A Lyapunov Exponent based Method for Online Transient Stability Assessment

A Lyapunov Exponent based Meod for Online Transient Stability Assessment P. Banerjee, S. C. Srivastava, Senior Member, IEEE Department of Electrical Engineering Indian Institute of Technology Kanpur Kanpur-208016, India param@iit.ac.in, scs@iit.ac.in K. N. Srivastava, Senior Member, IEEE ABB AB, Global Research Lab Forsargränd 7, 72178 Vasteras, Sweden ailash.srivastava@se.abb.com Abstract The post fault transient stability of e power system is assessed in is paper using Maximum Lyapunov Exponent (MLE). The phase angles, reported by e PMUs at e generator buses are used for estimating e MLE of e system. The sign and e magnitude of e MLE serve as indicators to e stability of e system. The meod utilized for estimation of e MLE is independent of e system model and is solely based on e measurement data of e system states, which can be obtained from e Phasor Measurement Units (PMUs). Different combination of fault duration and location are simulated to analyze e system stability using MLE on 3 machine 9 bus system and 10 machine New England (NE) 39 bus system. Keywords Transient stability, online detection, Lyapunov Exponent, PMU measurements I. INTRODUCTION The power system networ is possibly one of e most complex man made dynamical systems. It often experiences instability under sufficiently large disturbances. The assessment of e system stability is critical for reliable operation of a power system following disturbances. The early detection of e instability enables e operator to prevent major outages by taing preventive control actions. The modern power systems operate under competitive maret environment. In such a scenario, e power flow in e lines are dictated by e complex maret dynamics, which also leads to e increase in e power system oscillations, pushing e operating point to its stability limit. The large scale deployment of Phasor Measurement Unit (PMUs) in e power system networ has made it possible to estimate e bus voltage and branch current phasors, which are sent to e Phasor Data Concentrator (PDCs) located in e control centers at a rate of upto one phasor per cycle. The high refresh rates of PMUs are suitable for predicting online e impending instability. The classical stability analysis for power system applications is performed by modeling em in e form of Differential and Algebraic Equations (DAE) in a simulation framewor running on personal computer (PC). The stability of e simulated power system networ is evaluated for a preselected set of contingencies and preventive control actions are pre-computed and stored as a set of looup tables for e power system operators. However, e disturbances, experienced in e power system have uncertainty in magnitude and location, which might not have been envisaged in e predetermined set of contingencies. The inability of e operator to assess e stability online for a new contingency situation often drives e power system to cascaded tripping and large scale blacout. The online determination of e stability boundary for e power system transient studies was proposed as early as in [1], in which e estimation of Lyapunov energy function was carried out by using two simplifying assumptions, but e application was restricted to only small test systems. Extended equal area criteria based direct identification of e transient stability was proposed by Pavella et. al. [2], which used e first swing stability concept. The advent of synchrophasor technology opened a new paradigm of measurement based online stability prediction. The Lyapunov Exponents (LEs) have been used for e identification of unstable swing by Liu et. al. [3] [4] using real time PMU data. The multilayer perceptron and support vector machine based transient stability analysis have been proposed for large power system by Moulin et. al. [5]. Jonsson et. al. [6] had proposed a real time clustering of coherently swinging generators using Fourier analysis. Critical cutset based detection of loss of synchronism of swinging generators, based on computation of energy function, had been proposed by Padiyar et. al. [7]. Real time estimation of e stability margin using concept of Ball of Concave Surface was proposed by Sun et. al. [8]. The stability prediction for e post fault condition of e power system using LEs have been proposed in [9][10]. Determination of e voltage stability using Maximum Lyapunov Exponent (MLE) has been proposed in [11] for real time application. The estimation of LEs is performed by forming e state space equations of e power system, from which e variational equation is derived. The meod of integrating e variational equation for e post fault power system conditions depends on e size of e Jacobian matrix of e state space equation. The sign and e magnitude of e MLE are e indicating criteria for e divergence of e state space trajectories. This paper has proposed a measurement based estimation of e MLE using e PMU data wiout forming e variational equation of e state space representation. The effectiveness of e proposed meod for assessing e transient stability is demonstrated rough simulation of faults at different locations 978-1-4799-5141-3/14/$31.00 2014 IEEE

in 3 machine 9 bus system and 10 machine 39 bus New England (NE) system [12]. II. ESTIMATION OF MLE FROM PMU DATA A large power system consists of several generators, interconnected by long transmission lines. The steady state operation of e power system is characterized by e synchronous operation of all e rotating machines in e networ. The simulations have been carried out using PSCAD/EMTDC utilizing e built in synchronous machine model. The state space representation of a i synchronous machine in a power system networ, having m generators, at time instant is expressed as follows δi = ω0δωi (1) Δ ωi = hi( δ, Δωi ) where, δi is e i generator internal voltage in radian, ω 0 is e synchronous speed in radian/s, Δω i is e i generator per unit speed deviation from e synchronous speed, δ = ( δ 1... δ ) T m and hi is a nonlinear function describing e dynamics of e i generator. The state vector of e system at time instant is denoted by X = ( δ1... δm, Δω1... Δωm) During e steady state operation, e right side of (1) is zero. The system deviates from e steady state value on e occurrence of a fault, following which e state vector converges to a stable equilibrium point provided at e system is transiently stable under e post fault scenario. The state vector diverges and leads to instability if e post fault system is transiently unstable. The MLE is a definitive indicator for e stability of nonlinear systems. The Wolf s [13] meod has been generally used for e estimation of e MLE for e power system applications. The Jacobian matrix from e state space model of e multi machine power system is formed to estimate e evolution of e state vectors trajectories. The LEs estimate e exponential rate of convergence and divergence of e nearby trajectories in state space. The estimation of e MLE using Rosenstein [14] meod has e following advantage. (a) It does not require to construct e variational equation using Jacobian of e state space model. (b) It does not require to perform Gramm Schmidt Reorogonalization of e state vectors at each time step. (c) It is completely based on e measurement of e state variables. (d) It directly measures e MLE from e measured state variables. The time series data of e state variables of finite leng N measured by e PMUs are stored in a matrix, which is 1 denoted by (... N T Y = X X ). The LEs are defined as e average exponential rate of divergence of e nearby trajectories, which are separated initially by infinitesimally small distance in e phase plane. The meod proposed by Rosenstein approximates e separate nearby trajectories, by searching e nearest neighbor in e phase plane. The nearest neighbor for e point is denoted by X such at d (0) = min X X where, d (0) is e initial = 1: N, > 10 distance between e point and its nearest neighbor, and. denote e Euclidean norm. The nearest neighbor in e phase plane are also constrained by e mean period, which is considered as 0.0167s in is wor. This constraint is considered so at e nearest neighbors occur on e separate orbits of e attractor. The dynamics of e system is traced by monitoring e distance of e nearest neighbor points evolved after j time step. The distance between e nearest neighbor after j time step is denoted by j j d ( j) = X + X +. According to Rosenstein [14], e MLE of e system is calculated from e change in e distance of e nearest neighbor along e locus of e state space trajectory. The logarim of e distance between e nearest neighboring points are averaged over each point in time series of data of a finite window leng. The logarim of e average distance between e nearest neighbor at j time instant is related to e MLE, as given below ln d ( j) ln d (0) + λ ( jδ t) (2) where, λ 1 is e MLE and 1 Δ t is e sampling time of e time series data. In is wor 1 is 60, which is e data frame Δ t transmission rate of e PMU considered in e power system under study. The MLE at e time jδ t is given by rearranging e expression given in (2). III. SIMULATION RESULTS The MLE estimated from e time series data of e power system has been discussed in e previous section. The magnitude of e speed deviation vectors in a power system networ followed by a contingency is observed to be insignificant as compared to e phase angle of e generators. Hence, e phase angle of e generator buses are only considered in is wor for e state vectors and stored for consecutive 65 phasors, which is used as initial nearest neighbors. The MLE is estimated and reported at e time instant of e next phasors arriving after e initial 65 phasors. The accumulation of e initial 65 phasors starts soon after e fault is cleared in e system. The proposed meod for estimation of e MLE are used for predicting e transient stability for different cases of fault in 3 machine 9 bus system and 10 machine NE 39 bus system. The fault of different duration and location are created in bo e systems for transient stability prediction. The single line diagram of e 3 machine 9 bus system is shown in Fig. 1, in which e generators are connected at buses 1, 2 and 3, whereas, e loads are connected at buses 5, 7 and 9. The PMUs are assumed to be placed at e generator buses for obtaining e phasors of e voltage signal. The phase angle of e voltage signal at e generator buses are measured and aligned wi e time base for e estimation of

2 j0.0625 8 1.026-3.7 7 6 100W 35VAr 1.032-2.0 90W 30VAr j0.0586 3 163W (6.7VAr) 0.0085+j0.072 B/2=j0.0745 0.0119+j0.1004 B/2=j0.1045 85W (-10.9VAr) 1.025-9.3 18/230 1.016 0.7 230/13.8 1.025-4.7 0.996-4.0 125W 50VAr 9 5 1.013-3.7 (corresponding to 65 initial phasors) after e fault is cleared, which is shown in Fig 4 for e two cases of e fault clearing time. The MLE is observed to be positive for e diverging phase angles, leading to e unstable system at resulted from e clearing of fault at 1.30s. The evolution of e MLE over time for e fault clearing at 1.29s is observed to be negative, which is a case of system retaining stability after e fault is cleared at 1.29s. 4 1.026-2.2 1 1.040 0.0 71.6W (27VAr) Fig. 1. 3 machine 9 bus system. e MLE. A ree phase to ground fault is created between mid-point of e line joining buses 6 and 7 having fault resistance of 0.01 ohm at 1s. The system is observed to become unstable when e fault is cleared at 1.3s as e phase angle of e bus 2 diverges from e phase angle of e buses 1 and 3, which swing togeer as shown in Fig. 2. The same fault is also cleared at 1.29s resulting in retaining e stability of e system. The phase angles of all e ree generator buses swing togeer when fault is cleared at 1.29s, as shown in Fig 3. Fig. 4. MLE for a fault of 0.3s and 0.29s at line joining buses 6 and 7. The ree phase to ground fault is also created at e midpoint of e line joining buses 5 and 6 having fault resistance of 0.01ohm, at 1s. The fault is cleared at 1.49s, which maes e system unstable as e phase angle of e bus 2 diverges from e phase angle of e buses 1 and 3, which swings togeer as shown in Fig 5. The reduction of e fault duration from 0.49s to 0.48s retains e stability of e system and e phase angle of e buses 1, 2 and 3 swing togeer, as shown in Fig 6. The evolution of e respective MLE for fault clearing at 1.49s and at 1.48s are shown in Fig. 7. The MLE is observed to be positive for e fault clearing at 1.49s indicating instability of e system, whereas e negative MLE for e fault cleared at1.48s indicate e stable system. Fig. 2. Phase angle of generator buses for a fault of 0.3s at line joining buses 6 and 7. Fig. 5. Phase angle of generator buses for a fault of 0.49s at line joining buses 5 and 6. Fig. 3. Phase angle of generator buses for a fault of 0.29s at line joining buses 6 and 7. The MLE estimation starts at e arrival of e phasor after e initial 65 phasors are buffered following e fault clearing. The evolution of e MLE starts after around 1s Fig. 6. Phase angle of generator buses for a fault of 0.48s at line joining buses 5 and 6.

Fig. 7. MLE for a fault of 0.49s and 0.48s at line joining buses 5 and 6. The effect of e fault duration is also studied on e line joining buses 4 and 5. The ree phase to ground fault is created at e mid point of e line joining buses 4 and 5 at 1s for duration of 0.48s and 0.47 s. The system is observed to be unstable for e fault clearing at 1.48s from e inception, while wi e fault clearing at 1.47s after e inception, e system is observed to be stable. The evolution of e MLE for e two fault clearing times are shown in Fig. 8, where e MLE for e fault clearing at 1.48s is positive and, us, shows unstable scenario, where as e MLE for fault clearing at 1.47s is negative after remaining slightly positive for a short duration, which is a stable scenario. at 1.35s for e first case and at 1.34s for e second case, leading to instability of e system for e first case. The phase angle of e generator buses are shown in Fig. 10, where e phase angle of e generator bus 38 diverges from e phase angle of e oer generator buses swinging togeer, when e fault is cleared at 1.35s, while e fault clearing at 1.34s leads to e stable scenario, in which all e phase angle of e generator buses swing togeer, as shown in Fig 11. The MLE, estimated from e phase angle of e generator buses, is shown in Fig. 12, for bo e cases of fault clearing at 1.35s and 1.34s. The MLE estimated for e case of fault clearing at 1.35s is positive as e system becomes unstable whereas e MLE estimated for e fault clearing at 1.34s is Fig. 10. Phase angle of generator buses for a fault of 0.35s at line joining buses 25 and 26. Fig. 8. MLE for a fault of 0.48s and 0.47s at line joining buses 4 and 5. Fig. 11. Phase angle of generator buses for a fault of 0.34s at line joining buses 25 and 26. Fig. 12. MLE for a fault of 0.35s and 0.34s at line joining buses 25 and 26. Fig. 9. 10 machine New England (NE) 39 bus system. The online detection of transient stability by monitoring e evolution of e MLE is also studied for different locations of fault in e NE 39 bus system, shown in Fig 9. The MLE of e system is estimated by considering e phase angle of all e generator buses, assumed to be equipped wi PMUs. The ree phase to ground fault is simulated at e mid-point of e line joining buses 25 and 26 at 1s, and cleared negative as e system remains stable. The second scenario studied in e NE 39 bus system, is a line outage between buses 25 and 26, following a ree phase to ground fault at e mid-point of e line joining buses 25 and 26 at 1s. The line is discharged at 1.3s and e fault is not cleared till en. The phase angles of all e generator buses remain synchronized and swing togeer, which is shown in Fig. 13 (a). The system remains stable even after e outage of e

faulted line and e estimated MLE is negative, which is shown in Fig 13(b), estimated after e faulted line is isolated. The ree phase to ground fault is also created at e midpoint of e line joining buses 28 and 29 at 1s, which is cleared at 1.20s and 1.19s, considered as e two cases. The fault, when cleared at 1.20s, maes e system unstable as e phase angle of e generator bus 38 diverges from e phase angle of e rest of e generator buses in e system. The plot of e phase angle of all e generator buses after e fault is cleared at 1.20s and 1.19s are shown in Fig. 14 and Fig 15, respectively. Fig. 15. Phase angle of generator buses for a fault of 0.19s at line joining buses 28 and 29. Fig. 16. MLE for a fault of 0.2s and 0.19s at line joining buses 28 and 29. Fig. 13. (a) Phase angle of generator buses for line outage post fault of 0.3s at line joining buses 25 and 26. (b) MLE for line outage post fault of 0.3s at line joining buses 25 and 26. Fig. 17. MLE for a fault of 0.34s at bus 18. Fig. 14. Phase angle of generator buses for a fault of 0.2s at line joining buses 28 and 29. The unstable and e stable cases wi e two different fault clearing time are detected by e MLE, as shown in Fig. 16. The unstable case is detected by positive value of e MLE and e stable case has negative value of e MLE after remaining slightly positive for a very short duration. The ree phase to ground fault is also created at bus 18 in e NE 39 bus system, which is cleared at 1.34s after e inception of e fault. The stability of e system is not disturbed in is case and e phase angles of all e generator buses remain in synchronism after e fault is cleared. The plot of e MLE is shown in Fig 17, which is observed to be negative after e fault is cleared. Fig. 18. MLE for a fault of 0.14s and 0.13s at bus 4. The stability of e NE 39 bus system is also studied by creating a ree phase to ground fault at bus 4 and subsequently clearing it 0.14s and 0.13s after its inception. The system has been observed to be unstable when e fault is cleared after 0.14s after its inception. The system remains stable for e case of fault clearing after 0.13s from its inception. The MLE for e case of e fault clearing at 1.14s is observed to be positive indicating e unstable scenario, whereas e case of fault clearing at 1.13s retains e stability of e system and e MLE is negative, as shown in Fig 18.

I. CONCLUSION The online estimation of Maximum Lyapunov Exponent (MLE) for bo 9 bus and NE 39 bus systems have been demonstrated, in is wor, for detection of impending transient instability. The faults are created at different locations in e two system and subsequently cleared at different time durations maing e system stable for some cases and unstable for e remaining cases. The MLE is estimated from e phase angle of all e generator buses assumed to be measured by e PMUs after e fault is cleared and e transient stability of e system is determined wiin one second using e sign and e magnitude of e estimated MLE. The early indication of instability can be used by e system operator to initiate emergency controls or may trigger automatic sequence of protection mechanism to save e grid from possible blacout condition. ACKNOWLEDGMENT Auors an e Department of Science and Technology (DST), New Delhi, India for providing partial financial support, under e IIT Kanpur project no. DST/EE/20100258, to carry out is research wor. REFERENCES [1] M. Ribbens-Pavella and B. Lemal "Fast Determination of Stability Regions for On-Line Power System Studies", Proc. IEE, vol. 123, no. 7, pp.689-696 1976 [2] Y. Xue, T. Van Cutsem, and M. Ribbens-Pavella, "A simple direct meod for fast transient stability assessment of large power systems", IEEE Trans. Power Syst., vol. 3, pp.400-412 1988 [3] C. Liu, J. S. Thorp, J. Lu, R. J. Thomas and H. Chiang "Detection of transiently chaotic swings in power systems using real-time phasor measurements", IEEE Trans. Power Syst., vol. 9, no. 3, pp.1285-1292 1994 [4] C.-W. Liu and J.S. Thorp, "Application of Synchronized Phasor Measurements to Real-Time Transient Stability Prediction", IEE Proc.- Gener. Transm. Distrib., vol. 142, no. 4, pp.355-360 1995 [5] L. S. Moulin, A. P. A. da Silva, M. A. El-Sharawi and R. J. Mars II "Support vector machines for transient stability analysis of large-scale power systems", IEEE Trans. Power Syst., vol. 19, no. 2, pp.818-825 2004 [6] M. Johnson, M. Begovic and J. Daalder "A new meod suitable for real-time generator coherency determination", IEEE Trans. Power Syst., vol. 19, no. 3, pp.1473-1482 2004 [7] K. R. Padiyar and S. Krishna "Online detection of loss of synchronism using energy function criterion", IEEE Trans. Power Del., vol. 21, pp.46-55 1995 [8] K. Sun, S. Lee and P. Zhang "An adaptive power system equivalent for real-time estimationof stability margin using phase-plane trajectories", IEEE Trans. Power Syst., vol. 26, pp.915-923 2011 [9] J. Yan, C. Liu, and U. Vaidya, "PMU-based monitoring of rotor angle dynamics," IEEE Trans. Power Syst., vol. 26, pp. 2125-2133, 2011. [10] D. Prasad Wadduwage, Christine Qiong Wu, U.D. Annaage, Power system transient stability analysis via e concept of Lyapunov Exponents, Electric Power Systems Research, Volume 104, November 2013, Pages 183-192. [11] S.Dasgupta,M.Paramasivam,U.Vaidya,andV.Ajjarapu, Real-time monitoring of short-term voltage stability using PMU data, IEEE Trans. Power Syst., vol. 28, no. 4, pp. 3702 3711, Nov. 2013. [12] M. A. Pai, Energy Function Analysis for Power System Stability.Boston, MA: Kluwer, 1989. [13] A. Wolf, J. B. Swift, H. L. Swinnery and J. A. Vastano "Determining Lyapunov exponents from a time series", Physica, vol. 16D, pp.285-317, 1985 [14] M. T. Rosenstein, J. J. Collins, C. J. D. Luca, and D. Luca, A practical meod for calculating largest Lyapunov exponents from small data sets, Physica, vol. 65D, pp. 117 134, 1993.