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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 ... – PowerPoint PPT presentation

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Title: Chapter 5: Harmonic Analysis in Frequency and Time Domains


1
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

2
Overview
  • Introduction
  • Techniques for harmonic analysis
  • Conclusions

3
Introduction
  • Ideal operation conditions in power networks
  • Perfectly balanced
  • Unique and constant frequency
  • Sinusoidal voltage and current waveforms
  • Constant amplitude

4
  • However,network components (nonlinear and
    time-varying components and loads),distort the
    ideal sinusoidal waveform this distorting effect
    is known as harmonic distortion.

5
Digital Harmonic Analysis
  • Harmonic detection
  • Real time monitoring of harmonic content
  • Harmonic prediction
  • Harmonic simulation techniques

6
Harmonic Simulation Techniques
  • Frequency domain methods
  • Time domain methods
  • Hybrid time and frequency domain methods

7
Techniques for Harmonic Analysis
  • Frequency Domain.
  • Direct method
  • Iterative harmonic analysis
  • Harmonic power flow method

8
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)
9
Table 1. Power network harmonic representation
10
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.

11
  • 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.

12
  • Partitioning the matrix equation to separate the
    two types of busbars gives

(2)
13
  • The unknown voltage vector Vi is found by
    solving,

(3)
14
  • The harmonic currents injected by the harmonic
    voltage sources are found as,

(4)
15
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.

16
  • 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.

17
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.

18
  • The unified iterative solution for the system has
    the form,

(5)
19
  • 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.

20
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.

21
  • 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.

22
  • 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)
23
  • 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)
24
  • 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.

25
  • 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.

26
  • 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.

27
  • 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)
28
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.

29
  • 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.

30
  • 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.

31
Extrapolation to the Limit Cycle
32
  • 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)
33
  • Note that (9) allows the application of Newton
    type algorithms to extrapolate the solution to
    the Limit Cycle, obtained as 53,
  • where,

(10)
(11)
34
  • 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.

35
  • 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.

36
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.

37
  • The Fig. 1 illustrates the conceptual
    representation of the hybrid methodology.

Fig. 1 System seen from load nodes.
38
  • The iterative solution for the entire system has
    the form,

(12)
39
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.

40
  • 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.

41
  • 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.

42
  • 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.
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