Chapter 5: Harmonic Analysis in Frequency and Time Domains

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Chapter 5 Harmonic Analysis in Frequency and
Time Domains
Tutorial on Harmonics Modeling and Simulation

• Contributors A. Medina, N. R. Watson, P.
  Ribeiro, and C. Hatziadoniu

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Overview

• Introduction
• Techniques for harmonic analysis
• Conclusions

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Introduction

• Ideal operation conditions in power networks
• Perfectly balanced
• Unique and constant frequency
• Sinusoidal voltage and current waveforms
• Constant amplitude

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• However,network components (nonlinear and
  time-varying components and loads),distort the
  ideal sinusoidal waveform this distorting effect
  is known as harmonic distortion.

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Digital Harmonic Analysis

• Harmonic detection
• Real time monitoring of harmonic content
• Harmonic prediction
• Harmonic simulation techniques

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Harmonic Simulation Techniques

• Frequency domain methods
• Time domain methods
• Hybrid time and frequency domain methods

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Techniques for Harmonic Analysis

• Frequency Domain.
• Direct method
• Iterative harmonic analysis
• Harmonic power flow method

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Direct Method

• The frequency response of the power network, as
  seen by a particular bus, is obtained injecting a
  one per unit current or voltage at the bus of
  interest with discrete frequency steps for the
  particular range of frequencies.
• The process is based on the solution of the
  network equation,

(1)
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Table 1. Power network harmonic representation
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Hybrid voltage and current excitations

• Most power system nonlinearities manifest
  themselves as harmonic current sources, but
  sometimes harmonic voltage sources are used to
  represent the distortion background present in
  the network prior to the installation of the new
  nonlinear load.

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• A system containing harmonic voltages at some
  busbars and harmonic current injections at other
  busbars is solved by partitioning the admittance
  matrix and performing a partial inversion.
• This hybrid solution procedure allows the unknown
  busbar voltages and unknown harmonic currents to
  be found.

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• Partitioning the matrix equation to separate the
  two types of busbars gives

(2)
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• The unknown voltage vector Vi is found by
  solving,

(3)
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• The harmonic currents injected by the harmonic
  voltage sources are found as,

(4)
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Iterative Harmonic Analysis (IHA)

• The IHA is based on sequential substitutions of
  the Gauss type.
• The harmonic producing device is modeled as a
  supply voltage-dependent current source,
  represented by a fixed harmonic current source at
  each iteration.
• The harmonic currents are obtained by first
  solving the problem using an estimated supply
  voltage.

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• The harmonic currents are then used to obtain the
  harmonic voltages.
• These harmonic voltages in turn allow the
  computation of more accurate harmonic currents.
• The solution process stops once the changes in
  harmonic currents are sufficiently small.

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Harmonic Power Flow Method (HPF)

• The HPF method takes into account the
  voltage-dependent nature of power components.
• In general, the voltage and current harmonic
  equations are solved simultaneously using
  Newton-type algorithms.
• The harmonics produced by nonlinear and
  time-varying components are cross-coupled.

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• The unified iterative solution for the system has
  the form,

(5)
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• where ?I is the vector of incremental currents
  having the contribution of nonlinear components,
  ?V is the vector of incremental voltages and
  ?YJ is the admittance matrix of linear and
  nonlinear components.

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Time Domain

• In principle, the periodic behavior of an
  electric network can be obtained directly in the
  time domain by integration of the differential
  equations describing the dynamics of the system,
  once the transient response has died-out and the
  periodic steady state obtained.

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• This Brute Force (BF) procedure may require of
  the integration over considerable periods of time
  until the transient decreases to negligible
  proportions.
• It has been suggested only for the cases where
  the periodic steady state can be obtained rapidly
  in a few cycles.

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• In this formulation, the general description of
  nonlinear and time-varying elements is achieved
  in terms of the following differential equation,
• where x is the state vector of n elements

(6)
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• Practical nonlinear power systems can be
  appropriately solved in the time domain with a
  state space matrix equation representation based
  on non-autonomous ordinary differential equations
  having the form,
• where A is the square state matrix of size
  nn, B is the control or input matrix of size
  nr and u is the input vector of dimension r.

(7)
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• Widely accepted digital simulators for
  electromagnetic transient analysis, such as EMTP
  and PSCAD/EMTDCTM can be used for steady state
  analysis.
• However, the solution process can be potentially
  inefficient, as detailed before.

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• Here, a discrete time domain solution for any
  integration step length h is adopted, where the
  basic elements of the power network, e.g. R, L
  and C are represented with Norton equivalents
  depending on h.

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• Other power network elements are formed with the
  adequate combination of R, L and C, which are in
  turn combined together for a unified solution of
  the entire network in the time domain, e.g.

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• Where G is the conductance matrix, v(t) the
  unknown voltages at time t, i(t) the vector of
  nodal current sources and iH the vector of past
  history current sources.

(8)
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Fast Periodic Steady State Solutions

• A technique has been used to obtain the periodic
  steady state of the systems without the
  computation of the complete transient Aprille
  and Trick, 1972. This method is based on a
  solution process for the system based on Newton
  iterations.
• In a later contribution Semlyen and Medina,
  1995, techniques for the acceleration of the
  convergence of state variables to the Limit Cycle
  based on Newton methods in the time domain have
  been introduced.

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• Fundamentally, to derive these Newton methods it
  is assumed that the steady state solution of (6)
  is T-periodic and can be represented as a Limit
  Cycle for in terms of other periodic element of
  or in terms of an arbitrary T-periodic function,
  to form an orbit.

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• Before reaching the Limit Cycle the cycles of the
  transient orbit are very close to it. The
  location of these transient orbits are
  appropriately described by their individual
  position in the Poincaré Plane.

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Extrapolation to the Limit Cycle
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• It is possible to take advantage on the linearity
  taking place in the neighborhood of a Base Cycle
  if (6) is linearized around a solution x(t) from
  ti to tiT, yielding the variational problem,
• where is the T-periodic Jacobian matrix.

(9)
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• Note that (9) allows the application of Newton
  type algorithms to extrapolate the solution to
  the Limit Cycle, obtained as 53,
• where,

(10)
(11)
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• In (10) , and are the vectors of state variables
  at the Limit Cycle, beginning and end of the Base
  Cycle respectively, and in (11) C, I and are the
  iteration, unit and identification matrices,
  respectively.

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• It has been concluded from the analyzed case
  studies that the Newton methods based on a
  Numerical Differentiation (ND) and a Direct
  Approach (DA) process, respectively, require less
  than 43 of the total number of periods of time
  needed by the BF approach, substantially reducing
  the computation effort required by the ND and DA
  methods to obtain the periodic steady state
  solution.

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Hybrid Methods

• The fundamental advantages of the frequency and
  time domains are used in the hybrid methodology
  53, where the power components are represented
  in their natural frames of reference, e.g., the
  linear in the frequency domain and the nonlinear
  and time-varying in the time domain.

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• The Fig. 1 illustrates the conceptual
  representation of the hybrid methodology.

Fig. 1 System seen from load nodes.
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• The iterative solution for the entire system has
  the form,

(12)
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3. Conclusions

• A description has been given on the fundamentals
  of the techniques for the harmonic analysis in
  power systems, developed in the frames of
  reference of frequency, time and hybrid
  time-frequency domain, respectively. The details
  on their formulation, potential and iterative
  process have been given.

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• In general Harmonic Power Flow methods are
  numerically robust and have good convergence
  properties. However, their application to obtain
  the non-sinusoidal periodic solution of the power
  system may require the iterative process of a
  matrix equation problem of very high dimensions.

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• Conventional Brute Force methodologies in the
  time domain for the computation of the periodic
  steady state in the power system are in general
  an inefficient alternative which, in addition,
  may not be sufficiently reliable, in particular
  for the solution of poorly damped systems.

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• The potential of the Newton techniques for the
  convergence to the Limit Cycle has been
  illustrated.
• Their application yields efficient time domain
  periodic steady state solutions.