Spatial Branch and Bound Power Flow Optimization
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Solution Overview
Problem
Conventional methods for optimizing power flows in power grids, such as second-order cone programming, are not suitable for arbitrarily structured grids and fail to provide optimal solutions when sufficient conditions for relaxation are not met, especially when considering multiple time periods and storage devices.
Innovation Solution
The use of a spatial branch and bound framework with semi-definite programming (SDP) relaxation to determine lower bounds, combined with methods like Lagrangian dual subgradient and alternating direction method of multipliers (ADMM) to accelerate convergence and handle complex power grid structures, allowing for global optimization of power flows across multiple time periods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional second-order cone programming (SOCP) is used to solve OPF, then computational speed is improved, but solution accuracy deteriorates because sufficient conditions for relaxation are not satisfied in arbitrarily structured grids
Solution Approach 1:
The patent uses SDP relaxation to create a convex approximation (copy) of the non-convex OPF problem. This relaxed version provides computationally efficient lower bounds that can be solved quickly, while the branch-and-bound framework ensures the original problem's accuracy is recovered through systematic exploration of the solution space.
Solution Approach 2:
The patent segments the solution space into multiple regions using branch-and-bound, where each region is evaluated using SDP relaxation. This segmentation allows the method to achieve both computational efficiency (by evaluating regions systematically) and solution accuracy (by exploring enough regions to find the global optimum).
2Measurement precision
If SDP relaxation is used to determine lower bounds in branch and bound framework, then solution accuracy is improved through global optimization, but computational complexity increases
Solution Approach 1:
The patent applies SDP relaxation with different qualities to different parts of the solution process: using it for lower bound calculation where accuracy is critical, while using simpler upper bound methods where computational speed is more important. This local differentiation optimizes the overall balance between accuracy and complexity.
Solution Approach 2:
The patent performs preliminary SDP relaxation to obtain lower bounds before committing to detailed solution exploration. This preliminary action provides a foundation for the branch-and-bound process, allowing early pruning of suboptimal regions and reducing the overall computational complexity of finding the global optimum.
3Productivity
If linear under-approximation is used for cost function in branch and bound, then computational efficiency is improved, but optimization accuracy deteriorates due to shallow structure of cost function
Solution Approach 1:
The patent changes the approximation parameter from linear to quadratic (using SDP relaxation) for the cost function. This parameter change maintains computational tractability while significantly improving accuracy by capturing the curved structure of the cost function that linear approximation misses.
4Adaptability or versatility
If conventional OPF methods are applied to multi-period problems with storage devices, then problem scope is expanded, but solution feasibility deteriorates due to time coupling and ramp constraints
Solution Approach 1:
The patent segments the multi-period problem into individual time periods, solving each period's OPF separately using SDP relaxation and branch-and-bound. This segmentation breaks the time-coupling complexity while maintaining feasibility through systematic exploration of each period's constraints including ramp limits and storage operations.
Data Source
AI summary
A method determines a power flow of a power grid by optimizing an objective function representing an operation of the power grid using a spatial branch and bound (BB) framework for determining iteratively upper and lower bounds of the objective function. During the optimization, the lower bounds are determined using a semi-definite programming (SDP) relaxation of an optimal power flow (OPF) problem.