**1. Introduction**

Knowledge on actual overvoltages from transients in electrical transmission systems can be very helpful to estimate optimal measures for overvoltage protection, to understand the background of problems (if any) and/or to estimate related degradation of electrical insulation. Such transients arise e.g., from line energization and can cover a wide frequency spectrum [1]. The installation of equipment such as (R) C-dividers capable to capture high-frequency signals is costly, leaves a relatively big footprint and should be planned ahead. Dividers can also be prone to EMC (electromagnetic compatibility) disturbance [2], e.g., arising from ground currents upon a switching event. A non-invasive, cost-efficient, easy to (re-)install and EMC-proof measurement system would therefore be appealing to occasionally perform voltage monitoring.

This paper describes applications based on open-air capacitive sensors as part of a D/I (differentiating/integrating) measurement concept. The signal at the measurement cable end, characteristically terminated with 50 Ω, is basically the time-derivative of the phase-voltage as picked up with the open-air capacitive sensors. These sensors can be mounted in a short time without adaptation to the EHV (extra-high voltage) system. In other words, it is a non-invasive measuring system. The original waveform is restored by means of an analogue integrator. One of the earlier papers describing the method for application on high-voltage measurement is reference [3]. As the sensors basically consist of metal plates which form capacitances to high-voltage conductors nearby, the requirements related to costs, installation effort and being non-invasive are fulfilled. A drawback of the method relates to the prior unknown coupling strengths of open-air sensors and the limited

selectivity as they couple to any phase conductor in the vicinity. A matrix containing all couplings between sensors and phases needs to be established to entangle the cross-coupling.

A number of open-air sensor applications with solutions for the selectivity problem can be found in literature. In [4], line energization was studied. Decoupling could be achieved since different phases had distinct contact moments. This allowed to determine the responses of the sensors upon the three distinct energization moments. Each phase energization provided three sensor responses, enabling to establish the complete coupling matrix. In [5] the capacitive sensors were inside a GIS (gas insulated system) and in prior tests the phases could be energized separately. A harmonics study was published in [6] where the sensors were positioned halfway between two overhead line pylons. The capacitances were derived from equations for cylindrical line conductors above perfect conductive earth. Unknowns in distances (e.g., line heights) could be resolved by assuming symmetric power frequency voltages on the overhead lines and fitting the measured and calculated waveforms. For the transient overvoltage study in [7], a similar approach was taken. However, the sensors were placed near the terminations for high sensitivity. Due to the complex termination design, analytical field calculation was not possible, leaving the coupling matrix not fully determined. It was shown that with a few assumptions based on symmetry in the configuration the reconstruction could be realized.

This paper develops a methodology to analyse the consequence of uncertainties in the assumptions made to reconstruct the phase voltages. Section 2 briefly describes the methodology. Confidence bounds are needed to judge the accuracy of network simulations, which are implemented in PSCAD [8]. References [7] and [9] provide information on the modelled network detail and extension, respectively. In Section 3.1, these simulations are compared with measurements in relation to the experimental uncertainty. In [2] the merits of the D/I method in terms of EMC immunity are discussed. To verify the claims, simultaneous measurement of switching transients by the D/I system and a high voltage divider is performed. The reliability of the recordings from both systems is compared in Section 3.2. Section 4 reflects on the results in terms of accuracy as well as reliability and Section 5 concludes with situations for which D/I measurement is particularly of interest.

#### **2. Measurement Analysis and Uncertainty**

 *up*   = *ai*

Measured sensor signals *ui* relate to the phase voltages *Uj* according to a coupling matrix **M**: *u* = **M***U*. Reconstruction of the phase voltages from recorded signals can be achieved by inverting this relation. The matrix elements *Mij* are obtained by selecting the part from the recorded signals containing only the power frequency waveform (with angular frequency ω0). Sinusoidal fits *upf*,*<sup>i</sup>* are made and related with an assumed base of symmetric (per unit) phase voltages, i.e., equal amplitudes and 120◦ phase angle differences:

$$
\begin{pmatrix} u\_{pf,1}(t;q\_0) \\ u\_{pf,2}(t;q\_0) \\ u\_{pf,3}(t;q\_0) \end{pmatrix} = \begin{pmatrix} M\_{11} & M\_{12} & M\_{13} \\ M\_{21} & M\_{22} & M\_{23} \\ M\_{31} & M\_{32} & M\_{33} \end{pmatrix} \begin{pmatrix} \cos(\omega\_0 t + 2/3\pi) \\ \cos(\omega\_0 t) \\ \cos(\omega\_0 t - 2/3\pi) \end{pmatrix} \tag{1}
$$

$$
\text{with } u\_{pf,i}(t;q\_0) = a\_i \cos(\omega\_0 t + q\_0) + b\_i \sin(\omega\_0 t + q\_0)
$$

 +

 +

The reference time *t* = 0 in (1) corresponds with the central phase reaching its maximum value. If the trigger moment of the measurement device is not synchronized with (one of) the phase voltages, it can be implemented as a phase shift φ0. This additional unknown can be omitted by having the measurement time base related to a simultaneous recording of the phase voltage information, e.g., from a voltage transformer (synchronized), or it can be obtained from the measured waveforms themselves (unsynchronized). As (1) relates sinusoidal waves of a single frequency, each row equation determines only two parameters and the set of three equations is underdetermined. With ten unknowns (for unsynchronized measurements) and six fitted parameters also other sources of information must be employed. Reduction of the number of independent components *Mij* can be achieved by neglecting components when these are very small, or by having a symmetric positioning of sensors with respect

 + to the phase conductors meaning that various matrix components are equal [7,10]. Also parameter estimates can be made and their uncertainties can be retrieved, e.g., from comparison with numerical electrostatic field simulations on the system. Which assumptions are to be made and what the effects will be on the uncertainty margins are specific for the configuration at the measurement site. Measurements of energization transients are obtained from a double circuit 380 kV line-cable-line connection [7]. More specifically, measurements were taken at the underground cable to overhead line transition points (at both cable ends) and near an overhead line termination at one of the substations.

Uncertainty analysis concerns the determination on how input margins propagate into end result deviations according the designed measurement chain. In addition, reliability of the measurement can be impeded when also signal coupling along an unintended route takes place. This can be a serious problem when recording switching transients in (E)HV networks. Line energization causes steep switching fronts which, in particular at an overhead line to underground cable transition, may induce ground currents. These currents may partly find their way along the metallic screens of the measurement cables. Depending on the measurement system design and layout, this can disturb signal recordings.

In Figure 1 the configuration for a D/I system is compared to a capacitive divider configuration. The D/I sensor consisted of two 30 × 30 cm<sup>2</sup> plates, with the top plate sensing the phase voltages ( *CHV* = 0.1–1 pF) and the bottom plate connected to earth. For the D/I method, the capacitance between the plates is relatively low (here *CE* ≈ 300 pF). The measurement cable is characteristically terminated and its impedance *Rm* is far lower than |*ZE*| = 1/(ω *CE*). Up to the cut-o ff frequency, 1/(2 π*RmCE*), the time derivative *U*'(*t*) arises at the cable end. The signal waveform is restored by integration with an integrator time constant, τ*int*, depending on the integrator design. The active/passive integrator employed for the measurements is described in [11]. The over-all system response *U*(*t*) is given by:

$$\begin{array}{l} \text{d}I(t) = \frac{1}{\tau\_{\text{int}}} \int \text{d}I'(t) \text{d}t \text{ where } \text{d}I'(t) = \tau\_{\text{d}if} \frac{\text{d}I\_{HV}(t)}{\text{d}t} \\ \qquad \qquad \qquad \qquad \tau\_{\text{d}if} = R\_{\text{m}} \text{C}\_{HV} \end{array} \tag{2}$$

For a divider topology the situation is reversed and the impedance *Rm* is large compared to |*ZE*| in order not to load the low voltage branch of the divider. This di fference has repercussions on EMC aspects from ground currents as demonstrated in Figure 2. A part of the ground current will find its way along the measurement cable screens through magnetic induction. This common mode current *ICM* will cause a di fferential mode voltage *UDM* which adds to the intended signal. Its value depends on the quality of the measurement cable, which can be expressed as its transfer impedance *Zt* [12]:

$$
\mathcal{U}\_{DM} = I\_{CM} \mathcal{Z}\_t \tag{3}
$$

**Figure 1.** Measurement topologies for D/I and divider methods. Component values for D/I are such that the receiving cable end represents the time derivative of phase voltage *UHV* and the waveform is restored by integration (the open-air sensor design is shown on the left); For standard dividers this signal directly represents the phase voltage *UHV*.

The fraction of *UDM* arising at the receiving end of the cable is a division over the impedances at both ends as shown in the right side of Figure 2. For the D/I concept this is rather a small fraction, whereas for a divider it is the major part. In addition, the integration step in the D/I system reduces high frequency interference, which may have entered the measurement cable. Results will be presented comparing the D/I approach with results from pre-installed dividers near underground power cables at their transition points to overhead lines.

**Figure 2.** Effect of induced ground currents along the measurement cable (**left**); Through the cable transfer impedance the resulting common mode current *ICM* causes a differential mode voltage *UDM* which divides over the impedances at the cable ends (**right**).
