Transmission Topology Optimization

• Transmission topology optimization is a method that strategically adjusts network switching and generation dispatch to reduce operational costs and congestion.
• It employs both bus-branch and node-breaker formulations to enhance reliability, manage renewable variability, and meet security constraints.
• Advanced formulations using MILP, learning-assisted heuristics, and evolutionary methods accelerate solution times and improve system robustness.

Transmission topology optimization is the class of power-system optimization problems that selects and coordinates transmission-network switching actions to improve system performance. In its most common forms, it co-optimizes generation dispatch with line statuses, breaker configurations, busbar splits, or related substation actions in order to reduce production cost and congestion, relieve violations, reduce losses and renewable curtailment, improve reliability margins, and enable higher variable renewable energy penetration. The topic is variously framed as Optimal Transmission Switching (OTS), Network Topology Optimization (NTO), Transmission Network Topology Control (TNTC), or, when transition trajectories are modeled explicitly, topology transition or bumpless topology transition. Across these formulations, the central difficulty is the same: binary topological decisions are coupled to continuous network physics, producing non-convex hybrid optimization problems that are NP-hard and operationally constrained by security, connectivity, protection, and implementation requirements (Han et al., 5 Jun 2026).

1. Conceptual scope and control granularity

In transmission operations, topology control is used as either a preventive or a corrective action. Preventive topology control selects a topology before an operating hour to preempt expected congestion due to forecasted renewable variation, whereas corrective topology control changes topology after a contingency or after violations are identified by real-time contingency analysis. The historical literature began with corrective switching, but more recent work places topology decisions directly inside day-ahead or hour-ahead optimization, and increasingly studies the transition between topologies rather than only the terminal network state (Little et al., 2021).

Two network representations dominate the field. In bus-branch formulations, the decision variables are primarily line on/off statuses, and the optimization changes the graph by opening or closing selected transmission lines. In node-breaker formulations, the decision space is finer-grained: each substation can be represented with multiple busbars and breaker-controlled attachments of generators, loads, and line ends. This enables busbar splitting and merging, selective attachment of equipment to different busbars, and line switching within the same unified model. In that sense, node-breaker NTO is more granular than classical bus-branch OTS, and it is particularly natural when substations use breaker-and-a-half or similar arrangements (Park et al., 23 Feb 2026).

A second axis of variation concerns the role of network-side flexibility relative to other remedial measures. Topology optimization is frequently positioned as a low-direct-cost alternative or complement to redispatch, countertrading, FACTS, HVDC control, storage, and demand-side flexibility. This does not imply that topology is unconstrained; rather, it means that the optimization objective usually penalizes generation cost, load shedding, or curtailment more heavily than switching itself, unless explicit switching costs or switching-frequency limits are added (Han et al., 5 Jun 2026).

2. Canonical mathematical formulations

Under the linear DC approximation, the standard DC-OTS model uses buses $N$, lines $L \subseteq N \times N$, and a switchable subset $L_S \subseteq L$. For each node $i \in N$, the principal variables are generator output $P_i$, demand $d_i$, and angle $\theta_i$; for each line $(i,j) \in L$, the variables are flow $f_{ij}$ and, for switchable lines, binary status $x_{ij} \in \{0,1\}$. The objective is to minimize total generation cost,

$L \subseteq N \times N$0

subject to generator bounds, nodal power balance, DC power flow on non-switchable lines, thermal limits, a slack-angle constraint, and switching physics. A common mixed-integer linearization for switchable lines is

$L \subseteq N \times N$1

together with big-$L \subseteq N \times N$2 constraints

$L \subseteq N \times N$3

which enforce $L \subseteq N \times N$4 when the line is on and $L \subseteq N \times N$5 when it is open. This yields a mixed-integer linear program (Pineda et al., 2023).

Node-breaker NTO expands the decision space. In the co-optimization model for NTO with Dynamic Line Rating (DLR) and Variable Impedance Devices (VIDs), each substation $L \subseteq N \times N$6 has two busbars $L \subseteq N \times N$7 and breaker-status variables such as $L \subseteq N \times N$8 for busbar merge/split, $L \subseteq N \times N$9 and $L_S \subseteq L$0 for generator and load attachment, $L_S \subseteq L$1 for line-end attachment, and $L_S \subseteq L$2 for line open/closed status. The DC flow relation becomes

$L_S \subseteq L$3

while DLR updates thermal limits through

$L_S \subseteq L$4

and VIDs modify effective susceptance by

$L_S \subseteq L$5

Because substituting $L_S \subseteq L$6 into the DC flow equations introduces bilinear $L_S \subseteq L$7 terms, the resulting problem is a Mixed-Integer Nonlinear Programming (MINLP) model handled with McCormick envelopes (Park et al., 23 Feb 2026).

A third family targets congestion management via busbar splitting under a linearized AC model. There, each substation has a binary busbar-coupler variable $L_S \subseteq L$8 and connected elements $L_S \subseteq L$9 are assigned to busbars $i \in N$0 or $i \in N$1 through binaries $i \in N$2. The optimization minimizes a congestion penalty based on normalized apparent flows,

$i \in N$3

with piecewise linearization to obtain a MILP. The formulation imposes a maximum number of splits, minimum connectivity per busbar when split, and symmetry-breaking when the coupler is closed (Rajaei et al., 23 Oct 2025).

For day-ahead operation under renewable uncertainty, topology optimization has also been formulated as a stochastic Mixed-Integer Quadratic Convex Problem based on the LPAC approximation of AC power flow. In that setting, OTS and busbar splitting are combined for AC and hybrid AC/DC grids, with scenario-dependent injections, topology policies that can vary hourly or remain fixed over 24 hours, and explicit limits on the number of switching actions (Bastianel et al., 17 Sep 2025).

3. Computational structure, hardness, and exact strengthening

The core computational fact is that DC-OTS is NP-hard on general networks. The hardness persists under special structures, and even the problem of computing tight big-$i \in N$4 constants is itself NP-hard. This matters because large $i \in N$5 values weaken the LP relaxation, enlarge the feasible region in branch-and-bound, and degrade solver performance; conversely, valid but tighter bounds materially improve mixed-integer solution times (Pineda et al., 2023).

A widely used exact starting point is the shortest-path big-$i \in N$6 construction over a connected spanning subgraph of non-switchable lines. In the formulation summarized in the literature, if $i \in N$7 is the shortest path between the endpoints of a switchable line $i \in N$8 under edge costs $i \in N$9, then

$P_i$0

This is computationally cheap, but it can be loose and symmetric by construction. A stronger alternative solves linear bounding problems that incorporate an upper bound on total generation cost, yielding asymmetric cost-driven big-$P_i$1 values and, optionally, reduced line capacities. On 100 IEEE 118-bus instances, the baseline SP-OC method required about $P_i$2 seconds on average and left $P_i$3 instances unsolved within one hour, whereas the iterative BT-RC-H(4) variant reduced total average time to about $P_i$4 seconds, including about $P_i$5 seconds of bounding time and about $P_i$6 seconds of MILP solution time, with $P_i$7 unsolved instances (Pineda et al., 2023).

When all lines are switchable, exact modeling usually requires explicit connectivity constraints. One approach uses MTZ-type variables and directional line-selection variables to prevent islanding, while another exploits graph structure to compute valid big-$P_i$8 values from relaxed longest-path problems. In the all-line IEEE 118-bus setting, the simplified graph plus relaxed longest-path bounds were within $P_i$9 on average of exact longest-path bounds, yet were computed in about $d_i$0 seconds for the whole network, versus an average of $d_i$1 seconds for exact longest-path computation. Combined with an iterative incumbent-generation heuristic, this reduced average MILP time by $d_i$2 and unsolved instances by $d_i$3 compared to conventional methods (Aguilar-Moreno et al., 14 Feb 2025).

Exact sequential planning can also exploit temporal structure rather than only polyhedral strengthening. In the day-ahead multi-objective setting, the block algorithm partitions the 24-hour horizon into blocks of constant topology and enumerates the complete Pareto front over worst-case $d_i$4 loading, topological depth, switching count, and time away from the reference topology. For fixed bounds on depth and switch count, its evaluation count grows polynomially with the planning horizon, and on a highly congested day in the Dutch high-voltage grid it computed the full Pareto front in under three minutes (Groeneveld et al., 5 May 2026).

4. Sensitivity-based, heuristic, learning-assisted, and evolutionary methods

A large body of work addresses the practical gap between exact mixed-integer formulations and real-time or near-real-time operation by accelerating search rather than replacing physical constraints. In corrective TNTC for post-contingency overload relief, two classic screening factors are the Transmission Switching Distribution Factor (TSDF) and the Flow Transfer Distribution Factor (FTDF). Under the DC model, TSDF predicts the sensitivity of a monitored overloaded line to opening a candidate switching line, while FTDF multiplies that sensitivity by the post-contingency flow on the switching line, thereby estimating the actual MW redistribution. On the 2,383-bus Polish system, FTDF10 and FTDF20 achieved average violation reductions of $d_i$5 and $d_i$6, close to complete enumeration at $d_i$7, while finishing in $d_i$8–$d_i$9 seconds versus $\theta_i$0 seconds for complete enumeration (Li et al., 2019).

Solver-coupled heuristic architectures push this further. In asynchronous parallel OTSP, one process solves the full MILP while worker processes solve restricted subproblems and asynchronously inject improved incumbents into branch-and-bound. This architecture was tested on pglib-opf instances up to $\theta_i$1 buses and $\theta_i$2 branches, and on the 1354_pegase case it reached optimality in about $\theta_i$3 seconds with three workers, more than $\theta_i$4 faster in time-to-optimality than the best single-solver baseline (Hinneck et al., 2021).

Data-driven acceleration has recently become a dominant theme. In learning-assisted DC-OTS, historical solved instances are used through $\theta_i$5-nearest neighbors on load vectors to predict line-status consensus, fix high-confidence binaries, and tighten big-$\theta_i$6 constants either by updated shortest paths or by learned angle-difference bounds. On the IEEE 118-bus testbed, the exact baseline required about $\theta_i$7 seconds on average for Unif10 instances, while FixB-FatM achieved $\theta_i$8 optimal solutions in about $\theta_i$9 seconds and AngM with safety factor $(i,j) \in L$0 achieved $(i,j) \in L$1 optimal in about $(i,j) \in L$2 seconds, corresponding to about a $(i,j) \in L$3 speed-up. On correlated Normal-demand data, AngM with $(i,j) \in L$4 achieved $(i,j) \in L$5 optimal solutions in about $(i,j) \in L$6 seconds, about a $(i,j) \in L$7 speed-up versus the baseline (Pineda et al., 2023).

Graph learning pushes acceleration into substation-level reconfiguration. For congestion management via busbar splitting, a heterogeneous edge-aware message-passing network predicts effective split actions and then restricts the MILP to top candidates. On the GOC 2000-bus system, this produced AC-feasible solutions within about one minute, up to $(i,j) \in L$8 speed-up relative to the full MILP, and about a $(i,j) \in L$9 optimality gap in the accelerated pipeline (Rajaei et al., 23 Oct 2025). In a related breaker-level framework, OptiGridML combines a line-graph GNN that approximates DC flows with a heterogeneous GNN that predicts breaker states under structural and physical constraints. On synthetic networks with up to $f_{ij}$0 breakers, it delivered power export improvements of up to $f_{ij}$1 over baseline topologies while reducing inference time from hours to milliseconds (Meng et al., 3 Aug 2025).

Evolutionary quality-diversity methods address a different operational need: not only a single best topology, but a structured set of alternatives. In accelerated MapElites for TTO, a fully GPU-native DC load flow loop evaluates large batches of topologies and populates a repertoire over descriptors such as number of branch disconnections, number of substations split, and number of reassignment operations. The end-to-end pipeline, including importing and AC validation, runs in under 15 minutes; in the reported TSO studies, the DC optimizer evaluated 768,064 and 512,128 topologies in about three minutes, and the best AC-accepted topologies reduced overload energy from $f_{ij}$2 MW to about $f_{ij}$3 MW in one case and from $f_{ij}$4 MW to about $f_{ij}$5 MW in another (Westerbeck et al., 11 May 2026).

5. Security, uncertainty, and the transition between topologies

A recurrent operational issue is that the terminal optimum does not determine a safe implementation path. Optimal Topology Transition (OTT) therefore models a sequence of transitional topologies between an initial topology and a terminal topology, optimizing both transition feasibility and transition performance. In the DC formulation, each period has a binary line-status vector $f_{ij}$6, transitional steady states, intermediate topologies $f_{ij}$7, and connectedness constraints enforced through auxiliary flow variables. The objective combines boundedness, volatility, switching cost, slack penalties, and the number of batches. Numerical studies on IEEE 39-bus, IEEE 118-bus, and a German transmission network showed that ad hoc trajectories frequently violated operational constraints, whereas the optimal trajectories eliminated most of those violations; when no feasible trajectory existed to the steady-state-optimal topology, transition-embedded topology optimization found a transition-feasible optimal topology with dispatch-cost increases usually below a few per mille (Han et al., 2022).

Bumpless Topology Transition (BTT) extends this logic to dynamic performance. Each transition episode consists of an ACV-adjustment step followed by a single line-switching event, and the method co-optimizes line-switching sequence with fast control resources such as generator terminal voltages, ESS active-power orders, DVC setpoints, TCSC reactance, and converter virtual inertia and damping. The first stage is a convex mixed-integer program that minimizes a surrogate bumpiness metric; the second stage restores full AC feasibility through nonlinear programs; the third stage minimizes the $f_{ij}$8 norm of the linearized post-switching dynamics. On a modified IEEE 9-bus system, the BTT-optimal scheme reduced total bumpiness from $f_{ij}$9 and $x_{ij} \in \{0,1\}$0 in the best comparison schemes to $x_{ij} \in \{0,1\}$1, and on a modified IEEE 118-bus system the same qualitative superiority persisted across 200 scenarios (Han et al., 2022).

Security and uncertainty are equally central in day-ahead formulations. Scenario-based day-ahead topology optimization under renewable forecast uncertainty combines OTS and busbar splitting in AC and hybrid AC/DC grids, using real offshore wind forecast errors clustered into representative scenarios. The study compared hourly-optimized topology, one topology for 24 hours, and limited switching formulations, and reported that accounting for renewable uncertainty with at least 6 to 8 scenarios led to lower or comparable total costs relative to deterministic day-ahead forecasts, even when the frequency of topological actions was limited (Bastianel et al., 17 Sep 2025).

A closely related extension is the co-optimization of NTO with DLR and VIDs. In that framework, topology decisions, weather-driven thermal ratings, and controllable susceptance changes are optimized jointly. The formulation emphasizes that these technologies interact: DLR increases available capacity on specific corridors, NTO reroutes flows through topological changes, and VIDs reshape the electrical distribution of flows. This interaction is beneficial but also makes the model more nonlinear and more computationally demanding (Park et al., 23 Feb 2026).

6. Empirical impact, recurring limitations, and research directions

Across studies, the reported operational value of transmission topology optimization is large but context-dependent. In a modified RTS-96 with increasing wind penetration, annual gains from optimal topology control increased monotonically with wind penetration, reaching up to $x_{ij} \in \{0,1\}$2 million annual savings and exceeding $x_{ij} \in \{0,1\}$3 of total production cost at high wind; the network-induced cost gap between copper plate and the constrained base network rose to $x_{ij} \in \{0,1\}$4 million and as much as $x_{ij} \in \{0,1\}$5 of total production cost, and topology control reduced wind curtailment by up to $x_{ij} \in \{0,1\}$6 TWh relative to the no-topology case (Little et al., 2021). In the GET co-optimization study on RTS-24, NTO alone reduced base cost by $x_{ij} \in \{0,1\}$7, DLR plus NTO raised the reduction to about $x_{ij} \in \{0,1\}$8 under high wind, and the full NTO+DLR+VID configuration achieved up to $x_{ij} \in \{0,1\}$9 cost reduction relative to fixed topology with no GETs (Park et al., 23 Feb 2026).

At the same time, several limitations recur with unusual consistency. First, most scalable formulations still rely on DC or linearized AC approximations; post-solve AC power flow checks, AC OPF validation, or explicit AC recovery stages are therefore repeatedly recommended before implementation (Pineda et al., 2023). Second, topology decisions interact with protection coordination, observability, breaker wear, switching-frequency limits, and market design in ways that are only partially captured by standard objective functions. Third, security modeling is uneven across the literature: some studies are explicitly $L \subseteq N \times N$00 or contingency-aware, while others use base-case formulations and then discuss security as future work. Fourth, the transition problem is not ancillary. Work on OTT and BTT shows that even when a terminal topology is attractive in steady state, the path between the initial and terminal topologies can be infeasible, dynamically poor, or operationally unacceptable (Han et al., 2022).

This suggests a current research frontier defined less by whether topology matters than by how to make it reliable, scalable, and operator-usable. The monographic survey identifies steady-state transmission control, topology transition, and transient topology control as distinct but increasingly connected subfields, and the recent literature adds several convergent directions: stronger exact formulations and connectedness constraints; scenario-based and distributionally robust planning under VRE uncertainty; coordination with DLR, FACTS, HVDC, and storage; graph learning and solver-aware ML for candidate ranking, warm starts, and cut selection; and exact multi-objective planning tools that expose trade-offs among security, depth, switching count, and time away from reference topology rather than collapsing them into a single scalar objective (Han et al., 5 Jun 2026).