# Forecasting of Market Clearing Volume Using Wavelet Packet-Based Neural Networks with Tracking Signals

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## Abstract

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## 1. Introduction

## 2. Strategy of Proposed Model

#### 2.1. Input Selection

#### 2.2. Pre-Processing Using Wavelet Transform

#### 2.2.1. Conventional Wavelet Transform (WT)

#### 2.2.2. Wavelet Packet Based Decomposition (WPD)

#### 2.3. Linear Neural Network with Time Delay

- Step 1: From raw MCV data, an input time series vector has been formed, on the basis of ACF 17, time-lag data was chosen as the input variable for standalone NN models.
- Step 2: edcomposition of the original MCV time series into approximated (A1–A6) and detailed (D1–D6) subseries using $db10$.
- Step 3: The fourth level approximated and detailed MCV subseries with a 17 MCV time lag has been selected as an input variable for conventional WT-based models. The structure of the LNNTD model is shown in Figure 6 and the schematic flow diagram for the conventional WT-based MCV forecasting model is shown in Figure 8.
- Step 4: For the WPD-based model, a third-level decomposition is used [35,36,37,38,39] in which eight different high- and low-frequency component-based series are obtained. Two types of input selection criteria are adopted first, eight WPD-based decomposed series are used with 17 ACF-based time-lag series similar to that of the conventional WT-based model. In the second (proposed model), each WPD-based series has been forecasted individually, the schematic flow diagram is shown in Figure 9 and the TL’s are selected on the basis of ACF which are presented in Table 2.
- Step 5: For the forecasting, one-year MCV data was trained and tested for the next month, a similar process is continuously repeated up to the next 12 months, with a one-month moving window as shown in Figure 6. The epochs and performance goals have been chosen to be equal to 10,000 and 0.001, respectively.

#### 2.4. Accuracy Metrics

#### 2.5. Effect of Pre-Processing on the Performance of the Model

#### 2.5.1. Effect of Conventional WT

#### 2.5.2. Effect of WPD

## 3. Simulation Results

#### 3.1. Accuracy Analysis

- The performance in terms of accuracy of LNNTD is the best among all NN models.
- WT-based NN model accuracy is higher compared to the NN models.
- The accuracy level of the proposed model is found to be better amongst all others.
- In spite of that, FFNN is one of the toughest benchmarks to beat.

#### 3.2. Coefficient of Regression ${R}^{2}$ Analysis

## 4. Disscusion

#### 4.1. Validation of the Model Using Tracking Signals (TS’s)

#### 4.2. Multiple Steps-Ahead Forecasting

#### 4.3. Percentage of Improvement in Accuracy

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACF | Auto-correlation function |

ACO | Ant colony optimization |

AI | Artificial intelligence |

ANFIS | Adaptive neuro fuzzy inference system |

BP | Back propagation algorithm |

CEEMDAN | Complete ensemble empirical mode decomposition with adaptive noise |

CNN | Convolution neural network |

CWT | Continuous wavelet transform |

db10 | Daubechies wavelet |

DWT | Discrete wavelet transform |

ECPSO | Embedded chaotic particle swarm optimization |

ERNN | Elman recurrent neural network |

FFNN | Feed forward neural network |

FIR | Finite impulse response |

FNCI | Fuzzy neural network based on Choquet integral |

GA | Genetic algorithm |

GANN | Genetic algorithm based on neural network |

GMDH | Group method of data handling |

GSO | Gram-Shmidt orthogonalization |

IEX | Indian electricity exchange |

IMF | Intrinsic modes function |

LM | Levenberg marquardt algorithm |

LNN | Linear neural network |

LNNTD | Linear neural network with tapped delay |

LSTM | Long short-term memory network |

MAD | Mean absolute deviation |

MAE | Mean absolute error |

MAPE | Mean absolute percentage error |

MCV | Market clearing volume |

MLP | Multi layer perceptron |

NFN | Neural fuzzy network |

NN | Neural network |

PACF | Partial auto-correlation function |

PSO | Particle swarm optimization |

RBFNN | Radial basis function neural network |

SOM | Self organising map network |

TS | Tracking signal |

WFNN | Wavelet fuzzy neural network |

WPD | Wavelet packet-based decomposition |

WT | Wavelet transform |

VMD | Variational mode decomposition |

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Model | Learning Algorithms | Input Neurons | Transfer Function | Momentum Constant | Learning Rate |
---|---|---|---|---|---|

FFNN | LM | 17 | tansig, purelin | 0.06 | 0.001 |

ERNN | LM | 17 | tansig, purelin | 0.06 | 0.001 |

GANN | GA | 17 | tansig, purelin | 0.06 | 0.001 |

LNNTD | BP | 17 | purelin | 0.06 | 0.001 |

WT+LNNTD | BP | $17+5$ | purelin | 0.06 | 0.001 |

WPD+LNNTD | BP | $17+8$ | purelin | 0.06 | 0.001 |

Proposed | BP | - | purelin | 0.06 | 0.001 |

Wavelet Component | Selected Time Lags for Forecasting Target Series (T) |
---|---|

D30 | TL_10, TL_9, TL_8, TL_7, T_6, TL_5, TL_4, TL_3, TL_2, TL_1 |

D31 | TL_19, TL_18, TL_17, T_16, TL_15, TL_14, TL_13, TL_12, TL_11, |

TL_10, TL_9, TL_8, TL_7, T_6, TL_5, TL_4, TL_3, TL_2, TL_1 | |

D32 | TL_25, TL_12, TL_11, TL_10, TL_9, TL_8, TL_7, T_6, TL_5, TL_4, |

TL_3, TL_2, TL_1 | |

D33 | TL_28, TL_25, TL_21, TL_10, TL_9, TL_8, TL_7, T_6, TL_5, TL_4, |

TL_3, TL_2, TL_1 | |

D34 | TL_13, TL_12, TL_11, TL_10, TL_9, TL_8, TL_7, T_6, TL_5, TL_4, |

TL_3, TL_2, TL_1 | |

D35 | TL_7, T_6, TL_5, TL_4, TL_3, TL_2, TL_1 |

D36 | TL_8, TL_7, T_6, TL_5, TL_4, TL_3, TL_2, TL_1 |

D37 | TL_9, TL_8, TL_7, T_6, TL_5, TL_4, TL_3, TL_2, TL_1 |

Decomposition Level | Decomposed Signals | WT | Decomposed Signals | WPD |
---|---|---|---|---|

db10 | MAPE | db10 | MAPE | |

Level 1 | A1, D1 | 3.22 | A1:1, D1:2 | 3.115 |

Level 2 | A2, D1, D2 | 2.49 | A2:1, D2:2, A2:3, D2:4 | 1.882 |

Level 3 | A3, D1, D2, | 2.22 | A3:1, D3:2, A3:3, D3:4, | 1.258 |

D3 | A3:5, D3:6, A3:7, D3:8 | |||

Level 4 | A4, D1, D2, | 2.12 | A4:1, D4:2, A4:3, D4:4, A4:5, | 1.314 |

D3, D4 | D4:6, A4:7, D4:8, A4:9, D4:10, | |||

A4:11, D4:12, A4:13, D4:14, | ||||

A4:16, D4:16 | ||||

Level 5 | A5, D1, D2, | 2.13 | A5:1, D5:2, A5:3, D5:4, A5:5, | 6.43 |

D3, D4, D5 | D5:6, A5:7, D5:8, A5:9, D5:10, | |||

A5:11, D5:12, A5:13, D5:14, A5:16, | ||||

D5:16, A5:17, D5:18, A5:19, D5:20, | ||||

A5:21, D5:22, A5:23, D5:24, A5:25, | ||||

D5:26, A5:27, D5:28, A5:29, D5:30, | ||||

A5:31, D5:32 | ||||

Level 6 | A6, D1, D2, | 2.17 | - | - |

D3, D4, D5, D6 |

2016 | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD | Proposed |
---|---|---|---|---|---|---|---|

January | 2.794 | 3.686 | 2.783 | 2.763 | 1.641 | 1.047 | 0.164 |

February | 2.431 | 3.197 | 2.474 | 2.410 | 1.563 | 1.005 | 0.175 |

March | 3.613 | 3.969 | 3.597 | 3.569 | 1.848 | 1.214 | 0.220 |

April | 3.769 | 4.069 | 3.582 | 3.438 | 1.624 | 1.132 | 0.205 |

May | 3.646 | 4.555 | 3.559 | 3.572 | 1.788 | 1.152 | 0.198 |

June | 2.853 | 3.383 | 2.906 | 2.862 | 1.528 | 1.019 | 0.174 |

July | 2.961 | 3.047 | 2.856 | 2.789 | 1.331 | 0.854 | 0.170 |

August | 3.299 | 3.571 | 3.206 | 3.143 | 1.635 | 1.003 | 0.198 |

September | 3.843 | 4.634 | 3.919 | 3.782 | 1.658 | 1.034 | 0.192 |

October | 4.381 | 5.014 | 3.962 | 3.953 | 1.793 | 1.120 | 0.204 |

November | 4.207 | 5.469 | 4.352 | 4.140 | 2.410 | 1.741 | 0.285 |

December | 3.501 | 4.392 | 3.561 | 3.391 | 2.219 | 1.622 | 0.226 |

A.V. | 3.441 | 4.082 | 3.396 | 3.318 | 1.753 | 1.162 | 0.201 |

2016 | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD | Proposed |
---|---|---|---|---|---|---|---|

January | 115.253 | 151.121 | 115.221 | 113.850 | 67.0143 | 43.039 | 6.758 |

February | 103.007 | 138.273 | 105.280 | 102.680 | 65.7499 | 42.895 | 7.398 |

March | 161.492 | 176.189 | 161.020 | 158.840 | 82.0146 | 53.843 | 9.805 |

April | 198.377 | 214.583 | 188.657 | 178.610 | 83.9282 | 58.263 | 10.523 |

May | 149.702 | 185.845 | 146.430 | 146.830 | 74.4242 | 47.911 | 8.0387 |

June | 130.586 | 153.594 | 132.857 | 130.780 | 68.9155 | 45.504 | 7.704 |

July | 146.409 | 150.990 | 141.618 | 137.820 | 65.9655 | 41.784 | 8.249 |

August | 160.145 | 173.035 | 156.889 | 152.300 | 78.7189 | 47.937 | 9.366 |

September | 195.639 | 235.798 | 199.298 | 191.290 | 84.2253 | 52.282 | 9.811 |

October | 219.531 | 253.934 | 199.514 | 197.180 | 89.0009 | 54.529 | 9.973 |

November | 205.628 | 263.373 | 214.261 | 201.730 | 114.344 | 81.572 | 13.454 |

December | 152.716 | 192.387 | 155.27 | 147.560 | 93.9196 | 67.216 | 7.596 |

A.V. | 161.540 | 190.760 | 159.693 | 154.950 | 80.6851 | 53.065 | 9.056 |

Season | Period | Testing Period |
---|---|---|

Winter | December–March | week 1 |

Summer | April–May | week 1 |

Rainy | June–September | week 1 |

Dry | October–November | week 1 |

Models | Week 1 | Week 2 | Week 3 | Week 4 | A.V. |
---|---|---|---|---|---|

FFNN | 2.5142 | 3.6082 | 3.415 | 4.096 | 3.408 |

ERNN | 3.0571 | 3.9597 | 3.966 | 5.407 | 4.097 |

GANN | 2.5091 | 3.7614 | 3.39 | 4.127 | 3.447 |

LNNTD | 2.5345 | 3.7416 | 3.393 | 3.964 | 3.408 |

WT+LNNTD | 1.5189 | 1.5274 | 1.636 | 1.657 | 1.585 |

WPD+LNNTD | 1.8174 | 0.9913 | 1.0189 | 1.1304 | 1.2395 |

Proposed | 0.3008 | 0.1886 | 0.1973 | 0.2134 | 0.225 |

Models | Week 1 | Week 2 | Week 3 | Week 4 | A.V. |
---|---|---|---|---|---|

FFNN | 100.747 | 177.882 | 161.195 | 241.018 | 170.211 |

ERNN | 121.359 | 195.231 | 185.157 | 323.919 | 206.417 |

GANN | 100.576 | 185.298 | 159.586 | 244.208 | 172.417 |

LNNTD | 101.137 | 184.009 | 159.882 | 231.445 | 169.118 |

WT+LNNTD | 58.957 | 75.0393 | 74.9522 | 96.5705 | 76.3797 |

WPD+LNNTD | 70.9611 | 50.0129 | 48.34038 | 63.3021 | 58.1541 |

Proposed | 12.0576 | 9.45813 | 9.246981 | 11.9904 | 10.6883 |

Models | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD | Proposed |
---|---|---|---|---|---|---|---|

January | 0.959 | 0.97 | 0.968 | 0.923 | 0.977 | 0.990 | 0.9997 |

February | 0.917 | 0.855 | 0.912 | 0.916 | 0.970 | 0.987 | 0.9996 |

March | 0.873 | 0.851 | 0.873 | 0.875 | 0.972 | 0.987 | 0.9995 |

April | 0.859 | 0.833 | 0.866 | 0.868 | 0.973 | 0.986 | 0.9996 |

May | 0.873 | 0.811 | 0.876 | 0.876 | 0.971 | 0.988 | 0.9996 |

June | 0.916 | 0.891 | 0.912 | 0.913 | 0.980 | 0.991 | 0.9997 |

July | 0.908 | 0.898 | 0.912 | 0.915 | 0.982 | 0.993 | 0.9997 |

August | 0.925 | 0.913 | 0.926 | 0.928 | 0.983 | 0.994 | 0.9997 |

September | 0.851 | 0.803 | 0.843 | 0.853 | 0.975 | 0.990 | 0.9996 |

October | 0.654 | 0.875 | 0.916 | 0.919 | 0.984 | 0.994 | 0.9998 |

November | 0.912 | 0.865 | 0.901 | 0.913 | 0.977 | 0.988 | 0.9996 |

December | 0.933 | 0.904 | 0.927 | 0.937 | 0.979 | 0.989 | 0.9997 |

A.V. | 0.882 | 0.872 | 0.903 | 0.903 | 0.977 | 0.990 | 0.9996 |

Models | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD | Proposed |
---|---|---|---|---|---|---|---|

January | −34.78 | 68.77 | −23.3 | 7.92 | −4.12 | −15.11 | 0.57 |

February | −18.06 | 65.92 | −4.36 | 44.24 | 8.11 | −4.27 | 2.46 |

March | 88.32 | 60.51 | 99.98 | 80.86 | 99.48 | −2.01 | 2.62 |

April | 257.81 | 278.8 | 261.26 | 127.48 | 41.55 | −28.49 | 4.84 |

May | −97.35 | −83.9 | −68.25 | −31.01 | 39.91 | −5.29 | −1.00 |

June | 50.512 | 50.8 | 75.72 | 43.48 | 12.52 | −0.37 | 2.50 |

July | 104.03 | 57.02 | 87.04 | 83.65 | 37.69 | −16.34 | 3.05 |

August | −4.562 | 24.56 | 65.50 | 70.48 | 83.89 | −4.40 | 4.11 |

September | 49.949 | −32.7 | 43.63 | 72.82 | 133.80 | −15.19 | 6.83 |

October | −27 | 196.3 | 86.81 | 13.39 | 13.36 | −20.07 | −0.24 |

November | 43.307 | −30.9 | 55.79 | 35.45 | 3.65 | −5.75 | 0.90 |

December | −120.5 | −3.39 | −116.8 | −38.15 | 2.16 | 4.05 | −4.92 |

Models | Metrics | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | Step 6 |
---|---|---|---|---|---|---|---|

FFNN | MAPE | 2.78 | 3.99 | 4.80 | 6.57 | 8.68 | 9.89 |

FFNN | MAE | 114.97 | 165.64 | 196.21 | 270.32 | 354.35 | 404.99 |

ERNN | MAPE | 3.28 | 4.281 | 5.18 | 7.02 | 8.85 | 10.03 |

ERNN | MAE | 134.52 | 174.57 | 211.98 | 291.23 | 365.59 | 414.12 |

GANN | MAPE | 2.85 | 3.83 | 4.66 | 6.62 | 8.67 | 10.01 |

GANN | MAE | 117.44 | 157.51 | 191.88 | 272.66 | 353.79 | 410.55 |

LNNTD | MAPE | 2.76 | 3.87 | 4.62 | 6.35 | 8.31 | 9.57 |

LNNTD | MAE | 113.84 | 159.05 | 189.68 | 260.87 | 341.33 | 393.74 |

WTLNNTD | MAPE | 1.62 | 2.69 | 3.31 | 4.77 | 5.50 | 6.10 |

WTLNNTD | MAE | 66.19 | 110.32 | 137.06 | 196.34 | 226.94 | 249.91 |

WPD+LNNTD | MAPE | 1.04 | 2.32 | 3.04 | 4.27 | 5.17 | 5.81 |

WPD+LNNTD | MAE | 43.03 | 95.11 | 125.95 | 175.76 | 213.25 | 238.35 |

Proposed | MAPE | 0.19 | 0.37 | 0.59 | 1.07 | 1.34 | 1.57 |

Proposed | MAE | 7.07 | 15.22 | 24.23 | 44.02 | 54.76 | 63.29 |

Seasons | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD |
---|---|---|---|---|---|---|

Yearly | 94.15 | 95.07 | 94.08 | 93.94 | 88.53 | 82.70 |

Winter | 88.03 | 90.16 | 88.01 | 88.13 | 80.19 | 83.44 |

Summer | 94.77 | 95.23 | 94.98 | 94.95 | 87.65 | 80.97 |

Rainy | 94.22 | 95.02 | 94.17 | 94.18 | 87.94 | 80.63 |

Dry | 94.79 | 96.05 | 94.82 | 94.61 | 87.12 | 81.12 |

Seasons | FFNN | ERNN | GANN | LNNTD | WT+LNNTD | WPD+LNNTD |
---|---|---|---|---|---|---|

Yearly | 94.39 | 95.25 | 94.32 | 94.15 | 88.77 | 82.93 |

Winter | 88.03 | 90.06 | 88.01 | 88.07 | 79.54 | 83.00 |

Summer | 94.68 | 95.15 | 94.89 | 94.85 | 87.39 | 81.08 |

Rainy | 94.26 | 95.00 | 94.20 | 94.21 | 87.66 | 80.87 |

Dry | 95.02 | 96.29 | 95.09 | 94.81 | 87.58 | 81.05 |

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**MDPI and ACS Style**

Saroha, S.; Zurek-Mortka, M.; Szymanski, J.R.; Shekher, V.; Singla, P.
Forecasting of Market Clearing Volume Using Wavelet Packet-Based Neural Networks with Tracking Signals. *Energies* **2021**, *14*, 6065.
https://doi.org/10.3390/en14196065

**AMA Style**

Saroha S, Zurek-Mortka M, Szymanski JR, Shekher V, Singla P.
Forecasting of Market Clearing Volume Using Wavelet Packet-Based Neural Networks with Tracking Signals. *Energies*. 2021; 14(19):6065.
https://doi.org/10.3390/en14196065

**Chicago/Turabian Style**

Saroha, Sumit, Marta Zurek-Mortka, Jerzy Ryszard Szymanski, Vineet Shekher, and Pardeep Singla.
2021. "Forecasting of Market Clearing Volume Using Wavelet Packet-Based Neural Networks with Tracking Signals" *Energies* 14, no. 19: 6065.
https://doi.org/10.3390/en14196065