Tightening Quadratic Convex Relaxations for the AC Optimal Transmission Switching Problem

This archive is distributed in association with the INFORMS Journal on Computing under the MIT License. Portions of the code in the src/core directory are derived from PowerModelsAnnex.jl, which is licensed under the BSD License.

The software and data in this repository are a snapshot of the software and data that were used in the research reported on in the paper Tightening Quadratic Convex Relaxations for the AC Optimal Transmission Switching Problem by C. Guo, H. Nagarajan and M. Bodur.

Cite

To cite the contents of this repository, please cite both the paper and this repo, using their respective DOIs.

https://doi.org/10.1287/ijoc.2023.0236

https://doi.org/10.1287/ijoc.2023.0236.cd

Below is the BibTex for citing this snapshot of the repository.

@misc{guo2025acots,
  author =     {Guo, Cheng and Nagarajan, Harsha and Bodur, Merve},
  publisher =  {INFORMS Journal on Computing},
  title =      {Tightening Quadratic Convex Relaxations for the AC Optimal Transmission Switching Problem},
  year =       {2025},
  doi =        {10.1287/ijoc.2023.0236.cd},
  url = {https://github.com/INFORMSJoC/2023.0236},
  note =       {Available for download at https://github.com/INFORMSJoC/2023.0236},
}

Description

The goal of this repository is to demonstrate the efficiency and effectiveness of our strengthened Quadratic Convex (QC) relaxation formulation and algorithms for the AC Optimal Transmission Switching (ACOTS) problem.

To showcase the performance of our proposed method, we conducted three sets of comparative experiments:

• One against the state-of-the-art QC relaxation implemented in PowerModels.jl ("P" formulation),
• Another against the basic extreme-point enhanced formulation ("E" formulation),
• And a third using the lifted cycle constraints and optimization-based bound tightening (OBBT) ("ECB" and "ECB*" formulations).

The experiments were performed on a server running a 64-bit Linux operating system, equipped with an Intel Core i9-9900K CPU running at 3.6 GHz and 128 GB of RAM (for medium-scale instances), and a larger server with Intel CPUs and 250 GB of RAM (for large-scale instances). The test cases are based on the PGLib-OPF v20.07 benchmark library, covering instances from small networks (3 buses) to very large systems (up to 2,312 buses).

To ensure robustness in the presented results, each setup involved running the algorithms on a variety of test instances under different operating conditions (TYP, SAD, and API), with key metrics such as optimality gap and runtime recorded and compared.

In particular:

• We compared the basic PowerModels.jl relaxation ("P") against our extreme-point enhanced ("E") formulation, observing that "E" consistently provides tighter relaxations and closes larger optimality gaps, especially for congested (API) and small-angle (SAD) cases.
• We evaluated the impact of further strengthening through lifted cycle constraints and bound tightening ("ECB" formulation), which showed dramatic improvements, closing as much as 17.6% more gap compared with "E" alone.
• We also implemented a branch-and-cut variant ("ECB*") where lifted cycle constraints are dynamically generated during the branch-and-bound process. This approach retains nearly the same solution quality while significantly improving computational efficiency.
• When paired with the proposed maximum spanning tree heuristic, our solution methods find good-quality solutions for large-scale instances up to 2,312 buses.

The experimental results confirm that our strengthened formulations not only tighten the relaxation bounds compared to previous benchmarks but also allow for faster and more scalable solutions to realistic, large-scale ACOTS problems.

Replicating

Run the code and algorithm with the softwares and library of these versions:

• Julia - v1.6
• Juniper - v0.7.0
• Knitro - v13.0.1
• Ipopt - v3.13.4
• Gurobi - v9.0.0
• PGLib instances repo version v20.07 (provided in the data folder)

The Project.toml and Manifest.toml files specify package dependencies of the Julia code.

Figure 2

Run the "cyc_ext_mc_plot.jl" file.

UB in Table 2 and Table 4

Note that the code runs the 12 instances in the data folder by default. If you would like to run only the first instance, then in line 54 of "tests.jl" set args[2] = 1. If you would like to run all 72 instances in the paper, please download the additional instances from the PGLib instances repo version v20.07.

To obtain the values in the "UB" column of Table 2 and Table 4, we obtain the minimum of the following 4 values:

• Non-convex ACOTS model (1) with the Knitro solver: In "params.jl", set params["ots_exact_pm"] = true and params["minlp_solver"] = "knitro", then run the "test.jl" file.
• Non-convex ACOTS model (1) with the Juniper solver: In "params.jl", set params["ots_exact_pm"] = true and params["minlp_solver"] = "juniper", then run the "test.jl" file.
• Non-convex ACOPF model with all lines switched on: In "params.jl", set params["model"] = "opf" and params["opf_exact_pm"] = true, then run the "test.jl" file.
• Non-convex ACOPF model with the set of lines switched off, as indicated by the ACOTS-QC solutions: In "params.jl", set params["model"] = "opf", params["opf_exact_pm"] = true, and params["turn_off_lines"] = true, then run the "test.jl" file.

Other columns in Table 2 and Table 4

• Column "P": In "params.jl", set params["ots_qc_pm"] = true. Also, in line 54 of "tests.jl", set args[5] = true, then run the "test.jl" file.
• Column "E": In line 54 of "tests.jl", set args[4] = "tri" and args[5] = true, then run the "test.jl" file.
• Column "EC": In line 54 of "tests.jl", set args[3] = "1e-4", args[4] = "tri", args[5] = true, and args[9] = "true", then run the "test.jl" file.
• Column "ECB": In line 54 of "tests.jl", set args[4] = "tri", args[5] = true, args[8] = "true", and args[9] = "true", then run the "test.jl" file.
• Column "ECB*": In line 54 of "tests.jl", set args[3] = "1e-4", args[4] = "tri", args[5] = true, args[8] = "true", args[9] = "true", and args[10] = "true", then run the "test.jl" file.

Note that instances are run with warm-start files, which are located in results/opf_sol. The warm-start files can be generated as follows:

• For column "P": In "params.jl", set params["model"] = "opf" and params["write_sol"] = true. In line 54 of "tests.jl", set args[4] = "rmc". Then run the "test.jl" file.
• For column "E": In "params.jl", set params["model"] = "opf" and params["write_sol"] = true. In line 54 of "tests.jl", set args[4] = "tri". Then run the "test.jl" file.
• For column "EC", "ECB", and "ECB*": In "params.jl", set params["model"] = "opf" and params["write_sol"] = true. In line 54 of "tests.jl", set args[4] = "tri" and args[9] = "true". Then run the "test.jl" file.

Table 3 and Table 5

To obtain the results with the spanning tree heuristic: In "params.jl", set params["span_tree"] = true. Then in line 54 of "tests.jl", set args[3] = "1e-3" and args[7] = "3". The other settings for "E", "EC", "ECB", and "ECB*" instances are the same as in the previous section for Table 2 and Table 4.

Figure 4

In "params.jl", set params["load_multiplier"] to varying levels of multipliers from 1.0 to 1.16 with a step size 0.01. In line 54 of "tests.jl", set args[1] = 54, args[2] = 54, and args[4] = "tri", then run the "test.jl" file. The number of lines that are turned off is reported in the column "num_off_lines" of the printed csv file.

Table 6

• Column "E": In "params.jl", set params["model"] = "opf", then run the "test.jl" file.
• Column "EC": In "params.jl", set params["model"] = "opf". Then in line 54 of "tests.jl", set args[9] = "true", then run the "test.jl" file.