⚡ electrical engineering

On the Tightness of the Second-Order Cone Relaxation of the Optimal Power Flow with Angles Recovery in Meshed Networks

This letter investigates the tightness of the second-order cone relaxation for optimal power flow in meshed networks, focusing on voltage angle recovery and AC feasibility through theoretical analysis and numerical experiments on standard IEEE test cases.

Original authors: Ginevra Larroux, Matthieu Jacobs, Mario Paolone

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ginevra Larroux, Matthieu Jacobs, Mario Paolone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the electrical grid as a massive, complex city of roads where electricity is the traffic. The goal of the "Optimal Power Flow" (OPF) problem is to figure out the most efficient way to drive this traffic so that everyone gets to their destination without causing jams, while using the least amount of fuel (energy).

However, calculating the perfect route for this traffic is incredibly hard because the roads are curved, the speed limits change, and the cars interact in complicated ways. It's like trying to solve a 3D puzzle while blindfolded.

To make this easier, engineers use "relaxations." Think of a relaxation as drawing a simplified map of the city. Instead of worrying about every tiny curve and hill, you draw straight lines and flat roads. This makes the math easy to solve on a computer. The big question is: Is this simplified map accurate enough to actually drive on? If the solution you find on the flat map works in the real, bumpy world, the map is "tight." If not, you've found a route that looks good on paper but crashes in reality.

The Two Approaches

The paper compares two different ways of making these simplified maps for meshed networks (cities with lots of interconnected loops and shortcuts, like a spiderweb).

1. The "Blind" Approach (Reference [4])
This method ignores the direction of the traffic (voltage angles) while solving the problem. It solves the puzzle using only distances and speeds. Once it finds a solution, it tries to "recover" the directions afterward.

The Catch: In a city with loops (meshed networks), this is like trying to walk in a circle and expecting to end up exactly where you started without checking your compass. Sometimes, the math says you're back at the start, but in reality, you're a few feet off. The paper notes that this method often fails to guarantee a working solution in complex, looped cities.

2. The "Explicit" Approach (Reference [6])
This method tries to be smarter. It explicitly includes the direction (voltage angles) in the math from the very beginning. It claims that by doing this, it creates a perfect "relaxation"—a simplified map that is guaranteed to work in the real world, provided the angles aren't too extreme.

The Promise: The authors of the original paper [6] claimed that if their math says "zero gap" (meaning the simplified map and the real world match perfectly), then the solution is definitely safe and feasible.

The Big Reveal: The "Loop" Problem

The authors of this new paper (Larroux, Jacobs, and Paolone) say: "Hold on a minute. That promise isn't true for looped cities."

They argue that the "Explicit" approach (Reference [6]) isn't actually a perfect relaxation; it's more like a good approximation.

Here is the analogy:
Imagine you are walking through a park with a network of paths.

The Real World (AC-OPF): You walk a path, turn left, go around a lake, and come back to the start. You must end up exactly where you began.
The Simplified Map (SOC-OPF in [6]): The map assumes that if you walk a certain distance and turn a certain angle, you will return to the start.
The Flaw: In a complex looped network, the math in the simplified map can get "confused" by the loops. It might calculate a path that looks valid on the map, but when you try to walk it in the real world, the angles don't add up. You might end up 10 feet away from your starting point, or worse, the path might lead you into a wall.

The paper proves that even if the math says the "gap" is zero (the map looks perfect), the solution might still be impossible to drive in the real electrical grid because the angles around the loops don't close the circle correctly.

The Experiment: A Test Drive

To prove this, the researchers took two standard test cities (the IEEE 33-bus and 39-bus systems):

The Radial City (Tree-shaped): No loops. Just one way to get anywhere.
The Meshed City (Web-shaped): Lots of loops and shortcuts.

They took a real, working solution and tried to fit it into the "Explicit" simplified map.

In the Radial City: The map worked perfectly. The angles matched up, and the solution was valid.
In the Meshed City: The map failed. Even though the math looked "tight," the angles around the loops didn't add up to zero. It was like a GPS telling you to drive in a circle, but the roads actually forced you to drift off course.

What Does This Mean for the Real World?

This paper is a crucial warning for power grid planners and engineers:

Don't Trust the "Zero Gap" blindly: Just because a computer says a solution is mathematically perfect (tight) doesn't mean it will work in a real, looped power grid.
Approximation vs. Guarantee: The method in Reference [6] is a useful tool, but it is an approximation, not a guarantee. It's like using a GPS that is 99% accurate; it's great for getting close, but you still need to double-check the final turn.
The Need for a "Reality Check": No matter which method you use, you eventually need a final step to verify that the angles actually close the loops correctly. If you skip this step in a complex grid, you might design a power system that looks great on paper but fails when you flip the switch.

In short: The paper shows that in a complex, looped electrical grid, you can't just simplify the math and assume it works. You have to be careful about how the "directions" of electricity add up, or you might end up with a plan that looks perfect but is physically impossible to execute.

1. Problem Statement

The paper addresses a critical issue in power system optimization: the validity of Second-Order Cone (SOC) relaxations for the Alternating Current Optimal Power Flow (AC-OPF) problem in meshed networks.

Context: AC-OPF is non-convex and computationally difficult. SOC relaxations (SOCP) are widely used because they are convex and solvable efficiently.
The Conflict:
- In radial networks, SOC relaxations are often "exact" (tight), meaning the relaxed solution corresponds to a feasible AC solution.
- In meshed networks, voltage angles must satisfy cycle consistency constraints (Kirchhoff's Voltage Law around loops).
The Specific Gap: The authors investigate the formulation proposed in reference [6], which explicitly includes bus voltage angles in the SOC-OPF constraints. Reference [6] claims that under certain voltage and angle bounds, replacing a non-linear trigonometric constraint with a linear one constitutes a valid relaxation (i.e., the feasible set of the relaxed problem contains the original feasible set).
Core Question: Does the formulation in [6] truly provide a relaxation for meshed networks, or is it merely an approximation that may exclude valid AC-OPF solutions?

2. Methodology

The authors employ a combination of theoretical analysis and numerical experimentation to challenge the claims made in reference [6].

A. Theoretical Analysis

The paper compares two dominant SOC-OPF formulations:

The Baseline Model [4]: Eliminates voltage angles and branch current angles, relaxing non-convex power-loss equalities. Angle recovery is attempted a posteriori (after solving).
The Explicit Angle Model [6]: Retains bus voltage angles ( $\theta_n$ ) as decision variables. It replaces the non-linear branch voltage drop equation:
$V_{sl}V_{rl} \sin(\theta_l) = X_l P_{sl} - R_l Q_{sl}$
with a linearized constraint:
$\theta_l = X_l P_{sl} - R_l Q_{sl}$
(assuming $V \approx 1$ and $\theta \approx 0$ ).

Theoretical Argument:
The authors argue that while the linearization (9a) might appear to expand the feasible region locally, it fails to account for global cycle consistency in meshed networks.

In radial networks, nodal angles are uniquely determined by branch differences.
In meshed networks, branch angle differences must sum to zero (modulo $2\pi$ ) around any cycle.
The authors prove (Lemma 1) that if the non-linear constraint (5d) is replaced by the linear constraint (9a), the resulting branch angle differences do not guarantee satisfaction of the cycle constraint (11). Consequently, a solution feasible for the linearized model may not correspond to any valid set of nodal angles in the original AC-OPF.

B. Numerical Counterexample

To validate the theoretical claim, the authors conducted experiments on two standard IEEE test cases using the pandapower library:

IEEE 33-bus: A radial network.
IEEE 39-bus: A meshed network.

Experimental Steps:

Baseline Verification: They took a known feasible AC-PF solution from pandapower and verified it satisfied the non-relaxed formulation of [6] (o-ACOPF).
Consistency Test: They constructed the branch angle differences implied by the linearized SOC-OPF model (using the linear relation) and attempted to reconstruct the nodal angles ( $\theta_n$ ) by solving the linear system $(A_{sl} - A_{rl})\theta_n = \theta_l$ .
Residual Analysis: They calculated the residual norm of this system.
- If the residual is zero, the cycle consistency holds.
- If non-zero, the linearized angles cannot form a consistent set of nodal angles.

3. Key Contributions

Refutation of Theorem 1 in [6]: The paper provides a rigorous counterexample and theoretical proof that the formulation in [6] is not a relaxation of the AC-OPF for meshed networks. Instead, it is an approximation.
Clarification of Tightness: The authors demonstrate that even if the conic relaxation of the quadratic loss equality is "tight" (zero gap), the linearization of the angle constraint can still render the solution AC-infeasible in meshed networks.
Distinction between Relaxation and Approximation: The paper clarifies that a zero relaxation gap does not guarantee AC-OPF feasibility in meshed networks when angle recovery is involved, specifically when linearized angle constraints are used.
Comparative Analysis: It highlights the trade-offs between the "angle elimination" approach [4] (which requires complex a posteriori recovery) and the "explicit angle" approach [6] (which is an approximation).

4. Results

Radial Networks (IEEE 33-bus): The numerical experiment confirmed that for radial networks, the linear system is consistent (residuals $\approx 0$ ). The cycle constraints are naturally satisfied.
Meshed Networks (IEEE 39-bus):
- The linear system derived from the SOC-OPF model yielded non-negligible residuals (maximum residual of 0.0245 degrees).
- This deviation, when propagated through AC power flow equations, resulted in branch power-flow errors on the order of 0.5 p.u. (per unit).
- Conclusion: The solution obtained from the SOC-OPF model in [6] is infeasible for the original AC-OPF problem because the implied branch angles violate Kirchhoff's Voltage Law around loops.

5. Significance and Implications

Planning and Operations: Power system planners relying on the formulation in [6] for meshed networks may obtain solutions that appear optimal and "tight" but are physically unrealizable (infeasible) when implemented.
Theoretical Impact: The paper undermines the theoretical foundation (Theorems 1–4) of reference [6] regarding exactness and recovery in meshed networks.
Future Directions:
- The paper suggests that for meshed networks, an a posteriori procedure (like those in [4] or the phase-shifter approach in [5]) is still necessary to ensure AC feasibility, even if the relaxation gap is zero.
- It emphasizes that simply including angles in the constraints does not solve the non-convexity issue in meshed topologies; the cycle consistency constraint remains a fundamental barrier to exact convex relaxation without additional mechanisms (like phase shifters).

In summary, this letter serves as a crucial correction to the literature, warning that the explicit angle SOC-OPF formulation is an approximation for meshed grids, not a guaranteed relaxation, and that its solutions require careful feasibility checks before deployment.