STOCHASTIC SUBSPACE METHODS FOR LARGE POWER SYSTEM OSCILLATION MONITORING SEYED ARASH NEZAM SARMADI

Transcription

1 STOCHASTIC SUBSPACE METHODS FOR LARGE POWER SYSTEM OSCILLATION MONITORING By SEYED ARASH NEZAM SARMADI A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY School of Electrical Engineering and Computer Science AUGUST 2015

2 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of SEYED ARASH NEZAM SARMADI find it satisfactory and recommend that it be accepted. Mani Venkatasubramanian, Ph.D. Chair Anjan Bose, Ph.D. Chen Ching Liu, Ph.D. ii

3 ACKNOWLEDGEMENTS First and foremost, I would like to express my sincere gratitude to my advisor Dr. Mani Venkatasubramanian for giving me the opportunity to work under his supervision and the continuous support of my Ph.D. study and research. I am very impressed and grateful for his patience, deep knowledge, hard working, and motivation. I would also like to thank Dr. Anjan Bose and Dr. Chen-Ching Liu for their assistance and encouragement during my PhD coursework and research. Their enormous knowledge makes them sources of inspiration. I am thankful for the financial support from the U.S. Department of Energy, Power System Engineering Research Center, Entergy Inc. and Tennessee Valley Authority. I am specially honored to work with Mr. Floyd Galvan and his team from Entergy Inc. that gave me a valuable experience. I am also pleased to work with my colleagues Tianying Wu, Dr. Jiawei Ning, Zaid Tashman, Hamed Khalilinia, Ebrahim Rezaei and Dr. Chuanlin Zhao. Last but not the least; I would like to express my speechless gratitude to my family for their enduring love and support throughout my life that has been paramount in all my achievements. iii

4 STOCHASTIC SUBSPACE METHODS FOR LARGE POWER SYSTEM OSCILLATION MONITORING Abstract by Seyed Arash Nezam Sarmadi, Ph.D. Washington State University August 2015 Chair: Mani Venkatasubramanian Accurate knowledge and estimation of the low-frequency electromechanical oscillations are of vital importance for operational reliability of any power system. Such mode estimation basically can be done with two different approaches: using a power system model and by linearizing the equations about operating an equilibrium point or by using measurement based mode estimation methods. Measurement based algorithms for estimating low-frequency electromechanical modes serve as useful practical methods to monitor the modal properties of power system oscillations in real-time. There are two different types of measurement data: ring-down data and ambient data. A ringdown data is from system response to a sudden disturbance in power system. For ambient data, power system is assumed to be operating at its quasi-steady-state condition while the system input is from continuous small random fluctuations in loads and other related small variations which are assumed to be white noise. Stochastic subspace methods are effective algorithms for modal estimation of a system from ambient data. They have a relatively simple order selection technique and they are good choices for handling large data and system dynamic changes. They also are effective methods in estimating mode shapes. The main weakness of the subspace method is that of high computational burden because it requires Singular Value Decomposition iv

5 (SVD) of a large-dimensional matrix that makes it difficult to implement in real-time applications. Besides these natural electromechanical modes that are excited by load variations, forced oscillations can be introduced into power systems from external mechanisms such as cyclic loads or from mechanical aspects of generators. The existence of forced oscillations may affect the estimation accuracy of the natural modes from some measurement based estimation methods. They can also have a dangerous effect on generators caused by resonance between forced oscillations and local modes In the first part of this dissertation (chapters 2-3), two new stochastic subspace method are developed and introduced for the power system problem: Recursive Adaptive Stochastic Subspace Identification (RASSI) and Distributed Recursive Stochastic Subspace Identification (DRSSI). In the second part (chapter 4), the concept of resonance in power systems from Forced Oscillations (FO) is discussed and studied. v

6 TABLE OF CONTENTS ACKNOWLEDGEMENTS... iii ABSTRACT... iv LIST OF TABLES... x LIST OF FIGURES... xi ACKNOWLEDGEMENTS... iii 1. Introduction Introduction Recursive Adaptive Stochastic Subspace Identification [14] Decentralized Recursive Stochastic Subspace Identification [19] Resonance in Power Systems from Forced Oscillations [21] Dissertation Organization Recursive Stochastic Subspace Identification Stochastic Subspace Identification Recursive Stochastic Subspace Identification Recursive Adaptive Stochastic Subspace Identification RASSI Test Results Two-area Test System Parameter Selections Normal Study Cases Ramp Case vi

7 Step Case Extending RASSI to Multiple Parallel Engines WECC System Cases Event Event Eastern System Event WECC Breakup of Aug. 10, Decentralized Recursive Stochastic Subspace Identification Two-level Ambient Oscillation Monitoring Substation Level Computations Procedures at Control Center Level DRSSI DRSSI Test Results Two-Area Test System Detailed example for DRSSI Monte Carlo test results for two study cases Different Data Sharing Arrangements An Example of Closely Spaced Modes Western American Power System Data WECC Blackout of Aug Resonance in Power Systems From Forced Oscillations vii

8 4.1. Recent Event In Western Power System Resonance Theory from Physics Resonance In Kundur Test System Forced Oscillations and Inter-area Mode Forced Oscillations and Local Mode Summary of Resonance Cases SSI Estimation For The WECC Event Source of Forced Oscillations Conclusion and Future Work Conclusion Future Work References viii

9 LIST OF TABLES Table 2.1- Estimation Results for Case 1 of Two-area Network Table 2.2- Estimation Results for Kundur Test System Table 2.3- Average Execution Times for the Ramp Case Table 2.4- Mode estimation Results of WECC event Table 2.5- Mode estimation of WECC event Table 2.6- Mode estimation of Eastern System event Table 3.1- DRSSI results for the inter-area mode and local mode Table 3.2- DRSSI test results for Cases 1 and Table 3.3- DRSSI results seen by different areas Table 3.4- DFDD and DRSSI results for close modes case Table 3.5- Estimation results for the western system case Table 4.1- WECC FDD estimation results at two different times Table 4.2- SSI estimation results for Cases 3, 4 and Table 4.3- SSI estimation results for cases 6, 7 and Table 4.4- SSI estimation results for cases 9, 10 and Table 4.5- Effect of forced oscillation location on resonance Table 4.6- Effect of forced oscillation magnitude on resonance amplitude Table 4.7- SSI estimation results for Cases 12, 13 and Table 4.8- SSI estimation results for Cases 15, 16 and Table 4.9- Effect of forced oscillation location on resonance Table Summary of resonance effect on inter-area mode (cases 3 to 11) Table Summary of resonance effect on local mode (cases 12 to 17) Table SSI estimation results for the two WECC cases ix

10 LIST OF FIGURES Figure 2.1- One-line diagram of Kundur two-area network [1] Figure 2.2- Singular values for case 1 of two-area network Figure 2.3- Ramp test case for RASSI parameter Figure 2.4- Mode shape estimation for a) Inter-area mode b) Local mode Figure 2.5- Comparison of ramp case estimates Figure 2.6- Ramp case estimates for RASSI Figure 2.7- Comparison of step case estimates Figure 2.8- Flowchart of three-engine RASSI Figure 2.9- Step case estimates with RASSI double and triple engines with step switching Figure Step case estimates with RASSI double and triple engines with ramp switching Figure Ring-down responses for a) Event 1 b) Event Figure Mode shape estimation of WECC event 1: a) Prony b) RASSI Figure Mode shape estimation of WECC event 2: a) Prony b) RASSI Figure Active power in the Malin to Round Mountain #1 tie-line Figure Damping estimation of Aug. 10, 1996 using RASSI Figure 3.1- Two-level Oscillation Monitoring Structure Figure 3.2- Two-Level OMS Computational Structure Figure 3.3- Analysis of August 10, 1996 PMU data Figure 4.1- FDD mode shape estimates for 0.37 Hz mode: a) Case 1 b) Case Figure 4.2- MVA oscillations at PMU 1 for Case 1 and Case Figure 4.3- Mass-spring system with an undamped forced oscillation Figure 4.4- Oscillation amplitude for systems with different damping levels x

11 Figure 4.5- One-line diagram of modified Kundur two-area power system [1] Figure 4.6- Total MW flow on the tie-lines between buses 7 and 8 for a system with poorly damped inter-area mode Figure 4.7- Total MW flow on the tie-lines between buses 7 and 8 for a system with medium damped inter-area mode Figure 4.8- Total MW flow on the tie-lines between buses 7 and 8 for a system with well damped inter-area mode Figure 4.9- SSI mode shape of inter-area mode: a) Case 3 b) Case 4 c) Case Figure SSI mode shape of forced oscillation: a) Case 3 b) Case 4 c) Case Figure P78 Oscillations for poorly damped local mode Figure P78 Oscillations for well-damped local mode Figure SSI mode shape estimates of the WECC system: a) Forced oscillation for Case 1, b) Inter-area mode for Case 1, and c) inter-area mode for Case Figure SSI mode shape of forced oscillation for system of Cases 3-5: a) FO at 0.56 Hz b) FO at 0.66 Hz c) FO at 0.76 Hz xi

12 1. INTRODUCTION 1.1. INTRODUCTION Accurate knowledge and estimation of the low-frequency electromechanical oscillations are of vital importance for operational reliability of any power system. Such mode estimation basically can be done with two different approaches: using a power system model and by linearizing the equations about operating an equilibrium point [1] or by using measurement based mode estimation methods (e.g. [2],[3]). Measurement based approaches can be divided into two basically different categories: block processing and recursive methods. In the block processing methods, a fixed size time window is used to estimate modes and the window is updated to do a new estimation independently. In recursive methods, estimated modes are updated with each new sample or a fixed size of new samples by combining the new samples and previous estimations. There are two different types of measurement data: ring-down data and ambient data. A ring-down data is from system response to a sudden disturbance in power system such as outage of a generator or line tripping which results in a significant excitation of oscillatory modes. For ambient data, power system is assumed to be operating at its quasi-steady-state condition while the system input is from continuous small random fluctuations in loads and other related small variations which are assumed to be white noise. Since ambient data is always available through Wide Area Measurement Systems (WAMS) and they act as non-intrusive measurements, the algorithms which can be applied on the ambient data are quite useful (e.g.[2],[4]). Several methods have been introduced by researchers [5]. For block processing methods, [6] introduced a Modified Extended Yule Walker (MEYW) to identify modes in time domain. Subspace identification method (also in time domain) is used in [7]-[10] to estimate mode damping and frequency. Frequency Domain Decomposition (FDD) method has been developed 1

13 in [4] for power system mode estimation in frequency domain. A method based on wavelet transforms is proposed in [11]. Besides these block processing methods, recursive methods have been introduced recently to overcome some drawbacks of block processing methods. Robust RLS [12] and Regularized Robust RLS [13] are introduced based on recursive identification of ARMA blocks. The main advantage of the recursive methods is the speed of computation by updating previous estimations with new sampled data with a chosen forgetting factor. The other advantage of these methods is that they need less memory for data storage [12] RECURSIVE ADAPTIVE STOCHASTIC SUBSPACE IDENTIFICATION [14] Stochastic subspace methods are effective algorithms for modal estimation of a system from ambient data [8],[15]. They have a relatively simple order selection technique and they are good choices for handling large data and system dynamic changes [8]. It is also shown in [16] that subspace identification is an effective method in estimating mode shapes. The main weakness of the subspace method is that of high computational burden because it requires Singular Value Decomposition (SVD) of a large-dimensional matrix that makes it difficult to implement in real-time applications. Chapter 2 of this dissertation introduces a Recursive Adaptive Stochastic Subspace Identification (RASSI) method which overcomes the drawback of high computational burden by recursively updating the estimations that avoid computation of SVD in each instance of estimation. Therefore, the method provides both benefits, namely, ease of implementation of recursive methods as well as the effective estimation ability of subspace algorithms. It is shown that the proposed method gives fast accurate online estimation of mode frequency and damping as well as mode shapes from wide-area synchrophasor data. Also, an adaptive forgetting factor selection method is utilized to make the estimation more responsive 2

14 (under poorly damped conditions) or smoother (under well-damped conditions) when needed. The adaptive switching methodology in the proposed algorithm allows efficient tracking of mode damping levels from sudden changes in system conditions and from gradual variations. The adaptive switching is done independently for each oscillatory mode which is especially important for power system applications where the properties of different modes are related to different aspects of the system dynamics. For instance, the damping level of one inter-area mode may undergo a sudden change upon the loss of a critical tie-line while the other modes may not show any appreciable change. The proposed methodology adapts to changes in each mode independent of each other. Recursive subspace identification has been applied successfully in civil engineering and some other fields [17],[18]. This dissertation applies the recursive subspace algorithm to the power system problem, while also improving the methods by making the recursion adaptive. The test cases show that the adaptive recursive subspace algorithm is comparable to the block processing based subspace algorithms used earlier in power systems while being significantly faster and easier to implement. In short, the benefits of the proposed method are: It is a fast recursive method with short initialization window which can work with small refresh rates (e.g. 0.1 s). The short initialization window allows quick restart of the engine for any reason such as from a sudden loss of measurements. Small refresh rates enable fast tracking of damping changes that in turn helps associate damping changes with system changes in better root cause identification. Fast tracking and refresh rate are also important for automatic control actions in future designs. It is able to adapt independently to each mode in order to track the damping levels for all the modes efficiently. 3

15 The method provides a good balance between steady state performance with small standard deviations of estimations and fast transient response of the estimations under transient conditions DECENTRALIZED RECURSIVE STOCHASTIC SUBSPACE IDENTIFICATION [19] Recent paper [20] introduced a Distributed Frequency Domain Optimization (DFDO) method wherein the estimation task is divided between the substation level estimators and the control center level estimator in a closely coordinated fashion. The two-level estimation in [20] requires full-resolution PMU data (or equivalent FFT values) from substations, and conversely the substation level estimators need detrending angle reference and other supervisory guidance from the control center for local modal estimations. That is, the communication between substations and central processor in DFDO is bidirectional which requires higher network bandwidth and is rather expensive to implement. In the proposed two-level framework of [19], the modal estimations at the substation level are fully decoupled with no requirement of any detrending or supervisory guidance from the control center. That is, the communication is strictly unidirectional from the substations to the control center. Moreover, substation level modal estimates (rather than full fidelity PMU data) are sent to the control center at slower rates (say once a second), the communication requirement is much lighter in terms of network bandwidth. The substation modal estimation results are assembled and analyzed at the control center level estimation to derive overall modal properties of the system. Lack of centrally computed detrending signal makes the substation level estimation more challenging with respect to [20]. Similarly, the control center level estimation in this paper is much more complex compared to [20] because the control center does not supervise or 4

16 coordinate the substation level estimations, and it has to extract system modal information by careful analysis of substation modal estimates as shown in this paper. Given the complex dynamic interactions of large interconnected power systems, it is nontrivial to divide modal estimation between substations and the control center. The two-level framework developed in this paper is a major step toward fully analyzing system modal properties from a large number of available PMU measurements. The Decentralized Recursive Stochastic Subspace Identification (DRSSI) is introduced in this dissertation. DRSSI is derived by fitting RSSI [14] into the new two-level structure RESONANCE IN POWER SYSTEMS FROM FORCED OSCILLATIONS [21] Besides the natural electromechanical modes that are excited by load variations, forced oscillations can be introduced into power systems from external mechanisms such as cyclic loads or from mechanical aspects of generators [22],[23]. The existence of forced oscillations may affect the estimation accuracy of the natural modes from some measurement based estimation methods [24]. They can also have a dangerous effect on generators caused by resonance between forced oscillations and local modes [25],[26]. This dissertation studies the interactions of low frequency inter-area modes and forced oscillations. Owing to their geographically widespread nature, inter-area modes are shown to especially vulnerable to resonance from forced oscillations. When an inter-area mode is poorly damped, and when forced oscillations are injected at a frequency close to the system mode frequency at a location where the inter-area mode is participating, resonance can occur leading to oscillations that are much larger than that of the source. Typical inter-area modes such as the 0.22 Hz and 0.37 Hz North-South modes are seen all across the western American power system. Therefore, forced oscillation occurring anywhere in the western grid at a frequency close to 0.37 Hz could be a serious problem when the 0.37 Hz system mode is poorly damped. The 5

17 dissertation analyzes a recent event from summer 2013 when synchrophasor measurement based analysis detected a forced oscillation at 0.38 Hz that was injected from a hydro generator. Recent paper [27] reports a 140 MW tie-line oscillation event observed in China in 2005 that may be related to resonance between system modes and forced oscillations. These events point to the need for a systematic analysis presented in the paper on when resonance can occur between forced oscillations and inter-area modes. The concept of resonance is well known in physics. Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than the others [28]. An external forced oscillation with a frequency at one of the natural system frequencies for undamped or poorly damped modes will lead to resonance [28]. The destructive nature of resonance is wellrecognized in power systems in the context of sub-synchronous resonance [29] that arises from adverse interactions of electrical and mechanical torsional modes. Sustained oscillations with amplitude as high as 280 MW were encountered in the eastern American power system in June 1992 [30] and it lasted longer than 30 minutes. Oscillations cause problems for power quality and can potentially damage expensive power grid equipment. Detecting the presence of forced oscillations and their interactions with system modes is a challenging problem. Fortunately, with increasing number of installed Phasor Measurement Units (PMUs), it is feasible to detect forced oscillations, and predict resonance using measurement based framework. Some of the initial methods on detecting and locating forced oscillations can be seen in [31]. In this dissertation, the effects of forced oscillations on different system modes (such as local vs. inter-area), especially when resonance happens, are studied. The dissertation illustrates the effectiveness of a previously proposed ambient oscillation monitoring method, namely Stochastic Subspace Identification-Covariance, denoted SSI-Cov in [14], in detecting forced oscillations and in finding their locations. This dissertation carries out an extensive study of interactions of system modes and forced oscillations in the context of 6

18 resonance, and also analyzes an actual recent event from 2013 when a forced oscillation occurred at inter-area mode frequency in the western American power system DISSERTATION ORGANIZATION This dissertation is organized into 6 chapters. The first chapter contains an introduction to the work completed. Chapter 2 introduces a recursive time domain stochastic subspace methods [12], RASSI, after a brief introduction of the Subspace Identification and Stochastic Subspace Identification algorithms. The algorithm is tested using simulated data from Kundur two-area test power system as well as measured data from real systems. Chapter 3 introduces a two-level approach for oscillation monitoring and the Decentralized Recursive Stochastic Subspace Identification (DRSSI) [19] which is suitable for the two-level implementation. Chapter 4 studies the effect of Forced Oscillations (FO) on power systems [21]. It is shown that under certain conditions, a resonance may happen between FO and system modes resulting in large oscillations on tie-lines. Chapter 5 reviews the research contributions of this dissertation and suggests directions for future work to extend this research. Chapter 6 contains references used for this work. 7

19 2. RECURSIVE STOCHASTIC SUBSPACE IDENTIFICATION 2.1. STOCHASTIC SUBSPACE IDENTIFICATION When power system operates in the ambient condition, small variation of loads is assumed to be white noise inputs and stochastic subspace identification can be applied to estimate modal behavior of the system. Consider the n-th order linear system: = + = + (2.1) Where w is the process noise, is the measurement noise, R and R. The noises are assumed to be white, stationary, zero mean Gaussian noises. Differing in implementation aspects, SSI can be developed in two different ways: the covariance-driven stochastic subspace identification (SSI-COV) and the data-driven stochastic subspace identification (SSI-data) [17]. While the first one operates on the covariance of the output, the second one operates directly on the output data instead of calculating output covariance explicitly. MATLAB function N4SID is based on the data driven algorithm which has been applied to the power system in [6],[8],[16]. In this paper, SSI-COV is used for modal estimation of power system along with MATLAB routine N4SID for comparison. The covariance matrix H is defined as.. =.. (2.2) where! " and " denotes the expected value. It can be proven = $ (2.3) where $! ". Now if % and be the extended observability and controllability matrices of a system (,$,), then we have: =) *+$ $ $..,=% (2.4).. 8

20 is ideally defined for infinite number of measurements but since it is impossible to work with it in practice, an estimation of with finite set of measured covariances - - will be used as follows /= (2.5) of data In practical applications / can be found easily by defining two output sets over a window =).. * =).. * (2.6) 2 2 where2 I is the data window length. / then can be found by calculating /= 1 4 5! where 6 ranges over all sets of data and 4 is an optional normalization parameter and can be set to 1 since multiplying / by a constant does not change modal characteristics of the system. To find the state matrix and hence system modes we need to find the extended observability matrix. A finite dimensional estimate %- for % can be found from SVD of /corresponding to the n largest singular values where n is the system order. So if we take SVD of /, we get (2.7) /=789! =+7 7,: 8! <=9! 9 > (2.8) where 7 is the left singular vector, 8 is a diagonal matrix containing singular values and 9 is the right singular vector, and S R A A contains n dominant singular values. The extended observability matrix can be calculated by [15] %- =7 8 / (2.9) And, the extended observability matrix has the form 9

21 %- =.. R () (2.10) If we define %- the matrix O/ without the last l rows and O/ the matrix O/ without the first l rows, then we can find D and A/ as follows D=%-F O/, D = O/(1:H,:) (2.11) where F denotes the Moor-Penrose pseudo-inverse. Mode damping and frequency can be obtained from eigenvalue analysis with a given A/ using discrete to continuous transformation [32]. D I = 1 HLMD, D J I =D (2.12) K where J K is the sampling time. Mode shape can be obtained from right eigenvectors of D I, however these mode shapes are the mode shapes of the estimated model by algorithm. To obtain the system mode shape, algorithm's mode shapes should be multiplied by D I N K O =D I N (2.13) Where N K and N O are the system and model mode shapes related to the i-th mode RECURSIVE STOCHASTIC SUBSPACE IDENTIFICATION The subspace method without recursion is basically a block processing method and gives an independent estimation for each data window. To have an online estimation of the system, the recursive subspace method is proposed in [17],[18]. The first estimation should be like the non-recursive method using a pre-defined length for a window of data. After finding the initial using block processing method, can be 10

22 updated recursively using new sets of data. To do the recursive part, old data can be forgotten exponentially by multiplying by a forgetting factor P as follows / Q =P/ Q +! (2.14) Through the forgetting factor P in (2.14), the distant past information has increasingly negligible effect on the updating of coefficient. To make the recursive method much faster, O/ can be updated directly without taking SVD in each iteration. It is shown in [33] that for an unconstrained criterion 9+R(S),=T/ Q R(S)R! (S)/ Q T V (2.15) where W denotes Frobenius norm, R(S) is a stationary point of (2.15) if and only if R(S)=7J where 7 denotes any n left singular vectors. It is proven that all stationary points in (2.15) are saddle points except for 7=7 where (2.15) reaches its global minimum. Therefore, instead of taking SVD of / Q, a recursive estimation of O/ X can be obtained by minimizing (2.15) using the following least square problem: Y(Z)=[/ Z \Y ] (Z ^)[/ Z _ F (2.16) A faster implementation can be obtained by multiplying R! (S 1) with Q rather than / Q [17]. As it is mentioned earlier, the estimation process begins with a non-recursive initial estimation and after calculating / Q and 7. The next stage is calculating the following R(S 1)=7 J=7 (2.17) à Q =R! (S 1)/ Q b(s 1)=+à Q à! Q, F (2.18) (2.19) 11

23 After calculating (2.17)-(2.19) from initial estimation, where it is assumed in (2.17) that the unitary matrix J is an identity matrix, the following equations should be run in each iteration with a new set of data Q and Q R(S)= à Q Q R (2.20) h(s)=r! (S 1) Q R (2.21) N(S)=+(S) h(s), R (2.22) P d(s)=e Q! Q P P 0 f R (2.23) g(s)=+p d(s)+n! (S)b(S 1)N(S), N! (S)b(S 1) (2.24) R N(S)=+/ Q Q Q, R 2 (2.25) b(s)= 1 P +b(s 1) b(s 1)N(S)g(S), R (2.26) `(S)=P`(S 1)+h(S)! Q R 2 / Q =P/ Q + Q! Q R 2 2 R(S)=R(S 1)++(S) R(S 1)N(S),g(S) R 2 %- Q =R(S) (2.27) (2.28) (2.29) (2.30) Mode estimation can then be done using (2.30) and (2.11)-(2.13). Like other estimation methods, the performance of RSSI is dependent on the choice of parameters. These parameters are: 1. Initialization data length 2. Recursive window length 3. Model order 4. Forgetting factor 12

24 5. Estimation refresh rate Each of these parameters has different effect on the estimation accuracy. Hence it is important to study their effect and choose appropriate values for them. RSSI initializes with initialization of Ĥ matrix using (2.7) from SSI estimator. Assuming the system is in steady state operating conditions, the larger initialization data length, the more accurate the estimations are. However, the results may not get better by increasing the initialization window under typical quasi-steady-state conditions of real systems. Moreover, larger initialization data length comes with the usual drawback that there will be no estimation during this period. Our experiments show that an initial data length of 60 to 300 seconds will be long enough to get reasonably accurate estimations. Recursive window length is the size of data in each RSSI estimation. One benefit of recursive methods is that we are using knowledge of previous estimations, and this new data window length can be selected small enough to increase estimation speed and enabling faster implementation in real-time. Again, from our experiments, a recursive window length between 4 to 20 seconds of data can be selected to have a good estimation. Easy model order selection method is one of the benefits of RSSI. As it is mentioned in previous sections, system order can be selected using largest drop in singular values. This order selection in other methods can be done using trial and error (e.g. see [13]). Also, another useful property of SVD can be used to make estimations faster. SVD decomposes a matrix in such a way that singular values will be sorted from largest to smallest. Singular values and their corresponding singular vectors are related to eigenvalues and hence the system modes. Singular values are indicators of mode energy. Therefore, model order can be selected to be as small as needed to obtain the desired modes. When a mode is estimated with a system order as indicated by SVD, increasing this number further only affects the frequency and damping ratio in a limited way as shown in the next section. 13

25 Once a system order has been selected for the system, the nature of recursive method does not allow algorithm to change the order when RSSI is running. In real cases, when the number of excited modes in the measurement data may be changing, the full SVD can be calculated in parallel say every 10 or 15 minutes (along with the recursive RSSI) and if the SVD indicates a change in model order, the recursive engine RSSI can be reinitialized accordingly. Forgetting factor is an important parameter that balances the relative emphasis between prior mode histories versus new information in arriving at recursive estimations. More details will be discussed in the next section. The last tuning parameter is the estimation refresh rate and it determines how fast a new estimation should be done. A smaller refresh rate makes the estimator more responsive for realtime estimation. The refresh rate has a similar effect compared to the forgetting factor noted above. Smaller refresh rate tracks mode changes better while the estimation result with larger rate will be smoother for quasi-steady-state system conditions. A value between 0.1 to 1 second which is 3 to 30 new samples for each estimation has been effective in our experiments for RSSI for 30 Hz PMU sampling rate RECURSIVE ADAPTIVE STOCHASTIC SUBSPACE IDENTIFICATION As it is shown in (2.14), P is the forgetting factor which determines how fast previous estimations should be forgotten. Forgetting factor is usually chosen slightly smaller than 1.The closer the forgetting factor µ is to one, the more weighting there is on previous modal histories as opposed to new information. This relationship indicates that an estimator with higher forgetting factor gives smoother estimation whereas it is also slower to track changes in mode values. The choice of forgetting factor can be effectively managed using an adaptive approach as proposed below. There is a rich history of signal processing literature on variable forgetting factor designs in recursive least square methods [34],[35]. However, such methods are typically unsuited for 14

26 power system applications in our tests. The main reason for this is as follows: In power systems, we need to be tracking several modes simultaneously and the mode properties can change independent of each other as noted earlier from contingencies or from changes in generation schedules. Therefore, the approach of independent adaptive switching for each mode is wellsuited for power system applications. We propose a novel adaptive estimator that selects the modal estimates of multiple modes from the outputs of two parallel engines with different forgetting factors, and the method is shown to be effective in power system test cases in the next section. The adaptive method is based on the following practical facts: 1. Estimation methods are generally more accurate when the mode damping level is small since the oscillation damps down slowly and the mode is better excited. This is because the mode energy is higher for poorly damped modes and is the main motivation behind several frequency domain algorithms as discussed in the Appendix of [20]. 2. Lower damping is more important from operator's point of view and hence a more responsive estimator is needed to track the evolution of modes when the mode damping is small. In the proposed RASSI algorithm, two recursive estimations are done in parallel all the time, one using a higher forgetting factor (smooth estimation response) and the other using a lower forgetting factor (fast estimation response). The final RASSI estimate for each mode say h i is selected from either of the two respective estimates say h (slow estimate) and h (fast estimate) based on the logic shown below m h 1, no p 2 >p J stu J L >J k Lw p h = 2 <p J stu J y <J lh 2, no p 2 <p J stu J y >J k j Lw p 2 >p J stu J L <J (2.31) 15

27 whereh i is the final estimated mode, h and h are the mode estimates of two estimators with slower and faster tracking capability respectively. ξ 2 is the damping ratio value of the second (fast) estimator and ξ T is the damping ratio threshold. T u and T o are the total time durations that estimated damping ratio of the second fast estimator stays continuously below and over the damping ratio threshold respectively. And finally, T and T are defined as follows J ~ =J O J ~~ = J, no J <J (2.32) O J O, no J <J O where T umax and T omax are two tunable parameters that specify the maximum time considered as transient time when damping hits the threshold. When the damping ratio estimate of the fast engine goes below the threshold ξ T and stays below ξ T continuously for a pre-specified time, the logic decides that the mode is in low damping condition and uses the mode estimate of fast responsive estimator. Conversely, if the damping ratio of the fast estimator goes over threshold ξ T and stays over ξ T for a different pre-specified time, the mode is assumed to have recovered and the estimate of the slow smooth estimator is selected. The adaptive selection is done for estimate of each mode independently. A minimal exponential smoothing may be applied on h i to smooth out switching transitions as needed. Or, the transition from one engine estimate to the other can be done more gradually as discussed in section RASSI TEST RESULTS The algorithms are tested using simulated data from Kundur two-area test power system as well as measured data from real systems. Voltage phase angles are used as input signals in all test cases. Prony analysis is used as a common reference for both simulation and real cases to compare the estimation results of the ambient engine with the Prony results. 16

28 TWO-AREA TEST SYSTEM Figure (2.1) shows the outline of Kundur two-area test system. 1% of the two system loads at buses 7 and 9 is modeled to be random Gaussian noise to emulate the effect of random load fluctuations in the simulation. Network parameters are changed in such a way to vary the damping of the inter-area mode as well as one local mode for test purposes. Four cases of normal condition are studied where the damping ratios of the modes are intentionally changed by changing PSS parameters of the generators to test performance of RASSI. In order to validate tracking ability, RASSI is applied to a ramp case and a step case where the damping ratio changes in ramp and step like responses. Results are averaged over 10 Monte Carlo trials where each trial has 10 minutes of data for normal cases and 80 minutes for the ramp case. Figure 2.1- One-line diagram of Kundur two-area network [1] Parameter Selections A normal case that has a 0.62 Hz inter-area mode with 4.5% damping ratio and a 1.13 Hz local mode with 1.5% damping ratio is studied first to illustrate the effects of algorithm parameters on the estimation results. Figure (2.2) shows the result of SVD for singular values of this case. There is clearly a big drop from fourth to fifth singular value. Therefore, it can be concluded that the first and second modes have more energy compared to other modes. Accordingly, we set the base system order to be four. 17

29 Figure 2.2- Singular values for case 1 of two-area network Table (2.1) shows the estimation results of this case with different settings for parameters which were discussed in previous section. The first setting is selected to be the setting choice for later tests. The other parameter cases are defined in such a way that only one parameter is changed at each case to demonstrate its effect separately. Table I shows that estimation is reasonably well done for different settings. Although the second setting has the best estimation, it has a larger initialization window. The first setting is chosen to be the base case since it has a reasonably good performance and it is more appropriate for real cases because of the shorter initial window of 150 seconds compared to 210 seconds for the second setting. Since no estimates are available during the initialization window, shorter initialization window reduces the time-delay in arriving at the first estimation result for every instance of restarting the engine. Setting Initialization Data Length (sec) Table 2.1- Estimation Results for Case 1 of Two-area Network Recursive Window Length (sec) Model Order Refresh Rate (sec) Mode Frequency Mean(Hz) Std Damping Ratio Mean(%) Std

30 A ramp case is simulated next and is used for the adaptive parameter selection. The damping ratio of the inter-area mode with frequency around 0.6 Hz, changes from 7% at 10 minutes down to 2% at 30 minutes. Damping ratio of the mode stays at 2% for next 20 minutes. At 50 minutes, damping ratio starts to increase reaching 5.5% at 70 minutes and remains there from 70 to 80 minutes. Figure (2.3) shows one such case using and 0.99 as the slow and fast forgetting factors respectively. The selected damping ratio threshold is 4% and Tumax and Tomax are selected to be 1 and 10 minutes respectively. Figure 2.3- Ramp test case for RASSI parameter Normal Study Cases Four different cases for the two-area system with different (fixed) mode damping levels are considered as normal simulation cases. In the first case, both local and inter-area modes are poorly damped, while in the other cases, all the local modes are well-damped and hence the local modes are not excited. Mode damping levels are changed for different cases to see the effect of damping on the estimations. The base setting of the previous part (first setting in Table (2.1)) is taken as the parameter selection of the estimations excepting that the system order is changed to 2 for the cases 2 to 4 in Table (2.2). Table (2.2) shows the estimation results for these cases with Prony as the reference and RASSI, the proposed SSI-COV which will be called simply SSI throughout the paper and N4SID 19

31 (using MATLAB function) as a SSI-DATA method. These results show that how RASSI can estimate the modes with different damping level nicely in all the cases. A window length of 150 seconds and a refresh rate of 1 second are used for the SSI and N4SID in Table I. System order is taken to be 2 for case 1 while the order is 4 for cases 2 to 4 for SSI. Similarly, system orders are selected as 6 for case 1 and as 10 for cases 2 to 4 for MATLAB N4SID to get good estimations based on experiments. Table 2.2- Estimation Results for Kundur Test System Case Method Mode 1 Prony RASSI SSI N4SID Frequency (Hz) Mean STD Damping Ratio(%) Mean STD Prony RASSI SSI N4SID Prony RASSI SSI N4SID Prony RASSI SSI N4SID

32 Figure (2.4) shows mode shapes for the inter-area and local mode of the first case for only the generator buses. It can be easily seen that for the inter-area mode, generators 1 and 2 of area 1 oscillate against generators 3 and 4 of area 2. For the local mode, generator 3 oscillates against generator 4 while the mode shape magnitudes for generators in area 1 are small and that can be concluded the second mode is a local mode. These results are consistent with the analysis in [1]. In order to normalize mode shapes, each vector is divided by the maximum magnitude among all the vectors. Figure 2.4- Mode shape estimation for a) Inter-area mode b) Local mode Ramp Case The ramp case of part A.1 is used next to compare the tracking ability and estimation response of different methods. Figure (2.5) shows the identified mode with the four methods. Figure (2.6) shows the frequency and damping estimation of the same case with RASSI with more details. 21

33 Figure 2.5- Comparison of ramp case estimates Figure 2.6- Ramp case estimates for RASSI It can be seen from Figure (2.5) that both SSI and N4SID can track damping changes as well however at the cost of increased computational burden. Table (2.3) shows the average execution time of these methods for Monte Carlo cases of 80 min on a 3.40 GHz regular desktop computer. Table 2.3- Average Execution Times for the Ramp Case RSSI RASSI SSI N4SID Execution time (sec)

34 Step Case A step case is used next to show the tracking ability of RASSI for sudden changes of modal properties. Figure (2.7) shows the identified mode with the same four methods as in the ramp case. It can be seen from Figure 7 that tracking ability of RASSI for sudden changes is comparable to other block subspace methods. Figure 2.7- Comparison of step case estimates Extending RASSI to Multiple Parallel Engines The formulation of two parallel engines introduced in Section 2.3 can be easily extended to running multiple parallel recursive engines with different forgetting factors. With multi-core processing in modern computers, parallel recursive engines can be implemented at minimal additional computational cost. This section illustrates an extension to the case of three parallel recursive engines with the choice of 0.998, and 0.98 as the slow, medium and fast forgetting factors. The selected damping ratio thresholds are 3% (between fast medium) and 5% (between medium and slow) and Tumax and Tomax are selected to be 1 and 10 minutes respectively. The flowchart of Figure (2.8) shows how the three engine RASSI works. 23

35 Figure 2.8- Flowchart of three-engine RASSI Figure (2.9) shows a comparison of damping estimation results of RASSI with two and three parallel engines. It can be seen that the tracking response is a little better for the tripleengine RASSI compared to the double-engine RASSI, while the switching effects in transitioning between the internal engine results are also smaller. Damping estimations in Figure (2.9) show that the internal switching effect appears as a step change in both double and triple engines. To reduce this effect, switching can be done gradually over a pre-specified time window using a weighting factor. Figure (2.10) shows this "ramp switching" using a 1 minute transition window that avoids a sudden change between the engines. 24

36 Figure 2.9- Step case estimates with RASSI double and triple engines with step switching Figure Step case estimates with RASSI double and triple engines with ramp switching WECC SYSTEM CASES In this section, several events from the WECC system will be studied wherein a ringdown also happens such as shown in Figure (2.11). The ambient data after the ring-downs are used to estimate the inter-area mode and the ambient results are compared to the Prony result of the ring-down part. 25

37 Figure Ring-down responses for a) Event 1 b) Event Event 1 Prony analysis of the event shown in Figure (2.11-a) detected a 0.35 Hz mode with 12.89% damping. The ambient data from 100 sec to 600 seconds (after the ring-down) is used to estimate the mode using ambient methods. Eight buses are selected for the estimations for the ambient data after the ring-down. The resulting mode estimation results are shown in Table (2.4). Mode shapes are also estimated by RASSI and compared with Prony in Figure (2.12). The result shows the validity of the method and that buses 6 and 7 oscillate against the other buses. A window of 180 seconds is used as the analysis window for SSI and N4SID and as initial window for RSSI and RASSI and a window of 6 sec. is used as the recursive window for RSSI and RASSI. System order is selected to be 10 for RSSI, RASSI and SSI and 18 for N4SID. The adaptive parameters are selected the same as the two area cases and the refresh rate is 1 second for all cases. We can see that since the damping is higher than 4% threshold, RSSI and RASSI have the same result in Table (2.4). Table 2.4- Mode estimation Results of WECC event 1 Method Frequency Mean(Hz) Std Damping Ratio Mean(%) Std Prony RASSI RSSI SSI N4SID

38 Figure Mode shape estimation of WECC event 1: a) Prony b) RASSI Event 2 Prony analysis of the ring-down shown in Fig. 10.b detected a 0.38 Hz mode with 8.29% damping. Again, eight buses are selected for the RASSI estimation and the ambient data from 30 sec to 580 sec (after the ring-down) is used for ambient estimation. The resulting mode estimation is shown in Table (2.5) Figure (2.13) shows the mode shape estimation results by Prony and RASSI. They show that buses 6 and 7 oscillate against the rest for the inter-area 0.38 Hz mode. Similar to event 1, SSI and N4SID are also used for comparison reasons in Table V. While the SSI gives an estimation of the 0.38 Hz mode, N4SID could not estimate the mode properly and after increasing the model order to 30, it can find a 0.39 Hz mode but with very high damping ratio and damping standard deviation. The other parameters are the same as in the Event 1. Table 2.5- Mode estimation of WECC event 2 Method Frequency Mean(Hz) Std Damping Ratio Mean(%) Std Prony RASSI RSSI SSI N4SID

39 Figure Mode shape estimation of WECC event 2: a) Prony b) RASSI EASTERN SYSTEM EVENT The outage of a generation unit caused a ring-down (not shown to save space) in the eastern American power system. Prony analysis showed presence of a poorly damped 1.25 Hz mode at 1.5% damping. The disturbance was not strong enough to excite the inter-area mode at 0.45 Hz in the ring-down though it is seen in ambient data as shown in Table (2.6). Ambient data from 30 sec to 560 sec (after the ring-down) is used for mode estimation by RASSI, SSI and N4SID. The inter-area mode with the low energy appears to be problematic for detection by N4SID. All parameters are the same as in WECC cases except for system order which is 10 for all four methods. Table 2.6- Mode estimation of Eastern System event Method Mode Frequency Damping Ratio Mean(Hz) Std Mean(%) Std Prony RASSI RSSI SSI N4SID

40 WECC BREAKUP OF AUG. 10, 1996 On Aug. 10, 1996, a cascading outage in WECC system caused a major blackout. The system experienced growing oscillations during the event. Figure (2.14) shows the active power flow through the tile line from Malin to Round Mountain #1. There was an outage of a 500 kv line from Keeler to Allston that preceded the growing oscillations by about five minutes. This outage makes this a good case to test the performance of a mode identification method since the event has both ringdown and ambient data. RASSI is used to estimate the oscillating 0.27 Hz mode which changed to 0.25 Hz after the first outage. MW flows on two of the 500 kv tie lines (Malin to Round Mountain and BC Hydro to Custer) are selected as the inputs after removing the respective mean values. Initialization data length is selected to be 80 seconds. In order to track changes in the dominant mode, estimation refresh rate is selected 0.1 second or 2 new samples per estimation since the sampling frequency is 20 Hz. The 0.27 Hz mode damping is reported in [36] as 7% for before the first line trip, 3.46% after the Keeler-Allston line trip and decreases to negative damping values after Ross-Lexington line trip as the oscillations start to grow. Figure (2.15) shows damping estimation for the 0.27 Hz mode. It can be seen that RASSI handles both ringdown and ambient data automatically without having to reinitialize the algorithm. Estimation result in Figure (2.15) is comparable with other published papers and reports on the breakup case [4],[36-39]. Figure Active power in the Malin to Round Mountain #1 tie-line 29

41 Figure Damping estimation of Aug. 10, 1996 using RASSI 30

42 3. DECENTRALIZED RECURSIVE STOCHASTIC SUBSPACE IDENTIFICATION 3.1. TWO-LEVEL AMBIENT OSCILLATION MONITORING Here, a new fully decentralized two-level structure for ambient Oscillation Monitoring System (OMS) is introduced [19]. Each substation carries out local ambient modal analysis using DRSSI to be introduced later in this section for computing the preliminary estimates of local and inter-area modes. The local modal estimates including the estimates for dominant mode frequencies, their damping levels and energy levels are then sent to the control center periodically. Typically the local estimates can be sent to the control center say every one second or every five seconds and hence this is a big savings in terms of communication bandwidth compared to full fidelity PMU data. The main job of control center is to group and combine the modal estimation results obtained from substation level engines. Utility A Substation 1 OMS Substation k OMS Substation N OMS Substation 1 OMS Substation 2 OMS Substation 3 OMS Control Center OMS Substation 4 OMS Substation k-1 OMS ISO/RTO Control Center OMS Substation N-1 OMS Substation 4 OMS Control Center OMS Substation 3 OMS Substation 2 OMS Utility B Figure 3.1- Two-level Oscillation Monitoring Structure [19] As shown in Figure (3.1), the proposed architecture is easy to accommodate at the utility level as well as at the level of Independent System Operator (ISO) or Regional Transmission Operator (RTO). A utility shown by a cloud in Figure (3.1) can carry out its estimation by collecting estimations from substations belonging to the utility as well as possibly from few 31

43 substations of neighboring utilities. Similarly at ISO/RTO level, the estimation can be done by collecting estimates from the utilities as well as by directly gathering some critical substation modal estimates. The framework is inherently robust since loss of one or a few substation estimates can be easily handled at the control center level and will not significantly impact the overall estimation features at utility level or ISO/RTO level. The main advantages of the new two-level structure can be summarized as follows. 1) The architecture encourages use of all available PMU measurements by shifting bulk of the computational burden to the substations. Any problematic poorly damped local mode will be detected by a nearby PMU (if present) and problems with PSS being out-of-service or becoming defective can be monitored continuously. Moreover, inter-area modes will be detected by several of the substations providing higher redundancy in inter-area mode monitoring. 2) Heavy communication burden in sending high speed PMU data from all the substations to the control center is considerably alleviated (for the oscillation monitoring task). For the new structure, only substation modal estimation results need to be sent to the control center and at a much slower rate (say every few seconds). It may even be possible to send substation results to the control center using slower communication media such as used in Supervisory Control and Data Acquisition (SCADA). 3) The computational complexities of existing multi-dimensional algorithms such as SSI grow in exponential fashion as the number of PMU signals increases. Therefore, even when all the signals are available at the control center, the existing algorithms are not scalable for handling the large number of PMU measurements already available in the North American power grid. The two-level structure also enables multi-threaded parallel implementations of the new algorithm DRSSI. In the two-level architecture of Figure (3.1), most of the estimation related computations are done at the substation level. Figure (3.2) shows the computational steps at the substation level (Figure (3.2-a)) and at the control center level (Figure (3.2-b))). 32

44 Substation Local PMUs Substation 1 Estimation Results Substation k Estimation Results Time-Align Synchrophasor Data Time-Align Substation Estimation Results Detrending Identify Different System Modes by Grouping Results Ambient Modal Analysis Report Substation Modal Estimation Results Compute Modal Estimates for Each Mode by Weighting of Substation Results Compute Mode Shape for Each Mode Utility Control Center OMS ISO/RTO Control Center OMS Report System Modal Estimation Results OMS Historian a. Substation OMS b. Control Center OMS Figure 3.2- Two-Level OMS Computational Structure [19] SUBSTATION LEVEL COMPUTATIONS After time-aligning local PMU data, a detrending logic will be applied to relevant local - PMU signals to be used in ambient modal estimation. Based on tests on different choices of signals, bus voltage magnitudes are suggested to be good input signals for DFDD algorithm. On the other hand, phase angle differences are chosen as estimation input signals for DRSSI algorithm. Each algorithm calculates the mode frequency, damping ratio and mode energy for the dominant modes seen in local PMU measurements by applying the methodology of RSSI from section 2.2 respectively, and these estimates are sent to control center OMS. 33

45 PROCEDURES AT CONTROL CENTER LEVEL At the control center level OMS, the main tasks are 1) to group substation modal estimates into different system modes, and to compute overall estimates for each system mode. Figure (3.2-b) summarizes the computational steps within the control center OMS. Upon receiving results from substations, the estimates will be time-aligned. Then, the control center OMS will group all available substation estimates for the latest time together by frequency. Mode number can also be assigned for each grouped mode. Then the overall estimated mode frequency of each group will be calculated and updated for each refresh time. By doing this, it could follow the trend of dominant modes even while the mode frequencies may be drifting along. For each group of modes, the final result (frequency (o) and damping ratio (p)) of this system modal estimation will be obtained by a weighted average function, o =5 Š Š Œ Ž o (3.1) p = Š p (3.2) Œ Ž where is the number of reporting substations in this group, is the mode energy for mode n from substation 6, o and p are the estimated frequency and damping ratio for mode n form substation 6 respectively. Meanwhile, the total mode energy for mode n ( Š ) will also be recorded for comparing the strength of one mode to another. The concept of evaluating system modal estimates by weighted average of substation results was earlier introduced in [20]. However, the problem is more challenging in the decentralized framework of this paper because substation modal estimates are not coordinated. In the framework of [20], only a few dominant substations are selected by the control center for estimating the specific mode and their substation modal estimates are combined by the weighted averaging formulas (3.1) and (3.2). In this work, since there is no pre-selection of 34

46 dominant substations for each mode during the substation level estimation, the control center level estimator first selects the dominant substations for each mode from all the substation estimates by applying certain energy thresholds. The system modal estimates are then computed by only weighting the results from the dominant substation estimates rather than all available substation estimates for each mode. By recognizing the peak energy for each mode from all available substation estimates, the control center can recognize and discard the problematic substation estimates that have comparatively low energy (say less than 10% of the global peak) for each mode. Test results in sections 3.3 show that weighted average values from (3.1) and (3.2) with the screening procedure as described above capture the true values of mode frequency and mode damping ratio very well DRSSI A Recursive Stochastic Subspace Identification (RSSI) has been proposed in section 2.2 to estimate power system electromechanical modes. Stochastic Subspace Identification is used to estimate state space model of the system: = + = + (3.3) where R is the state of system, R is output or measurements, w is the process noise and is the measurement noise [14]. SSI gives estimations of A and C. Modal properties then can be obtained from eigenvalue analysis of the following continuous time matrices: D I = 1 HLM /, D J I =D (3.4) 35

47 where T is the sampling rate. SSI needs SVD calculation of a relatively large matrix at S each time window which is time consuming and is the well known drawback of this time domain identification method. To overcome this problem, RSSI is used in section 2.2 which needs SVD calculation only for its initialization step. Recursion can be done on SSI using equations (2.17)-(2.30). Distributed RSSI (DRSSI) introduced in this dissertation applies RSSI at each PMU enabled substation to estimate dominant modes seen at that substation. One or more local signals are used as the inputs of RSSI for each PMU. Once the modes are locally estimated at the substation, singular values for each substation for the identified modes are used as mode energy indicators for combining the substation results at the control center level OMS as noted in equations (3.1) and (3.2). New SSI SVD calculations are done periodically (say every three minutes in the examples of this paper) or after major disturbances to track the mode energy changes over time at different substations. Further details on RSSI can be seen in section 2.2. The focus in this chapter is to test how well the two-level formulation of Figure (3.2) works with RSSI algorithm DRSSI TEST RESULTS TWO-AREA TEST SYSTEM In this section, the new decentralized structure along with the algorithm DRSSI are tested on Kundur s two-area system [1]. Commercial program TSAT [40] has been used to generate simulation data. It is assumed that 1% of each load (real and reactive power) consists of Gaussian white noise at bus 7 and bus 9. The random noise represents ambient system load variations and they are assumed at be independent at each bus. It is assumed that there is a PMU installed at each bus. It is assumed that each PMU measures its bus voltage phasor and at least one line current 36

48 phasor on any of the lines connected to the bus. As discussed in Sections 3.1 and 3.2, bus voltage magnitudes (for mode damping level estimation) and bus voltage phase angles (for mode shape estimation) of 11 buses are used as inputs for DFDD. By knowing the transmission line network parameters, it is assumed that the neighboring bus voltage phasor can be calculated at each of the buses by standard circuit calculations from bus voltage phasor line current phasor available from the local PMU. Accordingly, phase angle difference between the measured bus voltage phase angle with respect to the respective calculated neighboring bus voltage phase angle at each of the 11 buses are used as inputs for DRSSI. There are two dominant modes in the Kundur s system as studied in this test case. There is an inter-area mode around 0.56 Hz and a local mode around 1.03 Hz related to receiving area (Bus 1 and Bus 2). To study different damping levels of the two modes, parameters of power system stabilizers of each generator are varied in the test system. Prior to the beginning of the simulation, a small disturbance is created to excite the modes so that Prony methods [3] can be used to estimate modal information of the system. The disturbance is simulated by removing one of the lines between buses 7 and 8 and by reconnecting it after 0.1 sec. SSAT [40] eigenvalue results from small-signal linearized models are also provided for comparison Detailed example for DRSSI A detailed example is presented for DRSSI to demonstrate how the two-level distributed version works. Small-signal model based analysis of the system by SSAT shows an inter-area mode with frequency Hz at 4.66% damping ratio, and a local mode at 1.03 Hz at 1.4% damping ratio. Based on a four minute time-domain simulation from TSAT that included 1% random noise at the two buses 7 and 9, we will estimate the system modes. DRSSI results for a random choice of recursive window say at time t=278 seconds from the TSAT simulation. At the 37

49 substation level, each engine generates its own results and reports the result along with its singular values to the control center. Table (3.1) shows DRSSI results for this time window. Table 3.1- DRSSI results for the inter-area mode and local mode Bus No. 1st Mode Sing. Value Freq. (Hz) Damp. Ratio (%) 2nd Mode Sing. Value Freq. (Hz) Damp. Ratio (%) Cont. Cent Table (3.1) shows that the inter-area mode is detected in local estimations at substations 5, 7, 8 and 9, while the local mode can be observed in substations 1, 2 and 5. When subspace identification is applied at the substation level, modes with small singular values yield biased estimations. To handle this, DRSSI rejects local substation modal estimates when the mode singular values are not at acceptable levels for the estimated damping ratio. Singular value thresholds of 50 for modes with damping ratio lower than 3% and 200 for modes with damping ratio higher than 3% are used in this paper. Accordingly, even though the inter-area mode is observed in all the substations, only buses 5, 7, 8 and 9 will report the inter-area mode estimates to the control center because the singular values at the other substations are below their respective thresholds. Similarly, buses 1, 2 and 5 are the only substations that report the local mode estimates. Control center OMS calculates the weighted overall estimates for both inter-area and local modes using equations (3.1) and (3.2) as reported in the last row of Table (3.1). Estimates of inert-area and local modes in Table (3.2) match well with SSAT results. 38

50 Monte Carlo test results for two study cases Two different operational scenarios are studied in two cases. In the first study case, the inter-area mode is designated to be poorly damped (with damping ratio around 2%) while the local mode is well-damped (with damping ratio around 8%). In the second study case, the interarea mode is designed to be at medium damping level while the local mode is poorly damped. The reporting rate is 30 Hz like from typical PMUs. For each of the two study cases, 10 sets of 10 minutes of ambient data with different random noise seeds are generated. For the recursive engine DRSSI, all combinations of two choices of initial time-window length, say Tin, (180 and 240 seconds), and two choices of analysis time-window length, say Tan, (6 and 10 seconds) have been used. SSAT [40] results and Prony [3] modal analysis results are used as reference values for comparison. Bus voltage phase angles from the eleven substations are used as inputs for the multi-dimensional algorithm FDD. From the results in Table (3.2), it is clear that DRSSI can estimate the dominant modes effectively for both poorly damped and medium damped modes. Table (3.2) shows the choices of initial window length of four minutes and analysis window of six seconds are noted as good choices for DRSSI. Case Method Table 3.2- DRSSI test results for Cases 1 and 2 T in (sec) T an (sec) Frequency (Hz) Mean STD Damping Ratio(%) Mean STD 1 SSAT Prony DRSSI DRSSI DRSSI DRSSI

51 2 SSAT Prony DRSSI DRSSI DRSSI DRSSI SSAT Prony DRSSI DRSSI DRSSI DRSSI Different Data Sharing Arrangements To further illustrate the new two-level structure proposed in this paper, let us consider three data sharing arrangements: 1) Area 1 utility receives and monitors substation results from buses 1, 2, 5, 6 and 7; 2) Area 2 utility receives substation results from buses 3, 4, 9, 10 and 11; 3) Regional regulator say an ISO monitors substation results from buses 7, 8 and 9 (tie-line buses). The control center level computations for each of the three entities, Area 1 utility, Area 2 utility, and ISO will be different as shown in Table (3.3) though the estimations are largely consistent with each other. In some test cases, DRSSI could not find modal results for certain modes when the buses available to that utility or ISO do not contain enough observable information about the mode. For instance, the poorly damped local mode is not detected by ISO 40

52 control center in Table (3.3) because this mode is not seen in any of the tie-line buses reporting to the ISO. Table 3.3- DRSSI results seen by different areas Case Method Area Frequency (Hz) Mean STD Damping Ratio(%) Mean STD 1 2 SSAT Prony DRSSI DRSSI DRSSI SSAT Prony DRSSI DRSSI DRSSI SSAT Prony DRSSI DRSSI DRSSI ISO ISO ISO An Example of Closely Spaced Modes When the system has two or more oscillatory modes with close mode frequency values, the ambient estimation problem becomes more challenging [4]. For illustration, the network and generator parameters of the Kundur system are modified so that the system has two different 41

53 modes with frequencies around 0.45 Hz. In Case 1, the inter-area mode at 0.45 Hz has damping ratio around 2% (poorly damped) while the local mode at Hz is well-damped with damping ratio near 6%. In case 2, the interarea mode has damping ratio at around 6% while the local mode has damping at around 2%. Table (3.4) shows the estimation results of DRSSI for this example. It can be seen that the method only detect the inter-area mode in this case and estimations match well with their linearized results from SSAT mode estimates. Even when the interarea mode has much higher damping compared to local mode in case 2, the energy associated with the interarea mode is high enough that the distributed methods detect the interarea mode well for this case in Table (3.4). Table 3.4- DFDD and DRSSI results for close modes case Case Method Frequency (Hz) Mean STD Damping Ratio(%) Mean STD 1 2 SSAT Prony DRSSI SSAT Prony DRSSI SSAT Prony DRSSI SSAT Prony DRSSI

54 WESTERN AMERICAN POWER SYSTEM DATA In order to show the applicability of the proposed method for large number of channels, 105 channels of voltage magnitude measurements are used over a ten minute time period when the system was operating under normal conditions. This is recent PMU data from the western American power system. Table (3.5) shows the estimation results of DRSSI and compares it with RSSI. Prony estimations are not shown for this case since there was no ring-down event to analyze for this data set. Table 3.5- Estimation results for the western system case Method Frequency (Hz) Mean STD Damping Ratio(%) Mean STD RSSI DRSSI WECC BLACKOUT OF AUG Recorded PMU data from August 10, 1996 blackout [41], [42] is used to test the algorithms. Time-plot of the active power generation of Grand Coulee generators during the event in Figure (3.3) shows poorly damped oscillations after Keller-Allston line tripping at around 400 seconds in Figure (3.3). During the blackout event, growing oscillations were seen on California-Oregon Inter-tie (COI) after Ross-Lexington line tripping after 700 seconds (not shown in Figure (3.3)). 43

55 Figure 3.3- Analysis of August 10, 1996 PMU data Because of lack of other measurement types, active power measurements from 9 PMUs have been taken as inputs for testing. In order to present more number of estimates, the initialization window of DRSSI have been shortened to be 180 seconds instead of 240 seconds used earlier in this chapter. Estimates from DRSSI and DFDD [19] algorithms presented in Figure (3.3) clearly show the dramatic change in 0.25 Hz mode damping level in the minutes before the blackout. The 0.25 Hz mode damping ratio was around 7% according to the estimates in Figure (3.3) before Keeler-Alston 500 kv line tripping. Prony analysis of the measurements over the time-period from 400 to 420 seconds shows the damping ratio of the 0.25 Hz mode at around 2.59% (shown by a red triangle in Figure (3.3)). DRSSI reinitializes around 400 seconds just after Keeler-Alston line tripping because the large disturbance could change the singular values. Figure (3.3) shows that the damping ratio estimates of the 0.25 hz mode from DRSSI engines to be close to zero just before 700 seconds that could have provided operator alarms or control triggers. Shortly after 700 seconds, Ross-Lexington line tripping occurred and growing oscillations were seen during the August 10, 1996 event that led to the large blackout. 44

56 4. RESONANCE IN POWER SYSTEMS FROM FORCED OSCILLATIONS 4.1. RECENT EVENT IN WESTERN POWER SYSTEM PMU based real-time framework for analyzing oscillatory modes based on the algorithm Frequency Domain Decomposition (FDD) has been implemented in several North American utilities [43]. FDD is a frequency domain algorithm that can estimate the mode frequency, mode damping ratio, mode energy, and mode shape for dominant oscillatory modes of a power system using ambient PMU data [44]. In one of the western American power system installations, during the summer season of 2013, FDD reported several instances when the damping level of the 0.37 Hz North-South inter-area mode of the western power grid was estimated to be unusually low below 3% for several hours. The analysis of one such operating condition is presented in this paper based on PMU historian data received from Western Electricity Coordinating Council (WECC). Table (4.1) shows FDD estimation results over two different ten minute time windows (denoted Cases 1 and 2) where both time windows were within the same one hour time period. The analysis is illustrated using five PMUs located widely across western system. Real-time implementation of FDD uses PMU bus voltage phase angle measurements as analysis inputs. However, because of historian issues, some of phase angles were not available for this archived data set. Therefore, product of bus voltage magnitude with a line current magnitude at the bus, which is the magnitude of the apparent power-flow on the line, is used as the analysis signal from each of the five PMUs. Results in this paper show that magnitudes of apparent power-flows serve as excellent signals for oscillation monitoring since they capture both aspects of active and reactive power-flows. FDD uses a four minute analysis window that is updated every one second in a moving window formulation. Table (4.1) shows the mean value and standard deviation results for the estimation of the 0.37 Hz mode for each of the two ten minute data sets. From Table I, it is clear that the mode damping is unacceptably low near zero for Case 1 while it is at a 45

57 healthy 12% level for Case 2. This is highly unusual that the damping changes by 12% in less than one hour. Further investigation of the FDD mode shapes shown in Figure (4.1) shows an important distinction between the two cases. Table 4.1- WECC FDD estimation results at two different times Case No. Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) Figure 4.1- FDD mode shape estimates for 0.37 Hz mode: a) Case 1 b) Case 2 The mode shape for Case 2 shows a typical mode shape of the 0.37 Hz inter-area mode for apparent power signals where the peak mode shape amplitude is seen at PMU 3 that is on a critical North to South tie-line. On the other hand, the mode shape is completely different for Case 1 when low damping is detected. For Case 1, the peak amplitude for mode shape is at PMU 1 which is at the terminal of a hydro power plant. Time-plots of the apparent power signal from PMU 1 shown in Figure (4.2) points to 10 MVA sustained oscillations that are observable for Case 1. 46

58 The project team contacted the utility owner of the power plant monitored by PMU 1, and that revealed an interesting phenomenon. On this specific day in summer of 2013, the wind power outputs in the system were high, and the utility had to back down on the hydro power output for power balancing. The difference between Cases 1 and 2 is that the hydro unit was operating in the rough zone in Case 1 while it was at the minimum capacity in Case 2. It is wellknown in turbine theory that hydro plants with Francis turbine blades undergo vortex related mechanical turbine oscillations [45] when the units are operated in the rough zone. Typically air compression systems are employed to keep the oscillations under control at low levels [45]. The frequency of the vortex oscillations depends on the nature of vortex (single versus double vortex etc.) and on the mechanical design of the turbine. In our case, the power plant monitored by PMU 1 experienced vortex related oscillations with frequency at around 0.37 Hz that were injected as MW oscillations into the western power grid in Case 1. The MW oscillations are observable in the time-plots of apparent power in Figure (4.2). Apparent Power (MVA) Time (sec) Apparent Power (MVA) Time (sec) (a) Case 1 (b) Case 2 Figure 4.2- MVA oscillations at PMU 1 for Case 1 and Case 2 In summary, PMU based analysis of the oscillations clearly pointed to the source of the oscillations to be PMU 1 which was at a hydro generator. Because of high wind conditions, the 47

59 hydro unit was operating in its rough zone that caused injection of 0.37 Hz forced MW oscillations into the grid. This is not a rare event because hydro generators must go through their rough zones (say for ten minutes) every time they are brought into and out of service. The vortex related oscillations can occur at different frequencies including at close to inter-area mode frequencies depending on the nature of the vortex rope and turbine properties. Therefore, the possibility of resonance between such forced oscillations and inter-area modes needs to be carefully studied. There is also a need to estimate the vortex related oscillation frequencies of different hydro units so that those with forced oscillation frequencies close to inter-area mode frequencies such as the generator at PMU 1 can be closely monitored in the future. The theory and nature of resonance in the context of inter-area modes are discussed in Section 4.3. The western system event is revisited in Section 5.4 by relating the observations of Section 4.3 to the real system event RESONANCE THEORY FROM PHYSICS Let us recall the theory of resonance from physics [46]. Consider the mass-spring system of Figure (4.3) with a forced undamped oscillation of W cos(`s). Then, from Newton's law, we get: +š +g =W œl (`S) (4.1) Figure 4.3- Mass-spring system with an undamped forced oscillation 48

60 where the system natural frequency ω0 and damping ž are: ` = g, ž= š (4.2) (4.2), we get: The damping ratio is given by ζ = Ÿ +ž +` = W Q (4.3) Equation (4.3) has a solution in the form of: = ( Q ) (4.4). Rewriting (4.1) in the complex plane and using Inserting (4.4) into (4.3) and solving for and gives: W m k = (` `) +(`ž) l ktan = `ž j ` ` (4.5) where is the amplitude of oscillation and is its phase difference with the forced oscillation. Resonance happens when A is at its maximum. From (4.5), for an undamped system (ž= 0), when the forced frequency is equal to the natural frequency (`=`), resonance occurs and the amplitude of the oscillation becomes infinity. However, for a damped system (ž 0), A will have a finite value from (4.5), and the maximum amplitude (called practical resonance) will happen at a frequency `O smaller than ` (based on the damping value). Three cases are of interest: 49

61 m` 0 k = W g 0 l` ` = W k g 2 j` =0 (4.6) where =. Next, the case of maximum amplitude can be obtained by taking derivative from Ÿ the first equality in (4.5) and making it equal zero. The max amplitude O and its frequency `O then will be: m O = W k g (1 1 4 ) l k`o =`(1 1 j 2 ) (4.7) Figure (4.4) shows the amplitude forced by the external oscillation for a system with different level of damping. The curve with the highest peak shows the lowest damped system (ζ = 2%) while the curve with the lowest peak shows the most damped system (ζ = 27%). The plot clearly shows that the effect of resonance can be ignored for well damped systems while it might be problematic for poorly damped systems. By definition, Q decreases as the damping γ increases. Therefore, for higher damped systems, the practical resonance is at peak frequencies `O that are lower than ` from equation (4.7). 120 A ω ω 0 max ω (rad/sec) Figure 4.4- Oscillation amplitude for systems with different damping levels 50

62 4.3. RESONANCE IN KUNDUR TEST SYSTEM This section uses the Kundur two-area test system [1] that is provided as a standard test case in [40]. A small generator induces a mechanical 10 MW sustained oscillation at a new bus 12 and is connected to bus 8 of the Kundur system (Figure (4.5)). The location of the forced oscillation will be changed to other buses later in this section. In order to simulate random load fluctuations, 1% of the loads at buses 7 and 9 are modeled by Gaussian white noise. Simulated measurements from the system for different combinations of forced oscillations and system modes are tested using the ambient modal estimation algorithm SSI-Cov (or SSI in short) from [14]. The objectives are twofold in this section: a) to understand when and how resonance can occur from forced oscillations, and b) to test how well SSI algorithm can detect and locate the source of forced oscillations. Figure 4.5- One-line diagram of modified Kundur two-area power system [1] SSI with analysis window length of 4 minutes, system order of 6, refresh rate of 1 second and 2I equal to 6 seconds [14] is used to detect the forced oscillation as well as system modes. Bus voltage phase angles from all 12 buses are used as measurement signals for SSI. Results are averaged over 10 Monte Carlo trials of 5 minutes each. Damping levels of the inter-area mode and one of the local modes are changed by adjusting the settings of the power system stabilizers. The frequency of forced oscillations is changed to values above and below THE system mode frequencies to test the performance of the SSI estimation in the spirit of tests introduced in [24]. Results in this section show the effectiveness of SSI-Cov [14] in simultaneous estimation of both the system mode and forced oscillations at different damping levels of the system mode 51

63 from ambient PMU data, even when their frequencies are close to each other. Earlier ambient modal analysis algorithms lack the ability for simultaneous estimation and also face other issues arising in estimation of system modes for such conditions: FDD (Section 4.1), Yule-Walker [24] and Welch-Half-Power-Point-Method [24] FORCED OSCILLATIONS AND INTER-AREA MODE This section analyzes the extent of resonance between forced oscillations and inter-area mode with respect to different sensitivities Resonance Let us start with a study case when the inter-area mode 0.56 Hz has a low damping ratio of 1.9%. Figure (4.6) shows the total MW oscillations on the tie-lines from bus 7 to 8 for three different frequencies of forced oscillations, namely, 0.53 Hz, 0.56 Hz, and 0.59 Hz. The tie-line fluctuations are very small for the case of no forced oscillation with the only changes coming from small random load fluctuations at buses 7 and 9. With forced oscillations, the tie-line MW oscillations are about 74 MW, 200 MW, and 70 MW for the forced oscillation frequency at 0.53 Hz, 0.56, and 0.59 Hz respectively. When the forced oscillation is exactly at the inter-area mode frequency of 0.56 Hz, resonance occurs resulting in 200 MW tie-line oscillations from a 10 MW forced oscillation at bus 12 nearby. This is consistent with the theory base in Section

64 MW No Forced Oscillation MW f 0 = 0.53 Hz Time (sec) Time (sec) MW f 0 = 0.56 Hz MW f 0 = 0.59 Hz Time (sec) Time (sec) Figure 4.6- Total MW flow on the tie-lines between buses 7 and 8 for a system with poorly damped inter-area mode Next, the study is repeated when the damping ratio of the inter-area mode is at a medium damping level of 5.2%. In Figure (4.7), the tie-line oscillations are at 65 MW, 90 MW and 56 MW for the forced oscillation frequencies at 0.53 Hz, 0.56 Hz and 0.59 Hz respectively. This shows that resonance effect is still present if only with reduced effect when compared to the lower damping case of Figure (4.6). Next, let us consider the system when the damping ratio of the 0.55 Hz inter-area mode is at a healthy 10.3%. In Figure (4.8), the tie-line oscillations are at 53 MW, 58 MW and 50 MW for the forced oscillation frequencies at 0.52 Hz, 0.55 Hz and 0.58 Hz respectively. Clearly the effect of resonance is much lesser in this case of well-damped system that is consistent with Figure (4.4). 53

65 700 No Forced Oscillation 700 f 0 = 0.53 Hz MW 600 MW 600 MW Time (sec) f 0 = 0.56 Hz MW Time (sec) f 0 = 0.59 Hz Time (sec) Time (sec) Figure 4.7- Total MW flow on the tie-lines between buses 7 and 8 for a system with medium MW No Forced Oscillation damped inter-area mode MW 700 f 0 = 0.52 Hz MW Time (sec) f 0 = 0.55 Hz Time (sec) MW Time (sec) f 0 = 0.58 Hz Time (sec) Figure 4.8- Total MW flow on the tie-lines between buses 7 and 8 for a system with well damped inter-area mode SSI estimation results For the first three cases in this section, the inter-area mode at 0.56 Hz stays at the low 1.9% damping ratio. Frequency of the forced oscillation (denoted FO in the tables) is close to the resonance point (0.56 Hz) in Case 4, at a nearby lower frequency (0.53 Hz) in Case 3 and at a nearby higher frequency (0.59 Hz) in Case 5. Table (4.1) shows the estimation results of these 54

66 cases by SSI and they are compared to model based SSAT eigenvalue results. Table (4.2) shows that SSI can estimate both the inter-area mode and forced oscillation at correct damping levels in all three cases. Mode shape plots of these cases from SSI are shown in Figure (4.9) for the inter-area mode and Figure (4.10) for the forced oscillation estimate. The mode shape of the inter-area mode in all three cases is as expected with buses in Area 1 swinging against those in Area 2. The amplitude of bus 12 in Figure (4.9-b) is higher because of resonance effect. For the forced oscillation, the source bus, bus 12, has the largest mode shape magnitude in Cases 3 and 5, making them somewhat easy to locate. Whereas for Case 4, the mode shape magnitude for bus 12 becomes small because of resonance effect. Interestingly, the mode shape phase for bus 12 becomes orthogonal to other buses. This is consistent with theory in equation (6) where δ is expected to be close to 90 degrees at resonance. The potential use of mode shape for locating the source of FO for resonant and non-resonant cases is discussed in more detail in Section 5.5. Figure 4.9- SSI mode shape of inter-area mode: a) Case 3 b) Case 4 c) Case 5 Table 4.2- SSI estimation results for Cases 3, 4 and 5 Case No. 3 4 Method Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) SSAT SSI FO actual SSI SSAT SSI FO actual SSI

67 5 SSAT SSI FO actual SSI Figure SSI mode shape of forced oscillation: a) Case 3 b) Case 4 c) Case 5 Next, the inter-area mode damping ratio is changed to 5.2%. Cases 6, 7 and 8, are used here to study the behavior of system when a forced oscillation exists at and close to the frequency of the 0.56 Hz system mode. Table (4.3) shows that SSI can nicely separate these modes and the effect of the forced oscillation on the estimation of the inter-area mode is minimal. The mode shape estimates of these cases are not discussed here to save space. Case No Table 4.3- SSI estimation results for cases 6, 7 and 8 Method Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) SSAT SSI FO actual SSI SSAT SSI FO actual SSI SSAT SSI FO actual SSI

68 Finally, the inter-area mode damping ratio is changed to 10.3% to see if resonance can still happen. Three cases, namely Cases 9, 10 and 11, are used like before. Results are shown in Table (4.4) where SSI can estimate both system mode and forced oscillation well. Again, mode shape estimates of these cases are not discussed here to save space. Western system example from Section 4.1 when the system experienced a forced oscillation close to the 0.37 Hz inter-area mode will be further analyzed in Section 4.4 as an example of a well-damped system mode. Table 4.4- SSI estimation results for cases 9, 10 and 11 Case No Method Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) SSAT SSI FO actual SSI SSAT SSI FO actual SSI SSAT SSI FO actual SSI Sensitivity to location of forced oscillation To study the resonance effect from different locations of forced oscillations, bus 12 in Figure (4.5) is connected to each of buses 1 through 11 respectively. Low damping conditions with the damping ratio of the inter-area mode 0.56 Hz at 1.9% are considered. Table V shows the amplitude of oscillations on various signals: a) MW flow P78 on the tie-lines between buses 7 and 8, b) bus voltage magnitude V9 at load bus 9, c) bus frequency f9 at load bus 9. From Table (4.5), it is clear that buses near the two ends of the system have larger effect on resonance compared to buses near the middle of the system. The strongest resonance happens when the forced oscillation is connected to bus 3 at the receiving end. In this case, the 10 MW forced oscillations at bus 3 57

69 cause 477 MW oscillations on the tie-lines. These relationships appear to be similar to the mode shape of the inter-area mode where the rotor angles at the two ends of the system show the largest swings. Accordingly, the last two columns in Table V show the mode shape magnitudes of each bus for the base case with no forced oscillation for the 0.56 Hz mode using SSI and FDD. The mode shape (denoted MS in Tables (4.5) and (4.9)) magnitudes show that the values are somewhat proportional to the MW oscillations if a forced oscillation exists at the bus. However, the relationship appears to be nonlinear that needs further investigation. Table 4.5- Effect of forced oscillation location on resonance FO source P 78 (MW) f9 (Hz) V9 (p.u) SSI MS mag. FDD MS mag

70 Sensitivity to amplitude of forced oscillations The theory of resonance discussed in Section 5.2 assumes linear dynamics. Accordingly, it is interesting to study the effect of forced oscillation magnitude on the amplitude of resonant oscillations and see if it grows linearly or not. To analyze this, forced oscillation at bus 12 has been connected to bus 8 as in Case 4 of Section with the forced oscillation at a resonant frequency of 0.56 Hz. Table (4.6) shows the amplitude of MW oscillations on P78 for different forced oscillation magnitudes. Table (4.6) shows that P78 oscillation amplitude does grow linearly with respect to forced oscillation amplitude for small values (up to 20 MW FO magnitude in Table (4.6)). At higher levels, the relationship becomes nonlinear and the system becomes unstable after 50 sec. for the 100 MW forced oscillation. Table (4.6) confirms that resonance as observed is a linear phenomenon for small values of forced oscillations. Table 4.6- Effect of forced oscillation magnitude on resonance amplitude FO magnitude (MW) P78 (MW) diverges FORCED OSCILLATIONS AND LOCAL MODE In this section, resonance between forced oscillations and a local mode of Kundur system associated with generators 1 and 2 is studied. Unlike the inter-area mode, it is shown that the 59

71 local mode will show resonance effect under very specific conditions and the effects are also relatively minor compared to the resonance with inter-area modes Resonance The local mode at 1.03 Hz has its damping ratio at 1.4% for this study case while the inter-area mode is well-damped. 10 MW forced oscillation is injected at bus 12 which is connected to bus 1 instead of bus 8 that was shown in Figure (4.5). The forced oscillation frequency is varied and the tie-line flow P78 for four different cases are shown in Figure (4.11). Oscillations on the tie-line are seen to be about 28 MW, 45 MW and 19 MW for the forced oscillation frequencies of 1.00 Hz, 1.03 Hz and 1.06 Hz respectively. Figure (4.11) shows that resonance does occur in this case as well from the higher MW oscillations at the resonant frequency compared to other frequencies. Next, the local mode is well-damped with frequency at 1.1 Hz and damping ratio at 9%. The tie-line oscillation amplitudes (Figure (4.12)) are now at 20 MW for the three forced oscillation frequencies of 1.07 Hz, 1.1 Hz and 1.13 Hz respectively. The resonance effect is minimal in this case. 700 No Forced Oscillation 700 f 0 = 1.00 Hz MW MW MW Time (sec) f 0 = 1.03 Hz Time (sec) MW Time (sec) f 0 = 1.06 Hz Time (sec) Figure P78 Oscillations for poorly damped local mode 60

72 700 No Forced Oscillation 700 f 0 = 1.07 Hz MW MW MW Time (sec) f 0 = 1.10 Hz Time (sec) MW Time (sec) f 0 = 1.13 Hz Time (sec) Figure P78 Oscillations for well-damped local mode SSI estimation results SSI estimation results for Cases 12, 13, and 14 when the local mode is poorly damped are shown in Table (4.7). Similarly, estimation results for well-damped cases, namely, Cases 15, 16 and 17 are presented in Table (4.8). Tables (4.7) and (4.8) show that SSI can estimate the damping levels of the local mode and the forced oscillation well with no adverse impact seen from the forced oscillation at a nearby frequency. Table 4.7- SSI estimation results for Cases 12, 13 and 14 Case No Method Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) SSAT SSI FO actual SSI SSAT SSI FO actual SSI SSAT SSI FO actual SSI

73 Table 4.8- SSI estimation results for Cases 15, 16 and 17 Case No Method Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) SSAT SSI FO actual SSI SSAT SSI FO actual SSI SSAT SSI FO actual SSI Sensitivity to location of forced oscillation In this subsection, the forced oscillation at bus 12 is connected to different buses in the system for the resonant case (Case 13) to study the impact of location on resonance. Table (4.9) is the local mode counterpart of Table (4.5) in Section which was for the inter-area mode. The results in Table (4.9) show that the local mode undergoes resonance with forced oscillation only when the forced oscillation is at a neighboring bus. In this case, bus 2 of Area 1 shows the most impact for resonance. As in the case of Table (4.5), the effects are compared with mode shape magnitudes calculated from FDD and SSI for the local mode when forced oscillations are not present and they serve as rough indicators. Further investigation is suggested. Table 4.9- Effect of forced oscillation location on resonance FO source P78 (MW) f9 (Hz) V9 (p.u) SSI MS mag. FDD MS mag

74 SUMMARY OF RESONANCE CASES Case studies on Kundur system in this section have shown that resonance can occur between a forced oscillation and an inter-area mode when the forced oscillation frequency is close to that of the inter-area mode if the inter-are mode is poorly damped and/or if the inter-area mode has strong participation at the location of forced oscillation. Resonance results in higher amplitudes of tie-line oscillations compared to otherwise while the resonance effect varies with location of forced oscillation. Highest vulnerability is seen when forced oscillation occurs at the distant ends of the inter-area mode. Resonance with local mode appears only for locations close to the generators involved in the local mode and the resonance effect on tie-line flows is much lesser compared to that from the inter-area modes. Tables (4.10) and (4.11) show the summary of features of cases 3 to 17 of Kundur system for comparison purposes. Given that the magnitude of the forced oscillation at source is 10 MW, the resonance effect is classified into very small, small, medium and large when the resulting tieline oscillation (TLO in short) is in the MW range, 10<TLO 20, 20<TLO 50, 50<TLO 100, and TLO>100, respectively in Table (4.10). It is interesting to note that there is some resonance effect on tie-line oscillations in all 9 cases of Table (4.10) for the inter-area mode while the amplitude varies depending on three factors: a) difference between the frequencies of forced 63

75 oscillation and inter-area mode, b) damping level of the inter-area mode, and c) participation level of the inter-area mode at the location of the forced oscillation. Comparatively, the resonance effects of local mode from forced oscillations on tie-line oscillations are small or mostly negligible in Table (4.11). Table Summary of resonance effect on inter-area mode (cases 3 to 11) Interaction with inter-area mode (FO frequency) minus (system mode frequency) Hz 0 Hz Hz Medium Large Medium Low resonance resonance resonance System mode damping level Medium (Case 3) Medium resonance (Case 6) Small (Case 4) Medium resonance (Case 7) Small (Case 5) Small resonance (Case 8) Small High resonance resonance resonance (Case 9) (Case 10) (Case 11) Table Summary of resonance effect on local mode (cases 12 to 17) Interaction with local mode (FO frequency) minus (system mode frequency) Hz 0 Hz Hz System mode Low Very Small resonance Small resonance No resonance (Case 14) 64

76 (Case 12) (Case 13) High No resonance (Case 15) No resonance (Case 16) No resonance (Case 17) 4.4. SSI ESTIMATION FOR THE WECC EVENT In this section, SSI is used to estimate the modes for Cases 1 and 2 of Section 5.1 that consist of archived WECC data from summer of System order for SSI is selected to be 16 and all other parameters are the same as in section 4.3. Like in the case of FDD, the magnitude of apparent power-flow computed from product of bus voltage and line current magnitudes from the same 5 PMUs are used for estimation. FDD as implemented in the western system utility is not able to estimate multiple modes that have nearby frequencies. FDD estimates whatever is seen as the dominant mode for the system at that frequency [44]. Nevertheless FDD estimation results clearly showed the change in mode shape while the damping dropped that helped identify the source of the forced oscillation. As discussed in Section 4.3, SSI has the ability to estimate nearby modes and forced oscillations. SSI estimation results for Cases 1 and 2 of Section 4.1 are presented in Table (4.12), and the mode shape estimates are shown in Figure (4.13). Table (4.12) and Figure (4.13) show that SSI estimates the presence of the well-damped inter-area mode for both cases. Specifically for Case 1, it estimates the simultaneous presence of forced oscillation at 0.38 Hz and a welldamped system mode at 0.4 Hz. The mode shape of the forced oscillation for Case 1 shows the dominant PMU to be PMU 1 (like FDD) which correctly points to the actual source of the forced 65

77 oscillation. Moreover, the mode shapes of the inter-area mode are as expected with highest presence from the tie-line PMU 3 for both Cases 1 and 2. The amplitude of the forced oscillation appears to have been small (10 MVA according to Figure (4.2)). Fortunately, the WECC inter-area mode at 0.4 Hz was well-damped during the occurrence of the 0.38 Hz forced oscillation, and there also was a 0.02 Hz difference between frequencies of two modes (according to SSI). Relating the event to the studies of Section 4.3, it is clear that resonance likely did not occur for this event. However, recurrence of such forced oscillations when one of the WECC inter-area modes is poorly damped could be problematic for the system especially when the forced oscillation occurs near the two ends of the WECC system say in British Columbia/Alberta or in California. Resonant oscillations in such circumstances would be felt all across WECC system and could lead to subsequent nonlinear effects. Large oscillations are especially problematic for nuclear generators, and sudden tripping of big generators from such oscillations can possibly lead to catastrophic blackouts. The interactions of forced oscillations and inter-area as well as local modes need to be further studied and wellunderstood. Case No. 1 Table SSI estimation results for the two WECC cases Mode Frequency Mean(Hz) Std(Hz) Damp. Ratio Mean(%) STD(%) Inter-area FO Inter-area

78 Figure SSI mode shape estimates of the WECC system: a) Forced oscillation for Case 1, b) Inter-area mode for Case 1, and c) inter-area mode for Case SOURCE OF FORCED OSCILLATIONS Locating the source of forced oscillations is a challenging problem and can possibly be done using estimated mode shapes. Examples in this section show that this location process highly depends on the absence or the presence of resonance effect RESONANCE NOT PRESENT When there is not a strong resonance effect, that is, when FO frequency is not near system mode frequencies, and/or system mode is well-damped, and/or FO source is at a location where the system mode of interest is not active, FO location can easily be found from mode shape magnitude. This can be seen in the real system example of Figure (4.1-a) or (4.13-a) where PMU 1 was confirmed to be the source of FO. Next, let us consider the mode shape estimates of the Kundur system example. Figures (4.10-a) and (4.10-c) show that mode shape magnitude of bus 12 (FO location) is only slightly larger than the mode shape magnitude at other buses. That is, the mode shape magnitude difference of bus 12 from other buses in Figures (4.10-a) and (4.10-c) is not as obvious as it was for the real system example in Figures (4.1-a) or (4.13-a). This is because the effect of resonance is present in Figures (4.10-a) and (4.10-c). The system mode is poorly damped and the FO frequency is still close to system mode frequency, and therefore, resonance is to be expected from Figure (4.6). These cases 3 and 5 were described as medium resonance cases in the summary table, Table (4.10). Interestingly, if the FO frequency moves farther away from system mode frequency of 0.56 Hz, the mode shape magnitude of bus 12 becomes larger because the 67

79 resonance effect reduces according to Figure (4.6). Figures (4.14-a), (4.14-b), and (4.14-c) show the mode shape estimates of the FO for the same system as in cases 3 to 5 where the FO is placed at 0.56 Hz, 0.66 Hz, and 0.76 Hz respectively. It can be seen that when the resonance effect becomes weak for the new cases in Figures (4.14-b) and (4.14-c), the FO location at bus 12 becomes clear. On the other hand, the resonant case of Figure (4.14-a) is difficult to analyze because the mode shape magnitude of FO location at bus 12 is small with respect to the generator buses. This is discussed next RESONANCE PRESENT In case of resonance between a system mode and FO, locating FO source is not straightforward. Physically, during resonance, generators which participate in the system mode are contributing significant energy into the mode that results in large power swings on the tie lines like in the cases studied in Table (4.5). For instance, when the FO location is at bus 8 in Table (4.5), the tie-line oscillation amplitude is 203 MW and the mode shape estimated by SSI for the forced oscillation is shown in Figure (4.14-a). In this case, the MW outputs of generators 1, 2, 3, and 4 are swinging with amplitudes 114, 100, 125, and 113 MW respectively which are clearly much higher than the 10 MW FO amplitude at generator 12 the FO location. In large power systems, there will be many generators which may have oscillation amplitudes comparable to the FO source amplitude and it will be difficult to pinpoint the FO location from among those many candidates. In such cases, the mode shape magnitude will not directly help in locating the source. 68

80 Figure SSI mode shape of forced oscillation for system of Cases 3-5: a) FO at 0.56 Hz b) FO at 0.66 Hz c) FO at 0.76 Hz On the other hand, as seen in Figure (4.14-a), mode shape phase at FO location should be close to 90 degrees off (perpendicular to) from other dominant system mode participants according to the theory of Section 4.1. Again, in a large power system, there may be many other generators with mode shape phases that are not aligned with the dominant participants of the system mode, and locating the FO among them is not straightforward. In summary, mode shape estimation may not help in locating the FO source in the case of resonance in large systems, and the topic needs further research. 69