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A Multi-Stage Linear Programming Framework for Three-Phase State Estimation in Low-Voltage Distribution Grids
Mohsen Banaei, Eghbal Hosseini, Ahlam Jameel, Konstantin Filonenko, Shahrzad Pour, Razgar Ebrahimy
TL;DR
Sparse and delayed measurements make nodal-voltage estimation difficult in low-voltage feeders. The paper uses iterative linear programming to estimate per-phase voltages in three-phase unbalanced grids, improving accuracy over a single-stage linear estimator on a real feeder.
Problem
LV feeders often lack real-time load measurements, have only coarse historical consumption records, and have high-rate voltage sensors at a handful of nodes, leaving full nodal-voltage estimation under limited observability.
Source: §1 · p. 1Method
Results
On a real 50-node Danish feeder, MSLP improved accuracy metrics over a single-stage linear estimator by roughly 16% on average, with an MAE of 0.641 V.
Source: §4.1 · p. 5, §5 · p. 7Takeaways & Limitations
For the studied feeder and meter configuration, historical averaging slightly outperformed forecasting, while halving estimation meters raised error metrics by about 68% on average.
Source: §4.3 · p. 5, §4.4 · p. 6The input-preparation result is specific to the studied grid and meter configuration and may differ for other feeders.
Source: §4.3 · p. 5
Abstract
from arXiv · showhide
Low-voltage (LV) distribution feeders are increasingly difficult to monitor because real-time load data are unavailable, historical measurements are sparsely sampled, and high-rate voltage sensors cover only a few nodes. This paper proposes a multi-stage linear programming (MSLP) estimator for three-phase unbalanced LV grids that reconstructs per-phase nodal voltages under such limited observability. The estimator linearises the three-phase power flow through voltage-to-power sensitivity matrices and adjusts the nodal active and reactive power injections so that the resulting voltages match the available measurements. To limit the error introduced by linearisation, the sensitivities are rebuilt and the power corrections refined over successive iterations, and power-balance constraints at intermediate metered nodes are added to tighten the feasible region. The method is validated on a real 50-node Danish LV feeder, where it attains a mean absolute error of 0.641V, a standard deviation of 0.816V, a root-mean-square error of 0.842~V, and a maximum error of 3.437V. Compared with a single-stage linear estimator, MSLP lowers the error metrics by roughly 16 percent on average, and additional studies quantify the influence of the input-preparation strategy and of the number of estimation meters.
1 Introduction
Low-voltage feeders are difficult to monitor because measurements are sparse and three-phase unbalance must be handled under tight runtime and data constraints. The paper proposes an iterative linear-programming estimator and evaluates its accuracy and sensitivity to input preparation and meter coverage.
1 Introduction
- Sparse real-time load data, coarse historical records, and voltage sensors at only a few nodes make full nodal-voltage estimation difficult in LV feeders.The paper identifies explicit treatment of three-phase unbalance under runtime and data-availability constraints as a recurring difficulty.Source: §1 · p. 1
- Single-stage linear estimation can accumulate error when large power corrections exceed the small-excursion range of its sensitivity coefficients.The proposed method addresses this limitation by repeatedly rebuilding sensitivities and refining corrections, with power-balance constraints at intermediate metered nodes.Source: §1 · p. 2
- The evaluation on a real 50-node Danish feeder measures accuracy gains over the single-stage estimator and examines input preparation and estimation-meter count.The paper builds on earlier single-stage linear-programming and hybrid-metaheuristic estimators.Source: §1 · p. 2
2 System Model and Problem Definition
The study formulates per-phase voltage reconstruction for a three-phase unbalanced feeder with limited metering. It uses prepared active and reactive injections alongside sparse voltage measurements, despite the nonlinear, nonconvex power-flow relationship.
2 System Model and Problem Definition
- The studied feeder has 50 radial nodes supplied by a 400 kVA, 10.4/0.4 kV Dyn5 transformer, with six estimation meters and three separate evaluation meters.Estimation meters are at nodes 1, 9, 16, 35, 40, and 50; evaluation meters at nodes 12, 21, and 38 are withheld from estimation.Source: §2 · p. 2
- Per-phase active and reactive injections are inferred from historical demand and disaggregated across phases using phase-share coefficients.The coefficients are estimated at metered buses and propagated to unmetered buses through impedance-based clustering.Source: §2 · p. 2
- The estimator reconstructs per-phase voltage magnitudes at every node from injections and sparse real-time voltage measurements, despite nonlinear, nonconvex three-phase power flow.Source: §2 · p. 2
3 Proposed Multi-Stage Linear Programming Method
The method adjusts nodal active and reactive power consumptions using linear voltage sensitivities to match metered voltages. It rebuilds sensitivities and refines corrections iteratively, while intermediate-meter balance constraints further restrict the feasible adjustments.
3 Proposed Multi-Stage Linear Programming Method
- OpenDSS supplies baseline voltages, while nodal active and reactive power perturbations are used to record their impacts on voltages at metered nodes.The grid model is created from topology information, and a power flow is run for the baseline.Source: §3 · pp. 2–3
- Repeatedly updating the operating point and sensitivity matrices lets the estimator limit linearisation error while keeping each optimization subproblem linear.The linear approximation is valid only for small power changes, and the method limits changes and updates the matrices across iterations.Source: §3 · pp. 3–4
- The optimization minimizes metered-voltage mismatch together with changes in active and reactive power.A penalty accounts for cases where power changes cannot reduce voltage mismatch to zero.Source: §3 · p. 4
- Power-balance constraints at intermediate metered nodes bound downstream consumption using the active and reactive power recorded through those meters.For node 16, the through-power must exceed the summed consumption of nodes supplied through it.Source: §3 · p. 4
- Bounds on active and reactive power changes keep corrections within ranges intended to avoid significant voltage-estimation errors.The optimization also penalizes the absolute sizes of the power changes.Source: §3 · p. 4
- At each iteration, the updated grid state determines new sensitivities and power corrections; the process repeats until changes fall below a threshold, then a final power flow yields estimated voltages.Source: §3 · p. 4
4 Results and Discussion
Across four dates, MSLP improves accuracy over the single-stage LP estimator at the cost of longer runtime. The evaluation also reports feeder-specific input-preparation results and sensitivity to meter coverage.
4 Results and Discussion
- The evaluation uses one representative day, 1 October 2025, at 15-minute resolution, with power-balance constraints active at middle node 16.The estimator is assessed against readings from evaluation nodes.Source: §4 · pp. 4–5
4.1 State Estimation Accuracy
- MAE is 0.641 V, STD 0.816 V, RMSE 0.842 V, and ME 3.437 V; 96% of errors stay below 1.632 V (0.7%).At evaluation meters, estimated profiles track measured voltages more closely than the raw power-flow curves across all phases.Source: §4.1 · p. 5, Figure 3 · p. 6, Figure 4 · p. 6
4.2 Comparison with the Single-Stage Estimator
- MSLP improves MAE, STD, RMSE, and ME over single-stage LP by 16.55%, 17.31%, 16.00%, and 9.81%, respectively, while running roughly 2.5 times slower.The comparison covers four dates.Source: §4.2 · p. 5, Table 1 · p. 6
4.3 Effect of the Input-Preparation Strategy
- For this feeder and meter configuration, historical averaging slightly outperforms forecasting for preparing real-time active and reactive powers.The result may differ for other feeders.Source: §4.3 · p. 5, Table 2 · p. 6
4.4 Effect of the Number of Estimation Meters
- Halving the number of estimation meters raises the error metrics by about 68% on average, showing sensitivity to metering coverage.The table reports metrics as estimation meters are progressively removed.Source: §4.4 · p. 6, Table 3 · p. 7
5 Conclusion
The paper presents MSLP for per-phase voltage estimation in three-phase unbalanced LV grids under limited observability. On a real 50-node Danish feeder it improves accuracy over a single-stage estimator, with longer runtime; future work targets lower computational cost and broader evaluation.
5 Conclusion
- MSLP estimates per-phase voltages in three-phase unbalanced LV grids under limited observability by re-linearising at each iteration and enforcing intermediate-node power balance while keeping subproblems linear.The iterative process progressively reduces linearisation error.Source: §5 · p. 7
- On a real 50-node Danish feeder, MSLP achieves an MAE of 0.641 V and improves accuracy metrics over a single-stage linear estimator by roughly 16% on average, at longer runtime.Source: §5 · p. 7
- For this feeder, historical averaging can be competitive with forecasting, while reduced metering coverage markedly degrades accuracy; future work targets lower computational cost and broader evaluation.The broader evaluation is to include larger feeders and more diverse operating conditions.Source: §5 · p. 7