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On the Radiality Constraints for Distribution System Restoration and Reconfiguration Problems
Ying Wang, Yin Xu, Jiaxu Li, Jinghan He, Xiaojun Wang
TL;DR
Distribution-system restoration and reconfiguration require radiality, but widely used spanning tree constraints can be insufficient. The paper analyzes this limitation, compares ST, SCF0, and combined SCF+ST constraints, and reports different computationally efficient choices for restoration and reconfiguration.
Problem
Widely used ST constraints may not ensure radiality, although radiality must be explicitly represented in distribution restoration and reconfiguration models.
Source: §I · p. 1Method
The paper analyzes ST constraints, tests SCF0 and SCF+ST, and combines SCF0 with ST constraints to form SCF+ST.
Source: §I · p. 1, B. Two Constraints Based on SCF · p. 2Results
SCF0 outperforms SCF+ST in restoration computation time, while SCF+ST significantly outperforms SCF0 in reconfiguration, especially for large systems.
Source: B. Results · p. 3Takeaways & Limitations
Both SCF0 and SCF+ST guarantee radial topology; the paper recommends SCF0 for restoration and SCF+ST for reconfiguration to achieve high computational efficiency.
Source: §VI · p. 3
Abstract
from arXiv · showhide
Radiality constraints are involved in both distribution system restoration and reconfiguration problems. However, a set of widely used radiality constraints, i.e., the spanning tree (ST) constraints, has its limitations which have not been well recognized. In this letter, the limitation of the ST constraints is analyzed and an effective set of constraints, referred to as the single-commodity flow constraints, is presented. Furthermore, a combined set of constraints is proposed and case studies indicate that the combined constraints can gain computational efficiency in the reconfiguration problem. Recommendations on the use of radiality constraints are also provided.
I. INTRODUCTION
Distribution networks are designed as meshed networks but operate radially, so restoration and reconfiguration models must explicitly impose radiality. This section introduces the ST constraints and the paper’s analysis of their limitations, alongside the problem formulations for reconfiguration and restoration.
I. INTRODUCTION
- ST constraints are widely used to impose radiality, but their validity depends on conditions that the paper analyzes.The letter also presents SCF0, a single-commodity flow constraint set, and proposes combining it with ST constraints as SCF+ST.Source: §I · p. 1
II. PROBLEM FORMULATION
- A radial distribution topology is represented as a forest with |R| trees, where each tree has one more node than line.The graph comprises buses and lines, and the topology is a subgraph of the distribution network.Source: §II · p. 1
- Reconfiguration minimizes power losses subject to power-system operating constraints and radiality, using line-status binaries and source outputs as decision variables.The operating constraints include power flow, bus voltage, line current, and generation or transformer capacity limits.Source: §II · p. 1
- Restoration prioritizes maximizing restored load and then minimizing power loss, with line-status and load-status binaries among its decision variables.The number of subsystems depends on fault locations and the restoration strategy.Source: §II · p. 1
III. LIMITATIONS OF ST CONSTRAINTS
The ST constraints encode parent-child relationships, assigning one parent direction to each connected line and no parent to roots. With multiple potential roots, however, they can admit disconnected loops rather than a radial topology.
III. LIMITATIONS OF ST CONSTRAINTS
- For each connected line, the ST constraints assign one parent-child direction; roots have no parent and other nodes have exactly one.The direction is represented by auxiliary variables associated with the line.Source: A. ST Constraints and the Resulted… · p. 1
- With multiple potential roots, ST constraints can admit disconnected loops, so they do not guarantee a radial topology.These loops are termed pseudo-roots.Source: A. ST Constraints and the Resulted… · p. 2
- In the illustrated two-source network, the ST constraints can produce two subnetworks instead of the desired single tree rooted at node 1.Nodes 4, 5, and 6 form a loop and serve circularly as one another’s parent, even though the ST constraints are satisfied.Source: A. ST Constraints and the Resulted… · p. 2, Figure 1 · p. 2
B. Analysis
A radial forest with |R| trees requires every non-root node to connect to a root and |N| = |E_topo| + |R|. The ST constraints satisfy the edge-count condition but do not guarantee connectivity.
B. Analysis
- A radial forest requires each non-root node to connect to a root and |N| = |E_topo| + |R|.Source: B. Analysis · p. 2
- The ST constraints satisfy the node-edge count condition but do not guarantee that each non-root node connects to a root.They are therefore necessary but insufficient for radial topology.Source: B. Analysis · p. 2
- Power-flow equations guarantee connectivity when the number of sources equals the number of roots and every node has load demand.The connected nodes then serve the load demand.Source: B. Analysis · p. 2
- Assigning a small artificial load, such as 0.001 pu, to zero-load nodes may introduce errors, and the source-root equality may fail in some restoration scenarios.The latter can occur when multiple sources connect within a sub-network to restore outage loads.Source: B. Analysis · p. 2
A. SCF Constraints for Condition 1
The single-commodity flow (SCF) formulation enforces connectivity by assigning fictitious demand to each non-root node and fictitious commodity sources to roots. Flow balance and line-status constraints ensure every node connects to a root.
A. SCF Constraints for Condition 1
- SCF assigns fictitious demand to each non-root node and treats roots as fictitious sources of a commodity.Each edge is given a direction for formulating the constraints.Source: A. SCF Constraints for Condition 1 · p. 2
- F_ij denotes fictitious flow on line (i, j), D_i is node i's fictitious demand, and M is a large positive number.D_i can be set to 1 for each non-root node, and M to |N|.Source: A. SCF Constraints for Condition 1 · p. 2
- Flow balance at every non-root node and zero flow on disconnected lines ensure that each node connects to at least one root.Source: A. SCF Constraints for Condition 1 · p. 2
B. Two Constraints Based on SCF
SCF0 combines flow constraints with an edge-count constraint to ensure radiality, while SCF+ST adds spanning-tree constraints. The added constraints can tighten reconfiguration subproblems and facilitate solution search.
B. Two Constraints Based on SCF
- SCF0 uses constraints (5)–(6) together with (4) to guarantee radial topology, and the constraint set proposed in is a compact form of SCF0.Source: B. Two Constraints Based on SCF · p. 2
- SCF+ST combines ST constraints (1)–(3) with SCF flow constraints (5)–(6) to ensure radiality.Source: B. Two Constraints Based on SCF · p. 2
- Table I compares the numbers of variables and constraints required by SCF0 and SCF+ST.Source: Table I · p. 2
- SCF+ST has more variables and constraints than SCF0, but its ST constraints add information about relationships among line-status binaries.The additional information may help narrow the search space through bounds tightening in reconfiguration subproblems.Source: B. Two Constraints Based on SCF · p. 2
- In the illustrated four-node subproblem, SCF+ST fixes a24 = 0, whereas SCF0 permits a24 to vary between 0 and 1.With a smaller feasible region, SCF+ST can obtain a higher lower bound and narrow the solution search space.Source: B. Two Constraints Based on SCF · p. 3, Figure 2 · p. 3
A. Case Information
The case information covers restoration tests on two systems and reconfiguration tests on four systems, with MISOCP models and two solvers. Restoration scenarios vary DG locations and critical-load priorities.
A. Case Information
- Restoration tests use modified 32-node and IEEE 123-node systems, with 36 and 124 line-status binaries and 32 and 85 load-status binaries, respectively.The 300 scenarios vary DG locations and critical-load priorities; restoration networks may contain multiple sources in a connected sub-network.Source: A. Case Information · p. 3
- Reconfiguration tests cover 32-, 83-, 135-, and 201-node systems, and both problem types use second-order-conic power-flow constraints in MISOCP models.Source: A. Case Information · p. 3
- The models are solved with MOSEK 9.1 and CPLEX 12.8 on a 3.6 GHz Intel Core i7 computer with 8 GB RAM.CVXPY with Python 3 is used to model the MISOCPs.Source: A. Case Information · p. 3
- Computation limits are 30 minutes for restoration and 10 hours for reconfiguration, with relative optimality-gap tolerances of 1e-4 for the 123-node system and 1e-8 otherwise.Source: A. Case Information · p. 3
B. Results
In restoration, SCF0 and SCF+ST ensure radiality in all tested scenarios, while ST fails in some; SCF0 is faster on average and has fewer ceiling-time violations. For reconfiguration, SCF+ST significantly outperforms SCF0, especially on larger systems.
B. Results
- ST fails to obtain a radial topology in 68 32-node scenarios and 39 123-node scenarios, while SCF0 and SCF+ST ensure radiality in all scenarios.Source: B. Results · p. 3
- SCF0 outperforms SCF+ST in restoration average computation time and percentage of scenarios violating the 30-minute ceiling.Average time covers scenarios that do not hit the ceiling.Source: B. Results · p. 3, Figure 3 · p. 3
- SCF+ST significantly outperforms SCF0 in reconfiguration computation time, especially for larger systems.Table II reports computation times for reconfiguration problems.Source: Table II · p. 3, B. Results · p. 3
- A possible explanation is that reconfiguration has only line-status binaries, whereas restoration also has load-status binaries, making binary relationships more complicated.SCF+ST's additional information about line-status binaries may accelerate solving when line binaries are the only integer variables.Source: B. Results · p. 3
VI. CONCLUSION
The conclusion is that spanning-tree constraints are necessary but insufficient for radiality, and that the preferred constraint set differs between restoration and reconfiguration.
VI. CONCLUSION
- Spanning-tree constraints are necessary but insufficient conditions for radial topology.Source: §VI · p. 3
- Both SCF0 and SCF+ST constraints guarantee radial topology, but their computation-time performance differs.Source: §VI · p. 3
- The paper recommends SCF0 for service restoration and SCF+ST for reconfiguration to achieve high computational efficiency.Source: §VI · p. 3