# Forecasting and Modelling the Uncertainty of Low Voltage Network Demand and the Effect of Renewable Energy Sources

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## Abstract

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## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Contributions

- A new ANN forecast model optimized by using the Golden Ratio Optimization Method (GROM) technique to examine household and small cities’ demand incorporating highly volatile renewable energy sources.
- Developing a realistic stochastic prediction model, which is a hybrid forecast model consisting of probabilistic and ARIMAX models. This hybrid forecast model and different rolling and point forecast models are developed in this paper to treat the stochasticity of LV and PV load profiles, taking into account the impact of uncertainty intervals on forecasting confidence bounds.
- This paper presents load forecasting for households and small cities using different forecasting methods. Smart meter data for ten household and PV systems were collected and used to predict induvial household demand, as presented in Appendix A. This work has developed forecast models to produce a potential demand profile for households and the PV system separately, in addition to net demand for up to one-day ahead. In addition, this research has provided an analysis of a typical household demand and PV system in Jordan within a real time period, supporting attempts to bridge the gap in the absence of comprehension demand behaviour data, especially in Middle Eastern countries like Jordan.

#### 1.3. Outline of Paper

## 2. Household and PV System Model Topology

#### PV System

## 3. Data Analysis

#### 3.1. PV System Data Analysis

#### 3.2. Weather Data

#### 3.3. Load Data Analysis

#### 3.3.1. Composition

#### 3.3.2. Time Series Analysis

- Analysis based on daily and weekly patterns, to examine, if applicable therein, hour/day–day/week–week demand and any formation of cycles.
- Analysis of autocorrelation and hourly energy consumption to investigate if there are any seasonal patterns, especially those not in day/week cycles.

## 4. Load Forecasting Models

#### 4.1. Probabilistic ARIMAX Forecast Model

#### Probabilistic ARIMAX Model

- Identify the forecast model variables and pre-sample data, where usually the training data set is used as the pre-sample data.
- A Monte Carlo sampling method is used to generate stochastic samples from the empirical joint distribution.
- Use the ARIMAX models from Section 4.1 to generate stochastic samples and obtain the model responses.

#### 4.2. ANN Forecast Model Optimized by Using Golden Ratio Optimization (GROM)

- 1-
- Variable selection:
- Output variables: the principal goal of this paper, future demand $\hat{\mathrm{L}}\left(\mathrm{t}+\mathrm{i}\right)$, and PV system output power $\hat{\mathrm{P}}\left(\mathrm{t}+\mathrm{i}\right)$.
- Input variables: initially, the external variables (temperature and wind speed) have been carefully chosen as key input variables, by reason of the robust link amongst them and the selected output variables. Furthermore, the experimental and error method was employed to choose extra input variables grounded in the relative historical profiles and current for household demand and PV system output power. In step 4, the results and analysis of the trial and error are provided for the purpose of checking parameters.

- 2-
- Data collection and pre-processing: the measured data is presented in Section 3. This step includes checking all data to avoid data waste. In addition, the step implies assaying the data to noise abatement, discerning trends, and finding any important link.
- 3-
- Dividing the data set: the collected data sets are separated into training, validation and testing data sets, as discussed in Section 3.
- 4-
- ANN model parameters selection: the capability of figuring out and alleviating the computation of complex correlations is the reason behind using parameter functions in this case study. Besides, the trial-and-error approaches apply as a consequence of identifying both numbers in hidden layers and neurons.
- Training function: Levenberg-Marquardt backpropagation.
- Transfer function: sigmoid function.
- Evaluation criteria: full squared errors.
- The stopping criteria: once there is no additional development in the error function.
- Input variables: in general, to improve the expected performance, a suitable external variable should be selected based on the objectives of the model and the availability of data. In Section 3.2, the analysis of data showed high correlation between the PV output and household demands and temperature. In addition, a positive strong correlation between the PV output and wind speed is presented. Therefore, weather conditions are recommended to be used as external variables: ${\mathrm{X}}_{1}$(t) is the hourly temperature and ${\mathrm{X}}_{2}$(t) is the hourly wind speed. In Section 3.3, the previous hour demand and the previous day demand at the same time showed a strong positive correlation with the current demand at Madaba; therefore, these two variables and hour of the day are recommended to be used as external variables${\mathrm{X}}_{3}$ to ${\mathrm{X}}_{5}$. In order to verify the impact of the proposed external variables on the forecast model accuracy, Section 5.3 presents a statistical analysis of the ANN forecast models with different external variables. The following exogenous variables are used in the PV power forecast model: ${\mathrm{X}}_{1}$: Temperature, ${\mathrm{X}}_{2}$: Wind Speed, ${\mathrm{X}}_{3}:$Hour of the day, ${\mathrm{X}}_{4}$: Former hour data and ${\mathrm{X}}_{5}$: Former day data in same hour. On the other hand, the following exogenous variable are used for the household and Madaba city demand forecast model: ${\mathrm{X}}_{1}$: Temperature, ${\mathrm{X}}_{2}$: Average of the previous two hours demand, ${\mathrm{X}}_{3}:$Hour of the day, ${\mathrm{X}}_{4}$: Former hour data and ${\mathrm{X}}_{5}$: Former day data in same hour.
- Number of hidden layers: two hidden layers.
- Number of hidden neurons: ten neurons in each hidden layer.

#### ANN-GROM Forecast Model

- Firstly, a number of random learning model parameters for the ANN forecast model, as population initialization is created and the mean value of the population is calculated.
- Secondly, the fitness of each model parameter is evaluated by using the learning cost function in ANN. Then, the fitness of the mean value of the population solution will be compared to the worst solution. In case the mean population solution has a better fitness result compared to the worst solution, the worst solution will be replaced by the mean population solution. This process in GROM aims to enhance the optimization speed to achieve convergence.
- Thirdly, a random solution vector is created in the population to determine and specify the new step direction and movement. The fitness of the new random solution and selected population will be compared to the mean solution. In this step, the random parameters solution aims to create a random movement towards the next step solution and to create the ability to search the whole space of the cost function. In order to select the size of movement towards the new solution and its direction, the Fibonacci formula (golden ratio) is used in this work as in [43]. The best parameters solution is the solution with the minimum objective function value. In GROM, the parameter solutions need to be updated and moved towards the best solution for the population [43].

## 5. Results and Discussion

- Comparing the forecast model performance over different data profiles: household demand, Madaba city demand, PV energy output and the net curve at the household, which is the difference between household demand and PV system output.
- Evaluating the impact of exogenous variables (weather conditions) on the prediction models.
- Evaluating the importance of designing a rolling load forecast model compared to a fixed forecast model, especially for volatile data profiles such as LV household demand.

#### 5.1. Overall Comparisons

#### 5.2. Forecast Error Analysis

#### 5.3. Effect of Exogenous Variables on Forecast Models

- Model A1: ARIMAX with two exogenous variables (${\mathrm{X}}_{1}$: Temperature and${\mathrm{X}}_{2}$: Wind Speed).
- Model A2: ARIMAX with one exogenous variable (${\mathrm{X}}_{1}$: Temperature).
- Model A3: ARIMAX with one exogenous variable (${\mathrm{X}}_{2}$: Wind Speed).
- Model A4: ARIMA (2,1,2) model for the household and Madaba city demand and ARIMA (1,1,2) for PV output. The ARIMA model has been derived by removing the external variables term from the ARIMAX model.
- Model NN1: ANN with the following exogenous variables (${\mathrm{X}}_{1}$: Temperature, ${\mathrm{X}}_{2}$: Wind Speed, ${\mathrm{X}}_{3}:$Hour of the day, ${\mathrm{X}}_{4}$: Previous hour data, ${\mathrm{X}}_{5}$: Previous day data in same hour).
- Model NN2: ANN model without exogenous variables that is related to weather conditions and includes the following variables (${\mathrm{X}}_{3}:$Hour of the day, ${\mathrm{X}}_{4}$: Previous hour data, ${\mathrm{X}}_{5}$: Previous day data in same hour).
- Model NN3: ANN model without exogenous variables that is related to time series and seasonality and includes only the variables related to weather condition (${\mathrm{X}}_{1}$: Temperature, ${\mathrm{X}}_{2}$: Wind Speed).

#### 5.4. Evaluating of the Importance of Designing a Rolling Load Forecast

#### 5.5. Evaluating the Impact of Demand Disaggregation

#### 5.6. Evaluation of the Proposlaictc Forecast

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Demand (kWh) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Time | House 1 | House 2 | House 3 | House 4 | House 5 | House 6 | House 7 | House 8 | House 9 | House 10 |

1 | 0.946 | 0.507 | 1.946 | 1.095 | 0.776 | 0.954 | 0.873 | 0.640 | 1.354 | 0.967 |

2 | 0.779 | 0.411 | 1.823 | 0.827 | 0.823 | 0.774 | 0.823 | 0.970 | 1.522 | 0.874 |

3 | 0.786 | 0.346 | 0.959 | 1.447 | 0.776 | 0.894 | 0.843 | 0.680 | 0.665 | 0.867 |

4 | 0.756 | 0.898 | 0.928 | 2.306 | 0.761 | 0.695 | 0.821 | 0.690 | 0.785 | 0.780 |

5 | 0.785 | 0.883 | 0.53 | 1.426 | 0.73 | 1.712 | 0.857 | 0.690 | 1.930 | 0.930 |

6 | 0.748 | 1.442 | 0.15 | 0.779 | 0.726 | 1.69 | 0.826 | 0.700 | 0.442 | 0.950 |

7 | 0.83 | 2.782 | 0.105 | 0.784 | 1.271 | 0.708 | 0.806 | 0.850 | 1.734 | 1.150 |

8 | 2.145 | 1.503 | 0.209 | 0.57 | 0.614 | 0.305 | 0.387 | 3.450 | 1.930 | 1.297 |

9 | 1.163 | 1.774 | 1.74 | 3.61 | 0.626 | 1.023 | 0.557 | 2.630 | 0.780 | 0.740 |

10 | 3.55 | 1.702 | 1.312 | 3.428 | 0.6 | 1.01 | 2.818 | 1.550 | 0.702 | 0.930 |

11 | 4.856 | 2.432 | 1.513 | 3.642 | 1.15 | 2.32 | 2.404 | 1.550 | 0.820 | 0.821 |

12 | 4.086 | 2.12 | 5.128 | 2.308 | 5.54 | 2.52 | 2.631 | 1.060 | 0.812 | 0.980 |

13 | 4.604 | 4.868 | 6.184 | 5.464 | 5.63 | 2.96 | 2.549 | 2.673 | 1.180 | 2.134 |

14 | 5.142 | 4.558 | 6.654 | 5.134 | 5.204 | 3.90 | 3.906 | 2.114 | 1.850 | 2.545 |

15 | 4.134 | 4.042 | 3.94 | 2.102 | 3.93 | 2.652 | 2.652 | 3.111 | 1.042 | 1.957 |

16 | 1.241 | 3.53 | 2.786 | 2.374 | 4.4 | 1.588 | 0.55 | 1.441 | 1.583 | 2.083 |

17 | 0.571 | 1.674 | 1.577 | 0.771 | 4.476 | 1.63 | 1.112 | 1.513 | 2.765 | 1.779 |

18 | 1.015 | 0.577 | 0.254 | 2.643 | 0.962 | 0.938 | 1.111 | 0.805 | 3.774 | 2.354 |

19 | 1.216 | 0.612 | 0.919 | 1.972 | 0.224 | 0.55 | 1.008 | 2.516 | 2.629 | 1.938 |

20 | 1.107 | 1.522 | 2.226 | 1.218 | 0.105 | 1.215 | 1.239 | 2.110 | 1.563 | 2.836 |

21 | 1.039 | 1.255 | 2.056 | 1.433 | 2.053 | 1.489 | 1.332 | 2.395 | 1.954 | 1.156 |

22 | 2.346 | 1.055 | 1.085 | 1.952 | 1.079 | 0.324 | 1.263 | 1.634 | 1.655 | 1.389 |

23 | 1.411 | 0.561 | 1.157 | 0.108 | 0.588 | 0.032 | 1.125 | 1.115 | 0.981 | 0.477 |

24 | 1.11 | 0.414 | 1.166 | 0.97 | 1.008 | 0.093 | 1.147 | 0.951 | 0.894 | 0.366 |

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**Figure 3.**An example for a single household PV system output (House 5) over one week with sunny day.

**Figure 5.**An example for a single household PV system output (House 7) over one week with unclear (cloudy) weather.

**Figure 6.**Partial Autocorrelation Function (PACF) plot for household PV output system within unclear sky days.

**Figure 7.**Illustration of distribution of: (

**a**) household demand and temperature (

**b**) PV system output and temperature in a 2D histogram.

**Figure 15.**General schematic of the short-term load forecasting procedure implemented in this article.

**Figure 16.**Methodology of the Autoregressive Integrated Moving Average (ARIMA) and ARIMAX forecasting models.

Content | Description | Quantity |
---|---|---|

PV panels | (Jinko) cells with 345 watt power, Mono type | 12 |

Inverter | 4 kW (ABB) | 1 |

panel area | 25 ${\mathrm{m}}^{2}$ | |

Electrical Wires | AC & DC Wires | --- |

Other electric components | Power Panels Circuit breakers | --- |

Month | Consumption kWh | Month | Consumption kWh |
---|---|---|---|

JAN | 1089 | JUL | 1012 |

FEB | 1080 | AGU | 1050 |

MAR | 574 | SEP | 784 |

APR | 544 | OCT | 510 |

MAY | 866 | NOV | 644 |

JUN | 870 | DEC | 900 |

**Table 3.**R-squared values for the relationship between hourly temperature and wind speed with the PV system output.

Correlated Variables | ${R}^{2}$ |
---|---|

$\mathrm{PV}\mathrm{output}$ vs. temperature | 94.5% |

$\mathrm{PV}\mathrm{output}$ vs. wind speed | 39.3% |

Household Demand Resolutions | μ (kWh) | $\mathbf{\Sigma}$ (kWh) | CV | Maximum Demand (kWh) | Minimum Demand (kWh) |
---|---|---|---|---|---|

Hourly | 1.6 | 1.40 | 87.2% | 6.9 | 0.0 |

Daily | 37.1 | 15.1 | 38.3% | 63.7 | 15.2 |

Correlated Variables | ${\mathbf{R}}^{2}$ |
---|---|

$(\mathrm{t})$ vs. $\mathrm{L}(\mathrm{t}-1)$ | 89.5% |

$\mathrm{L}(\mathrm{t})$ vs. $\mathrm{L}(\mathrm{t}-2)$ | 76.7% |

$\mathrm{L}(\mathrm{t})$ vs. $\mathrm{L}(\mathrm{t}-3)$ | 59.3% |

$\mathrm{L}(\mathrm{t})$ vs. $\mathrm{L}(\mathrm{t}-4)$ | 22.9% |

$\mathrm{L}(\mathrm{t})$ vs. $\mathrm{L}(\mathrm{t}-24)$ | 45.6% |

Traditional ANN | ANN-GROM | Probabilistic-ARIMAX | ARIMAX | |||||
---|---|---|---|---|---|---|---|---|

Mean Absolute Percentage Error (MAPE) | Root Mean Square Error (RMSE) | MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Household demand | 5.7% | 30.1 W | 3.4% | 20.1 W | 4.7% | 28.1 W | 6.2% | 39.8 W |

PV system output | 6.1% | 44.9 W | 4.8% | 25.8 W | 5.9% | 31.9 W | 7.1% | 50.0 W |

Net curve at the household | 7.5% | 59.8 W | 5.3% | 28.6 W | 6.5% | 40.8 W | 8.4% | 70.1 W |

Madaba City | 4.3% | 860 kW | 3.1% | 620 kW | 4.2% | 845 kW | 4.9% | 980 kW |

Forecast Models | |||||||||
---|---|---|---|---|---|---|---|---|---|

Model A1 | Model A2 | Model A3 | Model A4 | Model NN1 | Model NN2 | Model NN3 | ANN [47] | LSTM [47] | |

MAPE | 7% | 8.3% | 10.1% | 15.8% | 6.2% | 14.9% | 19.7% | 17.3% | 24.% |

RMSE | 49.2 W | 54.3 W | 61.7 W | 75.8 W | 45.1 W | 70.3 W | 108.9 W | 98.7 W | 153.2 W |

**Table 8.**The daily mean absolute percentage error (MAPE) for the rolling and fixed models started from 1 October 2020.

Traditional ANN | ARIMAX | ANN-GROM | Probabilistic-ARIMAX | |||||
---|---|---|---|---|---|---|---|---|

Rolling Forecast | Fixed Forecast | Rolling Forecast | Fixed Forecast | Rolling Forecast | Fixed Forecast | Rolling Forecast | Fixed Forecast | |

Day 1 | 4.2% | 5.8% | 5.9% | 6.6% | 3.1% | 4.0% | 4.3% | 5.7% |

Day 2 | 3.9% | 4.1% | 5.7% | 7.1% | 2.8% | 3.9% | 4.5% | 5.9% |

Day 3 | 4.8% | 6.1% | 5.3% | 6.9% | 3.3% | 5.1% | 4.2% | 6.0% |

Day 4 | 5.2% | 7.3% | 7.1% | 8.6% | 4.2% | 5.3% | 5.1% | 6.8% |

Day 5 | 5.4% | 6.1% | 6.7% | 7.9% | 4.5% | 5.5% | 5.4% | 6.1% |

Day 6 | 5.1% | 8.7% | 8.1% | 9.3% | 4.1% | 6.2% | 5.5% | 7.2% |

Day 7 | 3.8% | 4.1% | 3.8% | 4.2% | 2.9% | 3.9% | 4.0% | 4.1% |

Traditional ANN | ANN-GROM | Probabilistic-ARIMAX | ||||
---|---|---|---|---|---|---|

MAPE | RMSE | MAPE | RMSE | MAPE | RMSE | |

Single household demand | 5.7% | 30.1 W | 3.4% | 20.1 W | 4.7% | 28.1 W |

Aggregation of ten household demand | 4.9% | 250.9 W | 3.2% | 190.5 W | 4.5% | 250.3 W |

Madaba City | 4.3% | 860 kW | 3.1% | 620 kW | 4.2% | 845 kW |

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**MDPI and ACS Style**

Alasali, F.; Foudeh, H.; Ali, E.M.; Nusair, K.; Holderbaum, W.
Forecasting and Modelling the Uncertainty of Low Voltage Network Demand and the Effect of Renewable Energy Sources. *Energies* **2021**, *14*, 2151.
https://doi.org/10.3390/en14082151

**AMA Style**

Alasali F, Foudeh H, Ali EM, Nusair K, Holderbaum W.
Forecasting and Modelling the Uncertainty of Low Voltage Network Demand and the Effect of Renewable Energy Sources. *Energies*. 2021; 14(8):2151.
https://doi.org/10.3390/en14082151

**Chicago/Turabian Style**

Alasali, Feras, Husam Foudeh, Esraa Mousa Ali, Khaled Nusair, and William Holderbaum.
2021. "Forecasting and Modelling the Uncertainty of Low Voltage Network Demand and the Effect of Renewable Energy Sources" *Energies* 14, no. 8: 2151.
https://doi.org/10.3390/en14082151