A Parallelized Cutting-Plane Algorithm for Computationally Efficient Modelling to Generate Alternatives

TL;DR

This paper tackles the challenge of exploring the near-optimal feasible space in large-scale capacity expansion models under a budget slack by MGA, which is computationally demanding when hundreds of solutions are needed. It introduces a Parallelizable Cutting-Plane to Generate Alternatives (CGA) method, a tailored Benders-based reformulation that solves many small operational subproblems in parallel and uses cut sharing and objective partitioning to accelerate convergence. The authors prove the equivalence to monolithic MGA formulations, develop stopping criteria, and demonstrate that CGA significantly outperforms monolithic MGA in both speed and memory usage for linear problems, while enabling large mixed-integer MGA problems to be solved. The practical impact is a scalable, high-fidelity MGA framework that supports rapid generation of diverse investment portfolios for high-resolution energy system planning, enabling better stakeholder decision-making under uncertainty.

Abstract

Contemporary macro energy systems modelling is characterized by the need to represent strategic and operational decisions with high temporal and spatial resolution and represent discrete investment and retirement decisions. This drive towards greater fidelity, however, conflicts with a simultaneous push towards greater model representation of inherent complexity in decision making, including methods like Modelling to Generate Alternatives (MGA). MGA aims to map the feasible space of a model within a cost slack by varying investment parameters without changing the operational constraints, a process which frequently requires hundreds of solutions. For large, detailed energy system models this is impossible with traditional methods, leading researchers to reduce complexity with linearized investments and zonal or temporal aggregation. This research presents a new solution method for MGA type problems using cutting-plane methods based on a tailored reformulation of Benders Decomposition. We accelerate the algorithm by sharing cuts between MGA master problems and grouping MGA objectives. We find that our new solution method consistently solves MGA problems times faster and requires less memory than existing monolithic Modelling to Generate Alternatives solution methods on linear problems, enabling rapid computation of a greater number of solutions to highly resolved models. We also show that our novel cutting-plane algorithm enables the solution of very large MGA problems with integer investment decisions.

Figures (5)

• Figure 1: Convergence of primal feasibility in numerical examples for a 12-zone, continental US electricity system planning problem with one planning stage and 8,736 operational hours (an LP with 14.3 million variables, 25.6 million constraints) with least-cost and MGA formulations with $\beta =0.1$ and $2.0$. The MGA problem with the tighter budget converges much more slowly, indicating increased computational effort relative to least-cost solutions or loose budget constraints.
• Figure 2: Convergence of Total System Cost in example solutions of 26-zone, 8736-hour MGA problem with 30.1 million variables and 53.7 million constraints solved with CGA and no cuts retained between iterations. The problem converges slowly, indicating a series of unrealistic master problem solutions. Solution time per iteration also increases, as cutting-plane methods increase master problem size at each iteration.
• Figure 3: Scaling of 64-core parallelized monolithic MGA and 52-core cutting-plane algorithm average time per MGA solution and total runtime with zones. Each problem is run for a least-cost solution and 16 total MGA solutions. Runs tabulated in Table \ref{['tab:zone']}.
• Figure 4: Comparison of cut sharing methods. From the left, plot 1 shows the distribution of runtimes to solve one MGA problem for each method, plot 2 shows the distribution of cutting-plane iterations required to solve one MGA problem for each method, and plot 3 shows the distribution of master problem solve times for each method. Note that horizontal lines show maximum, minimum, and median values for each violin. The objective partitioned, first-n cuts method with 11000 cuts performs best in solution time and iterations required.
• Figure 5: Comparison of algorithm convergence, runtime, and master problem solve time characteristics for a single MGA solution for all included cut sharing and objective partitioning methodologies. From the left, plot 1 shows convergence as measured by system cost declines, plot 2 shows runtime as a function of cutting-plane iterations, and plot 3 shows the progression of master problem solve times with cutting-plane iterations. We note that all methods which share cuts from previous iterations have meaningful advantages in initial cost gap estimates and runtime, but take longer to solve master problems.

Theorems & Definitions (2)

• Theorem 1
• proof