## 3 Results
### Case Study: Faults in the Lithuanian Distribution Grid
#### 3.2.3 Case Studies of Two-Dimensional Probabilistic Analysis  
In anticipation of Smart Grids and the interoperability of their applications, the investigation of many events should be comprehensive (multidimensional). Two-dimensional probabilistic analysis is based on the conditional probability definition given by Equation (28). Obviously, such an investigation is possible only when information about the probability distribution of the second event is available. Let us begin with lightning events (Figure 71), which are among the six most common groups in the 10 kV grid (see Figure 69). The available division of the territory of Lithuania into lightning zones regarding annual thunderstorm duration can be found in [5,66]. This kind of analysis is beneficial for the field of lightning security (including the shielding of both substations and buildings), lightning arrester placement strategies, calculations of induced voltage in overhead power lines (including exceeding safe touch voltage). The cumulative distribution function of negative first stroke current can be found in [67]: a crest current up to 200 kA is expected. Usually, the first stroke is followed by 2-10 successive strokes with 0.03-0.05 s intervals between them [66]. When the cumulative distribution function is known, the probability density function is obtained as follows:  
\[F(I)=\frac{1}{1+\left(\frac{I}{24}\right)^{2.6}}\Rightarrow\frac{\mathrm{d}F} {\mathrm{d}I}=\frac{13\cdot\left(\frac{I}{24}\right)^{1.6}}{120\cdot\left( \left(\frac{I}{24}\right)^{2.6}+1\right)^{2}}. \tag{32}\]  
where: \(I\)--the crest current of the stroke, kA. Next, the analysis through a prism of repair duration is beneficial for outage management (including the economic consequences assessment), network modes planning, operational switching. The parameters of repair hour distribution can be determined from information given in [56]: currently, it should not be critical that power transformer repair data is from 1995-2000, but an update is desirable. The probabilistic surface, shown in Figure 72, depicts the average situation in the case of the voltage of the transformer's primary winding is either 110 kV or 35 kV: the Gaussian distribution is combined with the gamma distribution (shape parameter is 0.12 and scale parameter is 18.4). In 1995-2000, the most common fault types, determined by transformer oil testing, thermography, insulator testing, vibro-acoustic testing, and other methods, were the following: commutator or tap changer failure (24%), fan or pump failure (19%), partial discharge (16%) [56]. Figure 70: Convolutions of the Poisson and Gaussian models. (**a**) Case of shrub grow-in (overhead power lines); (**b**) Case of humidity, rain, snow, and dust (outdoor and modular substations). Legend: 1 (blue curve)—autoconvolution function of the Poisson distribution; 2 (blue curve)—autoconvolution function of the Gaussian distribution; red curve—convolution function. The last and the most widely understood example is the failure rate dependence on exploitation age. In general terms, the failure rate of a device can be represented with a bathtub curve [55; 56]. This U-shaped curve has three regions: (1) decreasing rate due to early failures (in particular manufacturing defects); (2) constant rate due to random failures; (3) increasing rate due to the ageing process. Despite being well-known, the curve does not have a universally agreed mathematical expression. Moreover, it is also not known whether a particular device will follow it (especially under proper maintenance). According to [56], the Weibull distribution, whose shape parameter is 0.5 and scale parameter (determined from given constants) is approximately equal to 280, must be used for 10-year period failure rate evaluation of power transformers (whose primary winding voltage is either 110 kV or 35 kV). However, as can be seen in Figure 73a, this model represents only the first region of the U-shaped curve.