**3. Results**

The challenge with open-air sensors is to entangle the cross-coupling. The approach advocated in this paper is to employ only the stationary power frequency component from symmetric phase voltages. This requires a minimal installation effort to set up the measurement system, which can be done in typically one to one and a half hour. The option to sequentially inject a signal in each phase and determine the responses as in [5] would provide sufficient information to derive the complete coupling matrix, but it takes time and requires extensive safety precautions. Using the first arriving travelling wave fronts as in [4] assumes that these remain sufficiently clear and recognizable after travelling along the lines and cables. The cross-bonding applied along the underground power cable system modifies the waveform further and therefore such a method is mainly feasible when measurements are conducted near the substation where the switching activities take place.

Different causes for uncertainty are analysed in [10]. The measurement accuracy was checked by determining the variation of the coupling matrix throughout a measurement session at a single location. As the sensors remain at fixed positions, variations are related to noise, stability of the D/I measurement system (in particular the active integrator part) and fitting technique. The overall variation remained within 2%. Deviation from assumed symmetric phase voltages will translate into similar magnitude variations in the recorded signals. However, it was shown that in terms of per unit values, i.e., the percentage of the overvoltage related to the amplitude of the power frequency component, such deviation is negligible. The uncertainty analysed in Section 3.1 concerns the assumptions made in order to reduce the number of independent matrix components in (1). Furthermore, imprecise sensor positioning or metal structures in the vicinity may contribute to deviations from model assumptions.

Section 3.1 details the methodology to establish the coupling matrix for two distinct measurement locations. Section 3.2 elaborates on comparison with system simulation and comparison between D/I and divider measurement.

#### *3.1. Accuracy of Decoupling Procedure*

The left side of each row equation in (1) is fitted with the steady state power frequency part of the recorded signal. A single frequency sinusoidal function is determined by two parameters, meaning each row equation in (1) is underdetermined. Since the phase voltages are symmetric, with their sum equal to zero, a constant Δ*i* added to each component in row *i* (*i* = 1, ... , 3) will result in the same function and therefore will still satisfy the fitted relation on the left hand side: all components within a row are determined up to the same constant. Usually, the far end couplings (*M*13 and *M*31) are relatively small with respect to the direct couplings (*Mii*). Neglecting these or assigning an estimated

(small) value will therefore only have a minor effect. For the middle row in (1), in case of a reflection symmetric configuration, the additive value is principally undetermined. However, deviations from symmetry should be accountable to provide for a margin in φ0. Therefore, a fourth uncertainty, Δ0, is introduced. The coupling matrix with four uncertainties related to the same number of lacking parameter information after fitting can be formulated as:

$$\mathbf{M} = \begin{pmatrix} M\_{11} & M\_{12} & M\_{13} \\ M\_{21} & M\_{22} & M\_{23} \\ M\_{31} & M\_{32} & M\_{33} \end{pmatrix} \pm \begin{pmatrix} \Delta\_{1} & \Delta\_{1} & \Delta\_{1} \\ \Delta\_{2} - 1/2\Delta\_{0} & \Delta\_{2} & \Delta\_{2} + 1/2\Delta\_{0} \\ \Delta\_{3} & \Delta\_{3} & \Delta\_{3} \end{pmatrix} \tag{4}$$

For distinct locations, site specific assumptions will be made with confidence intervals and the consequence on the reconstructed transient waveforms is determined as discussed below.

The components in coupling matrix *Mij* in (4) can be re-ordered such that they constitute a linear set of nine parameters indicated as *xk*=<sup>3</sup>(*<sup>i</sup>*−<sup>1</sup>)+*j* = *Mij*. For these parameters the error matrix **Ex** needs to be established, which contains all variances and covariances:

$$E\_{\mathbf{x},kl} = \langle \Delta \mathbf{x}\_k \Delta \mathbf{x}\_l \rangle = \left\langle (\mathbf{x} - \overline{\mathbf{x}})\_k (\mathbf{x} - \overline{\mathbf{x}})\_l \right\rangle \tag{5}$$

To this end, normal distributions are assigned to the four uncertainties in (4) and 1000 simulations are made to evaluate (5). The reconstructed phase voltages can be found by inverting matrix **M** and applying it to recorded waveforms containing the transient events. The components of the inverted coupling matrix **M**−<sup>1</sup> can be arranged in a linear set *yl*=<sup>3</sup>(*<sup>i</sup>*−<sup>1</sup>)+*j* = *M*−<sup>1</sup> *ij* as well, and its error matrix **Ey** can be evaluated according [13,14]:

$$\mathbf{E\_{y}} = \mathbf{O\_{1}}\mathbf{E\_{x}}\mathbf{O\_{1}^{T}}\quad\text{with}\quad O\_{1,lk} = \frac{\partial y\_{l}}{\partial x\_{k}}\tag{6}$$

The matrix **O1** represents a linearization of how component *yl* of the inverted matrix depends on a variation in the value *xk* of the original coupling matrix. Its calculation can conveniently be implemented by slightly varying numerically each coupling matrix component *Mij* separately. Next, the phase voltage waveforms are reconstructed by matrix multiplication for each sample in the measurement recordings. The propagation of the error is described by means of matrix **O2** which provides information on how each of the three reconstructed phase voltages varies upon variation in each of the inverted matrix components. The reconstructed phase waveforms are evaluated with [13,14]:

$$\mathbf{E}\_{\mathbf{U}} = \mathbf{O}\_{2} \mathbf{E}\_{\mathbf{y}} \mathbf{O}\_{2}^{T} \quad \text{with} \quad O\_{2,jl} = \frac{\partial \mathbf{U}\_{j}}{\partial y\_{l}} \implies$$

$$\mathbf{O}\_{2} = \begin{pmatrix} u\_{1} & u\_{2} & u\_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & u\_{1} & u\_{2} & u\_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & u\_{1} & u\_{2} & u\_{3} \end{pmatrix} \tag{7}$$

This equation is applied for each measured sample point. The square roots of the diagonal components in **EU** provide the standard deviations for each phase (per sample point). Adding and subtracting the deviations provide the confidence intervals of the reconstructed waveforms.
