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On the Concept of an Optimal Portfolio of Uncertain Flexible Loads

Julie Rousseau, Philipp Heer, Kristina Orehounig, Gabriela Hug

arXiv:2609.05176v1eess.SY

TL;DR

The paper addresses reserve quantification for energy-constrained, uncertain flexible loads under Danish reliability requirements and the resulting joint chance-constrained problem. It derives analytical reformulations and studies how adding loads with different stochastic behavior changes portfolio reserves. The reformulations closely match the exact formulation, while case studies identify optimal groupings and portfolios with highest marginal load value.

  • Problem

    Energy-constrained flexible loads have limited capacity and uncertain availability, making reliable aggregate reserve quantification a difficult joint chance-constrained problem.

  • Method

    The paper derives analytical reserve reformulations and theoretically analyzes the marginal reserve increase from adding a load to an existing portfolio.

  • Results

    Case studies demonstrate optimal load groupings and predict which portfolio gives an added load the highest marginal reserve value.

  • Takeaways & Limitations

    Grouping loads according to their stochastic behavior can change total reserves, and marginal-value analysis can identify beneficial portfolio assignments.

  • Takeaways & Limitations

    The theoretical framework assumes independent uncertainty sources and requires extension to correlated uncertainties and larger-scale case studies.

Abstract

from arXiv · show

Flexible loads can enhance power system stability by providing reserves, but their limited energy capacity and uncertain availability distinguish them from conventional generators. To accommodate these characteristics, the Danish Transmission System Operator (TSO) recently introduced new reserve market rules that incorporate energy constraints and relax reliability requirements. In this context, the optimal reserve quantification becomes a joint chance-constrained reserve quantification problem, which is difficult to solve. In this paper, we derive two analytical reformulations of this problem: an exact one when a reserve direction dominates and an approximate one, otherwise. Furthermore, when flexible loads must collectively satisfy a reliability requirement, we introduce the concept of an optimal portfolio of flexible loads: adding loads with similar expected values but different stochastic behaviors to an existing portfolio may change the total portfolio's reserves. To support this idea, we theoretically study the marginal increase in reserves resulting from adding a load. Numerical results show that our analytical reformulations closely match the exact formulation, with a mean absolute error of 2.5%. Case studies further demonstrate the existence of optimal load groupings and our ability to predict the portfolio in which a load's marginal value is highest, leveraging our theoretical analysis.

I. INTRODUCTION

Flexible loads can support grids with increasing renewable generation, but their energy limits and uncertain availability complicate reserve provision. The paper addresses this challenge by analytically quantifying portfolio reserves and studying how stochastic load groupings affect reserve capacity.

  • Flexible consumers can adapt electricity use for advance scheduling or real-time ancillary services as renewable generation replaces dispatchable plants.
  • Flexible loads are constrained in both power and energy, while their future flexibility depends on uncertain departures, weather, occupancy, and thermal-model accuracy.
  • Active energy management can offset prior activations, avoiding the conservatism of assuming full-power delivery throughout long reserve durations.
  • Danish reserve rules specify energy and opposite-direction power requirements and allow uncertain loads to participate with reduced reliability requirements.
  • The paper proposes analytical reserve quantification, studies optimal load groupings, and develops theory for the marginal reserve increase from adding a load.

1) EVs:

The paper models EV and HVAC flexibility through time-varying power and energy bounds. These bounds reflect technical constraints and uncertainty in vehicle availability, weather, occupancy, and thermal behavior.

  • 1) EVs:: EV flexibility is bounded by charging power and energy trajectories while guaranteeing each vehicle’s requested departure state of charge.
  • 1) EVs:: EV future flexibility is uncertain because departure times make its time-varying power and energy bounds stochastic.
  • 1) EVs:: HVAC consumption is modeled with a discretized first-order resistance-capacitance model linking power use to indoor temperature.
  • 1) EVs:: HVAC operation must keep indoor temperature within an acceptable comfort range while respecting power and energy bounds.
  • 1) EVs:: HVAC future energy and power bounds become stochastic because weather and other factors make future flexibility uncertain.

B. The Danish FCR-D Market

The Danish FCR-D market enables uncertain, energy-constrained loads to provide hourly asymmetric reserves under explicit energy and power requirements. The paper formulates the resulting revenue-maximization problem using load flexibility bounds.

  • B. The Danish FCR-D Market: FCR-D is an emergency reserve supporting regular FCR when frequency deviates more than 0.1 Hz, and the methodology is designed to comply with its requirements.
  • B. The Danish FCR-D Market: For reserves in one direction, providers must deliver full power for one third of an hour and reserve 20% of that power in the opposite direction.
  • B. The Danish FCR-D Market: The market optimization maximizes hourly reserve revenues using time-varying flexibility bounds, baseline consumption, and reserve prices.
  • B. The Danish FCR-D Market: For each load, the feasible region defined by constraints (7b)-(7e) is convex with at most six corners, so the subproblem can be solved by evaluating those corners.

2) Reliability Requirement:

The reliability requirement imposes a joint chance constraint on aggregate reserves, allowing other loads to compensate when one load underdelivers. The paper derives quantile-based reserve calculations and uses scenario methods when reserve directions are coupled.

  • 2) Reliability Requirement:: Danish rules allow uncertain flexible loads to offer reserves that are available with 90% reliability, regardless of activation.
  • 2) Reliability Requirement:: The formulation targets robust total reserves, allowing other loads to compensate when one load fails and thereby reducing conservativeness.
  • 2) Reliability Requirement:: When one reserve direction is clearly more valuable, the opposite direction is fixed at zero before solving the reliability-constrained reserve problem.
  • 2) Reliability Requirement:: The optimal upward reserve can be obtained as the (1 − R) quantile of the relevant aggregate stochastic reserve variable and evaluated using scenarios.
  • 2) Reliability Requirement:: When reserve prices are comparable, reserve directions are coupled through the joint chance constraint, so marginal quantiles are not generally optimal.
  • 2) Reliability Requirement:: The scenario-based joint-quantile method is sub-optimal because it optimizes each scenario before evaluating the joint quantile.

III. ON THE CONCEPT OF AN OPTIMAL PORTFOLIO

This section defines optimal portfolios as groupings of flexible loads designed to maximize total reserve capacity. Under reliability requirements, different groupings of the same loads can produce different total reserves.

  • Optimal portfolios group flexible loads to maximize their total reserve capabilities.
  • Without reliability requirements, grouping does not change total reserves because portfolio reserves equal the sum of individual reserves.
  • With reliability requirements, some groupings yield higher total reserves than others.

A. Portfolio’s Value Increase After Adding New Loads

The section analyzes how adding a flexible load changes a portfolio’s reserve quantile. A Taylor-von Mises expansion shows that the first-order effect depends on expected flexibility, while the second-order effect depends on variance and the portfolio’s log-density derivative.

  • The reserve increase from adding a small load is modeled as a quantile change in the perturbed portfolio distribution.
  • The Taylor-von Mises expansion evaluates how the portfolio’s quantile evolves around its initial distribution.
  • The first-order reserve increase equals the new load’s expected flexibility, while the second-order term depends on its variance and the initial portfolio’s log-density derivative.
  • A portfolio owner should favor loads with high expected flexibility and low variance when αX is positive.
  • A load’s first-order contribution is identical across portfolios, whereas its second-order contribution is larger in the portfolio with the smallest α factor.

IV. CASE STUDY

The case study evaluates EV and HVAC flexibility data, validates the analytical reformulations against established methods, and examines reserve differences across portfolio groupings. The experiments assess both computational accuracy and the existence of time-varying optimal portfolios.

  • The case study uses EV charging-event data and a dataset of 300 single-family houses to model uncertain flexible loads.
  • Fig. 2 compares one-day reserves for EV portfolios and HVAC portfolios with wide or tight indoor temperature comfort ranges.
  • The analytical reserve formulation is benchmarked against the ALSO-X method for portfolios of EVs under different reserve-price conditions.
  • 0.0195 kW mean absolute error and 0.063 kW maximum absolute error are reported when validating the first- and second-order terms of the quantile expansion.
  • Fig. 5 evaluates reserve totals from pairing two EV groups with one HVAC group across different portfolio configurations.
  • Fig. 6 evaluates pairings of one EV group with two HVAC groups, whose optimal configuration varies over time.

D. Optimal Portfolio of Flexible Loads

Under reliability requirements, grouping flexible loads can change total reserves, creating optimal portfolios. The paper analyzes portfolio composition, marginal load value, and boundaries on its conclusions.

  • Case studies: Pairing two EV groups while keeping the HVAC group separate produced the strongest configuration in the illustrated three-group case.The HVAC group’s reliability balanced EV uncertainty when paired with one EV group, but pairing both EV groups improved their combined reliability.
  • Case studies: The optimal configuration can vary over time as EV and HVAC uncertainty profiles change.At the worst hour, the best single-portfolio configuration offered 6% more reserves than the worst alternative pairing.
  • Marginal value: Fig. 7 evaluates the marginal reserve increase from adding EV or HVAC groups to different portfolios.EVs gained most from EV-only portfolios, while HVAC preferences varied by time; the proposed criterion uses the smallest product of log-density derivative and variance.
  • Scope: The paper does not prove that a single portfolio is generally superior, especially when loads are strongly correlated.Multiple portfolios may instead arise from contractual or operational constraints, with the single-portfolio result serving as a practical benchmark.
  • Contributions: The framework supports analytical reserve quantification, optimal load grouping, and identification of portfolios where individual loads have highest marginal value.The contribution is based on a quantile formulation that is exact in some cases and has small approximation errors in most practical settings.
  • Illustrative example: For four illustrative loads split into two portfolios, homogeneous grouping yielded more reliable upward reserves than heterogeneous grouping at a 70% reliability level.Loads C and D could provide more than 1.5 kW with 70% confidence, whereas heterogeneous groups each provided only 1 kW.

C. Von Mises-Taylor Expansion of the Quantile Function

The paper treats statistical quantities such as quantiles as functions of cumulative distributions. This perspective provides the conceptual basis for applying distributional Taylor expansions.

  • Statistical functions: Von Mises theory analyzes statistical functions through Taylor-like expansions in the space of probability distributions.This establishes the mathematical perspective needed to study how quantiles change when the underlying distribution is perturbed.
  • Statistical functions: The cumulative distribution function FX and a fixed reliability R define the statistical-function framework used for quantile analysis.The quantile function is presented as a statistical function of a stochastic cumulative distribution.
  • Statistical functions: The operator T is evaluated on a cumulative distribution and at a fixed reliability level.The framework connects the distribution FX with the quantile-related statistical function T.

2) Von Mises-Taylor Expansion:

The Von Mises-Taylor expansion extends ordinary Taylor analysis to statistical functions whose inputs are probability distributions. Its derivatives are directional and can include a second-order remainder.

  • Distributional expansion: Von Mises theory provides a Taylor-equivalent expansion for statistical functions in the space of probability distributions.The approach treats statistical functions as functions of cumulative distributions rather than ordinary finite-dimensional inputs.
  • Distributional expansion: The Von Mises derivative of T is defined through a function ϕ that describes sensitivity to a distributional perturbation.The derivative is taken relative to a change from FX toward another distribution FZ.
  • Distributional expansion: The derivative is directional because it depends on the chosen direction FZ.The notation T′(FX; FZ) records both the initial distribution and the perturbation direction.
  • Distributional expansion: The interpolated distribution FX,t=(1−t)FX+tFZ represents movement from the initial distribution toward the perturbing distribution.This path supplies the distributional analogue of a scalar Taylor perturbation.
  • Distributional expansion: Under twice-differentiability, the expansion includes first- and second-order terms plus a remainder.The second-order formulation requires the relevant derivatives to exist and leaves a remainder term for higher-order effects.

3) Application to the Quantile Function:

The paper applies distributional Taylor analysis to the quantile of a portfolio after adding a small random load. The resulting expansion links marginal reserve changes to the added load’s moments and the initial distribution’s density near the quantile.

  • Quantile application: The quantile function is differentiated by using the relationship between the perturbed distribution FZ and the initial distribution FX.The derivation applies the chain rule to the quantile-functional identity and obtains first- and second-order derivatives.
  • Quantile application: The notation qX is simplified to denote qX(1−R) in the derived quantile formulas.The fixed reliability level determines the quantile at which the expansion is evaluated.
  • Quantile application: The expansion is evaluated at the initial portfolio quantile qX to approximate the quantile of Z=X+εY.Here X is the existing portfolio and εY is the added load, treated as a small distributional perturbation.
  • Quantile application: The second-order expression contains terms involving E(Y)^2, Var(Y), and the density and density derivative evaluated at qX.These terms capture how the added load’s expected flexibility and variance interact with the initial portfolio distribution.
  • Quantile application: The derivation retains a second-order expansion and requires the remainder to be o(ε^2).The paper states that proving this behavior involves showing higher-order directional derivatives exist with order ε^k, while omitting the detailed proof.

4) Verification in the Gaussian Case:

The Gaussian case provides an explicit quantile formulation for verifying the proposed quantile expansion. The resulting quantile changes confirm the expansion's terms when a small independent Gaussian load is added.

  • 4) Verification in the Gaussian Case:: The verification assumes a Gaussian variable X and uses an explicit formulation of its (1−R)-quantile.The quantile is related to the corresponding quantile of the centered standard Gaussian.
  • 4) Verification in the Gaussian Case:: For independent Gaussian variables X and Y, the sum Z = X + εY is Gaussian, enabling an explicit expression for its (1−R)-quantile.The formulation uses the mean and variance of X and the perturbing load εY.
  • 4) Verification in the Gaussian Case:: The change in the (1−R)-quantile from adding εY to X is derived and expanded through second order.This directly evaluates the perturbation considered in the proposed quantile expansion.
  • 4) Verification in the Gaussian Case:: The Gaussian derivation identifies αX in the quantile expansion and confirms its value in this special case.The confirmation connects the general expansion to the explicit Gaussian calculation.
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