PMU-Based Power System Real-Time Stability Monitoring. Chen-Ching Liu Boeing Distinguished Professor Director, ESI Center

PMU-Based Power System Real-Time Stability Monitoring Chen-Ching Liu Boeing Distinguished Professor Director, ESI Center Dec. 2015

Real-Time Monitoring of System Dynamics EMS Real-Time Data Server Ethernet Proposed algorithm Goal is to improve the reliability of a power system against cascaded events to prevent widespread blackouts. GPS Satellite Data Concentrator State Estimator Communication Links Control Center Application of Phasor Measurement Units (PMUs) PMU can play a critical role in achieving real-time wide-area monitoring, protection and control, if the dynamic states can be tracked in real time. PMU... PMU PMU

Cascaded Events Cascaded events are identified as causes of recent large-scale blackouts. Date Location Effects 2003-08-14 U. S. and Canada 2003-09-28 Italy 2006-11-04 2009-11-10 West Europe Brazil Paraguay 1 province, 8 states and 50 million people were affected. 61.8GW of power was lost. Last for up to 29 hours. $6 billion economic loss. 180 GW of load was lost. 57 million people were affected. Restored after 20 hours. 8 countries and 50 million people were affected. 17 GW of load was lost. Restored after 1.5 hours. 60 million people were affected. 28.8GW of load was lost. Cascaded events in major blackouts follow a common process.

Cascaded Events Initiating events: natural calamities, power component failures, protection and control malfunctions, information and communication system failures, instability due to disturbances, human errors, gaming in electricity market, sabotage, intrusions, missing or uncertain information in decision making. Common phenomena: overload and over-excitation, loss of synchronism, and abnormal system voltage and frequency.

Main Results Develop State Calculator (SC) technique to estimate the dynamic states from available PMU measurements. Maximum Lyapunov Exponent (MLE) employed to predict loss of synchronism for power systems. Observability analysis for nonlinear dynamic power system in order to quantify the confidence level for dynamic monitoring.

Scope of Work Offline Study and Data Preparation Credible MLE window size T System static and dynamic model data Online Operation SC and MLE Real System t 0 t 1 Fault occurs Fault clear PMU Data T 1 Stability prediction T 2 PMU Data T 3 t 2 t 3 t 4 Stability prediction Time Observability Analysis Calculate system observability

PMU-Based Monitoring of Transient Stability Traditional SCADA data since 1960 s Voltage & Current Magnitudes Frequency Every 2-4 seconds PMU data Voltage & current phase angles Rate of change of frequency Time synchronized using GPS and 30-120 times per second

State-of-the-Art Transient stability approaches Time domain simulation Provide details of the dynamics High computational burden and lack of a stability index Energy function Relatively fast computational speed Provide a quantitative index Not incorporating sufficient details Decision tree Equal area criterion

State-of-the-Art (Cont.) PMU-based monitoring of power systems On-line time domain simulation Use PMU measurement to solve the initial states of post-fault system. Equal area criterion PMU-based out-of-step relay. The system is modeled as two interconnected equivalent generators and they are measured by two PMU sets to monitor the angular difference between them. Oscillation modal analysis PMU measurements at the substation level. Distributed frequency domain optimization (DFDO). Decision tree Energy function analysis

Problem Formulation Rotor angle stability is concerned with ability of power systems to maintain synchronism with respect to a small or large disturbance. Generator outputs change suddenly during a contingency period. Mechanical power inputs do not. These disturbances can cause severe oscillations in machine rotor angles and swings in power flows. Loss of synchronism occurs between one generator and the rest of the system, or between groups of generators, leading to instability.

Problem Formulation (cont.) Dynamic model of a power system: i 1 i Pmi Pei Dii 2H i i i 1,..., N Generators are modeled by swing equations; loads are represented by ZIP models. The system is said to be stable (rotor angle stability) if and only if x(t) approaches an asymptotically stable equilibrium point.

Maximum Lyapunov Exponent Lyapunov exponents are used to characterize whether a dynamical system is chaotic by measuring the exponential divergence or convergence of nearby trajectories. x 0 x 0 tx, 0 xt xt Given the solution of the N-dimensional system at time t can be computed by linear approximation as t x 0 x t T x t, x x t 0, the change t where is the Jacobian matrix of x t t, x at time t T x T matrix t, x / x t x i 0 j 0

Methodology: Maximum Lyapunov Exponent t The singular values of T, i.e., x 1... N 0, determine the expansion or contraction of the initial perturbation on each direction. x x 0 0 x 0 tx, 0 xt t x 0 x t T x The i-th Lyapunov exponent i is defined by the following limit 1 i lim ln it, x0 i 1,..., N t t 0 0 i Nearby trajectories diverge on this direction i Nearby trajectories converge on this direction

Maximum Lyapunov Exponent When time goes to infinity, the largest Lyapunov exponent exponentially. MLE of x(t) is used to monitor rotor angle stability after a disturbance. If MLE<0, the system is (asymptotically) stable. Otherwise, it is considered unstable., MLE, dominates MLE stability criterion provides a sufficient condition for system stability when MLE is negative. If MLE is positive, nearby system trajectories will diverge. System instability is likely to occur. 1

Computational Method There are difficulties in the calculation of MLE. 1) One can only calculate MLE over a finite time interval T. Solution: Spectrum analysis is used to select an appropriate time interval length T. 2) Observability of x(t) is required. Solution: Implicit Integration Method with Trapezoidal Rule is used to approximate unobservable state variables. 3) A fast way to calculate MLE is required for on-line applications. Solution: The standard method with GSR is used to calculate MLE after the approximation of the trajectory.

Spectrum Analysis Reasons: power swings exhibit strong periodic behaviors, and the frequency lies within a narrow frequency band under a given system scenario. One can simulate all the credible disturbances on a power system in an off-line study, and obtain power swing curves following the disturbances. Spectrum analysis is then applied to the power swing curves to identify a frequency band. T is set as the inverse ratio to the lower limit of the frequency band.

State Calculator (SC) Complete trajectories of x(t) are required. Given x f ( x), x( t)=[ x (t), x (t),, x (t)] T. x ( 1 t ),, x ( t ) are observed. x ( ),, ( ) +1 t x t 1 2 k n are estimated by x( t t)=x( t)+ f (x( t)) t 1 1 x( t t)=x( t)+ t f (x( t))+ f (x( t t)) 2 2 x 1 t + Δt = x 1 observed (t + Δt) x k t + Δt = x k observed (t + Δt) n k

Gram-Schmidt Reorthonormalization (GSR) In practice, the Gram-Schmidt Reorthonormalization (GSR) method is used to calculate the MLE within a finite time window T. 1. Take a small initial perturbation. The distance between two nearby trajectories after a small time interval Δt is ˆ xt 0 ln x 0 2. Reorthonormalize the separation ˆ xt xt = x0 ˆ xt 3. MLE is calculated as the time average of the rate of separation over a time window T ˆx 0 1... n t T n1 t x0 x ˆ t x2t x2t x0 x t x 2t

Flowchart Real Power System Off-Line Simulation of All Credible Disturbances Power Swing Curves x k Unmonitored State Variables,, 1 x n x ' 1, Monitored State Variables (Waveforms), x' k x ',, x ', x ',, x ' 1 k k1 n x,, 1 xk PMU State Calculator (SC) SC Calculates Unmonitored States Spectrum Analysis Time Window T Covering at Least One Cycle of the Power Swing Calculation of MLE over Finite Time Window T MLE t T Prediction of System Stability Using MLE

Case Study: 200-Bus WECC System 196 197 78 72 30 31 79 80 35 34 33 32 74 198 66 65 182 75 183 73 199 Legend PMU location 500kV 345-360kV 230-287kV 11-22kV Substation 200-bus system that resembles the structure of WECC power grid. 181 71 67 68 70 69 76 82 91 92 93 94 184 185 77 36 186 187 87 95 180 81 99 84 88 96 86 97 85 167 155 165 158 159 90 98 89 188 189 There are 31 generators and 8 of them are observable by PMU data. 83 162 161 157 164 190 118 119 172 173 111 120 121 122 123 168 169 114 124 125 115 128 129 126 127 130 131 170 171 100 106 101 112 113 105 37 64 156 45 44 139 138 200 195 166 160 12 194 193 163 191 192 5 17 21 6 11 Compare simulation results in PSS/E with SC and MLE results. 134 132 18 103 102 104 108 107 110 133 135 117 116 63 147 148 20 21 13 14 29 28 7 8 9 40 39 46 48 174 175 59 153 176 177 178 179 109 142 27 26 22 24 3 10 154 144 60 41 38 146 143 145 152 151 23 19 25 2 4 42 58 56 49 141 136 43 62 150 140 16 15 47 149 61 55 52 137 50 57 54 51 53

Summary of Simulation Results T is set to be 5s according to spectrum analysis. MLE t=5, calculated by the standard method with GSR, predicts losses of synchronism with a high level of accuracy. When the disturbance is generator tripping, the power swing is more likely to be unstable. On the other hand, when the disturbance of line tripping is applied to the power system, the power swing tends to be stable.

Relative Angular Speed of Generator 79 (radian/sec) Relative Rotor Angle of Generator 79 (radian) Estimation Results 118 119 103 102 30 31 196 79 80 78 197 72 71 69 76 82 91 92 67 68 93 70 94 168 172 169 173 114 111 124 126 120 125 127 121 115 122 128 130 123 129 131 134 132 104 133 107 135 110 108 39 46 48 174 178 176 40 175 179 177 59 153 154 60 38 41 49 58 56 42 43 150 62 47 149 55 52 Legend 35 34 33 PMU location 500kV 345-360kV 32 230-287kV 74 198 11-22kV Substation 66 65 75 183 73 199 182 181 184 185 77 36 186 187 87 95 180 81 99 84 88 96 86 188 97 85 167 155 165 158 159 90 98 189 89 83 162 161 157 164 190 170 5 156 163 171 21 166 112 100 45 44 160 6 113 139 138 101 191 11 106 105 64 200 17 12 194 193 192 37 195 18 117 13 116 63 8 9 20 147 148 14 29 28 7 21 109 27 26 142 10 22 3 24 144 145 23 151 25 2 4 146 152 19 143 136 141 140 16 15 61 137 1 Time-domain simulation result Approximation result 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 0.6 Time-domain simulation result Approximation result 0.4 0.2 0-0.2-0.4-0.6-0.8 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 50 57 54 51 53

Relative Angular Speed of Generator 79 (radian/sec) Relative Rotor Angle of Generator 79 (radian) Estimation Results 118 119 103 102 30 31 196 79 80 78 197 72 71 69 76 82 91 92 67 68 93 70 94 168 172 169 173 114 111 124 126 120 125 127 121 115 122 128 130 123 129 131 134 132 104 133 107 135 110 108 39 46 48 174 178 176 40 175 179 177 59 153 154 60 38 41 49 58 56 42 43 150 62 47 149 55 52 Legend 35 34 33 PMU location 500kV 345-360kV 32 230-287kV 74 198 11-22kV Substation 66 65 75 183 73 199 182 181 184 185 77 36 186 187 87 95 180 81 99 84 88 96 86 188 97 85 167 155 165 158 159 90 98 189 89 83 162 161 157 164 190 170 5 156 163 171 21 166 112 100 45 44 160 6 113 139 138 101 191 11 106 105 64 200 17 12 194 193 192 37 195 18 117 13 116 63 8 9 20 147 148 14 29 28 7 21 109 27 26 142 10 22 3 24 144 145 23 151 25 2 4 146 152 19 143 136 141 140 16 15 61 137 1 Time-domain simulation result Approximation result 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 0.4 Time-domain simulation result Approximation result 0.2 0-0.2-0.4-0.6-0.8 0 2 4 6 8 10 12 14 16 18 20 Time (sec) 50 57 54 51 53

MLE Computation Results 3-phase fault Clearing Time-domain simulation result MLE t=5 Bus 18 Generator 18 unstable 0.2958 Bus 30 Generator 30 unstable 0.1421 Bus 35 Generator 35 unstable 0.1559 Bus 36 Generator 36 unstable 0.4526 Bus 40 Generator 40 stable -0.1588 Bus 43 Generator 43 stable -0.0964 Bus 45 Generator 45 unstable 0.4102 Bus 47 Generator 47 stable -0.0816 Bus 65 Generator 65 unstable 0.1164 Bus 70 Generator 70 unstable 0.1065 Bus 32 Line 32-31 unstable 0.2782 Bus 64 Line 64-142 stable -0.1049 Bus 74 Line 74-78 unstable 0.4460 Bus 83 Line 83-168 stable -0.0434 Bus 104 Line 104-102 stable -0.3635 Bus 108 Line 108-174 stable -0.3144 Bus 114 Line 114-171 stable -0.1026 Bus 119 Line 119-131 stable -0.1919 Bus 122 Line 122-123 stable -0.2977 Bus 145 Line 145-151 stable -0.3150

MLE Computation Results 3-phase Time-domain Clearing fault simulation result MLE t=3 MLE t=4 MLE t=5 MLE t=6 MLE t=7 MLE t=10 Bus 116 Generator 116 unstable -0.0682 0.1751 0.1368 0.1745 0.1978 0.1748 Bus 138 Generator 138 unstable 0.2119 0.4644 0.4519 0.3234 0.3362 0.1785 Bus 144 Generator 144 unstable -0.1635 0.1416 0.2172 0.1501 0.1016 0.0778 Bus 149 Generator 149 unstable 0.0620 0.1209 0.2045 0.2679 0.4402 0.4350 Bus 198 Generator 198 unstable 0.2190 0.4344 0.3983 0.2815 0.3320 0.2537 Bus 58 Line 41-58 stable -0.2327-0.2199-0.2066-0.1890-0.1545-0.0220 Bus 142 Line 142-151 stable -0.2387-0.2242-0.2090-0.1839-0.1265-0.1514 Bus 132 Line 132-133 stable -0.2324-0.2180-0.2011-0.1636-0.1174-0.0830 Bus 182 Line 181-182 stable -0.1450-0.1401-0.1353-0.1306-0.1255-0.0862 Bus 155 Line 155-165 stable -0.1336-0.1293-0.1250-0.1210-0.1171-0.1060 MLE predicts system stability with a higher level of accuracy, as the value of T increases. The value of MLE varies with different disturbances. When the system is unstable, the value of MLE changes with respect to the time interval length T in a pattern that exhibits varying characteristics. When the system is stable, the value of MLE is negative but tends to increase as the window size increases.

MLE Technique Summary Attribute Underlying power system model required to implement in real-time? Can it work with limited PMU coverage? Can it predict both rotor angle stability and voltage stability? Can it predict instability while an event is in progress? System non-linearity considered? MLE Method Yes Yes No (Can predict voltage stability if system model is extended) Yes Yes

State-of-the-Art There were about 1000 PMUs installed across North America [1]. Key issue: Whether the measurements contain sufficient information for dynamic monitoring? This is an issue of observability. Different observability definition. Static observability Topological observability Dynamic observability [1] PMUs and synchrophasor data flows in North America, NASPI, 2014.

Static Observability The static state estimation based on DC power flow model: If the rank of H equals to H s column number, the system is observable. T H H is nonsingular. By least square algorithm, the states is solvable. z H * T = 1 H H T H z Observability Solvability

Topological Observability Static state estimation with PMU measurements Observed PMU Unobserved Directly Observed Observed Observed Observed PMU Directly Observed The ability of PMUs to measure line current phasors allows the calculation of the voltage at the other end of the line using Ohm s Law.

Topological Observability Level 1 bus: there is a PMU placed on this bus: Level 2 bus: there is at least one PMU placed on an adjacent bus j: If the entire network is covered by level-1 and level-2 buses, the system with this PMU placement scenario is called the system topological observability. Only 1/5 to 1/4 of the system buses are needed to be installed with PMUs to ensure the complete observability. Linear state estimation for voltage phasors. Static observability does not consider power system dynamic states.

Observability for Dynamic Monitoring Nonlinear dynamic system: x( t) f ( x( t)) z h( x) A dynamic system is observable if, at any time t, the current state x(t) can be determined using only the measurements h(x(t)). Criterion The system is locally observable at, if the observability matrix Ox ( ) has a full rank. The observability matrix is defined using Lie derivative. x t 0 Lf h h 1 h Lfh= Lfh f x x L h L ( L h) 2 f f f lx ( ) t 0 Lfh( x ) t 1 Lfh( xt)... n1 L f h( xt) 0 0 L fh x L fh x x1 xn lx ( ) Ox ( t ) x xx t n1 n1 L f h x L f hx x1 xn

Difficulty for Solving Observability For large scale systems, the computational effort for observability will increase significantly due to chain rules of higher order derivatives. Automatic differentiation : Apply elementary differentiation rules such as sum, product and chain rule to automatically calculate high order derivatives instead of using symbolic expressions. df dg dh For function f g h x, the chain rule gives Properties: Avoid truncation errors. Avoid huge memory consumption. Easier to understand. dx dh dx

Observability Index Defined as the ratio of smallest and largest singular value of Ox ( ) r / Normalization When all buses are placed with PMU measurements, the system is considered to be fully observable: r base Normalization: Power system dynamic equations: min r r r base max i 1 i Pmi Pei Dii 2H i z j i j j j i 1,..., n 1,...,k

Flowchart Real Power System Off-Line Simulation of All Credible Disturbances Power Swing Curves x k Unmonitored State Variables,, 1 x n x ' 1, Monitored State Variables (Waveforms), x' k x ',, x ', x ',, x ' 1 k k1 n x,, 1 xk PMU State Calculator (SC) SC Calculates Unmonitored States Spectrum Analysis Time Window T Covering at Least One Cycle of the Power Swing Calculation of MLE over Finite Time Window T MLE t T Prediction of System Stability Using MLE Observability Analysis along System Trajectory Calculation of Observability Index

Static observability Example: 3-Bus System G 1 Slack bus X13 X12 M21 X23 M23 G 2 G3 z21 B12 0 2 z B B 23 23 23 3 B rank B 12 0 B 23 23 2 System is observable.

Example: 3-Bus System Topological observability is not satisfied Bus 3 not observable by PMU at Bus 1. G1 PMU X12 observable G2 directly observable X23 Not observable G3 Nonlinear dynamic observability should be used. 36

Example: 3-Bus System Dynamic observability x 1 2 3 1 2 3 f1( x) 1 f2( x) 2 f3( x) 3 x f ( x) f4( x) P1 k12 sin( 1 2) k13 sin( 1 3) D1 1 f5( x) P2 k12 sin( 2 1) k23 sin( 2 3) D2 2 f ( x) P k sin( ) k sin( ) D The observability matrix 6 3 13 3 1 23 3 2 3 3 z h( x) L h h( x) 0 f 1 h Lf h f 1 x 2 1 Lfh Lfh f f4 P1 k12 sin( 1 2) k13 sin( 1 3) D1 1 x... 1 1 T

Example: 3-Bus System Different PMU placement scheme will yield different observability results. 60 0.12 Relative Rotor Angle of Generators (radian) 50 40 30 20 10 0 Delta12 Delta13 Delta23 Observability Index 0.1 0.08 0.06 0.04 0.02 PMU at Bus 1 PMU at Bus 2 PMU at Bus 3-10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Simulation result 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Observability with 1 PMU

Example: 3-Bus System When there is only one available PMU in the system, the level of observability is poor. As the number of PMUs increases, the level of observability is also improved. 0.12 0.7 Observability Index 0.1 0.08 0.06 0.04 PMU at Bus 1 PMU at Bus 2 PMU at Bus 3 Observability Index 0.6 0.5 0.4 0.3 0.2 PMU at Bus 1 and 2 PMU at Bus 1 and 3 PMU at Bus 2 and 3 0.02 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Observability with 1 PMU 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Observability with 2 PMUs

Example: 179-Bus WECC System Observability indices can be tracked along system trajectory 1 PMU measurement: Unobservable 3 x 10-11 Observability Index 2.5 2 1.5 1 0.5 The observability indices are essentially zero. The observability matrix is close to being singular. As a result, the system is considered unobservable. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s)

Example: 179-Bus WECC System (Cont.) 14 and 15 PMU measurements Observability Index 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 14 PMU measurements 15 PMU measurements 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (s) The observability indices are much higher and always above 0, but their values differ significantly. The observability of 15- PMU scenario is always higher than the 14-PMU scenario. Is system observable or not?

Observability and State Calculator (SC) Compare SC results for different PMU placement scenarios Relative Rotor Speed of Generators(Radian/s) 8 7 6 5 4 3 2 1 0 14 and 15 PMU measurements Simulation Result SC with 15 PMUs SC with 14 PMUs -1 0 2 4 6 8 10 12 14 16 18 20 Time (s) Choose 0.01 as a threshold value for observability index With 14 PMUs, the difference between SC results and simulation results is significant. The SC results with 15 PMUs are much closer to the simulation results. It indicates that 15 PMUs provides a sufficient level of confidence based on the system observability

Compared with Observability for Static Model Topological observability: the complete set of bus voltage phasors can be uniquely determined by PMU measurements. With 1/3 to 1/5 buses covered with PMUs, the system observability will be guaranteed. For dynamic observability, it can be concluded from the 179-bus case that only about 10% of the buses (15 buses) are needed to have PMU measurements to a satisfactory level of observability. The dynamics on one bus affect its neighboring buses. One measurement can be used to observe the dynamics of buses without direct connections.

PMU Deployment and Critical PMU Location 3.5 15 PMU measurements Placement scenario 1 Placement scenario 2 0.35 3 0.3 2.5 0.25 Observability Index 2 1.5 Observability Index 0.2 0.15 1 0.1 0.5 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time(sec) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time(sec) As the number of PMUs increases, the level of observability improves significantly. Different PMU placement scenario will affect the observability level.

PMU Deployment and Critical PMU Location Remove each PMU measurement respectively. Critical PMU location If the observability index of the system decreases dramatically, the bus from where the PMU is removed is considered to be a critical location for PMU placement.

Identify Best Location for PMU Installation Recursive method: The best deployment of the next PMU measurement can be investigated by checking all the remaining buses to make the system to have a highest observability level. 5 1 4 2 Existing PMU measurements 3

Conclusion PMU-based method for on-line dynamic security assessment Main idea is to calculate MLE in order to predict a loss of synchronism. The method is based on a solid analytical foundation. Computational efficient. Observability analysis for power system dynamic monitoring. An observability index is used to quantify the level of observability of a power grid under different PMU placement scenarios. A real-time dynamic monitoring technique called State Calculator is developed. Offline study to identify the most critical PMU locations.

Future Work Improve system dynamic model by considering the frequency-dependent characteristic of loads Structure-preserving model 0 P P D d i d i i Take into account both angle and voltage dynamics. Observability for large-scale power system. i The remedial control methodology is required to deal with complex instability scenarios. It should choose the control actions automatically to stabilize the system based on MLE results. i

References [1] J. Yan, C. C. Liu, and U. Vaidya, PMU Based Monitoring of Rotor Angle Dynamics, IEEE Trans. on Power Systems, 2011, 26(4), 2125-2133. [2] G. Wang, C. C. Liu, N. Bhatt, E. Farantatos, K. Sun. Observability for PMU-Based Monitoring of Nonlinear Power System Dynamics, Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid (IREP), Crete, Greece, August, 2013. [3] G. Wang, C. C. Liu, N. Bhatt, E. Farantatos, M. Patel. Observability of Nonlinear Power System Dynamics Using Synchrophasor Data, International Trans. on Electrical Energy Systems. DOI: 10.1002/etep.2116. [4] G. Wang, C. C. Liu, N. Bhatt, E. Farantatos. PMU-Based Vulnerability Assessment of Power Systems, To be published by the Institution of Engineering and Technology (IET), UK, 2016.

Thank you! Questions?