- freely available
Entropy 2018, 20(4), 283; https://doi.org/10.3390/e20040283
2. Modeling for TRSWA-BP Combined with EMD and PSR
2.1. Construction of the TRSWA-BP
2.1.1. Optimized Weight Iteration Calculation Based on the TRSWA
- Initialization is required at this stage, where the input data, including population size Dim, the maximum iteration Kmax, the temporary local network size ni, the search probability of local short-range connection Ps, the size of node neighborhood radius Rs and the maximum stored number of tabu list Ts, are defined. Furthermore, set the number of the current iterations as k = 1.
- Generate M (M > Dim) real-coded nodes by the Logistic chaotic map randomly, and calculate the fitness value of the objective function for each node. Find Dim optimal nodes among them as the initial population of node set for the TRSWA.
- Store the searched nodes in the tabu list.
- For each node of each generation in the node set of Dim population, there are ni searches of short-distance and random long-distance. Generate a random number Tm. If Tm < Ps, perform a local short-distance search, otherwise carry out a random long-distance search. Results are compared with the saved nodes in tabu list. If the results are in the list, search again. Calculate the updated objective function and find out the optimal node set.
- Generate a new node set, and calculate its objective function. The new node set is compared with the optimal one that has been generated in step (4), and then finds out the optimal set of the nodes. Record its location and the value of optimal fitness.
- Check the convergence criteria. If it is satisfied, end calculation. Otherwise, let k = k + 1 and return to step 3.
2.1.2. Modeling Process of the TRSWA-BP
- Determine the number of input and output nodes and the number of hidden layers and nodes in each hidden layer for the TRSWA-BP. Fix the set of training samples, and suppose k = 1.
- Set parameters for the BP, such as learning rate η and inertia coefficient α.
- Build an objective function through the training set, and the optimal weights of the BP are trained by the TRSWA.
- Set k = k + 1. Remove the earliest set from training samples, and add a newly acquired set into it. Repeat steps 3 and 4 until the termination condition is satisfied, and a TRSWA-BP with the best weights is established.
2.2. TRSWA-BP Combined with EMD
- Select data sequences, of which the length is N, including wind speed v(ti), wind direction d(ti), wind power p(ti), temperature tep(ti), and NWP wind speed denoted by vNWP(ti), i = 1, 2, …, N. Set k = 1.
- v(t), d(t), p(t), tep(t), and vNWP(t) are respectively decomposed by EMD first, whose principle is in the order of frequency from high to low. The intrinsic modal function (IMF) of nv, nd, np, nt, and nNWP layers, as well as a residual component r(·), are attained. They are denoted as IMF(v)1, …, IMF(v)nv, r(v), IMF(d)1, …, IMF(d)nd, r(d), IMF(p)1, …, IMF(p)np, r(p), IMF(t)1, …, IMF(t)nt, rt(t), and IMF(NWP)1, …, IMF(NWP)nNWP, r(NWP).
- As the number of nv, nd, np, nt, and nNWP may be unequal due to the decomposition, the minimum number nmin of them is taken as the number of the unified IMF layers of the five sequences.
- If nv > nmin in the data sequence of wind speed v(t), add all the IMF(v) behind the nminth layer together with IMF(v)nmin, to form a new IMF(v)’nmin, which means that IMF(v)’nmin = IMF(v)nmin + IMF(v)(nmin+1) + … + IMF(v)nv, (nv = 1, 2, 3, ..., nmin, …, nv). The other four sequences are treated the same way, except one (or those) when n* = nmin. Finally, the new unified sequences are obtained according to the decomposed layers. They are IMF(v)1, IMF(d)1, IMF(p)1, IMF(t)1, IMF(NWP)1; IMF(v)2, IMF(d)2, IMF(p)2, IMF(t)2, IMF(NWP)2; …, IMF(v)’nmin, IMF(d)’nmin, IMF(p)’nmin, IMF(t)’nmin, IMF(NWP)’nmin; and r(v), r(d), r(p), rt(t), r(NWP).
- Use the first N − 1 data of each layer’s new unified sequence, to train several forecasting models of TRSWA-BP, respectively. The best weights are obtained by the TRSWA (See Section 2.1.2), which are employed to predict the forecasted wind power P of each decomposed layer. They are denoted by P1, P2, P3, …, Pnmin, and Pr.
- To compose the predictions of each layer, we obtain the fitting wind power at the k moment, which is given by Pk = P1 + P2 + … + Pnmin + Pr.
- Set k = k + 1. A set of newly predicted wind power P is used as a known value to the training set, while the earliest set of data sequences is removed. Check whether it reaches the termination condition. If not, return to step (2), otherwise stop calculation.
2.3. TRSWA-BP Combined with PSR
- Select the same five sequences as step 1 in Section 2.2.
- Determine the m and τ of the above sequences through the C-C method.
- The sequences of v(ti), d(ti), p(ti), tep(ti), and vNWP(ti) are reconstructed, respectively, based on Equation (5), into the following sequences:
- Set k = k + 1. Add the new actual value into the time series and replace the earliest one to form scrolling sequences over time. Repeat steps 2 to 5 until the termination condition is satisfied.
2.4. TRSWA-BP Combined with EMD-Based PSR
3. Entropy Evaluation for Error Sequence on Non-Gaussian Property
3.1. Non-Gaussian Property of Wind Power Error Sequence
3.2. Evaluation Criterion Based on Normalized Renyi’s Quadratic Entropy
4. Simulation and Analysis
4.1. Prediction and Evaluation Based on the TRSWA-BP
4.2. Prediction Precision Based on the TRSWA-BP Combined with EMD & PSR
4.3. Training Times
Conflicts of Interest
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|Predictable Time Scales||1 h||4 h||6 h||24 h|
|Forecasting Time Scales||1 h||4 h||6 h||24 h|
|TRSWA-BP and EMD (one input)||NMAE||7.487%||9.891%||11.716%||13.245%|
|TRSWA-BP and EMD||NMAE||6.122%||8.325%||9.898%||11.652%|
|TRSWA-BP and PSR||NMAE||6.311%||7.359%||8.870%||10.543%|
|TRSWA-BP and EMD-based PSR||NMAE||5.257%||6.818%||8.131%||9.755%|
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