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2023 | Book

Fractal Tomography for Power Grids

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About this book

The book introduces a new concept of Fractal Tomograph (FT) to study large and small perturbations of power grids. A mathematical phrase called Active Network Twins (ANT) is discovered which bears astonishing similarity with the human DNA. ANT forms the backbone of analysis. Lineflows and load voltages are compositions of ANT, called fractals which constitute tomographs. First three chapters discuss these new concepts. A remarkable feature is that a minute observation of tomograph is sufficient to detect a malaise in the past, present, or future of the grid. Editing like in genes, to control flows and voltages, is explained in chapter 4. The baffling loop flows in the US-Canada region, a metropolitan blackout in India, and outages in Nepal are illustrated with notional networks. Causes are crisply identified (chapters 5 and 6). Frequency rendezvous and blockchain approach to electricity costs (chapters 7, 8) are radically new perspectives. The book is useful for grid controllers and researchers.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Similarities observed in trees, leaves, flowers, creatures, etc. have enticed researchers for long. Mandelbrot made formal efforts in 1960s and 70 s (Benoit B. Mandelbrot (Book), Fractals: Form, Chance, and Dimension, Echo Point Books & Media, Reprint ed., February 2020) to mathematically represent geometric similarities. A complex number was squared and another added to it. The computation was repeated recursively many times. The plots of numbers on the complex plane started showing repeated patterns after a large number of iterations. These plots were called Mandelbrot sets. Julia sets are similar but result from computation with the number that is added at each iteration. Plots of some natural fractals, Mandelbrot set and Julia set are shown below (List of fractals by Hausdorff dimension, Available online, (Accessed Feb. 2023) https://​en.​wikipedia.​org/​wiki/​List_​of_​fractals_​by_​Hausdorff_​dimension).
S. D. Varwandkar, M. V. Hariharan
Chapter 2. Active Network Twins
Abstract
Active Network Twins (ANT) is a mathematical phrase made up of parameters and topology of the power circuit. It is used as a linear multiplier in computation of lineflows and voltages in the power grid, and emulates the DNA in human body that determines shape and behaviours of various organs. This chapter discusses similarities between ANT and DNA.
S. D. Varwandkar, M. V. Hariharan
Chapter 3. Fractals and Tomographs
Abstract
Fractals represent patterns which can be visualized with an artist’s eye. MWs and voltages of circuit elements are numbers which have no physical shapes but, like an artist, one can, assign shapes to these variables and scale them as well. The shapes may not be useful for numeric manipulation; nevertheless, they lend themselves to comparison with natural fractals, especially when several of them come together to form a new one. Theory of fractals by Mandelbrot was inspired by natural repetitive shapes, but the theoretical development employed functions of a complex number which makes the interpretation more difficult. Natural reality is lost. Power system performance consists of real flows and real voltage magnitudes possess visibility and are meaningful to grid operators.
S. D. Varwandkar, M. V. Hariharan
Chapter 4. Editing
Abstract
System operators are required to control lineflows and load voltages during operation. Tomographs can help them to edit appropriate fractals for intended lineflows and voltages. Generator powers at specific buses are adjusted in editing. For every edit, a compensating power needs to be added or subtracted at another generator to preserve power balance. Tomograph fractals with a few elementary calculations are needed. This chapter demonstrates this application on the IEEE 39-bus New England system.
S. D. Varwandkar, M. V. Hariharan
Chapter 5. Power Reversals and Loop Flows
Abstract
Litigations sprung up between stakeholders and regulators in power markets in the first decade of this century as phenomenon of Lake Erie Circulation in the US–Canada region played truant with contracted flows in power markets. This invited attention of system experts even as the trading had to be banned over certain routes. The term “loop flow” was coined around that time.
S. D. Varwandkar, M. V. Hariharan
Chapter 6. Blackouts
Abstract
A tomographic study is presented in this chapter that provides a glimpse of intricacies of three important outage scenarios and provides valuable insights into shortcomings in the design of systems.
S. D. Varwandkar, M. V. Hariharan
Chapter 7. Frequency Rendezvous
Abstract
Significant frequency deviations occur in power systems during disturbances giving rise to synchronizing power flows. After an outage, the steady state is reached at a new frequency which we call frequency of rendezvous (FR).
S. D. Varwandkar, M. V. Hariharan
Chapter 8. MW-Blockchain
Abstract
Blockchain is a ledger which keeps systematic record of all transactions between various parties in the market. Same principle applies to MW-transactions from generator to bus to another bus, and finally to loads in power networks. A chain is thus formed by these transactions finally sinking into various loads. MW-transactions depend on network-impedance matrix. Fractals are used to compute MW-lineflows from the MWs of generators. When placed side-by-side, the chains of all generator form a blockchain. MW-Cost of electricity at user-end can be determined from the blockchain if MW-costs at the source are known.
S. D. Varwandkar, M. V. Hariharan
Metadata
Title
Fractal Tomography for Power Grids
Authors
S. D. Varwandkar
M. V. Hariharan
Copyright Year
2023
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-9934-43-0
Print ISBN
978-981-9934-42-3
DOI
https://doi.org/10.1007/978-981-99-3443-0

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