Normalized balanced magnitude

Parametrization of each residential site

The normalized balanced magnitude comprises two components, a deterministic and a stochas-tic component (c.f. equation 5.1). The parameters of the determinisstochas-tic component are obtained with the Fourier transform of the normalized balanced harmonic magnitude of each site. The parameters of the stochastic component are obtained with the residuals between the original values and the calculated deterministic component. Fig. 5.4 shows exemplary the original data (normalized balanced magnitude), the estimated deterministic component I⁽³⁾_{b norm D}(t)and the obtained residuals of the third harmonic of one site. For this example, the obtained determin-istic and stochastic components are:

I⁽³⁾_{b norm D}(t) = 0.35 + 0.32 · cos(2π · f_D· t + 67^◦) + 0.15 · cos(4π · f_D· t + 86^◦)+

0.05 · cos(6π · fD· t + 110^◦) I⁽³⁾_{b norm R}(t) = 0.95 · I⁽³⁾_{b norm R}(t − 1) + N (0 , 0.00029) (5.14) The deterministic component is described with the first four components (including the con-stant component) of the Fourier series. The number of components was selected based on an iterative procedure, where the number of components was increased step by step until the mean squared error between the original time-series and the deterministic component had a

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5. Model of residential low-voltage networks

Original D⁽³⁾(t) Residual

(3) b normI

Figure 5.4.: Example model fitting of one site

difference of less than 5%. The iterative procedure was applied to each site and each harmonic order, and the final selected number of components is the number of components that describes most of the sites and most of the harmonic orders appropriately.

The stochastic component is described with an autoregressive model of first order, which is fit-ted to the residuals of each site using maximum likelihood estimation [23]. The autoregressive model was selected based on the autocorrelation function (ACF) and the partial autocorrela-tion funcautocorrela-tion (PACF) of the residuals of each site and each harmonic order. Fig. 5.5 shows exemplary the ACF and PACF of the residuals of the example in Fig. 5.4. In this case, the ACF decays exponentially and the PACF has only one statistically significant spike at lag 1, which is the typical behavior of an autoregressive model of first order [110]. Similar graphs were obtained for each harmonic order and each site, and the autoregressive model of first order is selected as it is the best representation in most of the cases. Simple probability distributions are not recommended to describe the residuals, because they are not able to represent the depen-dence between successive values, a characteristic that should not be neglected with this type of data.

Figure 5.5.: Autocorrelation function and partial autocorrelation of the residuals of one site

The adequacy of the model to represent the normalized balanced magnitudes of each site is evaluated comparing the time-series and the probability distributions of the measured (dataset for modeling verification) and estimated data (data obtained with the model). The time-series are compared visually using time-series plots and the similarity measures introduced in section 4.4. The probability distributions are compared using Q-Q plots, which compare the quantiles of the original and estimated data sets, and if they come from the same distribution, then the Q-Q plot appears linear. Fig. 5.6 shows the time-series plots and the Q-Q plot of the exam-ple in Fig. 5.4. Both graphs show a very good similarity between the original and estimated data. Moreover, the Euclidean distance DME = 5.78 and Pearson’s correlation coefficient

5.2. Model parametrization

SM_P= 0.99indicate a very good similarity between real and estimated time-series, validating the model for this specific site.

0 0.2 0.4 0.6 0.8 1 X Quantiles

QQ-plot

0 1 2 3

-0.2 0 0.2 0.4 0.6 0.8 1

1.2 Time-series original data

0 1 2 3

(3) b normI

Time in days Time in days

Time-series estimated data

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

Y Quantiles

Figure 5.6.: Model evaluation results of one site, third harmonic, balanced component magnitude.

Here it is important to clarify, that the objective is to develop a generic model which is able to represent the behavior of typical residential sites, and not to develop accurate models of each site. In this case, approximate models of the sites are developed, which later can be compared to develop the generic model. For that reason, the number of Fourier coefficients and the order of the autoregressive model were fixed for all site models. Moreover, the models of each site can not be severally evaluated using formal fitting evaluation criteria or probabilistic tests, and the adequacy of the model can only be assessed by visual comparison of the time series and the probability functions.

Table 5.2 indicates the percentage of sites for which the model gives accurate results. The best results are obtained for the third, fifth and ninth harmonics, where more than 70% of the sites were correctly modeled. For the 13^thharmonic, only 19% of the sites could be correctly modeled with the time-series representation. This results are comparable with the time-series characterization presented in section 4.4. Comparing Table 5.2 with Fig. 4.14, it is clear that the model is successful only if the harmonic currents show a clear time-series. In general, the selected model can represent the harmonic behavior of the balanced harmonic magnitude of most residential sites. However, there are sites were there is no time-series, and the model may be not appropriate in those cases. To improve the model, first the characteristics of residential networks that lead to a time-series characteristic in the harmonic currents should be wholly comprehended, in order to define corresponding modeling methodologies for the sites where the time-series approach is not adequate.

Table 5.2.: Percentage of sites for which the time-series representation of I_b^(h) is adequate Harmonic order

3 5 7 9 11 13 15

Percentage of sites 97% 70% 55% 76% 49% 19% 51%

Parametrization of the generic model

The parameters of the models of the sites of the normalized balanced component are compared in order to determine the parameters of the generic model. Fig. 5.7 shows exemplary the com-parison of the parameters F C1,i and θ1,i of each of the sites, where a good similarity between the parameters of the sites is noticeable. For the comparison, only the results of the sites where the time series was successful are considered. Good similarity was also found for the other parameters of the deterministic component, which indicates a similar time-series between the

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5. Model of residential low-voltage networks

different sites. This result is also comparable with the time-series characterization made in the previous chapter (c.f. Table 4.1).

0 5 10 15 20 25 30 35 40

Figure 5.7.: Comparison parameters F C_1,i⁽³⁾ and θ_1,i⁽³⁾ and estimation of the parameters of the generic model

Normal distribution functions are selected to represent the parameters of the generic model, in such a way that the differences between different networks are also considered. For example, the parameter F C₁⁽³⁾of the generic model is a normal distribution of mean 0.295 and standard deviation 0.034, and the parameter θ⁽³⁾₁ is a normal distribution of mean 74.5^◦and standard de-viation 7.7^◦, as shown in Fig. 5.7. The mean and standard deviation of the normal distribution of each parameter are obtained by distribution fitting of the results of the different sites.

A correlation analysis between the parameters revealed that the parameters of the stochastic component, i.e. αb and σb, are highly correlated, and the correlation is almost the same for all harmonic orders. Fig. 5.8b shows the correlation between σ^(h)_b and α^(h)_b for all considered harmonic orders (h = 3, 5, ..., 15). The correlation between both parameters can be described with a linear regression. In this way, the parameter α^(h)_b is described with a normal distribution for each harmonic order, while the parameter σ_b^(h) is obtained with the regression. Fig. 5.8a shows exemplary the comparison of α_b⁽³⁾and its corresponding normal distribution

0 5 10 15 20 25 30 35 40

Figure 5.8.: Comparison parameters α⁽³⁾_b,i and σ_b,i⁽³⁾ and estimation of the parameters for the generic model

The mean and standard deviation of the normal distribution of each parameter of the generic model, including their confidence intervals, are listed in appendix C.

5.2. Model parametrization