Chapter 5: Harmonic Analysis in Frequency and Time Domains

1. Organized by Task Force on Harmonics Modeling & Simulation Adapted and Presented by Paulo F Ribeiro AMSC May 28-29, 2008 Chapter 5: Harmonic Analysis in Frequency and Time Domains 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 ΔI is the vector of incremental currents having the contribution of nonlinear components, ΔV is the vector of incremental voltages and [ΔYJ] 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 n×n, [B] is the control or input matrix of size n×r 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. P Limit Cycle ¥ x + D i 1 x + 1 i x D i Transient Orbit x i x A Cycle 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 ti+T, 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.