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Higher Order Statistical Techniques for Power System Signal Estimation and Identification

Higher Order Statistical Techniques for Power System Signal Estimation and Identification. Ekrem Gursoy, Huaiwei Liao and Dagmar Niebur. * Center for Electric Power Engineering Drexel University Philadelphia, PA 19104 Niebur@drexel.edu

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Higher Order Statistical Techniques for Power System Signal Estimation and Identification

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  1. Higher Order Statistical Techniques for Power System Signal Estimation and Identification Ekrem Gursoy, Huaiwei Liao and Dagmar Niebur *Center for Electric Power Engineering Drexel University Philadelphia, PA 19104 Niebur@drexel.edu The authors gratefully acknowledge the support of the project from the National Science Foundation under Grant # 9985119 Dagmar Niebur, NSF Workshop 11/03/03

  2. Outline • Motivation • Independent Component Analysis • Power System Applications • Active Load Profile Estimation • Reactive Load Profile Estimation • Harmonic Source Profile Estimation • Power System Topology Identification • Challenges • Conclusion Dagmar Niebur, NSF Workshop 11/03/03

  3. Power System Signal Processing With advances in telecommunication and measurement technology (GPS, wireless, distributed sensors etc), data storage and computing power, power systems will be monitored more extensively and at a shorter time-scale Current State of the Art in Traditional State, Load and Harmonic State Estimation • System snapshots taken at fixed point in time t • Usually neglects of correlation of signals in time • Assumes Gaussian distribution of measurement deviations thus ignoring higher order statistics • Measurement deviations usually assumed to be entirely due to measurement noise • Requires complete knowledge of system topology and parameters • Requires accurate mathematical system model • Assumptions on availability of measurements at each relevant network metering point • Suitable for centralized estimation as opposed to local hierarchical estimation Dagmar Niebur, NSF Workshop 11/03/03

  4. Study Case: Load Profile Estimation • Deregulation • Independent power producers will engage in • pool contracts with the load serving entities as well as • bilateral contracts with individual industrial or residential customers • Challenge: How to forecast individual demand profiles? • For many individual customers no historical hourly measurements exist • Only aggregate substation measurements are usually available • For spot-market pricing and energy trading only large industrial customers (>100 kW) are tariff metered on a half-hourly basis (Allera and Horsburgh, 1998) • Load profiles for smaller customer groups are estimated based on • Interpretation of operating data • Derivation from half-hourly tariff metering data (when available) • Estimation using statistical or simulation techniques • Direct monitoring of samples of customers Objective: Establish daily customer daily active, reactive and harmonicload profiles using branch power or non-load bus voltage measurements without any knowledge of system topology or parameters Dagmar Niebur, NSF Workshop 11/03/03

  5. Motivation: Cocktail Party Problem • Observed signals (party room sound) are linear mixtures of unknown statistically independent source signals (speakers) • Objective: Given xj(t), identify source signals sk(t) and mixing matrix Afor all t  [0,T] • Tool: Independent Component Analysis Dagmar Niebur, NSF Workshop 11/03/03

  6. Independent Component Analysis for Load Profile Estimation Voltage Deviation or Branch Power Deviation Measurements Lk(t1…tm) Original Sources Sk(t1…tm) Estimate of Original Sources Sk_es(t1…tm) V1(t1…tm) P1(t1…tm) S1(t1…tm) S1_es(t1…tm) S2(t1…tm) S2_es(t1…tm) V2(t1…tm) P2(t1…tm) Mixing Matrix A Demixing Matrix W . . . . . . . . . Vn(t1…tm) P1(t1…tm) Sn(t1…tm) Sn_es(t1…tm) Iterative Calculation of W Dagmar Niebur, NSF Workshop 11/03/03

  7. Problem Definition for Load Profile Separation Assumption: Statistical independence and non-Gaussian distribution of fast-varying component of the individual loads Linear superposition of load measurements Objective: Estimating individual active, reactive or harmonic load profiles through split of the aggregate substation data into its different contributions with respect to active, reactive or harmonic power consumption. Recognizing network topology and parameters from estimated mixing matrix. Approach: Statistical technique called Independent Component Analysis (ICA) whose technical application is also known as Blind Source Separation (BSS) Dagmar Niebur, NSF Workshop 11/03/03

  8. Independent Component Analysis Model Assumptions: 1. Linear superposition (mixing) of original loads Lj(ti) = Asj(ti) + ej for all instants ti, i=1…T and residual Gaussian noise ej 2. The ensemble of loads sj(ti) are statistically independent at time ti 3. At most one of the loads is Gaussian distributed. Given: Observed power or voltage measurements Lj(ti)  RMat time ti, i=1…T Unknown: Original individual active, reactive or harmonic loads sj(ti)  RN Mixing matrix A RM  N Goal: For M=N estimate original signals Sj_es(ti) = WLj(ti) Estimate de-mixing matrix W  A-1 RN N Dagmar Niebur, NSF Workshop 11/03/03

  9. Independent Component AnalysisOptimization Problem Goal: Estimate original signals Sj_es(ti) = WLj(ti)  Ses = WL Estimate de-mixing matrix W  A-1 RN N With less loads then time steps, i.e. (N<M)  over-determined Formulate as optimization problem with equality constraints Optimization objective: Achieve independence of  rows of Ses Optimize J(wi) = E(G(wiTYj)2) for i = 1, ..., N, subject to Sj_es= WLj and wjTwj = jj for all columns wj of W E denotes the statistical expectation G is a non-linear “contrast” function that measures statistical independence. Dagmar Niebur, NSF Workshop 11/03/03

  10. Contrast Functions Role of Contrast Function • to distinguish higher order statistics of components of L • to measure the degree of independence of the source signals. Choices of G include (Hyvärinen, 1999): 1. Higher order statistical cumulants: • Ex.: the 4th order kurtosis (Karhunen et al, 1998). 2. Minimal mutual information • Ex.: Kullback-Leibler divergence (Haykin, 1998) • Maximum Likelihood • Ex.: Maximum Entropy (Bell and Sejnowski, 1995) 4.  Single unit general odd contrast function separator for supergaussian distributions • G(y) = -exp(-y2/2) (Hyvärinen, 1999) Dagmar Niebur, NSF Workshop 11/03/03

  11. Comparison of ICA and PCA • PCA finds the directions of maximum variance • ICA finds the directions of maximum independence Dagmar Niebur, NSF Workshop 11/03/03

  12. ICA Algorithm • Start with linear mixture of two uniform distributed random variables • Whiten mixture to remove correlation • Search projections to maximize non-gaussianity • Dewhiten to restore original signals & mixing matrix Whitening Maximize Nongaussianity Restore Dagmar Niebur, NSF Workshop 11/03/03

  13. Linear Active Power Flow Model Active bus power deviations: P=[P2, …, Pns]T Angle deviations =[2, …, ns]T, where ns=# of busses. Active branch power flow deviation  PBranch = [P1_ij, ……, PN_i'j’] Linearized nodal “DC flow” equations: P =B (1) Branch flow for branch #k, k=1...N Pk_ij =Bij i - Bijj (2) Branch flow matrix equation (3) T is a rectangular matrix reflecting the imaginary part of the branch admittance (Im(Ybus) and the network topology (incidence matrix) • Active branch flow deviation measurements are linear mixtures of unknown active power load deviations Dagmar Niebur, NSF Workshop 11/03/03

  14. Linear Reactive Power Flow Model • Taylor expansion of power flow equations around operating point (V0,0) • Weak coupling of P and V • Constant power factor: • Linear reactive model: At times ti • Voltage deviation measurements are linear mixtures of unknown reactive power load deviations Dagmar Niebur, NSF Workshop 11/03/03

  15. Linear Harmonic Model Linear circuit under non-sinusoidal conditions: Ihj:phasor current at frequency h injected at node j and Vhj: phasor voltage at frequency h at node j, j=1…N. Yhi,j equivalent admittance at frequency h between nodes i and j. • Complex voltage measurements are linear mixtures of unknown complex currents Dagmar Niebur, NSF Workshop 11/03/03

  16. ICA Assumptions and Results Assumptions: 1. Linear superposition (mixing) of original loads Lj(ti) = Asj(ti) + ej for all instants ti , i=1…T and residual Gaussian noise ej 2. The ensemble of loads sj(ti) are statistically independent at time ti 3. At most one of the loads is Gaussian distributed. Results: • Estimated profiles in unknown order • Estimated profiles up to unknown scaling factor Dagmar Niebur, NSF Workshop 11/03/03

  17. Independence and Gaussianity • Using a 4-point moving average low-pass filter M does not change mixing matrix: • We verified for zonal NYISO data that the fast components of L and S are statistically independent. • We verified for zonal NYISO data that L and S are super-gaussian distributed. => Aggregate load data is super-gaussian => Individual load data is most likely not gaussian. Dagmar Niebur, NSF Workshop 11/03/03

  18. Eliminating Indeterminacy ICA determines load profiles up to a re-ordering and scaling • Re-ordering through matching of estimated load shapes with historic ones by calculation of correlation coefficients. • Re-scaling as Si_est(t) Estimated ith load shape corresponding to ith actual load, ciSi_est(t)+bi Scaled ith estimated load approximation to ith actual load, Wi Energy consumption of ith load from T0 to T1, i Power factor for ith load. Si,est_peak Peak of Si_est(t) , si,peak Peak of ith actual load, obtained from recording/ load forecasting, N Number of actual reactive loads. Dagmar Niebur, NSF Workshop 11/03/03

  19. Load Profiles Separation Algorithm Measure Voltage D|V| or Branch Power D|P| Fast-varying Component, Lf Filtering Mixed Signals L Slowly-varying Component, Lslow ICA Estimated Load Profiles Obtain Mixing Matrix and Estimated Source Signals DQ or DP Rescaling & Post-processing Estimated Topology & Parameters Prior Knowledge Dagmar Niebur, NSF Workshop 11/03/03

  20. PS Applications: Example 1Branch Active Power Flow as Mixed Signals Goal: Estimate active load profiles of bus 4, 5, 6, 9, 10, 11, 12, 13 and 14 from measurements of active power through branches ( circled ), 1-5, 4-5, 4-9, 6-12, 10-14, 13-14, 9-14 and 6-13. IEEE 14 Bus System Dagmar Niebur, NSF Workshop 11/03/03

  21. PS Applications: Example 1Branch Active Power Flow as Mixed Signals Source Signals: Active power drawn by load buses. Horizontal axis denotes time in hours, vertical axis denotes active power drawn by load bus in MW. Mixed signals: Active power flow through branches. The power flow injects the "From" end of branch. Dagmar Niebur, NSF Workshop 11/03/03

  22. PS Applications: Example 1Branch Active Power Flow as Mixed Signals Dagmar Niebur, NSF Workshop 11/03/03

  23. PS Applications: Example 2 Nodal voltage magnitude as Mixed Signals Goal: Estimate reactive load profiles of bus 4, 5, 9, 10, 11, 12, 13 and 14, from measurements of voltage magnitude of these buses. Dagmar Niebur, NSF Workshop 11/03/03

  24. PS Applications: Example 2 Nodal voltage magnitude as Mixed Signals Load profiles as source signals (active power and reactive power have same shape) Nodal voltage magnitude as mixed signals Dagmar Niebur, NSF Workshop 11/03/03

  25. PS Applications: Example 2 Results Absolute percentage errors and mean percentage errors between normalized original signals and estimated ones Estimated load profiles Dagmar Niebur, NSF Workshop 11/03/03

  26. Qualitative Comparison of Smoothed Original and Estimated Signals ______: Original reactive profiles; ………: Estimated reactive profiles Dagmar Niebur, NSF Workshop 11/03/03

  27. System Parameter EstimationResults & Analysis Example 2, Nodal voltage magnitude as mixed signals Estimated de-mixing matrix -178.4518 61.9988 57.0370 -7.6979 4.7301 4.1592 11.4703 4.8742 -615.6352 736.8732 -1.9885 8.0755 -24.6901 -20.8151 -56.5519 -11.8456 -1.3703 25.1752 -100.1960 20.4608 7.4773 0.2461 -11.0631 27.6159 -4.6217 9.3242 -196.1335 223.0159 -38.0283 2.7899 -3.2333 1.9740 0.6170 -9.6423 8.9327 -99.8930 112.5032 1.2090 4.9131 -1.4610 -28.3315 33.0683 16.1902 -8.9751 -2.5667 36.4178 -33.2520 -2.0882 14.4298 -23.3089 10.6519 -8.2563 6.0345 6.9161 -51.7313 22.5949 33.2886 -33.5190 14.7195 3.1457 0.5234 0.1404 15.8768 -61.8425 Used to estimate topology and parameters Corresponding B matrix (reduced) , 38.4851 -21.5786 -1.8500 0 0 0 0 0 -21.5786 35.2166 0 0 0 0 0 0 -1.8500 0 24.1502 -10.3654 0 0 0 -3.0291 0 0 -10.3654 14.7683 -4.4029 0 0 0 0 0 0 -4.4029 8.4970 0 0 0 0 0 0 0 0 5.4279 -2.2520 0 0 0 0 0 0 -2.2520 10.6697 -2.3150 0 0 -3.0291 0 0 0 -2.3150 5.3440 Dagmar Niebur, NSF Workshop 11/03/03

  28. PS Applications: Example 3Complex 5th Harmonic Voltage as Mixed Signals 3 similar harmonic sources (ASD) modeled as constant current source Goal: Estimate harmonic phasor current profiles of bus 3, 9 and 12 from measurements of harmonic voltage at bus 4, 6, 14. Dagmar Niebur, NSF Workshop 11/03/03

  29. PS Applications: Example 3Complex 5th Harmonic Voltage as Mixed Signals Dagmar Niebur, NSF Workshop 11/03/03

  30. Challenges • Investigate topology constraints for the mixing matrix • Investigate incomplete profiles and missing data • Identify topology and other parameters of electric power network • Develop and apply non-linear ICA for non-linear system identification and estimation • Study time-scales and real-time identification • Investigate minimum time windows for profile estimation, extend to state estimation • Bottleneck: Optimization procedure too slow for real-time computation Dagmar Niebur, NSF Workshop 11/03/03

  31. Conclusion • Feasibility study of application of Independent Component Analysis for active, reactive and harmonic load profile separation • Successful separations of load profiles from system measurements only without any prior knowledge about power electric network topology and branch impedances • Use of fast-varying component of load profiles to avoid statistical dependence between them • Estimating topology and parameters of electric power network by ICA separation process. Dagmar Niebur, NSF Workshop 11/03/03

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