Topology-Based OBBT in Power Systems

• Topology-based OBBT is a technique that enforces integrality on a selected subset of switching variables to tighten MILP bounds in DC-OTS problems.
• The method uses a graph-distance heuristic to apply localized integrality, balancing feasible region quality with computational efficiency.
• Computational results on the IEEE 118-bus system show significant improvements, including solve time reductions up to 64% and enhanced bound tightening.

Topology-based Optimization-Based Bound Tightening (OBBT) is an advanced computational technique for strengthening mixed-integer linear programming (MILP) relaxations in network topology optimization, particularly in power system operations. In the context of the direct current optimal transmission switching (DC-OTS) problem, topology-based OBBT leverages the network structure to enforce integrality only on a strategically selected subset of switching variables when tightening bounds. This selective integrality ensures strong local feasibility and substantial tightening, while maintaining the tractability of the underlying bounding linear programs (LPs). The method achieves a "sweet spot" between the extremes of full relaxation (all binaries relaxed) and full integrality (all binaries enforced), thereby substantially improving computational performance for topology optimization tasks (Pineda et al., 22 Jul 2025).

1. Mathematical Framework for DC-OTS and Big-M Formulation

The DC-OTS problem is defined over a power network comprising:

• $\mathcal{N}$: Buses,
• $\mathcal{L}$: Switchable lines,
• $p_n$: Generation at bus $n$,
• $d_n$: Demand at bus $n$,
• $\theta_n$: Voltage phase angle at $n$,
• $f_l$: Actual power flow on line $l$,
• $\tilde f_l = b_l(\theta_n-\theta_m)$: Dummy DC flow for $l=(n,m)$,
• $x_l\in\{0,1\}$: Line status (on/off),
• $\underline p_n, \overline p_n$: Generator limits,
• $\underline f_l, \overline f_l$: Thermal limits.

The standard big-M MILP formulation is: $$\min_{p,\theta,f,\tilde f,x}\;\sum_{n\in\mathcal N}c_n\,p_n$$ subject to: \begin{align*} &f_l = x_l\,\tilde f_l, \ &\tilde f_l = b_l(\theta_n-\theta_m), \ &\underline f_l \le f_l \le \overline f_l, \ &x_l\in{0,1} \quad (l \in \mathcal{L}), \ &p_n - d_n = \sum_{l\in\mathcal{L}(n,\cdot)}f_l - \sum_{l\in\mathcal{L}(\cdot,n)}f_l \quad (n \in \mathcal{N}), \ &\theta_{n_0}=0. \end{align*}

To linearize the bilinear $f_l = x_l\tilde f_l$ term, big-M constraints are introduced: $$\begin{aligned} (1-x_l)\,\underline M_l &\le -f_l + \tilde f_l \le (1-x_l)\,\overline M_l, \ x_l\,\underline f_l &\le f_l \le x_l\,\overline f_l. \end{aligned}$$

The joint feasible region is denoted as $\mathcal R(F,M)$.

2. Conventional Relaxation-Based OBBT and Its Shortcomings

OBBT aims to improve continuous variable bounds by solving two LPs per scalar bound—one minimizing and one maximizing each $f_l$ or $\tilde f_l$—subject to the big-M constraints, relaxed $x \in [0,1]^{|\mathcal L|}$, and an upper-bound cut on cost. In fully-switchable networks, relaxing all binary variables $x_l$ breaks Kirchhoff’s laws for $0 < x_l < 1$, which decouples $f$ and $\tilde f$ and generates excessively loose feasible sets. As a result, conventional OBBT provides little to no meaningful bound tightening for topology optimization (Pineda et al., 22 Jul 2025).

3. Topology-Aware Subproblems: Localized Integrality

Topology-aware OBBT introduces a proximity parameter $k$ to determine which binary variables retain their integrality in the bounding subproblem. For a target line $l$, the $k$-hop neighbor set is defined as: $$\mathcal L^k_l = \{\text{lines at graph distance} \leq k \text{ from } l\}, \quad \mathcal L^0_l = \emptyset.$$ Integrality is enforced only for $x_j$ with $j \in \mathcal L^k_l$; the remaining $x$ variables are relaxed to $[0,1]$. The domain for the bounding LPs becomes: $$\mathcal X^k_l := \{ x \in [0,1]^{|\mathcal L|} : x_j \in \{0,1\} \ \forall j \in \mathcal L^k_l \}.$$ For each line $l$,

$$\underline f_l = \min\{f_l : (p,\theta,f,\tilde f,x)\in\mathcal R(F,M),\ x\in\mathcal X^k_l, x_l=1, \sum_n c_n p_n \leq \overline C\}$$

and analogously for other bounds, including $\underline M_l$ and $\overline M_l$ for big-M tightening.

Local Kirchhoff laws are thereby enforced in regions proximate to each line, enabling significant bound improvement while bounding subproblem size remains moderate.

4. Graph-Distance Heuristic for Binary Selection

The central selection heuristic employs the line–line adjacency graph $G_L$, with nodes representing lines and edges joining any pair of lines sharing a bus. For each $k$:

• $\mathcal L^1_l$ = lines sharing a bus with $l$,
• $\mathcal L^{k}_l$ = union of all lines adjacent in $G_L$ to lines in $\mathcal L^{k-1}_l$.

By choosing a modest $k$, integrality is enforced only for lines that can meaningfully constrain angle differences in the immediate electrical neighborhood of line $l$. This local enforcement provides a balance between bound quality and LP complexity.

5. Algorithmic Procedure: TBT-$k$ Scheme

The topology-based OBBT algorithm, referred to as TBT-$k$ (for "Topology-based Bound Tightening at level $k$"), consists of four principal steps:

1. For chosen $k$, compute initial bounds $(F^0, M^0)$ using conservative surrogates (e.g., longest-path estimates).
2. Run a 10 s heuristic MILP solve to obtain a feasible cost $\overline C$.
3. For each $l \in \mathcal L$, solve four LPs (min/max $f_l$ and $\tilde f_l$) subject to $\mathcal X^k_l$ to update bounds $$\underline f_l, \overline f_l, \underline M_l, \overline M_l$$.
4. Solve the full DC-OTS MILP with the tightened bounds.

This process enables significant tightening while keeping the OBBT cost manageable.

6. Computational Results: IEEE 118-Bus Benchmark

Extensive computational evaluatation was performed on the IEEE 118-bus system (186 switchable lines; 300 random demand profiles). The following methodologies were compared:

• MIP with naïve $M$ (no OBBT)
• TBT-0 (all $x$ relaxed)
• TBT-$k$ with $k=1,\ldots,5$
• SBT-$t$ (“solver OBBT” with all $x$ binary but with a time cutoff $t$ ms)

The average improvements and computational costs are summarized below:

Method	Bound Tightening $(\Delta F,\ \Delta M)$	Total Time $T^T$ (s)	Time-outs
MIP (naïve)	N/A	327	20
TBT-0	(10.1%, 0%)	308	19
TBT-1	(12.6%, 1.5%)	--	--
TBT-2	(14.2%, 3.4%)	181	9
TBT-3	(17.9%, 4.8%)	--	--
TBT-4	(21.1%, 10%)	279	13

For hard instances, TBT-2 reduced average solve times from 757 s to 274 s (–64%) and reduced time-outs from 17 to 5.

Results indicate that increasing $k$ from 0 up to 2 yields substantial gains in both bound tightness and solve time reduction, with diminishing returns and increased cost for higher $k$ values.

7. Practical Consequences and Potential Extensions

Retaining integrality only for a topology-defined subset of switch variables achieves most of the bound-strengthening benefits of full OBBT at a fraction of the computational expense. The method demonstrates scalability to large practical test systems and can be routinely embedded within optimal transmission switching tools for improved reliability and reduced solve times.

Possible extensions include:

• Adaptive or line-specific choice of $k$ using sensitivity or centrality metrics,
• Integration of cycle- or cut-based relaxations,
• Generalization to AC-OTS problems under convex relaxation,
• Warm-starting and parallelization of bounding LPs.

The topology-aware OBBT framework establishes a controllable compromise between relaxation and full integrality, yielding solve time reductions of 40–60% on challenging benchmarks and setting a new standard for scalable bound tightening in power system topology optimization (Pineda et al., 22 Jul 2025).