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Extreme value distributions of peak loads for non-residential customer segments

Shaohong Shi, Eric A. Cator, Jacco Heres, Simon H. Tindemans

TL;DR

The paper addresses peak-load prediction for large non-residential customers by integrating extreme value theory with Velander's framework to model the distribution of peak loads. It develops an EVT-based peak-load distribution that reduces the qVF parameterization to four parameters: $\vartheta_0$, $\vartheta_1 a_K$, $\vartheta_1 b_K$, and $\gamma$, and demonstrates how this aligns with the qVF across quantiles. Via MQR and MLE on Liander data from three customer categories, the Fréchet class ($\gamma>0$) is found to best capture tail behavior, though estimates of $\gamma$ are near zero with small uncertainty. The resulting model offers a theoretically grounded, compact approach to peak-load forecasting with practical value for DSOs in planning and congestion management.

Abstract

Electrical grid congestion is a growing challenge in Europe, driving the need for accurate prediction of load, particularly of peak load. Non-time-resolved models of peak load offer the advantages of simplicity and compactness, and among them, Velander's formula (VF) is a traditional method that has been used for decades. Moreover, VF can be adapted into a quantile VF, which learns a truncated cumulative distribution function of peak load based on electricity consumption. This paper proposes a mathematical model based on extreme value theory to characterize the probability distribution of peak load for large non-residential customers. The model underpins the quantile VF as demonstrated through multiple quantile regression and reduces its representation to just four parameters without sacrificing predictive performance. Moreover, using maximum likelihood estimation and the likelihood ratio test, we validate that the probability distribution of peak load of analysed groups belongs to the heavy-tailed Fréchet class.

Extreme value distributions of peak loads for non-residential customer segments

TL;DR

The paper addresses peak-load prediction for large non-residential customers by integrating extreme value theory with Velander's framework to model the distribution of peak loads. It develops an EVT-based peak-load distribution that reduces the qVF parameterization to four parameters: , , , and , and demonstrates how this aligns with the qVF across quantiles. Via MQR and MLE on Liander data from three customer categories, the Fréchet class () is found to best capture tail behavior, though estimates of are near zero with small uncertainty. The resulting model offers a theoretically grounded, compact approach to peak-load forecasting with practical value for DSOs in planning and congestion management.

Abstract

Electrical grid congestion is a growing challenge in Europe, driving the need for accurate prediction of load, particularly of peak load. Non-time-resolved models of peak load offer the advantages of simplicity and compactness, and among them, Velander's formula (VF) is a traditional method that has been used for decades. Moreover, VF can be adapted into a quantile VF, which learns a truncated cumulative distribution function of peak load based on electricity consumption. This paper proposes a mathematical model based on extreme value theory to characterize the probability distribution of peak load for large non-residential customers. The model underpins the quantile VF as demonstrated through multiple quantile regression and reduces its representation to just four parameters without sacrificing predictive performance. Moreover, using maximum likelihood estimation and the likelihood ratio test, we validate that the probability distribution of peak load of analysed groups belongs to the heavy-tailed Fréchet class.
Paper Structure (12 sections, 2 theorems, 33 equations, 2 figures, 4 tables)

This paper contains 12 sections, 2 theorems, 33 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Extreme value distribution model of peak load
  3. Problem statement
  4. Model formulation
  5. Relation to the quantile Velander formula
  6. Parameter estimation methodology
  7. Multiple quantile regression
  8. Maximum likelihood estimation
  9. Data, results and analysis
  10. Parametric representation
  11. Gumbel or Fréchet
  12. Conclusion

Key Result

Theorem 1

Let $F$ be a distribution in $\mathcal{D}(G_\gamma)$ for a $\gamma \in \mathbb{R}$, and let $\vartheta_0$ and $\vartheta_1$ be two positive constants. Let $\left\{\, E_i \;\middle|\; i \in C \,\right\}$ be a set of non-negative numbers indexed by a set $C$. For $i \in C$, let $(X_i(t))_{t \in \mathb For $i \in C$ and $n \in \mathbb{N}_{>0}$, let $G^i_n(y)$ be the CDF of $X^i_n$, where Then there

Figures (2)

  • Figure 1: SBI code 8411 in 2022: peak loads and electricity consumption of customers alongside curves of quantile functions at different quantile levels, for the MLE-fitted Fréchet model.
  • Figure 2: Comparison of $\beta_\tau$ from the qVF under C4, from the Gumbel formulation and from the Fréchet formulation.

Theorems & Definitions (5)

  • Theorem 1
  • proof : Proof sketch
  • Theorem 2
  • proof : Proof sketch
  • Definition : EVD peak load model