Following the conventional ESN model, the DeepESN model is made up of multiple dynamical reservoir components. Specifically, the DeepESN reservoir is organized into stacked recurrent layers. For each layer, the output is the input of the next layer, as outlined in Figure 1 (Gallicchio and Micheli, 2017). In our work, NU indicates the number of the input measurements, NL indicates the reservoir layer count, NR denotes the number of the recurrent units, and t indicates time. Moreover, u(t) denotes the model input at time t, whereas x ⁽ⁱ⁾(t) represents the state for the ith reservoir layer at time t. The DeepESN reservoir dynamics are mathematically modeled as follows. The dynamics of the primary DeepESN layer can be expressed as x ( 1 ) ( t )    =   ( 1 - a ( 1 ) )   x ( 1 ) ( t − 1 ) +   a ( 1 )   f   ( W ( 1 )  u ( t )   + W ^ ( 1 )   x ( 1 ) ( t − 1 ) ) (1)

When i > 1, the DeepESN state is computed as x ( i ) ( t )   =   ( 1 -   a ( i ) )   x ( i ) ( t − 1 ) +   a ( i )   f   ( W ( i )  x ( i ) ( t )   + W ^ ( i )   x ( i ) ( t − 1 ) ) (2) where W⁽¹⁾ indicates the input weight matrix, W⁽ⁱ⁾ indicates the weights of the connections between the (i − 1) ^th and ith layers, W ^ ( i ) indicates the recurrent weights of the ith layer, a ⁽ⁱ⁾ ∈ [0, 1] indicates the leakage rate of the ith layer, and f represents the employed activation function (regularly chosen as the tanh function).

GWO could be a nature-inspired algorithm that imitates the chain of command of administration and daily routine (Mirjalili et al., 2014). The wolves have four conceivable sorts: alpha, beta, delta, or omega. The pioneers of the pack (called alphas), which may be recognized by the leading administration abilities instead of the most grounded body, make choices almost every day exercises for the whole pack. The beta wolf helps alpha to make a choice. The omega wolf position is most reduced among wolves, but it plays a key part in keeping up a prevailing structure. The delta wolf is auxiliary to the alpha and beta, but it has the upper hand over the omega within the previously mentioned chain of command. The GWO algorithm can be mathematically represented as follows: D → = | C ⋅ X p → ( t ) − X → ( t ) | (3) X → ( t + 1 ) = X p → ( t ) − A ⋅ D → (4) where t is utilized to mean the current iteration, X p → to mean the prey position vector, and X → to represent the wolf position vector. Meanwhile, A and C can be represented as A = 2 a ⋅ r 1 − a (5) C = 2 ⋅ r 2 (6) where a is diminished in a straight design from 2 to 0. r ₁, r ₂ are haphazardly created from the unit interval [0,1]. The wolf pack chasing design is driven by the alpha wolves and frequently by the beta and delta ones. This design may be scientifically sculptural as follows D α → = | C 1 ⋅ X α → − X → | D β → = | C 2 ⋅ X β → − X → | D δ → = | C 3 ⋅ X δ → − X → | (7) X 1 → = X α → − A 1 ⋅ D α → X 2 → = X β → − A 2 ⋅ D β → X 3 → = X δ → − A 3 ⋅ D δ → (8) X → ( t + 1 ) = ( X 1 → + X 2 → + X 3 → ) / 3 (9)

The Lorenz dynamical system (Wu et al., 2021) is a key benchmark of time series forecasting and is mathematically defined as d x d t   =   a ( - x + y ) d y d t =   b x - y - x z d z d t   =   x y - c z (14) where t expresses time, while the model coefficients a, b, and c are respectively chosen as 10, 28, and 8/3. Model training and testing were carried out with time series lengths of 4,000 and 1,000, respectively. For x-dimensional forecasting, past information of x(t −1), y(t −1), and z(t −1) is utilized in the prediction of the present x(t) values. In the arrangement to assess the viability and preferences of this proposed GWODESN, the conventional ESN, ELM, and the LSTM are chosen as benchmarks. The real value and the anticipated value of GWODESN, ESN, ELM, and the LSTM to begin with appeared in Figure 3, and the prediction accuracy is recorded in Table 1. It is clear that GWODESN is superior than others, showing the adequacy of this approach. In expansion, the yield of ELM cannot coordinate the real esteem, particularly at a few emphasis focuses. It moreover outlines the justification of RNN. The expectation mistakes of GWODESN, ESN, ELM, and the LSTM are advance compared in Figure 4. Figure 4 shows the box graph of absolute error recorded for 30 runs of diverse models. It can be seen that the GWODESN shows superior prediction accuracy and solidness than other models. The absolute error box graph of the ELM model is long, and it is known that the maximum error values are scattered, showing that the forecast performance of the ELM model is not as steady as in other models. The box chart of the GWODESN model is the most brief. Most of the absolute error values are smaller than other comparison models. Figure 5 gives the relative error distribution of the testing information by the GWODESN. Among the 1,000 testing cases, 93.3% of the relative errors were less than 1%. In general, the prediction accuracy of the GWODESN model is relatively high and relatively stable. In common, the expectation precision of the GWODESN model is moderately high and generally steady.

NARMA (Chouikhi et al., 2017), which is featured with a very high rate of chaos in its behavior, could also be an accepted studied benchmark. The flow of this benchmark is produced by Eq. 15: y ( t + 1 ) = c 1 y ( t ) + c 2 y ( t ) ∑ i = 1 k y ( t - i ) + c 3 x ( t - ( k - 1 ) ) x ( t ) + c 4 (15) where y(t) and x(t) are the output and input of the system at time t, separately. The consistent c is set as 0.3, 0.05, 1.5, and 0.1, separately. The k, which decides the intricacy of NARMA, is set to 10. As the same as the past simulation, the real value and the anticipated value of GWODESN are shown in Figure 6, and the desired precision is recorded in Table 2. It is evident that GWODESN can fit the real value ideally. The desired values of GWODESN, ESN, ELM, and the LSTM are developed and compared in Figure 7. It can be seen in Figure 7 that the GWODESN appears to have a more predominant estimate precision and solidness than other models. The absolute error box plot of the ELM is relatively long, and it is known that its significant error values are scattered. This indicates that the performance of the ELM is not as stable as other models. The box chart of the GWODESN is the shortest. Most of the absolute error values of GWODESN are smaller than those of other comparison models. Figure 8 gives the relative error dispersion of the testing data by the GWODESN. Among the 1,000 testing cases, 90.1% of the relative error distribution were less than 3.5% as appeared in Figure 8.

The MG (Mackey and Glass, 1977) may be a normal chaotic framework, which is known by its non-linear behavior. Thus, learning the patterns appears to be a challenging task. It is portrayed by Eq. 16. d x ( t ) d t = 0 . 2 x ( t - τ ) 1 + x 10 ( t - τ ) - 0 . 1 x ( t ) (16) where τ is a vital parameter of the MG system, which is regularly set to 17. In total, 4,000 tests were utilized as training data sets and 1,000 tests were utilized for testing. The forecast that comes about from the proposed GWODESN model appears in Figure 9 and Table 3. The conventionally used ESN has superior prediction performance to ELM and LSTM, and can fit the original industrial time series data well. Be that as it may, the performance of the GWODESN is superior to that of the conventional ESN. Figure 10 shows that the GWODESN has superior forecast precision and soundness to other models. Among the 1,000 testing cases, 99% of the relative mistakes were less than 1% in Figure 11.

CCPPs generally contain steam turbines (STs) and gas turbines (GTs), as well as heat recovery steam generators (HRSGs). For a CCPP, power generation is jointly performed by the steam and gas turbines, and is exchanged between each turbine and the others (Tüfekci, 2014). Here, we use CCPP data to evaluate the single-step prediction performance. The utilized dataset includes four input factors and one target variable, where this dataset was collected from 2006 to 2011. Figure 12 outlines both the GWODESN-predicted and measured electrical power outputs. The specked reddish straight line represents the ideal relationship of the predicted and measured values. The blue line demonstrates the GWODESN predicted outcomes. Almost all of the predictions are scattered around the ideal line. Figure 13 gives the relative error distribution of the GWODESN model on the test data. Among the 1,000 test samples, 90.3% of the relative errors are less than 1.6%. Obviously, the GWODESN model outperforms the other three competing models. The adequacy of the proposed model is shown by the results in Table 4.