# Multi-Step Energy Demand and Generation Forecasting with Confidence Used for Specification-Free Aggregate Demand Optimization

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## Abstract

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## 1. Introduction

- The end user is an energy consumer who views the recommendations about the time points in the near future when energy consumption should be preferred according to the desired weighting factors of the three optimization criteria and the unit prices of energy at every time point which are applicable to this user, and adjusts his/her consumption behavior accordingly.
- The end user is a Distribution Systems Operator that forwards the recommendations of these algorithms to its customers.
- The end user is someone authorized to determine the unit prices of energy consumption at every time point, so (s)he can examine to what extent the economic criterion contradicts the renewable energy penetration and power stability criteria and adjust unit prices accordingly, also dynamically.

## 2. Related Work and Our Contribution

## 3. High-Level Architecture

## 4. The Forecasting Algorithm

#### 4.1. Methodology of the Forecasting Algorithm

- Scaling each raw variable so that it has mean equal to 0 and standard deviation equal to 1. This is required by some of the non-linear regression architectures.
- In particular cases, the target variable is transformed to one of its features, but typically all raw variables are directly used in the following steps, just after aggregation.
- Selection of input variables based on linear regression.
- Grouping the historical dataset w.r.t. month and day of week by applying a clustering algorithm to the data of the target variable (i.e., the variable to be forecast). The selection of the clustering algorithm and respective clusters is automatic, based on the behavior of the target variable, and particularly the selection of algorithm is also based on training of linear regression approximations of the forecasting models.
- Supervised training of all regression architectures on the selected input variables. A separate training is performed for each group and forecasting horizon. The best architecture, based on 3-fold cross-validation, is automatically selected in each case, and the respective model is retrained within the whole historical dataset. (Cross-validation is a common method to evaluate the ability of a model to generalize to unknown data. Since usually about 50–70% of a dataset is used for a single training, considering 3 folds is a reasonable choice).

- t: time point for which the forecast is made
- s: forecasting horizon
- ${\widehat{y}}_{s}\left(t\right)$: estimation of the value that the target variable will have at the time point t based on the forecasting horizon s
- $G\left(t\right)$: the group of months and days of week that t belongs to
- ${x}_{G\left(t\right),s}\left(t\right)$: latest known measurement of the target variable within $G\left(t\right)$ when forecasting for the time point t with horizon s (“present value”)
- r: duration such that all past values of the variable to be forecast, within $G\left(t\right)$, obtained after final aggregation (see Section 4.3.1), with a delay up to this duration, are used as input to the model
- ${z}_{G\left(t\right),s}\left(t\right)$: latest known measurement of the target variable at the same time of the day as t within $G\left(t\right)$ when forecasting for the time point t with horizon s (“last value at same time”)
- ${\mathbf{p}}_{r,G\left(t\right),s}\left(t\right)$: past values of the variable to be forecast, within $G\left(t\right)$, obtained after final aggregation, with a delay up to duration r, to be used as input to the model, when forecasting for the time point t with horizon s
- ${h}_{G\left(t\right)}\left(t\right)$: historical mean of the target variable at the same time of the day as t within $G\left(t\right)$ (“average time profile”)
- $\mathbf{w}\left(t\right)$: weather variables at the time point t (forecast weather in case of real-time forecast of the target variable, or exact weather in case of training on a historical interval)
- ${\overline{\mathbf{w}}}_{G\left(t\right),s}\left(t\right)$: mean of weather variables within the interval from the time point of ${x}_{G\left(t\right),s}\left(t\right)$ to t—in practice, no variable from this vector proved to be useful (and it would not make sense in most of the cases), so it will not be discussed later

#### 4.1.1. Forecasting Regression Model Architectures and Tested Hyper-Parameters

- Multilayer Perceptron Regressor [MLP(hidden layer sizes)]—in case of absence of hidden layers [MLP()], this is reduced to linear regression
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- hidden layer sizes = (), (1), (2), (5), (1,1), (2,2), (5,5)
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- activation = “tanh”
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- solver = “lbfgs”
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- maximum number of iterations = ${10}^{8}$
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- alpha = 0

- Random Forest Regressor [RF(maximum depth)]
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- maximum depth = 1, 2, 5, 10
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- number of estimators = 100

- Nearest Neighbour Regressor [NN(number of neighbours, weights)]
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- number of neighbours = 5, 10, 20, 50, 100, 200
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- weights = “uniform”, “distance”

- Support Vector Regressor [SVR(penalty, kernel(degree), epsilon)]
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- penalty (C) = 0.1, 1, 10
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- kernel = “poly” (polynomial), “rbf” [Radial Basis Function (RBF)], “sigmoid”
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- degree = 1, 2 (applicable only to the “poly” kernel)
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- epsilon = 0.1, 0.01, ${10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$
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- maximum number of iterations = ${10}^{7}$

- Gaussian Process Regressor [GPR(alpha)] (when the number of data points is less than ${10}^{4}$, so that training is memory-efficient)
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- alpha = ${10}^{-10}$, ${10}^{-9}$, ${10}^{-8}$, ${10}^{-7}$, ${10}^{-6}$, ${10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$, 0.01, 0.1

- Kernel Regression [KR(kernel, gamma)]
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- kernel = “poly”, “rbf”, “sigmoid”
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- gamma = 0.01, 0.1, 1, 10, 100, ${10}^{3}$

- Exteme Learning Machines [ELM(hidden layer units, alpha, RBF width, activation function)]
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- hidden layer units = 1, 2, 5, 10, 20, 50, 100
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- alpha = 0, 0.5, 1
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- RBF width = ${10}^{-3}$, 1
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- activation function = “gaussian”, “tanh”

#### 4.1.2. Clustering Methodology and Relevant Architectures Tested

- Algorithms automatically selecting number of clusters:
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- Affinity Propagation (AP)
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- Mean Shift (MS)

- Algorithms requiring manual selection of number of clusters n (1, 2, 6 and 12 clusters are tested for month, whereas 1, 2 and 7 clusters are tested for day of week):
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- K-Means (KM(n))
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- Agglomerative Clustering (AC(n))

#### 4.1.3. Evaluation

- F: training-test set split of the cross-validation
- ${I}_{s,F}$: time interval for which forecasts are made based on horizon s, within all training/test sets obtained by split F when computing training/test NRMSE respectively and within all groups of months and days of week pairs resulting from clustering
- $y\left(t\right)$: exact value of the target variable at the time point t
- ${\widehat{y}}_{s}\left(t\right)$: forecast value of the target variable for the time point t
- $\overline{y}$: mean of the target variable within the whole historical dataset

#### 4.2. Raw Data Used from Each Demonstrator

- Orkney generation was frozen within some intervals, which were discarded from the analysis. After that, about 5% and 6% of ANM and non-ANM generation timestamps respectively appeared to have missing values. Furthermore, a few negative values of generation were replaced by 0.
- In Madeira case, about 1% of the timestamps were discarded, because the sum of registered total demand was not within the range $\pm 1\%$ of the total generation, renewable and non-renewable (thermal), although total demand and total generation should be equal.

#### 4.3. Experimental Results from the Forecasting Algorithm and Discussion

#### 4.3.1. Algorithmic Specifications and Initial Pre-Processing

#### 4.3.2. Input Variables Selection and Timestamps Clustering

#### 4.3.3. Results from Training the Regression Models

## 5. The Optimization Algorithm

- minimum cost (minimum mean product of demand and price per energy unit of demand)
- maximum usage of renewable energy (minimum variance of difference between demand and forecast generation)
- minimum power instability (minimum demand variance)

- Within the optimization interval, the total demand equals the total forecast demand (w.r.t. time). This means that demand can only be shifted, but not reduced overall.
- At each time point of the optimization interval, demand is within a time-point-dependent interval.

- peak: €0.1773/kWh
- half-peak: €0.1272/kWh
- off-peak: €0.0624/kWh

#### 5.1. The Mathematical Viewpoint of the Optimization Method

#### 5.2. Methodology of the Optimization Algorithm

- optimization interval
- weighting coefficients of the optimization criteria
- aggregate demand forecast with confidence, based on a configurable significance level
- aggregate generation forecast (when the renewables-related criterion is considered)
- demand price at each time of the day, considering also day of week and month

- ${I}_{opt}$: interval for which the optimization takes place
- $d\left(t\right)$: aggregate demand (power) at time t
- $\mathbf{d}$: vector of all $d\left(t\right)$, $t\in {I}_{opt}$, in ascending order of time
- $\widehat{d}\left(t\right)$: forecast aggregate demand at time t
- $[{\widehat{d}}_{l}\left(t\right),{\widehat{d}}_{u}\left(t\right)]$: 95% confidence interval for demand at time t (as explained in Section 4.1, the bounds depend on the predicted demand, but their difference from predicted demand depends only on the forecasting horizon, time of the day and clustering group)
- $KP{I}_{c}\left(\mathbf{d}\right)$, $KP{I}_{r}\left(\mathbf{d}\right)$, $KP{I}_{p}\left(\mathbf{d}\right)$: KPIs for cost, renewables and power stability criteria respectively
- ${w}_{c},{w}_{r},{w}_{p}\ge 0,{w}_{c}+{w}_{r}+{w}_{p}=1$: arbitrary weights of KPIs

- ${I}_{h}$: interval of training data for the demand forecasting models
- $c\left(t\right)$: expected aggregate demand cost per energy unit at the time point t (historical prices are set according to the same pricing policy of the current year)
- $g\left(t\right)$: aggregate generation at the time point t
- ${\sigma}_{d\left({I}_{opt}\right)}^{2}$: theoretical variance of $d\left(t\right)$ within ${I}_{opt}$ (assuming mean as constant equal to the historical mean demand)
- ${s}_{d\left({I}_{h}\right)}^{2}$: sample variance of $d\left(t\right)$ within ${I}_{h}$

#### 5.3. Flexibility Analysis

- x: type of KPI (cost, renewables or power criterion, or combination)
- $R{R}_{x,{I}_{opt}}$: reduction rate of $KP{I}_{x}$ within optimization interval ${I}_{opt}$
- $\widehat{\mathbf{d}}$: vector of forecast aggregate demand at all time points of ${I}_{opt}$ in ascending order of time
- ${\widehat{\mathbf{d}}}_{opt}$: vector of forecast optimal aggregate demand at all time points of ${I}_{opt}$ in ascending order of time

#### 5.4. Implementation Examples of the Optimization Algorithm and Discussion

## 6. Conclusions, Limitations and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Agglomerative Clustering |

AHP | analytical hierarchy process |

ANM | Active Network Management |

ANN | Artificial Neural Network |

AP | Affinity Propagation |

API | Application Programming Interface |

ARFIMA | Autoregressive Fractionally Integrated Moving Average |

ARIMA | Autoregressive Integrated Moving Average |

ARMA | Autoregressive Moving Average |

BATS | Box–Cox transformation, ARMA errors, Trend and Seasonal components |

BESS | Battery Energy Storage System |

CPP | Critical Peak Pricing |

DR | Demand Response |

ELM | Extreme Learning Machine |

EPSO | Evolutionary Particle Swarm Optimization |

EV | electric vehicle |

GPR | Gaussian Process Regressor |

GRBFN | Generalized Radial Basis Function Network |

KM | K-Means |

KPI | Key Performance Indicator |

KR | Kernel Regression |

MAUT | multi-attribute utility theory |

MAVT | multi-attribute value theory |

MCDA | multi-criteria decision analysis |

MLP | Multi-Layer Perceptron |

MS | Mean Shift |

NN | Nearest Neighbour Regressor |

NRMSE | Normalized Root Mean Square Error |

PSO | Particle Swarm Optimization |

PV | photovoltaic |

QPSO | Quantum PSO |

RBF | Radial Basis Function |

RF | Random Forest Regressor |

RTP | Real-Time Pricing |

SARIMA | Seasonal Autoregressive Integrated Moving Average |

SPEA | strength Pareto evolutionary algorithm |

SVR | Support Vector Regressor |

TOU | Time-of-Use |

TS | Tabu search |

## Appendix A. Access to Raw Data and Statistical Analysis

**Table A1.**Summary statistics of power data. Count is the number of timestamps when data exist, after resampling.

Demonstrator | Output | Count | Mean | Standard Deviation | Minimum | Median | Maximum |
---|---|---|---|---|---|---|---|

Orkney | generation—ANM | 10,520 | 8.21 MW | 6.28 MW | 0.00 MW | 7.21 MW | 20.79 MW |

generation—non-ANM | 10,440 | 11.16 MW | 6.59 MW | 0.19 MW | 11.90 MW | 26.04 MW | |

generation—total (wind) | 11,061 | 18.34 MW | 12.68 MW | 0.00 MW | 17.98 MW | 42.78 MW | |

demand | 11,060 | 16.99 MW | 3.99 MW | 2.68 MW | 17.22 MW | 33.75 MW | |

Samsø | generation—solar | 8769 | 6.33 KW | 11.63 KW | 0.00 KW | 0.04 KW | 50.34 KW |

demand—total | 9935 | 12.07 KW | 9.13 KW | 1.44 KW | 9.75 KW | 77.88 KW | |

demand—non-controllable | 9937 | 11.62 KW | 8.57 KW | 1.68 KW | 9.60 KW | 67.20 KW | |

demand—controllable | 828 | 0.47 KW | 0.54 KW | 0.00 KW | 0.23 KW | 3.02 KW | |

Madeira | generation—hydro | 15,017 | 7.84 MW | 8.47 MW | 0.83 MW | 3.89 MW | 42.95 MW |

generation—wind | 15,017 | 11.06 MW | 10.39 MW | 0.02 MW | 7.79 MW | 40.48 MW | |

generation—waste | 15,017 | 4.22 MW | 1.83 MW | 0.00 MW | 4.90 MW | 6.68 MW | |

generation—solar | 15,017 | 2.87 MW | 3.93 MW | 0.00 MW | 0.15 MW | 13.93 MW | |

demand | 15,017 | 96.47 MW | 17.77 MW | 62.09 MW | 100.64 MW | 136.74 MW |

**Figure A1.**Average time profiles of power data from Orkney. The horizontal axes represent time of the day (h), and the vertical ones power (MW). The legends explain the clustering groups in the form (months, days of week), where days are numbered from 0 (Monday) to 6 (Sunday).

**Figure A2.**Average time profiles of power data from Samsø. The horizontal axes represent time of the day (h), and the vertical ones power (KW). The legends explain the clustering groups in the form (months, days of week), where days are numbered from 0 (Monday) to 6 (Sunday).

**Figure A3.**Average time profiles of power data from Madeira. The horizontal axes represent time of the day (h), and the vertical ones power (MW). The legends explain the clustering groups in the form (months, days of week), where days are numbered from 0 (Monday) to 6 (Sunday).

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**Figure 2.**Number of group and forecasting horizon pairs for which each model architecture was the best.

**Figure 6.**Time series of exact and forecast values with 95% confidence interval for Madeira demand during the first 4 calendar weeks of 2021, based on 24-h (

**top**) and 1-h (

**bottom**) forecasting horizon. The effect of average time profile and the day of week clusters (business days, Saturday, Sunday) on all values is shown. The measurement unit of demand is MW and the time zone is 2 h ahead of Madeira’s.

**Figure 7.**Examples showing the effect of wind speed and sun on Madeira wind (

**left**) and solar (

**right**) generation respectively [exact and forecast generation values with 95% confidence interval based on 24-h (

**top**) and 1-h (

**middle**) forecasting horizon], during 2 consecutive days of the same clustering group with different weather and generation profiles, as shown by the scatter plots (

**bottom**). The measurement units of generation and wind speed are MW and mph respectively, and the time zone is 2 h ahead of Madeira’s.

**Figure 8.**Energy price zones for the tri-hour tariff in Madeira. The information in the figure has been provided by the experts of the demonstrator.

**Figure 9.**Low-level architecture of the optimization algorithm and relation with the forecasting algorithm.

**Figure 10.**Examples of the application of the optimization approach to Orkney. In the titles, the weights for cost, renewables and power KPIs respectively are shown. In the vertical axis, the measurement unit is MW.

**Figure 11.**Examples of the application of the optimization approach to Samsø. In the titles, the weights for cost, renewables and power KPIs respectively are shown. In the vertical axis, the measurement unit is KW.

**Figure 12.**Examples of the application of the optimization approach to Madeira. In the titles, the weights for cost, renewables and power KPIs respectively are shown. In the vertical axis, the measurement unit is MW.

Demonstrator | Demand | Generation |
---|---|---|

Orkney | total (3 December 2018–6 April 2020) | wind ≃ total (ANM, non-ANM) (3 December 2018–6 April 2020) |

Samsø | harbour (1 May 2016–19 June 2017) | PV estimation—harbour (2016) |

Madeira | total (1 January 2018–30 November 2019) | hydro, wind, waste (“bio”), solar (1 January 2018–November 2019) |

Demonstrator | Output | Input | Clustering Algorithms for Month/Day of Week | Month Groups & Respective Day of Week Groups (Sets) on the Left & Right Respectively of Slashes and Colons [Days are Numbered from 0 (Monday) to 6 (Sunday)] |
---|---|---|---|---|

Orkney | generation—ANM | present value, average time profile, wind speed, wind gust | singletons/- | singletons/- |

generation—non-ANM | present value, wind speed, wind gust | KM(6)/- | {{10,1}, {11,12,2–4}, {5}, {6}, {7,8}, {9}}/- | |

generation—total (wind) | present value, past values (≤23 h), wind speed, wind gust | MS/- | {{9–4}, {5}, {6–8}}/- | |

demand | present value, past values (≤23 h), apparent temperature | KM(6)/- | {{11–2}, {3}, {4}, {5,10}, {6,9}, {7,8}}/- | |

Samsø | generation—solar | present value, last value at same time, past values (1 h), average time profile, sun, humidity | singletons/KM(2) | {{1}:{{6–1,3,4},{2,5}}, {2}:{{4–1},{2,3}}, {3}:{{0,4,6},{1–3,5}}, {4}:{{5–1,3},{2,4}}, {5}:{{2–0},{1}}, {6}:{{0–3},{4–6}}, {7}:{{3–1},{2}}, {8}:{{0–3},{4–6}}, {9}:{{0,1,4},{2,3,5,6}}, {10}:{{0,4,6},{1–3,5}}, {11}:{{4–2},{3}}, {12}:{{0,1,4,5},{2,3,6}}} |

demand—total | present value, past values (≤23 h) | singletons/- | singletons/- | |

demand—non-controllable | present value, past values (≤23 h), average time profile | KM(6)/- | {{1–3,6}, {4,5}, {7}, {8}, {9–11}, {12}}/- | |

demand—controllable | last value at same time, average time profile | KM(6)/AP | {{1,2}:{{0,2,6},{1,3},{4,5}}, {3,4}:{{0,2,6},{1,3},{4,5}}, {5,6,9}:{{0–4},{5,6}}, {7}:{{0–2,4,5},{3},{6}}, {8}:{{5–0},{1–4}}, {10–12}:{{0,2,4,6},{1,3},{5}}}} | |

Madeira | generation—hydro | present value, past values (≤23 h) | -/- | -/- |

generation—wind | present value, past values (≤23 h), average time profile, wind speed | AC(2)/- | {{10,11,1–4}, {5–9,12}}/- | |

generation—waste | present value, last value at same time, average time profile | KM(2)/- | {{12,1,3–8}, {9–11,2}}/- | |

generation—solar | present value, last value at same time, past values (1 h), average time profile, sun, humidity | KM(2)/- | {{10–2}, {3–9}}/- | |

demand | present value, last value at same time, average time profile | singletons/MS | {{1}:{{0–4},{5},{6}}, {2}:{{0–4},{5},{6}}, {3}:{{0–4},{5},{6}}, {4}:{{0–4},{5},{6}}, {5}:{{0–4},{5},{6}}, {6}:{{0–4},{5},{6}}, {7}:{{0–4},{5},{6}}, {8}:{{0–4},{5},{6}}, {9}:{{0–4},{5},{6}}, {10}:{{0–4},{5},{6}}, {11}:{{0–4},{5},{6}}, {12}:{{0,3,4},{1,2},{5},{6}} |

Demonstrator | Output | Proposed Approach | No Clustering—All Candidate Regressors | Same Clusters—Linear Regression | Same Clusters—Benchmark Regressor |
---|---|---|---|---|---|

Orkney | generation—ANM | 0.3419 | 0.4164 | 0.4283 | 0.9217 |

generation—non-ANM | 0.4282 | 0.4727 | 0.5725 | 0.9351 | |

generation—total (wind) | 0.4022 | 0.4300 | 0.4877 | 0.8715 | |

demand | 0.3879 | 0.4141 | 0.4087 | 0.6715 | |

Samsø | generation—solar | 0.3678 | 0.4360 | 0.4335 | 1.3081 |

demand—total | 0.4108 | 0.4379 | 0.4209 | 0.5649 | |

demand—non-controllable | 0.3112 | 0.3263 | 0.3206 | 0.3959 | |

demand—controllable | 0.7122 | 0.9668 | 0.8152 | not computed | |

Madeira | generation—hydro | 0.3846 | 0.3846 | 0.3855 | 0.6025 |

generation—wind | 0.6862 | 0.6994 | 0.6880 | 0.8171 | |

generation—waste | 0.4524 | 0.4534 | 0.4567 | 0.4852 | |

generation—solar | 0.3371 | 0.3440 | 0.3396 | 1.3826 | |

demand | 0.1674 | 0.2781 | 0.1846 | 1.3447 |

**Table 4.**Unstandardized values of the optimization KPIs for standardized values equal to 1. Regarding Madeira cost, the unstandardized KPI is proportional to the standardized one. The unstandardized renewables and power stability KPIs are proportional to the square of the respective standardized ones.

Demonstrator | Cost | Renewables | Power |
---|---|---|---|

Orkney | not considered | 12.9285 MW | 3.9921 MW |

Samsø | not considered | 16.8783 KW | 9.1264 KW |

Madeira | 266,180/day | 73.5026 MW | 17.7684 MW |

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## Share and Cite

**MDPI and ACS Style**

Kolokas, N.; Ioannidis, D.; Tzovaras, D.
Multi-Step Energy Demand and Generation Forecasting with Confidence Used for Specification-Free Aggregate Demand Optimization. *Energies* **2021**, *14*, 3162.
https://doi.org/10.3390/en14113162

**AMA Style**

Kolokas N, Ioannidis D, Tzovaras D.
Multi-Step Energy Demand and Generation Forecasting with Confidence Used for Specification-Free Aggregate Demand Optimization. *Energies*. 2021; 14(11):3162.
https://doi.org/10.3390/en14113162

**Chicago/Turabian Style**

Kolokas, Nikolaos, Dimosthenis Ioannidis, and Dimitrios Tzovaras.
2021. "Multi-Step Energy Demand and Generation Forecasting with Confidence Used for Specification-Free Aggregate Demand Optimization" *Energies* 14, no. 11: 3162.
https://doi.org/10.3390/en14113162