Robust Steady-State Analysis of Power Grid using Equivalent Circuit Formulation with Circuit Simulation Methods Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Department of Electrical and Computer Engineering Amritanshu Pandey B.S., Electrical and Electronics Engineering, Visvesvaraya Technological University M.S., Electrical and Computer Engineering, Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA December 2018 © Amritanshu Pandey 2018 All rights reserved ACKNOWLEDGEMENTS To my advisors, Larry Pileggi and Gabriela Hug, without their guidance this research would not have materialized and this dissertation would be incomplete. To my committee members, Granger Morgan and Soummya Kar, for their guidance towards achieving the goals stipulated within this thesis. To my colleagues, Marko, Martin, David, Aayushya, Dimitrios, Joe and others, for being the most helpful corroborators and for enabling a welcoming and productive working environment. To my friends, Jolly, Naveen, Akhilesh, Panickos, and many others, for being the most amazing friends and company when the chips were low, and the research was hard. To my girlfriend, Deirdre, for being by my side throughout the whole journey and for uplifting my spirits when I needed it the most. Finally, to my wonderful family, Mom, Dad and Anshu, to whom I owe everything I have achieved in my life so far. Thesis Statement: Develop robust methods to obtain the steady-state operating point of the transmission and distribution power grid independently or jointly using equivalent circuit approach and circuit simulation methods Abstract v 1. Abstract A robust framework for steady-state analysis (power flow and three-phase power flow problem) of transmission as well as distribution networks is essential for operation and planning of the electric power grid. The critical nature of this analysis has led to this problem being one of the most actively researched topics in the energy field in the last few decades. This has produced significant advances in the related technologies; however, the present state-of-the-art methods still lack the general robustness needed to securely and reliably operate as well as plan for the ever-changing power grid. The reasons for this are manifold, but the most important ones are: i) lack of general assurance toward convergence of power flow and three-phase power flow problems to the correct physical solution when a good initial state is not available; ii) the use of disparate formulation and modeling frameworks for transmission and distribution steady-state analyses that has led to the two analyses being modeled and simulated separately. This thesis addresses the existing limitations in steady-state analysis of power grids to enable a more secure and reliable environment for power grid operation and planning. To that effect, we develop a generic framework based on equivalent circuit formulation that can model both the positive sequence network of the transmission grid and the three-phase network of the distribution grid without loss of generality. Furthermore, we demonstrate that when combined with novel as well as adapted circuit simulation techniques, the framework can robustly solve for the steady-state solution for both these network models (positive sequence and three-phase) by constraining the developed models in their physical space independent of the choice of initial conditions. Importantly, the developed framework treats the transmission grid no differently than the distribution grid and, therefore, allows for any further advances in the field to be directly applicable to the analysis of both. One of which is the ability to jointly simulate the positive sequence network of the transmission grid and three-phase network of the distribution grid robustly. Abstract vi To validate the applicability of our equivalent circuit formulation to realistic industry sized systems as well to demonstrate the robustness of the developed methods, we simulate large positive-sequence and three-phase networks individually and jointly from arbitrary initial conditions and show convergence to correct physical solution. Examples for positive sequence transmission networks include 75k+ nodes US Eastern Interconnection test cases and for three- phase networks include 8k+ nodes taxonomy distribution test cases. Contributions vii 2. Contributions The primary contributions of this thesis are as follows: I. This thesis develops a generic framework based on equivalent circuit formulation that can model the positive sequence transmission network and three-phase distribution network without loss of generality. II. Furthermore, it adapts and further develops novel circuit simulation methods for the field of power system analysis that can ensure robust convergence for positive-sequence power flow and three- phase power flow problems from arbitrary initial conditions. III. Finally, the developed equivalent circuit framework with circuit simulation methods is extended to model the joint transmission and distribution network while ensuring same robust convergence as in the case of power flow and three-phase power flow problems. viii TABLE OF CONTENTS 1. Abstract ............................................................................................................................................... v 2. Contributions ................................................................................................................................... vii 3. Introduction and Motivation .......................................................................................................... 15 4. Background and Literature Review .............................................................................................. 22 4.1 Positive Sequence and Three-Phase Power Flow Formulations .......................... 22 4.1.1 ‘PQV’ based Formulation for Positive Sequence Power Flow Problem ...... 22 4.1.2 Current Injection Method for Three-Phase Power Flow Problem ............... 23 4.1.3 Backward-Forward Sweep Method .................................................................. 24 4.1.4 Holomorphic embedding load flow method .................................................. 24 4.1.5 Continuation Power Flow Method ................................................................... 24 4.2 Circuit Simulation Methods ...................................................................................... 25 4.2.1 Limiting methods ................................................................................................ 26 4.2.2 Homotopy Methods ............................................................................................ 27 5. Equivalent Circuit Approach ......................................................................................................... 30 5.1 Split-Circuit Formulation due to Non-Analyticity of Power Flow Equations ... 30 5.2 Equivalent Circuit Models for the Positive Sequence Power Flow Problem ...... 32 5.2.1 PV Bus ................................................................................................................... 32 5.2.2 Voltage Regulation of the Bus ........................................................................... 34 5.2.3 Continuous Model for a Generator/PV Bus .................................................... 37 5.2.4 Slack Bus ............................................................................................................... 43 5.2.5 PQ Bus .................................................................................................................. 45 5.2.6 ZIP Model ............................................................................................................. 46 5.2.7 BIG Model ............................................................................................................ 48 5.2.8 Transformer ......................................................................................................... 49 5.2.9 Transmission Line ............................................................................................... 53 5.2.10 Preliminary Result for Positive Sequence Power Flow ................................. 55 5.3 Equivalent Circuit Models for Three-Phase Power Flow Problem ...................... 56 5.3.1 Slack Bus ............................................................................................................... 56 5.3.2 ZIP Load Model ................................................................................................... 57 5.3.3 Three-phase BIG load model ............................................................................. 59 ix 5.3.4 Transmission Line ............................................................................................... 59 5.3.5 Three-Phase Transformers ................................................................................. 61 5.4 Preliminary results for Three-phase power flow .................................................... 63 5.5 Physics Based Models ................................................................................................. 65 5.5.1 Physics based model for Induction Motor (IM) .............................................. 65 5.5.2 Steady-State Fundamental Frequency Model ................................................. 69 6. Circuit Simulation Methods for Power System Analyses .......................................................... 71 6.1 Limiting Methods .......................................................................................................