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Co-Optimization of Network Topology and Variable Impedance Devices under Dynamic Line Ratings Using Iterative McCormick Relaxation
Junseon Park, Hyeongon Park, Rahul K. gupta
TL;DR
The paper addresses limited coordination among grid-enhancing technologies by jointly optimizing NTO, DLR, and VIDs with iterative McCormick relaxation. Across benchmark simulations, the proposed method balances accuracy and computation time between the nonlinear and static McCormick approaches.
Problem
Existing work has largely optimized individual grid-enhancing technologies, with limited attention to coordinated operation that could leverage their combined flexibility.
Source: §1 · p. 4Method
The framework jointly models NTO, weather-dependent DLR, and VIDs, using iterative McCormick relaxation to handle bilinear constraints.
Source: §1 · p. 4Results
Across multiple benchmark networks, the proposed method offers a favorable accuracy-runtime trade-off relative to the more accurate but slower nonlinear approach and the less accurate but efficient static McCormick method.
Source: §6 · p. 10Takeaways & Limitations
The framework provides a way to assess coordinated operation of NTO, DLR, and VIDs under different operating and weather conditions.
Source: §1 · p. 4For hourly scheduling, conductor temperature is assumed to reach steady state within each time interval.
Source: §2.2 · p. 5
Abstract
from arXiv · showhide
Grid Enhancing Technologies (GETs), such as Network Topology Optimization (NTO), Dynamic Line Rating (DLR), and Variable Impedance Devices (VIDs) are increasingly deployed in power transmission grids to address congestion and improve grid utilization. However, their growing adoption necessitates coordinated operation to fully exploit their benefits. This paper presents a unified co-optimization framework that integrates NTO, DLR, and VID to alleviate congestion and minimize operational costs. NTO is modeled using a node-breaker formulation to provide enhanced switching granularity, while DLR incorporates weather-dependent parameters, including wind speed and ambient temperature, to dynamically adjust transmission limits. The resulting formulation is a mixed-integer nonlinear program due to power flow constraints with bilinear VID terms, and discrete topology decisions. To improve tractability, we propose using a McCormick relaxation approach to linearize bilinear terms. Furthermore, we propose an iterative refinement of the McCormick relaxation scheme to enhance solution accuracy. The proposed framework is validated through detailed analyses on the IEEE 118-bus system, along with additional case studies on other benchmark systems to assess scalability. Results demonstrate that the proposed approach improves solution accuracy compared to standard McCormick relaxation, achieves solution quality close to the nonlinear formulation, and maintains computational efficiency.
Nomenclature
The nomenclature defines the time, network, and component indices and sets, then identifies operating costs, ratings, and decision variables used in the model.
Indices and sets
- The indices identify time periods and network components, while the sets collect periods, substations, lines, generators, and loads.Additional sets distinguish generators and loads at substations and lines directed from or to a substation.Source: Nomenclature · p. 2
Variables
- Binary variables encode busbar, load, generator, line, and line-end connection or switching decisions.The nomenclature also defines voltage angles, line flows, load, and the DLR-to-SLR allowable-current ratio.Source: Nomenclature · p. 3
Parameters
- The model’s cost and operating parameters include generation and load-shedding costs, VOLL, conductor rating, switching-action limits, and generator limits.The nomenclature also defines conductor heat-balance quantities and the average conductor temperature.Source: Nomenclature · pp. 2–3
1. Introduction
The introduction identifies limited coordination among GETs as a research gap and presents an integrated framework for NTO, DLR, and VIDs. It also proposes iterative McCormick relaxation to address VID-related bilinear constraints.
1. Introduction
- Renewable integration and growing data-center loads add operational challenges, while new transmission construction and infrastructure upgrades can take years.The introduction also names congestion, voltage quality, and stability among the challenges.Source: §1 · p. 3
- Prior work largely optimized GETs individually, leaving limited attention to their coordinated operation and motivating a unified framework.The introduction cites examples combining DLR with NTO and VIDs with generator scheduling.Source: §1 · pp. 3–4
- The framework jointly models NTO, DLR, and VIDs, with node-breaker switching, controlled line-reactance adjustment, and weather-dependent transmission capacity.DLR uses real-time or forecasted weather, primarily wind speed and ambient temperature.Source: §1 · p. 4
- Iterative McCormick relaxation repeats the relaxation over smaller intervals to reduce computational burden and improve accuracy over standard McCormick relaxation.Flexible line-reactance adjustment creates bilinear constraints, making the optimization difficult to solve directly.Source: §1 · p. 4
- The framework is presented as a way to assess coordinated GET control’s potential to improve flexibility, renewable accommodation, and reliability compared with independent optimization.The introduction states these outcomes are difficult to achieve when technologies are optimized independently.Source: §1 · p. 4
2. Modeling Grid Enhancing Technologies
The modeling section combines node-breaker topology decisions, weather-based dynamic line ratings, and bounded variable line susceptance within the transmission-network model.
2.1. Network Topology Optimization Model
- The node-breaker NTO model allows busbars to be connected or split, assigns generators and loads to either busbar, and represents line switching.The model constrains the total number of bus-splitting and line-switching actions over the horizon and balances power at each busbar.Source: §2.1 · pp. 4–5
- The network uses a DC power-flow approximation, with line capacities represented over time and line susceptance fixed when impedance control is absent.The approximation assumes small voltage-angle differences, negligible line resistance relative to reactance, and voltage magnitudes near 1 per unit.Source: §2.1 · p. 5
2.2. Dynamic Line Rating Model
- DLR adjusts transmission capacity using weather-dependent conductor ratings, with allowable current computed from forecasted ambient temperature and wind speed relative to the SLR reference.For hourly scheduling, conductor temperature is assumed to reach steady state within each interval; the SLR reference uses 40°C and 0.5 m/s wind speed.Source: §2.2 · pp. 5–6
2.3. Variable Impedance Devices Model
- VIDs vary line susceptance within a bound set by a factor of nominal susceptance, enabling controlled changes to line impedance.The variable impedance is included to reroute power flows and help avoid congestion and reduce costs.Source: §2.3 · p. 6
3. Co-optimization Framework
The co-optimization framework minimizes total operational cost while coordinating NTO, DLR, and VID subject to their technology-specific constraints.
3. Co-optimization Framework
- The objective minimizes total operational cost, comprising generation cost and load-shedding cost.Load shedding is penalized according to the value of lost load.Source: §3 · p. 6
- The framework formulates coordinated operation of NTO, DLR, and VID as an optimization problem subject to each technology’s constraints.The final formulation is nonlinear because VID-related bilinear constraints enter when the VID model is substituted into the topology model.Source: §3 · p. 6
- Generation cost is modeled as a linear function of generator output, and the formulation can represent piecewise linear approximations of quadratic cost curves.The model defines generator output across the time horizon.Source: §3 · p. 6
4. Solution Methodologies
The section compares direct solution of the bilinear formulation with static and iterative McCormick relaxations. Iterative refinement tightens variable bounds to reduce the relaxation gap while retaining an MILP framework.
4. Solution Methodologies
- The section evaluates Nonlinear, Static McCormick, and Iterative McCormick to compare numerical precision and computational tractability for the bilinear VID constraints.The VID bilinear constraints make the optimization problem nonlinear, motivating comparison of solution strategies.Source: §4 · p. 6
4.1. Nonlinear Approach
- The Nonlinear approach directly solves the original bilinear formulation and serves as a solution-quality benchmark, but can be computationally intensive as networks or topology decisions grow.It uses an optimization solver capable of handling non-convexity.Source: §4.1 · p. 6
4.2. Static McCormick Approach
- Static McCormick introduces auxiliary variables and McCormick envelopes for VID bilinear terms, converting the relaxed non-convex constraints into linear inequalities and an MILP.It applies the standard relaxation noniteratively.Source: §4.2 · pp. 6–7
4.3. Iterative McCormick Approach
- The Iterative McCormick approach incrementally tightens variable bounds to reduce the relaxation gap while preserving MILP tractability and improving accuracy without directly solving the bilinear formulation.The algorithm repeatedly solves with McCormick relaxation and updates bounds toward a target relaxation level.Source: §4.3 · p. 7
5.1. Simulation Setup
The simulations use multiple benchmark transmission systems, time-varying load and weather inputs, and common operating settings. They compare fixed and optimized topology while varying DLR and VID deployment.
5.1.1. Test Systems and Data
- The study analyzes IEEE 118-bus in detail and tests scalability on case24_ieee_rts, pglib_case588, and case1354pegase, using 24-hour load profiles and Austin weather data for DLR ratings.Hourly loads scale nominal bus demand; ambient temperature and wind speed inform time-varying thermal rating coefficients.Source: §5.1.1 · p. 7
5.1.2. Parameter Settings
- Generator minimum output is set to zero, VID susceptance adjustment is ±10%, VOLL is 2,000 $/MWh, and SLR and DLR share parameter settings except line ratings.Source: §5.1.2 · p. 7
5.1.3. Implementation and Solution Methods
- The models are implemented in MATLAB and solved with Gurobi 12.0.31; Iterative McCormick uses a 5% incremental step size.Simulations ran on an Apple M4 Pro 14-core system with 24 GB unified memory.Source: §5.1.3 · p. 7
5.1.4. Simulation Scenarios
- Fixed-topology cases optimize DLR and VID without switching, while optimized-topology cases co-optimize topology and GETs and stop at a 0.5% gap or 3,600 seconds.Source: §5.1.4 · p. 7
- Both scenarios impose line-flow limits at 80% of base-case loading and incrementally deploy DLR and VID on lines ranked by average utilization; Table 1 reports fixed-topology nonlinear operational costs by deployment level.The prioritization is based on a base-case optimal power flow without VID, DLR, or NTO.Source: §5.1.4 · p. 7, Table 1 · p. 8
5.2. Simulation Results
In fixed-topology tests, DLR drives most of the operational-cost and line-loading reductions, while VID adds flow redistribution and greater savings when combined with DLR. With optimized topology, NTO reduces costs, and Iterative McCormick reaches the target optimality gap in scenarios where the nonlinear approach does not.
5.2.1. Fixed Topology
- 24.53% is the maximum total cost reduction from combined DLR and VID deployment; at 40% DLR, the absolute reduction from 40% VID rises from 12.95 × 10^4 $ to 15.23 × 10^4 $.The reported increase indicates that VID's cost-saving efficacy is enhanced when paired with DLR.Source: Figure 3 · p. 8, §5.2.1 · p. 8
- 21.31% versus 2.74%: at 40% deployment, DLR alone reduces operational cost more than VID alone.Increasing DLR penetration consistently decreases cost, while cost responds relatively marginally to VID penetration.Source: Figure 3 · p. 8, §5.2.1 · p. 8
- 0.653 to 0.575 (mean Δ̄u = −0.078) is the system-wide average-loading change with DLR; VID-only changes it by +0.020, and DLR+VID by −0.072.DLR raises capacity, while VID changes flow distribution by modifying line reactance; combined loading relief is close to DLR-only.Source: Figure 4 · p. 8, §5.2.1 · p. 9
- More than two orders of magnitude lower computation time than Nonlinear is reported for McCormick methods; Iterative McCormick takes longer than Static but has lower relative errors.The iterative method's runtime remains on the same order as Static McCormick, and its errors are measured against the Nonlinear solution.Source: Table 2 · p. 9, Figure 5 · p. 9, §5.2.1 · p. 9
5.2.2. Optimized Topology through NTO
- 2.33% in the base case and 3.84% in the (20, 20) scenario are the reported cost reductions from NTO setup (5, 2) relative to fixed topology.The comparison uses Nonlinear results for fixed topology and Iterative McCormick results for optimized topology.Source: Table 3 · p. 9, §5.2.2 · pp. 9–10
- At the 3,600s limit, Nonlinear fails to reach the target optimality gap in several optimized-topology scenarios, whereas Iterative McCormick always reaches it.The comparison assesses computational performance under optimized topology.Source: Table 4 · p. 10, §5.2.2 · p. 10
5.3. Validation for Other Benchmark Test Cases
Across the evaluated networks and scenarios, NTO consistently reduces operational cost, while combined DLR and VID generally yields the lowest cost. The proposed approach solves the full 24-hour case24_ieee_rts NTO problem within a few minutes and obtains case1354pegase solutions within 30 minutes for selected operating periods.
5.3. Validation for Other Benchmark Test Cases
- NTO consistently reduces operational cost across the evaluated networks and scenarios, and combined DLR and VID generally produces the lowest cost.The relative effectiveness of DLR and VID varies across systems with network topology, congestion patterns, and device placement.Source: Table 5 · p. 10, §5.3 · p. 10
- Within a few minutes for the full 24-hour case24_ieee_rts profile and within 30 minutes for case1354pegase, Iterative McCormick solves the reported NTO problems.The large-scale test systems use representative selected operating periods rather than the full daily horizon.Source: §5.3 · p. 10
6. Conclusion
The paper presents a co-optimization framework for NTO, VID, and DLR, using iterative McCormick relaxation to address bilinear constraints. Benchmark simulations report a balance between solution accuracy and computational efficiency compared with nonlinear and static McCormick approaches.
6. Conclusion
- The framework co-optimizes NTO, VIDs, and DLR to minimize operational cost, with bilinear constraints arising from variable impedance in the topology optimization model.The problem is mixed-integer nonlinear, and the paper addresses its nonlinearity with McCormick relaxation enhanced by an iterative approach.Source: §6 · p. 10
- Across multiple benchmark networks, Iterative McCormick offers a favorable accuracy-runtime trade-off against exact but slower Nonlinear and efficient but less accurate Static McCormick methods.The reported validation is through simulations on benchmark networks.Source: §6 · p. 10
- Future work will extend the framework to other controllable resources, including energy storage systems.Source: §6 · p. 10