Distribution Grid Reconfiguration and Restoration via Linearizable Multi-phase Power Flow Modeling

I. INTRODUCTION

• Topology optimization combines discrete branch decisions with nonlinear AC power flow, while multi-period restoration further increases the search space.Existing data-driven approximations also face limited topology adaptivity and parameter growth with feeder size.Source: §I · p. 1
• Prior neural PF proxies map optimization inputs to solutions, whereas linear and piecewise-linear surrogates approximate PF equations while retaining operating constraints.End-to-end topology inputs do not guarantee representation of unseen topologies, and dense PWL layers have quadratic parameter complexity.Source: §I · p. 1
• The N-PWL model incorporates branch statuses into the branch-flow-to-injection mapping, preserving branch-node consistency and representing unseen topologies without retraining.Area-based sparsification makes parameter growth linear with feeder size, and the model supports MILP reformulations of DNR and ROP.Source: §I · pp. 1–2

II. DISTRIBUTION GRID TOPOLOGY OPTIMIZATION

• The AC grid formulation is specialized to single-period DNR and multi-period ROP optimization tasks.Source: §II · p. 2
• The model also supports unbalanced multi-phase feeders, including unequal phase loading, missing phases, and mutual coupling.The paper validates this extension on the unbalanced IEEE 123-bus feeder.Source: §II · p. 2
• The topology model uses binary open/closed line statuses to gate potential AC branch flows, which connect to nodal injections through power balance.Source: §II · p. 2

A. Distribution Network Reconfiguration (DNR)

• DNR selects a radial feeder configuration to minimize active power losses while maintaining full load service.Source: A. Distribution Network… · p. 3
• Controllable sectionalizing and tie switches transfer loads among alternative feeder paths, while the optimization jointly determines switch statuses and a feasible operating point.Source: A. Distribution Network… · p. 3
• Binary switch decisions and nonlinear AC power-flow relations make the DNR formulation a mixed-integer nonlinear program.The conventional objective focuses on loss minimization, though DG costs and switching penalties can also be added.Source: A. Distribution Network… · p. 3

B. Restoration Ordering Problem (ROP)

• ROP maximizes total energy served over a multi-period restoration horizon after a disruptive event.Limited repair resources can require temporary load curtailment and reduce network transferability.Source: B. Restoration Ordering Problem (ROP) · p. 3
• Each damaged line’s status indicates whether it has been restored and energized, with restoration assumed immediate upon repair and service continuing thereafter.Only damaged-line statuses are optimized; other line statuses remain fixed.Source: B. Restoration Ordering Problem (ROP) · p. 3
• The model tracks the fraction of nominal demand served at each bus and period, expressing served demand under a fixed power factor.Source: B. Restoration Ordering Problem (ROP) · p. 3
• Initial post-event statuses and a per-period restoration budget constrain the plan; time coupling increases complexity over DNR.Constraints also require monotonic load recovery, radial operation, and complete restoration within the horizon.Source: B. Restoration Ordering Problem (ROP) · p. 3

III. PIECEWISE LINEAR (PWL) PF MODELING VIA NEURAL APPROXIMATION

• The first two trainable layers approximate quadratic voltage-product terms, while fixed linear layers form potential branch flows and topology-dependent nodal injections.Only the trainable parameters require learning; the subsequent layers are determined by branch parameters and the line-status vector.Source: §III · pp. 3–4, Figure 1 · p. 4
• The final fixed layer incorporates branch statuses into nodal injections, allowing a trained model to represent any feasible unseen topology without retraining.The training data must cover the full operating range for candidate topologies; representative training topologies can be selected so each branch is energized at least once.Source: §III · p. 4
• A first-order baseline at a voltage reference gives a local approximation, while the neural correction targets nonlinear residuals as operating points or topologies shift.The model also uses a first-order approximation to form the voltage-square term, described as accurate in practice due to voltage regulation.Source: §III · p. 4
• ReLU activation makes the voltage-state-to-PF mapping continuous piecewise-linear, enabling a mixed-integer linear reformulation.Without nonlinear activation, the model reduces to a first-order approximation with trainable parameters.Source: §III · p. 4

A. Training of the N-PWL PF Model

• The training loss uses target labels for potential branch flows and nodal injections, with λ balancing the two error terms and larger λ emphasizing PF outputs.The loss is averaged over training samples, and λ can be selected through cross-validation and hyperparameter tuning.Source: A. Training of the N-PWL PF Model · p. 4
• ReLU neurons partition the input space into affine regions, and their activation patterns produce continuous piecewise-linear PF outputs for a fixed topology.A layer with K ReLU neurons admits at most 2^K activation patterns.Source: A. Training of the N-PWL PF Model · pp. 4–5
• For the IEEE 33-bus feeder, K = 35 was adequate, while K = 275 was adequate for the IEEE 123-bus feeder, suggesting K ≊ Nϕ.The reported systems have Nϕ = 33 and 274, respectively; larger K increases expressiveness but also trainable parameters and may affect generalizability with limited training data.Source: A. Training of the N-PWL PF Model · p. 5
• The fully connected model has quadratic parameter growth with phase-node count, motivating a scalable design for large feeders.The parameter count depends on K scaling with Nϕ and the number of phase-specific branches scaling linearly with Nϕ.Source: A. Training of the N-PWL PF Model · p. 5

B. Low-Complexity Modeling for Scalability

• The model sparsifies its first two trainable layers by partitioning the feeder into areas, while leaving the topology-dependent final layer unchanged.Each area predicts its PF outputs from a subset of voltage states; boundary-branch inputs can be augmented with the neighboring bus voltage variables.Source: B. Low-Complexity Modeling for… · p. 5, Figure 4 · p. 5
• Areas group electrically close buses using branch affinity λij = 1/|zij|, with spectral clustering producing partitions whose intra-area affinities exceed inter-area affinities.The number of areas is chosen near Nϕ/N′ϕ so that each area remains much smaller than the full feeder.Source: B. Low-Complexity Modeling for… · p. 5
• With N′ϕ ≪ Nϕ phase nodes per area, the sparse N-PWL model has O(NϕN′ϕ) trainable parameters and linear complexity scaling with Nϕ.This linear scaling reduces the previous quadratic order and can enhance scalability for large feeder systems.Source: B. Low-Complexity Modeling for… · p. 5

IV. N-PWL MODEL FOR TOPOLOGY OPTIMIZATION

• Sign-based pre-activation bounds yield a tighter ReLU formulation than conventional big-M constraints.The bounds are constructed from system operating limits, and the formulation uses the input range for the voltage-state deviations.Source: §IV · p. 6
• Fixing neurons whose activation status is constant across the input range removes inactive inequalities without affecting optimality.Numerical experiments report an approximately 5% reduction in neurons.Source: §IV · p. 6
• The trained N-PWL mappings admit an equivalent mixed-integer representation over a bounded input range, enabling MILP formulations of DNR and ROP.The ReLU representation is exact, and subsequent mappings are affine once activation indicators are fixed.Source: §IV · pp. 5–6
• McCormick constraints linearize branch-status products, and polyhedral line-rating constraints complete the MILP reformulations.The line-rating approximation incurs reported error below 0.625%.Source: §IV · p. 6

V. NUMERICAL STUDIES

• The IEEE 123-bus feeder is used for PF modeling and topology optimization, with OpenDSS evaluating the resulting network decisions.The study formulates N-PWL and LinDistFlow optimization models in Python, solves MILPs with Gurobi, and uses BONMIN for nonconvex AC counterparts.Source: §V · p. 6
• Nine controllable switches and up to six branch switchings yield 63 admissible configurations, including the nominal network.The nine include five original switches and four normally open three-phase ties; SW1 and SW6 are fixed closed and SW8 is fixed open.Source: §V · pp. 6–7, Table II · p. 7
• 20,000 valid samples combine 2,000 load profiles with 10 representative radial topologies and retain only converged OpenDSS solutions satisfying operating limits.Loading cases are normal (40%), heavy (30%), and phase-imbalanced (30%).Source: §V · p. 7
• Testing extends beyond training topologies to cover all 63 admissible network configurations in the feeder.Training uses a 90%-10% split of the 2,000 load profiles.Source: §V · p. 7

A. N-PWL Power Flow Modeling Performance

• Comparisons use DirectNN, N-PWL_Full, and N-PWL_Sparse with K = 275 hidden neurons; the sparse model distributes neurons across five areas.DirectNN maps multi-phase voltage state directly to PF outputs and requires a separately trained model for each of the 10 representative topologies.Source: A. N-PWL Power Flow Modeling Performance · p. 7
• Both N-PWL models outperform DirectNN on branch-flow approximation and nodal-injection prediction, with especially significant improvement for injections.N-PWL_Sparse achieves slightly lower errors than N-PWL_Full while retaining linear parameter scaling.Source: A. N-PWL Power Flow Modeling Performance · p. 8, Figure 6 · p. 8
• Across 53 unseen network configurations, average errors remain at the same level, while maximum errors increase slightly.The models transfer to unseen topologies by updating Wψ(ϵ) in the final layer; N-PWL_Sparse is used for later optimization studies.Source: A. N-PWL Power Flow Modeling Performance · p. 8, Figure 6 · p. 8

B. N-PWL PF for Solving DNR Problem

• N-PWL and LinDistFlow produce MILP formulations, whereas AC-DNR retains nonlinear PF equations in a nonconvex mixed-integer nonlinear program.The study evaluates optimality, feasibility, and optimization solve time, excluding model building and AC re-evaluation from solve time.Source: B. N-PWL PF for Solving DNR Problem · p. 8
• The comparison averages performance over 80 normal-loading profiles and switching budgets α ∈ {2, 4, 6}.Each method’s selected topology is re-evaluated with multi-phase PF solutions from OpenDSS.Source: B. N-PWL PF for Solving DNR Problem · p. 8
• PWL-DNR achieves an optimality ratio above 97%, improves on LDF-DNR in optimality and feasibility, and needs approximately one-tenth of AC-DNR solve time at larger switching budgets.The feasibility advantage over LDF-DNR becomes more pronounced as the switching budget α increases.Source: B. N-PWL PF for Solving DNR Problem · p. 8, Figure 7 · p. 8

C. Applications to the ROP

• N-PWL and LinDistFlow yield multi-period MILPs, while AC-ROP is a nonconvex MINLP and restoration decisions are assessed through AC re-evaluation.Fixing restoration orders reduces the re-evaluation to an NLP over continuous operating and load-service variables.Source: C. Applications to the ROP · pp. 8–9
• The study evaluates 2-, 4-, and 6-damaged-line cases with restoration budget η = 1 per period and horizons T ∈ {2, 4, 6}.Performance is averaged over the same 80 normal-loading profiles used in the DNR study.Source: C. Applications to the ROP · p. 9
• PWL-ROP achieves an optimality ratio above 96%, outperforms LDF-ROP in optimality and feasibility, and reduces AC-ROP computation time by an order of magnitude.In the 6-DL case, AC-ROP takes approximately three times as long as AC-DNR.Source: C. Applications to the ROP · p. 9, Figure 9 · p. 9, Figure 8 · p. 9

• On the unbalanced, multi-phase IEEE 123-bus feeder, the model improved PF prediction over end-to-end baselines and accurately represented unseen topologies.Source: §VI · p. 9
• For DNR and ROP, the model improved AC feasibility over linearized PF models and reduced computation time versus AC-PF-based nonlinear formulations, particularly with many topology decisions.Source: §VI · p. 9
• The sequential layer design incorporates branch statuses into the mapping from branch flows to nodal injections, preserving PF branch-node consistency and supporting unseen topologies without retraining.Source: §VI · p. 9
• Area-based sparsification achieves linear scaling of trainable NN parameters with feeder size, and the resulting model supports MILP reformulations of DNR and ROP.Source: §VI · p. 9
• Future directions include GPU-based MILP acceleration, multi-energy applications, and broader distribution monitoring and operations.Source: §VI · p. 9