An Efficient Hybrid Heuristic for the Transmission Expansion Planning under Uncertainties

We address the stochastic transmission expansion planning (STEP) problem considering uncertainties in renewable generation capacity and demand. STEP’s objective is to minimize the total investment cost of new transmission lines and generation cost. To tackle the computational challenges of large-scale systems, we propose a heuristic approach that combines the progressive hedging (PH) algorithm for scenario-wise decomposition with an integrated framework for solving the resulting subproblems. The latter combines a destroy-and-repair operator, a beam search procedure, and a mixed-integer programming approach. The proposed framework is evaluated on large-scale systems from the literature, containing up to 10000 nodes, adapted to multiple scenarios based on parameters from the California test system (CATS). Compared with a non-trivial baseline algorithm that includes the integrated MIP and heuristics, the proposed PH-based framework consistently improved solution quality for the six systems considered (including CATS), achieving an average optimality gap reduction of 16.23% within a 2-hour time limit.

💡 Research Summary

The paper tackles the stochastic transmission expansion planning (STEP) problem, which seeks to minimize the combined investment cost of new transmission lines and the expected operational cost of generation under uncertainties in renewable generation capacity and demand. Traditional mixed‑integer programming (MIP) formulations become intractable for large‑scale power systems (thousands to tens of thousands of buses) because the number of binary investment decisions and scenario‑specific constraints explode. To overcome this, the authors propose a hybrid solution framework that couples a scenario‑wise decomposition method—Progressive Hedging (PH)—with a powerful heuristic engine for solving each scenario subproblem.

The PH component follows the cost‑proportional approach of Watson and Woodruff. It starts from a feasible incumbent that includes all candidate lines, solves each scenario independently, and then aggregates the scenario solutions by enforcing a consensus on the binary line‑building decisions. After each PH iteration two candidate global solutions are generated: a β‑intersection solution (lines built in at least β % of scenarios) and a union solution (lines built in any scenario). Their costs are evaluated, and if power‑flow violations appear, a dedicated repair routine is invoked. The algorithm iterates until a time limit is reached, always keeping the best penalized‑cost solution.

Within each scenario, the authors introduce an integrated heuristic consisting of three layers:

1. Destroy‑and‑Repair (D&R) – Starting from an initial set of candidate lines K′, the algorithm computes residual flows on those lines from an LP solution. A proportion rt (initially 0.5) of the lines with the highest residual flows is selected for removal. The LP is resolved, violations are measured, and if infeasibility occurs a repair step attempts to re‑insert some of the removed lines. The removal proportion rt is adaptively increased or decreased (by 0.5 × dt) based on whether the new solution improves cost and satisfies constraints. A binary‑search‑like scheme refines rt until a stopping condition (max iterations, time limit, or convergence) is met. D&R quickly yields a feasible, reasonably good solution for a single‑scenario TEP.

2. Iterative Beam Search (BS) – To escape local minima, BS maintains a beam of promising partial solutions. At each level it expands a limited number of branches (controlled by parameters itbs, η, ω, N, γ), evaluates them with the LP model, and keeps only the best‑scoring ones. The beam width and depth are dynamically adjusted based on improvement stagnation. This multi‑solution exploration improves the quality of the D&R output without excessive computational burden.

3. Exact MIP refinement – After the heuristic phase, any remaining time up to the per‑scenario limit tl is spent solving the subproblem with a commercial MIP solver, using the heuristic solution as a warm start. This step guarantees that the final solution is at least as good as the heuristic one and often closes the optimality gap further.

The complete workflow is: (i) decompose the stochastic STEP problem with PH, (ii) for each scenario apply D&R → BS → MIP, (iii) aggregate scenario solutions and iterate PH. The authors evaluate the approach on six large‑scale test systems derived from the California Test System (CATS), ranging from a few thousand to 10,000 buses, with multiple scenarios generated from realistic renewable and demand data. A 2‑hour computational budget is imposed for each instance. Compared with a strong baseline that already combines an integrated MIP with heuristics, the proposed PH‑based hybrid framework consistently yields better solutions, achieving an average optimality‑gap reduction of 16.23 % across all cases. The results also demonstrate good scalability: even when the number of scenarios grows, the PH decomposition keeps memory and runtime manageable, while the heuristic efficiently handles each subproblem.

Key contributions of the work are:

• A simple yet effective D&R operator that leverages residual line flows and binary‑search‑style adjustment of removal ratios.
• An iterative beam‑search heuristic that simultaneously explores multiple promising solutions, enhancing the D&R output.
• A hybrid PH‑heuristic‑MIP architecture that decomposes large stochastic TEP problems and solves them with high quality within practical time limits.
• Empirical evidence on realistic, large‑scale networks showing substantial gap reductions over existing methods.

The paper concludes that the proposed hybrid approach offers a practical pathway for utilities and planners to address stochastic transmission expansion in modern power systems with high renewable penetration. Future research directions include automated parameter tuning, integration of reinforcement‑learning‑based line selection, and validation on real operational data and multi‑year planning horizons.