Spatial Branch and Bound for Global Power Flow Optimization
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Solution Overview
Problem
Conventional methods for optimizing power flows in electric power networks often rely on simplifying assumptions, nonlinear programming, or relaxation techniques that fail to guarantee globally optimal solutions or are computationally inefficient.
Innovation Solution
A spatial branch and bound procedure combined with semi-definite programming is used to globally optimize power flows, partitioning the feasible region and determining a minimal lower bound to speed up convergence and ensure global optimality.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of energy
If conventional nonlinear programming (NLP) is used to determine lowest cost per kilowatt hour, then the objective function is minimized, but global optimality of voltages and generator levels cannot be guaranteed
Solution Approach 1:
The feasible region is partitioned into multiple sub-regions using spatial branch and bound methodology. Each sub-region is explored separately to ensure that the global optimum is not missed, thereby guaranteeing reliability of the solution while minimizing energy loss.
Solution Approach 2:
Semi-definite programming is used to determine a minimal lower bound before the main optimization process. This preliminary action provides a benchmark that ensures the final solution achieves global optimality, resolving the contradiction between cost minimization and optimality guarantee.
2Device complexity
If simplifying assumptions are used to reduce quadratic equalities and inequalities to linear forms, then computational complexity is reduced, but validity for all networks is compromised
Solution Approach 1:
The patent changes the mathematical parameters and formulation of the optimization problem using semi-definite programming and spatial branch and bound. This allows the solution to handle the full quadratic nature of power flow equations without simplifying assumptions, maintaining validity for all network configurations while managing computational complexity through structured algorithms.
3Reliability
If branch & bound procedure with Lagrangian duality is used, then lower bounds are determined, but convergence speed is considerably slow due to irregular problem nature
Solution Approach 1:
The patent introduces a spatial dimension to the branch and bound procedure by partitioning the feasible region in multiple dimensions simultaneously. This dimensional approach accelerates convergence by reducing the search space more effectively than traditional one-dimensional branching, while still determining reliable lower bounds through semi-definite programming.
Data Source
AI summary
A power flow problem (OPF) in an electric power network is globally optimized using a branch and bound tree of nodes connected by edges. The BB initially includes at least a root node, and each node represents a feasible region of limits on voltages and powers. An upper bound on the OPF problem is solved for selected nodes using nonlinear programming, while a lower bound is solved using a convex relaxation. The lowest upper and lower bounds are updated using the current upper and lower bound. If a difference between the lowest upper and lowest lower bound is less than a threshold, then outputting the voltages and the powers for the electric power network as represented by the feasibility region for the selected node. Otherwise, the feasible region of the node is partitioned to replace the node. The process is repeated until the tree is empty.