To validate the multi-objective optimization model for PI construction, the wind power data sampled every 10 min from the Alberta interconnected electric system in 2015 are applied. A total of 6000 datasets from 1 January are chosen as the research data with 80% for training and the other 20% for validating and testing, respectively. Firstly, the datasets are normalized in

$[0,1]$ to adjust the parameters of WNN and avoid the influence from different magnitudes of the datasets. Then the structure of WNN using the Morlet wavelet as the basic wavelet is determined. It is necessary to balance the complexity and the learning capacity of WNN. The C-C phase space reconstruction and Kolmogorov theorem are adopted to choose the optimal WNN structure as 4-8-2. Finally, the parameters of WNN and EKMOABC will be set as mentioned in

Section 4.

The CWC for PI construction is a single-objective optimization problem, while PIMOC is a multi-objective optimization problem. To valid the experiment results, the basic ABC algorithm in [

21] is used to optimize the CWC, and the MOABC algorithm in [

22] to optimize the WNN for PIMOC. The food source number of ABC and MOABC are both set to 40, the maximum iteration time is set to 200, the maximum number of solutions in the archive is 20, and the maximum number of obsolete scout bees is 50.

#### 5.1. Performance Comparisons between CWC with Different Parameters and PIMOC

To validate the influences of parameters on CWC, the control parameter

$\eta $ in Equation (5) is chosen as

$\eta =10,\text{}50,\text{}100$ and

$\mu $ is set as 80%, 85%, 90%, 95%, and 99%. Each experiment is repeated 10 times with different

$\eta $ and

$\mu $. The average results are shown in

Table 1.

From

Table 1, it is obvious that PICP increases with the increase of

$\mu $ when

$\eta $ is fixed, and PICAW also becomes larger, which makes the accuracy of the PIs worse. On the other hand,

$\eta $ has a great effect on the quality of PIs with the fixed

$\mu $. A small

$\eta $ is helpful to improve the accuracy of PIs with a small width of PIs, but it is difficult to ensure that PICP reaches its expected

$\mu $ and meets the reliability requirements of the PIs. A large

$\eta $ can enhance the reliability of the PIs, and the PICP is always higher than its expected value. However, this will lead to a large width of the PIs, which then makes the solving process into a local optimum. Thus, the controlling parameter

$\eta $ is an uncertain factor in PI construction with CWC, which is unfavorable for solving this optimization problem in high quality.

In order to compare the prediction results between CWC (with the ABC-WNN as the prediction model) and PIMOC (with the MOABC-WNN as the model), the experiments with a group of

$\mu $ (from 80–99%) in different

$\eta $ are performed and the results are shown in

Figure 2.

As is shown in

Figure 2, the prediction results of PIMOC are better than CWC both in the criteria of PICP and PINAW. With different

$\eta $, the PICP and PICAW of CWC change greatly. During the optimization of CWC, the solutions that cannot satisfy the given confidence level

$\mu $ will be penalized by

$\eta $ and determined whether to be saved or not according to

$\eta $. Perhaps a large

$\eta $ can ensure that the requirement of the confidence level

$\mu $ should be met, but some good solutions which have a PICP slightly less than its expected value, but a good PI width, will be weeded out. However, if a small

$\eta $ is chosen and then the solutions with a slightly worse confidence level, but a small width of the PIs is saved, those solutions dissatisfy the requirement of the confidence level and will not be penalized sufficiently. As a result, the confidence level of all the solutions will be lowered. Thus, a suitable

$\eta $ is of great significance to CWC, but it is always chosen empirically. In PIMOC, this multi-objective optimization problem is solved by adopting a multi-objective evolution algorithm directly instead of transforming it into a single-objective optimization problem, which avoids the choosing of

$\eta $ and it is a benefit for improving the quality of PIs both in accuracy and reliability.

For a better explanation, the PIs of CWC are shown in

Figure 3,

Figure 4 and

Figure 5 with three different

$\eta $ (the value of

$\eta $ is same as it in

Table 1) and the PIs of PIMOC are shown in

Figure 6, where Pareto optimal sets are sorted in ascending order according to PICP and the first, fifth, tenth, fifteenth, and twentieth PIs are plotted to make a comparison with the PIs of CWC.

As is shown in

Figure 3,

Figure 4 and

Figure 5, with the control parameter

$\eta $ increasing, the prediction accuracy of the PIs is improved, while the width is larger and the reliability becomes worse, which is just the same as in

Table 1. An unreasonable choice of

$\eta $ may easily cause some excellent solutions unreserved in the next iteration. From Equation (5), it can be seen that, in CWC, the quality of the PIs is assessed impartially only based on a single synthetical criterion. While in PIMOC, some excellent solutions are reserved by the Pareto dominance strategy avoiding the choice of

$\eta $, and the assessment is carried out according to both PIMOC, both in accuracy and reliability. Then we can see that the performances of PIMOC in

Figure 6 are better than all those of CWC in

Figure 3,

Figure 4 and

Figure 5.

#### 5.2. Performances of the Multi-Objective Evolution Algorithm in PI Construction

To evaluate the proposed EKMOABC for solving the multi-objective optimization problem with PIMOC, three classic multi-objective optimization algorithms, including the basic MOABC, NSGAII (non-dominated sorting genetic algorithm II) [

23], and MOPSO (multi-objective particle swarm optimization) [

24], are used to optimize the parameters of WNN for contrast experiments. The population size, the maximum number of iterations, and the maximum number of solutions in the archive are set to 40, 200, 20, respectively, for all algorithms. The crossover probability and the mutation probability of NSGAII are set to 0.8 and 0.2, respectively. The inertia weight and the learning factor of MOPSO are set to 0.8 and 2, respectively. The probability choices parameter

${\eta}_{1}$ and

${\eta}_{2}$ of EKMOABC are set to 50 and 10. The maximum obsolete number of scout bees in MOABC and EKMOABC are all set to 50. The comparative results for PI construction with four different multi-objective evolutionary optimization algorithms are shown in

Figure 7.

As shown in

Figure 7, Compared with the other three evolutionary optimization algorithms (NSGAII, MOPSO, and MOABC) in WNN optimization, the proposed EKMOABC-WNN can ensure a higher confidence level and narrower width of PIs, especially when the PICP is above 97%, where there is no suitable width of PIs that can be chosen by the other three algorithms. The Pareto optimal set with EKMOABC-WNN has not only better convergent performance, but also distributing performance. One of the reasons is that NSGAII, MOPSO, and MOABC may be easily trapped into a local optimum and their searching capacities will become so weak that the better solutions cannot be searched in the constraint range. Another reason is that the distribution of the Pareto set is not considered sufficiently in the process of iteration in NSGAII, MOPSO, and MOABC, which results in the Pareto set non-uniform distribution. In addition, the EKMOABC is based on the relationship between Pareto domination and distribution. The advanced probability choice equation of EKMOABC can avoid the solutions being trapped into local optimum effectively and it is a benefit for the distribution of the Pareto optimal set. The strategy of the guidance with elite population knowledge plays an important role in searching for better solutions and improving the convergent performance of the Pareto optimal set.