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Radiality Constraints for Resilient Reconfiguration of Distribution Systems: Formulation and Application to Microgrid Formation
Shunbo Lei, Chen Chen, Yue Song, Yunhe Hou
TL;DR
Existing radiality formulations can underutilize the expanded topological flexibility of distribution-system reconfiguration. This paper introduces graph-theoretically justified two-step radiality constraints and applies them to post-disaster microgrid formation, whose model supports more flexible sub-grid merging and separation and shows broader feasibility and improved restoration-related outcomes than literature models.
Problem
Existing radiality formulations describe only a subset of the feasible region, underutilizing DS flexibilities in resilient reconfiguration.
Method
The paper uses auxiliary fictitious spanning trees and actual spanning-forest subgraphs, then applies the constraints to post-disaster MG formation.
Results
The proposed model allows more flexible sub-grid merging and separation, yields fewer infeasible cases, and reports higher restored loads than models in and.
Takeaways & Limitations
The formulation supports more coordinated use of DS flexibilities for resilient MG formation and reduces computational complexity in the reported case studies.
Abstract
from arXiv · showhide
Network reconfiguration is an effective strategy for different purposes of distribution systems (DSs), e.g., resilience enhancement. In particular, DS automation, distributed generation integration and microgrid (MG) technology development, etc., are empowering much more flexible reconfiguration and operation of the system, e.g., DSs or MGs with flexible boundaries. However, the formulation of DS reconfiguration-related optimization problems to include those new flexibilities is non-trivial, especially for the issue of topology, which has to be radial. That is, existing methods of formulating radiality constraints can cause underutilization of DS flexibilities. Thus, this work proposes a new method for radiality constraints formulation fully enabling the topological and some other related flexibilities of DSs, so that the reconfiguration-related optimization problems can have extended feasibility and enhanced optimality. Graph-theoretic supports are provided to certify its theoretical validity. As integer variables are involved, we also analyze the tightness and compactness issues. The proposed radiality constraints are specifically applied to post-disaster MG formation, which is involved in many DS resilience-oriented service restoration and/or infrastructure recovery problems. The resulting new MG formation model, which allows more flexible merge and/or separation of sub-grids, etc., establishes superiority over the models in the literature. Case studies are conducted on two test systems.
I. INTRODUCTION
New DS flexibilities make radiality-constrained reconfiguration harder to formulate, while existing methods can underutilize feasible topologies. The paper proposes a more flexible formulation and applies it to post-disaster MG formation.
- Automation, distributed generation, and MG technologies enable more adaptive DS reconfiguration, including flexible DS and MG boundaries.
- Existing radiality formulations cover only a subset of the actual feasible region, limiting coordination and utilization of DS flexibilities.This is particularly problematic for post-disaster reconfiguration and co-optimization with evolving repairs.
- The proposed radiality formulation enables topological and related flexibilities, with graph-theoretic validity, tightness, and compactness analyses.The authors state that it can provide extended feasibility and enhanced optimality.
- The formulation is applied to post-disaster MG formation for DS restoration and recovery problems.The model forms MGs energized by DGs and/or other power sources.
- Compared with models in and, the proposed model allows more flexible merging and separation of sub-grids.
II. PROPOSED RADIALITY CONSTRAINTS
The proposed method regulates DS topology in two steps: a fictitious spanning tree contains the actual topology, which is selected as a spanning-forest subgraph. This structure supports flexible resilient reconfiguration.
- Graph-theoretic concepts: A spanning tree connects all vertices without cycles, while a spanning forest has no cycles and consists of connected components that are spanning trees.
- Graph-theoretic concepts: Normal reconfiguration forms a κ1-tree with each component containing a substation, whereas resilient reconfiguration forms an optimized κ2-tree with κ2 ≥ κ1.
- Two-step radiality formulation: Any subgraph of a spanning tree is a spanning forest, and any spanning forest of a connected graph is a subgraph of a spanning tree.These relationships provide the graph-theoretic basis for the formulation.
- Two-step radiality formulation: The method first makes auxiliary β variables form a fictitious spanning tree, then restricts actual α variables to a subgraph of that tree.α determines the DS topology; β is auxiliary and does not determine the physical topology.
- Flexibility relative to prior formulations: Unlike one-step formulations that directly enforce a spanning tree or forest, the two-step method enables more adaptive sub-grid merging, separation, and power-source allocation.
1) Validity:
The proposed constraints are theoretically certified and provide a tighter spanning-forest formulation than the compared methods.
- Theoretical proofs establish the validity of the proposed radiality constraints, whereas references and lack analytical proofs for their formulations.
- The proposed model attains the tightest formulation of the DS spanning forest polytope, while methods in and are relatively less tight.
3) Compactness:
The proposed radiality constraints can be implemented by adding one constraint to common formulations, while offering compactness and flexibility advantages over prior methods.
- 3) Compactness:: The proposed topology constraints are generally more compact than those of and, while remaining as tight or tighter.Less-tight but more compact formulations are also available for constraint (1).
- 3) Compactness:: Application requires simply adding constraint (2) to commonly used single-commodity flow-based radiality constraints.By contrast, and introduce virtual nodes or branches and additional optimization modeling components.
- 3) Compactness:: The constraints enable topological and related flexibilities in reconfiguration optimization, including both merging and possible separation of sub-grids.The method in allows merging but not possible separation.
- 3) Compactness:: A tight formulation can reduce the gap between mixed-integer and linear-programming relaxations, whereas compactness reduces computation per explored branch-and-cut node.Tightness and compactness are often conflicting formulation objectives.
- 3) Compactness:: The spanning tree polytope is the convex hull of incidence vectors representing possible spanning-tree topologies.For the 3-node example, the incidence vectors are [1,1,0], [1,0,1], and [0,1,1].
1) Loop-eliminating method [4]:
The loop-eliminating method enforces radiality by opening every loop, achieving a tight formulation at the cost of exponentially many constraints and NP-hard loop enumeration.
- 1) Loop-eliminating method [4]:: The method enumerates all distribution-system loops and enforces each loop to be open.It is equivalent to the subtour-elimination formulation of spanning-tree constraints.
- 1) Loop-eliminating method [4]:: Finding all loops in a graph is NP-hard, limiting the method’s practicality.Its formulation contains an exponential number of constraints.
- 1) Loop-eliminating method [4]:: Its LP relaxation defines the spanning tree polytope and therefore has integer extreme points.The formulation uses only |L| variables.
4) Parent-child relation-based method [7]:
The parent-child relation-based approach assigns each non-substation node a parent, but can create disconnected graphs with loops; the proposed formulation instead characterizes spanning forests through a two-step construction.
- 4) Parent-child relation-based method [7]:: The method instructs every node except the substation to have one parent, yet it can produce a disconnected graph with loops.Thus, the parent assignment alone does not guarantee the desired radial topology.
- 4) Parent-child relation-based method [7]:: A multicommodity-flow formulation sends one unit of each node-specific commodity from the substation to its destination, using only included directed-tree arcs.The formulation uses variables λij and has a polynomial number of variables and constraints.
- 4) Parent-child relation-based method [7]:: The LP relaxation of constraints (3)–(9) defines the spanning tree polytope with a polynomial number of variables and constraints.Among formulations that are tight and applicable to planar and non-planar graphs, it is generally the most compact.
- 4) Parent-child relation-based method [7]:: If constraint (1) defines the spanning tree polytope in β-space, constraints (1)–(2) define the spanning forest polytope in α-space.The resulting formulation’s vertices are 0-1 incidence vectors of spanning forests.
- 4) Parent-child relation-based method [7]:: The tightness and compactness of constraints (1)–(2) follow from the explicit formulation chosen for constraint (1).This permits selecting among formulations across a spectrum of tightness and compactness levels.
IV. APPLICATION TO RESILIENT MG FORMATION
The paper applies its radiality formulation to resilient MG formation, producing a model that preserves radiality while allowing flexible MG boundaries, source allocation, and islanding decisions.
- Application and formulation: The proposed radiality model regulates DS topology in resilient MG formation and is intended for restoration and recovery problems.The formulation is applied to construct an optimization model for post-disaster MG formation.
- Application and formulation: The model maximizes weighted restored loads while enforcing radiality, power balance, source capacities, voltage limits, branch limits, and faulted-branch status.Constraints (1)–(2) ensure radiality and topological flexibility; equations (11)–(19) represent operational and fault-related requirements.
- Comparison with prior models: The proposed formulation handles meshed DSs through radiality constraints, whereas the model in is designed for radial systems and its extension may not eliminate loops.The comparison identifies radiality constraints as the mechanism enabling meshed-DS reconfiguration without loops.
- Flexibility: Unlike prior models, it permits flexible DG allocation and an optimized number of MGs, supporting adaptive merging and separation as damaged network parts are repaired.The literature models allocate one DG to each MG, whereas the proposed model allows the number of MGs and DG assignments to vary.
- Flexibility: The model can represent load islands, mobile-source allocation, and intentionally unenergized islands, avoiding forced energization or pickup of all nodes and loads.These options are especially relevant when faulted branches isolate source-free areas or many loads lack switches.
V. CASE STUDIES
The case studies evaluate the proposed resilient MG formation model on two test systems using mixed-integer programming solved with GuroBI.
- Experimental setup: Two test systems are used to demonstrate the proposed resilient MG formation model.The involved mixed-integer programming problems are solved with Gurobi 7.5.2 using default settings on an Intel i5-4278U computer with 8GB memory.
A. IEEE 33-Node Test System [1]
On the IEEE 33-node system, the proposed model forms more flexible MG and load-island structures, restores more load, and uses DG capacity more effectively than models in and. Across 10,000 random fault scenarios, it also achieves broader feasibility and stronger restored-load results, while a tighter variant improves search consistency.
- Illustrative MG formation: The proposed model forms a 7-tree with 4 MGs and 3 load islands, merging two prior MGs and separating another to avoid forcing node 23’s load pickup.Models and form six MGs, while the proposed model restores nodes 31 and 33 and leaves node 23 unenergized.
- Illustrative MG formation: 55.6%, 67.4% and 95.4% are the DG capacity utilization rates for models, and the proposed model, respectively.The corresponding restored loads are 1500kW, 1820kW and 2575kW.
- Random-fault evaluation: The proposed model has far fewer infeasible cases across 10,000 randomly generated DS-fault scenarios and finds feasible operating points when models and return infeasibility.The authors attribute the extended feasibility to radiality constraints that fully enable topological and related flexibilities.
- Random-fault evaluation: 15.0% and 8.4% more loads are restored on average than by models and, respectively, with equal or better performance in all cases.The proposed model also has the highest average, median, and minimum restored loads and the smallest standard deviation.
- Tightness and compactness: The directed multicommodity-flow variant is tighter but less compact, exploring fewer branch-and-cut nodes on average and producing more consistent computation times.In 10.9% of cases, average time decreases from 1.92 s to 0.69 s and explored nodes from 815 to 186 relative to the less tight variant.
B. IEEE 123-Node Test System [28]
On the IEEE 123-node system, the proposed MG formation model demonstrates flexible post-disaster formation, improved restoration coordination, and computational benefits from tighter radiality constraints.
- MG formation: The model’s extended feasibility and enhanced optimality are attributed to radiality constraints that enable topological and related DS flexibilities.The reported advantage includes more coordinated matching of DGs and loads for service restoration.
- Comparative evaluation: The study compares extra restored loads against models in [18] and [19], and compares explored branch-and-cut nodes between single- and directed multicommodity-flow formulations.These comparisons are presented through histograms for the IEEE 123-node system.
- Computational performance: The tighter MG formation model has shorter and more consistent computation times than the less tight version.The tighter formulation can reduce branch-and-cut exploration while retaining more flexible DS operation.
VI. CONCLUSION
The paper proposes radiality constraints that enable topological and related distribution-system flexibilities, and applies them to resilient post-disaster MG formation. Case studies report higher resilience enhancement through more coordinated utilization of system resources than existing MG formation models.
- Conclusion: The proposed radiality constraints fully enable topological and related flexibilities in DS reconfiguration optimization problems.The formulation is specifically applied to resilient post-disaster MG formation.
- Conclusion: Case studies show higher resilience enhancement than existing MG formation models through more coordinated utilization.