northeastern universitym... · 2019. 12. 20. · contents list of figures vi list of tables ix list...

Robust Multi-Area State Estimation for Large Scale Power Systems

A Dissertation Presented

by

Pengxiang Ren

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Electrical Engineering

Northeastern University

Boston, Massachusetts

November 2019

To my family.

i

Contents

List of Figures vi

List of Tables ix

List of Acronyms x

Acknowledgments xii

Abstract of the Dissertation xiii

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Boundary Measurements Modification to Avoid Spreading of Errors 102.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Review of Conventional State Estimator . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 State Estimation Problem Formulation . . . . . . . . . . . . . . . . . . . . 132.2.2 WLS State Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Build Measurement Function and Jacobian . . . . . . . . . . . . . . . . . 152.2.4 Bad Data Detection and Identification . . . . . . . . . . . . . . . . . . . . 182.2.5 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Sensitivity Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Residual and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Measurement Classification . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Sensitivity Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Algorithm to Modify the Measurement Set . . . . . . . . . . . . . . . . . . . . . . 262.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Simulation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.2 Measurement Modification Results on IEEE 118 Bus System . . . . . . . . 312.5.3 Simulation on IEEE 118 Bus System . . . . . . . . . . . . . . . . . . . . 332.5.4 Simulation on Large Scale Power System . . . . . . . . . . . . . . . . . . 36

2.6 Further Analysis of the Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . 39

ii

2.6.1 Redundancy of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6.2 Algorithm for Diverged State Estimation . . . . . . . . . . . . . . . . . . 402.6.3 Limitations of the Proposed Algorithm . . . . . . . . . . . . . . . . . . . 42

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Multi-Area State Estimator (MASE) 443.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Divergence of State Estimator . . . . . . . . . . . . . . . . . . . . . . . . 443.1.2 Proposed Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Literature Review of MASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Area Definition and Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 Zonal Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 Connectivity of the System . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.3 Area Modification Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Area Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 All Pseudo Injection Placement . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Critical Pseudo Injection Placement . . . . . . . . . . . . . . . . . . . . . 573.4.3 Area Boundary Manipulation to Reduce Pseudo Injections . . . . . . . . . 58

3.5 Two-Level State Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5.1 First and Second Level Formulation . . . . . . . . . . . . . . . . . . . . . 623.5.2 First Level Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5.3 Second Level Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5.4 Procedure of Two-Level State Estimator . . . . . . . . . . . . . . . . . . . 65

3.6 Recursively Partitioned State Estimator . . . . . . . . . . . . . . . . . . . . . . . 663.6.1 Partition Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.6.2 Area Partition Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.6.3 Recursively Partitioned State Estimation (RPSE) Algorithm . . . . . . . . 71

3.7 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.7.1 Install MATLAB App . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.7.2 Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.8 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.8.1 Procedure of Two-Level State Estimation . . . . . . . . . . . . . . . . . . 793.8.2 Procedure of Recursively Partitioned State Estimation . . . . . . . . . . . 793.8.3 Validation of the Proposed Estimators . . . . . . . . . . . . . . . . . . . . 833.8.4 Simulations on IEEE 118 Bus System . . . . . . . . . . . . . . . . . . . . 883.8.5 Simulations on Large Scale Power System . . . . . . . . . . . . . . . . . . 903.8.6 Examples of GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Transmission Line Parameter Estimation 1014.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 Literature Review of Transmission Line Parameter Estimation . . . . . . . . . . . 1034.3 Transmission Line Model of a Single Line . . . . . . . . . . . . . . . . . . . . . . 111

4.3.1 Transmission Line Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.2 Three Phase Transmission Line Model . . . . . . . . . . . . . . . . . . . . 112

iii

4.3.3 Unknowns of an Untransposed Transmission Line . . . . . . . . . . . . . 1154.3.4 Unknowns of a Partially Transposed Transmission Line . . . . . . . . . . . 1164.3.5 Unknowns of a Transposed Transmission Line . . . . . . . . . . . . . . . 1174.3.6 Unknowns in Positive Sequence of a Transmission Line . . . . . . . . . . 119

4.4 Static Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4.1 Static Parameter Estimation on Untransposed Line . . . . . . . . . . . . . 1214.4.2 Static Parameter Estimation on Partially Untransposed Line . . . . . . . . 1234.4.3 Static Parameter Estimation on Transposed Line . . . . . . . . . . . . . . 1244.4.4 Static Estimation of Positive Sequence Parameters . . . . . . . . . . . . . 125

4.5 Joint State Estimation and Parameter Tracking for Untransposed Lines . . . . . . . 1254.5.1 Three Phase State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 1264.5.2 Parameter Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5.3 Joint State Estimation and Parameter Tracking . . . . . . . . . . . . . . . 1284.5.4 Initialization of Parameter Tracking . . . . . . . . . . . . . . . . . . . . . 1314.5.5 Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.6 Simulation of JSEPT on Parameters of Three Phase Untransposed Line . . . . . . 1324.6.1 Simulation based on ATP model . . . . . . . . . . . . . . . . . . . . . . . 1334.6.2 Estimation based on actual PMU measurements . . . . . . . . . . . . . . . 136

4.7 Transmission Line Model of a Power Grid . . . . . . . . . . . . . . . . . . . . . . 1384.7.1 Formulation of Transmission Line Model of a Power Grid . . . . . . . . . 1384.7.2 PMU Placement for Tracking Parameters of Multiple Lines in a Power Grid 142

4.8 Joint State Estimation and Parameter Tracking for System (JSEPTS) . . . . . . . . 1434.8.1 State Estimation Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1444.8.2 Parameter Tracking Formulation . . . . . . . . . . . . . . . . . . . . . . . 1454.8.3 Proposed JSEPTS Formulation . . . . . . . . . . . . . . . . . . . . . . . . 147

4.9 Simulation of JSEPTS on Parameters of a Power Grid . . . . . . . . . . . . . . . . 1484.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Estimation of Machine States with Model Uncertainties 1515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.2.1 Linearization and Discretization of State Equation . . . . . . . . . . . . . 1545.2.2 Linearization of Measurement Equation . . . . . . . . . . . . . . . . . . . 1555.2.3 Construction of Structured Uncertainties . . . . . . . . . . . . . . . . . . . 155

5.3 Robust Extended Kalman Filter (REKF) . . . . . . . . . . . . . . . . . . . . . . . 1575.3.1 Ordinary Regularized Least Squares . . . . . . . . . . . . . . . . . . . . . 1575.3.2 Regularized Least Squares with Uncertainties . . . . . . . . . . . . . . . . 1595.3.3 Procedure of REKF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.4.1 Dynamic Model of Synchronized Generator . . . . . . . . . . . . . . . . . 1645.4.2 Implementation of Proposed Robust Kalman filter . . . . . . . . . . . . . 166

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

iv

6 Conclusion and Future Work 1736.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Bibliography 176

A Conventional Data Format 187A.1 IEEE Common Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.2 PSSE Input Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

B Nomenclature of Operators 191

v

List of Figures

1.1 Global Renewable Power Capacity 2007-2017 . . . . . . . . . . . . . . . . . . . . 21.2 Renewable electricity Generation in US . . . . . . . . . . . . . . . . . . . . . . . 21.3 EMS Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Functions of State Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Two port equivalent π-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 IEEE 14 Bus Power Flow Test Case . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Diagram Illustrating Relationship of the Measurements . . . . . . . . . . . . . . . 242.5 IEEE 14 Bus Power Flow Test Case with Zonal Information . . . . . . . . . . . . 252.6 Diagram illustrating measurements to be eliminated . . . . . . . . . . . . . . . . . 272.7 Diagram illustrating measurements to be eliminated . . . . . . . . . . . . . . . . . 302.8 118 Bus System with three zones and specified measurements . . . . . . . . . . . 322.9 118 Bus System with three zones and specified measurements after modification . . 322.10 Sparsity Structure of the Sensitivity Matrix of 118 Bus System . . . . . . . . . . . 332.11 c©PET SE result for part of 118 bus system without any errors . . . . . . . . . . . 342.12 c©PET SE results using original measurement configuration and introducing an error

in injection at bus 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.13 c©PET SE results using modified measurement configuration and introducing an

error in injection at bus 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Example illustrating bus definitions . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Disconnected bus groups of the same zone . . . . . . . . . . . . . . . . . . . . . . 533.3 Modified areas for the network of Figure 3.2 . . . . . . . . . . . . . . . . . . . . . 553.4 Augmented areas for the network of Figure 3.3 . . . . . . . . . . . . . . . . . . . 563.5 Example of boundary topology and measurements . . . . . . . . . . . . . . . . . . 573.6 Area Boundary Manipulation: original . . . . . . . . . . . . . . . . . . . . . . . . 613.7 Area Boundary Manipulation: expansion . . . . . . . . . . . . . . . . . . . . . . . 623.8 Layout of two-level state estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 653.9 Illustration of three partitions of a network . . . . . . . . . . . . . . . . . . . . . . 683.10 Illustration of augmented area A . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.11 Illustration of augmented area B . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.12 Illustration of augmented area C . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.13 Flowchart of RPSE algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vi

3.14 GUI static illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.15 IEEE 118 bus system with three predefined areas . . . . . . . . . . . . . . . . . . 803.16 First level state estimation on IEEE 118 bus system . . . . . . . . . . . . . . . . . 803.17 Second level state estimation on IEEE 118 bus system . . . . . . . . . . . . . . . . 813.18 First level state estimation for IEEE 118 bus system . . . . . . . . . . . . . . . . . 813.19 RPSE second partitioning of IEEE 118 bus system . . . . . . . . . . . . . . . . . 823.20 RPSE third partitioning of IEEE 118 bus system . . . . . . . . . . . . . . . . . . . 823.21 RPSE fourth partitioning of IEEE 118 bus system . . . . . . . . . . . . . . . . . . 833.22 Running time comparison of different methods . . . . . . . . . . . . . . . . . . . 853.23 Running time comparison of different methods . . . . . . . . . . . . . . . . . . . 863.24 Running time comparison of different methods . . . . . . . . . . . . . . . . . . . 873.25 MSE of estimated states for all areas . . . . . . . . . . . . . . . . . . . . . . . . . 873.26 Breaker topology of Bus 110 in IEEE 118 bus system . . . . . . . . . . . . . . . . 903.27 Breaker topology of Bus 110 with a3 open in IEEE 118 bus system . . . . . . . . . 903.28 SE solutions of large scale system diverged case 1 . . . . . . . . . . . . . . . . . . 943.29 SE solutions of large scale system diverged case 2 . . . . . . . . . . . . . . . . . . 943.30 SE solutions of large scale system diverged case 3 . . . . . . . . . . . . . . . . . . 953.31 GUI illustration: error in IEEE 118 bus system . . . . . . . . . . . . . . . . . . . 963.32 GUI illustration: error in 16348 bus system . . . . . . . . . . . . . . . . . . . . . 973.33 GUI illustration: error in 16216 bus system . . . . . . . . . . . . . . . . . . . . . 973.34 GUI illustration: error in 16794 bus system . . . . . . . . . . . . . . . . . . . . . 98

4.1 Model of a three phase transmission line . . . . . . . . . . . . . . . . . . . . . . . 1124.2 π model of three phase transmission line . . . . . . . . . . . . . . . . . . . . . . . 1134.3 Singular value of coefficient matrix using single snapshot measurement . . . . . . 1234.4 Singular value of coefficient matrix using multiple snapshot measurements . . . . . 1234.5 Flowchart of JSEPT overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 ATP Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.7 Estimation results of constant parameter case A1 . . . . . . . . . . . . . . . . . . 1344.8 Estimation results of varying parameter case A2 . . . . . . . . . . . . . . . . . . . 1354.9 Estimation results of varying parameter case A3 . . . . . . . . . . . . . . . . . . . 1364.10 Estimation results of erroneous parameter initial case A4 . . . . . . . . . . . . . . 1374.11 Estimation results of actual PMU measurements . . . . . . . . . . . . . . . . . . . 1374.12 Three bus system to illustrate the construction of branch-bus incidence matrix . . . 1394.13 Constant Parameters of branch 13-14 . . . . . . . . . . . . . . . . . . . . . . . . . 1484.14 Varying parameters of branch 1-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.15 Varying Parameters of branch 7-8 . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.1 Comparison of state estimate without uncertainty . . . . . . . . . . . . . . . . . . 1675.2 Error Variance curves for EKF and REKF without uncertainty . . . . . . . . . . . 1675.3 Comparison of state estimate with uncertain Xd . . . . . . . . . . . . . . . . . . . 1685.4 Error Variance curves for EKF and REKF with uncertain Xd . . . . . . . . . . . . 1685.5 Comparison of state estimate with uncertain D . . . . . . . . . . . . . . . . . . . 1705.6 Error Variance curves for EKF and REKF with uncertain D . . . . . . . . . . . . . 1705.7 Comparison of state estimate with uncertain Xd with transient . . . . . . . . . . . 171

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5.8 Error Variance curves for EKF and REKF with uncertain Xd with transient . . . . 171

viii

List of Tables

2.1 Comparison of Estimated States for 118 Bus System with Error in Inj 46 . . . . . . 352.2 Monte Carlo Simulation of Measurement Residuals for 118 Bus System with error

in a single zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Zonal Information Before and After Measurement Set Modification . . . . . . . . 372.4 Measurement Residuals Comparison with Single Error in Zone 15 . . . . . . . . . 372.5 Measurement Residuals Comparison with Single Error in Zone 34 . . . . . . . . . 382.6 Measurement Residuals Comparison with Errors in Zone 14 and 17 . . . . . . . . 382.7 Measurement Residuals Comparison with Errors in Zone 15 and 35 . . . . . . . . 382.8 Measurement Information about System Redundancy . . . . . . . . . . . . . . . . 392.9 Simulation Results of 118 Bus System Divergent Case 1 . . . . . . . . . . . . . . 412.10 Simulation Results of 14143 Bus System Divergent Case with Error in Zone 15 . . 412.11 Simulation Results of 14143 Bus System Divergent Case with Error in Zone 20 . . 412.12 Simulation Results of 14143 Bus System Divergent Case with Error in Zone 35 . . 42

3.1 Comparison of area boundary cut-set objectives . . . . . . . . . . . . . . . . . . . 833.2 Comparison of methods on area 3 of IEEE 118 bus system with measurement noise 843.3 Comparison of methods on area 3 of IEEE 118 bus system without measurement noise 853.4 Comparison of methods on area 31 of 17014 bus system without measurement noise 853.5 Simulation of IEEE 118 bus system with gross error in real injection at bus 54 . . . 893.6 Simulation of IEEE 118 bus system with gross error in real flow at branch 38 to 65 893.7 Simulation of IEEE 118 bus system with topology error at bus 110 . . . . . . . . . 913.8 Simulation of 16348 bus system with gross error in real injection at bus 54 . . . . . 913.9 Simulation of 16216 bus system with gross error in real flow at branch 9408 to 8547 923.10 Simulation of 16794 bus system with gross error in reactive injection at bus 5432 . 933.11 Simulation of topology error in 16348 bus system with missing branch 14591 – 14621 96

4.1 Classification of line parameter estimation methods . . . . . . . . . . . . . . . . . 1054.2 MPE and NRMSE of constant parameters case A1 . . . . . . . . . . . . . . . . . . 1344.3 CC and NRMSE of varying parameter case A2 . . . . . . . . . . . . . . . . . . . 1354.4 CC and NRMSE of varying parameter case A3 . . . . . . . . . . . . . . . . . . . 1364.5 CC and NRMSE of actual case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.6 Results of PMU placement for parameter tracking of a power grid . . . . . . . . . 143

5.1 Robust extended Kalman filter algorithm . . . . . . . . . . . . . . . . . . . . . . . 163

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List of Acronyms

2LvSE Two-Level State Estimator.

AC Alternating Current.

AGC Automatic Generation Control

ATP Alternative Transients Program.

CC Correlation Coefficient.

CDF Common Data Format.

CT Current Transformer.

CtrlSE Centralized State Estimator.

DC Direct Current.

DER Distributed Energy Resource.

EKF Extended Kalman Filter.

EMS Energy Management System.

EMTP Electromagnetic Transients Program.

EV Error Variance.

GPS Global Positioning System.

GUI Graphical User Interface.

IEEE Institute of Electrical and Electronics Engineers.

ISO Independent System Operator.

JSEPT Joint State Estimation and Parameter Tracking.

JSEPTS Joint State Estimation and Parameter Tracking for System.

KF Kalman Filter.

x

LAV Least Absolute Value.

LMS Least Median Squares

LS Least Squares.

MASE Multi-Area State Estimation.

MPE Mean Percentage Error.

NP Nondeterministic Polynomial problem.

NRMSE Normalized Root Mean Squared Error.

OPF Optimal Power Flow.

OPP Optimal PMU Placement.

PET Power Education Toolbox.

PMU Phasor Measurement Unit.

PT Potential Transformer.

REKF Robust Extended Kalman Filter.

RTO Regional Transmission Organization.

SCADA Supervisory Control And Data Acquisition.

SCOPT Security Constrained Optimal Power Flow.

SE State Estimator.

RMS Root Mean Square.

RPSE Recursively Partitioned State Estimator.

UKF Unscented Kalman Filter.

WLS Weighted Least Squares.

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Acknowledgments

Foremost, I would like to express my sincere gratitude to my advisor Prof. Ali Abur andProf. Hanoch Lev-Ari for the continuous support of my Ph.D study and research. Their patience,dedication and immense knowledge guided me all the time during my research, writing papers,reports and this dissertation. And this will also keep me go forward in the following of my life. Icould not have imagined having better advisors and mentors for my Ph.D study.

Besides my advisor, I would like to thank Prof. Bahram Shafai, who is also in mydissertation committee, for his interesting classes, and encouragement and insightful comments.

My sincere appreciation also goes to Dr. Jianzhong Tong, Dr. Emanuel Bernabeu and Mr.Joseph Ciabattoni, for sponsoring me the projects I’ve done during my Ph.D study. And also foroffering me the summer internship opportunities in their group. Also, many thanks to Mr. Eric Hsia,Julan Feng, Daniel, Sheshanth and Mahesh who gave me a lot of help during the internship.

I thank my labmates in Northeastern University: Ahmet Oner, Alireza Rouhani, AndreLangner, Arthur Mouco, Bilgehan Donmez, Cesar Antonio Galvez Nunez, Chenxi Xu, David Kelle,Eduardo Werley, Ramtin Khalil and Yuzhang Lin, for the stimulating discussions and for all the funwe have had in the last several years. Also, I thank for the CURENT colleagues in University ofTennessee, Knoxville: Denis Osipov, Hantao Cui, Nan Duan, Jingxin Wang and many others, forcollaborative works we had done and the hospitality during annual SVT.

Last but not the least, I would like to thank my girl friend Xiaomeng Peng who supportsme all the time during my PhD life and gives me more than love. And I would like to thank myparents Zemin Ren and Zhiying Zhang, for giving birth to me at the first place and supporting mespiritually throughout my life.

xii

Abstract of the Dissertation

Robust Multi-Area State Estimation for Large Scale Power Systems

by

Pengxiang Ren

Doctor of Philosophy in Electrical Engineering

Northeastern University, November 2019

Dr. Ali Abur, Dr. Hanoch Lev-Ari, Advisor

Renewable energy technologies, such as wind and solar power, are becoming increasinglyattractive complementary resources to the existing energy supplies, because of climate changeand energy diversification concerns. The increased percentages of power generated by stochasticrenewable energy sources require new system operation strategies dealing with the issues relatedreliability of the monitoring, control, generation and transmission system.

State estimator (SE), as a crucial component of energy management system (EMS), pro-vides the most likely state of a power system based on the measurements received from variousequipment and substations. The robustness of state estimators is critical in order to serve otheroperation and market applications in power system control centers, such as security assessment, unitcommitment, economic dispatch, etc. There are a number of reasons which may result in unreliablestate estimation solutions, including equipment failures, human factors, not updated network, mali-cious cyber attack, variations of ambient conditions, etc. From the engineering point of view, due tothe potential errors in measurements, parameters or network model, even the most reliable estimatorsare vulnerable to occasional divergence.

This dissertation addresses these issues and develops practical solutions to be implementedin large scale power systems. From the perspective of static estimator, a boundary measurementmanipulation algorithm is developed first to isolate the errors within individual zones. Two morerobust multi-area state estimators are developed in order to ensure a converged state estimationsolution for the largest possible portion of the system. They can also facilitate the implementation ofCPU-intensive solution algorithms on the isolated small area in order to detect and identify specificerrors and source of divergence. Afterwards, to deal with the parameter inaccuracies in transmissionlines, algorithms are developed to dynamically track the parameters of a three phase untransposed lineas well as multiple lines in a power grid. Last but not least, the robustness of dynamic state estimator

xiii

is improved by reformulating the Kalman filter with the consideration of parameter inaccuraciesin electric machines. By applying all the developed algorithms, the state estimator becomes morerobust against divergence and estimation solutions are more reliable.

xiv

Chapter 1

Introduction

1.1 Motivation

In recent years, motivated by global warming, as well as other ecological and economic

concerns, there is a widespread support for renewable energy. It encompasses a broad, diverse

array of technologies, including solar photovoltaics, solar thermal power plants and heating/cooling

systems, wind farms, hydroelectricity, geothermal power plants, and ocean power systems and the

use of biomass. Globally, renewable energy systems are rapidly becoming more efficient and cheaper,

consequently their share of total energy consumption is increasing. As of 2017 worldwide, more than

two thirds of all new electricity capacity installed was renewable. Among them, wind and hydropower

accounted for most of the remaining renewable capacity additions, contributing more than 29% and

nearly 11%, respectively. In Figure 1.1, we can see that total renewable power capacity more than

doubled in the decade 2007-2017, and the capacity of non-hydropower renewables increased more

than six times [1]. Specifically for US, the increase of wind and solar energy sources are more

significant, as shown in Figure 1.2. In 20 years, the hydropower generation remains around 300

GWh, and the capacity is about 100 GW. Overall, the electricity generated from renewable sources

in US are around 18% of total generation in 2018, whereas the share is 15% in 2016 and only 10% in

2010. And it is estimated that in 2050, the share will reach 50%.

Another prevailing topic of future power system is distributed energy sources (DERs).

Different from the traditional centralized electricity generation such as natural gas power plants,

hydropower plants and utility scale wind or solar farms etc., DERs are electricity-producing resources

or controllable loads that are directly connected to a local distribution system. DERs can include

solar photovoltaic panels, small wind turbines, combined heat and power plants, electricity storage,

1

CHAPTER 1. INTRODUCTION

Figure 1.1: Global Renewable Power Capacity 2007-2017

Figure 1.2: Renewable electricity Generation in US

small natural gas-fueled generators, electric vehicles and etc. In recent years, DER installations

have increased significantly in some regions of the United States due in part to technology advances

and state energy policies. What’s more, DER is facing continued growth due to customer desire for

self-supply, environmental considerations, decreasing acquisition costs of DER technologies and

regional and federal policies encouragement.

Increasing amounts of renewables in transmission and distribution levels can change how

the nowadays power system works and will require new strategies of operation. Attention must

be paid to potential reliability impacts, the time frame required to address reliability concerns,

coordination and system protection of transmission and distribution system, and the cyber-security

concerns. To ensure the system operates economically and reliably, a crucial tool for system operators

is the energy management system (EMS).

The energy management system is designed to monitor, control and optimize the operation

of the whole system, in order to reduce energy consumption, improve the utilization of the system,

2


increase reliability, predict electrical system performance, and optimize energy usage to reduce cost.

It includes but not limited to the following applications.

• State estimation. State estimation processes the real time measurement acquired from Supervi-

sory Control and Data Acquisition (SCADA) system or Phasor Measurement Units (PMUs),

and calculates the the statistically most probable set of states (voltage magnitudes, phase

angles, transformer taps and etc.) existing on the network. In addition, the state estimator has

the ability to filter out small noises and detect and identify gross measurement errors.

• Contingency Analysis. After the state estimation process, it is necessary to test that model

for a large number of outages to determine if the system can recover from the outage without

problems. The contingencies can be modeled using a power flow program by running the

contingencies one at a time. Considering the numerous combinations of multiple contingencies,

some contingency selection algorithms are needed.

A concise configuration of EMS is provided in Figure 1.3

Figure 1.3: EMS Configuration

Therefore, since almost all of the applications depend on the accurate state estimates, the

robustness of state estimator is a critical aspect to evaluate the performance of EMS and thus of

3


the reliable operation of power system. And with the penetration of renewable sources and new

technologies, there is a need for more accurate state estimates, in order to support more advanced

power grid operation strategies for independent system operators (ISOs), utilities and consumers.

However, there are a number of reasons which may result in unreliable or inaccurate state estimation

solutions, including but not limited to the followings.

• Failures to update network data. This is one of the most common situations that would lead

to the failure of state estimator. Usually the changing network data, including circuit breaker

status, control system status and other topology status, should be reported promptly to the

system operator. However, this is not always the case due to communication issues, such as

failure of communication equipment. Thus, the inaccurate network model will lead the state

estimates biased.

• Equipment failures. Different from the failures above, in this case the measurement equipment

such as potential transformer (PT) and current transformer (CT) fail to provide the correct

values of the measured voltages, currents, power flows or frequency.

• Malicious cyber attack. With the rapid development and deployment of communication and

information technologies in power systems, the concerns regarding cyber security have been

significantly growing. Any of the systems’ principal elements, power generation, transmission,

or distribution, could be targeted for a cyber attack. In Ukraine, December 2015, attackers

targeted substations that lower transmission voltages for distribution to consumers. As a result,

30 substations were switched off, and about 230 thousand people were left without electricity

of a period from 1 to 6 hours.

• Variations of ambient conditions. The impact of ambient conditions can not be ignored for

modern power grid. For example the transmission line parameters largely depend on the

temperature of conductors. The variation of conductor temperature is related to both internal

and external factors such as ambient temperature, wind speed and direction, and solar radiation.

Neglecting the varying ambient conditions may lead to inaccurate line parameters and thus

biased state estimates.

• Inadvertent human entry factors.

Technically speaking, in the sense of numerical formulation of state estimation problem,

all of the above factors may result in:

4


• incorrect network model,

• gross errors in various measurements,

• inaccurate network or electric machine parameters.

Therefore, it is necessary to investigate and develop the algorithms to obtain unbiased

estimation solutions within a reasonable time. Commonly, power system state estimators can

be broadly classified under two categories:: static state estimator for transmission or distribution

system performs real-time monitoring of the entire system based on network model and available

measurements; and dynamic state estimator that focuses on the dynamic behavior of electric machines

such as generators and exciters, and the dynamics of control functions. Note that there are some

other state estimators which are beyond the scope of this dissertation, such as forecasting-aided state

estimator which incorporates the state and measurement forecasts to enhance the performance of

static state estimation.

From the point view of static state estimation, though bad data processing can detect and

eliminate bad measurements, it may still suffer from the failures due to gross measurement errors

and topology errors. Under extreme circumstances, the state estimator may fail to converge and the

operators no longer observe the system conditions. Currently, there is very few research focusing on

such scenarios. Thus, some algorithms are proposed in this dissertation to address the problem.

Another critical aspect of static state estimation is network parameters, i.e. line parameters.

Line parameter is critical due to its role in determining network model, system loss and dynamic

ratings of lines, which are the important quantities for operations and power markets. Though there

are numerous publications discussing the calculation and estimation of line parameters. the real-time

estimation for parameters of three phase lines is still an active topic of investigations. Therefore,

accurate tracking of line parameters is also studied in this dissertation.

Last but not least, the results of electric machine state estimates by dynamic state estimation

can also be biased due to possibly inaccurate machine parameters. Such bias can deteriorate the

accuracy of dynamic of electric machines, impacting AGC related applications. A new algorithm is

developed in this dissertation to address this issue.

An outline of the chapter will be presented next.

1.2 Outline

The dissertation focuses on the following issues:

5


• Measurement design to avoid spreading of measurement errors.

• Multi-area state estimator (MASE).

• Transmission line parameter estimation.

• Robust extend Kalman filter (REKF) for estimating electric machine states with inaccuracy in

parameters.

When the state estimator fails to converge it is usually due to an error in a specific part of

the grid, yet due to the integrated formulation of the state estimation problem, it will cause divergence

of the overall state estimator. Hence, in a multi-area large scale power grid, the system-wide state

estimator’s performance will be limited by the area with the least reliable and accurate model and

measurements in the overall system. We focus on the scenarios when SE suffers from gross errors or

even fails to converge because a certain area experiences convergence issues. The spread of errors can

be predicted by using the sensitivity of these errors to various measurement residuals. If one has the

ability to manipulate the measurement design, these error residual spread areas can be intentionally

made to coincide with the operational zones and thus, state estimation results of individual zones

can be made insensitive to the errors in the measurements of other zones while maintaining a fully

observable large scale interconnected system with several zones. The main advantage of this design

will be isolation of errors within individual zones thereby allowing an unbiased state estimation

solution to be obtained for a large portion of the overall system even when errors exist in one or a

few system zones. In Chapter 2, this scheme will be explained in detail along with theoretical proofs

and numerical simulations.

However, such modification requires elimination of some redundant measurements at zone

boundaries to make sure the remaining boundary measurements become critical. It decreases the level

of measurement redundancy, and consequently reduces the robustness of the system. Therefore, the

multi-area state estimator (MASE) is proposed to mitigate the problem. A two-level state estimator

(2lvSE) involves a two stage solution where the first stage obtains individual area solutions, followed

by a coordination stage where these solutions are combined and synchronized by a central processor.

The main idea which is to use the first stage of the 2LvSE algorithm in order to identify and isolate

the area which contains the root cause of divergence, is presented briefly in 3. What’s more, the

isolated area can be further partitioned into even smaller systems, each of which can conduct the same

process as the isolated area. This estimator, named recursively partitioned state estimator (RPSE),

has more advantages than the centralized state estimator and two-level state estimator. It will not only

6


ensure a converged state estimation solution for the largest subset of buses in the system, but also

facilitate possible implementation of CPU-intensive solution algorithms on the isolated small area in

order to detect and identify the source of divergence. The developed multi-area state estimators will

be described in Chapter 3 thoroughly.

To deal with the parameter inaccuracies in transmission lines, a number of methods have

been proposed for identification and correction of transmission line parameter errors. But most of the

them assume a fully transposed transmission line, which limits their applicability in more general

settings, especially for the distribution level. Among the algorithms focusing on untransposed line

parameters, a prevailing way to deal with rank deficiency of the measurement-parameter coefficient

matrix is using multi-scan measurements. More specifically, one can use either a sliding window or

several consecutive measurement snapshots in a single estimation step. However, the interval from

which measurements are collected has to be long enough to generate a well-conditioned coefficient

matrix, yet short enough to justify the assumption of negligible variation of parameters across this

interval. These conflicting constraints limit the accuracy and applicability of parameter estimation

techniques based on multiple measurements. The problem to estimate parameters in untransposed

transmission line can be better addressed by a dynamic parameter estimation technique, as described

in Chapter 4. The proposed method alternates between state estimation, which relies on the most

recent parameter estimates and serves to suppress voltage and current measurement noise, and

parameter tracking, which relies on the most recent state estimates and serves to farther suppress

current measurement noise. By iteratively processing state estimation and parameter tracking, the

mismatch between actual and estimated parameters can be reduced. The proposed dynamic approach

works well for both time-invariant and time-variant line parameters under considerable measurement

noise. It is also modified and implemented on a power grid observed by PMUs to track the parameters

of all lines in the system.

To deal with the parameter inaccuracies in electric machines, specifically in generators and

exciters, a different scheme is required. When accurate prior knowledge of the rotating machine

nonlinear state-space model is available, standard dynamic state estimators, such as the extended

Kalman filter can provide accurate state estimates both in steady state and in transients. A robust

continuous-time discrete-measurement extended Kalman filter for non-linear state estimation in the

presence of modeling uncertainties is developed in Chapter 5. The proposed filter is designed to map

the continuous-time nonlinear equation with unstructured uncertainty into a discrete-time linearized

state equation with structured parameter uncertainty. The proposed filter will provide a state estimate

for the nonlinear system, mitigating the bounded parametric uncertainties that standard extended

7


Kalman filter cant.

1.3 Major Contributions

The major contributions of this dissertation are as follows:

• A boundary measurement manipulation algorithm to avoid spreading of errors is developed. It

not only provides a way to block the errors between different control zones, but also gives an

insight into how different kinds of measurements will impact the error spreading.

• A two-level multi-area state estimator is developed in order to address the non-convergent state

estimation cases. The new estimator is implemented and tested using a very large scale actual

utility system data and measurements.

• A customized partitioning algorithm is developed in order to implement the multi-area state

estimator yet maintain connectivity and observability of individual areas.

• The observability issue related to the boundary of areas is addressed. When system is decoupled

into several areas, the observability of each area may be violated due to the topology change.

Thus, several algorithms are designed and compared to ensure observability particularly under

the multi-area framework.

• The development of recursively partitioned state estimator (RPSE). RPSE aims to further

partition the problematic areas detected by two-level state estimator, and proceed recursively

to find and isolate a small enough partitions from the rest of the system. A specific area

partitioning technique is developed with new area definitions. Then the performance of RPSE

and 2lvSE are evaluated based on actual convergent and divergent cases of several large scale

systems. Results prove the effectiveness of real world implementation.

• Three phase transmission line parameter estimation algorithm is developed. First, the linear

formulation of three phase line parameters to the terminal measurements is constructed.

Then, with appropriate initialization, the developed algorithm JSEPT overcomes the rank

deficiency problem by introducing an iterative process between three phase state estimation

and line parameter estimation. This procedure is in real-time, i.e. it uses only one snapshot of

measurement to estimate the parameters at that time.

8


• The algorithm of estimating parameters in multiple transmission lines is developed then. Based

on the idea of JSEPT, JSEPTS is developed by alternating between state estimation for the

whole grid and the parameter estimation for one or multiple lines. These line parameter

estimation algorithms can be implemented to the lines whose parameters are varying or

suspicious to errors, both in transmission and distribution systems.

• The robust extended Kalman filter is developed for estimating machine states even with the

uncertainties in parameters. A detailed interpretation of Kalman filter is described first. The

uncertainties in the continuous-discrete measurement model are then structured. By solving a

min-max problem, the robust version of extended Kalman filter is developed at last.

9

Chapter 2

Boundary Measurements Modification

to Avoid Spreading of Errors

2.1 Background

Static state estimators not only provide the best estimates of the system states but also the

capability to detect and identify bad measurements. Given a measurement set, if a single bad data

exists in a non-critical (redundant) measurement, one or more measurement residuals will become

significantly large. A traditional way to mitigate the impact of errors in measurements is bad data

processing. Based on largest normalized residual test or other techniques, the existence of gross

errors in the measurement set can be detected, identified and eliminated. This procedure, along with

a detailed review of traditional state estimation technique, is described in detail in Section 2.2.

However, when facing a very large scale power system, such bad data process is quite

time-consuming [2]. Considering the fact that such bad data may occur in only one or several

predefined control zones, rather than dispersed in all of them, one may think the possibility to isolate

the erroneous zones and keep the state estimates from the rest.

The sensitivity matrix, as will be discussed in Section 2.3, can help to provide a theoretical

basis. The spread of errors can be predicted by using the sensitivity of these errors to various

measurement residuals. This issue is well investigated and systematic procedure for identifying the

network measurements that will be impacted as a result of a specific measurement error is developed

in [3] and [4]. Further work presented in [5] and [6] provide insights and practical algorithms to

detect the so called error residual spread areas for a measured power grid. These studies elegantly

10

CHAPTER 2. BOUNDARY MEASUREMENTS MODIFICATION TO AVOID SPREADING OF ERRORS

consider the situation from the operators point of view where the information on the measurement

configuration and the network topology are used to sketch the error residual spread areas. If on the

other hand, one has the ability to manipulate the measurement design, these error residual spread

areas can be intentionally made to coincide with the operational zones and thus, state estimation

results of individual zones can be made insensitive to the errors in the measurements of other zones

while maintaining a fully observable large scale interconnected system with several zones. The main

advantage of this design will be the isolation of errors within individual zones thereby allowing an

unbiased state estimation solution to be obtained for a large portion of the overall system even when

errors exist in one or a few system zones.

In this chapter, existing measurement design for a given large scale system having multiple

zones will be modified in order to enable isolation of error residuals in different zones of the

network [7]. The main idea is based on the observation that critical measurements have the ability to

block residual error spreading between error spread zones [8]. Hence, by strategically modifying

measurement design, each topological zone can be made to also behave as a residual error spread

zone. This modification requires elimination of some redundant measurements at zone boundaries

to make sure the remaining boundary measurements become critical and block the spread of error

residuals to neighboring zones.

More specific, the principle theory of state estimation residual analysis and sensitivity

analysis is described in Section 2.3. It first analyzes the residual sensitivity as an extension of the

weighted least squares (WLS) state estimation discussed in Section 2.2.2. Then, the measurements

are classified and the definition of redundant and critical measurements is provided. According to

such definition, the sensitivity matrix is further defined and analyzed.

In Section 2.4, the boundary measurement modification algorithm is proposed and de-

scribed in detail. Computational aspects of the proposed algorithm will be discussed and results from

test cases will be reported in Section 2.5. The measurement modification results are described and

discussed first. Then, based on IEEE 118 bus system and a large scale system, several scenarios are

simulated to testify the performance against bad data. The results include the figures for specific

cases, estimated states and residuals of several representative zones. By comparing those results, the

proposed algorithm is proved to be effective.

In Section 2.6, some further analysis of the proposed algorithm are provided, including

the impact of measurement redundancy, scenarios of diverged state estimator and limitations of the

algorithm, followed by the conclusion in Section 2.7.

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2.2 Review of Conventional State Estimator

Static state estimation, as first developed in late 1960s by Schweppe [9], is a crucial

component of modern energy management system. The main objective of static state estimation is to

monitor the steady-state operating conditions of the power grid based on the available measurements.

A detailed flowchart is provided in Figure 2.1, illustrating all the functions related to a typical state

estimator.

Figure 2.1: Functions of State Estimator

As shown in the above figure, the measurements from state estimator can either be provided

by the supervisory control and data acquisition (SCADA) system through remote terminal units

(RTUs) and other kinds of sensors, or from phasor measurement units (PMUs). SCADA measure-

ments are received typically every few seconds and include real/reactive power injections such as

generator outputs and loads, real/reactive power flows measured at the terminals of transmission

lines and transformers as well as voltage magnitude measurements at system buses. Measurements

obtained from PMUs are more frequent, usually at a rate of 30 times per second, and more accurate.

However, PMU measurements are not considered in this section since they are not integrated in

conventional SE. But they will be analyzed and utilized in later chapters.

The state estimators also include the following functions:

• Topology processor gathers status data about the circuit breakers and switches.

• Observability analysis determines if a state estimation solution can be obtained by detecting

12


the adequacy of measurement configuration.

• Bad data processor detects the existence of measurement errors.

In this section, the problem formulation of power system state estimation is reviewed,

followed by the weighted least squares (WLS) method for obtaining the state estimation solution. The

issues about building measurement Jacobian matrix, bad data processor and observability analysis

are briefly discussed then.

2.2.1 State Estimation Problem Formulation

The state estimates can be obtained by minimizing the weighted sum of squares of the

measurement residuals, or equivalently solving the following optimization problem:

minx

J =

m∑i=1

Wir2i (2.1)

subject to zi = hi(x) + ri, i = 1, · · · ,m. (2.2)

whereJ : objective;

Wi : weight of ith measurement;

ri : residual of ith measurement;

m : the number of measurements.

The solution of the above optimization problem is called the weighted least squares (WLS)

estimator for the state vector x. Consider the measurement model:

z =

z1

z2

...

zm

=

h1(x1, x2, · · · , xn)

h2(x1, x2, · · · , xn)...

hm(x1, x2, · · · , xn)

+

e1

e2

...

em

= h(x) + e (2.3)

where

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z : known measurement vector;

x : the state vector to be estimated, x = [x1, x2, · · · , xn];

n : the number of states;

h : h(x)T = [h1(x), h2(x), · · · , hm(x)], hi(x) is a linear or nonlinear func-

tion relating the system state vector x to the ith measurement;

e : vector of measurement errors, e = [e1, e2, · · · , em];

m : the number of measurements.

The measurement errors are assumed to be Gaussian with

• E[ei] = 0.

• E[eiej ] = 0 if i 6= j

σ2i if i = j

• Cov(e) = R = diagσ21, σ

22, · · · , σ2

m.

We can then rewrite the objective function as:

J(x) =m∑i=1

Wir2i

=m∑i=1

(zi − hi(x))2/Rii

= [z − h(x)]TR−1[z − h(x)]. (2.4)

At the minimum, the first-order optimality conditions will have to be satisfied. These can be expressed

in compact form as

g(x) =∂J(x)

∂x= −HT (x)R−1[z − h(x)] = 0 (2.5)

where H(x) is the measurement Jacobian matrix that

H(x) =∂h(x)

∂x.

Expanding the nonlinear function g(x) into its Taylor series around the state vector xk yields

g(x) = g(xk) +G(xk)(x− xk) + · · · = 0.

Neglecting the higher order terms lead to an iterative solution scheme known as the Gauss-Newton

method as shown below

xk+1 = xk −[G(xk)

]−1g(xk), (2.6)

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wherek : iteration index;

xk : the state vector solved at iteration k;

G(xk) : gain matrix, with G(xk) =∂g(xk)

∂x= HT (xk)R−1H(xk);

g(xk) : first derivative of cost function, g(xk) = −HT (xk)R−1[z − h(xk)].

Now define ∆xk = xk+1 − xk, thus (2.6) can be written as the Normal Equation that

G(xk)∆xk = −HT (xk)R−1[z − h(xk)]. (2.7)

2.2.2 WLS State Estimation Algorithm

Weighted least squares state estimation involves the iterative solution of the Normal

equation given by (2.7). A traditional WLS state estimation algorithm is described as follows:

Step 1: Start iterations, set the iteration index k = 0.

Step 2: Initialize the state vector xk(k = 0), typically as a flat start, i.e. voltage magnitude to be 1.0

per unit and phase angle to be 0.

Step 3: Calculate the gain matrix G(xk).

Step 4: Solve for ∆xk according to (2.7), usually by forward-back substitutions of sparse linear set

of equations derived from G(xk).

Step 5: Test for convergence, check whether max |∆xk| < ε. If no, update xk+1 = xk + ∆xk,

k = k + 1, and go back to step 3. Else stop.

The above algorithm involves the formulation of measurement Jacobian matrix, bad data detection,

observability analysis and etc., which will be discussed in the following sections.

2.2.3 Build Measurement Function and Jacobian

The measurement function h(x) contains the line power flows, bus power injections, bus

voltage magnitudes and etc. These measurements can be expressed in terms of the state variables.

It is easy to obtain the Jacobian incident to measured voltages. Thus, in this section, the Jacobian

incident to injections and line flows are focused. For an N bus power system, if using the polar

15


coordinates, x will be a (2N − 1) × 1 state vector including (N − 1) phase angles θ and N bus

voltage magnitudes V :

xT = [θ2 θ3 · · · θN V1 V2 · · · VN ].

gft+jbft

f t

vf vtif it

gsf+jbsf gst+jbst

Figure 2.2: Two port equivalent π-model

Such measurement function h(x) can be formulated based on the two-port π-model as

shown in Figure 2.2. The bus power injection at bus k can be formulated as

Sk = Pk + jQk = vk (Ybusv)∗k (2.8)

whereSk : power injection at bus k;

Pk : real power injection at bus k;

Qk : reactive power injection at bus k;

Ybus : Nb ×Nb bus admittance matrix;

v : Nb × 1 bus voltage vector, v =

v1

v2

...

vk...

vNb

=

V1∠θ1

V2∠θ2

...

Vk∠θk...

VN∠θNb

;

k : index indicates kth bus. vk indicates the kth element of the vector v;

Nb : number of buses in the system.

To calculate line power flows, we first take the π-model of branch l in Figure 2.2 as an example. We

can write if,lit,l

=

Yff,l Yft,l

Ytf,l Ytt,l

vfvt

, (2.9)

16


where the elements Yff,l, Yft,l, Ytf,l, Ytt,l in branch admittance matrix can be obtained from the

parameters in equivalent π model. Note that all variables in (2.9) are scalars. We can expand the

current and voltage variables in (2.9) to vector as

iF = YF v (2.10)

iT = YT v (2.11)

where

iF : Nbr × 1 vector consisting of all branch currents from the ’from’ side.

iF =[if,1 if,2 · · · if,l · · · if,Nbr

]T;

iT : Nbr × 1 vector consisting of all branch currents from the ’to’ side.

iT =[it,1 it,2 · · · it,l · · · it,Nbr

]T;

YF : Nbr×Nb coefficient matrix of all branch ’from’ side currents to all voltages;

YT : Nbr ×Nb coefficient matrix of all branch ’to’ side currents to all voltages;

l : index indicates lth branch;

Nbr : number of branches in the system.

Note that AT and A∗ indicates the transpose and conjugate transpose of vector or matrix

A respectively. Therefore, the power flow on branch l can be obtained by

Sf,l = Pf,l + jQf,l = vk (YF v)∗l, (2.12)

St,l = Pt,l + jQt,l = vk (YT v)∗l, (2.13)

where

Sf,l : power flow on branch l from f side to t side;

Pf,l : real power flow on branch l from f side to t side;

Qf,l : reactive power flow on branch l from f side to t side;

St,l : power flow on branch l from t side to f side;

Pt,l : real power flow on branch l from t side to f side;

Qt,l : reactive power flow on branch l from t side to f side;

l : index indicates lth branch. ·l indicates the lth element of the vector inside

parentheses or braces;

17


k : index indicates kth bus. ·k indicates the kth element of the vector inside

parentheses or braces. Note that bus k is one of the terminals of branch l. In

(2.12) and (2.13), bus k is the f and t side of branch l, respectively.

For the bus voltage measurements, usually only voltage magnitude are measured in SCADA

system. Thus, the voltage magnitude measurement on bus i is vi = |V i|. The measurement

Jacobian H(x) is defined as the partial derivative of the measurement function:

H(x) =∂h(x)

∂x. (2.14)

Therefore, based on (2.8), (2.12) and (2.13), all the bus injections and line flows can be listed as

Sinj = v (Ybusv)∗, (2.15)

Sflow,f = v(f) (YF v)∗, (2.16)

Sflow,t = v(t) (YT v)∗. (2.17)

Note that the operator indicates the Hadamard product that takes two matrices (or vectors) of the

same dimensions and produces another matrix where each element i, j is the product of elements

i,j of the original two matrices. Given a line list, all the from side bus voltages consist the Nbr × 1

vector v(f), and the to side bus voltages consist the v(t). Therefore, by taking the partial derivative

of (2.15), (2.16) and (2.17) to the state vector, and selecting the ones with enabled measurements,

the Jacobian matrix incident to measured injections and power flows can be obtained.

2.2.4 Bad Data Detection and Identification

One of the essential functions of a state estimator is to detect measurement errors, and to

identify and eliminate them if possible. Measurements may contain errors due to various reasons.

Random errors usually exist in measurements due to the finite accuracy of the meters and the noise in

telecommunication medium. Provided that there is sufficient redundancy among measurements, such

errors are expected to be filtered by the state estimator. Large measurement errors can also occur

when the meters have biases, drifts or wrong connections. Telecommunication system failures or

noise caused by unexpected interference also lead to large deviations in recorded measurements.

For static state estimator, bad data processing techniques and algorithms have been inves-

tigated for decades right after the development of state estimation. A number of earlier literatures

have been published to solve this problem in different ways: plausibility check, largest normalized

18


measurement residual test, chi-squares test, hypothesis testing identification, etc. There are two

prevailing techniques to identify bad data as described in [2]: largest normalized residual test and

hypothesis testing identification method. For the former one, normalized value of the residual for

measurement i can be obtained by simply dividing its absolute value by the corresponding diagonal

entry in the residual covariance matrix

rNi =|ri|√Ωii

If the largest value among all of normalized residuals is larger than a chosen identification threshold,

for instance 3.0, this measurement is identified as bad data and need to be eliminated from the

measurement set. A main weakness of this method is that it’s based on the residuals which may be

strongly correlated. Hence, in case of multiple bad data, this correlation may lead to comparable size

residuals for good as well as bad measurements. Moreover, if there exists multiple bad data, they

have to be eliminated one by one. For the latter one, consider that the WLS estimator is run and the

normalized residuals are calculated. A set of measurements with the largest normalized residuals are

then picked making sure that they are linearly independent and non-critical. A test for the estimated

errors of the suspect set can be devised then.

Some relatively new techniques have been studied to improve the performance of bad data

identification from two aspects: increase the level of measurement redundancy, and utilize alterna-

tive cost function formulations rather than traditional weighted least squares technique. Besides,

phasor-aided state estimation [10–12] is developed to enhance the robustness of state estimation via

introducing phasor measurements. Forecasting-aided state estimation [13, 14] takes advantage of

using historical database to extract information necessary to produce state/measurement forecasts

which can be incorporated into the SE process to enrich it and improve the capability for bad data

identification. And the dynamics of measurement variations can also be a useful complement to

detect false data attacks [15].

2.2.5 Observability Analysis

Power system state estimator uses the set of available measurements in order to estimate

the system state. Given a set of measurements and their locations, the network observability analysis

will determine whether a unique estimate can be found for the system state. Telecommunication

errors, topology changes or meter failures may occasionally lead to cases where the state of the entire

system can not be estimated. Then, the system will contain several isolated observable islands, each

one having its own phase angle reference that is independent of the rest [2]. Network observability

19


analysis allows detection of such cases and identifies all the existing observable islands prior to the

execution of the state estimator.

Observability analysis can be carried out using the fully coupled or decoupled measurement

equations. A general numerical method using nodal variables to determine network observability is

described as follows. Consider the linearized measurement model

∆z = H∆x, (2.18)

the numerical observability analysis method can be outlined as follows:

Step 1: Read the network and measurement information.

Step 2: Determine the set of unobservable buses Sbusunob, set of observable island Sisleob, set of

island need to be checked Sislewaiting. As the initialization, move the whole system into

Sislewaiting.

Step 3: Choose the largest network in Sislewaiting and labeled as Sislepre.

Step 4: Based on the topology of Sislepre, formulate measurement Jacobian H , branch to bus

incident matrix A and gain matrix as G = HT H .

Step 5: Incomplete Cholesky factorization of the gain matrix G yields L that

L =

X 0 0 · · · 0

X X 0 · · · 0...

.... . .

...

X X · · · X 0

X X · · · X X

,

where X stands for nonzero or zero entries. For a nonsingular matrix G, all diagonal

elements of its decomposition L are not zero.

Step 6: Modify the zero pivots of the decomposition L to different non-zero values if there are. The

corresponding entry of the right hand side vector tA will be assigned an arbitrary value.

The arbitrary values assigned in this manner should be distinct from each other and this is

accomplished by assigning integer numbers in increasing order.

Step 7: Calculate estimated state x = (L LT )−1tA and branch flow estimates that Pb = A x. The

unobservable branches are identified as non-zero elements in Pb.

20


Step 8: Delete all the unobservable branches and modify the system Sislepre to Sislepost. The network

Sislepost may have one or several islands which are not topologically interconnected.

• If the system Sislepost only have one island, Sislepost = Sislepre, delete Sislepost in

Sislewaiting and move it to Sisleob. Then go to step 9.

• If there are isolated single buses, move them to Sbusunob.

• If there are several islands whose number of buses are larger than 1, delete them from

Sislepost and move these islands to Sislewaiting.

Step 9: Check if Sislewaiting is empty or not:

• If it is empty, go to step 10.

• If it is not empty, go to step 3.

Step 10: Terminate.

By following the above procedure, all the unobservable buses are obtained in Sbusunob. All

observable islands are in Sisleob.

2.3 Sensitivity Matrix Analysis

2.3.1 Residual and Sensitivity Analysis

As discussed in Section 2.2.1, state estimator provides the most likely state of bus voltage

magnitudes and phase angles based on the nonlinear vector function h relating the measurement z

and state x vectors. We can simplify (2.3) as

z = h(x) + e (2.19)

where e is the vector of measurement error with zero mean and Cov(e) = R. The state vector x

can be estimated using the well-known weighted least square method. This can be accomplished by

obtaining a linear approximation of (2.19) that

∆z = z − h(x0)

= H∆x+ e

= H(x− x0) + e

= H∆x+ e

(2.20)

21


where H is the gradient of function h with respect to x. Thus the incremental change in the state x

which will minimize the objective function:

J(∆x) = (∆z −H∆x)TR−1(∆z −H∆x) (2.21)

will be given by:

∆x = (HTR−1H)−1HTR−1∆z. (2.22)

Then the estimated measurement vector z will be given by:

∆z = H∆x = K∆z (2.23)

where

K = H(HTR−1H)−1HTR−1 (2.24)

is the hat matrix. The hat matrix also holds the following properties

K K = K

K H = H

(I−K)H = 0

Thus the measurement residual r holds

r = ∆z −∆z

= ∆z −K∆z

= (I−K)∆z

= (I−K)(H∆x+ e)

= (I−K)e

= Se

(2.25)

where I is the identity matrix with appropriate size.

Given the property of hat matrix that K H = H , now we can define the relation between

measurement residual r and measurement error e as the residual sensitivity matrix S

S = I−H(HTR−1H)−1HTR−1. (2.26)

The sensitivity matrix S is symmetric if and only if the covariance of the errors are all equal.

22


WLS estimation is based on the assumption that the measurement errors are distributed

according to a Gaussian distribution given as ei ∼ N(0, Rii) for all i. Using the linear relation

between the measurement residuals and measurement errors given by (2.25), the mean and the

covariance of the residuals can be derived as follows:

E[r] = E[Se] = SE[e] = 0 (2.27)

Cov(r) = Ω

= E[rrT ]

= SE[eeT ]ST

= SRST

= SR.

(2.28)

Therefore, it holds that r ∼ N (0,Ω).

2.3.2 Measurement Classification

Power systems contain various types of measurements spread out in the system. These

measurements have different properties and affect the outcome of the state estimation accordingly,

depending on not only their values but also their locations. For explicit illustration, IEEE 14 bus

power flow test case is provided in Figure 2.3 as an example to show the measurement classification.

Figure 2.3: IEEE 14 Bus Power Flow Test Case

In Figure 2.3, the black rectangular bar indicates the bus, and blue line indicates the

transmission line. The red circle on the line stands for power flow measurement through the line. The

23


red circle with arrow at bus indicates the injection measurement. These two kinds of measurements,

i.e. flow and injection measurements, are metered by appropriate equipment, usually potential

transformer (PT) and current transformer (CT). The notation of measurements are as follows:

• Inj i : real or reactive power injection on bus i.

• Flow i-j : real or reactive power flow on the branch from bus i to bus j.

Also note that in the following discussion, the injection and flow represent both real and reactive

power measurements.

There are three major classifications for the measurements in power systems: criti-

cal/redundant measurement, actual/pseudo measurement and zonal/boundary measurement. A

diagram is provided in Figure 2.4, indicating the overall relations between actual and pseudo, zonal

and boundary, critical and redundant measurements.

Figure 2.4: Diagram Illustrating Relationship of the Measurements

Classification 1: Critical/Redundant Measurement. As discussed in Section 2.2.5, the

definition of critical measurement is crucial for the observability analysis of a power system in sense

of state estimation. A critical measurement is the one whose elimination from the measurement

set will result in an unobservable system, therefore not all the states of the system can be observed.

The column of the matrix H , corresponding to a critical measurement will be identically equal to

zero. Furthermore, the measurement residual of a critical measurement will always be zero. Take

Figure 2.3 as an example, flow 7-8 is a critical measurement because without it, state of bus 8 can

not be observed. To the contrary, a redundant measurement is the measurement which is not critical.

Only redundant measurements may have nonzero measurement residuals.

Classification 2: Actual/Pseudo Measurement. An actual measurement is the measurement

that physically exists in the power system. A pseudo measurement is usually an injection, obtained

24


from various ways such as load forecasting. In real world operation, for the purpose of ensuring

state estimation observability, pseudo injections are always added to all the buses who do not have

incident injection measurements. A major difference between actual and pseudo measurement is the

weight: pseudo measurements have very small weights compared to the actual ones’. In Figure 2.3,

all measurements are actual measurements.

Classification 3: Zonal/Boundary Measurement. Large scale power grids typically have

well defined control zones in order to meet the requirement of different power system planning and

operation procedures. It is possible that some zones are not interconnected, but such circumstance

will be discussed in Section 3.3. In this chapter, we assume these zones are interconnected. With

proper definition of zones, the boundary buses in one zone are defined as the buses directly connected

to the buses in other zones. And the boundary branches are the lines connecting two buses from

different zones. The boundary measurements are the injection measurements that on the boundary

buses and the flow measurements on the boundary branches. And the internal measurements are the

rest measurements sorted by zones. Take the IEEE 14 system as an example, it is divided into two

zones: 1, 2, 3, 4, 5, 7, 8, 9 and 6, 10, 11, 12, 13, 14, as shown in Figure 2.5. Thus, the boundary

buses are 5, 9 for zone one and 6, 10, 14 for zone two. And the boundary branches are 5-6, 9-10

and 9-14.

Figure 2.5: IEEE 14 Bus Power Flow Test Case with Zonal Information

25


2.3.3 Sensitivity Matrix

The sensitivity matrix S is a square matrix with the dimension equals to the number of

measurements. According to (2.25), the diagonal and off-diagonal entries of S correspond to the

following sensitivities.

• Sii: ith measurement residual to ith measurement error;

• Sij : ith measurement residual to jth measurement error.

Every measurement can belong to one of two categories, it will either be critical or

redundant [2]. It is usually sufficient to inspect only the diagonal entries in identifying critical

measurements as shown below,

• if Sii = 0: ith measurement is critical;

• if Sii 6= 0: ith measurement is redundant.

Measurements of each zone can be categorized into two subsets: internal measurements

and boundary measurements. Boundary measurements include injection measurements at the

boundary buses, and flow measurements on the branches that connect two neighboring areas. Internal

measurements are incident to buses which belong to the same area. Based on such classification,

the sensitivity matrix can be rearranged in the following form where internal measurements are first

listed followed by its boundary measurements.

S =

Szone1

Sb1. . .

Szone2. . .

SzoneN

(2.29)

2.4 Algorithm to Modify the Measurement Set

The main purpose of this work is to isolate bad measurements inside power system zones

such that the influence of bad data in one zone will only impact the measurement residuals inside

this zone. In order to achieve this goal, the concept of critical measurements will be used as

shields between neighboring zones. It should be noted that the measurement residual of a critical

26


measurement will always be zero, i.e., bad data appearing in a critical measurement cannot be

detected. However, on the positive side, a critical measurement at the zone boundary will act as a

shield blocking the spread of measurement error residuals to neighboring areas.

The proposed approach is based on restructuring the sensitivity matrix in such a way that it

has null row/columns separating diagonal blocks representing measurement sets belonging to various

network zones. The algorithm will primarily manipulate the boundary measurements forcing them to

be critical since they will be the ones acting as blockers of error residual spread between individual

zones.

The diagram given in Figure 2.6 indicates the relation between actual and pseudo, zone

and boundary, critical and redundant measurements. The shaded area in the figure indicates the

measurements that need to be eliminated. It is noted that removal of measurements will modify the

composition of the remaining measurement set. Thus, the redundancy of some of measurements may

change due to the elimination of others. In such case, Jacobian matrix H and sensitivity matrix S

must be updated every time a redundant measurement is removed. This presents a computational

challenge for large scale systems since the computation of sensitivity matrix S requires extensive

effort to invert gain matrix G. Thus, sparse matrix methods are employed for this part of the

computations.

Figure 2.6: Diagram illustrating measurements to be eliminated

Hence, implementation of the main idea of removing all redundant measurements at zonal

boundaries can be accomplished by the following algorithm:

Step 1: Obtain the network data and measurements for the entire system. Define zones if there is no

such definition.

Step 2: Add pseudo injection measurements to all buses that lack injection measurements. This step

27


is optional. It is included due to its common use by commercial state estimation software to

ensure observability for every state estimation run.

Step 3: Identify zonal boundary measurements, including boundary flows and boundary injections.

Sort the measurements according to their zone number, and order the boundary measurements

based on their zone numbers as shown in the sensitivity matrix structure described above.

Step 4: Form the Jacobian matrix H and sensitivity matrix S according to the sorted measurement

set.

Step 5: Find all the pseudo measurements that belong to the boundary measurements, then check

their sensitivities and choose measurement i with the largest |Sii|. If |Sii| < ε (ε is small

enough, usually 10−9), and all the column elements |S·i| < ε and row elements |Si·| < ε,

then go to next step. Else delete measurement i and go to step 4.

Step 6: Find boundary measurements that are not pseudo measurements and repeat the procedure of

step 5.

Step 7: Check if all the boundary measurements are critical. If true, terminate the procedure. Else,

go to step 4.

When the above procedure is executed, all the redundant measurements incident to the

boundary buses will be eliminated and the sensitivity matrix will be converted to the desired block

diagonal form as described below.

S =

Szone1 0 0 · · ·

0 0 0 · · ·0 0 Szone2...

.... . .

(2.30)

A flowchart of the algorithm is given in Figure 2.7. In the flowchart, the cylinder shape

block stands for the initial information. The rectangular block stands for the process, the diamond

block for the condition statement and the parallelogram for the variables. The flowchart begins with

the initial information: Psse.out which contains system bus and branch information, and HC-output

which contains the one time shot raw measurement information. The bus and branch information

is processed first with connectivity examination and zone redefinition, and then written into IEEE

common data format. The raw measurement information is processed with rearranging the sequence

28


and adding pseudo measurement. Using the information available now, the different class of buses,

branches and measurements incident to boundary can be figured out. The next step is to build the

Jacobian matrix H and sensitivity matrix S using the equations discussed before. Till now, all

the preparation work for the main iterative approach is finished. With the ’if’ condition statement

executed, the number of redundant boundary measurements gets one less every time, until the

termination condition fulfilled. The final results are saved in appropriate format, waiting for the

examination of random measurement errors.

29


Figure 2.7: Diagram illustrating measurements to be eliminated

30


2.5 Simulation

2.5.1 Simulation Approach

The numerical simulation is based on Monte Carlo method to simulate the cases under

random measurement errors. The detailed steps to process this are as follows:

Step 1: Add random value single error to a random measurement in the measurement set.

Step 2: Run the state estimator.

Step 3: record estimated states and measurement residuals.

Step 4: Repeat steps 1, 2 and 3 for a large number of cases.

Once the Monte Carlo simulations are finished, the recorded results can be used to see if

the measurement errors in one zone can spread and influence the residuals in other zones. Power

Education Toolbox ( c©PET [16]) is used for obtaining the state estimation solutions corresponding

to randomly introduced bad data in various zones.

2.5.2 Measurement Modification Results on IEEE 118 Bus System

The proposed approach is tested on the well-known and documented IEEE 118 bus test

system. It consists of 118 buses and 179 branches. The system is intentionally considered to have

3 zones. In Figure 2.8, the bus numbers are indicated for each zone, boundary buses are identified

by showing the tie-line connections to them for each zone. The solid circles indicate injection

measurements and solid lines indicate flow measurements. Among injection measurements, injection

70, 75, 44 and 118 are actual measurements and the rest are pseudo measurements.

Applying the measurement modification algorithm described above, a number of redundant

measurements are eliminated.

• Critical measurements: flow 75-118, 68-81

• Redundant measurements: injection 38, 42, 43, 44, 49, 65, 68, 69, 70, 75, 77, 81, 118; flow

75-77, 43-44, 38-65.

The remaining critical boundary measurements are shown in Figure 2.9.

31


Figure 2.8: 118 Bus System with three zones and specified measurements

Figure 2.9: 118 Bus System with three zones and specified measurements after modification

There are only two critical boundary measurements left, connecting the three islands. In

Figure 2.10, the visualized block diagonal form of sensitivity matrix is shown with several null rows

and columns. Note the three blue blocks representing the three zones separated by null rows/columns.

32


Figure 2.10: Sparsity Structure of the Sensitivity Matrix of 118 Bus System

2.5.3 Simulation on IEEE 118 Bus System

An example of the 118 bus system with single measurement error exists in a random zone

is discussed specifically in this section. Figure 2.11, Figure 2.12 and Figure 2.13 show the state

estimation results, including the residuals of real power flow and reactive power flow, under different

circumstances. More specifically, Figure 2.11 shows the results without any measurement errors.

Figure 2.12 shows how single error in injection 46 influences the residuals inside and outside its own

zone, namely zone 2. As a comparison, Figure 2.13 shows the results of the same error after applying

the proposed measurement manipulation algorithm.

Figure 2.11 shows part of the state estimation results of 118 bus system using the original

measurement set. All the normalized residuals are shown next to the measurements, such as 0.003P

and 0.013Q of the flow 43-44 indicate no bad data as they are insignificant (i.e. less than 3.0) to be

suspected as bad data. Figure 2.12 shows results when an error is intentionally introduced in injection

measurement at bus 46 of zone 2. A number of its adjacent measurements, including measurements

belongs to zone 1, are impacted with very large normalized residuals (in red) and therefore impacted

by the bad measurement in zone 2. In Figure 2.13, the normalized residuals of the measurements of

zone 1 are back to their low or insignificant level, while the normalized residuals of zone 2 remain

large indicating suspect bad data in that zone. This indicates the error in the injection measurement

46 in zone 2 is successfully confined within the zone.

33


Figure 2.11: c©PET SE result for part of 118 bus system without any errors

Figure 2.12: c©PET SE results using original measurement configuration and introducing an error in

injection at bus 46

Numerical results of estimated states are shown in Table 2.1. Since the error is in zone

2, the results show that only the estimated states in zone 2 changed drastically after modification,

compared to the results before modification. In the table, ’Mag.’ and ’Ang.’ stands for magnitude

and the phase angle of state estimates respectively. ’T’, ’B’ and ’A’ indicates the values of actual

states, state estimates before manipulation and estimates after, respectively.

Results obtained by applying Monte Carlo simulations of a total of 100 runs on 118

34


Figure 2.13: c©PET SE results using modified measurement configuration and introducing an error

in injection at bus 46

Zone Bus Mag.(T) Ang.(T) Mag.Diff(B) Ang.Diff(B) Mag.Diff(A) Ang.Diff(A)

1 41 0.9668 -22.92 -0.0126 -1.7 0 0.02

1 35 0.9807 -18.92 -0.006 -1.12 0 0.01

2 46 1.005 -11.42 0.1158 15.71 0.1711 -1.84

2 45 0.9867 -14.22 0.535 5.66 0.1191 -13.09

3 110 0.9730 -11.86 -0.0002 -0.49 0 0.01

Table 2.1: Comparison of Estimated States for 118 Bus System with Error in Inj 46

bus system also show that both the maximum absolute residual and the sum of the residuals of

measurements in each zone that does not contain any bad measurement remain below 0.01 p.u. which

is much smaller than the same metrics for the zone containing the error. In Table 2.2, Monte Carlo

simulation results of ten test cases are shown. The maximum absolute measurement residual of each

zone is computed and reported in the table. By comparing the residuals before and after applying

the proposed modification process, it is possible to see if the errors remain confined to the zone or

spread all over the system. In the table, ’Loc.’ stands for the location of erroneous measurement,

’Res. Zone’ stands for zonal measurement residuals, ’B’ and ’A’ stands for the estimation before and

after modification respectively.

35


Case Zone Loc. Status Res. Zone1 Res. Zone2 Res. Zone3 Blocked?

1 2 Inj 46 B 0.087 0.489 0.021

1 2 Inj 46 A 0.002 0.448 0.004 Y

2 1 Inj 34 B 0.242 0.033 0.007

2 1 Inj 34 A 0.232 0.001 0.001 Y

3 3 Flow 77-80 B 1.310 2.018 6.414

3 3 Flow 77-80 A 0.003 0.004 4.791 Y

4 2 Flow 54-55 B 0.102 0.625 0.083

4 2 Flow 54-55 A 0.003 0.625 0.001 Y

5 1 Inj 27 B 3.392 0.874 0.301

5 1 Inj 27 A 2.715 0.002 0.001 Y

6 3 Flow 82-96 B 0.093 0.178 0.770

6 3 Flow 82-96 A 0.001 0.001 0.726 Y

7 1 Flow 30-38 B 3.975 1.267 0.675

7 1 Flow 30-38 A 3.064 0.003 0.003 Y

8 3 Inj 106 B 0.674 0.626 2.418

8 3 Inj 106 A 0.002 0.001 2.413 Y

9 1 Flow 24-70 B 0.540 0.078 0.092

9 1 Flow 24-70 A 0.495 0.002 0.001 Y

10 2 Flow 49-51 B 1.066 4.984 0.583

10 2 Flow 49-51 A 0.004 0.513 0.001 Y

Table 2.2: Monte Carlo Simulation of Measurement Residuals for 118 Bus System with error in a

single zone

2.5.4 Simulation on Large Scale Power System

The developed algorithm is also applied to a large scale utility system to check the

effectiveness of the measurement design algorithm. This large scale system has 51 designated zones

and thousands of measurements. Given the space limitations, simulation results for only the ten

largest zones of the system will be shown. Table 2.3 shows the detailed information of these ten

zones. Again, ’B’ and ’A’ indicate the before and after manipulation respectively.

After the elimination of redundant measurements based on the above described algorithm,

36


Zone No. Buses No. Inj(B) No. Flow(B) No. Inj(A) No. Flow(A)

15 1687 106 120 6 7

17 1588 23 24 2 2

14 1512 24 28 3 0

20 775 28 39 4 0

35 753 24 6 2 0

34 633 39 22 2 2

41 551 44 8 11 0

13 482 46 55 10 3

22 460 19 21 5 3

5 409 9 10 5 0

Table 2.3: Zonal Information Before and After Measurement Set Modification

a rather small number of boundary measurements are left. Monte Carlo simulations are carried

out under random errors and different cases. Four examples in the following tables are shown to

illustrate how the proposed strategic modification of the measurements can block error spread. In

these tables, the values of maximum absolute measurement residuals (MAMR) of each zone are

shown in descending order, both for the original as well as the modified measurement systems.

’Detected’ in the third and fifth columns of the tables stands for whether the error(s) can be detected

from the corresponding zone. If the answer is true, denoted as ’Y’, it indicates that the error will

impact the corresponding zone, and vice versa.

Table 2.4 and Table 2.5 show the results for the case of single error in zone 15 and zone 34

respectively.

Zone MAMR (B) Detected (B)? MAMR (A) Detected (A)?

15 1.206 Y 1.198 Y

20 0.274 Y 0.003 N

13 0.018 N 0.005 N

17 0.001 N 0.001 N

25 0.001 N 0.001 N

Table 2.4: Measurement Residuals Comparison with Single Error in Zone 15

37



34 1.402 Y 0.718 Y

14 0.017 Y 0.004 N

41 0.012 Y 0.001 N

36 0.001 N 0.001 N

15 0.001 N <0.001 N

Table 2.5: Measurement Residuals Comparison with Single Error in Zone 34

Table 2.6 provides results of the case of simultaneous errors in two different zones, 14 and

17. Similarly, in Table 2.7, errors are in zone 15 and 35. All the results validate that the spread of

errors in one or multiple zones to other zones is avoided by the proposed measurement design.


14 1.380 Y 1.380 Y

17 0.585 Y 0.582 Y

1 0.341 Y 0.005 N

39 0.053 Y 0.003 N

40 0.047 Y 0.002 N

15 0.003 N 0.003 N

Table 2.6: Measurement Residuals Comparison with Errors in Zone 14 and 17


15 2.214 Y 2.088 Y

35 1.222 Y 1.117 Y

41 0.202 Y 0.001 N

13 0.018 Y 0.001 N

14 0.001 N 0.001 N

Table 2.7: Measurement Residuals Comparison with Errors in Zone 15 and 35

38


2.6 Further Analysis of the Proposed Algorithm

2.6.1 Redundancy of the System

There is no doubt that the redundancy of the system will be decreased because of the

elimination of boundary redundant measurements. To analyze the level of redundancy lost, two

crucial ratios are the evaluated: measurement-to-state ratio, and the ratio of number of eliminated

measurements to all the measurements.

Note that the number of eliminated measurements largely depends on the number of zones.

For example, if there is only one zone, such modification process would be senseless since there is no

so-called boundary measurements and thus there is no eliminated measurement. If there are too many

zones, nearly all the redundant measurements would be eliminated with only critical measurements

left. The following table gives such information of 14 bus system, 118 bus system and an 14143 bus

system. Note that for the original system, denoted as ’B’, all measurements are redundant.

14 Bus System 118 Bus System 14143 Bus System

Zone No. 2 3 48

Whole Meas.(B) 57 456 63813

Whole Meas.(A) 47 424 61365

Redundant Meas (A) 43 410 59999

Critical Meas.(A) 4 14 1366

Eliminated Meas.(A) 10 32 2448

Ratio 1 17.54% 7.02% 3.83%

Ratio 2 8.51% 3.30% 2.22%

Table 2.8: Measurement Information about System Redundancy

In Table 2.8, Ratio 1 indicates the proportion of eliminated measurements to whole mea-

surements before modification. Ratio 2 compares the number of eliminated redundant measurements

with the number of critical measurements newly generated. The ratio results in Table 2.8 contain two

important messages:

• The percentage of eliminated measurements among whole measurements continue decrease

with the increase of bus number, which to some extend illustrates the impact of modification

to the original system keeps reducing.

39


• The eliminated redundant boundary measurements create nearly half number of critical mea-

surements near the boundary of each zone. This indicates that the level of redundancy of the

original system, which is added with enough number of pseudo measurements, is high enough

to tolerate the elimination of redundant measurements without producing too many critical

measurements which are not acceptable for a mature system.

2.6.2 Algorithm for Diverged State Estimation

The proposed algorithm aims at solving the state estimation problem when the system

contains highly corrupted measurements, which may drastically bias the state estimates and make

the system divergent. In the previous discussion, all the simulations are based on the assumption that

the state estimation converged. In this section, the diverged scenarios are simulated and discussed

based on numerical solutions.

Now take the IEEE 118 bus system as an example. Suppose a gross error occurred at real

injection at bus 59 in zone 2, when conducting state estimation based on the original measurement

set, the iterative process can not stop because it will meet the termination criteria, as discussed in

Section 2.2.2, that max |∆xk| < ε. Thus, the WLS procedure will finally exceed the iteration limit

and diverge.

By applying the proposed algorithm, all the redundant measurements incident to the

boundary of zones are eliminated. When solving the weighted least squares problem based on the

modified measurement set, according to (2.7), it holds that

∆xk =[G(xk)

]−1HT (xk)R−1r. (2.31)

Note that the measurement error e can spread to measurement residual r through r = Se. Since

after boundary measurement modification, the block matrix S can isolate the errors in one zone from

spreading to other zones, and according to (2.31), only the ∆xk incident to the erroneous zones can

not converge.

The following Table 2.9 show three test cases with divergent simulation results. To show

the divergence, the iteration results of the state estimator are also provided, along with the largest

residuals of each zone and some representative buses’ states.

Table 2.9 indicate that even if the system is under divergence, the modification of boundary

measurements can still block the errors from spreading in 118 bus system. The following three cases

shown in Table 2.10, Table 2.11 and Table 2.12 provide three simulations based on the 14143 bus

40


Table 2.9: Simulation Results of 118 Bus System Divergent Case 1

Case Zone Loc. Status MAMR Z.1 MAMR Z.2 MAMR Z.3 Blocked?

1 2 Inj 59 B 131.954 1218.701 26.277

1 2 Inj 59 A 0.00 1433.767 0.00 Y

2 1 Inj 32 B 488.946 1.813 0.450

2 1 Inj 32 A 816.348 0.001 0.000 Y

3 3 Flow 93-94 B 0.728 1.185 18.861

3 3 Flow 93-94 A 0.005 0.003 32.234 Y

system. Since it is not easy to show all the states of 14143 bus system, only the large residuals of

certain zones are given.

Table 2.10: Simulation Results of 14143 Bus System Divergent Case with Error in Zone 15


15 45.901 Y 41.401 Y

29 2.926 Y 0.013 N

32 1.172 Y 0.003 N

23 0.325 Y 0.002 N

20 0.240 Y 0.001 N



20 283.38 Y 787.79 Y

13 4.018 Y 0.007 N

18 3.610 Y 0.004 N

15 2.816 Y 0.002 N

8 0.508 Y 0.001 N

41




35 100.13 Y 102.26 Y

41 5.263 Y 0.003 N

14 0.088 Y 0.001 N

36 0.056 Y 0.001 N

34 0.028 Y 0.001 N

2.6.3 Limitations of the Proposed Algorithm

From all of the above discussions and simulations, we can see that the goal to isolate the

erroneous zone state estimation is achieved by manipulating the boundary measurements. However,

there are still limitations and drawbacks which can not be ignored:

• The proposed algorithm generates a number of critical measurements, which makes the incident

state estimates more vulnerable facing bad data on the boundary critical measurements. In

other words, the proposed algorithm sacrifices the robustness of state estimation against bad

data on boundary critical measurements, for improving the robustness of these on internal

measurements.

• The number of critical measurements is increasing with the increase of system size, which

leads to a decrease of state estimation robustness.

• The process to identify and delete redundant boundary measurements one at a time is compu-

tationally expensive.

2.7 Conclusion

For large interconnected systems, errors in a certain zone can affect not only this zone

itself, but also several zones nearby or even the whole system. This impact is even worse if the

measurements are highly corrupted that can make the state estimator diverge. Based on the principle

theory of state estimation and residual analysis, a strategic measurement reconfiguration and design

approach is proposed in this chapter. The main goal of this work is to allow isolation of the impact of

measurement errors in different zones so that zonal results of state estimation for very large scale

42


networks can be obtained in a manner remaining insensitive to errors in other zones of the system.

Monte Carlo simulations are used to verify that the use of proposed measurement design will ensure

that errors in measurements of a given zone will not influence measurement residuals in other zones.

However, as discussed in Section 2.6.3, there are still a number of drawbacks and limitations

of the proposed algorithm. Therefore, in next chapter, several more robust algorithms are proposed

to address these problems, especially when the state estimator fails to converge.

43

Chapter 3

Multi-Area State Estimator (MASE)

3.1 Background

3.1.1 Divergence of State Estimator

Increased penetration of renewable sources coupled with connection of unconventional

loads to power grids force the operators to model and monitor the grid more closely and reliably. Large

scale system state estimators (SE), as discussed in Section 2.2, have been successfully implemented

and used by numerous control center operators for this purpose. However, even the most reliable

estimators are vulnerable to occasional failure in detecting measurement, parameter and model

errors. Even worse, it is also possible that state estimator fails to converge due to gross errors in

network model, parameters or measurements. This chapter focuses on such cases when SE fails to

converge because a certain area experiences convergence issues which may be due to bad data, loss

of measurements, topology errors, network parameter errors and extreme operation conditions etc.

For the Gauss-Newton algorithm which is one of the traditional algorithms to solve WLS

state estimation problem, the convergence is not guaranteed, not even local convergence. A converged

solution may fail to be reached when the starting point is far away from the globally optimal solution

which may be severely biased by very large residuals. Besides, under severe operation conditions

when the gain matrix is ill-conditioned, it is also possible that SE diverges. What worse, the traditional

bad data detection techniques, such as the ones described in Section 2.2.4, will also fail because no

state estimate will be available when the estimator fails to converge. In order to obtain a solution

even when the conventional estimator fails to converge, there are several methods proposed in the

literatures.

44

CHAPTER 3. MULTI-AREA STATE ESTIMATOR (MASE)

In Section 3.1.1.1 and 3.1.1.2, two existing methods to deal with convergence issues of the

Gauss-Newton algorithm in state estimation, named step size control and trust region method, are

described and analyzed. Some other methods such as least absolute value (LAV) and least median

squares (LMS) are robust regression methods which are not sensitive to outliers in the measurement

set. However, due to high nonlinearity and potential inaccuracy of the measurement model for a

large scale system, it is not guaranteed to filter out all errors, find the global minimum and obtain

converged solutions.

As a consequence, considering the drawbacks of existing methods, one still needs a reliable

and computationally robust alternative to deal with the divergence issues of the state estimator.

3.1.1.1 Step size control

For the least squares problem, the Gauss-Newton step can be modified to satisfy an

acceptability criteria. The update equation given by (2.6) can be modified as:

xk+1 = xk + α ∆xk. (3.1)

The choice of α ensures the convergence criteria J(xk+1 < Jxk) and determines the convergence

behavior. α should be within the interval αmin < α < αmax, that the upper bound ensures the

backtracking (inner) loop will terminate with an acceptable step, and lower bound ensures that steps

will not be excessively small. The choice of αmax and αmin is problem dependent, typical choices

being 0.5 and 0.1. Some possible choices are described below:

• α can be chosen as a fixed small number, usually 0.1, to reduce the step size.

• α can be chosen as α = 1− e−βk, where β could be a fixed number such as 3, where k is the

iteration index.

• As provided in [17], α can also be obtained by calculating the algebra of some polynomial

constraints, as

α =−g(xk)∆xk

2[J(xk + ∆xk)− J(xk)− g(xk)∆xk].

Regretfully, all the methods discussed above have shortcomings as listed below:

• They do not guarantee convergence all the time.

• They may require a very large number of iterations to converge.

45


• The chosen step may achieve relatively little reduction in the objective function, compared to

other steps of the same length but in different directions.

3.1.1.2 Trust Region Method

As discussed in the previous section, the methods based on modification of Newton’s step

do not guarantee convergence of the state estimator. Trust region method, to the contrary, ensures

convergence of the least squares problem. It is a robust implementation of the algorithm whose origin

lies in the work of Levenberg [18] and Marquardt [19]. The trust region is a region in the problem

space in which we can trust that a quadratic model is an adequate model of the objective function.

The details of implementation procedure can be found in [17].

In finding a solution for the state estimation problem, the trust region method-based state

estimator is found to be very reliable under severe conditions. However, the estimation solutions

may carry significant biases, which will depend on operating conditions and types of bad data. Thus,

despite the availability of a converged estimation solution, this may not have any practical use for

the operator. Besides, its excessive computational burden and bias of the estimated states limit its

practical implementation.

3.1.2 Proposed Algorithms

In Chapter 2, a boundary measurement modification algorithm is proposed to address

the error spreading problem. However, it has limited applicability due to its drawbacks. But the

idea of decoupling the system into areas provides an alternative insight to deal with errors in zones.

Considering the fact that when the state estimator fails to converge it is usually due to such an error

in a specific part of the grid, yet due to the integrated formulation of the state estimation problem, it

will cause divergence of the overall state estimator. Hence, in a multi-area large scale power grid,

the system-wide state estimator’s performance will be limited by the area with the least reliable and

accurate model and measurements in the overall system.

Different from the global algorithms as described in Section 3.1.1.1 and 3.1.1.2, the

algorithms proposed in the following sections focus on an alternative scheme based on the earlier

work about multi-area two-level state estimator (2LvSE) [20, 21]. This method involves a two stage

solution where the first level obtains individual area solutions, followed by a coordination level where

these solutions are combined and synchronized by a central processor. The purpose of the second

46


level is to synchronize individual area reference angles as well as to detect and remove any errors

that were missed by individual area solvers.

The main idea which is to use the first stage of the 2LvSE algorithm in order to identify

and isolate the area containing the root cause of divergence, is presented in [22, 23]. In the proposed

approach, the system is strategically partitioned into areas according to some predefined criteria.

Then, each of the partitioned area SE solutions are independently attempted to be obtained. If any

of the area SEs fail to converge, these areas are removed from the network model and the solutions

from all other areas are reconciled and synchronized to reach a partial solution for the overall system.

Efficient implementation of this approach involves some challenges related to strategic partitioning

while maintaining area observability, details of which will be explained later.

Alternatively, a state estimator that recursively partitions zones can also be implemented,

which has more advantages than the centralized state estimator and two-level state estimator. The

erroneous zone which is detected from the first stage of two-level state estimator can be split into

two smaller zones. By conducting state estimation on these two zones separately, such error can be

located again. By repeatedly applying this divide and estimate process, a small enough partition can

finally be reached which will fail to converge. This will be referred as the ’recursive partitioning state

estimator (RPSE)’ which will provide the state estimation solution for the largest possible sub-area in

the entire system. This approach not only always ensures a converged state estimation solution for the

largest subset of buses in the system, but also facilitates possible implementation of CPU-intensive

solution algorithms on the isolated small area in order to detect and identify the source of divergence.

There are several technical issues that need to be addressed before such a scheme can

be implemented. These include but not limited to: partitioning method, system observability, area

synchronization, optimality of the solutions, information security, etc. These are addressed in various

different ways which will be described in the sequel. This chapter will introduce the developed

algorithms and the simulation results obtained by implementing them on some test systems as well

as several large scale systems. This chapter is arranged as follows:

Section 3.2 reviews a number of literatures discussing about the multi-area state estimation.

The hierarchical and the decentralized MASE approaches are described, discussed and classified

based on the criteria proposed in [24].

Section 3.3 is concerned about proper definition of the areas so that each area will be

composed of topologically connected buses. The area definitions existing in network data files do

not always have this property i.e. buses in a certain area may not form a connected graph. This issue

is addressed by strategically assigning boundary buses and branches to neighboring areas.

47


When a system is partitioned in to several interconnected areas, the state estimation

observability criteria may be violated due to altered measurement allocation. In Section 3.4, the

observability issue is described and discussed. Three schemes, including all pseudo injection

placement, critical injection placement and area boundary manipulation, are proposed, analyzed and

compared.

After the solving the problem of area manipulation and observability, the two-level state

estimator is proposed in Section 3.5. It first starts with the nomenclature and then provides a detailed

description of the algorithmic steps involved in the two-level state estimator.

In Section 3.6, the recursive partitioning state estimator (RPSE) is introduced. The

proposed partitioning technique for RPSE is presented first, followed by the description of the overall

estimation scheme.

Section 3.7 describes the graphical user interface which is developed in order to facilitate

repeated execution of the software using different save cases. It also serves as a user manual for the

developed software.

In Section 3.8, the proposed two-level state estimator and recursively partitioned state

estimator are validated at first. Then, several divergent scenarios including measurement and topology

error are simulated to testify the algorithms. At last, several practical scenarios where commercial

state estimator fails to converge are tested and the results are discussed. Followed by the remarks and

conclusions for the MASE in Section 3.9.

3.2 Literature Review of MASE

Multi-area state estimation (MASE) is certainly not new as evident from numerous publi-

cations since 1970s. MASE has drawn the attention of researchers [25] first mainly to address the

issues of computational efficiency. Since various methods proposed in the literature published before

year 2010 are discussed in an excellent review in [24], thus in this section, only the publications after

are discussed in detail.

As stated in [24], methods proposed for MASE can be differentiated with respect to the

following implementation categories:

• Area overlapping level: non-overlapping, tie-line overlapping and extended overlapping.

• Computing architecture: hierarchical and decentralized.

• Coordination scheme: at the SE level, at the iteration level and hybrid.

48


• Measurement synchronization: with or without using phasor measurement units (PMUs).

• Solution methodology.

Recently, MASE was investigated further by several researchers [26–34]. Most of them

assume availability of PMU measurements to synchronize the results of each area [26, 28–32].

In [26], the authors proposed a two-level scheme that removes the bad data and topology

errors, which are major problems today at the substation level. The first part of the paper describes

the layered architecture of databases, communications, and the application programs that are required

to support this two-level linear state estimator. And the second part describes the mathematical

algorithms that are different from those in the existing literature. As the availability of phasor

measurements at substations will increase gradually, this paper describes how the state estimator can

be enhanced to handle both the traditional state estimator and the proposed linear state estimator

simultaneously.

[27] presents a fully distributed state estimation algorithm for wide-area monitoring in

power systems. Through iterative information exchange with designated neighboring control areas,

all the balancing authorities (control areas) can achieve an unbiased estimate of the entire power

system’s state. In comparison with existing hierarchical or distributed state estimation methods, the

novelty of the proposed approach lies in that: 1) the assumption of local observability of all the

control areas is no longer needed; 2) the communication topology can be different than the physical

topology of the power interconnection; and 3) for DC state estimation, no coordinator is required for

each local control area to achieve provable convergence of the entire power system’s states to those

of the centralized estimation.

In [28], distributed state estimation methods are treated under a unified and systematic

framework. The proposed algorithm is based on the alternating direction method of multipliers. It

leverages existing state estimation solvers, respects privacy policies, exhibits low communication

load, and its convergence to the centralized estimates is guaranteed even in the absence of local

observability. Beyond the conventional least-squares based SE, the decentralized framework ac-

commodates a robust state estimator. By exploiting interesting links to the compressive sampling

advances, the latter jointly estimates the state and identifies corrupted measurements.

The authors of [29] proposed a method to obtain a composed single system state of different

monitored areas with local estimations and external equivalent models, which neither require the use

of a coordination agent nor requires extensive data interchange. Phasor measurement unit (PMU)

measurements taken from system boundaries are used to provide updated equivalent networks as

49


well as an inter-area synchronism. The approach allows testing erroneous external models directly in

the state estimation framework. The detectability of bad data in interconnection measurements are

also preserved.

In [30], the authors present a fast state estimator and a corresponding bad data processing

architecture aimed at improving computational efficiency and maintaining high estimation accuracy

of existing state estimation algorithms, simultaneously. The conventional and phasor measurements

are separately processed by a three-stage SE method and a linear estimator, respectively. Then,

the derived estimates are combined using estimation fusion theory. To eliminate computational

bottlenecks of the conventional bad data processing scheme, bad data identification is moved before

the second stage of supervisory control and data acquisition based SE, and bad phasor measurements

or bad conventional measurements in the phasor measurement units observable area are identified

and processed all at once, which can dramatically reduce the implementation time, especially for

large-scale networks with multiple bad data.

[31] proposes a fully distributed robust bilinear state-estimation method that is applicable

to multi-area power systems with nonlinear measurements. The distributed bilinear formulation

of state estimation problems is developed. In both linear stages, the state estimation problem

in each area is solved locally, with minimal data exchange with its neighbors. The intermediate

nonlinear transformation can be performed by all areas in parallel without any need of inter-regional

communication. This algorithm does not require a central coordinator and can compress bad

measurements by introducing a robust state estimation model.

In [32], an efficient and accurate method for updating measurements weight in a distributed

multi-area power system state estimation is proposed. In the proposed scheme, which includes pseudo

measurements to enhance the measurement redundancy at the local and the global levels, the devel-

oped weight update approach along with the weight adjustment equations for pseudo measurements

can improve the accuracy and convergence speed of the measurements weight updating.

In [33], a matrix splitting technique is used to carry out the matrix inversion needed for

calculating the Gauss-Newton step in a distributed fashion. In detail, the authors proposed a fully

distributed Gauss-Newton algorithm for state estimation of electric power systems. At each Gauss-

Newton iteration, matrix-splitting techniques are utilized to carry out the matrix inversion needed for

calculating the Gauss-Newton step in a distributed fashion. In order to reduce the communication

burden as well as increase robustness of state estimation, the proposed distributed scheme relies

only on local information and a limited amount of information from neighboring areas. The matrix-

splitting scheme is designed to calculate the Gauss-Newton step with exponential convergence

50


speed.

In [34], a hierarchical MASE method is proposed by exchanging the sensitivity functions

of local state estimator rather than boundary measurements. Instead of exchanging boundary

measurements or state estimates, the proposed technique is based on exchanging the sensitivity

functions of local state estimators. The main benefit of the proposed scheme is the improved

convergence speed, which also reduces the amount of information exchange required.

3.3 Area Definition and Manipulation

3.3.1 Zonal Framework

Consider a system with N buses which is composed of n interconnected areas. It is

assumed that each bus belongs to one and only one area, whereas a system branch may be inside an

area or may be connecting two areas. This will lead to the definition of the set of all system buses S

which will contain the sets of Si each denoting the buses for the ith area. Areas are interconnected

by tie lines whose terminal buses belong to different zones. Those terminal buses belonging to an

area i will form the set of boundary buses of the ith zone denoted by Sbi . All other buses in zone i

will be called internal buses and denoted by Sinti . The set of all the boundary buses in a system will

be denoted by Sb. All buses which are connected to the buses in set Sbi but do not belong to set Sinti

will be defined as the external buses for the area i and will form the set Sexti .

The following relationship holds for i = 1...n :

• S = S1 ∪ S2 ∪ ... ∪ Sn

• Si ∩ Sj = ∅ for all j 6= i and j = 1...n

• Si = Sbi ∪ Sinti

• Sb = Sb1 ∪ Sb2 ∪ ... ∪ Sbn = Sext1 ∪ Sext2 ∪ ... ∪ Sextn

Based on the above decomposition, the state variables of the ith area can be defined as follows

• The vector xbi at the boundary buses in set Sbi .

• The vector xinti at the boundary buses in set Sinti .

• The vector xexti at the boundary buses in set Sexti .

51


The state vector of the ith area is now defined as the composite vector xi = [xbi , xinti , xexti ],

whose dimension is ni. This decomposition scheme includes not only the states of that zone but also

part of the states belonging to its immediate neighbors. Hence, some of the states will be estimated

simultaneously by two neighboring zone estimators. One specific example is shown in Figure 3.1.

Figure 3.1: Example illustrating bus definitions

3.3.2 Connectivity of the System

Consider a power network and its associated graph defined by G = (V,E), with V buses

and E branches. Partitioning of G into smaller graphs with certain desired properties is a problem

well investigated in the literature of NP-hard problems. In the field of power systems, partitioning

techniques are widely applied, some examples of which are given below:

• multi-stage state estimation [28, 34];

• decentralized optimal power flow [35, 36];

• unit commitment [37];

• parallel system restoration and back protection;

• dynamic behavior analysis and model reduction [38, 39].

Most large scale power grids already have well defined control areas, which can be used as

a starting point for partitioning the system. The main problem with the existing membership of buses

52


to areas is that due to geometrical, electrical, or security reasons, buses belonging to the same areas

are not always interconnected.

A given power system can be partitioned into several areas based on different criteria. This

task can be done manually or by state of art techniques such as machine learning. One can also

use of the already defined areas and zones of the utility systems based on geography and company

ownership of substations. ISOs or regional transmission organizations (RTOs) usually coordinate

and monitor the operation of large numbers of electric utilities within a single or multiple US states.

Therefore, a large scale system is composed of several control zones, each having its own data base,

application programs and measurement scheme. This makes the partitioning of the whole system

easier, because one can follow the predefined zones and make minor modifications. However, due

to geometrical, electrical, or security reasons, buses belonging to the same zone are not always

interconnected, i.e. the network may not always be a connected graph.

As an example, a small part of the 16794-bus system which has disconnected bus groups

belonging to the same zone is shown in Figure 3.2.

Figure 3.2: Disconnected bus groups of the same zone

To avoid confusion, in the sequel, ’zone’ will be used for the subsystems predefined in

network raw file and ’area’ will be used for the subsystems after manipulation and to be solved by

the multi-area state estimator. These zones in Figure 3.2 have the following properties:

• A zone consists of several disconnected subgraphs, for example the 334-bus subgraph of Zone

1.

• There are some isolated small islands, but they are not very far from each other, for example

the 6-bus subgraph of Zone 1.

53


• Some zones only contain a very small number of buses.

Since individual zonal state estimator can only solve SE for area without physical islands,

some of the zones need to be expanded, shrunk or even eliminated. Therefore, an area modification

procedure with the following objectives needs to be developed:

• Ensure network connectivity of every modified area by merging buses of small subgraph into

nearby zones.

• Reduce the number of buses that need to be assigned.

• Reduce the number of buses in larger zones and increase the number of buses in smaller zones.

3.3.3 Area Modification Algorithm

Steps of the proposed area modification algorithm are given below:

Step 1: Identify the existing zones by the set of buses belonging to each zone for all n zones

Z1, Z2, ..., Zn.

Step 2: Order the zones starting with the largest as A1, A2, ..., An. Initialize area index k = 1 and

connectivity index for each area as c1 = c2 = ... = cn = 0.

Step 3: For area k, identify all disconnected subgraphs as follows:

3.1: begin at any arbitrary bus of the graph;

3.2: proceed from that bus using breadth-first search, counting all nodes reached;

3.3: once the graph is fully traversed, if the bus count is equal to the number of buses of the

graph, declare the graph is connected. Otherwise store the counted buses as a subgraph

and then select a bus that has not been counted and go back to the step 3.2.

If the area is connected, let ck = 1 and go to Step 7. Else go to Step 4.

Step 4: For nk subgraphs in area Ak, sort by their sizes in decreasing order and denote as Ak,1, Ak,2,

. . . , Ak,nk. If the size of Ak,1 is larger than an area size threshold NATH , then let subgraph

index j = 2 (this indicates that all subgraphs need to be processed except Ak,1; otherwise

let j = 1 and ck = 1.

54


Step 5: Obtain all the buses that directly connect but not belong to the buses in the subgraph Ak,j .

These neighboring buses belong to other areas. Select the area that contains the most number

of these buses and merge subgraph Ak,j to this area.

Step 6: If j < nk, let j = j + 1 and go back to Step 5. Else go to Step 7.

Step 7: If k < n, let k = k + 1 and go back to Step 3. Else go to Step 8.

Step 8: If all areas are connected, i.e.∑n

i=1 ci = n, go to Step 9. Otherwise go back to Step 2.

Step 9: Store the resulting areas and stop.

The above algorithm enables a simple partitioning of the system and creation of a connected

graph. In Step 2 and 3, areas are ordered according to their size and the modification begins from

the largest area, yielding evenly distributed area sizes. Once the disconnected subgraphs of this area

are merged by nearby areas, these areas may be automatically re-connected, thus may no longer be

modified.

For a very small area, the change of its boundary topology is relatively significant, and

may consequently influence state estimation results much. Therefore, in Step 4, an arbitrary area size

limit is necessary to control the size of smallest area.

In Step 5, neighboring buses of one subgraph are specified and the areas they belonged to

are recorded. These areas are candidates that subgraph will be merged into and the area with most

connections to the subgraph is the desired one. This scheme ensures that each assignment makes

least change of the local topology, i.e. least interconnections between modified areas.

Figure 3.3: Modified areas for the network of Figure 3.2

In Figure 3.2, a small part of a large system is shown as an example. Bus 319 in Area

1 and bus 2440 in Area 11 are connected to several other buses but not shown. Area 1 contains 5

disconnected parts: the largest subsystem has 334 buses and rest subsystems only have a few buses.

55


Figure 3.4: Augmented areas for the network of Figure 3.3

According to the modification rules above, the modified areas are shown in Figure 3.3. And the

augmented area definition is shown in Figure 3.4.

3.4 Area Observability Analysis

Prior to the execution of individual area state estimation, network observability needs to be

checked for each area. Given a set of measurements and their locations, the network observability

analysis will determine if a unique set of estimates can be found for the system states [2]. When

a network is decomposed into several areas, the boundary information of each area needs to be

defined. For the first level state estimation as discussed above, injection measurements incident to

the external buses in an area can no longer be used and have to be removed, which may lead to the

loss of observability. Take area 20 in Figure 3.5 as an example, the states of internal buses, along

with its external buses 3197 and 2967, are estimated by first level state estimation. The injections at

bus 3197 and 2967 however are unusable and have to be removed. This causes branch 8811 to 2967

to become an unobservable branch. Since the entire system is observable, this loss of observability

for individual areas after partitioning is mainly due to the removal of injections at external buses.

There are several methods to mitigate this problem, which are introduced and discussed in

this section. Comparisons between different methods based on numerical simulations are provided in

Section 3.8.

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Figure 3.5: Example of boundary topology and measurements

3.4.1 All Pseudo Injection Placement

One straightforward way to ensure area observability is to blindly assign pseudo injections

to all the external buses of this area whose original injections, if exist, have to be removed. This

method is abbreviated as API. The values and weights of the pseudo injections need to be assigned

carefully. Using the forecast loads may improve the quality of state estimation solutions. The weights

should not be larger than the smallest weight used for all actual measurements. Though the smaller

the better, excessively small weights may increase the singularity level of the gain matrix in state

estimation, thus leading to ill-conditioning problem. Numerical simulations are provided in Section

3.7.2 to evaluate the impact of different pseudo weights.

Since the introduced pseudo injections are not all critical, the states estimated by first

level may have different level of biases depending on the number and assigned weights of pseudo

injections. In practice, considering the measurement noise, the impact of pseudo injections can

be ignored when their weights are relatively small. Also note that the states of all external buses

estimated in first level are used as inputs to the second level, with low weights.

3.4.2 Critical Pseudo Injection Placement

Exploiting the property that critical measurements do not spread measurement errors [40],

one can strategically introduce only critical pseudo injections to ensure observability. The algorithm

is briefly outlined below and more detailed description of numerical observability restoration method

can be found in [2, 41].

57


Step 1: Construct real power measurement Jacobian matrix H and corresponding gain matrix

G = HTH .

Step 2: Apply Cholesky factorization to G and find first zero pivot during factorization.

Step 3: Add a pseudo injection at the bus corresponding to the zero pivot. A zero value with an

arbitrary weight (preferably very small) will be used for the pseudo measurement.

Step 4: Repeat Step 2 and 3 until no more zero pivots are encountered and factorization is completed.

The above algorithm uses the P -θ observability which considers the decoupled linearized

measurement model. It assumes that real (P ) and reactive (Q) measurements are paired. Note

that the pseudo injections to be added in Step 3 are also added in pairs. By introducing pseudo

injections following the above procedure, the resulting system is barely observable where all pseudo

injections are critical in sense of P -θ observability. Such scheme is denoted as critical pseudo

injection placement considering decoupled model observability (CPID).

When considering complete measurement model, the assigned pseudo injections following

CPID may no longer be critical due to the coupled real and reactive power model. To consider the

full model observability, one can construct Jacobian matrix using all measurements instead of only

real ones in Step 1. For Step 3, a real or reactive pseudo injection is added one at a time. Usually,

critical pseudo injection placement considering full model observability (CPIF) requires less or equal

number of pseudo injections than CPID.

A similar method has been presented in [21]. It avoids adding pseudo injections instead of

manipulating the decomposition of the gain matrix. When the inverse of gain matrix is required for

the WLS estimation, an incomplete Cholesky factorization is carried out to deal with the zero pivots.

However, the complexity of matrix factorization is N3 where N is the size of the matrix. Thus, such

scheme is computationally expensive when the system size is relatively large. CPID and CPIF also

suffer from the computational burden as discussed above. However, if the network measurements

and topology do not change, the set of critical pseudo injections will be fixed and can be determined

only once off-line.

3.4.3 Area Boundary Manipulation to Reduce Pseudo Injections

Some methods to assign pseudo injection measurements based on existing area partition

are discussed above. It is also possible to manipulate the boundary topology of two neighboring

areas to reduce the number of required pseudo measurements.

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Before introducing the area manipulation algorithms, an important property of boundary

observability is discussed first. Consider the boundary buses between areas 20 and 14 in Figure 3.5 as

an example. States of bus 3197 and bus 2967 are to be estimated in the first level state estimation for

area 20 where injection measurements incident to these two buses will be unusable and thus removed.

Since there is a flow measurement along the tie-line 8808 to 3197, this branch will still be observable

and state of bus 3197 can be determined. However, this will not be the case for bus 2967 without any

incident flow measurement. It is safe to say that, if the external bus of a partition of an observable

system has an incident flow measurement, state of this bus can be estimated. For a real power system,

most of the buses have incident injection measurements, either ordinary injections, pseudo loads or

zero injections. In such sense, the following discussion is based on the assumption that all buses have

incident injections. The observability of boundary branches can be then determined according to the

existence of incident flow measurements.

Using the existing area definitions and proper manipulation, an optimized boundary topol-

ogy can be obtained. The key idea is to find possible cut-sets containing branches with flow mea-

surements. The following approaches can be used to modify the boundary between two neighboring

areas:

• Area boundary manipulation with expansion scheme (ABME): for each area, expand its exter-

nal tier till most of branches incident to the buses in outermost tier have flow measurements.

• Area boundary manipulation with redefinition scheme (ABMR): find a new boundary of two

areas such that boundary branches contain more flow measurements.

The expansion scheme can be transformed into a minimum cut problem in the optimization

theory. One parameter M needs to be chosen in advance. It represents the number of external tiers

outside the boundary tier whose buses are candidates for the expansion of the area. Furthermore,

it limits the number of buses to be checked, as well as the size of expansion. Consequently it

avoids searching the new boundary exhaustively for the whole system, and terminates the search

appropriately. As a payoff, the resulting area definition is not globally optimal, but the best within a

limited number of external tiers. The steps of ABME algorithm for one area are given below:

Step 1: For an area i, the external tiers can be defined as:

• Sext:1i consists of the remote end buses of tie-lines incident to the area, i.e. buses that

are directly connected to the boundary buses Sbi but not belong to area i. It is identical

to the definition Sexti in Section 3.3.1.

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• Sext:Mi consists of the buses that are connected to the buses in Sext:M−1i but not in

Sext:M−2i , · · · , Sext:1i and boundary tiers.

Set the default value of M to be 4.

Step 2: Introduce a pseudo source bus ps that it is directly connected to all boundary buses of area i.

Also introduce a pseudo sink bus ps that it is connected to all the buses in Sext:Mi . Branches

connecting pseudo source or sink to other buses are denoted as pseudo branches. Then,

construct an undirected graph G = (V,E) that

• V consists of pseudo source, Sbi , Sext:1i , · · · , Sext:M−1

i , Sext:Mi and pseudo sink.

• E consists of all the branches and pseudo branches incident to buses in V .

Step 3: Assign weights to all the branches. The weight of a branch from bus u to bus v, denoted as

cu,v, equals to

• Wpb for pseudo branch;

• Wwf for the branch with flow measurement;

• Wwof for the other branches.

A typical choice of the weights is Wpb = 10000,Wwf = 1 and Wwof = 100.

Step 4: Formulate the minimum cut problem from ps to pt. A cut C = (V S, V T ) is a partition

of V such that ps ∈ V S and pt ∈ V T . The total weights of the branches that connect

the source part V S to the sink part V T is defined as c(V S, V T ) =∑

(i,j)∈E ci,jdi,j where

di,j = 1 if i ∈ V S and j ∈ V T , and 0 otherwise. Therefore, the minimum cut problem can

be formulated [42] as to minimize the total weights, shown in follows:

minimize c(V S, V T )

subject to du,v − du + dv ≥ 0, (u, v) ∈ Edv + dps,v ≥ 1, (ps, v) ∈ E−du + du,pt ≥ 0, (u, pt) ∈ E

where du = 1 if u ∈ V S, and 0 otherwise.

Step 5: Use linear programming to solve the minimum cut problem. Solution consists of the

branches whose removal will split the subsystem into two parts. These branches are also

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called minimum cut branches. Since the graph is connected and undirected, the weights

are positive and rational, such branches can always be found to split the graph. If there

are multiple solutions with the same objectives, the one with the least number of buses in

expansion will be chosen.

By applying the above procedure to an area, minimum cut branches, all buses belonging

to pseudo source side V S and sink side V T can be obtained. The buses of minimum cut branches

in the sink side are defined as the new outermost external tier, denoted as xext:oi . The definitions of

internal and boundary buses of the area remain the same. All other buses in the pseudo source side,

along with xext:oi , constitute a new external bus set, denoted by xexti . An example of a subsystem

is shown in Figure 3.6. By manipulating the boundary and external buses of area 1, results of new

area topology are given in Figure 3.7. The minimum cut-set is highlighted in blue dashed line in

Figure 3.7. All the buses to the left of red dashed line and outside the original area 1 are external

buses. A numerical validation of the proposed ABME method is provided in Section 3.7.2.

Figure 3.6: Area Boundary Manipulation: original

For the area redefinition scheme (ABMR), which is to define a new boundary of two

neighboring areas, it can set the new ABMR boundary according to the minimum cut-set branches

obtained from ABME. For the subsystem in Figure 3.7, the minimum cut-set branches can be chosen

as the new boundary branches between the two areas. As a result, area 1 expands and area 2 shrinks.

This procedure can be repeated for all areas in the system, resulting an area definition whose boundary

branches have more flow measurements. However, for second level state estimation, ABME has

more overlapped buses whose states are estimated twice, improving the quality of estimates. Results

of ABME will be discussed in the simulation section.

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Figure 3.7: Area Boundary Manipulation: expansion

3.5 Two-Level State Estimator

3.5.1 First and Second Level Formulation

There are several ways to implement multi-area state estimator as classified in [24]. The

two-level state estimator formulation is a hierarchical scheme based on overlapping tie-lines and

SCADA measurements. It consists of two part: first level state estimation and second level state

estimation.

In the first level, each area will have its own state estimator, which will process locally

acquired measurements along with any available boundary measurements. Estimated states from

each area along with its own boundary measurements will be telemetered to the control center where

the second level estimator will be executed. The second level state estimator will be responsible for

the estimation of the coordination vector including states of boundary buses and zonal reference

angles.

3.5.2 First Level Formulation

In the first level estimation, each area process state estimation individually. The vector of

available measurements zi in area i includes injection, flow and voltage measurements incident to

internal buses, boundary buses and external buses of zone i, except for the injection measurements

incident to the external buses. Similar to centralized state estimator as discussed in Section 2.2, by

introducing weighted least squares method to solve the above equation, the estimation problem for

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area k can be stated as

minxk

Jk =

mk∑i=1

Wk,ir2k,i, (3.2)

subject to zk,i = hk,i(xk) + rk,i, i = 1, · · · , mk, (3.3)

whereJk : objective function of area k;

xk : state variables of area k, including internal buses xintk , boundary buses xbkand external buses xextk ;

zk,i : ith measurement of area k;

Wk,i : weight of ith measurement of area k;

rk,i : residual of ith measurement of area k;

mk : the number of measurements in area k.

We also have the measurement function for area k:

zk = hk(xk) + ek (3.4)

wherezk : mk × 1 measurement vector of area k, including injection, flow and voltage

measurements incident to the buses in this area;

e : mk × 1 measurement error vector of area k, having a Normal distribution

with zero mean and covariance Rk;

h : nonlinear measurement function of are k.

It is assumed that it is the responsibility of individual areas to make sure that there is

enough redundancy in the area measurement set to allow bad data identification and elimination

for all internal area measurements. Actually, since almost every bus has injection measurement or

pseudo load, the internal part of each zone is observable. This means that at the completion of the

first level estimation step, the internal state estimate xint for each area can be assumed to be unbiased.

On the other hand, as discussed in Section 3.4, if there are not sufficient measurements incident to

the external buses, then the associated states will not be observable and will be ignored.

3.5.3 Second Level Formulation

The second level estimator will coordinate the individual zonal estimates and ensure that

all bad data associated with the boundary measurements will be identified and corrected. The states,

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which are to be estimated in the second level, will be defined as

xs =

xb1...

xbn

u2

...

un

where

xbi : boundary bus states of ith area;

ui : phase angle of the slack bus of the ith area with respect to the slack bus of

area 1, where area 1 is arbitrarily chosen to be the reference with u1 = 0.

For the measurement vector zs, it holds

zs =

zu

xb

xint

xext

where

zu : boundary measurement vector, which includes the tie-line flow measure-

ments and injection measurements at all boundary buses;

xb : first level state estimates for the boundary buses. These are treated as pseudo-

measurements by the second level state estimator. The covariance of these

pseudo-measurements is obtained from the covariance matrix of the states

Rx,b,i;

xint : state estimates of the internal buses which are directly connected to the

boundary buses. These are also treated as pseudo-measurements by the

second level state estimator. The covariance of these pseudo-measurements

is obtained from the covariance matrix of the states Rx,int,i;

xext : first level state estimates for the external buses. These are treated as pseudo-

measurements by the second level state estimator. The covariance of these

pseudo-measurements is obtained from the covariance matrix of the states

Rx,ext,i.

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Therefore, the measurement model for second level state estimator is

zs = hs(xs) + es. (3.5)

Similarly, we can formulate WLS problem as

minimizex

J = rsWirs, (3.6)

subject to zs = hs(xs) + rs. (3.7)

Since the boundary injections will be used also in the second level estimation, the topology

information around those boundary nodes should be provided to the control center. This is the only

’raw’ information that needs to transmit to the coordinator, and in addition, the results of first level

state estimation. This scheme is quite suitable since it meets the security requirements for each

area without forcing them to release details of their internal system topology. As expected, the

effectiveness of second level estimation strongly depends on the incident measurement redundancy

and quality. Note that synchronized phasor measurements in the second level can increase this

redundancy very effectively.

3.5.4 Procedure of Two-Level State Estimator

The proposed layout of the two-level state estimator (2LvSE) is shown in Figure 3.8.

Figure 3.8: Layout of two-level state estimator

The following procedure is implemented to test the performance of multi-area approach

under divergent conditions:

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Step 1: read and parse power flow files and measurement files

Step 2: (optional) add a severe bad data to any measurements

Step 3: (optional) conduct centralized state estimation (CtrlSE) for the entire system

Step 4: parse every zone and apply area modification algorithm

Step 5: add pseudo measurements to ensure observability

Step 6: execute two-level state estimator (2LvSE)

• run first level state estimation on each area in sequential or parallel manner

• collect the results of first level and run second level state estimation

• combine the solutions provided by first and second level

Step 7: (optional) compare results from CtrlSE and 2LvSE.

In both the first and the second level state estimation, to identify the bad data, one can either

carry out the Largest Normalized Residual Test [2] or introduce PMU measurements [43]. Finally, it

should be noted that due to the absence of iterations between the first and second level estimators,

this two-level algorithm will yield identical results as a single integrated estimator only in the absence

of any measurement errors. When there are bad data, even if they are missed by individual area

first level estimators, they will be detected, identified and corrected by the coordinating second

level estimator. Furthermore, if bad data in a given area is so severe that it prohibits convergence of

the area estimator, solutions of such area will automatically be dormant in the integrated solutions,

resulting in a coordination of the converged solutions from the rest areas.

3.6 Recursively Partitioned State Estimator

This section will describe the development of a recursively partitioned state estimator

(RPSE). This is one of the novel achievements and will be shown to have advantages over the

centralized and two-level state estimators developed and discussed above. The main approach is to

detect and identify the erroneous area using the results of the first stage of two-level state estimator,

and then split this area into two smaller subsystems. By conducting state estimation on each of

two subsystems separately, the region containing erroneous factors can be detected. We can then

further split it to two subsystems. By recursively applying this process that partition the erroneous

66


area size by a half once at a time, a small enough subsystem will finally be identified and isolated.

This way, a converged and reliable solution can be salvaged for the rest of system buses except for

a relatively small area which is identified to contain the error. This RPSE will thus provide the

solutions for largest possible portion of the entire system. Besides, it can also facilitate the use of

complex CPU-intensive methods to deal with the divergent subsystem since its size is reduced down

to a very small and manageable level.

3.6.1 Partition Framework

Given a power network, partitioning problem can be defined on a graph G = (V,E),

with V buses and E branches, such that it is possible to partition G into smaller components with

specific properties. This task can be done based on modification of predefined zones, which has been

discussed in Section 3.3.3. It solves the several problems that occurred in the modification

• Buses in a single zone may not be connected

• Some zones are excessively small

• Some zones are not observable.

For the proposed recursively partitioned state estimator, area partitioning will be repeated

multiple times. At the beginning, whole system is partitioned in a way similar to the first level of

two-level state estimator. To obtain larger solvable area, we need to partition the system further into

smaller subsystems. To make the problem simple and easily understood, bi-partitioning is chosen

which will partition the system S into subsystem A and B. Different to the definitions of zonal

information in Section 3.3.1, boundary buses, internal-boundary buses, purely-internal buses and

external buses of each partition are defined as follows.

• Ab1 is the set of boundary buses in A.

• Ab2 is the set of buses that are inside A, directly connected to but not belong to the boundary

bus set Ab1.

• Apint is the set of buses in A but not belongs to Ab1 and Ab2.

• Ab1 ∪Ab2 ∪Apint = A with Ab1 ∩Ab2 = Ø, Ab1 ∩Apint = Ø and Ab2 ∩Apint = Ø.

A similar definition holds for subsystemB. What’s more, since there are only two partitions

in S, the following relation holds:

67


• A ∪B = S.

• A ∩B = Ø.

• Aext is the set of external buses of A and Aext = Bb1.

• Bext = Ab1.

We can define a new partition C as C = Ab1 ∪Bb1. Figure 3.9 illustrates these partitions.

Figure 3.9: Illustration of three partitions of a network

The one tier out systems of subsystems A, B and C are denoted as SA, SB and SC, respectively,

where

• SA = A ∪Aext,

• SB = B ∪Bext,

• SC = Ab1 ∪Ab2 ∪Bb1 ∪Bb2,

• S = Apint ∪Ab2 ∪Ab1 ∪Bb1 ∪Bb2 ∪Bpint

= SA ∪Bb2 ∪Bpint

= SB ∪Ab2 ∪Apint

= SC ∪Apint ∪Bpint

.

Figure 3.10, Figure 3.11 and Figure 3.12 indicate the augmented area of each partition respectively.

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Figure 3.10: Illustration of augmented area A

Figure 3.11: Illustration of augmented area B

Figure 3.12: Illustration of augmented area C

3.6.2 Area Partition Algorithm

Implementation of the proposed state estimator necessitates iterative partitioning of the

network. At each step, the system is to be split into two subsystems with the following requirements

• Two subsystems are needed.

69


• The number of buses in one subsystem is close to that of another, i.e.∣∣∣No. of buses in partA− No. of buses in partB∣∣∣ < bus threshold.

Usually, the threshold is set equal to 10.

• It is preferred that there are more flow measurements on the boundary branches which connect

the two partitions.

There are many approaches proposed for graph partitioning, as listed below.

• Kernighan-Lin algorithm [44]. It is a heuristic algorithm for finding partitions of graphs. The

algorithm attempts to find an optimal series of interchange operations between elements of A

and B that minimize the sum of weights of the subset of edges that cross from A to B.

• Fiduccia-Mattheyses algorithm [45]. This is an improvement for KL algorithm. It solves the

Hypergraph bi-partitioning problem with consideration of unbalanced partitions and weights

of vertexes.

• Methods solving Balanced Connected Partition (BCP) problem. However, as discussed

in [46, 47], the BCP problem which aims to equalize the sum of vertexes’ weights of each

partition, is an NP-complete problem that can not be solved in polynomial time.

• METIS algorithm from Karypis and Kumar. It works well on low dimensional graphs, but for

large scale system, it may fail.

• Spectral method. It has very good performance on some specific type of networks, such as

dumbbell graph or other weakly connected graph. For the graph with high connectivity, it may

not provide the desired solution unless some modifications are made to the method.

• Genetic algorithm. As a popular general heuristic method, it applies to any type of problems.

However, it does not guarantee the quality of the solutions, as well as the running time.

The first two algorithms do not guarantee every partitioned area to be connected, while the

BCP problem aiming to equalize the sum of vertices’ weights for each partition, is NP-complete,

hence can not be solved in polynomial time. Thus, these methods are not computationally practical

for partitioning large scale power grids.

In this section, we use Kernighan-Lin algorithm with weighted branches (edges). A brief

description of this approach is given below. Given an undirected weighted graph G, with vertex set

70


V , the goal of the algorithm is to partition V into two disjoint subsets A and B of equal or nearly

equal size, in a way that minimizes the sum of the weights of the subsets of edges that cross from

A to B. The algorithm maintains and improves a partition, in each pass using a greedy algorithm

to pair up vertices of A with vertices of B, so that moving the paired vertices from one side of the

partition to the other will improve the partition. After matching the vertices, it then performs a subset

of the pairs chosen to have the best overall effect on the solution quality. Note that the KL algorithm

does not guarantee the connectivity of the partitioned subsets, therefore, a modification of the buses

are necessary, which is similar to the solution of the first problem described in Section 3.3.3.

According to experience based on simulations, the weights of branches with flow measure-

ments are chosen to be 0.1 and the rest are 10.

3.6.3 Recursively Partitioned State Estimation (RPSE) Algorithm

The developed algorithm for recursively partitioned state estimation (RPSE) will be de-

scribed in this section. The approach is based on the premise that state estimators typically fail to

converge due to one or several local divergent factors. If areas which are strongly affected by the

divergent factors can be identified and removed from the network model, a converged solution for

the remaining part of the system can still be obtained. So, let us assume that the centralized state

estimator fails to converge for a certain case, and one of the area state estimators also diverges during

the first stage of executing a two-level state estimator. Let the diverged area be area S. Then, the

recursive partitioning procedure will be implemented as follows:

Step 1: Let the diverged zone be S. If the size of S is smaller than the termination threshold, go to

step 5 with Starget = S, else go to step 2.

Step 2: Partition area S into two areas A and B according to the techniques discussed in Section

3.6.2. Augment A and B with one-tier-out buses to SA and SB. Construct area C and its

one-tier-out augmentation SC.

Step 3: Solve SE for SA and SB that:

• If SE diverges for SA but not for SB, go to step 1 and substitute S with SA.

• If SE diverges for SB but not for SA, go to step 1 and substitute S with SB.

• If SE diverges for both SA and SB, go to step 4.

• If SE converges for both SA and SB, go to step 5, set Starget = SA⋃SB.

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Step 4: Solve SE for area SC.

• If SE diverges, go to step 1 and substitute S with SC.

• If SE converges, go to step 5 and set Starget = SA⋃SB.

Step 5: Terminate the ’Partition - SE’ loop from step 2 to step 4. The target area Starget will be the

smallest unsolvable area.

Step 6: The largest solvable area can be determined by eliminating the unsolvable area Starget from

the entire system.

Step 7: Solve SE for the largest solvable area.

Therefore, overall recursively partitioned state estimation algorithm can be illustrated as

follows.

Step 1: Solve the centralized state estimation for the entire system as discussed in Section 2.2. If the

centralized SE converges, terminate. Else go to step 2.

Step 2: Obtain zonal system and measurement information as discussed in Section 3.3.1.

Step 3: Solve the first level estimation (first stage of two-level state estimator) for each individual

area.

Step 4: Identify all divergent areas.

• If there is no diverged area, do the second level estimation of 2LvSE.

• If there are more than one divergent area, merge the topologically connected areas, and

repeat the following procedures for each island.

• If there is only one area (or one of the areas discussed above), go to step 5.

Step 5: Mark this area as area S. Follow the partitioning procedure step 1 t step 7 discussed in this

section.

Step 6: State estimates can be obtained from the partitioning procedure.

A flowchart for the overall algorithm is given in Figure 3.13.

72


Centralized State Estimator

convergediverge

Obtain SE solution of each individual

zone (similar to the first stage task of

two-level state estimator)

Combine and

synchronize

SE solutions

of each area

(second stage

of two-level

state

estimator)

All of areas converge

One zone

diverges

More than 1 area diverge

Denote this zone to be

S

Combine these areas

to be a new area

If (number of buses in S )

< N0

Yes

No

Split S into two parts:

A and B.

Obtain one-tier-out:

SA and SB

Conduct state

estimation on SA and

SB in parallel

SA diverge

SB converge

SA converge

SB diverge

SA diverge

SB diverge

SA converge

SB converge

Let S = SA Let S = SB

Starget = S

Starget = S

Construct

area SC,

do state

estimation on

SC Let S = SC

diverge converge

Eliminating Starget

from the whole

system, do state

estimation on the rest

system

SE solution obtained

Figure 3.13: Flowchart of RPSE algorithm

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3.7 Graphical User Interface

3.7.1 Install MATLAB App

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment. A

proprietary programming language developed by MathWorks, MATLAB allows matrix manipu-

lations, plotting of functions and data, implementation of algorithms, creation of user interfaces,

and interfacing with programs written in other languages, including C, C++, C#, Java, Fortran and

Python.

Graphical user interface, also known as GUI or UI, provides point-and-click control of

software applications. The GUI discussed in this chapter is based on MATLAB c© R2016b. The

GUI typically contains controls such as menus, toolbars, buttons, and sliders. An application with

graphical user interfaces is developed based on the algorithms and techniques discussed previously

in this report.

For the computers that have MATLAB R2016b installed, it is possible to run the executable

file independently without installing anything or opening MATLAB software. The name of the

executable file is MultiAreaStateEstimator. Please double click to run it. However, due to the

compatibility between different versions of MATLAB, it may fail to run the app because of missing

correct version of MATLAB Runtime (version 9.1). The MATLAB Runtime is a standalone set of

shared libraries that enables the execution of compiled MATLAB applications or components on

computers that do not have MATLAB installed. In such sense, the installation of the app is required.

For the computers that do not have MATLAB R2018b or MATLAB Runtime version 9.2

installed, please follow the instructions below.

1. Locate the MyAppInstaller web executable.

2. Double click the installer to run it.

3. If you connect to the internet using a proxy server, enter the server’s settings.

4. Click Next to advance to the Installation Options page.

5. Click Next to advance to the Required Software page.

6. Click Next to advance to the License Agreement page.

7. Read the license agreement. Check Yes to accept the license.

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8. Click Next to advance to the Confirmation page. Usually it downloads the MATLAB Runtime

v9.1 and the download size is about 537 MB.

9. Click Install.

10. Click Finish.

11. Run the standalone application.

It is also worth to note that, in order to display the interface without issue, the resolution of

the monitor should not be smaller than 1200 ×800.

Some examples using the interface are provided in Section 3.8.

3.7.2 Graphical User Interface

The developed graphical user interface is shown in Figure 3.14. It contains 8 major

functions.

Figure 3.14: GUI static illustration

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3.7.2.1 Parse Test Case

In this function, the 4 test cases can be selected:

• IEEE 118 bus system;

• 16348 bus system, this is a summer system;

• 16216 bus system, this is a winter system;

• 16794 bus system, this is a summer system.

When click the push button ’Parse rawfile measurements’, the test cases that selected

previously will be parsed. By clicking ’Creat input file in CDF’, one can choose to output the power

flow data into text file in IEEE common data format, which is illustrated in Appendix A.

3.7.2.2 System Specifications

This is an optional function. The major information of the system, including number of

buses, branches, zones and measurements are specified and provided. The corresponding numbers

will be shown automatically right after clicking the push button ’Parse rawfile measurements’ and

finishing the parse procedure. For IEEE 118 bus system, it takes less than 1ms. For large scale

systems, this may take 2-3 seconds. User can also check other system information by choosing one

from the list, including

• number of loads;

• number of generators;

• number of shunts;

• number of tap transformers;

• number of phase shifters;

• number of real injection measurements;

• number of reactive injection measurements;

• number of real flow measurements;

• number of reactive flow measurements;

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• number of voltage magnitude measurements;

• number of voltage phase angle measurements;

• number of tap measurements;

• number of phase shifter angle measurements.

3.7.2.3 Specify Bus

This is an optional function. One can search for information corresponding to a specific

bus. It supports to input the number of bus or name of bus. Once the bus is identified, the information

of incident buses, generators, loads, shunts, branches, transformers are shown in the text blocks.

Besides, the buses that are directly connected to the specific bus will also be identified and shown.

This function can be used to check the type of a bus, the number of loads and generators, and its

position in the area.

3.7.2.4 Measurement Error

This is an optional function. One can search and locate a specific bus. The principle

information including its area and bus type will be shown after clicking the search button. One can

choose in the radio button and push button group to specify the candidate measurement that going to

be added with gross error. All the measurements corresponding to the input bus will be identified.

The radio button will be automatically disabled if there is no such type of measurement.

After selecting the desired measurement in the list, one can specify how to manage this

measurement. Three choices are provided: change value to an input value, delete this measurement or

restore all the measurements back to original status. If choose to change measured value, a relatively

large value can be typed into the text block and the measurement set will be updated by clicking

’update measurement’.

3.7.2.5 State Estimator

The state estimation parameters are to be specified first, the default settings are with

tolerance 10−4 and iteration limit 20. This is valid for all three state estimators.

For the centralized state estimator, after click the push button, solutions will be reported

into a text file named ’output ctrlSE.txt’. The status of the state estimation, i.e. converged or diverged,

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will be shown right to the push button. Also note that the bad data identification process is available

by checking the box.

For the two-level state estimator, one more option is given: the external tiers used by each

zone can be changed. Besides, the divergent area, if exists, will be specified right to the push button.

For the recursively partitioned state estimator, the partition bus limit can be changed. This

is the threshold that the partitioning procedure will stop if the area reaches this number of buses.

3.7.2.6 SE solution Display

After the solutions of state estimators are obtained, one can choose to display the numerical

results. The default buses are selected according to the bus with erroneous measurements. More than

half of the buses are within the same area as that bus. The positions of the buses to their areas, i.e.

internal, boundary or external, are different to diverse the results to be shown.

It is also possible to input user’s choice of buses. Note that the delimiter between two input

buses is comma.

3.7.2.7 Numerical Results Table

The table in the interface provides the numerical results of eleven selected buses. The

solutions include:

• Bus number;

• Bus name if exists;

• Zone that the bus belongs to;

• Actual voltage magnitude, marked as Actual Vm;

• Actual phase angle, marked as Actual Va;

• Voltage magnitude estimates of centralized state estimator, marked as CtrlSE Vm;

• Phase angle estimates of centralized state estimator, marked as CtrlSE Va;

• Voltage magnitude estimates of two-level state estimator, marked as 2LvSE Vm;

• Phase angle estimates of two-level state estimator, marked as 2LvSE Va;

• First level state estimation objective values of the area ;

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• Voltage magnitude estimates of recursively partitioned state estimator, marked as RPSE Vm;

• Phase angle estimates of recursively partitioned state estimator, marked as RPSE Va.

The numeric values that are not available will be substituted by -999. If the there is no

corresponding bus name, empty entry will be shown.

3.7.2.8 Pie Chart of Percentage of Buses in the Solved Area

The two-level state estimator and recursively partitioned state estimator solves for the

largest solvable area of the whole system. It is important to show the percentage of buses solved as

an comparison of the performance. When clicking the ’Display solutions’ button, such pie charts

will be displayed with solved area in green and unsolved area in red.

3.8 Simulation

In this section, several numeric simulations are conducted and solutions are analyzed.

At first, the two-level state estimator is validated from three prospectives: procedure of 2LvSE,

validation of ABME and comparison of different methods to ensure observability.

3.8.1 Procedure of Two-Level State Estimation

The two-level state estimation is implemented and tested using the IEEE 118 bus system.

The system is partitioned into three interconnected areas, as shown in Figure 3.15. The first level

estimation on area 3 provides the state estimates of red buses in Figure 3.16. The states of boundary

and external buses of all areas are synchronized by second level state estimation, as shown in

Figure 3.17, highlighted in blue circles.

3.8.2 Procedure of Recursively Partitioned State Estimation

When there is no divergence factors such as measurement error or topology error, the

RPSE will follow exactly the same procedure as 2LvSE. However,

One simulation example based on IEEE 118 bus system is provided to illustrate the

performance of RPSE. Suppose one gross error is added to the injection measurement at bus 2. The

solutions of first level state estimation of 2LvSE are given in Figure 3.18. Bus 2 is marked in the plot.

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Figure 3.15: IEEE 118 bus system with three predefined areas

Figure 3.16: First level state estimation on IEEE 118 bus system

The red circles in the top middle plot indicates the buses to be estimated in area 1. The blue circles in

left bottom plot indicates the area 2 and green circles in right bottom are area 3.

The area 1 diverges due to the gross error in injection measurement at bus 2. Therefore,

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Figure 3.17: Second level state estimation on IEEE 118 bus system

Figure 3.18: First level state estimation for IEEE 118 bus system

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the red area in Figure 3.18 is to be partitioned further, as shown in Figure 3.19.

Figure 3.19: RPSE second partitioning of IEEE 118 bus system

The state estimation for the area designated with yellow circles on the right side of

Figure 3.19 fails to converge again. Further splitting results in the two areas shown in Figure 3.20.

Figure 3.20: RPSE third partitioning of IEEE 118 bus system

This time the purple area which is shown on left side of Figure 3.21 fails to converge.

Splitting it into two smaller partitions yields the area with red circles on right side of Figure 3.21

which fails to converge and the size of this area is now smaller than the specified partitioning

threshold. Thus it is concluded that the measurement of gross errors are located in this area and the

rest of the system constitutes the largest solvable area. By conducting state estimation for the rest of

system, 87% of buses are solved by RPSE, whereas 2LvSE provides only 68%.

There are several numerical results for RPSE provided in the following sections.

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Figure 3.21: RPSE fourth partitioning of IEEE 118 bus system

3.8.3 Validation of the Proposed Estimators

3.8.3.1 Validation of ABME

A numerical comparison of the area boundary cut-set objectives with and without proposed

ABME algorithm is provided in Table 3.1. The weights are the same as defined in Step 3, and

2×Wpb is subtracted from the objectives of the minimum cut-set. The objectives without proposed

algorithm is calculated based on the original boundary branches and weights.

Area 1 6 15 18 51

Obj. without ABME 30 218 25 322 1128

Obj. with ABME 27 22 25 22 134

Table 3.1: Comparison of area boundary cut-set objectives

Results in Table 3.1 show that some of the areas have noticeable improvements. Observ-

ability of the areas where all minimum cut-set branches have flow measurements, will be guaranteed.

Thus, no pseudo injections will be needed. For the rest of the areas, an observability check based on

CPIF technique will still be needed, however the required number of critical pseudo injections will

be no more than that of CPIF only.

3.8.3.2 Comparison of Methods to Ensure Area Observability

Several different pseudo injection placement algorithms to ensure area observability, includ-

ing all pseudo injection (API), critical pseudo injections considering decoupled model observability

(CPID), critical pseudo injections considering full model observability (CPIF) and area boundary

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expansion (ABME) are discussed in Section 3.4.3. However, as described before, ABME expands

the original area to a larger one by augmenting some neighboring buses. Therefore, there are more

states to be estimated. To make a fair comparison, a modified metric ABME* is introduced and

defined as the weighted squares of the measurement residuals corresponding to the original area.

Three scenarios are simulated:

• Area 3 of IEEE 118 bus system containing 51 buses. The weights of all actual measurements

are chosen as 104. Measurements are without any noise. Three different level of pseudo

injection weights are provided to evaluate their impacts: 104, 1 and 10−4. Estimation results

of first level SE on area 3 are provided in Table 3.2.

• Area 3 of IEEE 118 bus system . White Gaussian noise with zero mean and variance 10−4 are

added to all measurements. Three different pseudo weights are compared: 104, 1 and 10−4.

Estimation results of first level SE on area 3 are provided in Table 3.3.

• Area 31 of an 17014 bus real world interconnected system containing 890 buses. The mea-

surements are synthesized without any noise for better illustration of the comparison. The

weights of measurements vary from 0.01 to 100, depending on the type of measurements.

Three different pseudo weights are compared: 10−2, 10−5 and 10−8. Estimation results of

first level SE on area 31 are provided in Table 3.4.

Note that Npi stands for the number of pseudo injections to be assigned, and wp for the

weight of pseudo injections. CtrlSE indicates the conventional centralized state estimation.

Table 3.2: Comparison of methods on area 3 of IEEE 118 bus system with measurement noise

Estimation ObjectivesNpi Time(s)

wp 104 1 10−4

API 1541.4 91.6 90.8 8 0.0116

CPID 286.9 91.3 90.8 4 0.0258

CPIF 90.9 91.1 90.8 3 0.0301

ABME* 90.9 91.1 90.8 0 0.0354

The average running time of different methods for several different areas are plotted in

Figure 3.22. Thus, the following conclusions can be drawn from the above figure and tables:

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Table 3.3: Comparison of methods on area 3 of IEEE 118 bus system without measurement noise


wp 104 1 10−4

API 130.6 0.13 1.3× 10−4 8 0.0122

CPID 1.3× 10−3 1.6× 10−4 1.7× 10−5 4 0.0290

CPIF 1.4× 10−8 1.4× 10−8 1.4× 10−8 3 0.0287

ABME* 1.4× 10−8 1.4× 10−8 1.4× 10−8 0 0.0352

Table 3.4: Comparison of methods on area 31 of 17014 bus system without measurement noise


wp 10−2 10−5 10−8

API 5.2 0.0052 5.2× 10−5 202 0.2483

CPID 0.0947 9.4× 10−3 9.2× 10−6 4 0.4956

CPIF 3.4× 10−9 3.4× 10−9 3.4× 10−9 2 1.2224

ABME* 3.4× 10−9 3.4× 10−9 3.4× 10−9 1 0.2679

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

Bus number

0

0.2

0.4

0.6

0.8

1

1.2

Tim

e (

s)

Running time comparison of different methods

CPIF

CPID

ABME

API

Figure 3.22: Running time comparison of different methods

• When pseudo weights are large, the performance can be ranked as CPIF ≈ ABME > CPID >

API.

• When pseudo weights are small, CPID, CPIF and ABME have similar performances and API’s

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is slightly worse.

• Regarding computational efficiency, the following is always true: CPIF ≥ CPID > API.

Comparing to CPIF and CPID, ABME requires more time to initialize when the size of the

area is small, but comparatively much less with increasing area sizes, as verified by the results

shown in Table 3.3 and Table 3.4.

3.8.3.3 SE Solution Validation of 2LvSE and RPSE using GUI

The numerical validation of 2LvSE and RPSE is conducted by running the state estimators

without any measurement or topology error, and comparing the estimation solutions with the actual

states. Using the developed graphical user interface, estimated states from three state estimators,

i.e. centralized SE, 2LvSE and RPSE are provided. Two systems are tested using the developed

GUI, as shown in Figure 3.23 and Figure 3.24. Results indicate that both 2LvSE and RPSE performs

well when there is no error. When there is no gross error or other factors to make the state estimator

diverge, RPSE will have identical solutions to 2LvSE.


86



Besides, the mean squared error (MSE) of estimated states with respect to their actual

values for all areas are plotted in Figure 3.25. Note that three areas are empty and state estimates of

area 20 and 22 are not available. It verifies the accuracy of the first level state estimation that the

MSE values are all smaller than 0.01. Note that the MSE for the estimates of CtrlSE is 0.0046.

0 10 20 30 40 50 60

Area No.

-7

-6

-5

-4

-3

-2

-1

log

10(M

SE

of estim

ate

d s

tate

s)

MSE of area estimated states

Figure 3.25: MSE of estimated states for all areas

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3.8.4 Simulations on IEEE 118 Bus System

In this section, some simulations are conducted on IEEE 118 bus system. Firstly, a

graphical illustration of the 118 bus system is provided and described. The scenarios of gross

measurement errors and topology errors are simulated then. All the simulation results indicate that

the proposed 2LvSE and RPSE can achieve the desired goal that the diverged area is isolated.

3.8.4.1 Gross Measurement Error

To analyze the performance of the proposed two-level state estimator, several simulations

based on IEEE 118 bus system are provided in this section. Very large errors are added to the

measurements to make the centralized state estimator diverge, Whereas the two-level state estimator

and recursively partitioned state estimator can provide a rational solution for the majority portion of

the system.

In the following tables, ’Position’ indicates the bus location within an area. ’V ’ and ’θ’

stand for the estimated voltage magnitude and phase angle respectively. The ’actual’ values represent

the power flow solution, i.e. true states. The results that are not available marked by ’NA’.

(a) Gross error in a real injection measurement at bus 59 in area 2. Bus 59 is an internal

bus of area 2. Results in Table 3.5 indicates that area 2 is isolated by two-level state estimator. The

percentage of buses solved by RPSE is more than 2LvSE.

(b) Gross error in a real flow measurement at branch 38 to 65. Bus 65 is a boundary bus of

area 2 and bus 38 is a boundary bus of area 1. Results in Table 3.6 indicates that area 1 and 2 are

isolated by two-level state estimator. The percentage of buses solved by RPSE is more than 2LvSE.

3.8.4.2 Topology Error

A simulated topology error occurs at bus 110, which is an internal bus of area 3. This bus

is connected to 4 buses: 103, 109, 111 and 112, with 4 breakers, as shown in Figure 3.26. Breaker a1,

a2 and a3 are closed and a4 is open, but the control center does not have such information. Suppose

there is a plan to open breaker a3 for maintenance, thus bus 110 will be split into two parts: one is

connected to bus 103, 109 and 111, and the other is connected to bus 112, as shown in Figure 3.27 .

Therefore, a topology error occurs at this area, which will make CtrlSE and first level SE in

area 3 diverge. By introducing proposed algorithm, results of rest zones can be obtained and shown

in the following Table 3.7. Results indicate that area 3 is isolated by two-level state estimator.

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Table 3.5: Simulation of IEEE 118 bus system with gross error in real injection at bus 54

Bus 1 38 41 58 65 109

Area 1 1 2 2 2 3

Position I B I I B I

Actual V 0.9550 0.9620 0.9670 0.9590 1.0050 0.9670

Actual θ 0 6.2400 -3.9000 4.8400 16.9800 8.2600

CtrlSE V&θ NA NA NA NA NA NA

2LvSE V 0.9550 0.9620 NA NA 1.0050 0.9670

2LvSE θ 0 6.2400 NA NA 16.9800 8.2600

RPSE 0 0 0.9670 NA 0 0

RPSE 0 0 -3.9000 NA 0 0

Percentage of buses that 2LvSE provides solutions 81.0%

Percentage of buses that RPSE provides solutions 86.4%

Table 3.6: Simulation of IEEE 118 bus system with gross error in real flow at branch 38 to 65

Bus 1 38 58 65 68 109

Area 1 1 2 2 3 3

Bus Position I B I B B I

Actual V 0.9550 0.9620 0.9590 1.0050 1.0030 0.9670

Actual θ 0 6.2400 4.8400 -16.9800 16.8800 8.2600


2LvSE V NA NA NA 1.0050 1.0030 0.9670

2LvSE θ NA NA NA 0.0997 -0.0003 -8.6203

RPSE V NA NA 0.9590 1.0050 1.0030 0.9670

RPSE θ NA NA 4.8400 -16.9800 16.8800 8.2600



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Figure 3.26: Breaker topology of Bus 110 in IEEE 118 bus system

Figure 3.27: Breaker topology of Bus 110 with a3 open in IEEE 118 bus system

3.8.5 Simulations on Large Scale Power System

This is a large scale power system operated by an ISO of US. The areas are defined and

modified according to the previously proposed algorithms. Several simulations are then provided.

First, numerical results of three scenarios based on three different save cases are given. Then, several

real divergent cases of the commercial state estimator are regenerated and simulated by the developed

algorithms. There are three different save cases: 16348 bus system, 16216 bus system and 16794 bus

system.

3.8.5.1 Numerical Results of Divergent Scenarios

According to the given results below, the proposed two-level state estimator successfully

isolates the divergent area and provides the accurate state estimates for the rest of the system. And

as expected, the proposed recursively partitioned state estimator provides more state estimates than

2LvSE.

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Table 3.7: Simulation of IEEE 118 bus system with topology error at bus 110

Bus 38 58 70 97 105 109

Area 1 2 3 3 3 3

Bus position B I B I I I

Actual V 0.9620 0.9590 0.9840 1.010 0.9650 0.9680

Actual θ 6.2400 4.8400 11.7600 17.0700 9.76 8.2600


2LvSE V 0.9620 0.9590 0.9840 NA NA NA

2LvSE θ 6.2400 4.8400 11.7600 NA NA NA

RPSE V 0.9620 0.9590 0.9840 1.010 0.9650 NA

RPSE θ 6.2400 4.8400 11.7600 17.0700 9.76 NA



a) A single gross error is introduced in real power injection measurement at an internal PV

bus 54 in area 1.

Table 3.8: Simulation of 16348 bus system with gross error in real injection at bus 54

Bus 53 78 240 357 8000 13069

Area 1 1 1 1 19 37

Bus position I I I B I B

Actual V 1.030 1.020 1.030 1.030 0.999 1.001

Actual θ -0.347 0.158 -4.730 -8.270 16.900 78.800


2LvSE V NA NA NA 1.030 0.999 1.001

2LvSE θ NA NA NA -8.670 16.500 78.300

RPSE V NA 1.020 1.030 1.030 0.999 1.001

RPSE θ NA 0.158 -4.730 -8.270 16.900 78.800

Percentage of buses that 2LvSE provides solutions 98%

Percentage of buses that RPSE provides solutions >99.5%

Table 3.8 shows that both CtrlSE and area 1 SE diverge. The states of bus 357 can still be

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estimated in 2LvSE because this bus is not only a boundary bus of area 1, but also an external bus of

a neighboring area. Its states can be estimated twice by each neighbor area SE and then synchronized

in the second level state estimation. Since first level state estimation for area 1 diverges, only the

estimates from the neighboring area are used in the second level. Also note that because the reference

bus is different, there is a fixed difference between the phase angle estimates θ provided by 2LvSE

and the true θ.

b) A single gross error is introduced in real flow measurement on branch 9408 to 8547. In

16216 bus system, bus 8547 is a load bus in area 20. Bus 9408 is load bus in area 22. Thus branch

9408 to 8547 is a boundary branch. Results show that both area 20 and area 22 are diverged. RPSE

will merge this two areas together and do the partition-SE recursively. Results show that RPSE solves

over 99% of the system, while 2LvSE just solves 95%.

Table 3.9: Simulation of 16216 bus system with gross error in real flow at branch 9408 to 8547

Bus 1356 8500 8547 9230 9299 12082

Area 5 19 20 22 22 36

Bus position I I I I B B

Actual V 1.010 0.989 1.020 1.020 1.010 1.020

Actual θ -2.280 -0.410 17.10 1.770 0.738 44.800


2LvSE V 1.010 0.989 NA NA 1.010 1.020

2LvSE θ -2.280 -0.410 NA NA 0.738 44.800

RPSE V 1.010 0.989 NA 1.020 1.010 1.020

RPSE θ -2.280 -0.410 NA 1.770 0.738 44.800


Percentage of buses that RPSE provides solutions >99%

c) A single gross error is introduced in a reactive injection measurement at bus 5432. Bus

5432 is a load bus in area 17 in the 16794 bus system. Results in Table 3.10 shows that the RPSE

still get over 99% of the whole system solved, whereas the 2LvSE just has 90%.

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Table 3.10: Simulation of 16794 bus system with gross error in reactive injection at bus 5432

Bus 968 4953 4963 6127 6253 12345

Area 3 17 17 17 17 35

Bus position B I I I B I

Actual V 1.01 1.03 0.988 1.02 0.999 1.01

Actual θ 14.1 40.4 33.9 27.8 34.4 16.4


2LvSE V 1.01 NA NA NA 0.999 1.01

2LvSE θ 13.2 NA NA NA 33.5 15.5

RPSE V 1.01 NA 0.988 1.02 0.999 1.01

RPSE θ 14.1 NA 33.9 27.8 34.4 16.4


Percentage of buses that RPSE provides solutions >99%

3.8.5.2 Real Divergent Scenarios

There are several cases for which the commercial state estimator fails to converge. Some

of these cases are regenerated in order to simulate and test the corresponding scenarios using our

developed estimators. As expected similar to the commercial estimator, our centralized state estimator

also fails to converge. On the other hand, the 2LvSE and RPSE manage to work well and provide

solutions for a large portion of the system.

(a) Real divergent case 1

In this case, the control center received bad telemetry from the substation with very high

MVAR value, thus the commercial state estimator diverged. To simulate this scenario, gross error

is added on the incident injection measurements. Results of simulations using the proposed state

estimators are shown in Figure 3.28.

(b) Real divergent case 2

In this case, the telemetry is in an external area that suffers from lack of telemetry equip-

ment. The measured value may be largely biased from the actual values. Results of simulations using

the proposed state estimators are shown in Figure 3.29.

(c) Real divergent case 3

In this case, bad telemetry data from the whole substation leads the state estimator diverge.

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Figure 3.28: SE solutions of large scale system diverged case 1


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To simulate such circumstance, gross error is added to one injection at either bus of this substation.

Results of simulations using the proposed state estimators are provided in Figure 3.30.


(d) Real divergent case 4

This case is a little different from the previous ones. It involves a topology error where the

breaker status hasn’t been reported promptly. Therefore, one branch is missing from the system. The

centralized state estimator diverged due to such topology error. Results of simulations are shown in

Table 3.11.

3.8.6 Examples of GUI

In this section, four more examples of running GUI, Figure 3.31, Figure 3.32, Figure 3.33

and Figure 3.34 ,are provided to validate the effectiveness of the proposed state estimators.

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Table 3.11: Simulation of topology error in 16348 bus system with missing branch 14591 – 14621

Bus 345 14534 14559 14612 14621 15054

Area 1 43 43 43 43 44

Bus position I I I B I I

Actual V 1.03 1.11 1.08 1.05 1.04 1.01

Actual θ -2.02 52.8 49.4 44.9 48.3 81.4


2LvSE V 1.03 NA NA 1.05 NA 1.01

2LvSE θ -2.88 NA NA 44.1 NA 80.7

RPSE V 1.03 1.11 1.08 1.05 NA 1.01

RPSE θ -2.02 52.8 49.4 44.9 NA 81.4



Figure 3.31: GUI illustration: error in IEEE 118 bus system

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Figure 3.32: GUI illustration: error in 16348 bus system


97



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3.9 Conclusion

When solving the state estimation problem for very large interconnected systems, it is

possible for certain zones to experience convergence issues due to various reasons such as bad

data, loss of measurements, topology errors. In such cases, a solution may not be reached by the

integrated centralized state estimator. In this chapter, two alternative approaches are developed,

implemented and tested. This chapter describes these algorithms and illustrates their effectiveness by

simulation results obtained using several test and actual systems. The proposed approaches are shown

to effectively detect and isolate the subsystem containing divergent factors, hence to successfully

provide accurate estimates for the remaining parts of the system.

The following tasks are undertaken and successfully completed:

• Literature review: several recent literatures about avoiding state estimation divergence and

multi-area state estimator are reviewed in the first and second section.

• Area definition and manipulation: zones and areas of a given system are defined and topologi-

cally analyzed. In order to avoid disconnected subsystems belonging to the same zone, an area

manipulation method is proposed to ensure that every area is a connected graph.

• Area observability analysis: a modified algorithm to check system observability, as well as

three ways to ensure area observability are proposed in this section. Simulations are provided

to compare these algorithms and conclusions are drawn.

• Two-level state estimator: The general framework for the first and second level state estimation

are reviewed. A customized algorithm for two-level state estimation is then proposed, followed

by the illustrations and simulations based on the IEEE 118 bus system.

• Recursively partitioned state estimator: the RPSE is developed in order to further reduce the

unsolvable portion of the overall system. A partitioning algorithm is proposed and implemented,

followed by the development of a recursively partitioned state estimator.

• Graphical user interface: a graphical user interface (GUI) is developed to facilitate the compar-

ison of the developed state estimators. A concise users manual is provided to help with the

installation and usage of the developed programs. Various functions are described in detail

using examples of screen shots.

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• Simulations: several validations of the proposed estimators are provided. based on IEEE 118

bus system and some large scale power systems. Solutions indicate that the proposed two-level

state estimator and recursively partitioned state estimator both have satisfactory performance.

More studies are carried out where several actual divergent SE cases are reproduced, simulated

and solved successfully. Finally, results on graphical user interface are provided.

In next chapter, we turn our focus to the line parameters, which is another critical aspect to

make the state estimation process robust.

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Chapter 4

Transmission Line Parameter

Estimation

4.1 Introduction

In the previous chapters, the robustness of static state estimator against divergent factors

is discussed. A critical aspect that leads to the divergence of SE is the errors in line parameters.

Therefore, in this chapter, we turn our focus to the accurate estimation of line parameters.

As discussed in Chapter 1, renewable energy technologies are becoming increasingly

attractive complementary resources to the existing energy supplies. One critical aspect to successful

deployment of renewable resources is the consideration of transmission capacity. Transmission

capacity of the power system must be sufficient to handle the unplanned dynamics caused by variable

outputs of renewable sources and transport the generated energy for long distances from renewable

sources to load centers without creating congestion [48]. Many of the proposed methods for managing

the impact of variable power output of renewable sources require accurate real time information on

transmission line parameters. Since transmission systems are often operated close to their limits,

accurate relay protection settings based on accurate line parameters become very important. Other

network applications, such as fault location and state estimation [49], also strongly rely on accurate

and timing knowledge of the parameters of all transmission lines. In summary, accurate knowledge

of transmission line parameters is a crucial requirement for reliable and efficient operation of future

power grids.

Transmission line parameters, including series resistance, series inductance, shunt capac-

101

CHAPTER 4. TRANSMISSION LINE PARAMETER ESTIMATION

itance and shunt conductance, are used to build the steady state line model used in most network

applications. Traditional methods use physical characteristics of the lines such as cable type and

tower geometry to calculate line parameters [50–52]. While mathematically sound, this approach

is prone to errors, since it ignores continuously changing operational factors, such as skin effect,

ambient temperature and other weather conditions. In contrast, measurement-based methods can

provide reliable estimates of line parameters, as well as track variations in their values. These

measurements can be obtained from SCADA system, phasor measurement units (PMUs), or fault

records of protective relays [53–86]. A thorough literature review for them are given in the Section

4.2.

Among these algorithms, a prevailing way to deal with rank deficiency of the measurement-

parameter coefficient matrix is using multi-scan measurements: [64, 74, 80] for transposed line,

and [60, 63, 65, 66, 69] for untransposed transmission line. More specifically, one can use either a

sliding window or several consecutive measurement snapshots in a single estimation step. However,

the interval from which measurements are collected has to be long enough to generate a well-

conditioned coefficient matrix, yet short enough to justify the assumption of negligible variation of

parameters across this interval. These conflicting constraints limit the accuracy and applicability

of parameter estimation techniques based on multiple measurements. This problem can be better

addressed by a dynamic parameter estimation technique, especially for the future grids where

real-time information about the varying line parameters will be needed.

In this chapter, transmission line parameter tracking algorithms are proposed, both for a

single untransposed transmission line and several lines in a power grid. The major advantage of the

proposed algorithms is to dynamically track the line parameters and avoid using multiple scan of

measurements.

In Section 4.3, the theoretic background of transmission line model is provided first. Then,

the parameters and unknowns to be estimated for different kinds of line transpositions are determined

and analyzed.

The static parameter estimation methods are then proposed in Section 4.4. The advantages

and limitations of static methods are analyzed. It can be found that static estimation method works

for most type of line transpositions except untransposed lines. What’s more, the static method can

provide a comparatively accurate set of initial parameter estimates for the proposed algorithms in the

following sections.

In Section 4.5, a joint state estimation and parameter tracking algorithm is proposed for

tracking untransposed transmission line parameters in real-time. The core idea of the proposed

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algorithms is to alternate between state estimation, which relies on the most recent parameter

estimates and serves to suppress voltage and current measurement noise, and parameter tracking,

which relies on the most recent state estimates and serves to farther suppress current measurement

noise. By iteratively processing state estimation and parameter tracking, the mismatch between

actual and estimated parameters can be reduced. The proposed dynamic approach works well for

both time-invariant and time-variant line parameters under considerable measurement noise. The

corresponding simulations are provided in Section 4.6.

The idea of JSEPT is extended for the whole grid implementation. In Section 4.7, the model

for multiple transmission lines in a power grid is described and analyzed. The parameter tracking

algorithm is then proposed in Section 4.8, named joint state estimation and parameter tracking for

system (JSEPTS). The simulations of JSEPTS are provided in Section 4.9. Final conclusions are

provided at last.

4.2 Literature Review of Transmission Line Parameter Estimation

In one of the earliest papers on the subject [87], Merrill and Schweppe proposed estimation

of network parameters using a single data scan (one time point) of measurement data. A difficulty

with this approach is that there are rarely enough measurements in the vicinity of uncertain parameters

to enable their estimation with a single scan of measurements.

An alternative approach has been proposed in which a set of measurements at multiple

time points is used. A batch processing algorithm was suggested in [88] to solve for parameter

estimates. Batch processing algorithms are well suited for off-line studies but are not amenable to

on-line applications.

Debs proposed a recursive algorithm based on the Kalman filter in his 1974 paper [89].

In his work, he modeled the bus voltage and angle variables as Markov processes and the network

parameters as constants. The use of dynamic models allows one to use a recursive estimation in which

apriori information about the state and parameter estimates is combined with current measurement

data in order to update the parameter estimates. Computational experience has indicated that the

problem, as formulated by Debs has the potential for convergence problems when it is applied to

problems with large networks and/or several uncertain parameters. Furthermore, Debs’ formulation

treats network parameters as constants. This limits the algorithms flexibility, since some network

parameters, such as corona losses, are time varying.

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The presented method in [55] utilizes a recursive maximum likelihood estimation and a

Kalman filter to accurately obtain estimates of branch parameters by continuously refining parameter

values over many measurement samples. The method estimates all branch parameters and can

be implemented in off-line as well as on-line mode. It was shown to provide accurate results in

the presence of noise in measurements and has the ability to detect and reject gross measurement

errors. In the on-line mode, the method can track parameter values as they change in time providing

continues monitoring and refinement of data base parameter values. It carefully utilizes available

measurement redundancy by first estimating parameters of well metered branches and then adaptively

expanding the solution scope to include less metered branches.

Through 1980s till now, there are a few literatures using physical characteristics of the

lines such as cable type and tower geometry to calculate line parameters, such as [50–52]. In [51]

and [50], the handbook formulas are applied and analyzed. The authors of [52] compared different

approximation methods and develops a new method for simplification of Carsons equations to model

distribution lines for unbalanced power flow and short-circuit analysis. The comparisons are made

for real and imaginary parts of self and mutual impedances.

For the prevailing measurement based methods, there is a significant body of literature

discussing the identification of parameters in transposed transmission lines [53–55, 57–86, 90].

Many of them assume a fully transposed transmission line, such as in [55, 64, 67, 68, 73, 74, 78, 80–

83], which limits their applicability in more general settings. The asymmetrical spacing of many

transmission lines leads to different self and mutual inductances, which then corresponds to an

unsymmetrical system. Thus, a more general and comprehensive treatment of line parameters for a

general untransposed transmission line appears necessary, such as [60, 61, 63, 65, 66, 69].

In terms of methodology, most of the methods proposed can be categorized into 12 types,

listed in Table 4.1.

In the following, these literatures will be reviewed one by one.

Some early research work focused on state estimation residual sensitivity analysis. In [53],

the method consists in exploiting the information contained in the residuals in order to compute

the parameter error responsible of the observed (and supposedly unacceptable) residuals. This

computation is done within the context of estimation theory, using a linearized sensitivity relationship

between measurement residuals and parameter errors. Liu, Wu, and Lun proposed parameter

estimation in [54] based on the analysis of state estimator measurement residuals. In this approach

the state estimation and parameter estimation problems are solved separately. The approach is based

on a single scan of measurements and therefore suffers from the same observability problems as the

104


Table 4.1: Classification of line parameter estimation methods

Method to estimate parameters literature

Recursive filter [89], [55]

Parameter calculated from geometry [50–52], [52]

Parameter error identification based on state

estimation residual sensitivity analysis

[53], [54]

Parameter calculated from measured

frequency-dependent impedance

[61]

Modal transformation [60], [61], [82]

Parameter calculated from fault record [63], [66], [73], [82]

Data-Driven methods [85]

Lagrange multipliers [62], [72], [86]

State vector augmentation [55], [64], [71], [74], [75], [84], [86]

Parameter estimation algorithm - Kalman Fil-

ter

[67, 68], [76], [79] [90]

Parameter estimation algorithm - WLS [60], [63], [65], [66], [69], [80], [81]

work of Merrill and Schweppe.

The utilization of fault record is a popular way to calculate or estimate line parameters,

such as [63, 66, 73, 82]. Authors of [63] present a theory for the identification of transmission line

parameters based on time-varying phasors. By assuming equal admittances at both ends of a π section

model it is possible to calculate the currents through the line using the measured currents of the fault

records. Afterwards, the line parameters are estimated using the least squares method. It is shown

that only measurements including transients can be used for the identification. In [66], a special

feature of the proposed method is the consideration of unsymmetrical transmission lines and the

application of a high-accuracy signal modeling technique. Derived from a line model, an estimation

equation for determining the different impedances is developed. This equation contains the signal

models of measurements as well as their time derivatives, which can be estimated very precisely

compared to the classical approach based on filtering. This approach has some disadvantages which

can be overcome by fitting the data within the fault segment to certain signal models using an

advanced Prony method. In [73], the authors proposed a parameter estimation method that relies

on fault records captured by digital relays and utilizes the wave propagation speed equation. The

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fault records are the unsynchronized voltage and current waveforms recorded at both ends of a line.

One of the main contributions of this work is the use of the wave propagation speed equation to

simplify the problem and to improve the estimation accuracy. As a result, both the positive and zero

sequence parameters can be estimated using simple algorithms in a decoupled solution process. [82]

presents a method for estimating on-line parameters of series compensated line using synchronized

time-domain data captured by the intelligent electronic devices at both ends of the line which is free

of the compensation model. The method uses traveling waves generated during disturbance to obtain

the propagation constant of the line, which is used to estimate the resistance and the characteristic

impedance of the line. Subsequently, the inductance and capacitance of the line are calculated.

[62, 72, 86] are three representatives for the use of Lagrange multipliers to estimate line

parameters. In [62], a parameter error identification method is proposed which is based on Lagrange

multipliers corresponding to a single measurement scan. The proposed method has the desired

property of distinguishing between bad analog measurements and incorrect network parameters, even

when they appear simultaneously. [72] This paper investigates the problem of network parameter

error detection and identification in power systems. This paper provides an improvement of method

via the use of multiple measurement scans which increases the local redundancy at no additional

cost. And [86] proposed the method to identify bad parameters as well as measurement errors based

on normalized Lagrange multiplier test and normalized residual test.

The state vector augmentation method is a popular way to deal with parameter and measure-

ment errors simultaneously for decades. In [64], the proposed algorithm aims to estimate the positive

sequence parameters under the scenarios when PMUs exhibit frequent impulsive noises induced by

improper hardware wiring, unavailability of the GPS time reference or communication interferences.

The author introduces an optimal estimator which minimizes the impacts of unsynchronized mea-

surements and measurement errors. And in [71], the authors proposed a three-stage off-line approach

to detect, identify, and correct series and shunt branch parameter errors. In Stage 1 the branches

suspected of having parameter errors are identified through an identification index. Such index of a

branch is the ratio between the number of measurements adjacent to that branch, whose normalized

residuals are higher than a specified threshold value, and the total number of measurements adjacent

to that branch. Using several measurement snapshots, in Stage 2 the suspicious parameters are

estimated, in a simultaneous multiple-state-and-parameter estimation, via an augmented state and

parameter estimator which increases the state vector for the inclusion of suspicious parameters. Stage

3 enables the validation of the estimation obtained in Stage 2, and is performed via a conventional

weighted least squares estimator. In [74], a practical technique is proposed for the estimation of

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transmission line parameters while the required data are phasor measurements at one end of a given

line and conventional magnitude measurements at the other end. A non-linear weighted least-square

error (NWLSE) algorithm is employed for the maximum-likelihood estimation of parameters.

What’s more, in [84], the systematic errors in the input measurements from the line

ends are considered. The proposed method aims to increase the accuracy of impedance parameter

estimates by estimating the correction factors for the systematic errors. Thus, an augmentation

method is proposed in this paper to combine system states and line parameters and formulate a

nonlinear WLS problem. [75] presents a novel method of estimating distribution line parameters

using only root mean square voltage and power measurements under consideration of measurement

tolerances, noise, and asynchronous timestamps. A measurement tolerance compensation model and

an alternative representation of the power flow equations without voltage phase angles are introduced

then. Later on, the proposed approach in [76] exploits the absence of the time-variant voltage angle

in the transformed set of power flow equations from [75], as the remaining unknown parameters

to be estimated are all constants. This converts the estimation into a pure parameter identification

problem which is overdetermined with respect to the number of multiple scans of measurements.

The measurement noise covariance matrix of the EKF is modified consequently in order to properly

account for all noisy quantities. To construct measurement model, some measurements are used as

inputs. The noise of inputs are handled as part of the measurement noise covariance matrix. And

for nonlinear system, a linearization (first-order Taylor approximation) is introduced. While state

augmentation may work well for transposed lines, our experiments show that for untransposed lines,

where there are more unknown parameters, algorithm will not always converge.

There is a number of literatures using the Kalman filter to estimate the parameters dynam-

ically, such as in [67, 68], [76], [79] [90]. In [79], the proposed method applies a digital dynamic

filter technique to perform the estimation based on minimization of residual least absolute value.

The proposed technique can be used on-line to account for any dynamics changes in line constants.

Synchronized phasor measurements are used to feed the dynamic filter. The synchronized samples of

voltage and current waveforms, at both the receiving and sending end terminals, are used to set an

over-determined system of equations, and then formulated as an estimation problem.

The authors of [67] and [68] present a Kalman filter-based approach for tracking states

and line parameters simultaneously. The state-and-parameter estimation in parameter estimation is

reformulated as two loosely-coupled linear subproblems of state tracking and parameter tracking.

For state tracking, which can be used to determine bus voltages in parameter estimation or to track

the system state in (dynamic) state estimation. Dynamic behavior of bus voltages under possible

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abrupt changes is studied, using a novel and accurate prediction model. The measurement model is

also improved. An adaptive filter based on optimal tracking with correlated prediction-measurement

errors, including the module for abrupt-change detection and estimation, is developed. For parameter

tracking, a new prediction model for parameters with moving means is adopted. The uncertainty

in the voltages is covered by pseudo measurement errors resulting in prediction-measurement-

error correlation. An error-ensemble-evolution method is proposed to evaluate the correlation. An

adaptive filter based on the optimal filtering with the evaluated correlation is then developed, where a

sliding-window method is used to detect and adapt the moving tendency of parameters.

The main difference between the method in [90] and the one in [67] is the iterative process

within each measurement scan. More specific, the authors of [67] proposed a modified version of

Kalman filter with specific terms added to the innovation in measurement update step: there are

no other loops besides the ordinary Kalman filter. Whereas the algorithm proposed in this paper

recursively processes the state estimation and parameter tracking for a single measurement scan.

When it obtains converged solutions from the recursive process, the algorithm moves to the next

measurement scan.

The literature on untransposed line parameter estimation includes several useful contri-

butions. There are several methods based on modal transformation [60, 61, 63, 66, 69]. In [61],

the authors presented a procedure to derive longitudinal frequency-dependent transmission line

parameters directly from the measured impedances of the line. Based on the Laplace transform,

untransposed line parameters are identified using synchronized phasors at both terminals [60]. And as

discussed before, Shulze et al. proposed a phase domain method to identify parameters by analyzing

fault records, using WLS technique in [63] and relying on Prony forecasting method in [66].

In [61], the authors presented a procedure to derive longitudinal frequency-dependent

transmission line parameters directly from the measured impedances of the line. It is to calculate

transmission line parameters per unit length. With this methodology, the transmission-line parameters

can be obtained starting from impedances measured in one terminal of the line. First, it shows the

classical methodology to calculate frequency-dependent transmission-line parameters by using

Carsons and Pollaczecks equations for representing the ground effect and Bessels functions to

represent the skin effect. After that, a new procedure is shown to calculate frequency-dependent

transmission-line parameters directly from currents and voltages of an existing line. In [60], it solves

for the line parameters based on Laplace transform technique by utilizing three sets of synchronized

voltage and current phasors.

Authors of [69] analyzed the limitations of the positive sequence transmission line model

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when used for parameter estimation of transmission lines that are untransposed and proposed a

novel method using linear estimation theory to identify the parameters more reliably. This method

can be used for the most general case: short/long lines that are fully transposed or untransposed

and have balanced/unbalance loads. Besides the positive/negative sequence impedance parameters,

the proposed method can also be used to estimate the zero sequence parameters and the mutual

impedances between different sequences. This paper also examines the influence of noise in the

PMU data on the calculation of parameters.

The authors of [65] apply a Gauss-Newton method for estimating distributed line parame-

ters. In detail, it describes a method of live line measurement of the parameters of an untransposed

three phase one circuit transmission line and untransposed two parallel transmission lines and neces-

sary conditions to measure line parameters. A mathematical approach is proposed using synchronous

voltages and line currents at both ends of the line in different states, where two models, the equivalent

π circuit and distributed constant line are employed. For two parallel transmission lines, applying a

mode decomposition, we show that two parallel transmission lines can be transformed into double

independent three phase circuits.

Rest of papers have different perspective to estimate line parameters [57–59, 70, 77, 78, 80,

81, 83, 85].

In [57], a preprocessing method that identifies both multiple topology errors and bad

measurements is described. The method determines the branch statuses by testing their real and

reactive power flow estimates of all the branches of the network, irrespective of their assumed statuses.

The power flows are the state variables of two decoupled real and reactive power models that stem

from both a detailed substation representation and a super-node modeling. They are estimated by

means of the iteratively reweighted least-squares algorithm that implements the Huber M-estimator.

The procedure is not prone to divergence problems, which is of great value in a real time environment.

In [58], two sets of synchronized voltage and current phasors from the two terminals of

the line are utilized to obtain the ABCD parameters of the line. How distributed parameter per unit

length can be obtained is not covered.

The authors of [59] suggest a method for deriving the line characteristic impedance and

propagation constant by making use of on-line voltage and current phasors captured at sending-end

and receiving-end of the line.

In [70], three phase line parameters are estimated based on real and reactive power

measurements. It is shown that the resulting system of equations is extremely ill-conditioned in

nature and to solve this ill-conditioned system of equations, an optimization-based procedure is

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developed. If a large number of solution vectors are calculated for the same measurement vector from

randomly chosen initial vectors, and the final solution is taken to be the average of these solution

vectors, then the random errors in the individual elements of all these solution vectors are expected to

cancel each other and, thus, the final solution would be quite close to the true solution. The proposed

technique is based on the fact that if a large number of solutions are calculated, then the average of

these solutions would be very close to the actual, true solution.

[83] focus on the data quality and selection. It proposes a multi-point transmission

line parameter estimation model with an adaptive data selection scheme based on measured data.

Data selection scheme, defined with time window and number of data points, is introduced in the

estimation model as additional variables to optimize. The data selection scheme is adaptively adjusted

to minimize the relative standard deviation (RSD) of estimated parameters. An iterative technique

derived from the Newton method is adopted to solve the proposed model by fitting the relationship

between the RSD and data selection scheme with exponential functions.

[77] a method is proposed with the aim of obtaining accurate estimates of potentially

variable impedance parameters, in the presence of systematic errors in voltage and current measure-

ments. It is assumed that the behavior of the resistance and reactance is approximately linear over

short periods relative to the thermal time constant of overhead line conductors. Conductance and

susceptance are assumed to be constant. The method is based on optimization to identify correction

constants for the phasors. More specific, parameters are first calculated based on measured voltages

and currents, whose errors are also modeled. Then the calculated parameters are assumed to be new

measurements to formulate LS problem based on the parameter changing model.

[78] introduces the propagation speed equation besides typical Kirchoff’s equations. It

shows that the active power, reactive power, and voltage magnitude data measured at the two ends

of a transmission line are sufficient to determine the positive-sequence line parameters. The phase-

angle information is not essential. The proposed method can be easily implemented using SCADA

measurements only, with the consideration of propagation speed equation.

[80] proposes a methodology for identifying and estimating the erroneous transmission

line parameters using measurements provided by phasor measurement units (PMUs) and estimated

states provided by a state estimator. The main advantage of the proposed methodology is that for the

identification of the erroneous transmission lines and the estimation of the line parameters only one

PMU is required for the monitoring of the transmission line

In [81], the authors present a state estimation technique for three-phase power systems

where not only bus voltage phasors but also the temperature of transmission line conductors is

110


considered as states. Transmission line admittance parameters depending on line conductor and

ambient temperature are approximated from the precomputed data based on polynomial interpolations.

Finally, WLS estimation problem is formulated and solved.

A data-driven method is proposed in [85]. Considering the fact that the regression-based

method is very sensitive to even a small error in measurements, the authors propose the error-

in-variables model in a maximum-likelihood estimation framework for joint line parameter and

parameter estimation. While directly solving the problem is NP-hard, the authors successfully adapt

the problem into a generalized low-rank approximation problem via variable transformation and

noise de-correlation. For accurate topology estimation, we let it interact with parameter estimation in

a fashion that is similar to expectation-maximization algorithm in machine learning. The proposed

PaToPa approach does not require a radial network setting and works for mesh networks.

4.3 Transmission Line Model of a Single Line

4.3.1 Transmission Line Geometry

Transmission line parameters and the associated circuit model are crucial to the operation

of power system. Usually, the parameters of transmission lines, including resistance, inductance,

capacitance and conductance, are decided by the following geometric information:

• Conductor characters include wire size, conductor material, area of conductor, number of

aluminum strands, number of steel strands, number of aluminum layers, DC resistance, AC

resistance, inductive reactance and capacitive reactance etc.

• Tower configuration including tower height, conductor spacing, phase spacing and conductors

per bundle.

• Transposition.

• Line length.

• Power base, voltage base, impedance base, admittance base and impedance base.

Transmission lines can be modeled as transposed or untransposed lines based on their

installation. At the transmission level, transposition can be partly and easily accomplished by

manipulating the relative position of the conductors of individual phases, or via transposition tower.

The transposition ensures the symmetric capacitance of a three phase line, and to some extent, ensures

111


the balanced operation of the power system by physically manipulating the three phase transmission

line to be symmetric over a long distance. Also, transposing is an effective measure for the reduction

of inductively linked normal mode interference.

While most high voltage lines tend to be transposed, there are also those which are

untransposed for one or more of the following reasons:

• Not designed to have such transposition structure.

• Aging problem.

• Geometry reasons such as ’T’ point of three transmission lines.

• The installation of new lines or dismantling of old lines.

According to the length of the transmission line, three types of line models can be used:

short line, medium length line and long line. They all are based on a π-equivalent circuit, which will

be provided in detail in the following section. Since most transmission lines are three-phase, their

detailed modeling will be described first in the following section.

4.3.2 Three Phase Transmission Line Model

A three phase transmission line model with mutual coupling impedances and admittances

is shown in Figure 4.1 [91]. A typical equivalent π model with lumped parameters can be constructed

then, as shown in Figure 4.2.

Figure 4.1: Model of a three phase transmission line

It is assumed that the shunt admittances are equal at the sending and receiving end of the

transmission line. In the general case of a three phase untransposed transmission line, the series

impedance matrix Zseries and shunt admittance matrix Yshunt of the π circuit should be symmetric

112


Figure 4.2: π model of three phase transmission line

according to [91]. The line parameters can be defined as in (4.1) and (4.2). This is the most general

case which involves no assumptions about the possible similarities between elements of the matrix,

even though most line configurations may present such simplifications due to the choice of conductors

and/or existing conductor configuration and tower geometry.

Zseries =

Zaa Zab Zac

Zab Zbb Zbc

Zac Zbc Zcc

(4.1)

Yshunt =

Yaa Yab Yac

Yab Ybb Ybc

Yac Ybc Ycc

(4.2)

All of the following 6 complex entries in Zseries are considered to be unknown: Zaa, Zab,

Zac, Zbb, Zbc and Zcc. The number of unknowns can alternatively be determined by considering the

distributed parameters of the three phase transmission line, which are the entries of the resistance

matrix r, inductance matrix l, capacitance matrix c and conductance matrix g, as shown below:

r, l, c, g =

r, l, c, gaa r, l, c, gab r, l, c, gacr, l, c, gba r, l, c, gbb r, l, c, gbcr, l, c, gca r, l, c, gcb r, l, c, gcc

. (4.3)

That it holds

Zseries = r + jωl (4.4)

Yshunt = g + jωc (4.5)

where j is the imaginary unit, ω = 2πf , f is the system frequency.

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The GPS synchronized voltage and current phasor measurements at both terminals can be

written in vector form as

Vs =

Vsa

Vsb

Vsc

, Vm =

Vma

Vmb

Vmc

, V =

Vs

Vm

(4.6)

Is =

Isa

Isb

Isc

, Im =

Ima

Imb

Imc

, I =

Is

Im

. (4.7)

According to the equivalent π model, the relationship between currents and voltages of

both terminals is

Is + Im =Yshunt

2(Vs + Vm) (4.8)

Vs − Vm = Zseries Izpi (4.9)

Izpi = Is −Yshunt

2Vs = −Im +

Yshunt2

Vm. (4.10)

Thus, a new equation can be derived from (4.8), (4.9) and (4.10) with the introduction of two-port

nodal admittance matrix Y that

I = Y V (4.11)

where

Y =

Yshunt/2 + Z−1series −Z−1

series

−Z−1series Yshunt/2 + Z−1

series

. (4.12)

Note that with the consideration of ground wire, a transmission model is built to be a

4-by-4 matrix. But it is possible to reduce the dimension of the matrix to 3-by-3 by assuming that the

voltage of ground wire is zero. Therefore, three phase model in Figure 4.1 can be obtained.

The symmetry of Zseries and Yshunt implies that all four block elements of Y ∈ C6×6

in (4.12) are symmetric as well. Note C indicates the complex coordinate space. Thus, Y can be

expressed as

Y =

Y1 Y2 Y3 Y4 Y5 Y6

Y2 Y7 Y8 Y5 Y9 Y10

Y3 Y8 Y11 Y6 Y10 Y12

Y4 Y5 Y6 Y1 Y2 Y3

Y5 Y9 Y10 Y2 Y7 Y8

Y6 Y10 Y12 Y3 Y8 Y11

. (4.13)

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4.3.3 Unknowns of an Untransposed Transmission Line

For untransposed transmission line, the complex entries in the upper triangular part of

Zseries (also true for Yshunt) are different from each other, due to the non-identical and asymmet-

rically configured phase conductors. However, the conductances of most high voltage overhead

transmission lines are usually negligible (unless they are operated under highly stressed electric field

conditions with significant corona) so disregarding them will have little effect on the calculation of

the parameters. Therefore, Yshunt will be purely imaginary.

Now considering the rectangular coordinate representation, i.e. decoupling the real and

imaginary part of the matrices, there will be 18 real unknowns in Y of (4.13). This can also be

validated using (4.3). Ignoring the conductance g, there will be 6 unknowns in r, l and c each.

These 18 unknowns p1...18 will constitute the parameter vector p(UT ) ∈ R18×1 given as:

p(UT ) =[p1 p2 · · · p18

]T. (4.14)

Note that p1...18 indicates the combination of 18 parameters from p1 to p18. R indicates the real

coordinate space. Therefore, once the unknowns are available, the series impedance Zseries and

shunt admittance Yshunt can be obtained as

[Zseries]−1 =

−p1 + jp7 −p3 + jp8 −p5 + jp9

−p3 + jp8 −p10 + jp14 −p12 + jp15

−p5 + jp9 −p12 + jp15 −p16 + jp18

(4.15)

Yshunt = 2j

p2 + p7 p4 + p8 p6 + p9

p4 + p8 p11 + p14 p13 + p15

p6 + p9 p13 + p15 p17 + p18

(4.16)

Note that the expression of Zseries is consistent to the one in (4.1): the inverse of such

symmetric matrix is also symmetric and there is a nonlinear relationship between the elements

Zaa, Zab, Zac, Zbb, Zbc, Zcc in (4.1) and p1, p3, p5, p7, p8, p9, p10, p12, p14, p15, p16, p18. The

same is true for Yshunt in (4.2).

According to (4.12) and the discussions above, the structure of the 18 unknowns p1...18 in

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Y can be found in (4.14).

Y =

p1 + jp2 p3 + jp4 p5 + jp6 −p1 + jp7 −p3 + jp8 −p5 + jp9



−p1 + jp7 −p3 + jp8 −p5 + jp9 p1 + jp2 p3 + jp4 p5 + jp6



(4.17)

4.3.4 Unknowns of a Partially Transposed Transmission Line

In general, the number of distinct unknown parameters depends on the geometry of

conductors. Practically, there are only a few geometric configurations of high voltage overhead

lines. For instance, a 345-KV transmission system has only four possible configurations: lattice-type,

pole-type, H-frame-type and Y-type. In these four cases, two of the three mutual impedances are

very close to each other due to symmetric geometry. Therefore, it is possible to approximate the

transposition partially, as

• self resistances located on the main diagonal are equal to each other, and it is also true for self

inductance and capacitance, i.e. r, l, caa = r, l, cbb = r, l, ccc;

• mutual resistances located on a sub-diagonal are equal to each other, and the same holds for

mutual inductances and capacitances, i.e. r, l, cab = r, l, cbc,

the following Toeplitz matrices can be obtained.

r, l, c =

r, l, caa r, l, cab r, l, cacr, l, cab r, l, caa r, l, cabr, l, cac r, l, cab r, l, caa

. (4.18)

In a partially transposed line, Zseries and Yshunt are Toeplitz and the relationship between

them is given in (4.20) and (4.21). The parameter vector p(PT ) ∈ R11×1 is formulated as

p(PT ) =[p1 p2 · · · p11

]T. (4.19)

Note that the parameters p1...11 here are different from the ones in (4.14).

For Zseries, though there are only 6 independent unknowns, its inverse will have 8 parame-

ters that hold nonlinear relationship with the 6 unknowns. Therefore, ensure the linear relationship

116


between the currents and voltages in (4.11), we use 8 unknowns instead of 6. When conducting static

parameter estimation aiming for initial values, numerical results show little difference of estimated

parameters whether using equality constraints for the nonlinear mapping between 6 unknowns and 8

parameters or not. Therefore, Zseries is formulated as shown in (4.20).

[Zseries]−1 =

Zaa Zab Zac

Zab Zaa Zab

Zac Zab Zaa

−1

=1

S

Z2aa − Z2

ab ZabZac − ZabZaa Z2ab − ZaaZac

ZabZac − ZabZaa Z2aa − Z2

ac ZabZac − ZaaZabZ2ab − ZaaZac ZabZac − ZaaZab Z2

aa − Z2ab

=

p1 − jp2 p3 − jp4 p5 − jp6

p3 − jp4 p7 − jp8 p3 − jp4

p5 − jp6 p3 − jp4 p1 − jp2

, (4.20)

where S is the determinant of Zseries. And for Yshunt,

Yshunt = 2j

p9 p10 p11

p10 p9 p10

p11 p10 p9

. (4.21)

Therefore, the nodal admittance matrix Y can be expressed as

Y =

p1 − jp2 + jp9 p3 − jp4 + jp10 p5 − jp6 + jp11

p3 − jp4 + jp10 p7 − jp8 + jp9 p3 − jp4 + jp10 −[Zseries]−1

p5 − jp6 + jp11 p3 − jp4 + p10 p1 − jp2 + jp9

−p1 + jp2 −p3 + jp4 −p5 + jp6

−p3 + jp4 −p7 + jp8 −p3 + jp41

2Yshunt + [Zseries]

−1

−p5 + jp6 −p3 + jp4 −p1 + jp2

.

(4.22)

4.3.5 Unknowns of a Transposed Transmission Line

For transposed transmission line, parameters in each phase are identical, indicating the

resistance matrix r (also true for inductance matrix l, capacitance matrix c) in (4.3) have only two

independent unknowns, i.e.,

r, l, caa = r, l, c, gbb = r, l, ccc (4.23)

117


r, l, cab = r, l, c, gbc = r, l, cac. (4.24)

Therefore, there will be 6 unknowns to be estimated, shown as follows.

p(TT ) =[p1 p2 p3 p4 p5 p6

]T. (4.25)

Then it holds that all the diagonal elements of Zseries are equal, and all the other elements are equal,

as shown in (4.26). This property also holds for Yshunt, as shown in (4.27).

[Zseries]−1 =

Zaa Zab Zab

Zab Zaa Zab

Zab Zab Zaa

−1

=1

S

Z2aa − Z2

ab Z2ab − ZabZaa Z2

ab − ZaaZabZ2ab − ZabZaa Z2

aa − Z2ab Z2

ab − ZaaZabZ2ab − ZaaZab Z2

ab − ZaaZab Z2aa − Z2

ab

=

p1 − jp2 p3 − jp4 p3 − jp4

p3 − jp4 p1 − jp2 p3 − jp4

p3 − jp4 p3 − jp4 p1 − jp2

. (4.26)

Yshunt = 2j

Yaa Yab Yab

Yab Yaa Yab

Yab Yab Yaa

= 2j

p5 p6 p6

p6 p5 p6

p6 p6 p5

. (4.27)

And the corresponding nodal admittance matrix Y is

Y =


p3 − jp4 + jp6 p1 − jp2 + jp5 p3 − jp4 + jp6 −[Zseries]−1


−p1 + jp2 −p3 + jp4 −p3 + jp4

−p3 + jp4 −p1 + jp2 −p3 + jp41

2Yshunt + [Zseries]

−1

−p3 + jp4 −p3 + jp4 −p1 + jp2

.

(4.28)

118


4.3.6 Unknowns in Positive Sequence of a Transmission Line

In the analysis of transposed transmission line parameters, especially for a power grids

with a considerable number of lines, usually only the positive sequence parameters are concerned,

including resistance, inductance and capacitance. Therefore, there are 3 unknowns as follows.

p(PS) =[p1 p2 p3

]T(4.29)

[Zpos]−1 = p1 − jp2 (4.30)

Yshunt = 2jp3. (4.31)

4.4 Static Parameter Estimation

One straight forward way to estimate the transmission line parameters is to formulate the

measurement Jacobian based on single or multiple snapshots of measurements and then solve for

estimates using weighted least squares technique. But for different kind of line transpositions, such

formulation is different, and needs to be analyzed separately in detail.

To simplify the state estimation processes, it is good to decouple the complex variables

into their real (with subscript r) and imaginary (with subscript i) components. The rectangular

formulation also gives better performance of state estimation and convergence characteristics, com-

paring with complex variables or magnitude-phase angle formulation [92]. Take the complex current

measurement vector I in (4.7) and complex voltage measurement vector V in (4.6) as an example,

the derived real parameter vector I ∈ R12×1 and V ∈ R12×1 will be

I =[Isar Isai Isbr Isbi Iscr Isci Imar Imai Imbr Imbi Imcr Imci

]T, (4.32)

V =[Vsar Vsai Vsbr Vsbi Vscr Vsci Vmar Vmai Vmbr Vmbi Vmcr Vmci

]T.

(4.33)

Therefore, the measurement equation can then be formulated as

I = Hv p+ e, (4.34)

where I is the vector of current measurement, p is the vector of unknown parameters, e represents

the current measurement noise with zero mean and covariance R. The voltages V are assumed to be

noise free and can be used directly in building the measurement Jacobian matrix Hv.

119


Solutions for (4.34) can be obtained using the weighted least squares (WLS) method [2].

Such problem can be formulated as to minimize

J(p) = (I −Hv p)TR−1(I −Hv p), (4.35)

and solved as

p = (HTv R−1Hv)

−1HTv R−1I (4.36)

However, for different kinds of lines as discussed in Section 4.3, the number of unknowns

are quite different. Therefore, in the following sections, the structures of p and Hv are discussed for

each kind of line separately and explicitly. A general conclusion is made at first here, that if using

only one snapshot of measurements, static estimation works for all kinds of lines except untransposed

line.

It is also worth to illustrating the general structure of Hv at first, since it is the key

coefficient in static parameter estimation. To reduce the work of formulating Hv for different kinds

of lines, one can formulate a ’generalized’ version H(G),v at first. H(G),v is a 12-by-24 real matrix

and can be constructed by two smaller real matrices H(G),v,A ∈ R6×12 and H(G),v,B ∈ R6×12, as

given in (4.37) and (4.38).

H(G),v,A =

Vsar −Vsai Vsbr −Vsbi Vscr −Vsci 0 0 0 0 0 0

Vsai Vsar Vsbi Vsbr Vsci Vscr 0 0 0 0 0 0

0 0 Vsar −Vsai 0 0 Vsbr −Vsbi Vscr −Vsci 0 0

0 0 Vsai Vsar 0 0 Vsbi Vsbr Vsci Vscr 0 0

0 0 0 0 Vsar −Vsai 0 0 Vsbr −Vsbi Vscr −Vsci0 0 0 0 Vsai Vsar 0 0 Vsbi Vsbr Vsci Vscr

(4.37)

120


H(G),v,B =

Vmar −Vmai Vmbr −Vmbi Vmcr −Vmci 0 0 0 0 0 0

Vmai Vmar Vmbi Vmbr Vmci Vmcr 0 0 0 0 0 0

0 0 Vmar −Vmai 0 0 Vmbr −Vmbi Vmcr −Vmci 0 0

0 0 Vmai Vmar 0 0 Vmbi Vmbr Vmci Vmcr 0 0

0 0 0 0 Vmar −Vmai 0 0 Vmbr −Vmbi Vmcr −Vmci0 0 0 0 Vmai Vmar 0 0 Vmbi Vmbr Vmci Vmcr

(4.38)

Therefore,

H(G),v =

H(G),v,A H(G),v,B

H(G),v,B H(G),v,A

(4.39)

In the following discussion, by manipulation the columns ofH(G),v, the coefficient matrices

for each kind of lines can be obtained easily.

4.4.1 Static Parameter Estimation on Untransposed Line

According to the definition of unknowns in untransposed line (4.15), the following linear

measurement model can be derived from (4.11)

I = H(UT ),v p(UT ) + e, (4.40)

In (4.40), the size of vector p(UT ) is 18, indicating that there are 18 unknowns for an

untransposed line. Therefore, H(UT ),v is a 12-by-18 matrix and can be built in a way that its each

column equals to a combination of the columns in H(G),v.

H(UT ),v =[H(G),v,1 −H(G),v,13, H(G),v,2, H(G),v,3 −H(G),v,15, H(G),v,4,

H(G),v,5 −H(G),v,17, H(G),v,6, H(G),v,14, H(G),v,16, H(G),v,18,

H(G),v,7 −H(G),v,19, H(G),v,8, H(G),v,9 −H(G),v,21, H(G),v,10,

H(G),v,20, H(G),v,11 −H(G),v,23, H(G),v,12, H(G),v,24

](4.41)

After formulating H(UT ),v, one can find that the rank deficiency of the measurement-

parameter coefficient matrix H(UT ),v will make the traditional weighted least squares technique fail

to estimate the parameters. As discussed before, several methods proposed previously use more

than two snapshots, i.e. multi-scan measurements to mitigate such rank deficiency problem. The

121


number of snapshots may vary due to different scenarios: if the sampling time is short and the

measurements do not vary much, more snapshots will be required to provide enough information for

estimating parameters. One way is to shrink the time scale by using multi-scan measurements in

one estimation. As an alternative, a sliding window method will capture the dynamic variation of

parameters more precisely. It constructs a window that contains T consecutive currents measured

from time k−T + 1 to time k. The index T is referred to as the footstep of the estimation. Therefore,

using the measurements inside this sliding window, a new measurement model can be constructed,

viz. Ik−T+1

Ik−T+2

...

Ik

=

H(UT ),v,k−T+1

H(UT ),v,k−T+2

...

H(UT ),v,k

p(UT ) +

ek−T+1

ek−T+2

...

ek

(4.42)

However, there are significant drawbacks of the least-squares method in calculating param-

eters of an untransposed transmission line, especially when utilizing PMU measurements:

• Noise in measured voltages are not considered. Though (4.40) and (4.42) model the noise in

current measurements properly, the influence of voltage measurement noise can not be ignored.

And this is also a general drawback of the static parameter estimation.

• It is computationally expensive for the evaluation of the coefficient matrix H(UT ),v.

• The rank deficiency problem still holds for some scenarios. Rank deficiency of H(UT ),v is

the motivation for using multi-scan measurements. However, under normal operation of the

power system, or when the time interval used to acquire the multi-scan snapshots is very

short, consecutive measurement are all nearly the same, so that the coefficient matrix H(UT ),v

in (4.42) is still very nearly rank-deficient. Figure 4.3 and Figure 4.4 present a comparison

of the singular values of the coefficient matrices in (4.40) and (4.42), using actual PMU

measurements. Clearly, using more snapshots does not improve the singular value profile of

the coefficient matrix H(UT ),v: it still consists of a few dominant singular values and a very

high σmax/σmin ratio. A longer multi-scan interval is needed to get a meaningful increase

in the smallest singular value. For measurement acquired by PMUs, this could translate to

hundreds of snapshots.

• The previous rank deficiency problem leads to a new problem: it is of large difficulty and

randomness to decide the time range which contains enough information for the parameter

122


1 2 3 4 5 6

Singular Value Index

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

log

10(S

ing

ula

r V

alu

e o

f C

oe

ffic

ien

t M

atr

ix)

Single Snapshot

Figure 4.3: Singular value of coefficient matrix using single snapshot measurement

1 2 3 4 5 6

Singular Value Index

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

log

10(S

ing

ula

r V

alu

e o

f C

oe

ffic

ien

t M

atr

ix)

100 Snapshots

Figure 4.4: Singular value of coefficient matrix using multiple snapshot measurements

estimation and does not violate the assumption of slowly changing operating conditions.

These observations motivate us to adopt a dynamic approach, based on a combination of

state estimation and parameter tracking, which will be presented in Section 4.5.

4.4.2 Static Parameter Estimation on Partially Untransposed Line

Although the static parameter estimation approach has limited accuracy in the untransposed

case, it can be used to initialize our parameter tracking procedure. In this section, we deviate the

rank deficiency problem by applying the formulation (4.34) to a partially transposed line and thus

reducing the number of distinct unknown parameters to 11, which allows us to use a single snapshot

of voltage and current measurements.

As discussed in Section 4.3.4, the vector of unknown parameters is p(PT ), thus, (4.34) can

be rewritten as

I = H(PT ),v p(PT ) + e. (4.43)

123


Here I is a 12-by-1 vector denoting currents in real-imaginary decoupled format as in (4.32), and

H(PT ),v is a 12-by-11 current-to-parameter coefficient matrix consisting of entries in units of voltages.

The formulation of H(PT ),v can be derived by matrix manipulation according to (4.11), (4.22) and

(4.43).

An alternative way to derive H(PT ),v is by manipulating H(G),v in (4.39). Similar to the

(4.41), H(PT ),v can be built as a combination of the columns in H(G),v.

H(PT ),v =[H(G),v,1 +H(G),v,1 −H(G),v,13 −H(G),v,23,

H(G),v,2 +H(G),v,12 −H(G),v,14 −H(G),v,24,

H(G),v,3 +H(G),v,9 −H(G),v,15 −H(G),v,21,

H(G),v,4 +H(G),v,10 −H(G),v,16 −H(G),v,22,

H(G),v,5 −H(G),v,17, H(G),v,6 −H(G),v,18,

H(G),v,7 −H(G),v,19, H(G),v,8 −H(G),v,20,

−H(G),v,14 −H(G),v,20 −H(G),v,24,

−H(G),v,16 −H(G),v,22, −H(G),v,18

]. (4.44)

4.4.3 Static Parameter Estimation on Transposed Line

Since for a transposed line, there are only six unknowns to be estimated. Thus the parameter

estimation problem in a transposed line can easily be formulated as

I = H(TT ),v p(TT ) + e, (4.45)

where H(TT ),v is a 12-by-6 current measurement to parameter coefficient matrix and can be formu-

lated according to (4.28). Also, it can be formulated as following.

H(TT ),v =[H(G),v,1 +H(G),v,7 +H(G),v,11 −H(G),v,13 −H(G),v,19 −H(G),v,23,

H(G),v,2 +H(G),v,8 +H(G),v,12,

H(G),v,3 +H(G),v,5 +H(G),v,9 −H(G),v,15 −H(G),v,17 −H(G),v,21,

H(G),v,4 +H(G),v,6 +H(G),v,10,

H(G),v,14 +H(G),v,20 +H(G),v,24,

−H(G),v,16 +H(G),v,18 +H(G),v,22

]. (4.46)

124


4.4.4 Static Estimation of Positive Sequence Parameters

For the positive sequence parameters, it is easy to construct the measurement equation as

I = H(PS),v p(PS) + e, (4.47)

where p(PS) is a 3-by-1 parameter vector, I is a 4-by-1 current measurement vector indicating the

rectangular form of positive sequence current of both ends of the line.

I =

Isr

Isi

Imr

Imi

. (4.48)

H(PS),v is a 4-by-3 coefficient matrix, as shown in (4.49)

I =

Isr

Isi

Imr

Imi

=

p1 p2 − p3 −p1 −p2

p3 − p2 p1 p2 −p1

−p1 −p2 p1 p2 − p3

p2 −p1 p3 − p2 p1

Vsr

Vsi

Vmr

Vmi

+ e

= H(PS),vp(PS) + e

=

Vsr − Vmr Vsi − Vmi −VsiVsi − Vmi Vmr − Vsr Vsr

Vmr − Vsr Vmi − Vsi −VmiVmi − Vsi Vsr − Vmr Vmr

p1

p2

p3

+ e. (4.49)

4.5 Joint State Estimation and Parameter Tracking for Untransposed

Lines

Since the parameters of an untransposed transmission line can not be determined by the

static state estimator, a joint state estimation and parameter tracking method is proposed in this

section. Considering the measurement noise, a generic measurement model (at time k) can be written

as

zk = h(xk, pk) + ek (4.50)

125


wherezk : vector including all voltage and current measurements at time k, zk = Vk

Ik

;

xk : state vector containing voltages at both ends of the transmission line at time

k;

pk : line parameter vector at time k, it is an 18-by-1 vector;

ek : vector of measurement noise at time k.Usually, h(xk, pk) is highly nonlinear if xk is augmented by pk. The proposed method is

to formulate (4.50) as two loosely-coupled sub-problems, namely, state estimation and parameter

tracking. We formulate state estimation as a WLS problem, which avoids the appreciable errors

by introducing state dynamics. We track parameters using Kalman filter where dynamics of the

parameters is assumed, considering parameters vary as a random walk process during a short period.

Using this method, xk is first estimated using the most recent estimated pk, because it is

uncertain rather than completely unknown. Then using the estimated xk , pk can be updated by

parameter tracking. Finally, using the newly updated pk, xk can be estimated again. This iterative

process will finally converge to the desired state and parameter estimates.

4.5.1 Three Phase State Estimation

The measurement model for state estimation is

zx,k = Hp,k xk + vx,k (4.51)

wherezx,k : measurement vector for state estimation at time k,zx,k = zk ;

vx,k : vector of measurement noise with zero mean and covariance Rx;

Hp,k : measurement-state coefficient matrix at time k, defined as Hp,k =

[I12×12, Hp,k] ∈ R24×12 where I12×12 indicates an identity matrix and

Hp,k ∈ R12×12 is constructed by the most recent parameter estimates pk.In (4.51), it shows a linear function relating the measurements to the state variables. The number of

measurements here is 24, while the number of states to be estimated is 12. WLS state estimation

procedure can make use of the redundant measurements in order to filter out the measurement noise.

According to the basic knowledge of WLS state estimation [2], xk can be determined as

xk =(HTp,kR

−1x Hp,k

)−1HTp,kR

−1x zx,k . (4.52)

126


The above formulation is basically a three phase static state estimator for an isolated single

line where measurements are acquired from raw PMU data. However, if there is three phase state

estimator for the whole system, one can use the real-time state estimates as the measurements instead

of raw data. In such case, the impact of gross errors and synchronization errors among raw data can

be eliminated, and therefore improves the reliability of our proposed algorithm.

4.5.2 Parameter Tracking

Once the states are estimated, the next step is to estimate the parameters. In this section,

the measurement model for parameter tracking is constructed as

zp,k = Hx,k pk + vp,k (4.53)

wherezp,k : measurement vector for parameter tracking at time k, zp,k = Ik;

vp,k : vector of measurement noise with zero mean and covariance Rp;

Hx,k : current-parameter coefficient matrix consisted of xk, Hx,k ∈ R12×18.

The

dynamic behavior of line parameters can be modeled as

pk = pk−1 + wp,k (4.54)

where wp,k is the process noise caused by disturbances (wind, rain, etc.) with zero mean and

covariance Qp. (4.54) is referred as the prediction model of parameters, which implicitly assumes

that the parameters stay unchanged except for the effect of zero-mean disturbance. Nevertheless, it

can be replaced by other models such as:

pk = f(pk−1) + wp,k, (4.55)

if enough information to predict transmission line conditions (e.g. weather forecast) is somehow

available.

The Kalman filter, also known as linear quadratic filter, is applied for solving parameter

tracking. Kalman filter is implemented using an algorithm that uses a series of measurements

observed over time containing noise, and estimates the unknown variables with the consideration of

both measurement model and prediction model. The recursive steps of the Kalman filter using the

state dynamic model (4.54) and measurement model (4.53) are outlined below.

State predict:

pk|k−1 = pk−1|k−1 (4.56)

127


Pk|k−1 = Pk−1|k−1 +Qp (4.57)

Measurement update:

Kk = Pk|k−1HTx,k

(Hx,kPk|k−1H

Tx,k +Rp

)−1(4.58)

pk|k = pk|k−1 +Kk

(zp,k −Hx,kpk|k−1

)(4.59)

Pk|k = Pk|k−1 −KkHx,kPk|k−1 (4.60)

wherepk|k−1 : estimate at time k given measurements up to and including time k − 1;

pk|k : estimate at time k given measurements up to and including time k;

Pk|k−1 : covariance matrix of pk|k−1;

Pk|k : covariance matrix of pk|k;

Kk : Kalman gain.

4.5.3 Joint State Estimation and Parameter Tracking

Details of the two major stages of the proposed joint state estimation and parameter tracking

(JSEPT) approach are presented in the previous sections. The overall procedure of the proposed

method can now be expressed as below:

Step 1: Initialize p0|0, P0|0 and set k = 1.

Step 2: For each measurement instant k(k ≥ 1), Let j = 1, pk−1|k−1,j = pk−1|k−1 andPk−1|k−1,j =

Pk−1|k−1. The subscript j indicates the jth iteration.

Step 3: Process the State Predict part of parameter tracking by introducing the subscript j and

deriving pk|k−1,j and Pk|k−1,j .

Step 4: For state estimation, formulate Hp,k,j using the predicted parameters pk|k−1,j , then process

state estimation and derive xk,j .

Step 5: Back to the Measurement Update part of parameter tracking, formulate Hx,k,j using xk,j ,

then derive pk|k,j and Pk|k,j .

Step 6: Termination criterion include the following:

• j > 1 and ‖pk|k,j − pk|k,j−1‖ < ε;

128


• j > N .

Either of the above criterion is satisfied, go to Step 7. Otherwise, repeat Step 3 to Step 5

with j = j + 1 and set pk−1|k−1,j = pk|k,j−1 and Pk−1|k−1,j = Pk|k,j−1.

Step 7: Iteration of j is terminated. The estimated values at this instant k are pk|k = pk|k,j ,

Pk|k = Pk|k,j and xk = xk,j . If k ≤ M , let k = k + 1, go back to Step 2 to process next

instant; else go to Step 8.

Step 8: Terminate.

Note the following quantities in the above procedure:

ε : predetermined threshold to decide the termination of internal iteration;

N : the iteration limit of j in each instant k;

M : the number of measurements.The proposed method gives a generic approach to solve the problem iteratively. From

the viewpoint of state estimation, the estimated voltages depend on not only the measurements,

but also the most recently estimated parameters. The introduction of redundant measurements, i.e.

voltages, ensures the innovation of the Kalman filter does not equal to zero. For parameter tracking,

since the measurements are only currents which are not enough to run WLS estimation, historical

record is also taken into consideration. The Kalman filter is needed to produce statistically optimal

estimate of the parameters. The termination criterion in Step 6 ensures the convergence and stability

of the algorithm. The desired estimate of parameters can be obtained when two successive iterations

provide close enough solutions. The iteration limit ensures the parameter estimates will not become

outliers. Empirical simulations show that one hundred iterations are enough for tracking varying

parameters. Figure 4.5 illustrates the overall iterative procedure.

129


Figure 4.5: Flowchart of JSEPT overall Procedure

130


4.5.4 Initialization of Parameter Tracking

For the Kalman filter of parameter tracking, there are two sets of values need to be

predetermined: noise covariance and the initial value of the states and parameters. The first one is

related to Kalman filter tuning problem and there are a lot of literature about this topic. usually, the

covariance could be chosen as P0|0 = p0|0 pT0|0 if no elements in p0|0 equals to 0, or identity matrix

otherwise. In the simulations provided in the Section 4.6, the ratio of process noise covariance to

measurement noise covariance is one thousand.

For the initialization, parameters that are not far from their actual values are more appre-

ciable. According to simulation results, zero initials require much more iterations to converge. It

may sometimes fail to converge to actual ones within available measurements because of multiple

solution problem. However, considering for normal operations, one would have an approximate

knowledge of the parameters, the initial value requirement can be achieved. In addition, the static

parameter estimation algorithm proposed in Section 4.4.2 approximates the structure of parameters

to reduce the size of unknowns, which will provide a helpful initialization to mitigate the problem.

Simulations show that such estimates are accurate enough to use as initial values for the proposed

JSEPT procedure.

It is also necessary to mention that the size of initial parameters is 11, which is different to

the number of parameters in JSEPT. There is a need to augment the vector using equality properties

of the elements. Such transformation is listed below.

p(UT ),1 = p(PT ),1

p(UT ),2 = p(PT ),9 − p(PT ),2

p(UT ),3 = p(PT ),3

p(UT ),4 = p(PT ),10 − p(PT ),4

p(UT ),5 = p(PT ),5

p(UT ),6 = p(PT ),11 − p(PT ),6

p(UT ),7 = p(PT ),2

p(UT ),8 = p(PT ),4

p(UT ),9 = p(PT ),6

p(UT ),10 = p(PT ),7

p(UT ),11 = p(PT ),9 − p(PT ),8

p(UT ),12 = p(PT ),3

p(UT ),13 = p(PT ),10 − p(PT ),4

131


p(UT ),14 = p(PT ),8

p(UT ),15 = p(PT ),4

p(UT ),16 = p(PT ),1

p(UT ),17 = p(PT ),9 − p(PT ),2

p(UT ),18 = p(PT ),2

4.5.5 Practical Applications

There are several implementations of the proposed algorithm. A transmission line may

suffer from potential parameter change due to repair, maintenance or weather change. It is also

necessary to calculate the parameters of a new transmission line and verify parameters for aged

lines. The proposed algorithm can be implemented for such lines, relying on existing or temporarily

installed PMUs, to estimate parameters precisely.

4.6 Simulation of JSEPT on Parameters of Three Phase Untransposed

Line

To demonstrate the performance of proposed method, the problem of untransposed transmis-

sion line parameter identification is tested under two scenarios: ATP based experimental simulation

and actual PMU measurements based calculation.

To evaluate the results numerically, mean-percentage-error (MPE), normalized root-mean-

squared-error (NRMSE) and correlation coefficient (CC) are introduced. The superscript i of p

indicates ith parameter, and M indicates the number of measurements.

MPE(i) =100%

M

M∑k=1

p(i)k − p

(i)actual,k

p(i)actual,k

, (4.61)

NRMSE(i) =RMSE(i)∣∣∣∣ 1

M

M∑k=1

p(i)actual,k

∣∣∣∣ =

√1M

M∑k=1

(p(i)k − p

(i)actual,k)

2∣∣∣∣ 1M

M∑k=1

p(i)actual,k

∣∣∣∣ . (4.62)

132


Correlation coefficient (CC) can evaluate the capability of tracking time-variant parameters.

The closer CC is to 1, the more similar the two trajectories. It is defined as

CC(i) =cov(p(i), p

(i)actual)

σp(i)σp(i)actual

, (4.63)

where cov indicates the covariance of two variables and σ stands for standard deviation.

4.6.1 Simulation based on ATP model

This scenario is based on the simulation in ATP, which is a universal program system for

digital simulation of transient electromagnetic. An lumped line model of 50km long, 3 phase, 8

circuits overhead untransposed transmission line is built in ATP, as shown in Figure 4.6 It lies between

Figure 4.6: ATP Simulation Model

SEND end and RECV end that highlighted in red. Nodal voltage and line current measurements are

on both terminals of the transmission line. These measurements are then modified by adding white

noise with zero mean and 0.001 p.u standard deviation. Since the actual parameters can be calculated

from the predetermined simulation model, a comparison between estimated and actual parameters is

provided, denoted as blue circles and red dashed lines in the following graphs respectively. Only two

representative parameters, p1 and p10 are plotted. Besides, categorized by the different variations of

line parameters, three cases are presented and discussed to illustrate the performance of proposed

algorithm. Note that in the following figures, y-axis indicates the per-unit value of the corresponding

parameters and x-axis indicates the number of measurement scans.

1) Constant parameter case. In this case parameters are time-invariant. Figure 4.7 shows

the estimated results of 2 parameters given 20 measurements. Plots indicate that all the estimated

parameters are very close to the actual ones.The values of MPE and NRMSE are shown in Table 4.2.

2) Varying parameter case A2. To simulate time-variant line parameters, transmission line

distance are manipulated in ATP. In this case A, parameters are changing slowly and continuously to

imitate continuously changing circumstances, such as increasing temperature. Simulation results are

presented in Figure 4.8, indicating the estimates are close to actual values.

133


0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

3.8

3.9

4

4.1

Valu

e (

p.u

.)

p1

0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

4

4.2

4.4

Valu

e (

p.u

.)p10

Figure 4.7: Estimation results of constant parameter case A1

Table 4.2: MPE and NRMSE of constant parameters case A1

Parameter p1 p2 p3 p4 p5 p6

MPE(%) 0.01 -0.01 -0.04 -0.01 0.03 -0.01

NRMSE(×10−4) 2.14 1.09 5.19 1.03 2.65 1.45


MPE(%) -0.01 -0.00 0.01 -0.02 0.01 0.05

NRMSE(×10−4) 1.12 3.49 1.63 2.54 1.14 4.78


MPE(%) -0.01 -0.01 0.01 -0.02 -0.01 -0.01

NRMSE(×10−4) 0.99 1.23 1.32 3.04 1.07 1.02

Since in this case, parameters are changing along with time, therefore MPE will no longer

be helpful to evaluate the performance. Thus the correlation coefficient (CC) is provided, as shown in

Table 4.3. Figure 4.8, along with Table 4.3, indicates that proposed algorithm has a good performance

when the parameters are changing.

3) Varying parameter case A3. In this case, parameters are set to be varying at some

specific measurement points. This is aim to simulate the situation when transmission line suffers

from sudden shortage or climate change, such as rain, wind. Figure 4.9 and Table 4.4 show that

proposed algorithm also performs well under the varying parameter scenario.

4) Erroneous initial parameter case (A4). Erroneous initial parameter values will have

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0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

3.85

3.9

3.95

4

Valu

e (

p.u

.)

p1

0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

4.15

4.2

4.25

4.3

Valu

e (

p.u

.)p10

Figure 4.8: Estimation results of varying parameter case A2

Table 4.3: CC and NRMSE of varying parameter case A2


CC 0.99 1.00 0.99 0.95 0.98 1.00

NRMSE(×10−2) 0.31 0.09 0.27 1.52 0.24 0.10


CC 0.96 1.00 0.96 1.00 1.00 0.99

NRMSE(×10−2) 0.73 0.05 0.56 0.14 0.04 0.29


CC 0.97 0.97 1.00 0.99 0.98 0.99

NRMSE(×10−2) 0.56 0.62 0.11 0.03 0.37 0.38

different impact under different circumstances, as shown in Figure 4.10 For small errors, the algorithm

can rapidly adjust the estimates by placing more emphasis on measurement update process and

thus the estimates can be corrected quickly. Actually this is the simulation case of actual PMU

measurements where the initialization is not perfect. For large errors, it will take longer for the

proposed algorithm to reach a good solution and the convergence rate will largely depend on the

variation of measurements.

Simulations based on ATP are presented and discussed above. Though the time-varying

case may not be realistic, the promising results indicate that proposed algorithm can handle and

perform well under different variations of line parameters. And the following application to actual

135


0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

4

4.2

4.4

Valu

e (

p.u

.)

p1

0 2 4 6 8 10 12 14 16 18 20

Scan of Measurement

4.1

4.2

4.3

4.4

Valu

e (

p.u

.)p10

Figure 4.9: Estimation results of varying parameter case A3

Table 4.4: CC and NRMSE of varying parameter case A3


CC 0.99 0.91 0.99 1.00 1.00 0.90

NRMSE(×10−2) 0.09 0.84 0.48 0.05 0.21 0.40


CC 1.00 0.95 0.92 1.00 0.99 0.99

NRMSE(×10−2) 0.07 0.75 0.56 0.12 0.13 0.41


CC 0.97 0.99 0.95 0.99 1.00 1.00

NRMSE(×10−2) 0.56 0.09 0.74 0.10 0.07 0.08

PMU data will provide another proof.

4.6.2 Estimation based on actual PMU measurements

The second scenario is based on actual PMU measurements on both sides of one 345KV

transmission line. Since we don’t have access to the three phase parameters of this line but just

positive sequence values, the static algorithm to calculate parameters is implemented first. Then,

using these estimated parameters as initial values, the proposed joint state estimation and parameter

tracking algorithm is implemented. Simulations are based on 9800 measurements during nearly

six minutes (30 Hz sample rate). Results are presented in Figure 4.11 where x-axis indicates the

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0 20 40 60 80 100 120 140 160 180 200

Scan of Measurement

3.8

3.9

4

4.1

Valu

e (

p.u

.)

p1

0 20 40 60 80 100 120 140 160 180 200

Scan of Measurement

4

4.2

4.4

Valu

e (

p.u

.)p10

Figure 4.10: Estimation results of erroneous parameter initial case A4

measurement snapshots. Note that parameters here are in per unit values.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Scan of Measurement

11.216

11.218

11.22

11.222

Valu

e (

p.u

.)

p1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Scan of Measurement

17.605

17.61

17.615

17.62

Valu

e (

p.u

.)

p10

Figure 4.11: Estimation results of actual PMU measurements

It can be seen from the plot that line parameters do not vary much during the experiment

period. Positive sequence parameters from utility database are provided in Table 4.5. As a comparison,

the mean value of estimated parameters are transformed into positive sequence values. Results show

that the estimated values match the database closely.

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Table 4.5: CC and NRMSE of actual case

Parameter R X B

Database value (p.u.) 0.000481 0.006223 0.11114

Estimated value mean 0.000498 0.006475 0.11440

4.7 Transmission Line Model of a Power Grid

The static method proposed in Section 4.4 can solve for the positive sequence parameters

as well as the parameters of a partially transposed line or a transposed line. And in Section 4.5,

JSEPT aims for solving parameters in an untransposed line. All the above discussions are focusing

on a single transmission line, assuming PMUs on both terminals.

It is obvious that the parameter tracking can be implemented into a power grid, with PMUs

installed on every buses. However, a new problem is carried out that whether it is possible to reduce

the number of PMUs needed for the parameter tracking. In this section, the transmission lines of

a power grid are considered. The positive sequence model for the transmission lines in a power

grid is described and analyzed along with a generic technique to build the branch current to bus

voltage coefficient matrix. Afterwards, a PMU placement technique is proposed to reduce the number

of measurements needed for tracking the parameters all lines. A concise simulation results are

provided at last to compare the number of PMUs needed for parameter tracking with the one for state

estimation. The algorithm for tracking line parameters in a power grid, i.e. JSEPTS, is described in

the next section.

4.7.1 Formulation of Transmission Line Model of a Power Grid

In this section, the formulation of transmission line model and the power grid model are

illustrated. Recall that for a single transmission line l1 (with two terminals, i.e. ’from’ side f : sm

and ’to’ side t : sn), an equivalent π model with lumped parameters can be constructed as (4.11) and

(4.12). One can rewrite (4.11) using the notation specially for a power grid, as shown in (4.64).Il1:f−t

Il1:t−f

=

Yl1 + YBl1 −Yl1−Yl1 Yl1 + YBl1

VsmVsn

(4.64)

where Il1:f−t indicates the branch current from the f side to t side of branch l1, which are sm and sn

respectively. Yl1 = gl1 + jbl1 is the series admittance and YBl1 = jBl1 is the shunt admittance. It

138


is assumed that the shunt admittances are equal at both end of the transmission line and the shunt

conductance is neglected. Compared to (4.12), it holds that Yl1 = Z−1series and YBl1 =

1

2Yshunt.

For a given power system with Nb buses and Nbr branches, a branch-bus incidence matrix

A1 can be constructed as a real Nbr-by-Nb matrix. The elements of each row of A1 will have three

values

• If the corresponding bus is the f side of the branch, then the value of the element will be 1.

• If the corresponding bus is the t side of the branch, then the value of the element will be -1.

• All the rest elements are 0.

Then, two augmented branch-bus incidence matrices, denoted as A1 and A2, can be constructed as

A1 =

A1

−A1

, (4.65)

and A2 can be obtained by substituting all -1 in A1 with 0. An example is provided for a 3-bus

system shown in Figure 4.12. Note that for the branch-bus incidence matrix, 1 is assigned to f side

and -1 is assigned to t side.

Figure 4.12: Three bus system to illustrate the construction of branch-bus incidence matrix

A1 =

1 −1 0

0 1 −1

, A1 =

1 −1 0

0 1 −1

−1 1 0

0 −1 1

, A2 =

1 0 0

0 1 0

0 1 0

0 0 1

,

Given the series admittances and shunt admittances of all branches, one can construct the series

admittance matrix Yc ∈ C2Nbr×2Nbr and shunt admittance matrix YB,c ∈ C2Nbr×2Nbr as

Yc = diag(Yl1 , Yl2 , . . . , YlNbr, Yl1 , Yl2 , . . . , YlNbr

) (4.66)

YB,c = diag(YBl1 , YBl2 , . . . , YBlNbr, YBl1 , YBl2 , . . . , YBlNbr

), (4.67)

139


where diag(X) indicates to formulate a square matrix with the elements of X on the main diagonal.

Note that the admittance of each branch is repeated twice in the diagonal of Y and YB .

Then, we can construct the branch current vector Ic ∈ C2Nbr×1 and Vc ∈ C2Nb×1 as

Ic =

Il1:f−t

Il2:f−t...

IlNbr:f−t

Il1:t−f...

IlNbr:t−f

, Vc =

Vs1

Vs2...

VsNb

, (4.68)

and the relationship between all branch currents and bus voltages of the system can be written as

Ic = YcA1Vc + YB,cA2Vc (4.69)

Since all the variables and matrices are complex, it is good to decouple the complex

variables into their real (r) and imaginary (i) components to simplify the formulation and calculation.

Now consider branch series conductance g ∈ RNbr×1 and diagonal matrix g ∈ R2Nbr×2Nbr as

g = [gl1 , gl2 , · · · , glNbr]T , (4.70)

g =

diag(g) 0

0 diag(g)

. (4.71)

Similarly, series susceptance b ∈ RNbr×1, shunt B ∈ RNbr×1 and the corresponding matrices b and

B can also be constructed as

b = [bl1 , bl2 , · · · , blNbr]T , (4.72)

b =

diag(b) 0

0 diag(b)

, (4.73)

B = [Bl1 , Bl2 , · · · , BlNbr]T , (4.74)

B =

diag(B) 0

0 diag(B)

. (4.75)

Thus, we can have the series admittance matrix

Yc = g + jb (4.76)

140


and the shunt admittance matrix

YB,c = jB (4.77)

The rectangular form of all branch currents I ∈ R4Nbr×1 and all bus voltages V ∈ R2Nb×1

can be expressed as

I =

Ic,rIc,i

=

Il1:f−t,r...

IlNbr:f−t,r

Il1:t−f,r...

IlNbr:t−f,r

Il1:f−t,i...

IlNbr:f−t,i

Il1:t−f,i...

IlNbr:t−f,i

(4.78)

V =

Vc,rVc,i

=

Vs1,r

Vs2,r...

VsNb,r

Vs1,i

Vs2,i...

VsNb,i

. (4.79)

Note that the sequence of real and imaginary components in current and voltage vector of (4.78) and

(4.79) are different from the ones in (4.32) and (4.33).

Therefore, (4.69) can be rewritten in rectangular form as

I =

Ic,rIc,i

= Y V =

gA1 −bA1 − BA2

bA1 + BA2 gA1

Vc,rVc,i

. (4.80)

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4.7.2 PMU Placement for Tracking Parameters of Multiple Lines in a Power Grid

Commonly considered objective of the optimal PMU placement (OPP) problem [93, 94]

is to find the minimum number of PMUs and their locations in order to render a fully observable

network. However, the required number of PMUs for the proposed parameter tracking method will be

higher in order to not only ensure observability, but also to track parameters of every network branch.

In order to simplify the placement procedure, installed PMUs are assumed to measure voltage phasor

at the bus and current phasors at all branches incident to that bus. For purposes of this study, zero

injection buses are not considered, but they can be easily incorporated if needed.

To estimate the parameters of a line, at least one current measurement located on either

terminal is required. This is because if without any measurement, such branch will become irrelevant

in sense of state estimation. Furthermore, the topology of the system needs to be considered carefully.

One can categorize the buses into two:

• terminal bus: bus with a single incident branch;

• internal bus: a bus which is not a terminal bus.

Note that the bus set of all terminal buses is denoted as TB, and that of all internal buses is IB. All

the buses of the system consist the set B that B = TB ∪ IB. Note that for a given system, it is

possible that TB = ∅ or IB = ∅.Consider again the 3-bus system in Figure 4.12. Buses 1 and 3 are terminal buses and

bus 2 is an internal bus. One PMU can be placed at bus 2 to achieve state estimation observability.

However, the parameters of branch l1 and l2 still cannot be determined and tracked. This is because

the currents (Il1:B2−B1 and Il1:B2−B3) are critical measurements, observing the states associated

with buses 1 and 3. On the other hand, if two PMUs are installed at buses 1 and 3, two currents

Il1:B1−B2 and Il1:B3−B2 will become redundant since the system will now remain observable upon

elimination of either current. Therefore, the objectives of optimal PMU placement for tracking

parameters of all branches (OPPTP) can be stated as:

O1 : at least one current measurement on every branch;

O2 : all current measurements must be redundant (in the state estimation sense);

O3 : install a minimum number of PMUs.It can be easily proved that for a system with TB = ∅, O2 can be satisfied immediately for a PMU

placement scheme satisfying O1. For a system with terminal buses, current redundancy can be

achieved by placing PMUs at the terminal buses. Therefore, the following optimization problem for

142


OPPTP can be formulated:

minNb∑i=1

qi

s.t.∣∣A1

∣∣q ≥ 1

qj = 1, for all j ∈ TB

qi ∈ 0, 1

(4.81)

In (4.81), qi is a binary scalar indicating whether a PMU is installed at bus i. Nb is the

number of buses in the system. A1 is the branch-bus incident matrix defined in Section 4.7.1.∣∣A1

∣∣indicates that all entries of A1 are replaced by their absolute values. q is a vector of all qi’s and 1 is a

vector of Nbr 1’s.

Solving the optimization problem (OPPTP) of (4.81), the optimal number of PMUs can be

determined. In Table 4.6, solutions are provided for different systems, along with the results obtained

by solving traditional OPP as a comparison. Based on the simulated cases it is observed that in order

to satisfy OPPTP requirements PMUs need to be installed at around 60% of the buses, whereas this

number will be around 30% for the case of OPP carried out strictly for observability of states.

Table 4.6: Results of PMU placement for parameter tracking of a power grid

System Num. OPPTP Num. OPP

IEEE 14 bus system 8 4




2071 bus system 1207 634

4.8 Joint State Estimation and Parameter Tracking for System (JSEPTS)

In this section, a method to estimate positive sequence line parameters of a power grid

using a few PMUs located on selected buses is proposed. Extending the idea of iterative processing

developed in Section 4.5, a method of joint state estimation and parameter tracking for systems

(JSEPTS) is developed. JSEPTS has two parts: a phasor-only linear state estimator for the power grid,

and a linear Kalman filter based parameter estimator for one or several lines in the system. Similarly,

143


the JSEPTS alternates between state estimation, which depends on the most recent parameter

estimates, and parameter tracking, which depends on the most recent state estimates. By iteratively

conducting state estimation and parameter tracking, the mismatch between actual and estimated

parameters is minimized.

4.8.1 State Estimation Formulation

Given a system with PMUs strategically installed at some buses, one can define the current

measurement vector Iz and voltage measurement vector Vz as

Iz = [· · · , Ism−sn,r, Ism−sn,i, . . . ]T , (4.82)

Vz = [· · · , Vsj ,r, Vsj ,i, . . . ]T , (4.83)

where

sm − sn ∈ CL : Current Phasor Location List

sj ∈ V L : Voltage Phasor Location List.

Note that Iz ∈ R2NzI×1 and Vz ∈ R2NzV ×1, where NzI is the number of current measurements, and

NzV is the number of voltage measurements. Besides, Nzinj zero injections formulates the vector

0 = [0, 0, . . . , 0]T consists of 2Nzinj zeros.

The measurement model for state estimation based on phasor measurement at time k can

be formulated as

zx,k =

Vz,k

Iz,k

0

=

I

Hp,I,k

Hp,inj,k

xk + vx,k = Hp,k xk + vx,k (4.84)

where zx,k ∈ R(2NzI+2NzV +2Nzinj)×1 is the vector of all measurements, consisting of measured

voltage V , current I at time k, and zero injection 0. xk ∈ R2Nb×1 indicates the states of the system

to be estimated at time k. vx,k is the vector of measurement noise with zero mean and covariance Rx.

Note that for zero injections, a very small value is assigned to the corresponding positions in Rx.

The subscript p in Hp,I,k and Hp,inj,k indicates the vector containing all the branch

parameters of the system, i.e.

p = [gT , bT , BT ]T ∈ R3Nbr×1.

At time k, according to (4.80), Y can be formulated by pk, denoted as Yk. The coefficient of

measurements to states Hp,k is linear and contains three parts:

144


• I ∈ R2NzV ×2Nb consists of 1’s and 0’s, indicating the coefficient between voltage measure-

ments and states;

• Hp,I,k ∈ R2NzI×2Nb can be formulated as part of rows of Yk, indicating the coefficient between

current measurements and states;

• Hp,inj,k ∈ R2Nzinj×2Nb can be formulated by adding some rows of Yk.

The weighted least squares (WLS) state estimation procedure can make use of the redundant

measurements in order to filter out the measurement noise. By solving the minimization problem

minimizex

J = (zx,k −Hp,k xk)TR−1

x (zx,k −Hp,k xk) (4.85)

xk can therefore be determined as

xk =(HTp,kR

−1x Hp,k

)−1HTp,kR

−1x zx,k . (4.86)

4.8.2 Parameter Tracking Formulation

Once the states are estimated, the next step is to estimate the parameters. The measurement

model for tracking the parameters of branch l1 can be constructed as

zp,l1,k = Hx,l1,k pl1,k + vp,l1,k (4.87)

where pl1,k ∈ R3×1 is the parameter vector consists of three parameters, gl1 , bl1 and Bl1 , at time k.

There could be one or two current measurements incident to branch l1, designated by zp,l1,k. vp,l1,k

is the vector of measurement noise. Hx,l1,k is the current-parameter coefficient matrix consists of the

most recent available state estimates xk.

According to (4.80), and assuming only one current measurement Il1:f−t is incident to

branch l1, Hx,l1,k can be formulated asIl1:f−t,r

Il1:f−t,i

=Hx,l1,k pl1,k

=

xsm,r − xsn,r −xsm,i + xsn,i −xsm,ixsm,i − xsn,i xsm,r − xsn,r xsm,r

pl1,k.(4.88)

145


Since Hx,l,k may suffer from rank deficiency problem with different choice of target

branches and measurements, the dynamic behavior of parameters is modeled as (4.89)

pl,k = pl,k−1 + wp,l,k, (4.89)

where wp,l,k is the process noise caused by variations of ambient conditions such as temperature or

wind. It is with zero mean and covariance Qp,l. Such prediction model implicitly assumes that the

parameters stay unchanged except for the effect of zero-mean disturbance.

To track the parameters dynamically, the Kalman filter is needed to produce statistically

optimal estimate of the parameters. It can estimate the unknown variables with the consideration

of both prediction model and measurement model. Similar to the description in Section 4.5.2, the

recursive steps of the Kalman filter using the state dynamics (4.89) and measurement model (4.87)

are outlined below.

State Prediction:

pl,k|k−1 = pl,k−1|k−1 (4.90)

Pk|k−1 = Pk−1|k−1 +Qp,l (4.91)

Measurement Update:

Kk = Pk|k−1HTx,l,k

(Hx,l,kPk|k−1H

Tx,l,k +Rp,l

)−1(4.92)

νp,l,k = zp,l,k −Hx,l,kpl,k|k−1 (4.93)

pl,k|k = pl,k|k−1 +Kkνp,l,k (4.94)

Pk|k = Pk|k−1 −KkHx,l,kPk|k−1 (4.95)

wherepl,k|k−1 : parameter estimate at time k given measurements up to and including time

k − 1;

pl,k|k : parameter estimate at time k given measurements up to and including time

k;

Pk|k−1 : covariance matrix of pl,k|k−1;

Pk|k : covariance matrix of pl,k|k;

Kk : Kalman gain;

νp,l,k : innovation of the Kalman filter.

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4.8.3 Proposed JSEPTS Formulation

Details of the two major stages of the proposed joint state estimation and parameter tracking

for systems (JSEPTS) approach are presented in the previous sections. The overall procedure of the

proposed method can now be expressed as below:

Step 1: Initialize pl,0|0, P0|0 and set k = 1

Step 2: For each measurement instant k(k ≥ 1), Let j = 1, pl,k−1|k−1,j = pl,k−1|k−1 and

Pk−1|k−1,j = Pk−1|k−1. The subscript j indicates the jth iteration.

Step 3: Process the State Predict part of parameter tracking by introducing the subscript j and

deriving pl,k|k−1,j and Pk|k−1,j .

Step 4: For state estimation, form pk,j using the predicted parameters pl,k|k−1,j and the rest constant

parameters prest, then formulate Hp,k,j . Process state estimation and derive xk,j .

Step 5: Process the Measurement Update part of parameter tracking, formulate Hx,l,k,j using part

of the estimates xk,j , then derive pl,k|k,j and Pk|k,j .

Step 6: Termination criterion include the following:

• j > 1 and ‖pl,k|k,j − pl,k|k,j−1‖ < ε;

• j > M1.

Either of the above criterion is satisfied, go to Step 7. Otherwise, repeat Step 3 to Step 5

with j = j + 1 and set pl,k−1|k−1,j = pl,k|k,j−1 and Pk−1|k−1,j = Pk|k,j−1.

Step 7: Iteration of j is terminated. The estimated values at this instant k are pl,k|k = pl,k|k,j ,

Pk|k = Pk|k,j and xk = xk,j . If k ≤Mz , let k = k + 1, go back to Step 2 to process next

instant; else stop.

Note the following quantities in the above procedure:

ε : predetermined threshold to decide the termination of internal iteration;

M1 : the iteration limit of j in each instant k;

Mz : the number of measurement scans.The termination criterion in Step 6 ensures the convergence and stability of the algorithm.

The desired estimate of parameters can be obtained when two successive iterations provide close

enough solutions. The flowchart of JSEPTS is similar to the one in Section 4.5.3.

147


4.9 Simulation of JSEPTS on Parameters of a Power Grid

Simulations are carried out based on IEEE 14 bus system with varying operation conditions.

PMUs are installed at bus 2, 4, 5, 8, 9, 11, 12 and 13 and can provide 120 sets of time-synchronized

voltage and current measurements per second. Gaussian white noise with standard deviation 10−4

are added to the data captured within 20 seconds. This indicates that maximum measurement

uncertainties are within 0.05%. Parameters of only one branch are tracked and results for three

scenarios are provided. Rest of parameters are set to be Rp,l = 10−4, Qp,l = 10−2, ε = 10−5,

M1 = 10.

Case a. Constant Parameters of branch 13-14 are tracked with random initial values of

parameters. Simulation results are provided in Figure 4.13. Plots indicate that the parameters can be

precisely tracked after the initialization period.

0 5 10 15 20

1

1.2

1.4Line Parameter g

0 5 10 15 20

-2.4

-2.2

Line Parameter b

0 5 10 15 20-0.01

0

0.01Line Parameter B

Estimated

Actual

JSEPTS: IEEE 14 Bus system Branch 13 - 14

Initialization of PT : Random

CPU time 4.4615 second

x-axis label: (second)

Figure 4.13: Constant Parameters of branch 13-14

Case b. Varying parameters of branch 1-2 with abrupt changes are tracked with random

initial values of parameters. Simulations results are presented in Figure 4.14, indicating the estimation

results are close to actual values.

Case c. Varying Parameters of branch 7-8 are tracked with perfect initial values of

parameters. Figure 4.15 indicates that the proposed algorithm also performs well for the branch

148


0 5 10 15 20

4.55

5.56

6.5

Line Parameter g

0 5 10 15 20

1

1.5

Line Parameter b

0 5 10 15 20

0.2

0.4

0.6

Line Parameter B

Estimated

Actual


Initialization of PT : Random



Figure 4.14: Varying parameters of branch 1-2

incident to terminal bus 8.

0 5 10 15 20

-0.1

0

0.1

Line Parameter g

0 5 10 15 20

-6

-5

-4

Line Parameter b

0 5 10 15 20-0.01

0

0.01Line Parameter B

Estimated

Actual


Initialization of PT : Perfect



Figure 4.15: Varying Parameters of branch 7-8

149


The performance of the proposed JSEPTS algorithm is found to be highly satisfactory. The

recorded CPU times are commensurate with on-line implementation of the algorithm for real-time

tracking of the line parameters.

4.10 Conclusion

In this chapter, the accuracy of transmission line parameters are focused. Two transmission

line parameter tracking algorithms are proposed, one for a single untransposed transmission line and

the other for multiple lines in a power grid.

In detail, the transmission line model is analyzed and a linear relationship between the PMU

measurements and the unknown parameters is constructed at first. Particularly for untransposed lines,

due to the observability issue and rank deficiency of the measurement Jacobian matrix, traditional

methods will fail to provide real-time parameter estimates. Therefore, a joint state estimation and

parameter tracking (JSEPT) approach is proposed, involving the alternation between state estimation

and parameter tracking processes, finally having the estimated parameters converge to their best

estimates. Furthermore, the algorithm is modified and applied for multiple lines in a power grid. The

corresponding problems such as grid transmission line modeling, PMU placement are discussed,

followed by the description of the proposed JSEPTS. Numerical simulations are conducted on both

simulated and actual data, indicating that parameters can be closely tracked. The major advantage of

the proposed algorithms is to dynamically track the line parameters and avoid using multiple scan of

measurements.

There are several implementations of the proposed algorithm, either for a single transmis-

sion line or for the whole system. A transmission line may be experiencing parameter variations

due to frequent maintenance, repair or weather changes. It may also be necessary to calculate the

parameters of a new transmission line and verify existing line parameters quarterly or yearly. For

JSEPT, it can be implemented for such lines, relying on existing or temporarily installed PMUs, to

estimate parameters precisely. For JSEPTS, a period execution scheme can be adopted to track the

parameters of all lines one by one, in a sequential manner. Alternatively, by using conventional error

detection techniques such as normalized residual test, a list of lines with suspicious parameters can

be singled out and tracked simultaneously by the proposed JSEPTS.

The parameters of transmission lines are analyzed in this chapter. The last part of this

dissertation will be about parameters of electric machines. In next chapter, the estimation of electric

machine states with inaccuracies in machine parameters is discussed and analyzed.

150

Chapter 5

Estimation of Machine States with

Model Uncertainties

5.1 Introduction

Static state estimation, as first developed in the 1960s by Schweppe [9], is a crucial

component of modern energy management systems (EMS). In the previous chapters, the traditional

static state estimator is reviewed first in Section 2.2, followed by a description of a multi-area static

state estimator in Chapter 3. As stated in [95], one of the significant roles of static state estimation

in the EMS is to facilitate static security assessment. However, with the increased penetration of

renewable and distributed energy sources, dynamic security assessment becomes more and more

important to the system operators. This requires accurate knowledge of the dynamic state of the

system such as the states of generators, loads and any other dynamic elements.

Numerous researchers have studied dynamic estimation of machine states, such as [96–103].

In early studies, the authors of [96] mention the possibility of using an augmented state vector which

will include in addition to the system bus voltage phasors, dynamics of the power plants, loads and

the effects of various controls. Such state vector augmentation idea is reviewed and modified in a

more recent work [101]: the state vector of the dynamic state estimator is augmented by the variables

representing the bus voltages as well as the machine states, including the rotor angle and rotor speed.

Besides, [97] investigates the feasibility of applying Kalman filtering techniques to include dynamic

state variables in the state estimation process. In [98], the authors focused on the scenario when

some inputs for estimating the machine states are not available. The extended Kalman filter with

151

CHAPTER 5. ESTIMATION OF MACHINE STATES WITH MODEL UNCERTAINTIES

unknown inputs is proposed for identifying and estimating the states and the unknown inputs of

the synchronous machine simultaneously. In [99], a deterministic sampling technique known as

the unscented transform is used to calculate the mean and covariance of the nonlinear functions of

machine state transition and observation models. An extended particle filter is proposed in [100]

to estimate the dynamic states of a synchronous machine using PMU data. The proposed extended

PF improves robustness of the basic PF against bad data through iterative sampling and inflation of

particle dispersion. Authors of [102] present a two stage dynamic state that the first stage involves a

robust network static state estimator providing bus voltage estimates and the second stage involves

independently and simultaneously executed dynamic estimators for individual generator-turbine

systems. The idea is further extended in [103], a linear phasor based LAV estimator is used for

each zone in addition to a dynamic state estimator in order to simultaneously estimate the dynamic

state variables of the machine. When an accurate prior knowledge of the rotating machine nonlinear

state-space model is available, standard dynamic state estimators, such as the extended Kalman filter

(EKF) or the unscented Kalman filter (UKF) can provide accurate state estimates both in steady state

and in transients.

However, a critical premise of Kalman filter theory (including ordinary KF, EKF and

UKF) is that the underlying state-space model is accurate. When this assumption is violated, the

performance of the filter can deteriorate. In practice, parameters of rotating machines are known only

with limited accuracy, and their values may vary with time. For instance, it is shown in [104] that

some machine parameters are readily correlated with actual resistances or inductances, which may

vary due to different operating conditions. Several methods have been proposed in the literature to

mitigate the effect of model uncertainties on the overall performance, such as the H∞ approach, the

set-valued estimation approach and the guaranteed-cost paradigm. In particular, a powerful approach

to the design of robust dynamic state estimators was presented in [105]. This approach involves a

min-max criterion, designed to minimize in each iteration the worst-possible regularized least squares

residual norm over the range of all possible uncertainties. Several extensions and modifications of

the regularized least squares approach are presented in [106–109]. In [106] and [107], the filter

is extended to time-delay systems and time varying systems respectively. In [108], it provides a

direct implementation of [105] with the discussion of correlated dynamics and measurement noise.

In [109], a robust continuous-discrete Kalman filter is developed for a linear system by transforming

the discrete-time procedure in [105] and applying it to reduce the effect of network delay. However,

these previous applications of the regularized least squares approach were limited to linear state-space

models: the resulting robust dynamic state estimation techniques do not perform well in a nonlinear

152


setting.

In this chapter, a robust continuous-time discrete-measurement extended Kalman filter

is developed for non-linear state estimation in the presence of modeling uncertainties [110]. The

derivation relies on a general min-max regularized least squares optimization framework, which was

developed in [105]. It is modified to accommodate the continuous-discrete, nonlinear system setting,

and present the corresponding robust filter solution. Section 5.2 introduces the nonlinear state-space

model as well as the structure of the uncertainties, followed by the implementation aforementioned

linearization and discretization procedure. In addition, the proposed filter is designed to map the

continuous-time nonlinear equation with unstructured uncertainty into a discrete-time linearized state

equation with structured parameter uncertainty. The proposed robust extended Kalman filter (REKF)

is derived and presented in a format similar to the Kalman filter in Section 5.3. It can provide a

state estimate for the nonlinear system, mitigating the bounded parametric uncertainties that standard

EKF cannot. In Section 5.4, a typical rotating machine model is provided and numerical results are

presented to demonstrate the performance of the robust filter for dynamic state estimation.

5.2 Problem Formulation

A typical continuous-time discrete-measurement dynamic nonlinear system model can be

described by

x(t) = f(x(t)) + w(t) (5.1)

y(tn) = h(x(tn)) + ηn. (5.2)

where x(t) is the state of the system. w(t) and ηn are the process noise and measurement noise,

respectively. Both types of noise are assumed to be i.i.d. with E[w(t)wT (s)] = Q(t)δ(t− s) and

E[ηnηTn ] = R(tn), where δ(·) denotes the Dirac delta function.

The functions f and h are nonlinear functions of the state vector x(t). For simplicity, we

assume uniformly spaced sampling times, i.e. tn = nT1, where T1 is known as the sampling interval.

However, the proposed algorithm can be applied even if the sampling times are not uniformly spaced,

which will be discussed later.

153


5.2.1 Linearization and Discretization of State Equation

Based on a first-order Taylor series approximation, the nonlinear term f(x(t)) can be

linearized around a state estimate x(ti), as shown below:

f(x(t)) = f(x(ti)) +∂f

∂x

∣∣∣∣x=x(ti)

[x(t)− x(ti)

]+O

(([x(t)− x(ti)

]2))

(5.3)

By defining the Jacobian matrix

Ai =∂f

∂x

∣∣∣∣x=x(ti)

(5.4)

and

Ui = f(x(ti))−Aix(ti) (5.5)

and neglecting the higher order terms in (5.3), the state equation (5.1) can be approximated as

x(t) = Aix(t) + Ui + w(t). (5.6)

Now one can employ a discrete-time approximation of (5.6), using a refined time grid

ti = iT2, to integrate this differential equation, so that

x(ti+1) = eAiT2x(ti) +

∫ T2

0eAiτ Uidτ +

∫ T2

0eAiτw(ti − τ)dτ (5.7)

The value of T2 determines the accuracy of the discretized model. If T2 is small enough, the state

dynamics from x(ti) to x(ti+1) can be regarded as linear. Thus the error arises from linearization

can be significantly reduced.

According to [111], the second integral term in (5.7) can be approximated by a discrete-time

process noise, viz.,∫ T2

0eAiτw(ti − τ)dτ ≈

∫ T2

0w(ti − τ)dτ = µ(ti)− µ(ti − T2) (5.8)

where µ(t) is the integral of w(·), i.e. a Brownian motion process. By defining

wi∆= µ(ti)− µ(ti − T2), (5.9)

we can then derive the discrete-time process noise covariance:

Qi = EwiwTi = Q(ti)T2. (5.10)

Also, the first integral term can be approximated by T2Ui. Therefore, with proper substitution of

subscript xi = x(ti) and xi|i = x(ti), (5.6) can be approximated in a discrete-time format, viz.

xi+1 = Fixi +Giwi + Ui (5.11)

154


where xi, wi, Ui ∈ Rd×1, Fi, Gi,Ai ∈ Rd×d, d is the number of the states, and

Fi = eAiT2 (5.12)

Gi = T2I (5.13)

Ui =

∫ T2

0eAiτ Uidτ =

(∫ T2

0eAiτdτ

)Ui ≈ T2Ui (5.14)

where I is an identity matrix of appropriate size.

5.2.2 Linearization of Measurement Equation

The measurement model (5.2) is in discrete format with sampling instant tn and sampling

interval T1. Note that the subscript of (5.2) is different from that of (5.11) since the state equation is

discretized based on the discretization instant T2 which may be much smaller than the measurement

sampling interval T1. In order to unify the discrete-time grid of the state equation (5.11) with that

of the measurement equation (5.2), we will describe the measurement process as occurring at the

time instants ti = iT2, but with missing measurements. Whenever ti 6= nT1 the corresponding

measurement is not available, so that we skip the measurement update step, as detailed in Section

5.3. Thus, we replace (5.2) by the discrete-time version

yi = h(xi) + ηi (5.15)

We will also need to introduce the observation matrix Hi, defined by the Jacobian

Hi =∂h

∂x

∣∣∣x=xi|i−1

(5.16)

where yi, ηi ∈ Rm×1, Hi ∈ Rm×d and m is the number of measurements of yi.

5.2.3 Construction of Structured Uncertainties

The construction of the REKF algorithm relies on the assumption that the system state

equation involves the modified function fδ , due to model parameter inaccuracies, so that the true

state equation is, in fact

x(t) = fδ(x(t)) + w(t) (5.17)

The symbol δ is used to denote perturbations caused by model parameter inaccuracy. Thus (5.12)

has to be replaced by Fi + δFi = e(Ai+δAi)T2 , where

δAi =∂fδ∂x

∣∣∣∣x=xi|i

− ∂f

∂x

∣∣∣∣x=xi|i

. (5.18)

155


The effect of parameter inaccuracy on the term Ui of (5.11) depends on the perturbation

δBi = fδ(xi|i)− f(xi|i). (5.19)

Taking into account the perturbations caused by parameter inaccuracy, we need to modify

(5.11) to obtain

xi+1 = (Fi + δFi)xi + (Gi + δGi)wi + (Ui + δUi) (5.20)

in which the uncertainties δFi, δGi and δUi can be structured as a function of δAi and δBi by

introducing the uncertainty terms into (5.12) and (5.14), viz.,

δFi = e(Ai+δAi)T2 − eAiT2 ≈ δAi T2 (5.21)

δUi = −δAi xi|i T2 + δBi T2, (5.22)

and since Gi does not depend on system parameters as illustrated in (5.13), then

δGi = 0. (5.23)

An approximation is made here that

δBi ≈ δAiαxi|i (5.24)

for the rotating machine system which will be discussed later. Here the uncertainty scale factor,

denoted as α, is defined by

α = maxi

( norm(δBi)norm(δAi xi|i)

)(5.25)

In practice, α is decided empirically according to different systems and different operating conditions.

Actually, if the uncertainties are bounded and not too large, and the system operates in pseudo steady

state, α will be relatively small ( α < 0.1). One explanation is that the states of the machine remain

fairly constant under steady state conditions leading to the value of f to be very small. Since the

uncertainty discussed in this paper will not deteriorate the stability of the system but only bias the

states, fδ is also close to zero.

We also assume ‖δAi‖ ≤ ε by choosing an upper bound ε for the norm of perturbations to

the A matrix. Therefore, we can now define an arbitrary contraction matrix ∆i, viz.

∆i =δAiε, (5.26)

156


so that ‖∆i‖ ≤ 1. This allows us to introduce the ’structured uncertainty’ relation with respect to

δFi and δUi as [δFi δUi

]= Mi∆i

[Ef,i Eu,i

](5.27)

where Mi, Ef,i ∈ Rd×d, Eu,i ∈ Rd×1 and

Mi = ε T2 I (5.28)

Ef,i = I (5.29)

Eu,i = −(1− α)xi|i. (5.30)

We omit δGi from this relation since it is zero according to (5.23). Notice that the relation (5.27)-

(5.30) relies only on the assumption that ‖δA‖ ≤ ε. In contrast, the derivation in [105] relies on a

much more restricted model of parameter uncertainty.

5.3 Robust Extended Kalman Filter (REKF)

5.3.1 Ordinary Regularized Least Squares

As stated in [112], many estimation techniques rely on solving regularized least squares

problem of the form

minx

[xTQx+ (Ax− b)TW (Ax− b)

](5.31)

where xTQx is a regularization term with the weighting matrix Q = QT and W = W T are positive

definite. Since this cost function is convex, it has a unique global minimum, which can be determined

by equating the derivative of the cost to zero. The resulting optimal solution is

x =[Q+ATWA

]−1ATWb. (5.32)

Now consider the following state space model

xi+1 = Fixi +Giui (5.33)

yi = Hixi + vi (5.34)

where ui and vi are zero mean noise processes with covariances Qi and Ri, respectively. In order to

map (5.33) and (5.34) into a regularized least squares problem of the form (5.31), we now introduce

the mappings

x←

xi − xi|iui

(5.35)

157


b← yi+1 −Hi+1Fixi|i (5.36)

A← Hi+1

[Fi Gi

](5.37)

Q←

P−1i|i 0

0 Q−1i

(5.38)

so that we can rewrite the problem of (5.31) as

minxi,ui

[ ∥∥xi − xi|i∥∥2

P−1i|i

+ ‖ui‖2Q−1i

+ ‖yi+1 −Hi+1xi+1‖2R−1i+1

]. (5.39)

The minimizing solution of this deterministic regularized least squares problem turns out to be the

standard Kalman filter for the linear state-space model (5.33) and (5.34). This observation provides a

convenient starting point for the derivation of a robust version of the EKF, which we will carry out in

Sections 5.3.2 and 5.3.3.

We can now apply the mappings (5.35)-(5.38) to the explicit expression (5.32). Denoting

the minimizing arguments of (5.39) as xi|i+1 and ui|i+1, then they can be expressed as

xi|i+1 = xi|i + Pi|iFTi H

Ti+1R

−1i+1(yi+1 −Hi+1xi+1|i+1), (5.40)

ui|i+1 = QiGTi H

Ti+1R

−1i+1(yi+1 −Hi+1xi+1|i+1). (5.41)

Thus, if we introduce the quantity xi+1|i+1 and substitute (5.40) and (5.41) into it, we can obtain

xi+1|i+1 = Fixi|i+1 +Giui|i+1

= Fixi|i + FiPi|iFTi H

Ti+1R

−1i+1(yi+1 −Hi+1xi+1|i+1)

+GiQiGTi H

Ti+1R

−1i+1(yi+1 −Hi+1xi+1|i+1)

= Fixi|i + (FiPi|iFTi +GiQiG

Ti )HT

i+1R−1i+1(yi+1 −Hi+1xi+1|i+1)

(5.42)

Next, introduce the shorthand notation

xi+1|i∆= Fixi|i (5.43)

and

Pi+1|i∆= FiPi|iF

Ti +GiQiG

Ti , (5.44)

so that (5.42) can then be further manipulated into the form

xi+1|i+1 = xi+1|i + Pi+1|iHTi+1R

−1i+1(yi+1 −Hi+1xi+1|i+1) (5.45)

158


and thus

xi+1|i+1 =[I + Pi+1|iH

Ti+1R

−1i+1Hi+1

]−1[xi+1|i + Pi+1|iH

Ti+1R

−1i+1yi+1

]= xi+1|i +

[I + Pi+1|iH

Ti+1R

−1i+1Hi+1

]−1[Pi+1|iH

Ti+1R

−1i+1(yi+1 −HT

i+1xi+1|i)]

= xi+1|i + Pi+1|iHTi+1R

−1i+1

[I +Hi+1Pi+1|iH

Ti+1R

−1i+1

]−1(yi+1 −HT

i+1xi+1|i)

= xi+1|i + Pi+1|iHTi+1

[Ri+1 +Hi+1Pi+1|iH

Ti+1

]−1(yi+1 −HT

i+1xi+1|i).

(5.46)

This expression is precisely the measurement update at time i + 1, mapping xi+1|i into xi+1|i+1,

using the Kalman gain

Ki+1 = Pi+1|iHTi+1

[Ri+1 +Hi+1Pi+1|iH

Ti+1

]−1. (5.47)

In addition, the covariance is updated as

Pi+1|i+1 = (I−Ki+1Hi+1)Pi+1|i (5.48)

Till now, the Kalman filter is reestablished after the algebra described above. A brief

conclusion can be drawn from this section that the minimization problem of (5.31) is exactly the

same as the ordinary Kalman filter. And this provides the theoretical background of the proposed

robust version of the Kalman filter.

5.3.2 Regularized Least Squares with Uncertainties

Now consider the case when the nominal data A, b may be subject to disturbances

δA, δb in (5.31). Following the approach proposed in [105], we consider the min-max (i.e., worst

case) cost function

minx

maxδA,δb

[‖x‖2Q + ‖(A+ δA)x− (b+ δb)‖2W

]. (5.49)

whereδA : perturbation matrix to the nominal matrix A, it satisfies a model of the form

δA = H ∆Ea;

δb : perturbation vector to the nominal vector b, it satisfies a model of the form

δb = H ∆Eb;

H,Ea, Eb: known quantities of appropriate dimension;

∆ : a contraction that ‖∆‖ ≤ 1.By defining

y∆= ∆(Eax− Eb), (5.50)

159


we can derive

Hy = H∆(Eax− Eb) = δAx− δb. (5.51)

Thus, (5.49) can be rewritten as

minx

max‖y‖≤‖Eax−Eb‖

[‖x‖2Q + ‖Ax− b+Hy‖2W

]. (5.52)

We review the solution of (5.52) given in [105] as

x =[Q+AT WA

]−1[AT W b+ λETa Eb

](5.53)

where

Q = Q+ λETa Eb (5.54)

W = W +WH(λI−HTWH)†HTW (5.55)

and the non-negative scalar parameter λ is determined as

λ = arg minλ>‖HTWH‖

[‖x(λ)‖2Q + λ ‖Eax(λ)− Eb‖2 + ‖Ax(λ)− b‖2W (λ)

](5.56)

where

W (λ) = W +WH(λI−HTWH)†HTW (5.57)

Q(λ) = Q+ λETa Ea (5.58)

x(λ) =[Q(λ) +ATW (λ)A

]−1[ATW (λ)b+ λETa Eb

](5.59)

The lower bound on λ, as illustrated in [105], is

λl =∥∥HTWH

∥∥ (5.60)

Now turn back to the original state space estimation problem. Similar to the expressions

from (5.35) to (5.38), and according to the structured state space model with state equation (5.20)

and the measurement equation (5.15), the optimal robust filtering problem is defined as follows: at

time step i, given an estimate of the state xi denoted as xi|i, a positive-definite weighting matrix Pi|iwhich represents the covariance for state estimation error, and the measurement at time i + 1, i.e.

yi+1, obtain a new estimate at time xi+1|i+1 by solving the min-max problem

minxi,wi

maxδFi,δUi

[ ∥∥xi − xi|i∥∥2

P−1i|i

+ ‖wi‖2Q−1i

+ ‖yi+1 − h(xi+1)‖2R−1

i+1

](5.61)

160


subject to the constraints on δFi and δUi specified in (5.27). We can map the robust filtering problem

(5.61) into a format of (5.49) by introducing the identifications from (5.62) to (5.73)

x←

xi − xi|iwi

(5.62)

Q←

P−1i|i 0

0 Q−1i

(5.63)

A← Hi+1

[Fi Gi

](5.64)

b← yi+1 −Hi+1Fixi|i −Hi+1Ui (5.65)

W ← R−1i+1 (5.66)

δA← Hi+1

[δFi 0

](5.67)

δb← −Hi+1δFixi|i −Hi+1δUi (5.68)

Ea ←[Ef,i 0

](5.69)

Eb ← −Ef,ixi|i − Eu,i (5.70)

H ← Hi+1Mi (5.71)

∆← ∆i (5.72)

yi+1 = yi+1 − h(xi+1|i) +Hi+1xi+1|i. (5.73)

The optimization problem of (5.61) is in the form that allows us to obtain a solution by

relying on the explicit expressions specified by (5.53) to (5.60). Solutions xi+1, wi+1 of the

optimization problem (5.61) can be obtained by introducing

xi+1|i+1 = Fixi+1 +Giwi+1 + Ui (5.74)

with the modified weighting matrices

Q =

P−1i|i + λiE

Tf,iEf,i 0

0 Q−1i

(5.75)

W = R−1i+1 =

(Ri+1 − λiHi+1MiM

Ti H

Ti+1

)−1. (5.76)

161


Moreover, λi should be within the interval

λi >∥∥MT

i HTi+1R

−1i+1Hi+1Mi

∥∥ = λl,i. (5.77)

The time-dependent parameter λi has to be determined by solving an optimization problem at each

step of the recursion. However, as pointed out in [105], the optimal value of λi usually lies near to its

lower bound. Thus, a simple choice for λi is

λi = (1 + β)λl,i (5.78)

where β can be chosen in practice as an arbitrarily small number, for example 0.05.

5.3.3 Procedure of REKF

In the above sections, the formulation of extended Kalman filter as well as the structured

uncertainties are discussed. The original continuous-time-discrete-measurement model is linearized

and discretized first, and the resulting new model is provided in (5.20) and (5.15) and revisited here

as

xi+1 = (Fi + δFi)xi +Giwi + (Ui + δUi) (5.79)

yi+1 = h(xi+1) + ηi+1 (5.80)

The perturbations in the state transition model is structured as (5.27). Thus, the robust extended

Kalman filter (REKF) can be derived after some lengthy algebraic manipulations, and is summarized

in Table 5.1.

The REKF algorithm generates the robust estimate xi+1|i+1 given xi|i in step 3, which

is very similar in nature to the time- and measurement- update expressions of a standard extended

Kalman filter. The main difference is that the parameters are modified for robust estimation, as

presented in steps 1 and 2.

What’s more, in step 3, the criteria that whether or not to proceed the measurement update

is specified. Note that the state transition model is discretized with the time interval T2, and the

measurement sampling interval is T1. During the iterative process of REKF, the measurement update

step is executed only when a sensor measurement is available, i.e., when the measurement time-stamp

tn coincides with one of the discrete-time grid points ti. However, when a measurement is not

available at a given ti, the measurement-update step is skipped, as shown in step 3 of the REKF.

Therefore, the proposed REKF can also work for the case that sampling times are not uniformly

spaced, as long as each sampling instant coincides with a discrete-time point ti for some i. What’s

162


Table 5.1: Robust extended Kalman filter algorithm

more, the interval T1 could be chosen as the precision of sampling interval in order to catch up with

all the measurements, i.e., T2/T1 is integer.

It is easy to verify that if the system is without uncertainty, by setting Mi = 0 and λi = 0,

the parameter modification in step 2 is not used, namely, the modified parameters coincide with the

original parameters. In this case, the proposed REKF algorithm reduces to the ordinary EKF. Also

163


note that in step 2, the † indicates that

λ†i =

λ−1i λi 6= 0

0 λi = 0.

5.4 Simulation

5.4.1 Dynamic Model of Synchronized Generator

To simplify the model, we focus in this section only on the generator part of the rotating

machine. The dynamic equations related to the two-axis model of a synchronous generator [113] are

expressed as follows:

θ = ω − ω0 (5.81)

H

πf0ω = PM − Pe −D(ω − ω0)/ω0 (5.82)

T ′doE′q = −E′q − (Xd −X ′d)Id + Efd (5.83)

T ′qoE′d = −E′d + (Xq −X ′q)Iq (5.84)

where VdVq

=

sin(θ) −cos(θ)cos(θ) sin(θ)

V cos(θ0)

V sin(θ0)

(5.85)

Id =E′q − VqX ′d

(5.86)

Iq =−E′d + Vd

X ′q(5.87)

Pe = VdId + VqIq (5.88)

Qe = −VdIq + VqId (5.89)

Note that (5.81) to (5.84) are the generator dynamic equations. θ is the rotor angle, ω is

the rotor speed, E′d and E′q are d-axis and q-axis transient voltages respectively, Efd is the field

voltage. Pe and Qe are the active and the reactive power delivered by the generator. V and θ0 are the

generators terminal voltage magnitude and phase angle. We assume that Efd, V and θ0 are known

time-variant variables. For other parameters, the definitions and typical values can be found in [113]

164


and [114]. The general form of the machine dynamic equations can be expressed as in (5.1) and (5.2)

where

x =

θ

ω

E′q

E′d

(5.90)

y =

PeQe

(5.91)

The corresponding A of (5.4), F of (5.12), U of (5.14) and H of (5.16) are given in the

following. Note that the subscripts are neglected for simplicity.

A =

0 1 0 0

−Z1πf0

H−Dπf0

Hω0−Vdπf0

HX ′d

Vqπf0

HX ′q(X ′d −Xd)Vd

T ′doX′d

0 − Xd

T ′doX′d

0

(Xq −X ′q)VqT ′qoX

′q

0 0 − Xq

T ′qoX′q

(5.92)

where

Z1 =E′qVq

X ′d+E′dVdX ′q

+

(V 2

X ′q− V 2

X ′d

)cos(2θ − 2θ0) (5.93)

Vd = sin(θ − θ0)V (5.94)

Vq = cos(θ − θ0)V. (5.95)

Also

F = eAT2 , (5.96)

U = (f(x)−Ax)T2, (5.97)

H =∂h(x)

∂x=

Z1 0VdX ′d

− VqX ′q

Z2 0VqX ′d

VdX ′q

(5.98)

where

Z2 =E′dVqX ′q

−E′qVd

X ′d+

(V 2

X ′d− V 2

X ′q

)sin(2θ − 2θ0). (5.99)

165


5.4.2 Implementation of Proposed Robust Kalman filter

Four cases are provided here to show the performance of the proposed robust extended

Kalman filter, with a comparison to conventional EKF. The linearization constant in the algorithm

is T1/T2 = 50 , the uncertainty scale is α = 0.5, β = 0.05 and ε = 10. To acquire the actual

states and the measurements, the machine system is simulated in PowerWorld Simulator with proper

parameters. Then, the measurements are modified by adding white noise with zero mean and 0.02

p.u. standard deviation. Besides the comparison of the state estimates of EKF and REKF, the error

variances (EV) of EKF and REKF are also calculated as

EV (xi) = (xi|i − xactual,i)T (xi|i − xactual,i) (5.100)

for each case to further illustrate the performance.

Case 1: No uncertainty. To test the performance of the proposed algorithm, the case without

uncertainty is first simulated to show the accuracy of linearization and discretization. Figure 5.1

shows eight plots with 2 trajectories each: the dashed blue for the actual state of the generator rotor

speed, the red solid line in left plots indicates the state estimate of the REKF, and the right ones are of

EKF. The plots demonstrate that without uncertainty, EKF and REKF have very similar performance.

The curves of error variance of EKF and REKF nearly overlap each other in Figure 5.2.

Moreover, the value of error variance is very small, which indicates that linearization and discretiza-

tion have a minor impact on the performance of the REKF, as shown in Figure 5.2.

Case 2: Uncertainty in Xd. The parameter Xd denotes the synchronous reactance of the

generator, whose value cannot be exactly inferred due to the changing operation conditions. The

uncertainty in Xd is set to be 30%.

In Figure 5.3, the rotor speed and q-axis transient voltage in the second and third row

illustrate obvious improvement of the accuracy of estimated states. For the rotor angle in the first row

and d-axis transient voltage in the fourth row, REKF doesnt offer much improvement, but they are

still not worse than EKF. The results in Figure 5.4 give a straightforward illustration that the REKF

performs better than EKF, since the error variance of the former is much smaller than the latter.

166


Figure 5.1: Comparison of state estimate without uncertainty

Figure 5.2: Error Variance curves for EKF and REKF without uncertainty

167


Figure 5.3: Comparison of state estimate with uncertain Xd

Figure 5.4: Error Variance curves for EKF and REKF with uncertain Xd

168


Case 3: Uncertainty in D. The damping constant is usually ignored in many simplified

calculations. However, it is not always equal to 0. When this parametric uncertainty is taken into

consideration, we have the following results in Figure 5.5 and Figure 5.6. According to the results

of EKF in Figure 5.5, it is shown that damping constant influence the rotor angle and rotor speed

rather than the transient voltages. Comparing the performance of EKF and REKF, our robust EKF

algorithm achieves a smaller estimation error than the EKF when there is uncertainty in the damping

coefficient.

Case 4: Uncertainty inDd with transient. In this case, a transient of the system is simulated.

The transient event is caused by an open circuit of one line in the system and will finally cause a bias

to the generator states. Note that the initialization transient is neglected for better illustration of the

system transient.

Figure 5.7 and Figure 5.8 show that the designed REKF can still improve the state estimate

toward the actual one even in the presence of transients.

169


Figure 5.5: Comparison of state estimate with uncertain D

Figure 5.6: Error Variance curves for EKF and REKF with uncertain D

170


Figure 5.7: Comparison of state estimate with uncertain Xd with transient

Figure 5.8: Error Variance curves for EKF and REKF with uncertain Xd with transient

171


5.5 Conclusions

In this chapter, a robust extended Kalman filter is developed for a continuous-time discrete-

measurement system, and applied to estimate dynamic machine states. In particular, the nonlinear

state-space model is first linearized and discretized strategically with little change of the estimation

results. The uncertainties are then structured to a consistent form and the optimization problem

of minimizing the worst-possible squared residual norm was formulated accordingly. By solving

this problem, the resulting REKF recursions are derived. The results of simulation show that the

proposed filter outperforms the conventional extended Kalman filter in the presence of parametric

uncertainties.

172

Chapter 6

Conclusion and Future Work

6.1 Conclusion

With the increased penetration of renewable and distributed energy sources, the robustness

of state estimators becomes increasingly crucial in order to reliably serve other operation and market

applications in power system control center. There are various reasons for obtaining unreliable

state estimation solutions, including equipment failures, human factors, incorrect network topology,

malicious cyber attack, variations of ambient conditions, etc. Technically speaking, due to errors

in measurements, parameters or network model, even the most reliable estimators are vulnerable to

occasional failure.

This dissertation addresses these issues and develops practical solutions to be implemented

in large scale power systems.

Considering the fact that when solving the power system state estimation problem for very

large interconnected systems, it is possible for certain zones to experience convergence issues due to

various reasons such as bad data, loss of measurements, topology errors. In Chapter 2, a strategic

measurement reconfiguration and design approach is proposed. The main goal of this work is to

allow isolation of the impact of measurement errors in different zones so that zonal results of state

estimation for very large scale networks can be obtained in a manner remaining insensitive to errors

in other zones of the system.

In Chapter 3, several algorithms are reviewed and developed to provide an alternative

approach which will automatically detect issues associated with the affected subsystem and isolate

its solution from the rest of the system solution. The proposed two-level state estimator approach

is shown to effectively isolate the subsystem containing the cause of divergence, hence enabling

173

CHAPTER 6. CONCLUSION AND FUTURE WORK

the estimator to provide accurate estimates for the remaining parts of the system. The recursively

partitioned state estimator is carried out to further partition the divergent area to obtain more state

estimates. A new partition framework is developed with the consideration of area connectivity and

observability. Performance of the proposed method is verified provide solutions for the largest

solvable part of the system and isolate those areas which are impacted. The algorithm can be

improved by developing better partitioning technique, utilizing multiple partition snapshots and

coordinating with techniques to deal with divergent causes.

Chapter 4 considers the problem of inaccurate transmission line parameters. The joint

state estimation and parameter tracking approach is proposed for tracking the parameters of a

three phase untransposed transmission line. It involves alternation between state estimation and

parameter tracking processes, finally having the estimated parameters converge to their best estimates.

Numerical simulations are conducted on both simulated and actual data, indicating that parameters

can be closely tracked. Then, the algorithm is generalized to apply for tracking the parameters of

multiple lines in a power grid with PMUs installed on strategically selected buses. The proposed

algorithms can be implemented for lines who may suffer from potential parameter change, relying

on existing or temporarily installed PMUs, to estimate parameters precisely.

The parameters in electric machines may also suffer from uncertainties due to inadvertent

failures to update network data, variations of ambient conditions, etc. In Chapter 5, to obtain a

robust estimate of machine states, a robust extended Kalman filter for a continuous-time discrete-

measurement system is developed and applied. In particular, the nonlinear state-space model was first

linearized and discretized strategically with little change of the estimation results. The uncertainties

were then structured to a consistent form and the optimization problem of minimizing the worst-

possible squared residual norm was formulated accordingly. The results of numerical simulations

showed that the proposed filter outperforms the conventional extended Kalman filter in the presence

of parametric uncertainties.

The main contributions of this dissertation are as follows:

• A boundary measurement manipulation algorithm to avoid spreading of errors is developed. It

not only provides a way to block the errors between different control zones, but also gives an

insight into how different kinds of measurements will impact the error spreading.

• A two-level multi-area state estimator is developed in order to address the non-convergent state

estimation cases. The new estimator is implemented and tested using a very large scale actual

utility system data and measurements.

174

CHAPTER 6. CONCLUSION AND FUTURE WORK

• A customized partitioning algorithm is developed in order to implement the multi-area state

estimator yet maintain connectivity and observability of individual areas.

• The state estimation observability issues related to the boundary of areas are addressed.

• A recursively partitioned state estimator (RPSE) is developed to further partition the diverged

areas. The performance of RPSE and 2lvSE are evaluated based on actual convergent and

divergent cases of several large scale systems.

• Three phase transmission line parameter estimation algorithm is developed. The algorithm of

estimating parameters in multiple transmission lines is developed then. These procedures cam

track the line parameters in real-time, with the utilization of only one snapshot of measurement

to estimate the parameters at that time. Also, these algorithms can be implemented to the lines

whose parameters are varying or suspicious to errors, both in transmission and distribution

systems.

• The robust extended Kalman filter is developed for estimating machine states even with the

uncertainties in parameters.

6.2 Future Work

While the dissertation successfully addressed several issues in detail, there is still room to

further extend the results to develop the following applications:

• The implementation of 2lvSE and RPSE in real world, for long term testing using actual data

from utilities, and finally commercializing it as an application to the energy management

system.

• For the line parameter estimation, an implementation in a mixed-phase distribution system

would be possible by minor modification of the algorithm. Therefore, a mixed-phase distributed

level state estimator along with a PMU placement method need to be developed for this purpose.

• The robust extended Kalman filter can be generalized for a generic continuous-time discrete-

measurement system with uncertainties in the nonlinear parameters.

175

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186

Appendix A

Conventional Data Format

A.1 IEEE Common Data Format

The complete description of IEEE common data format (CDF) can be found in [115]. A

compact form of the bus and branch data format are provided below.

IEEE CDF Bus Data

1 Columns 1- 4 (I) Bus number (I)

Columns 7-17 (NAME) Name (A) (left justify)

2 Columns 19-20 (AREA) Load flow area number (I)

3 Columns 21-23 (ZONE) Loss zone number (I)

4 Columns 25-26 (IDE) Type (I)

5 Columns 28-33 (VM) Final voltage, p.u. (F)

6 Columns 34-40 (VA) Final angle, degrees (F)

7 Columns 41-49 (PL) Load MW (F)

8 Columns 50-59 (QL) Load MVAR (F)

9 Columns 60-67 (PG) Generation MW (F)

10 Columns 68-75 (QG) Generation MVAR (F)

11 Columns 77-83 (KV) Base KV (F)

12 Columns 85-90 (VS) Desired volts (pu) (F)

(if this bus is controlling another bus)

13 Columns 91-98 (QT) Maximum MVAR or voltage limit (F)

14 Columns 107-114 (GL) Shunt conductance G (per unit) (F)

187

APPENDIX A. CONVENTIONAL DATA FORMAT

15 Columns 115-122 (BL) Shunt susceptance B (per unit) (F)

16 Columns 124-127 (IREG) Remote controlled bus number

IEEE CDF Branch Data

1 Columns 1- 4 (I) Tap bus number (I) *For transformers or

phase shifters, the side of the model the

non-unity tap is on

2 Columns 6- 9 (J) Z bus number (I) *For transformers and

phase shifters the side of the model the

device impedance is on

3 Columns 11-12 (AREA) Load flow area (I)

4 Columns 13-14 (ZONE) Loss zone (I)

5 Column 17 (CKT) Circuit (I) * (Use 1 for single lines)

6 Column 19 (TYPE) Type (I)

0 - Transmission line

1 - Fixed tap

2 - Variable tap for voltage control

(TCUL, LTC)

3 - Variable tap (turns ratio) for MVAR

control

4 - Variable phase angle for MW control

(phase shifter)

7 Columns 20-29 (R) Branch resistance R, per unit (F)

8 Columns 30-40 (X) Branch reactance X, per unit (F)

9 Columns 41-50 (B) Line charging B, per unit (F)

10 Columns 51-55 (RATEA) Line MVA rating No 1 (I)

11 Columns 57-61 (RATEB) Line MVA rating No 2 (I)

12 Columns 63-67 (RATEC) Line MVA rating No 3 (I)

13 Columns 69-72 (ICONT) Control bus number

14 Column 74 (CSIDE) Side (I)

0 - Controlled bus is one of the terminals

188


1 - Controlled bus is near the tap side

2 - Controlled bus is near the impedance

side (Z bus)

15 Columns 77-82 (RATIO) Transformer final turns ratio (F)

16 Columns 84-90 (ANGLE) Transformer (phase shifter) final angle

(F)

17 Columns 91-97 (RMI) Minimum tap or phase shift (F)

18 Columns 98-104 (RMA) Maximum tap or phase shift (F)

19 Columns 106-111 (STEP) Step size (F)

20 Columns 113-119 (VMI) Minimum voltage, MVAR or MW limit

(F)

21 Columns 120-126 (VMA) Maximum voltage, MVAR or MW limit

(F)

A.2 PSSE Input Data File

A raw file is a collection of unprocessed data. This means the file has not been altered,

compressed, or manipulated in any way by the computer. Raw files are often used as data files by

software programs that load and process the data. These files contain power flow system specification

data for the establishment of an initial working case. Several of these files may be read when a new

power flow case is being built up from subsystem data being provided by several different power

companies or organizations. A compact form of major components are listed as follows.

• ID : System and Base

• BUS : Bus Data

• LOAD : Load Data

• GENERATOR : Generator Data

• BRANCH : Branch Data

• TRANSFORMER ADJUSTMENT : Transformer Adjustment Data

• AREA INTERCHANGE : Area Interchange Data

189


• TWO TERMINAL DC LINE : Two-Terminal dc Line Data

• SWITCHED SHUNT : Switched Shunt Data

• TRANSFORMER IMPEDANCE CORRECTION TABLES

• MULTI-TERMINAL DC LINE DATA

• MULTI-SECTION LINE GROUPING DATA

• ZONE : Zone Data

• INTER-AREA TRANSFER DATA

• OWNER : Owner Data

• FACTS CONTROL DEVICE DATA

190

Appendix B

Nomenclature of Operators

C Complex coordinate space

Cn Complex coordinate space of n dimensions. It can also be regarded as an

n× 1 column vector with complex elements.

Cn×m Symbolic notation of a complex matrix with dimension n×mCov(x) Covariance (matrix) of the random variable vector x.

E[x] The expected value of the random variable or vector x.

I Identity matrix

N (µ, σ2) Normal distribution (or Gaussian distribution) with the expectation µ and

standard deviation σ.

R Real coordinate space

Rn Real coordinate space of n dimensions. It can also be regarded as a real

n× 1 column vector.

Rn×m Symbolic notation of a real matrix with dimension n×mO The big O notation that describes the limiting behavior of a function when

the argument tends towards a particular value or infinity. It can also be

used to describe the error term in an approximation to a mathematical

function.

191

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