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Hierarchical Agglomerative Clustering for Efficient Annual Voltage Security Assessment in Very-High RES Penetrated Power Systems

Rock Agon, Robin Preece, Jovica V. Milanovic

arXiv:2608.28296v1eess.SY

TL;DR

The paper addresses the difficulty of reducing costly full-year voltage-security simulations when injection-profile similarity does not reliably reflect voltage behavior. It clusters operating points using AC-derived voltage-risk responses, PCA, and Ward-linkage HAC, then evaluates normal and contingency reproduction. On the IEEE Voltage Test System, the approach compresses the annual set by 99.66% while achieving 98.3% steady-state and 93.4% post-contingency reconstruction accuracy.

  • Problem

    Full-year simulations are computationally costly, while demand and generation profile similarity may fail to preserve voltage-security characteristics under high RES penetration.

  • Method

    The framework represents operating points with system-wide voltage-risk indices from AC power-flow responses, reduces dimensionality with PCA, and selects regimes using Ward-linkage HAC.

  • Results

    99.66% annual operating-point reduction preserves voltage behavior with 98.3% steady-state and 93.4% post-contingency accuracy.

  • Takeaways & Limitations

    Voltage-response clustering remains accurate and reliable under disturbances while providing an efficient and physically meaningful representation of voltage regimes.

Abstract

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Voltage security assessment in power systems with high renewable energy source (RES) penetration requires analyzing many operating conditions to capture variability and uncertainty, but simulating a full year of operating points is computationally costly - motivating the selection of representative operating points (ROPs). Most existing methods cluster demand and generation profiles, but similarity in these profiles does not guarantee similarity in voltage behavior, since reactive power limits, voltage-control actions, and nonlinear network interactions shape voltage response in ways that cannot be inferred from power profile patterns. This paper proposes an unsupervised learning framework that selects ROPs based on the system's actual voltage response: each operating point is represented by system-wide voltage-risk indices from AC power-flow solutions, Principal Component Analysis reduces dimensionality, and Hierarchical Agglomerative Clustering with Ward linkage identifies representative voltage regimes. A comprehensive set of evaluation criteria then measures how well the selected ROPs reproduce the full year's voltage-security characteristics under normal and contingency conditions. On the IEEE Voltage Test System under very high RES penetration, the framework reduces the annual operating point set by 99.66 percent while reproducing full-year voltage behavior with 98.3 percent reconstruction accuracy in steady state and 93.4 percent in post-contingency response, outperforming existing injection-space clustering and heuristic sampling.

I. INTRODUCTION

Very-high RES penetration makes annual voltage-security assessment computationally demanding and weakens injection-profile similarity as a proxy for voltage behavior. The paper therefore selects representative operating points in voltage-response space and evaluates their coverage of normal and contingency behavior.

  • Full-year probabilistic and sequential AC simulations capture stochastic operating conditions but are computationally costly, motivating representative operating point selection.
  • High RES penetration expands operating conditions beyond a few seasonal patterns, limiting heuristic typical-day selection.
  • Similar net-injection profiles can produce different voltage distributions, so injection-space clustering may fail to preserve voltage-security characteristics.
  • 99.66% scenario-set compression preserves voltage regimes with 98.3% steady-state accuracy and 93.4% contingency-analysis accuracy.
  • The evaluation includes statistical, quantile, distribution-shape, extreme-tail, and post-contingency criteria for representative operating points.
  • The proposed framework clusters operating points using actual AC voltage responses and physics-guided voltage-risk indicators rather than demand and generation profiles.

B. Stage II: Voltage-response Space: Feature Representation

The framework represents each operating point with seven system-wide voltage-risk indices that summarize voltage behavior across the network, including limits, dispersion, violations, and reactive capability.

  • Seven system-wide voltage-risk indices summarize the main characteristics of each operating point’s voltage profile.System-wide indicators are used instead of analyzing individual bus voltages separately.
  • Voltage bounds use V_low = 0.95 and V_high = 1.05 p.u. to identify undervoltage and overvoltage conditions.The indices are defined from bus voltages relative to these lower and upper limits.
  • Voltage spread is measured across the system to quantify how widely bus voltages vary.
  • Voltage violation indices quantify violation magnitude, RMS violation severity, and the proportion of buses experiencing violations.These measures capture both the severity and extent of voltage-limit violations.

5) Reactive power reserves:

Reactive-power-reserve information is included among the voltage-response features, after which correlation analysis and PCA produce a compact, decorrelated representation for clustering.

  • 5) Reactive power reserves:: The reactive-power-reserve index quantifies the remaining reactive capability of generators relative to their reactive-power limits.It uses generator reactive output and the corresponding reactive-power limits.
  • Correlations among voltage-risk indices can create redundancy and distort Euclidean distances used for clustering.Strongly correlated indices may carry overlapping information and receive excessive weight in distance calculations.
  • PCA converts correlated voltage-response variables into orthogonal components while preserving dominant variance.The transformation provides a linear, compact feature space for clustering.
  • PCA is applied to the centered dataset of T operating points and m indices, projecting observations onto the first k principal components.The retained components are associated with the largest covariance-matrix eigenvalues.
  • At least 99% of the original feature-space variance is preserved while reducing variable redundancy.This lower-dimensional, decorrelated representation is intended to improve clustering robustness.

D. Stage IV: Clustering and ROP Selection

The framework clusters operating points in PCA voltage-response space using hierarchical agglomeration and selects one representative operating point from each voltage-behavior regime.

  • Each cluster groups operating points with comparable voltage-security characteristics, and one representative operating point is selected per cluster.
  • HAC begins with singleton operating-point clusters and repeatedly merges the pair with the smallest inter-cluster distance.The process produces a complete bottom-up merge hierarchy.
  • The hierarchy exposes nested voltage operating regimes at different resolution levels without requiring the cluster count beforehand.
  • Ward linkage is selected because it minimizes the increase in within-cluster variance and forms compact, statistically coherent clusters.It accounts for the full membership of each cluster when evaluating merges.
  • Ward merging greedily minimizes total within-cluster sum of squares, encouraging low-variance and coherent voltage regimes.

2) Representative Operating Point Selection:

The framework evaluates whether cluster representatives preserve full-year voltage-security behavior across distributions, geometry, extreme tails, and reconstruction accuracy.

  • Representative selection: Each cluster's representative operating point is the observation nearest its centroid in PCA space.This selection is intended to represent the cluster's average voltage-response characteristics.
  • Evaluation framework: The weighted ROP distribution repeats each representative according to its cluster size, enabling direct comparison with the full dataset.The resulting weighted distribution has the same number of operating points as the full dataset.
  • Moment and quantile preservation: Moment and quantile preservation compares mean, standard deviation, and 95% VaR for every voltage-risk index, normalized by each index's interquartile range.Normalization places errors from indices with different units and scales onto a common bounded scale.
  • Distributional geometry: Distributional geometry compares full and weighted-ROP empirical CDFs using Kolmogorov–Smirnov and first-order Wasserstein distances.The KS distance measures the largest vertical gap, while Wasserstein distance measures the average horizontal gap; Wasserstein values use the same interquantile normalization.
  • Extreme-tail coverage: Tail-risk evaluation uses CVaR magnitude accuracy and Tail Coverage Ratio to assess whether severe RMS VVS conditions are preserved and represented.The overall tail-risk accuracy averages these two indicators, while reconstruction accuracy combines moment/quantile, geometry, and tail-risk components with equal weight.

B. Post-Contingency Performance Evaluation

Post-contingency evaluation applies the most critical line contingency and compares representative-point voltage responses with full-year simulations using VSPI.

  • Post-contingency performance evaluation: The most critical line contingency is applied to evaluate the framework under disturbances.The resulting voltage responses are compared against those from full-year simulations.
  • Post-contingency performance evaluation: VSPI aggregates voltage violations across all buses to capture both voltage-deviation severity and their distribution.The index is derived from the commonly used Performance Index.
  • Comparative analysis: The comparative analysis evaluates voltage-response-space ROPs against injection-space clustering and heuristic sampling.These alternatives include the industry-practice heuristic approach.

A. Injection Space: Feature Representation

The injection-space baseline represents each operating hour through net active-power injections at buses containing loads or generation, then applies dimensionality reduction and K-Medoids++.

  • Feature representation: Injection-space clustering uses each hour's net active-power injection at every bus as its feature vector.The feature space differs from voltage-response clustering by using injections rather than voltage-response indices.
  • Feature representation: The injection set includes buses hosting at least one conventional generator, RES unit, or load.The net injection definition distinguishes these connected-unit sets.
  • Baseline performance: Reactive-power augmentation can slightly improve injection-space clustering but still underperforms voltage-response-space clustering.Comparable performance requires substantially more ROPs, increasing computational cost and potentially biasing comparisons.
  • Clustering procedure: The 74-dimensional injection vector is normalized, reduced with PCA, and clustered using K-Medoids++.K-Medoids++ is selected for robustness to outliers and uses actual data points as cluster centers.

B. Heuristic Sampling: Industry Benchmark

The heuristic benchmark partitions hours by broad temporal categories and selects extrema of aggregate load and renewable-generation features within each category.

  • Feature representation: Each hourly operating point is described by total system load, aggregated photovoltaic generation, and aggregated wind generation.These three quantities form the heuristic feature vector.
  • Temporal partitioning: Hours are partitioned by season, day type, and time of day to provide temporal diversity.Categories include winter through autumn, weekday/weekend/holiday, and morning, peak-load, and nighttime periods.
  • Representative selection: Within each category, the heuristic selects operating points with the lowest and highest values of each feature.The resulting set is intended to cover typical and extreme demand and generation conditions.

V. CASE STUDY AND NUMERICAL RESULTS

The case study validates the voltage-response-based clustering framework on an IEEE Voltage Test System adapted to very-high RES penetration, with comprehensive comparisons under normal and contingency conditions.

  • Evaluation design: The evaluation compares the proposed approach with injection-space clustering and heuristic sampling under normal and contingency conditions.The comparison assesses whether voltage-response structuring better preserves voltage-security characteristics.
  • Case study: The framework is evaluated in DIgSILENT PowerFactory on an IEEE Voltage Test System adapted to emulate very-high RES penetration.The case study includes test-system modeling, operating-data generation, voltage-response representation, and performance evaluation.
  • System modeling: Central-area loads use WECC composite load and DER A models, while wind, PV, and behind-the-meter DERs are distributed across specified buses.Wind plants are placed on g6, g7, g14, and g16; PV plants are placed on g15, g17, and g18.

B. Voltage-response Space Representation

The voltage-response representation combines correlated voltage-risk indices, PCA compression, and clustering-based operating-point selection, with 30 HAC clusters performing well in reconstruction tests.

  • Voltage-risk representation: Seven voltage-risk indices capture interdependent violation, voltage-extrema, dispersion, and reactive-reserve characteristics.Violation-related metrics are strongly coupled, while reactive-reserve indices show physically consistent relationships with violations.
  • Dimensionality reduction: PCA compresses the standardized voltage-risk space, with four components preserving 99.24% cumulative variance.PC1, PC2, and PC3 explain 54.13%, 33.04%, and 11.26% of total variance, respectively.
  • Clustering: K-Medoids++ selects 30 representative operating points in injection space, while HAC is evaluated for voltage-response-space clustering.The injection-space baseline uses 30 medoids, and HAC cluster-count sensitivity is assessed separately.
  • Cluster-count selection: 30 clusters provide a favorable HAC trade-off because reconstruction improvements beyond 30 are marginal or negative.The internal Cophenetic Correlation Coefficient measures how faithfully the hierarchical tree preserves pairwise distances.
  • Evaluation results: The voltage-response HAC method outperforms injection-space clustering and heuristic sampling across the evaluation components.It achieves 1.7% reconstruction error versus 4.3% for injection-space K-Medoids++ and 12.5% for heuristic sampling, with 98.3% steady-state representativeness.

F. Post-Contingency ROPs Evaluation

Post-contingency evaluation shows that voltage-response clustering reconstructs annual voltage risk more accurately and conservatively than heuristic or injection-space approaches, including after synchronous-condenser removal.

  • 93.4% reconstruction accuracy was achieved by voltage-space clustering for the full-year post-contingency VSPI, with approximately 0.02 error.The small positive deviation supports conservative security analysis.
  • After removing the system’s only synchronous condenser, heuristic and injection-space methods consistently underestimated VSPI relative to the full-year reference.The heuristic method exhibited the largest negative bias, indicating underrepresentation of voltage-critical operating conditions.
  • Near-zero error with a slight positive deviation was obtained by voltage-space clustering after synchronous-condenser removal.The marginal positive bias avoids underestimating system vulnerability and provides a balanced approximation.
  • Voltage behavior forms nested regimes of increasing stress severity, so preserving voltage-regime structure is essential for reliable disturbance assessment.Injection-space clustering and heuristic sampling miss critical regimes despite acceptable performance under normal conditions.
  • The comparison pairs each feature space with its best-performing algorithm: K-Medoids++ for injection space and HAC–Ward for voltage-response space.This design separates feature-space effects from clustering-algorithm choice in the reported comparison.
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