1 = max og f (0) (X j ) γ 0 <γ 1 < <γ i 1 k 1 f j=γ 0 (X j ) + k 1 0 = max{ω (i) [k 1], Ω (i 1) [k 1]}. k og f (i) (X j ) f j=γ 0 (X j ) i og f (i) (X j ) f j=γ 0 (X j ) i og f (i) (X j ) f j=γ 0 (X j ) i } og f (i) (X j ) f j=γ 0 (X j ) i } = } + ] og f ]} (i 1) (X j ) f j=γ 0 (X j ) i 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 fina D-CuSum test statistic is defined as foows: where Ω (i) k { W k = max Ω (0) k,..., Ω(T ) k },, 0 (4.4) = max{ω (i) k 1, Ω(i 1) k 1 } + og f (i) (X k ) f 0 (X k ), (4.5) for i = 0,..., T, Ω ( 1) k := 0 for a k Z and Ω (i) 0 := 0 for a i. The corresponding stopping time is given by comparing W k against a predetermined positive threshod: τ = min{k 1 : W k > A}. To demonstrate the performance of the agorithm we generate a process 20

6 5 4 3 2 X k 1 0-1 -2-3 0 10 20 30 40 50 60 70 80 90 100 Time Figure 4.1: A typica reaization of a sequence of i.i.d. Gaussian variabes. The statistics of the process are characterized by (4.6). 120 100 A=100 80 W k 60 40 20 0 0 10 20 30 40 50 60 70 80 Time Figure 4.2: D-CuSum test run for data of Figure 4.1. with the foowing statistica behavior: X k N (0, 1), if 1 k 19, N (0, 1.5), if 20 k 39, N (0, 2), if 40 k 59, N (0, 2.5), if 60 k 79, N (0, 3), if 80 k. (4.6) A reaization of this process up to sampe 100 is shown in Fig. 4.1. The evoution of the D-CuSum agorithm for this process reaization is shown in Fig. 4.2. We see that the D-CuSum statistic starts to grow after γ 0 = 20, eventuay crossing a threshod of A = 100. 21

CHAPTER 5 QCD ALGORITHMS FOR NON-TRANSIENT POWER SYSTEM LINE OUTAGE DETECTION In this Chapter, we study the ine outage detection probem in the case of no transient dynamics. We start by presenting the compete underying power system mode, which incudes an arbitrary number of transient periods. Next, we propose a statistica agorithm for detecting ine outages in a power system, for the specia case of one post-change stage, and show that it has better performance than other schemes proposed in the iterature. Our agorithm is based on the Generaized Cumuative Sum (G-CuSum) test from the quickest change detection (QCD) iterature, a test which is formuated by using the CuSum statistic in a generaized manner, i.e., by cacuating a test statistic for each post-change distribution, and comparing each statistic with a corresponding threshod. Different methods of seecting the test threshods, incuding using the notion of KL divergence, are examined. Our agorithm expoits the statistica properties of the measured votage phase anges before, during, and after a ine outage, whereas other methods in the iterature ony utiize the change in statistics that occurs at the instant of outage. From now on, the time indexes for every process wi be paced in braces and not as a subscript, so that reading is made easier. 5.1 Power System Mode Let L = {1,..., L} denote the set of ines in a system with N buses. A transmission ine can be denoted either by an integer, or by a coupe of integers (m, n) denoting that this ine connects bus m to bus n. 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 (see e.g., [23]), which 22

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)), (5.1) 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). 5.1.1 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 the power fow equations of (5.1) around (θ i [k], V i [k], P i [k], Q i [k]), i = 1,..., N, and use the DC power fow assumptions (see e.g., [23]), namey, (i) fat votage profie, (ii) negigibe ine resistances, and (iii) sma phase ange differences, 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], (5.2) 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 23

the reference bus removed. In an actua power system, random fuctuations in the oad drive the generator response. Therefore, in this thesis, we use the so-caed governor power fow mode (see e.g., [24]), 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], (5.3) 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 during the pre-outage period. Then, we can substitute (5.3) into (5.2) 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 g [k] = M 0 P d [k] [ = [M0 1 M0 2 B 0 P d [k] ] P d [k] = (M 1 0 B 0 + M 2 0 ) P d [k] = M 0 P d [k], ] (5.4) where M 0 = M 1 0 B 0 + M 2 0. 5.1.2 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 24

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 preoutage ine fow across ine from m to n. Foowing a simiar approach as that 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, (5.5) 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 0. Furthermore, by using the governor power fow mode of (5.3) and substituting into (5.5), and simpifying, we obtain: θ[γ 0 ] M 0 P d [γ 0 ] P [γ 0 + 1]M 0 r. (5.6) 5.1.3 Post-Outage 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 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., H = H 0 + H. Then, by etting M := H 1 = [M 1 M 2 ], and deriving in the same manner as the pre-outage mode of (5.4), 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, (5.7) where M,i = M 1B i + M 2, i = 1, 2,..., T. 25

5.1.4 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 ) p possibe ocations to pace the PMUs. In this thesis, we assume that the PMU ocations are fixed; in genera, the probem of optima PMU pacement is NP-hard and its treatment is beyond the scope of this thesis. Let M = M 0, if 1 k < γ 0,. M,T, if k γ T. (5.8) 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 (5.8) as foows: ˆM = C M, (5.9) 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]. (5.10) 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 [25], it can be argued 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] are roughy independent. Since ˆθ[k] depends on P d [k] through the inear reationship given in (5.10), we have that: 26

ˆθ[k] f 0 := N (0, ˆM 0 Λ ˆM T 0 ), if 1 k < γ 0, f (0) := N ( P [γ + 1]CM 0 r, ˆM 0 Λ ˆM T 0 ), if k = γ 0, (5.11). f (T ) := N (0, ˆM,T Λ ˆM,T T ), if k γ T, 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. 5.1.5 Non-transient Statistica Mode In this chapter, we focus on the specia case where T = 1 (aso note that due to the meanshift γ 1 = γ 0 + 1), i.e., after an outage the distribution changes from pre-change to a post-change distribution after going through the meanshift phase. In particuar, in this chapter we wi deveop ine outage detection techniques for the case that { ˆθ[k]} k=1 is characterized by the foowing statistica behavior: ˆθ[k] f 0 := N (0, ˆM 0 Λ ˆM T 0 ), if 1 k < γ 0, f (0) := N ( P [γ + 1]CM 0 r, ˆM 0 Λ ˆM T 0 ), if k = γ 0, f (1) := N (0, ˆM,1 Λ ˆM 1,1 ), if k > γ 0. (5.12) 5.2 Line Outage Detection Using QCD In the ine outage detection probem setting of the present chapter, the outage induces a change in the statistica characteristics of the observed sequence { ˆθ[k]} k 1 which is summarized by (5.12). The goa is to detect the outage in ine as quicky as possibe subject to fase aarm constraints. It is quite apparent that the present statistica mode is amost identica to the QCD setting studied in Chapter 3. Thus, we wi use QCD-based stopping rues to detect the statistica behavior shift that occurs after the outage. 27

5.2.1 Probem Setup The goa in ine outage detection is to design stopping rues that wi detect ine outages 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 wi be using Lorden s deay metric: [ ] WADD (τ) = sup ess sup E γ0, (τ γ 0 ) + ˆθ[1],..., ˆθ[γ 0 1]. (5.13) γ 0 1 The difference between the metric of (5.13) and (3.1) is that here we suffer a different deay depending on which ine is outaged. 5.3 QCD-based Line Outage Detection Agorithms With the statistica mode for { ˆθ[k]} k 1 in pace, the probem of detecting a ine outage was formuated as a probem of detecting a change in the probabiity distribution of the sequence of observations { ˆθ[k]} k 1 as quicky as possibe given fase aarm constraints. As a resut, we can use the QCD theory presented in this work to design agorithms for ine outage detection. 5.3.1 The Generaized CuSum Agorithm In our setting, the ine in which the outage occurs is unknown, i.e., the postchange distribution induced on the observation sequence { ˆθ[k]} k 1 is unknown. Since there are L ines, we have L different post-change scenarios, i.e., we have L possibe post-change distributions. The probem of detecting an abrupt change in the distribution of a process, in which the post-change distribution beongs to a known set of distributions is caed the composite QCD probem. Since we have a tota of L post change scenarios, we use the generaized version of the CuSum agorithm, namey, the Generaized CuSum (G-CuSum) agorithm. This agorithm invoves cacuating a CuSum statistic for each possibe post-change scenario. We present two versions of this agorithm: one proposed in [3], in which a common threshod is used for a CuSum statistics, and another in which the threshods are seected based on each ine s KL divergence. 28

We define the G-CuSum statistic corresponding to ine outage recursivey as: { W GC [k] = max W GC [k 1] + og f (1) ( ˆθ[k]) f 0 ( ˆθ[k]), og f (0) } ( ˆθ[k]) f 0 ( ˆθ[k]), 0, (5.14) with W GC [0] = 0 for a L. Note that an extra og-ikeihood ratio is added inside the maximum. This term is used to capture the instantaneous meanshift that occurs at the time of the outage. The G-CuSum stopping time is defined as: { } τ GC = min inf{k 1 : W GC [k] > A GC }. (5.15) L We now present different ways of choosing the threshods for the G-CuSum test. It can be shown (see, e.g., [26]) that by choosing A GC = og β og ξ, (5.16) with ξ being a positive constant independent of β, the expected deay for each possibe outage differs from the corresponding minimum deay among the cass of stopping times C β = {τ : E (τ) β}, as β, by a bounded constant. A choice of threshods for the G-CuSum agorithm is obtained by setting β = 1 L for a L. This way we get a common threshod, i.e., A GC = A GC = og(βl) for a L. It can be shown (see, e.g., [27]) that by choosing the threshods this way, we can guarantee that E [τ GC ] β. Using the resuts in [26], another choice of the threshods coud be based on a reative performance oss criterion, i.e., β = 1 D(f (1) f 0 )L(ζ ), (5.17) 2 where with ] ζ = im E (1) b [e { (S [τ b ] b)}, (5.18) τ b = inf{k 1 : S [k] b}, (5.19) and S [k] = k j=1 og f (1) ( ˆθ[j]) f 0 ( ˆθ[j]). (5.20) Note that this choice of threshod depends on the asymptotic overshoot of an SPRT-based test. As we show ater through case studies in Chapter 7, the threshod choice of (5.17)-(5.20) resuts in performance gains compared to choosing a 29

common threshod for a the ines. 5.4 Other Line Outage Detection Agorithms In this section, we present some other change detection agorithms that can be shown to be equivaent to other techniques proposed in the iterature. For exampe, the ine outage detection agorithm proposed in [10] can be shown to be equivaent to a og-ikeihood ratio test that ony uses the most recent measurements. 5.4.1 Meanshift Test The meanshift test is a one-shot detection scheme in that the agorithm uses ony the most recent observation to decide whether a change in the mean has occurred and ignores a past observations. The meanshift statistic corresponding to ine is defined as foows: [k] = og f (0) ( ˆθ[k]) f 0 ( ˆθ[k]). (5.21) W MS The decision maker decares a change when one of the L statistics crosses a corresponding threshod, A MS. The stopping time for this agorithm is defined as: { } τ MS = min inf{k 1 : W MS [k] > A MS }. (5.22) L The meanshift test ignores the persistent covariance change that occurs after the outage. In particuar, note that the meanshift test is using the ikeihood ratio between the distribution of the observations before and at the changepoint. More specificay, assuming that an outage occurs in ine, the expected vaue of the statistic at the changepoint is given by [ E (0) og f (0) ] ( ˆθ[k]) f 0 ( ˆθ[k]) = D(f (0) f 0 ) > 0, (5.23) where E (0) denotes the expectation under distribution f (0). On the other hand, after the changepoint (k > γ 0 ), the expected vaue of the statistic is given by [ E (1) og f (0) ] ( ˆθ[k]) f 0 ( ˆθ[k]) = D(f (1) f 0 ) D(f (1) f (0) ), (5.24) 30