*5.2. Learning-Aided Model Construction*

As mentioned before, generation prior distribution is assumed to be a uniform distribution over generators' nominal limits. Regarding generation and load balance constrained, the total load is determined as the sum of generation. Nodal load is then acquired by sampling from historical load distributions. As for the settings of time-domain simulation, fault start time is 0.1 s, simulation period is 2 s, and timestep is 0.05 s. Following the above preconditions, 10,000 samples are generated, of which the ratios to the training set, validation set, and test set are 80%, 10%, and 10%.

**Figure 2.** The modified IEEE 39-bus system for case study.

The DBN structure from the input layer to the output layer is {93-40-20-10-5-4}, where the elements stand for neuron quantity. Sigmoid is selected as the activation function. After training the samples, the DBN is forwarded to be tested on out-of-sample sets (i.e., test set). The scatter of estimates vs. actual values and error distribution is shown in Figure 3. The coefficient of determination (R2) is 0.9814, and the mean square error is 5.84 <sup>×</sup> <sup>10</sup>−4. As shown in Figure 3b, the error distribution approximately obeys a normal distribution, and the mean and standard deviation of the estimation error are 0 and 0.023. It can be found that 95% of the samples are in the interval of [−0.042, 0.046] through statistics, and the error with the 95% confidence level of the normal distribution is 0.004. Figure 3 demonstrates that the proposed learning model can render accurate TSA and strongly generalizes.

**Figure 3.** The visualization of testing the trained learning model: (**a**) estimate vs. true; (**b**) error distribution.

To further verify the performance of DBN, comparisons against back propagation neural network (BPNN), support vector regression (SVR), and regression tree (RT) are carried out, and the outcomes are given in Table 2. Mean square error (MSE) and square correlation coefficient (SCC) are used to evaluate the performance. You can see clearly that the DBNs beat other learning methods; thus, it can be concluded that DBN is the best one in TSA tasks.

**Table 2.** Accuracy comparison of each TSI surrogate model on test sets.

