# Forecasting of Day-Ahead Natural Gas Consumption Demand in Greece Using Adaptive Neuro-Fuzzy Inference System

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## Abstract

**:**

## 1. Introduction

#### 1.1. Indicative Related Work on AI Applied in Natural Gas Consumption Forecasting

#### 1.2. Related Work on ANFIS in Energy Consumption Forecasting

#### 1.3. Related Work on ANFIS in Natural Gas Consumption Forecasting

#### 1.4. Related Work on Fuzzy Cognitive Maps (FCMs) in Energy and Natural Gas Consumption Forecasting

#### 1.5. Research Gap and the Novelty of This Study

- The creation and demonstration of a simple, fast, robust ANFIS prediction tool to forecast NG demand using historical time series data. The proposed model is characterized by high flexibility, especially in large datasets, easiness of use and low execution time requirements.
- The rigorous ANFIS fine-tuning for determining the most appropriate architecture for an enhanced prediction performance.

#### 1.6. Aim of This Research Work

- (a)
- To develop a robust ANFIS model to provide accurate short-term forecasts for a number of cities in Greece, using a relatively large dataset. At the same time, the authors perform model fine-tuning that can lead to high accuracy in most distribution points. The proposed model is characterized by high flexibility, easiness of use and low execution time requirements.
- (b)
- To apply FCMs, ANNs and hybrid combinations of them to forecast NG demand in the same dataset, since these approaches have been proved as efficient techniques for NG demand forecasting according to the relevant literature.
- (c)
- To assess the performance of these soft computing methods in terms of prediction accuracy using well-known evaluation metrics.
- (d)
- To compare forecasting accuracy results of the proposed approach with those of the other soft computing and ANN methods that were examined, and finally decide on which model offers the best forecasting accuracy.

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.2. Methods

#### 2.2.1. Adaptive Neuro-Fuzzy Inference System (ANFIS)

#### 2.2.2. Proposed ANFIS Architecture Applied in Natural Gas Consumption Forecasting

^{10}= 1024 specific areas, representing one rule for each specific area, and the total number of rules is 1024, which is a very complicated structure. Therefore, the grid partitioning method is mainly used when the number of input variables is small.

_{1}, x

_{2}, …, x

_{n}are the n inputs. In this case, ANFIS needs to define k

_{0}, k

_{1}, k

_{2}up to k

_{n}, and it is very time consuming to efficiently calculate the outputs when a large number of parameters are considered. On the other hand, when a constant MF is selected, the algorithm needs to define only one parameter to provide a reliable forecasted value. Thus, the computational time is really low.

#### 2.2.3. Testing and Evaluation

^{2}). The mathematical equations of the statistical indicators are described below.

- Mean squared error:$$\mathrm{MSE}=\frac{1}{T}{\displaystyle \sum}_{t=1}^{T}{\left(Z\left(t\right)-X\left(t\right)\right)}^{2}$$
- Root mean squared error:$$\mathrm{R}\mathrm{MSE}=\sqrt{\mathrm{MSE}}$$
- Mean absolute error:$$\mathrm{MAE}=\frac{1}{T}{\displaystyle \sum}_{t=1}^{T}\left|Z\left(t\right)-X\left(t\right)\right|$$
- Mean absolute percentage error:$$\mathrm{MAPE}=\frac{1}{T}{\displaystyle \sum}_{t=1}^{T}\left|\frac{Z\left(t\right)-X\left(t\right)}{Z\left(t\right)}\right|$$
- Coefficient of determination:$$\mathrm{R}=\frac{T{{\displaystyle \sum}}_{t=1}^{T}Z\left(t\right)\xb7X\left(t\right)-\left({{\displaystyle \sum}}_{t=1}^{T}Z\left(t\right)\right)\left({{\displaystyle \sum}}_{t=1}^{T}X\left(t\right)\right)}{\sqrt{T{{\displaystyle \sum}}_{t=1}^{T}{\left(Z\left(t\right)\right)}^{2}-{\left({{\displaystyle \sum}}_{t=1}^{T}Z\left(t\right)\right)}^{2}}\xb7\sqrt{T{{\displaystyle \sum}}_{t=1}^{T}{\left(X\left(t\right)\right)}^{2}-{\left({{\displaystyle \sum}}_{t=1}^{T}X\left(t\right)\right)}^{2}}}$$

^{2}, i.e., closer to 1, mean better model performance and the regression line fits the data well. A coefficient of determination value of 1.0 points out that the regression curve fits the data perfectly.

## 3. Results

#### 3.1. Comparison with ANNs, FCMs and Hybrid FCM-ANN

#### 3.2. Discussion of Results

- The proposed ANFIS method exhibits the best performance when certain configuration settings are selected for the examined datasets which are linked to ten cities of Greece. The authors concluded that a certain configuration is best for the examined ANFIS model, after having conducted a number of experiments and following a trial-and error approach. The best ANFIS model is based on a distinct architecture that features a 2-2-2-2-2 triangular or gaussian MF.
- The proposed ANFIS architecture is superior to the four benchmark and well-known ANN and FCM methods (ANN, SOGA-FCM, RCGA-FCM, Hybrid FCM-ANN), which have been efficiently used in NG consumption forecasting. The results presented in Table 7, which gathers various error indicators and the R
^{2}, as prediction accuracy indices for all five architectures, show that the best ANFIS model holds the best prediction accuracy among all the methods that were included in this comparative analysis.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Type of Input MF | Number of MFs | Type of Output MF | Number of Rules | MSE | RMSE | MAE | MAPE | R^{2} | Time (s) |
---|---|---|---|---|---|---|---|---|---|

trimf | 2-2-2-2-2 | Linear | 32 | 0.001195 | 0.034572 | 0.019426 | 11.72180 | 0.982121 | 148 |

trapmf | 2-2-2-2-2 | Linear | 32 | 0.001358 | 0.036859 | 0.020861 | 12.12378 | 0.979559 | 148 |

gbellmf | 2-2-2-2-2 | Linear | 32 | 0.001267 | 0.035603 | 0.019921 | 11.46446 | 0.980963 | 148 |

Gaussmf | 2-2-2-2-2 | Linear | 32 | 0.001298 | 0.036038 | 0.020259 | 11.97794 | 0.980468 | 148 |

Gauss2mf | 2-2-2-2-2 | Linear | 32 | 0.001406 | 0.037496 | 0.020878 | 11.26382 | 0.978860 | 148 |

pimf | 2-2-2-2-2 | Linear | 32 | 0.001635 | 0.040442 | 0.022176 | 12.08298 | 0.975405 | 148 |

dsigmf | 2-2-2-2-2 | Linear | 32 | 0.001423 | 0.037733 | 0.021062 | 11.26721 | 0.978592 | 148 |

psigmf | 2-2-2-2-2 | Linear | 32 | 0.001423 | 0.037733 | 0.021062 | 11.26722 | 0.978592 | 148 |

trimf | 2-2-3-3-3 | Linear | 108 | 0.001476 | 0.038430 | 0.020941 | 11.17862 | 0.977773 | 328 |

Gaussmf | 2-2-3-3-3 | Linear | 108 | 0.002038 | 0.045149 | 0.023286 | 12.71720 | 0.969241 | 328 |

Type of Input MF | Number of MFs | Type of Output MF | Number of Epochs | Optimization | Number of Rules | Time Run |
---|---|---|---|---|---|---|

trimf, trapmf, gbell, gauss, pim, sigm | 2-2-2-2-2 | Constant | 10 | Hybrid | 32 | 7 s |

trimf, trapmf, gbell, gauss, pim, sigm | 2-2-3-3-3 | Constant | 10 | Hybrid | 108 | 11 s |

trimf, trapmf, gbell, gauss, pim, sigm | 3-3-3-2-2 | Constant | 10 | Hybrid | 108 | 19 s |

trimf, trapmf, gbell | 3-3-3-3-3 | Constant | 10 | Hybrid | 243 | 68 s |

trimf | 3-3-4-4-4 | Constant | 10 | Hybrid | 576 | 10 min 10 s |

trimf | 3-3-5-5-5 | Constant | 10 | Hybrid | 1125 | 40 min |

trapmf | 3-3-4-4-4 | Constant | 10 | Hybrid | 576 | 12min |

trapmf | 3-3-5-5-5 | Constant | 10 | Hybrid | 1125 | 70 min |

gbellmf | 3-3-4-4-4 | Constant | 10 | Hybrid | 576 | 12 min 35 s |

gbellmf | 3-3-5-5-5 | Constant | 10 | Hybrid | 1125 | 50 min |

gaussmf | 3-3-3-3-3 | Constant | 10 | Hybrid | 243 | 4 min |

gaussmf | 3-3-4-4-4 | Constant | 10 | Hybrid | 576 | 25 min |

gaussmf | 3-3-5-5-5 | Constant | 10 | Hybrid | 1125 | 47 min |

gauss2mf | 3-3-3-3-3 | Constant | 10 | Hybrid | 243 | 4 min |

gauss2mf | 3-3-4-4-4 | Constant | 10 | Hybrid | 576 | 25 min |

gauss2mf | 3-3-5-5-5 | Constant | 10 | Hybrid | 1125 | 47 min |

pimf | 3-3-3-3-3 | Constant | 10 | Hybrid | 243 | 3.5 min |

pimf | 3-3-4-4-4 | Constant | 10 | hybrid | 576 | 20 min |

pimf | 3-3-5-5-5 | Constant | 10 | hybrid | 1125 | 42 min |

**Figure A1.**Comparison of forecasting results for each city considering all examined methods. (

**a**) Testing for Karditsa, (

**b**) testing for Larissa, (

**c**) testing for Markopoulo, (

**d**) testing for Serres, (

**e**) testing for Thessaloniki, (

**f**) testing for Trikala, (

**g**) testing for Volos.

## Appendix B

#### Appendix B.1. Fuzzy Cognitive Maps

_{i}(t) is the value of the i-th concept at the t-th iteration, w

_{j,i}is the weight of the causal relationship between concepts X

_{j}and X

_{i}taking values from the range [−1,1], t is discrete time, i,j = 1, 2, …, n, n is the number of concepts, and F is the transformation function normalizing the factor values to the range [0,1] or [−1,1]. Fuzzy cognitive maps can be constructed based on expert knowledge or with the use of machine learning algorithms. The aim of fuzzy cognitive map learning is to determine the weights of the causal relationships between concepts on the basis of available time series.

_{1}, b

_{2}are the learning parameters, n

_{c}is the number of the concepts in the candidate FCM model, n

_{r}is the number of the non-zero relationships between concepts, n is the number of all possible concepts, and $eror{r}_{l}$ is the learning error type (Equation (A3)).

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**Figure 1.**The TSK ANFIS architecture [107].

**Figure 3.**Screenshots regarding (

**a**) the training dataset configuration, (

**b**) the number and type of MF configuration.

**Figure 7.**Forecasting results for three cities considering the best ANFIS method. (

**a**) Testing Alexandroupoli, (

**b**) testing Athens, (

**c**) testing Drama, (

**d**) testing Karditsa, (

**e**) testing Larissa, (

**f**) testing Markopoulo, (

**g**) testing Serres, (

**h**) testing Thessaloniki, (

**i**) testing Trikala, (

**j**) testing Volos.

**Figure 9.**Comparison of forecasting results for each city considering all examined methods. (

**a**) Testing for Alexandroupoli, (

**b**) Testing for Athens, (

**c**) Testing for Drama.

**Figure 10.**Comparison of results between machine learning and soft computing methods for three benchmark cities.

City | Time Period of the Examined Data | City | Time Period of the Examined Data |
---|---|---|---|

Alexandroupoli | 2/2013–10/2018 | Markopoulo | 3/2010–10/2018 |

Athens | 3/2010–10/2018 | Serres | 6/2013–10/2018 |

Drama | 9/2011–10/2018 | Thessaloniki | 3/2012–10/2018 |

Karditsa | 5/2014–10/2018 | Trikala | 9/2012–10/2018 |

Larissa | 3/2010–10/2018 | Volos | 3/2010–10/2018 |

Type | Parameter | Unit |
---|---|---|

Input | Demand of a day before | MWh |

Input | Current day demand | MWh |

Input | Daily average temperature | Celsius degrees |

Input | Month indicator | K = 1/12, 2/12, …, 1 |

Input | Day indicator | l = 1/7, 2/7, …, 1 |

Output | A day ahead NG demand | MWh |

**Table 3.**Different configurations of the selected ANFIS architectures regarding constant output membership function (MF).

ANFIS Run | Type of Input MF | Number of MFs | Type of Output MF | Number of Epochs | Learning Method |
---|---|---|---|---|---|

1 | trimf | 2-2-2-2-2 | Constant | 10 | Hybrid |

2 | trapmf | 2-2-2-2-2 | Constant | 10 | Hybrid |

3 | gbellmf | 2-2-2-2-2 | Constant | 10 | Hybrid |

4 | Gaussmf | 2-2-2-2-2 | Constant | 10 | Hybrid |

5 | Gauss2mf | 2-2-2-2-2 | Constant | 10 | Hybrid |

6 | pimf | 2-2-2-2-2 | Constant | 10 | Hybrid |

7 | dsigmf | 2-2-2-2-2 | Constant | 10 | Hybrid |

8 | psigmf | 2-2-2-2-2 | Constant | 10 | Hybrid |

9 | trimf | 2-2-3-3-3 | Constant | 10 | Hybrid |

10 | trapmf | 2-2-3-3-3 | Constant | 10 | Hybrid |

11 | gbellmf | 2-2-3-3-3 | Constant | 10 | Hybrid |

12 | Gaussmf | 2-2-3-3-3 | Constant | 10 | Hybrid |

13 | Gauss2mf | 2-2-3-3-3 | Constant | 10 | Hybrid |

14 | pimf | 2-2-3-3-3 | Constant | 10 | Hybrid |

15 | dsigmf | 2-2-3-3-3 | Constant | 10 | Hybrid |

16 | psigmf | 2-2-3-3-3 | Constant | 10 | Hybrid |

17 | trimf | 3-3-3-2-2 | Constant | 10 | Hybrid |

18 | trapmf | 3-3-3-2-2 | Constant | 10 | Hybrid |

19 | gbellmf | 3-3-3-2-2 | Constant | 10 | Hybrid |

20 | Gaussmf | 3-3-3-2-2 | Constant | 10 | Hybrid |

21 | trimf | 3-3-3-3-3 | Constant | 10 | hybrid |

22 | trimf | 3-3-3-3-3 | Constant | 10 | backpropa |

23 | trapmf | 3-3-3-3-3 | Constant | 10 | hybrid |

24 | trapmf | 3-3-3-3-3 | Constant | 10 | backpropa |

25 | gbellmf | 3-3-3-3-3 | Constant | 10 | hybrid |

26 | gbellmf | 3-3-3-3-3 | Constant | 10 | backpropa |

27 | trimf | 3-3-3-3-3 | Constant | 30 | hybrid |

28 | trimf | 3-3-3-3-3 | Constant | 50 | hybrid |

29 | trapmf | 3-3-3-3-3 | Constant | 30 | hybrid |

30 | trapmf | 3-3-3-3-3 | Constant | 50 | hybrid |

31 | gbellmf | 3-3-3-3-3 | Constant | 30 | hybrid |

32 | gbellmf | 3-3-3-3-3 | Constant | 50 | hybrid |

33 | trimf | 3-3-4-4-4 | Constant | 10 | hybrid |

34 | trimf | 3-3-5-5-5 | Constant | 10 | hybrid |

35 | trapmf | 3-3-4-4-4 | Constant | 10 | hybrid |

36 | trapmf | 3-3-5-5-5 | Constant | 10 | hybrid |

37 | gbellmf | 3-3-4-4-4 | Constant | 10 | hybrid |

38 | gbellmf | 3-3-5-5-5 | Constant | 10 | hybrid |

39 | gaussmf | 3-3-3-3-3 | Constant | 10 | hybrid |

40 | gaussmf | 3-3-4-4-4 | Constant | 10 | hybrid |

41 | gaussmf | 3-3-5-5-5 | Constant | 10 | hybrid |

42 | gauss2mf | 3-3-3-3-3 | Constant | 10 | hybrid |

43 | gauss2mf | 3-3-4-4-4 | Constant | 10 | hybrid |

44 | gauss2mf | 3-3-5-5-5 | Constant | 10 | hybrid |

45 | pimf | 3-3-3-3-3 | Constant | 10 | hybrid |

46 | pimf | 3-3-4-4-4 | Constant | 10 | hybrid |

47 | pimf | 3-3-5-5-5 | Constant | 10 | hybrid |

Anfis Run | Type of Input MF | Number of MFs | Type of Output MF | Number of Epochs | Optimization | MSE | RMSE | MAE | MAPE | R^{2} |
---|---|---|---|---|---|---|---|---|---|---|

1 | trimf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0010 | 0.0320 | 0.0192 | 12.6882 | 0.9849 |

2 | trapmf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0013 | 0.0366 | 0.0245 | 19.8878 | 0.9806 |

3 | gbellmf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0011 | 0.0335 | 0.0209 | 14.7498 | 0.9834 |

4 | Gaussmf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0011 | 0.0326 | 0.0201 | 13.8422 | 0.9842 |

5 | Gauss2mf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0011 | 0.0324 | 0.0197 | 13.5785 | 0.9845 |

6 | pimf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0015 | 0.0389 | 0.0254 | 19.5486 | 0.9782 |

7 | dsigmf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0014 | 0.0378 | 0.0244 | 18.7851 | 0.9794 |

8 | psigmf | 2-2-2-2-2 | Constant | 10 | Hybrid | 0.0014 | 0.0378 | 0.0244 | 18.7851 | 0.9794 |

9 | trimf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0015 | 0.0388 | 0.0232 | 15.8840 | 0.9774 |

10 | trapmf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0020 | 0.0448 | 0.0269 | 19.2727 | 0.9698 |

11 | gbellmf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0014 | 0.0379 | 0.0227 | 15.5056 | 0.9785 |

12 | Gaussmf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0014 | 0.0379 | 0.0226 | 15.7640 | 0.9784 |

13 | Gauss2mf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0017 | 0.0410 | 0.0241 | 15.8227 | 0.9747 |

14 | pimf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0130 | 0.1141 | 0.0347 | 21.6717 | 0.8552 |

15 | dsigmf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0020 | 0.0448 | 0.0254 | 16.7809 | 0.9698 |

16 | psigmf | 2-2-3-3-3 | Constant | 10 | Hybrid | 0.0020 | 0.0448 | 0.0254 | 16.7809 | 0.9698 |

17 | trimf | 3-3-3-2-2 | Constant | 10 | Hybrid | 0.0012 | 0.0348 | 0.0210 | 14.6116 | 0.9819 |

18 | trapmf | 3-3-3-2-2 | Constant | 10 | Hybrid | 0.0018 | 0.0430 | 0.0297 | 27.0255 | 0.9723 |

19 | gbellmf | 3-3-3-2-2 | Constant | 10 | Hybrid | 0.0013 | 0.0355 | 0.0212 | 14.4247 | 0.9810 |

20 | Gaussmf | 3-3-3-2-2 | Constant | 10 | Hybrid | 0.0011 | 0.0337 | 0.0198 | 12.8988 | 0.9829 |

21 | trimf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.0021 | 0.0455 | 0.0242 | 15.4964 | 0.9698 |

22 | trimf | 3-3-3-3-3 | Constant | 10 | backpropa | 0.0559 | 0.2365 | 0.1610 | 74.9654 | 0.7447 |

23 | trapmf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.0031 | 0.0556 | 0.0281 | 21.5814 | 0.9562 |

24 | trapmf | 3-3-3-3-3 | Constant | 10 | backpropa | 0.0501 | 0.2238 | 0.1527 | 72.2629 | 0.7404 |

25 | gbellmf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.0014 | 0.0374 | 0.0217 | 14.6538 | 0.9791 |

26 | gbellmf | 3-3-3-3-3 | Constant | 10 | backpropa | 0.0015 | 0.0392 | 0.0265 | 25.1796 | 0.9793 |

27 | trimf | 3-3-3-3-3 | Constant | 30 | hybrid | 0.0016 | 0.0403 | 0.0224 | 13.3194 | 0.9759 |

28 | trimf | 3-3-3-3-3 | Constant | 50 | hybrid | 0.0017 | 0.0417 | 0.0224 | 13.2276 | 0.9745 |

29 | trapmf | 3-3-3-3-3 | Constant | 30 | hybrid | 0.0029 | 0.0539 | 0.0245 | 17.2238 | 0.9590 |

30 | trapmf | 3-3-3-3-3 | Constant | 50 | hybrid | 0.0017 | 0.0416 | 0.0233 | 16.4612 | 0.9745 |

31 | gbellmf | 3-3-3-3-3 | Constant | 30 | hybrid | 0.0013 | 0.0366 | 0.0213 | 13.2077 | 0.9799 |

32 | gbellmf | 3-3-3-3-3 | Constant | 50 | hybrid | 0.0019 | 0.0432 | 0.0236 | 13.4445 | 0.9724 |

33 | trimf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0023 | 0.0479 | 0.0251 | 15.4225 | 0.9662 |

34 | trimf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0078 | 0.0884 | 0.0320 | 17.3158 | 0.9006 |

35 | trapmf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0021 | 0.0454 | 0.0275 | 23.1769 | 0.9695 |

36 | trapmf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0098 | 0.1084 | 0.0450 | 19.3158 | 0.8806 |

37 | gbellmf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0022 | 0.0472 | 0.0256 | 16.1637 | 0.9669 |

38 | gbellmf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0044 | 0.0660 | 0.0307 | 18.0977 | 0.9376 |

39 | gaussmf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.0013 | 0.0365 | 0.0212 | 13.8235 | 0.9800 |

40 | gaussmf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0019 | 0.0431 | 0.0241 | 14.7715 | 0.9720 |

41 | gaussmf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0056 | 0.0746 | 0.0314 | 17.5307 | 0.9185 |

42 | gauss2mf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.0017 | 0.0409 | 0.0235 | 16.2626 | 0.9755 |

43 | gauss2mf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0040 | 0.0632 | 0.0260 | 17.3863 | 0.9407 |

44 | gauss2mf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0072 | 0.0847 | 0.0290 | 17.7331 | 0.9048 |

45 | pimf | 3-3-3-3-3 | Constant | 10 | hybrid | 0.1224 | 0.3499 | 0.0482 | 26.0901 | 0.3608 |

46 | pimf | 3-3-4-4-4 | Constant | 10 | hybrid | 0.0026 | 0.0510 | 0.0307 | 24.7626 | 0.9615 |

47 | pimf | 3-3-5-5-5 | Constant | 10 | hybrid | 0.0022 | 0.0466 | 0.0285 | 22.3553 | 0.9678 |

City | Anfis Run | Type of Input MF | Number of MFs | Number of Rules | Time (s) | MSE | RMSE | MAE | MAPE | R^{2} |
---|---|---|---|---|---|---|---|---|---|---|

Alexandroupoli | 17 | trimf | 3-3-3-2-2 | 72 | 5 | 0.0024 | 0.0494 | 0.0351 | 10.5278 | 0.9638 |

39 | gaussmf | 3-3-3-3-3 | 243 | 47 | 0.0031 | 0.0557 | 0.0355 | 10.1556 | 0.9538 | |

20 | gaussmf | 3-3-3-2-2 | 72 | 5 | 0.0023 | 0.0480 | 0.0341 | 10.1123 | 0.9659 | |

Athens | 1 | trimf | 2-2-2-2-2 | 32 | 7 | 0.0021 | 0.0457 | 0.0295 | 20.1799 | 0.9825 |

17 | trimf | 3-3-3-2-2 | 108 | 19 | 0.0026 | 0.0511 | 0.0315 | 19.7972 | 0.9786 | |

20 | gaussmf | 3-3-3-2-2 | 108 | 19 | 0.0022 | 0.0467 | 0.0306 | 21.2929 | 0.9818 | |

Drama | 17 | trimf | 3-3-3-2-2 | 108 | 19 | 0.0026 | 0.0511 | 0.0363 | 6.2547 | 0.8997 |

1 | trimf | 2-2-2-2-2 | 32 | 5 | 0.0026 | 0.0513 | 0.0361 | 6.2235 | 0.8975 | |

20 | gaussmf | 3-3-3-2-2 | 108 | 13 | 0.0026 | 0.0508 | 0.0371 | 6.4071 | 0.8995 | |

Karditsa | 17 | trimf | 3-3-3-2-2 | 108 | 12 | 0.0019 | 0.0434 | 0.0242 | 13.8394 | 0.9789 |

1 | trimf | 2-2-2-2-2 | 32 | 4 | 0.0018 | 0.0421 | 0.0236 | 11.6196 | 0.9801 | |

4 | gaussmf | 2-2-2-2-2 | 32 | 4 | 0.0019 | 0.0431 | 0.0248 | 13.3841 | 0.9792 | |

Larissa | 1 | trimf | 2-2-2-2-2 | 32 | 4 | 0.0012 | 0.0352 | 0.0203 | 10.9568 | 0.9817 |

4 | gaussmf | 2-2-2-2-2 | 32 | 4 | 0.0012 | 0.0352 | 0.0204 | 10.9833 | 0.9817 | |

20 | gaussmf | 3-3-3-2-2 | 108 | 19 | 0.0010 | 0.0314 | 0.0184 | 10.5236 | 0.9858 | |

Markopoulo | 1 | trimf | 2-2-2-2-2 | 32 | 5 | 0.0091 | 0.0956 | 0.0728 | 25.0887 | 0.6593 |

4 | gaussmf | 2-2-2-2-2 | 32 | 5 | 0.0096 | 0.0980 | 0.0755 | 26.7510 | 0.6364 | |

17 | trimf | 3-3-3-2-2 | 108 | 19 | 0.0259 | 0.1609 | 0.1087 | 36.7174 | 0.5126 | |

Serres | 1 | trimf | 2-2-2-2-2 | 32 | 5 | 0.0007 | 0.0271 | 0.0176 | 10.4721 | 0.9839 |

4 | gaussmf | 2-2-2-2-2 | 32 | 5 | 0.0008 | 0.0279 | 0.0185 | 11.2421 | 0.9831 | |

39 | gaussmf | 3-3-3-3-3 | 243 | 45 | 0.0008 | 0.0285 | 0.0194 | 12.1163 | 0.9824 | |

Thessaloniki | 17 | trimf | 3-3-3-2-2 | 108 | 13 | 0.0015 | 0.0382 | 0.0229 | 16.1046 | 0.9773 |

20 | gaussmf | 3-3-3-2-2 | 108 | 13 | 0.0013 | 0.0363 | 0.0219 | 14.1944 | 0.9795 | |

39 | gaussmf | 3-3-3-3-3 | 243 | 45 | 0.0021 | 0.0459 | 0.0256 | 15.2032 | 0.9672 | |

Trikala | 1 | trimf | 2-2-2-2-2 | 32 | 4 | 0.0019 | 0.0433 | 0.0232 | 10.5817 | 0.9815 |

4 | gaussmf | 2-2-2-2-2 | 32 | 4 | 0.0020 | 0.0450 | 0.0245 | 11.1412 | 0.9800 | |

20 | gaussmf | 3-3-3-2-2 | 108 | 13 | 0.0028 | 0.0530 | 0.0271 | 11.7631 | 0.9708 | |

Volos | 1 | trimf | 2-2-2-2-2 | 32 | 4 | 0.0021 | 0.0459 | 0.0317 | 13.2520 | 0.9564 |

4 | gaussmf | 2-2-2-2-2 | 32 | 4 | 0.0021 | 0.0460 | 0.0314 | 13.1629 | 0.9563 | |

20 | gaussmf | 3-3-3-2-2 | 108 | 12 | 0.0020 | 0.0445 | 0.0323 | 13.9710 | 0.9588 |

**Table 6.**The best ANFIS models for all the cities under investigation (10 epochs and hybrid optimization).

Title 1 | Anfis Run | Type of Input MF | Number of MFs | Type of Output MF | Optimization | MSE | RMSE | MAE | MAPE | R^{2} |
---|---|---|---|---|---|---|---|---|---|---|

Alexandroupoli | 20 | gaussmf | 3-3-3-2-2 | Constant | Hybrid | 0.0023 | 0.0480 | 0.0341 | 10.1123 | 0.9659 |

Athens | 17 | trimf | 3-3-3-2-2 | Constant | Hybrid | 0.0026 | 0.0511 | 0.0315 | 19.7972 | 0.9786 |

Drama | 1 | trimf | 2-2-2-2-2 | Constant | Hybrid | 0.0026 | 0.0513 | 0.0361 | 6.2235 | 0.8975 |

Karditsa | 1 | trimf | 2-2-2-2-2 | Constant | Hybrid | 0.0018 | 0.0421 | 0.0236 | 11.6196 | 0.9801 |

Larissa | 20 | gaussmf | 3-3-3-2-2 | Constant | Hybrid | 0.0010 | 0.0314 | 0.0184 | 10.5236 | 0.9858 |

Markopoulo | 1 | trimf | 2-2-2-2-2 | Constant | Hybrid | 0.0091 | 0.0956 | 0.0728 | 25.0887 | 0.6593 |

Serres | 4 | gaussmf | 2-2-2-2-2 | Constant | Hybrid | 0.0008 | 0.0279 | 0.0185 | 11.2421 | 0.9831 |

Thessaloniki | 20 | gaussmf | 3-3-3-2-2 | Constant | Hybrid | 0.0013 | 0.0363 | 0.0219 | 14.1944 | 0.9795 |

Trikala | 4 | gaussmf | 2-2-2-2-2 | Constant | Hybrid | 0.0020 | 0.0450 | 0.0245 | 11.1412 | 0.9800 |

Volos | 4 | gaussmf | 2-2-2-2-2 | Constant | Hybrid | 0.0021 | 0.0460 | 0.0314 | 13.1629 | 0.9563 |

**Table 7.**Comparison results among the artificial neural network (ANN), fuzzy cognitive map (FCM), hybrid FCM-ANN and best ANFIS architectures of each city.

City | Method | MSE | RMSE | MAE | MAPE | R^{2} |
---|---|---|---|---|---|---|

Alexandroupoli | RCGA-FCM | 0.0047 | 0.0684 | 0.0538 | 17.6233 | 0.9450 |

SOGA-FCM | 0.0045 | 0.0672 | 0.0526 | 17.1707 | 0.9484 | |

ANN | 0.0042 | 0.0645 | 0.0505 | 16.1131 | 0.9439 | |

Hybrid FCM-ANN | 0.0034 | 0.0579 | 0.0427 | 14.3034 | 0.9498 | |

Best ANFIS | 0.0023 | 0.0480 | 0.0341 | 10.1123 | 0.9659 | |

Athens | RCGA-FCM | 0.0022 | 0.0473 | 0.0303 | 23.5985 | 0.9676 |

SOGA-FCM | 0.0029 | 0.0539 | 0.0337 | 22.7453 | 0.9646 | |

ANN | 0.0010 | 0.0323 | 0.0198 | 14.2464 | 0.9844 | |

Hybrid FCM-ANN | 0.0014 | 0.0374 | 0.0230 | 17.5418 | 0.9790 | |

Best ANFIS | 0.0026 | 0.0511 | 0.0315 | 19.7972 | 0.9786 | |

Drama | RCGA-FCM | 0.0080 | 0.0894 | 0.0749 | 12.9942 | 0.8691 |

SOGA-FCM | 0.0056 | 0.0748 | 0.0600 | 10.1766 | 0.8796 | |

ANN | 0.0025 | 0.0501 | 0.0357 | 6.1657 | 0.9025 | |

Hybrid FCM-ANN | 0.0028 | 0.0526 | 0.0363 | 6.2502 | 0.8941 | |

Best ANFIS | 0.0026 | 0.0513 | 0.0361 | 6.2235 | 0.8975 | |

Karditsa | RCGA-FCM | 0.0039 | 0.0624 | 0.0379 | 27.5914 | 0.9591 |

SOGA-FCM | 0.0488 | 0.2210 | 0.1397 | 50.2112 | 0.9711 | |

ANN | 0.0016 | 0.0405 | 0.0245 | 17.4579 | 0.9819 | |

Hybrid FCM-ANN | 0.0017 | 0.0407 | 0.0245 | 18.4095 | 0.9817 | |

Best ANFIS | 0.0018 | 0.0421 | 0.0236 | 11.6196 | 0.9801 | |

Larissa | RCGA-FCM | 0.0027 | 0.0515 | 0.0331 | 22.2481 | 0.9638 |

SOGA-FCM | 0.0025 | 0.0505 | 0.0328 | 22.9579 | 0.9649 | |

ANN | 0.0013 | 0.0355 | 0.0209 | 13.2479 | 0.9812 | |

Hybrid FCM-ANN | 0.0013 | 0.0356 | 0.0215 | 13.1974 | 0.9811 | |

Best ANFIS | 0.0010 | 0.0314 | 0.0184 | 10.5236 | 0.9858 | |

Markopoulo | RCGA-FCM | 0.0075 | 0.0868 | 0.0726 | 26.0003 | 0.6975 |

SOGA-FCM | 0.0078 | 0.0883 | 0.0739 | 26.3345 | 0.6955 | |

ANN | 0.0172 | 0.1310 | 0.1048 | 34.8594 | 0.4765 | |

Hybrid FCM-ANN | 0.0070 | 0.0836 | 0.0667 | 23.7166 | 0.7094 | |

Best ANFIS | 0.0091 | 0.0956 | 0.0728 | 25.0887 | 0.6593 | |

Serres | RCGA-FCM | 0.0017 | 0.0409 | 0.0274 | 16.5199 | 0.9648 |

SOGA-FCM | 0.0495 | 0.2225 | 0.1632 | 72.9785 | 0.9772 | |

ANN | 0.0008 | 0.0275 | 0.0179 | 10.9948 | 0.9842 | |

Hybrid FCM-ANN | 0.0008 | 0.0289 | 0.0190 | 11.5000 | 0.9821 | |

Best ANFIS | 0.0008 | 0.0279 | 0.0185 | 11.2421 | 0.9831 | |

Thessaloniki | RCGA-FCM | 0.0029 | 0.0541 | 0.0339 | 29.9713 | 0.9565 |

SOGA-FCM | 0.0029 | 0.0539 | 0.0340 | 30.1471 | 0.9568 | |

ANN | 0.0017 | 0.0412 | 0.0262 | 23.8748 | 0.9735 | |

Hybrid FCM-ANN | 0.0019 | 0.0441 | 0.0266 | 23.8835 | 0.9696 | |

Best ANFIS | 0.0013 | 0.0363 | 0.0219 | 14.1944 | 0.9795 | |

Trikala | RCGA-FCM | 0.0059 | 0.0770 | 0.0453 | 21.9722 | 0.9528 |

SOGA-FCM | 0.0433 | 0.2082 | 0.1287 | 42.7427 | 0.9715 | |

ANN | 0.0020 | 0.0443 | 0.0258 | 14.1183 | 0.9804 | |

Hybrid FCM-ANN | 0.0019 | 0.0432 | 0.0251 | 13.9034 | 0.9815 | |

Best ANFIS | 0.0020 | 0.0450 | 0.0245 | 11.1412 | 0.9800 | |

Volos | RCGA-FCM | 0.0028 | 0.0526 | 0.0397 | 17.8195 | 0.9436 |

SOGA-FCM | 0.0027 | 0.0520 | 0.0395 | 17.8988 | 0.9445 | |

ANN | 0.0020 | 0.0444 | 0.0319 | 13.2504 | 0.9588 | |

Hybrid FCM-ANN | 0.0020 | 0.0446 | 0.0307 | 12.7881 | 0.9587 | |

Best ANFIS | 0.0021 | 0.0460 | 0.0314 | 13.1629 | 0.9563 |

Architectures | Parameters for Athens City | Average Running Time |
---|---|---|

ANN | Multilayer feed forward network, six inputs, 10 neurons, one output, sigmoidal activation function, Levenberg-Marquardt learning, epochs = 20 | 16–20 s |

RCGA-FCM | Uniform crossover with probability 0.4, Mühlenbein’s mutation with probability 0.4, ranking selection, elite strategy, population size 200, maximum number of generations 200 | 808 s |

SOGA-FCM | Uniform crossover with probability 0.4, Mühlenbein’s mutation with probability 0.4, ranking selection, elite strategy, population size 200, maximum number of generations 200, learning parameters b_{1} = b_{2} = 0.01 | 799 s |

Hybrid FCM-ANN | Multilayer feed forward network, four inputs selected by SOGA-FCM (month, temperature, demand of a day before, current demand), one hidden layer with 10 neurons, one output, sigmoidal activation function, Levenberg-Marquardt learning, epochs = 20 | 811 s |

Best ANFIS | Triangular mf, 2-2-2-2-2 or 3-3-3-2-2, Constant output, epochs = 10, Hybrid optimization | 4–19 s |

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**MDPI and ACS Style**

Papageorgiou, K.; I. Papageorgiou, E.; Poczeta, K.; Bochtis, D.; Stamoulis, G.
Forecasting of Day-Ahead Natural Gas Consumption Demand in Greece Using Adaptive Neuro-Fuzzy Inference System. *Energies* **2020**, *13*, 2317.
https://doi.org/10.3390/en13092317

**AMA Style**

Papageorgiou K, I. Papageorgiou E, Poczeta K, Bochtis D, Stamoulis G.
Forecasting of Day-Ahead Natural Gas Consumption Demand in Greece Using Adaptive Neuro-Fuzzy Inference System. *Energies*. 2020; 13(9):2317.
https://doi.org/10.3390/en13092317

**Chicago/Turabian Style**

Papageorgiou, Konstantinos, Elpiniki I. Papageorgiou, Katarzyna Poczeta, Dionysis Bochtis, and George Stamoulis.
2020. "Forecasting of Day-Ahead Natural Gas Consumption Demand in Greece Using Adaptive Neuro-Fuzzy Inference System" *Energies* 13, no. 9: 2317.
https://doi.org/10.3390/en13092317