Power System Dynamics Prediction with Measurement-Based AutoRegressive Model

Transcription

1 21, rue d Artois, F PARIS CIGRE US National Committee htt : // Grid of the Future Symosium Power System Dynamics Prediction with Measurement-Based AutoRegressive Model C. LI 1*, J. CHAI 1, Y. LIU 1 N. BHA 2, A. DEL ROSSO 2, E. FARANAOS 2 1 University of ennessee, Knoxville 2 Electric Power Research Institute Knoxville, N USA Knoxville, N USA SUMMARY he raidly growing deloyment of Phasor Measurement Units (PMU) makes it feasible to analyze ower system dynamic behavior in a real-time environment, and even rediction or early detection of instability, using synchrohasor data. Autoregression is an imortant technique to extract relationshis among measurement signals and is a owerful tool of system identification. In this aer, a measurement-based Multivariate AutoRegressive (MAR) model is resented. It is designed to build a mathematical model for several measurement signals. he MAR model is a linear model. It is suitable for a linear condition which holds the majority of the time with ower system oerations. With a given model order and study signals, the MAR model can be trained to minimize overall error. Once it is trained, it can be used to redict the dynamics of study signals recursively with some initial dynamics data oints. he main challenge in training the MAR model is that the number of unknowns is roortional to the roduct of the signal number and the model order. hough more signals and higher order usually give a better rediction result, too many unknowns cannot be solved with a limited number of measurement data oints. his roblem is addressed in this aer with the delayed correlation coefficient technique. By analyzing the structure of the MAR model, the contribution of all inut signals can be measured with a delayed correlation coefficient. hose signals with high correlation coefficients contribute more than signals with lower coefficients. o reduce the number of inut terms, only the signals with high coefficients are chosen as inuts. o verify the accuracy of the MAR model and model reduction technique, a 23-bus model is examined. Simulations show that the MAR model can rovide an accurate rediction for events of a similar tye. And model rediction can be dramatically reduced with the correlation coefficient technique while keeing model accuracy. o conclude, the MAR model roosed in this aer is an attractive aroach for ower system dynamics rediction. It can be trained with ure measurement data. hough the MAR model is comlicated for too many study signals, its comlexity can be reduced with the delayed correlation coefficient technique without losing accuracy. KEYWORDS Power systems, Synchrohasor measurement, Multivariate AutoRegressive (MAR) model, Model reduction, Correlation coefficient cli27@utk.edu

2 I. INRODUCION A ower system is subjected to small or large disturbances at all times. According to ower system reliability standards [1], major disturbances should be limited in a small area and cleared with relay devices. However, a ower system can lose synchronism in some extreme cases such as cascading failures which may lead to disastrous social and economic consequences [2]. Predicting system dynamics after an event haens can hel to assess and maintain the security of the grid. Physical model-based time-domain simulation is the rimary method to assess system dynamic resonse under events[3]. hough the simulationbased method hass more detailed reresentation of the grid (comared to a measurementbased technique), and the ability to simulate what-if scenarios, including contingencies, it is difficult to udate the model to the continuously evolving oerating conditions, and simulation models, esecially load models, are not accurate for on-line alications[4]. With the growing deloyment of Phasor Measurement Units (PMU) [5] and Frequency Disturbance Recorders (FDRs) [6], a massive amount of measurement data of ower system dynamics are recorded. he measured hasor data are more reresentative of actual system conditions than time-domain simulation results because they are obtained from directly measured voltage and current waveforms. he advantage of synchrohsor data reveals a ossible alication of studying ower system with ure measurement data. here are mainly two aroaches to redict ower system dynamics with measurement data. One is combining measurement data with artial model arameters, e.g., generator imedance, inertia, etc[7]-[8], the other is redicting dynamics with ure measurement data by treating a ower system as a black-box with finite measurement oints of PMU or FDR[9]- [11]. Artificial intelligence based methods also fall in the latter category[12]. his aer deals the dynamics rediction roblem with system identification, and is organized as follows. In section II, a Multivariate AutoRegressive (MAR) model is introduced and the dynamics rediction rocedure with the MAR model is roosed. Model training and reduction issues are discussed in section III. In section IV, some rediction examles are given and the model reduction technique is verified. Conclusions are drawn in section V. II. HE PREDICION MODEL WIH HE MULIVARIAE AUOREGRESSIVE MODEL A. Multivariate AutoRegressive Model System identification (SI) is an imortant method of building mathematical models for a dynamic system[13]. For a linear time-invariant system samled at a time interval of h, a general SI transfer function structure is Bq Cq Aq yt ut et F q D q (1) where 1 na 1 nb 1 nc 1 na 1 nb 1 nc 1 nd 1 n f 1 n 1 n Aq 1 aq... a q, Bq bq... bq, Cq 1 cq... c q D q 1 d q... d q, F q 1 f q... f q d Here, y(t) is the outut signal which is the observable (measurable) system resonse of interest, u(t) is the inut signal which is a disturbance signal of the system or a stimuli maniulated by the observer, and e(t) is a sequence of indeendent random variables. n a, n b, n c, n d, and n f are orders of each art. q -1 is a backward shift oerator and q -1 y(t)= y(t-h). For ower systems, dynamics resonse of frequency, voltage magnitude and hase angle is of key interest and thus can be chosen as outut signals of an observation model (1). hough they can be directly measured with PMUs or FDRs, the disturbance signals of the event, i.e., inut signals, are hard to measure due to the diversity of events. Since ower f. 2

3 system dynamic quantities affect and are affected by other measurement signals, an observation model can be develoed to model the interaction between those measurement signals by choosing one signal as an outut signal and treating other signals as inut signals. With interactional measurement signals y 1 (t),, y (t), if y i (t) is chosen as an outut signal and all other signals are treated as inut signals, a Multi-Inut Single-Outut (MISO) MAR model with C(q)=1, D(q)=1 and F(q)=1 can be develoed as Exanding (2) yields A q y t B q y t e t i i ij j i j1, ji n n ai bji + ( ) y t a y t kh b y t kh e t i ik i ijk j i k=1 j1, jik=1. (2), (3) where n ai is the order of signal y i (t) and n bji is the order of signal y j (t) when y i (t) is the outut signal and y j (t) is the inut signal. k is the index of time delay. In the MAR model, the orders of each signal can be different. For simlicity, a uniform order n can be chosen for all signals. With this maniulation, (3) can be rewritten as y t n a y ( tkh)+ n b y tkh e t. (4) or Let b iik = -a ik, we have i ik i ijk j i k=1 j1, jik=1 n y t b y tkh e t. (5) i ijk j i j1k1 Equation (5) can be further exressed in vector form as y t by t e t (6) i ij j i j1 i j ij i j1 y t y t b e t, (7) where suerscrit indicates transose, and 1,...,,,..., bij bij bijn y j t yj th yj tnh. Choosing each measurement signal as outut signal, we have y1 t b1jy j t e1 t j1.... (8) y t bjy j t e t j1 Let y t y t y t 1,...,, we have a Multi-Inut Multi-Outut (MIMO) MAR model y t b11... b1 y1 t e1 t (9) b... b y t e t 1 Since the MISO MAR model (7) is art of the MIMO MAR model (9), (7) can be treated as a sub-model of MIMO MAR model (9) and is denoted as sub-model i. B. Prediction Procedure In (3), the time delay index k starts from 1 and y i (t) is totally determined by historical data, i.e., data at time rior to time t. So, equation (9) give a one-ste rediction of a dynamic system. In equation (9), the random art e(t) corresonds to the unmeasured disturbance 3

4 signals and is difficult to model. For simlicity, the e i (t) art is neglected and the MAR model becomes b11... b1 y1 t y t (10) 1... t b b y or y t BY t (11) where b11... b1 B , Y t y1 t... y t b b Once the MAR model is obtained, dynamics can be redicted recursively based on the flowchart in Fig. 1 where t max is the length of a study time window. Fig.1 Flowchart of dynamics rediction with MAR model III. MODEL RAINING AND REDUCION C. raining of the MAR Model he MIMO MAR model (9) is comrised of sub-models, and can be develoed by training each sub-model searately. For sub-model i of (7), m-n equations can be written as (12) with an event of m measurement data oints. yin1h y jn1h bij ein1h j1... (12) It can be re-written as i j ij i j1 y b y mh mh e mh 4

5 where 1... y 1 Y A B E, (13) i i i i y1 n h n h Ai , Yi yin1 h... yimh y1 mh... y mh Bi bi 1... bi, Ei ein1 h... eimh o get the best MAR model with (13), the error art E i should be minimized. Parameter B i can be estimated with least square error estimator to minimize the error art E i, = -1 i i i i i B A A A Y. (14) D. Reduction of MAR Model he MISO MAR model develoed with (7) includes n arameters for each inut signal y j. It is called full MAR model with n unknowns. With more and more measurement units deloyed, the number of unknowns increases linearly with measurement number. hough the MAR model is linear, too many unknowns may cause huge comutation burden and model comlexity, and thus revent the full MAR model from being alicable for online alications. he increase of number of unknowns leads to a ossible roblem of under-determinacy. here are m-n equations (constraints) but n unknowns in (7) for a MISO MAR model. If m- n<n (or m-n n and A i is singular), (12) is underdetermined and mathematically unsolvable. o overcome this difficulty, either the data oint number should be increased or the number of unknowns should be reduced. However, m cannot be increased arbitrarily. In most cases, a disturbance is damed in 20s or shorter. he time window length for model training is usually 10s to 20s. oo long of a time window may include data oints of the new steady state and make A i singular. Under a secific PMU reorting rate, the data length m is limited. So, m cannot be increased unless PMU is reorting at a higher rate. It is more ractical to reduce the number of unknowns than to increase the measurement data length. Since the number of unknowns is n, there are two direct ways to reduce the comlexity of a MISO MAR model. One way is to reduce the model order n. However, n is a key arameter of the MAR model, and dynamics rediction accuracy is greatly affected by the model order. In general, a higher order usually yields better training accuracy and rediction accuracy. o kee high rediction accuracy, the model order cannot be dramatically reduced. he other way is to reduce the number of inut signals. he relationshi between the outut signal y i and different inut signals y j is different. hose inut signals which are more related to the outut signal contribute more to the outut signal. o measure the contribution of each inut signal, a zero-delay Correlation Coefficient (CC) is introduced in [14] to select the inut signals. he zero-delay CC is defined as r ij = m yilhyjlh l1 m 2 m 2 yilh yjlh l1 l1. (15) With the method roosed in [14], inut signals with high zero-delay CC are selected as effective inut signals and other signals are comletely removed from (7). he main roblem of this method is that it does not differentiate the inut signals and secific inut terms of each inut signal. For each inut signal y j, there are n inut terms of. 5

6 outut signal y i in (7), i.e., b ij1 y j (t-h), b ij2 y j (t-2h),,and b ijn y j (t-nh). hose inut terms are the delayed time series of original signal y j. If the zero-delay CC r ij is high, it is not necessary that all the n delayed inut terms of y j contribute significantly to outut signal y i, and vice versa. It is referable to study the contribution of each inut term secifically, instead of the overall contribution of each inut signal. o overcome the drawbacks of the zero-delay CC, the delayed CC of each inut term is used in this aer to find a better combination of inut terms. ake a k-delay inut term of signal y j, i.e., y j (t-kh) of (7), for examle, the k-delayed CC between the inut signal y j and outut signal y i is r ijk = j m i lk1 m i lk1 l1 y lh y l k h 2 m j y lh y l k h 2. (16) he delayed CC examines the correlation of each delayed inut term with the outut signal. Comaring (15) with (16), it can be found that the zero-delay CC is symmetric, i.e., r ij =r ji, however, the delayed CC is asymmetric, i.e., r ijk r jik. his leads to the main difference between the inut selections with the two CCs. With zero-delay CC, if signal y j is highly correlated with signal y i, signal y j is the inut signal of signal y i, and vice versa. For delayed CC, if k-delay inut term of signal y j is highly correlated with signal y i, y j (t-kh) can be selected as the inut term of signal y i. However, if the k-delay inut term of signal y i is not highly correlated with signal y j, it does not have to be selected as an inut term of y j. So, B of (11) does not necessarily have symmetric structure when delayed CC is used for inut term selection. With the delayed CC, it is not necessary to find the otimal model order n. With a higher model n, all inut terms can be examined with the delayed CC and only those with the delayed CC higher than a user-defined threshold are selected for develoing the sub-model of y i. he delayed CC lies in the range of [0, 1] and a threshold higher than 0.9 is usually suggested for inut term selection. his will be further examined in section IV. IV. EXAMPLES A 23-bus model is chosen as a test system. he system data can be found in the PSS/E manual[15]. he simulation time ste is set as half cycle (1/120s). Frequency, voltage and hase angle of all 17 PQ buses are monitored. In this section, a full MAR model of those 51 signals is develoed with a 15s-long data, under 1% load increase event and rediction is made for a 5% load increase event. he order of each signal is 13. Additionally, suose the data reorting rate of 23-bus model is 30Hz, which can be achieved by down samling the 120Hz simulation data, another reduced MAR model is also develoed with 15s-long dynamic data with threshold of delayed CC of 0.9. Prediction of voltage and frequency of bus 151 are shown in Fig. 2 and 3. In Fig. 2 and 3, diamonds on the dashed lines show where rediction of the full MAR model starts when data rate is 120Hz. Comaring the dashed lines with actual resonse, it can be concluded that the full MAR model can redict the dynamics with high accuracy. he squares in Fig. 2 and 3 show where rediction of the reduced MAR model starts when data rate is 30Hz. In the reduced MAR model, there are only 44 unknowns for the submodel of signal 1 (frequency of bus 151). he submodel with the most unknowns is signal 15 (hase angle of bus 201), which is constituted by 190 unknowns. he submodel with the most unknowns is signal 14 (voltage of bus 201), which consists of 35 unknowns. he number of unknowns of the reduced MIMO MAR model is 5030 which is about 15% of the full MAR model. hough the MAR model is dramatically reduced, it still rovides good rediction result in Fig. 2 and 3. 6

7 Fig.2 Frequency dynamics rediction with reduced model Fig.3 Frequency rediction with reduced model V. CONCLUSIONS With more and more PMUs, massive amounts of measurement data give an alternative aroach to study ower system dynamics with real resonse. he MAR model roosed in this aer is romising for ower system study of linear conditions with ure measurement data. It can be easily udated online with tyical events. o imrove its feasibility, a model reduction technique with delayed correlation coefficient rovides a good guideline for inut selections. Simulations show that the accuracy of rediction is high for test cases and the model reduction technique works well to select the most effective inut terms. ACKNOWLEDGEMEN his work was funded by DOE and Electric Power Reseach Institute (EPRI) under roject DE-FOA and suorted in art by the Engineering Research Center Program of the National Science Foundation (NSF) and Deartment of Energy (DOE) under NSF Award Number EEC and the CUREN Industry Partnershi Program. BIOGRAPHIES C. Li is a research associate at the University of ennessee, Knoxville. He received his B.S. and Ph.D. degrees from Shandong University, China in 2006 and 2012, resectively. His current research interests include ower system dynamics analysis, wide-area monitoring, and stability control. J. Chai is a Ph.D. student in Electrical Engineering at the University of ennessee, Knoxville. He received his B.S degree from ianjin University, China, in His research interests include ower system wide-area monitoring and control, and ower system dynamic analysis. Y. Liu is currently the Governor s Chair at the University of ennessee, Knoxville and Oak Ridge National Laboratory. Prior to joining UK/ORNL, she was a Professor at Virginia ech. She received her MS and Ph.D. degrees from the Ohio State University, Columbus, in 1986 and She received the BS degree from Xian Jiaotong University. She led the effort to create the North America ower grid monitoring network (FNE) at Virginia ech which is now oerated at UK and ORNL as GridEye. Her current research interests include ower system wide-area monitoring and control, large interconnection level dynamic simulations, electromagnetic transient analysis, and ower transformer modelling and diagnosis. N. Bhatt is a Fellow of IEEE and has worked at Electric Power Research Institute (EPRI) since July Before joining EPRI, Dr. Bhatt worked at American Electric Power (AEP) for 33 years. He received a BSEE degree from India, and MSEE and PhD degrees in electric ower engineering from the West Virginia University. He was a co-author of an IEEE working grou aer that received in 2009 an award as an Outstanding echnical Paer. A. Del Rosso is a Project Manager at the Electric Power Research Institute (EPRI). Formerly, he served as Senior Consultant Engineer at Mercados Energéticos & Energy Market Grou. He received his PhD degree from the National University of San Juan, Argentina, and his electrical engineer diloma from the National echnological University, Mendoza, Argentina. His major areas include 7

8 ower system transmission efficient, ower system dynamic and control, transmission lanning, integration of renewable, system reliability, energy storage, and on-line dynamic security assessment. E. Farantatos is currently a Sr. Project Engineer/Scientist at EPRI. He received the Diloma in Electrical and Comuter Engineering from the National echnical University of Athens, Greece, in 2006 and the M.S. in E.C.E. and Ph.D. degrees from the Georgia Institute of echnology in 2009 and 2012 resectively. His research interests include ower systems state estimation, rotection, stability, oerations, control, synchrohasor alications, renewables integration and smart grid technologies. VI. REFERENCES [1] NERC ransmission Planning (PL) Standard PL System Performance Following Loss of a Single Bulk Electric System Element (Category B) [Online]. Available: htt:// [2] IEEE/CIGRE Joint F on Stability erms and Definitions, Definition and Classification of Power System Stability, IEEE rans. on Power Systems, vol. 19, no. 3, , [3] Prabha Kundur, Power System Stability and Control, New York: McGraw-Hill, Inc [4] US-Canada Power System Outage ask Force. Final Reort on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations.[Online], Available: htt://energy.gov/sites/rod/files/oerod/documentsandmedia/blackoutfinal-web.df [5] J. De La Ree, V. Centeno, J. S. hor, A. G. Phadke. Synchronized Phasor Measurement Alications in Power Systems. IEEE rans. on Smart Grid, vol. 1, no. 1, , [6] Zhian Zhong, Chunchun Xu, Li Zhang, etc, Power System Frequency Monitoring Network(FNE) Imlementation, IEEE rans. on Power Systems, vol. 20, no. 4, , [7] F. F. Song,. S. Bi, Q. X. Yang. Perturbed rajectory Prediction Method based on Simlified Power System Model, in Proc. of IEEE PES Power System Conference and Exosition, , [8] Yujing Wang, Jilai Yu, Real ime ransient Stability Prediction of Multi-Machine System Based on Wide Area Measurement, in Proc. of Asia-Pacific Power and Energy Engineering Conference,. 1-4, [9] J. H. Sun, K. L. Lo, ransient Stability Real-ime Prediction for Multi-Machine Power Systems by Using Observation, in Proc. of IEEE Region 10 Conference on Comuter, Communication, Control and Power Engineering, vol. 5, , [10] Guoqing Li, Fujun Sun, Qiang Ren, Real-ime Prediction and Control Method for ransient Stability of Multi-Machine Power System Based on Outside Observation, Power System echnology, vol. 19, no. 1, , 1995 (in Chinese). [11] F. F. Sun,. S. Bi, H. Y. Li, and etc. he Perturbed rajectories Prediction Method Based on Wide-Area Measurement System, in Proc. of IEEE PES ransmission and Distribution Conference and Exhibition, , [12] A. G. Bahbah, A. A. Girgis, New Method for Generators Angles and Angular Velocities Prediction for ransient Stability Assessment of Multimachine Power Systems Using Recurrent Artificial Neural Network. IEEE rans. on Power Systems, vol. 19, no. 2, , [13] Lennart Ljung, System Identification: heory for the User, Englewood Cliffs: Prentice-Hall, Inc [14] Feifei Bai, Yilu Liu, Kai Sun, et al, Inut Signals Selection for Measurement-Based Power System ARX Dynamic Model Resonse Estimation, 2014 IEEE PES &D Conference, Chicago, IL, US, [15] Siemens, PSS/E User Manual of Version 33,