relay characteristics.Figure 5.3 Performance of the distance protection, relay I, with a phase A t...Figure 5.4 Example microgrid circuit, legacy line differential protection.Figure 5.5 Line differential relay characteristics (alpha plane method).Figure 5.6 Performance of the line differential protection, relay I, with a ...Figure 5.7 Performance of the line differential protection, relay I, with a ...Figure 5.8 Example topology of a microgrid, adaptive protection scheme.Figure 5.9 An example microgrid system, differential energy‐based protection...Figure 5.10 Phase A current measurement at bus B‐1 of the line DL‐1, grid‐co...Figure 5.11 Phase A current measurement at bus B‐2 of the line DL‐1, grid‐co...Figure 5.12 Phase A current spectral energy at bus B‐1 of the line DL‐1, gri...Figure 5.13 Phase A current spectral energy at bus B‐2 of the line DL‐1, gri...Figure 5.14 Phase A differential energy of the line DL‐1, grid‐connected mod...Figure 5.15 Phase A differential energy of the line DL‐1, islanded mode.Figure 5.16 Control architecture of an example microgrid.Figure 5.17 Two voltage sources and the inductive circuit connecting them.Figure 5.18 Active power/frequency droop and reactive power/voltage droop.Figure 5.19 Active power/frequency droop diagram, method 1.Figure 5.20 Reactive power/voltage droop diagram, method 1.Figure 5.21 Droop control diagram, method 2 [15].Figure 5.22 Secondary control, active power/frequency droop.Figure 5.23 Secondary control, reactive power/voltage droop.Figure 5.24 Two voltage sources and the resistive circuit connecting them.Figure 5.25 Two voltage sources (the inverter and the grid) and the resistiv...Figure 5.26 π‐Equivalent microgrid circuit model.Figure 5.27 Example microgrid system, dynamic state estimation‐based protect...Figure 5.28 Dynamic state estimation‐based protection results: low impedance...Figure 5.29 Dynamic state estimation‐based protection results: high impedanc...Figure 5.30 Adaptive setting‐less protection scheme.Figure 5.31 UMPCU installation and its functions.Figure 5.32 Test case for a distribution feeder. The acronyms WG and L repre...Figure 5.33 Expected errors of states (test case 1).Figure 5.34 Actual absolute errors of states (test case 1).Figure 5.35 Computation time (test case 1).Figure 5.36 Expected errors of states (test case 2).Figure 5.37 Actual absolute errors of states (test case 2).Figure 5.38 Computation time (test case 2).Figure 5.39 Equivalent circuit for a three‐phase induction machine.Figure 5.40 Confidence levels for test cases 3–7.Figure 5.41 Terminal currents for test case 6 (high impedance fault).Figure 5.42 Top: terminal currents for test case 4 (three‐phase‐to‐ground fa...
5 Chapter 6Figure 6.1 Left: Voltage magnitude and angle estimation errors per bus for t...Figure 6.2 Left: The IEEE 14‐bus system partitioned into four areas [25, 31]...Figure 6.3 Average error curves
6 Chapter 7Figure 7.1 Shortest propagation paths from Sensor “k” to the fault on Line “...Figure 7.2 Virtual nodes generated at the points “βk,ℓ Dℓ” ...Figure 7.3 (a) Location of a fault occurring at 99 miles away from Bus 63 th...Figure 7.4 Phasor measurements provided by a PMU.Figure 7.5 7‐Bus system for illustration.Figure 7.6 Placement of 3 two‐channel PMUs in the 7‐bus system.Figure 7.7 Optimal PMU deployment on a 4‐bus system (a) ignoring zero‐inject...Figure 7.8 Configuration of 3 single‐channel PMUs in the 7‐bus system (the d...Figure 7.9 Placement of 5 two‐channel PMUs in the 7‐bus system.
7 Chapter 8Figure 8.1 Schematic of the proposed robust hybrid state estimation framewor...Figure 8.2 The absolute estimation error of the bus voltage magnitude with G...Figure 8.3 The absolute estimation error of the bus voltage angles with Gaus...Figure 8.4 The absolute estimation error of the bus voltage magnitude with G...Figure 8.5 The absolute estimation error of the bus voltage angles with non‐...Figure 8.6 The absolute estimation error of the bus voltage magnitude with L...Figure 8.7 The absolute estimation error of the bus voltage angles with Lapl...Figure 8.8 The estimated states of the proposed method under different bad d...
8 Chapter 9Figure 9.1 χ2 probability density function.Figure 9.2 Least square objective function.Figure 9.3 Least‐absolute‐value objective function.Figure 9.4 Least trimmed square objective function.Figure 9.5 Least‐trimmed‐absolute‐value objective function.
9 Chapter 10Figure 10.1 The 68, 95, and 99.7% confidence interval of a Gaussian‐distribu...Figure 10.2 The 13‐bus feeder.Figure 10.3 The voltage profile of the 13‐bus feeder (case 1).Figure 10.4 PDF of the voltage estimate at bus 6 of the 13‐bus feeder (case ...Figure 10.5 The voltage profile of the 13‐bus feeder including confidence va...Figure 10.6 Voltage profile of the 13‐bus feeder confidence values (case 2)....Figure 10.7 Voltage profile of the 13‐bus feeder confidence values (case 3)....Figure 10.8 The 145‐bus test feeder.Figure 10.9 The voltage profile of the 145‐bus feeder.Figure 10.10 The voltage of the 145‐bus feeder with a heat map overlay.Figure 10.11 Flowchart of the probabilistic observability assessment.Figure 10.12 Modified IEEE test feeder.Figure 10.13 Voltage profile for worst‐case scenario one (maximum load and n...Figure 10.14 Voltage profile for worst‐case scenario two (minimum load and m...Figure 10.15 The convergence behavior of the compliance ratio at bus 33.Figure 10.16 Modified IEEE test network with the voltage control devices.Figure 10.17 Voltage profile for worst‐case one (maximum load and no distrib...Figure 10.18 Voltage profile for worst‐case two (minimum load and maximum di...
10 Chapter 11Figure 11.1 Multi‐area partition strategies.Figure 11.2 Multi‐area state estimation: in‐series and in‐parallel execution...Figure 11.3 MASE computing architecture.Figure 11.4 Example of equivalent power injection creation at a shared bus....Figure 11.5 Flowchart of the MASE first step.Figure 11.6 Sub‐areas without measurement points at the shared node.Figure 11.7 Sub‐areas with measurement point installed at the shared node.Figure 11.8 Flowchart of the MASE second step.Figure 11.9 95‐bus network.Figure 11.10 Current magnitude estimation in Scenario 1.Figure 11.11 Voltage magnitude estimation in Scenario 1.Figure 11.12 Current magnitude estimation in Scenario 2.Figure 11.13 Voltage magnitude estimation in Scenario 2.Figure 11.14 Voltage magnitude estimation in Scenario 3.Figure 11.15 Voltage magnitude estimation in Scenario 4.Figure 11.16 Voltage phase angle estimation in Scenario 4.
11 Chapter 12Figure 12.1 The interconnection of the power system in North America [1].Figure 12.2 Architectures of multi‐area state estimators: hierarchical versu...Figure 12.3 Illustration of topological methods for observability analysis. ...Figure 12.4 A three‐area power system.Figure 12.5 Architecture of the method in [10].Figure 12.6 Illustration of the phase angle rotation method.Figure 12.7 Complete scheme of SFHSE.Figure 12.8 IEEE 118‐bus three area system [46].Figure 12.9 Four‐area real power system.
12 Chapter 13Figure 13.1 State estimation process block diagram.Figure 13.2 State estimation flowchart.Figure 13.3 Standard transmission line π model.Figure 13.4 Steps of parallel algorithm generation.Figure 13.5 CPU, GPU, CUDATM, and OpenMP resources.Figure 13.6 Gauss–Jacobi iterative method for two subsystems.Figure 13.7 Flowchart of ASM method with time stem τ. i, current subsys...Figure 13.8 The ASM‐based Jacobi WLS algorithm with BDD. k, time step; i, th...Figure 13.9 Domain decomposition: (a) interconnection of two subsystems and ...Figure 13.10 Original power system decomposed into J subsystems for RJDSE im...Figure 13.11 IEEE 39‐bus power system used to build large‐scale test cases....Figure 13.12 Fermi GPU architecture.Figure 13.13 Voltage magnitudes for Case 1 with respect to system size.Figure 13.14 Phase angles for Case 1 with respect to system size.Figure 13.15 Decomposing a Case 1 into four subsystems to apply the ASM algo...Figure 13.16 Percentage of execution time breakdown with respect to system s...Figure 13.17 Hierarchy of parallelism. τ, integration time step; t, sim...Figure 13.18 Estimation errors in GPU‐based ASM for Case 1 compared with PSS...Figure 13.19 Snapshot of estimation error for Case 1 at bus numbers 10, 11, ...Figure 13.20 Percentage of time used for various steps in GPU‐based ASM.Figure 13.21 Execution time (TEx) and speedup (Sp) comparisons of multithrea...
13 Chapter 14Figure 14.1 The working principle of Gauss–Newton method: (a) honest and (b)...Figure 14.2 Two major ways of convergence of the dishonest method on a linea...Figure 14.3 Simplified structure of a GPU.Figure 14.4 Accuracy of the dishonest Gauss–Newton method compared with