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Efficient Quantile-Resolved Hosting Capacity Assessment on Nodal Level for Low-Voltage Grids

Maximilian Köhler, Edwin Mora, Mathias Duckheim, Stefan Niessen

arXiv:2609.09060v1eess.SYcs.SY

TL;DR

Quantile-resolved hosting capacity supports risk-aware planning, but scenario-based calculations are computationally demanding for periodic or large-scale assessment. The paper introduces pHCA, which models load uncertainty and uses an AC-OPF formulation; it reports runtimes falling from minutes or hours to seconds while achieving comparable accuracy to Monte Carlo sampling.

  • Problem

    Monte Carlo methods construct quantile-resolved HC distributions by solving an OPF for each scenario, creating a computational challenge for periodic calculations and large-scale planning.

    Source: §1 · p. 2

  • Method

    pHCA represents load samples as a multivariate normal distribution propagated through a linearized power-flow model and uses a single AC-OPF formulation incorporating uncertainty and correlations.

    Source: §1 · p. 2, §2.1 · p. 3

  • Results

    11 min to 2 s for the 10-bus network, and an estimated 38 h to 50 s for the realistic LV feeder, with pHCA reported to achieve accuracy comparable to Monte Carlo sampling.

    Source: §1 · p. 2, §4 · p. 8

  • Takeaways & Limitations

    On realistic LV grids, pHCA is presented as suitable for fast calculations when load hypotheses are well represented by normal distributions.

    Source: §2.1 · p. 3, §4 · p. 8

    Distribution-tail differences may arise because normal fitting cannot capture the original load tails; linearization errors and omitted covariance between the load hypothesis and added DER capacity can also affect results.

    Source: §2.1 · p. 3, §4 · p. 7

Abstract

from arXiv · show

Hosting capacity - the maximum additional capacity a network can accommodate without violating operational limits - is a key metric in distribution system planning and operation. Decisions on grid reinforcements and the deployment of flexibility management require not only the worst-case HC but also an understanding of the distribution of HC under different likelihoods in load and generation patterns. Quantile-resolved HC distributions provide this view by expressing HC as a function of an acceptable operational limit exceedance likelihood. Monte Carlo sampling is the established approach for computing such distributions but demands large computational resources. Approximations sacrifice either accuracy, the ability to capture uncertainty correlations, or scalability when assessing real-world networks. This paper introduces a computationally efficient method for calculating distributions for quantile-resolved HC. It uses a representation of load samples as multivariate normal distribution, propagated through a linearized power flow model. This allows for leveraging a re-parametrized AC-OPF problem for each hosting capacity quantile. Benchmarking against Monte Carlo-based methods on realistic LV networks demonstrates that the proposed method achieves comparable accuracy with a mean deviation of approx. 3%, while reducing computational time by orders of magnitude. For the exemplary networks the computational time decreases from 11 min to 2 s, and 38 h to 50 s, respectively. The method's scalability is also suitable for recalculation in 15-minute cycles encountered in DSO practice for e.g., real-time grid management.

1 Introduction

Risk-aware hosting-capacity assessment requires distributions across load and generation conditions, but scenario-based computation is costly. The paper introduces pHCA to calculate quantile-resolved distributions while incorporating uncertainty and correlations.

1 Introduction

  • Traditional worst-case planning assumes simultaneous peak loading across all DERs and ensures operational safety but leads to systematic over-investment; scenario-based planning accounting for load and generation diversity can reduce grid reinforcement costs by up to 45%.Source: §1 · p. 1
  • Monte Carlo methods construct HC distributions by solving an OPF for each scenario, making repeated calculations and large-scale planning computationally demanding.Quantile-resolved HC supports decisions at different risk levels, such as the 95th or 99th percentile.Source: §1 · p. 2
  • pHCA incorporates load and generation uncertainties and correlations in a single AC-OPF formulation to determine the complete HC distribution with accuracy comparable to Monte Carlo sampling.The introduction contrasts this approach with approximations that may incur substantial errors or fail to preserve correlations, and robust optimization that does not provide quantile-resolved distributions.Source: §1 · p. 2

2 Problem Formulation and Solution Approaches

The section formulates hosting-capacity assessment around nodal power samples, network states, and operational constraints. It sets out sample-based assessment and introduces the pHCA procedure.

2 Problem Formulation and Solution Approaches

  • The section covers grid representation, benchmark and pHCA formulations, and metrics for comparing HC distributions.These are the four components identified in the section overview.Source: §2 · p. 2

2.1 Grid Modeling

  • The network is represented by node and edge sets, with loads and generators assigned to the network.Nodes and edges correspond to busbars and branch elements, respectively.Source: §2.1 · p. 2
  • The reference load hypothesis uses selected load samples, treats generators as negative loads, and maps element powers to nodal loads using a load-node incidence matrix.The matrix C represents the incidence of load and generator elements at nodes; multiple DERs may share a node, and some nodes may have none.Source: §2.1 · p. 2
  • Load samples may come from historical data or forecasts, or be represented by a normal distribution; a linearized power-flow model maps nodal powers to voltage and current states.The state variables are voltage magnitudes at non-slack nodes and current magnitudes.Source: §2.1 · p. 3

2.2 Hosting Capacity Analysis as Optimization Problem

  • For each node and sample, HCA maximizes additional load subject to voltage and current feasibility constraints.The stated voltage limits are 0.9–1.1 p.u.; current limits follow the maximum branch currents in the network data.Source: §2.2 · p. 3

2.2.1 Sample-based Hosting Capacity Assessment (sHCA)

  • sHCA treats the sample-wise hosting-capacity values as a distribution and obtains quantiles from its inverse CDF, but requires |T| OPFs per node.A 5% quantile of 10 kW means that 10 kW can be connected without exceeding operational limits in 95% of cases.Source: §2.2.1 · pp. 3–4

2.2.2 Probabilistic Hosting Capacity Assessment (pHCA)

  • The next subsection introduces pHCA as a sequence of steps for quantile-resolved hosting-capacity assessment.The supplied passage announces the procedure but does not specify its steps.Source: §2.2.2 · p. 4

4 Determine tightened constraints similar to Eq. (1),

The pHCA derives tightened constraints from load and state distributions, then optimizes expected added nodal power in a reparameterized OPF. The section also defines distribution-comparison metrics, including a quantile error scaled to the sHCA distribution's spread.

4 Determine tightened constraints similar to Eq. (1),

  • The method propagates expected nodal powers and covariances through a linearized power-flow model to obtain voltage and current distributions used to reformulate constraints by quantile.The resulting voltage and current amplitude differences are calculated with the quantile function.Source: §4 · p. 4
  • The pHCA optimizes expected additional nodal power under tightened grid constraints, while retaining full AC power-flow equations for the optimization.Linearization is used only to calculate state distributions and reformulate constraints.Source: §4 · p. 4
  • The probabilistic approach reduces the required OPF calculations for a network to |N′|.Source: §4 · p. 4

2.3 Evaluation Metrics

  • MRE_n measures expected-value bias, while qRE_n,x measures quantile discrepancy relative to the sHCA distribution's standard deviation.The scaling makes qRE reflect how HC error relates to likelihood sensitivity: equal HC errors imply greater likelihood sensitivity for narrower distributions.Source: §2.3 · p. 5

3 Results

The results section evaluates the approaches on a modified 10-bus network and describes the data and computational setup used for the comparisons.

3 Results

  • The evaluations use consumption and generation profiles selected from one year of real, time-resolved measurements sampled at 15-minute intervals.Source: §3 · p. 5
  • Computational effort is measured on an Apple M3 Pro MacBook Pro, and the authors note that the Julia sHCA implementation may favor sHCA over the Python pHCA implementation.Source: §3 · p. 5

3.1 10-bus Distribution System

  • The 10-bus test network has four loads, 10 nodes, one transformer, and eight lines, with selected line lengths modified to distinguish load-specific HC behavior.The modified lengths are l4 = 0.1 km, l5 = 1.0 km, l6 = 0.05 km, and l7 = 0.3 km.Source: §3.1 · p. 5

1. Normalized Sampling

For normalized-sampling validation, the load profiles are represented by a fitted multivariate normal distribution and used in both probabilistic optimization and sample-based assessment.

1. Normalized Sampling

  • The load profiles are fitted with a multivariate normal distribution for validation.Source: §1 · p. 6
  • The fitted distribution's expected values and covariances provide inputs to the probabilistic optimization approach.Source: §1 · p. 6
  • Samples drawn from the fitted normal distributions are used as inputs to the sHCA.Source: §1 · p. 6

2. Direct Sampling

The direct-sampling section compares pHCA with sample-based results and reports a capacity-risk point for the analyzed bus. It also describes application to a realistic German LV feeder.

2. Direct Sampling

  • Approximately 50 kW can be added at the analyzed bus at a 20% probability of exceeding operational limits.Source: Figure 1 · p. 6, §2 · p. 6
  • Figure 1 compares pHCA with sHCA results using direct and normalized sampling for the node connected to load 1.The text identifies purple as sHCA, blue as pHCA, and red as direct-sampling results.Source: Figure 1 · p. 6, §2 · p. 6

3.2 Demonstration on a Realistic LV Feeder

  • The realistic German DSO feeder has 105 nodes, 38 loads, 102 lines, and one transformer.Source: §3.2 · p. 6

4 Discussion

pHCA closely matches sHCA across nodes and quantiles, with reported errors on both test networks and much shorter computation times. Differences are most pronounced in distribution tails, where the authors identify several possible error sources.

4 Discussion

  • The pHCA CDFs closely matched sHCA across nodes and quantiles, although slight shifts and different distribution widths appeared, especially in the tails.Figure 3 presents the 5% HC quantile CDFs for all nodes, with selected nodes compared across the three approaches.Source: §4 · pp. 6–7, Figure 3 · p. 8
  • Possible sources of tail errors include multivariate-normal fitting, power-flow linearization, and omitted covariances between the load hypothesis and added DER capacity.The authors say these sources influence the tails especially.Source: §4 · p. 7
  • For the 10-bus grid, MRE was below or around 0.3%; on the realistic LV feeder, MRE did not exceed 3.5% and qRE was below 0.8 for almost all nodes.The 10-bus qRE was below 0.6 except for about 0.8 in the direct-sampled comparison at node 7.Source: §4 · pp. 7–8
  • 11 min versus 2 s for the 10-bus grid, and an estimated 38 h versus 50 s for the realistic LV feeder, were the sHCA and pHCA runtimes, respectively.The 10-bus calculation used around 35k time steps; the feeder comparison used the same sample setup.Source: §4 · p. 8
  • The authors consider pHCA promising for large networks and fast applications, with 3–4% error on realistic LV grids when the load hypothesis is well represented by normal distributions.They also identify integration of grid-parameter, measurement-error, and forecast uncertainties as an application.Source: §4 · p. 8

5 Summary and Outlook

The pHCA closely matches sHCA across nodes and quantiles while significantly reducing computational effort. Future work targets operational planning and flexibility management, broader load-distribution modeling, and flexibility representation in hosting-capacity assessment.

5 Summary and Outlook

  • The pHCA closely matches sHCA for all nodes and quantiles while significantly reducing computational effort.The authors describe it as promising for large networks, large sample counts, and applications requiring repeated fast calculations.Source: §5 · p. 8
  • Future work will extend pHCA to proactive operational planning for congestion reduction and explore integration with curative LV grid or flexibility management.Source: §5 · p. 8
  • The method may be extended to Gaussian Mixture Models to better represent original load behavior and potentially improve accuracy over multivariate-normal fitting.Source: §5 · pp. 8–9
  • Further development will focus on adequately representing flexibility in hosting-capacity methods, which can benefit DER integration and management and support grid-connection assessments.Source: §5 · p. 9
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