1. Implementation Principles
The IMTB applies the frequency scan technique based on time-domain simulations in an electromagnetic transient (EMT) simulation tool, to extract the frequency-domain small-signal immittance or transfer function model at the point of connection (PoC) of an electrical circuit. The immittance is by default provided in the impedance formats, and the definitions are described in the following section. If users require admittance models, further manual processing using the toolbox’s library is needed. The frequency scan principles are briefly described in last section.
The impedance definitions for AC and DC PoCs both assume three-terminal network connections, and the circuit configurations using single-line diagram are provided in figures below. Table below listed the measured responses from the IMTB for AC and DC scans. If unclear, more detailed definitions for AC scan can be found in the relevant references, as provided in the table below. For the DC scan, the impedance is defined based on pole-to-neutral voltages and currents flowing into poles.
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Figure 1. Terminal definitions of IMTB for AC and DC scans. (a) AC scan; (b) DC scan.
Table 1. Impedance or transfer function definition of IMTB frequency scan results
Note:1 The Jacobian matrix is defined using the power flowing out of the test objective (current flowing out of the test objective) as the positive sign convention, following the common definition in power system analysis. This is different from the impedance calculation, where the current flowing into the test object is defined as the positive sign convention.
The IMTB applies the frequency scan technique to extract the frequency-domain small-signal models at the PoC of a test objective. The frequency scan concept is elaborated in figures below using a single-input single-output impedance calculation based on circuit and phasor illustrations. To measure the small-signal impedance model of a test objective, the original system can be perturbed by a voltage or current source at its PoC, as shown in figure below. The perturbation source can be designed as a sinusoidal perturbation at the angular frequency of ωp as small enough, so it guarantees the perturbed system is still capturing the linear approximation of the original system. Then by measuring the small-disturbed voltage and current wave-forms and analyzing their responses at the perturbed frequency ωp, it is possible to calculate the impedance at this perturbed frequency, as illustrated by figure below using time-domain waveforms and phasor plots. If one sweeps the perturbed frequency ωp in the specified frequency range, the small-signal impedance can be calculated in the frequency domain. At each perturbed frequency, the calculated impedance is a complex number. Such a frequency scan concept can be extended to other transfer function calculations if the input and output are defined accordingly. It can also be extended to higher-dimensional systems, then the measured frequency response can be a multi-input multi-output transfer function matrix.
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Figure 2. Frequency scan concept illustration using impedance measurement.(a) Perturbation injection at the PoC of the test object; (b) Impedance calculation at a perturbed frequency.
In the IMTB, the frequency scan technique is applied to calculate different types of impedance models and transfer functions. A high-level flowchart is provided in figure below to illustrate the implementation of frequency scan.
- The frequency scan setting is firstly configured based on user inputs.
- Then through coordinate transformations, the needed perturbations are directly applied in EMT simulations using the three-terminal controlled voltage or current perturbation sources, and the simulation is carried out at each perturbed frequency. Here single-tone perturbation is applied each time for the simulation.
- After the simulation is done, the voltage and current responses at the PoC are recorded, and the coordinate transformations are applied on time-domain waveforms to the relevant reference frames for frequency-domain calculation. For each perturbed frequency, the Discrete Fourier Transformation is then applied for frequency-domain response analysis and then the impedance or transfer function calculation.
- Finally, the data is exported, and plots are created using Bode diagrams.
Figure 3. Flow chart of the frequency scan implementation in IMTB
[1] CIGRE WG B4.67, AC side harmonics and appropriate harmonic limits for VSC HVDC, 2019.
[2] Y. Liao and X. Wang, "Stationary-Frame Complex-Valued Frequency-Domain Modeling of Three-Phase Power Converters," IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1922-1933, June 2020.
[3] Z. Shen, “Online measurement of three-phase AC power system impedance in synchronous coordinates,” Ph.D. dissertation, Dept. Elect. Eng., Virginia Polytech. Inst. State Univ., Blacksburg, VA, USA, 2012.
[4] Z. Yang, C. Mei, S. Cheng and M. Zhan, "Comparison of Impedance Model and Amplitude–Phase Model for Power- Electronics-Based Power System," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 3, pp. 2546-2558, Sept. 2020.