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Adaptive State Estimation Under Topological Uncertainty in Unobservable Primary Distribution Systems Using Strategically Placed Sensors
Farah Elsherif, Behrouz Azimian, Anamitra Pal
TL;DR
DER integration and sparse, costly monitoring complicate real-time awareness in primary distribution networks. The paper combines correlation-driven sensor placement, specialized DNN estimation, and fine-tuning for topology changes; it reports improved estimation and adaptation outcomes across evaluated conditions.
Problem
DER-related operating variability and limited real-time monitoring challenge situational awareness in primary distribution networks, where SMD deployment is costly and sparse.
Source: §I · p. 1Method
The framework jointly places SMDs using temporal and spatial correlations, uses separate DNN models for voltage magnitude and angle, and fine-tunes DSSE models after topology changes.
Source: §I · p. 1, §III · p. 3, §IV · p. 4Results
67-71% MAPE and 37-80% MAE improvements over no adaptation were reported across three topology transitions; adaptation took under one minute versus 15-25 minutes for complete retraining.
Source: §VI · p. 6Takeaways & Limitations
The evaluated framework maintained TI accuracy above 99.9% and DSSE errors within operational bounds under Gaussian, GMM, and Laplacian noise while adapting to topology changes.
Source: §VI · p. 6
Abstract
from arXiv · showhide
The rapid integration of distributed energy resources is fundamentally altering power flow patterns in primary distribution networks and intensifying operational uncertainty. These problems are further compounded by lack of real-time situational awareness and frequent topology changes. To address these problems, this paper proposes an integrated deep learning framework for simultaneous topology identification (TI) and distribution system state estimation (DSSE) in real-time unobservable primary distribution networks instrumented by a minimal set of synchronized measurement devices (SMDs). A correlation-driven SMD placement algorithm is introduced first that jointly satisfies TI accuracy and DSSE performance requirements by exploiting temporal and spatial correlations in nodal voltage measurements. A dual deep neural network (DNN)-based DSSE model is developed next to estimate three-phase voltage magnitudes and angles across diverse operating conditions. To extend the framework beyond the base topology, fine-tuning-based transfer learning is employed to adapt the DSSE model to reconfigured topologies using limited retraining data. The framework is validated under both Gaussian and non-Gaussian measurement noise and benchmarked against a conventional estimation approach and a single DNN model.
I. INTRODUCTION
DER integration increases distribution-network uncertainty, while sparse or costly monitoring limits real-time awareness in primary circuits. The paper combines correlation-driven sensor placement, specialized DNN estimators, and topology-triggered adaptation.
I. INTRODUCTION
- DER-driven voltage fluctuations, reverse flows, and localized violations make real-time situational awareness important for reliable grid operation.The introduction notes these conditions can propagate into feeders and remain undetected without adequate monitoring,.Source: §I · p. 1
- Smart meters are delayed and located on secondary circuits, SCADA observability is largely feeder-head-only, and SMD deployment remains sparse because of cost.Field measurements also indicate SMD noise is better modeled as non-Gaussian than by the Gaussian assumption used in conventional estimators.Source: §I · p. 1
- The framework combines temporally and spatially informed SMD placement, separate magnitude and angle DNN estimators, and fine-tuning to adapt DSSE after topology changes.The stated contributions include performance across hours and Gaussian and non-Gaussian noise, with topology-triggered transfer learning for reconfigured networks.Source: §I · pp. 1–2
II. TEMPORALLY-AWARE CORRELATION-DRIVEN SMD PLACEMENT
The placement strategy accounts for diurnal voltage variation by selecting representative time slots before sensor placement. It uses hourly voltage statistics to group operating conditions and support placement across the day.
II. TEMPORALLY-AWARE CORRELATION-DRIVEN SMD PLACEMENT
- Sensor placement must balance locations informative for topology identification with locations that represent network-wide voltages across changing daily conditions.The paper notes that a strategy derived from a single hour may not provide adequate coverage during other hours.Source: §II · p. 2
A. Temporal Correlation Analysis
- Historical hourly load injections are used to solve power flows and generate nodal voltage-phasor time series for temporal analysis.The voltage profiles are denoted by M.Source: A. Temporal Correlation Analysis · p. 2
- Kernel density estimates and pairwise Jensen-Shannon divergences compare hourly voltage-magnitude distributions at each node, then aggregate dissimilarities across nodes.The method uses KDE to represent non-Gaussian, multimodal residential load distributions without imposing a parametric form.Source: A. Temporal Correlation Analysis · p. 2
- Average-linkage hierarchical clustering of the systemwide dissimilarity matrix uses a 0.5 merge-gap threshold and yields two representative time slots.A 0.4 threshold would instead produce a third cluster containing only two hours, considered insufficient to capture that regime's diversity.Source: A. Temporal Correlation Analysis · p. 2
B. SMD Placement for TI and DSSE
The placement procedure takes representative time slots and operational constraints as inputs, then selects sensors to meet topology-identification and state-estimation requirements. Its first placement stage uses current phasors to select locations for topology identification.
B. SMD Placement for TI and DSSE
- The placement procedure uses representative slots, sensor budget, TI accuracy and DSSE correlation thresholds, and node count to produce the required SMD locations.The representative slots are one of the inputs to this second stage.Source: B. SMD Placement for TI and DSSE · p. 2
1) Part A: Placement for TI:
- Part A greedily adds locations through sequential forward selection until the DNN-TI accuracy threshold is met across all representative time slots.The formulation seeks the minimum SMD set whose current-phasor measurements enable the required accuracy.Source: 1) Part A: Placement for TI · p. 2
B. SMD Placement for TI and DSSE
- The algorithm begins by identifying representative slots through JSD-based hierarchical clustering before selecting TI sensor locations.Algorithm 1 lists the clustering step as the first stage of the placement procedure.Source: 1) Part A: Placement for TI · p. 3
A. SMD Placement for TI:
The TI placement stage iteratively adds features and locations to satisfy accuracy and budget conditions. The DSSE stage augments the selected set to provide correlation-based coverage of unmonitored nodes.
A. SMD Placement for TI:
- TI placement iterates sequential feature selection until the sensor budget is reached or the accuracy threshold is met, then advances through representative slots.If no switches exist, the algorithm moves directly to the DSSE placement stage.Source: A. SMD Placement for TI · p. 3
- Current phasors are used for TI because switching changes the admittance matrix and redistributes power flows, affecting branch currents near switching events.The described changes are dominant in branch-current magnitudes and angles at electrically nearby locations.Source: B. SMD Placement for DSSE · p. 3
B. SMD Placement for DSSE:
- DSSE placement adds sensors until every unmonitored node has an SCC of at least DSSEcorr with a selected node in its feeder or voltage-regulated zone.The algorithm clusters the SCC matrix and adds a high-average-intra-cluster-correlation node where the threshold is not met.Source: 2) Part B: Placement for DSSE · p. 3, B. SMD Placement for DSSE · p. 3
2) Part B: Placement for DSSE:
The placement procedure couples TI and DSSE objectives across representative time slots, while separate DNN models map SMD measurements to topology probabilities or estimated voltage phasors.
2) Part B: Placement for DSSE:
- Iterative coupling ensures both TI and DSSE placement objectives are satisfied across all representative time slots once their thresholds are met.Source: 2) Part B: Placement for DSSE · p. 3
III. DUAL-MODEL DEEP INFERENCE FOR TOPOLOGY AND VOLTAGE PHASOR ESTIMATION
- TI and DSSE use separate models because classification and continuous voltage-phasor regression require task-specific learned representations.Source: §III · p. 3
- The shared DNN takes an SMD measurement vector as input and outputs either topology-class probabilities or estimated voltage phasors.Source: §III · p. 3
A. DNN-Based Topology Identification (TI)
The TI model classifies current phasor measurements across feasible network topologies, while DSSE uses separate regression models for three-phase voltage magnitudes and angles.
A. DNN-Based Topology Identification (TI)
- The TI model classifies current phasor measurements from SMD locations across feasible network topologies.Each topology represents a unique combination of switch statuses and receives an integer label.Source: A. DNN-Based Topology Identification… · p. 3
- The shared DNN architecture provides the general model structure used for both TI and DSSE.Source: Figure 3 · p. 3
B. Dual DNN-Based State Estimation
- DSSE takes voltage and current phasor measurements as inputs and trains separate regression models for three-phase magnitudes and angles across all nodes.The models are separated because magnitude and angle have different spatial correlations and dynamic ranges.Source: B. Dual DNN-Based State Estimation · p. 3
- Each DSSE model is a fully connected feedforward DNN with batch normalization and dropout, with architecture parameters tuned independently by model and time slot.Source: B. Dual DNN-Based State Estimation · p. 3
IV. TOPOLOGY-TRIGGERED MODEL ADAPTATION THROUGH FINE-TUNING (FT)
Topology changes shift SMD measurement distributions, making full retraining impractical; the framework instead sequentially fine-tunes the active DSSE model using limited data after detected changes.
IV. TOPOLOGY-TRIGGERED MODEL ADAPTATION THROUGH FINE-TUNING (FT)
- Topology changes redistribute branch currents and shift SMD measurements, while retraining from scratch for each topology requires impractical amounts of data and computation.Source: §IV · p. 4
- Fine-tuning reuses the most recently trained model’s weights and updates them on limited data from the new topology rather than starting from random initialization.The approach relies on overlapping electrical statistics across topologies, with higher layers encoding topology-specific mappings.Source: §IV · p. 4
- When TI detects a topology change, the active DSSE model is fine-tuned sequentially using Nt scenarios, where Nt is much smaller than the original training-set size Ns.Source: §IV · p. 4
- During operation, TI monitors measurements and classifies topology; detected changes trigger fine-tuning on stored scenarios, after which DSSE inference resumes with the updated model.The procedure is evaluated for four consecutive topology transitions.Source: §IV · p. 4
A. Test System and Data Generation
The evaluation combines a 240-node, 84-topology test system with joint sensor placement and DNN estimation under multiple noise models. The proposed placement meets TI and DSSE targets with eight SMDs, while dual DSSE models outperform a unified model and use far fewer sensors than LSE.
A. Test System and Data Generation
- The 240-node system has three feeders, nine sectionalizing switches, 84 feasible topologies, and one year of hourly smart-meter data for 193 load nodes.The system is the US Midwest primary distribution test system described in.Source: A. Test System and Data Generation · p. 4
- Training data use node- and hour-specific KDE load-injection distributions, Monte Carlo scenarios, OpenDSS power flows, and Gaussian, GMM, and Laplacian measurement noise.The GMM and Laplacian models represent non-Gaussian characteristics consistent with field SMD measurements.Source: A. Test System and Data Generation · p. 4
B. Proposed DNN-Based TI and DSSE
- JSD clustering yields two representative time slots; Algorithm 1 selects eight SMD locations using TIacc = 99% and DSSEcorr = 0.98.The slots separate daytime and evening hours with higher voltage variability from overnight hours with lower variability.Source: B. Proposed DNN-Based TI and DSSE · pp. 4–5
- Integrated placement meets both objectives across all noise types and time slots: TI accuracy above 99.9%, magnitude MAPE within 0.04%, and angle MAE within 0.03°.The TI-only and DSSE-only placements each fail to meet the other task's requirements.Source: B. Proposed DNN-Based TI and DSSE · p. 5
- Dual models reduce Mag. MAPE by 52% and Ang. MAE by 47% in Slot 1, and by 58% and 48%, respectively, in Slot 2.The comparison is against a single unified DNN under GMM noise.Source: Table I · p. 5, B. Proposed DNN-Based TI and DSSE · p. 5
- The TI model uses six layers with 800 neurons per layer, while independently tuned magnitude and angle DSSE models use shallower architectures.The fine-tuned model inherits the base architecture and uses a 10^-4 learning rate for 50 epochs.Source: B. Proposed DNN-Based TI and DSSE · p. 5, Table II · p. 5
- With 8 SMDs, the proposed DNN is slightly worse than LSE, which requires 113 SMDs, and performs similarly under Gaussian and GMM noise.This corresponds to about a 93% reduction in instrumentation needs.Source: Table III · p. 5, B. Proposed DNN-Based TI and DSSE · p. 5
C. TL for Topology Adaptation
For topology transitions under GMM noise, fine-tuning maintains estimation accuracy near complete retraining at lower computational cost. Without adaptation, errors worsen; sequential fine-tuning also recovers when returning to the base topology.
C. TL for Topology Adaptation
- Without adaptation, MAPE degrades to 0.072–0.090% and angle MAE reaches up to 0.098° across the tested transitions.The sequence is Tbase → T36 → T2 → Tbase.Source: C. TL for Topology Adaptation · p. 5
- Sequential fine-tuning accurately recovers on return to Tbase from an intermediate model state without storing model checkpoints.T36 differs from Tbase in five switch positions, and T2 differs in two.Source: C. TL for Topology Adaptation · p. 5
- Fine-tuning maintains 0.025–0.027% MAPE and 0.018–0.020° MAE, approaching complete-retraining accuracy at a fraction of its computational cost.The sequence tests Slot 1 under GMM noise.Source: C. TL for Topology Adaptation · p. 5, Figure 6 · p. 6, Table IV · p. 6
VI. CONCLUSION
The framework combines sparse SMD placement with joint topology identification and three-phase state estimation, then adapts the estimator to reconfigured topologies. Results include reduced sensor requirements, gains from separate magnitude and angle models, and rapid adaptation across topology transitions.
VI. CONCLUSION
- 8 SMD locations met the 99% TI accuracy and 0.98 SCC thresholds across noise conditions, compared with 113 locations required by conventional LSE.The placement algorithm used two representative time slots and combined temporal variability with spatial voltage correlation.Source: §VI · p. 6
- Under GMM noise, dual magnitude and angle DNN-DSSE models outperformed a single unified estimator by up to 52% in magnitude MAPE and 48% in angle MAE.Gains were consistent across both time slots.Source: §VI · p. 6
- Across three consecutive topology transitions, FT-based transfer learning improved MAPE by 67–71% and MAE by 37–80% over no adaptation, adapting in under one minute versus 15–25 minutes for complete retraining.Source: §VI · p. 6
- TI accuracy remained above 99.9% and DSSE errors stayed within operational bounds under Gaussian, GMM, and Laplacian noise.The reported robustness spans all three tested noise conditions.Source: §VI · p. 6