## 2 A Proposed ASPC Approach for OCRs in a DG-Integrated Distribution Network
### Coordination Constraints of OCRs
#### 2.3.3. Calculating the Min-Max Confidence Interval of Fault Currents in the DN  
Having DGs  
Figure 2 shows a flow diagram of forecasting a min-max confidence interval of load power at each load bus in the DN. According to the work [82], there are seven main steps to forecast a min-max confidence interval of load power at each load bus on a feeder of distribution network as the following. * Step 1--Input the historical load profiles at each load bus and perform the wrangling of the input data. * Step 2--Calculate Probability Density Function (PDF) of load data and check whether the load data are in a Gaussian distribution or not. If the input load data have a Gaussian distribution, it is continued to Step 6. * Step 3--If the input load data are not normally distributed, then the \"differencing\" method is used to eliminate the data trend by creating a new differential load data series in the one-day-ahead basis and the PDF of new differential load data series will be next calculated. When its PDF is in a Gaussian distribution, Step 6 will be implemented. * Step 4--If the new differential data series is still not normally distributed, a Principal Components Analysis (PCA) method is conducted; and then the PDF is re-calculated for new differential load sub-datasets. If the new differential load sub-datasets are normally distributed, then it is continued with Step 6; otherwise, it goes to Step 5. * Step 5--A 'dendrogram' method is conducted over the whole new differential load data series to explore the relationship among 15-min, 30-min, or hourly load data points, and then cluster them into many smaller differential load sub-datasets. Subsequently, it will go to Step 6. * Step 6--The input load data at Step 2, the new differential load data series at Step 3, the new differential load sub-datasets at Step 4, or the smaller differential load sub-datasets at Step 5 are filtered-out with a possible confidence range of 13 levels, specifically from 90% to 99% with an increasing interval of 1%, 4.5-sigma (~99.73%), 5.5-sigma (~99.9937%), and 6-sigma (~99.99966%). Then, it is shown that the best confidence interval of load data will be determined by an artificial neural network (ANN) forecasting model with the lowest Mean Absolute Percentage Error (MAPE) metric. * Step 7--Applying the best confidence interval at Step 6 to eliminate unexpected outliers/noises of the original input load data; and then a min-max confidence interval of load power, [\(P_{load\_min}\), \(P_{load\_max}\)], at each load bus will be calculated by Equation (10). \[\left\{ \begin{array}{l}P_{load\_max}=\mu_{P}+\frac{Z}{\sqrt{N}}\sigma_{P}\\ P_{load\_min}=\mu_{P}-\frac{Z}{\sqrt{N}}\sigma_{P}\end{array} \right.\] (10) where a \(Z\) factor is taken from a normal/Gaussian distribution table corresponding to the best confidence level; \(\mu_{P}\) is the mean value of load power at each load bus; \(N\) is the number of observed load data points; and \(\sigma_{P}\) is the standard deviation of the analyzed load data. Figure 2: A flow diagram of forecasting a min-max confidence interval of load power at each load bus in the DN. 2. Calculating a Min-Max Confidence Interval of Load Current at Each Load Bus  
Based on the min-max confidence interval of load power at each load bus in the DN and the nominal bus-voltage value, a min-max confidence interval of load current \([I_{\text{load\_min}},I_{\text{load\_max}}]\) at each load bus can be calculated by:  
\[\left\{ \begin{array}{l}I_{\text{load\_min}}=\frac{P_{\text{load\_min}}}{P_{ \text{nom}}}\\ I_{\text{load\_max}}=\frac{P_{\text{load\_max}}}{P_{\text{nom}}}\end{array} \right. \tag{11}\]  
where \(V_{nom}\) is the nominal bus-voltage value. The minimum and maximum load current values, \(I_{\text{load\_min}}\) and \(I_{\text{load\_max}}\), at each bus are used for the current injection-based power flow analysis in the distribution network with DGs. 3.