*2.1. Impedance Identification*

The used signal generation is a sink: A load resistor is switched on and off in a random pattern. Hence, the load current is a random pulse pattern. Figure 1 shows the principle connection scheme of the network impedance measurement (NIM) device and its corresponding complex equivalent circuit. The voltage in off and on state (*<sup>v</sup>*1(*t*) and *<sup>v</sup>*2(*t*)) as well as the current in the on state *<sup>i</sup>*2(*t*) (off state current *<sup>i</sup>*1(*t*) is zero) are measured. The frequency-dependent complex values *<sup>V</sup>*1(*ω*), *<sup>V</sup>*2(*ω*) and *<sup>I</sup>*2(*ω*), derived from the fast Fourier transform, are used to calculate the complex frequency-dependent impedance *Z* N(*ω*) (1).

$$\underline{Z}\_{\rm N}(\omega) = \frac{\underline{V}\_{2}(\omega) - \underline{V}\_{1}(\omega)}{\underline{I}\_{2}(\omega)}\tag{1}$$

#### *2.2. Impedance Identification of Three Phase Systems*

In this section, the measurement system for the determination of the frequency dependency of the grid impedance is presented. The device includes highly accurate sensors for the measurement of voltage and current wave forms. Figure 2 shows a simplified scheme of the measurement setup to evaluate the impedance of the three-phase mid voltage PCC [10,11]. A resistive load is switched by an insulated gate bipolar transistor (IGBT), while the measurement loop (e.g., L1–L2) is selected by a B6 thyristor bridge. A 3D model of the measurement device and its main components are depicted in Figure 3. The measurement device is portable and may be connected to connection points in medium voltage grids via its own medium voltage switch gear (SF6 circuit breaker). To determine the line impedances of a three-phase system four measurements are required:


**Figure 1.** Principle of the network impedance measurement/identification.

**Figure 2.** Impedance identification circuit for 20 kV medium voltage level.

For each measurement the following parameters are recorded over at least one period of 50 Hz:


The loop impedances *<sup>Z</sup>*ab(*ω*), *<sup>Z</sup>*bc(*ω*), *<sup>Z</sup>*ca(*ω*) are derived from the recorded parameters with (1). These loop impedances can be rearranged into the line impedances *<sup>Z</sup>*a(*ω*), *<sup>Z</sup>*b(*ω*), and *<sup>Z</sup>*c(*ω*) [10]:

$$\underline{Z}\_{\rm a}(\omega) = \frac{1}{\underline{2}} \cdot \left[ \underline{Z}\_{\rm ab}(\omega) - \underline{Z}\_{\rm bc}(\omega) + \underline{Z}\_{\rm ca}(\omega) \right] \tag{2}$$

$$\underline{Z}\_{\rm b}(\omega) = \frac{1}{2} \cdot \left[ \underline{Z}\_{\rm ab}(\omega) + \underline{Z}\_{\rm bc}(\omega) - \underline{Z}\_{\rm ca}(\omega) \right] \tag{3}$$

$$\underline{Z}\_{\rm c}(\omega) = \frac{1}{2} \cdot \left[ -\underline{Z}\_{\rm ab}(\omega) + \underline{Z}\_{\rm bc}(\omega) + \underline{Z}\_{\rm ca}(\omega) \right] \tag{4}$$

**Figure 3.** Impedance identification measurement device for 20 kV medium voltage level.

Typically, the impedances are average values over ten 50 Hz periods measured every five minutes over a month or more to identify impedance changes over daytime. Figure 4 shows a sample measurement. The compression approach, explained in Section 4, is applied to the voltage and current data, which is recorded during grid excitation and which is used for the calculation of the grid impedance. The authors want to determine the effect of data compression on the grid impedance calculation with this compressed raw dataset.

This setting yields 288 measurements per day or 8928 per month for each recorded voltage and current parameter. The sample rate is 500 kHz. Overall, the authors use a dataset with d = 206,744 data points (sample size *d*) for each recorded parameter, voltage and current. For an easy representation of the following figures and the compression dataset, the number of measuring points *d* was used (instead of the time vector *t*). Theoretically, the respective t-vector would have to be multiplied by the reciprocal of the sampling rate (500 kHz).

As the measurement device is remotely controlled and the data is transferred via the cellular network a compression is beneficial to optimize the data transfer speed and costs. Especially when the measurement device's installation site is in areas with low network coverage exhibiting low transfer rates. The transients in the recorded voltage and current parameters are the essential part of the data, as they determine the frequency dependency of the impedance values.

**Figure 4.** Measurement results with *U*DS (**a**) and *I*DS (**b**).

#### **3. Lossy Compression Techniques**

Various meta-analyses, types and overviews of data compression approaches can be found in [12–16]. The compression techniques are divided into lossy and lossless methods. Lossy ones generate better results by losing (preferably irrelevant) information. This can be explained by the fact that the result of the decompression is not identical to the starting dataset. In contrast, lossless methods produce an identical decompressed dataset [13].

The combination of impedance measurements and data compression can hitherto be found only in other fields of research. As an example serves the medical area where an extensive comparison of compression methods adapted to the impedance of cardiomyocytes is presented. The approach uses the wavelet transformation technique to analyse the effect of compression on sensitive data coming from cardiomyocytes and generating compression ratio of round about 5:1 [9].

Hereinafter follows a short description of the lossy compression methods that are compared in this paper. All of these approaches are frequently used for other types of data (SVD, WT) and appear interesting for the paper approach (TFA) [17–22]. The two well-known, widely used approaches WT and SVD are only briefly described. For further explanation, please consult the references that are listed in the Sections 3.1 and 3.2. TFA is explained in more detail but can be found in [17,22] if necessary.

## *3.1. SVD—Singular Value Decomposition*

The so-called Singular Value Decomposition (SVD) splits a m × n set of data (voltage/current × time stamp) **DS** into three different matrices (5). The diagonal matrix **Σ** contains the singular values (SVs), see also Figure 5.

$$\mathbf{DS}\_{m \times n} \quad = \quad \mathbf{U}\_{m \times m} \Sigma\_{m \times n} \mathbf{V}\_{n \times n}^{\mathbf{T}} \tag{5}$$

The data compression takes advantage of the fact that a close approximation of **DS** can be achieved by keeping the significant SVs of matrix **Σ**. The compression success depends on the amount of reduction of singular values in **Σ**.

**Figure 5.** SVD of data matrix **DS**.
