Electric Grid Control via Lifted QP Decomposition
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Solution Overview
Problem
The computational complexity of solving quadratic programs for electric grid control increases cubically with the number of generators, and existing methods struggle with non-convexity and infeasibility due to generator constraints, leading to instability and high operational costs.
Innovation Solution
A system and method that reformulates the quadratic program into a lifted space with additional non-negative variables, transforming constraints and objective functions to create a homogeneous quadratic program, allowing for efficient detection of infeasibility and solution projection using decomposition methods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional quadratic programming methods are used to control electric grid operations, then operational constraints can be enforced, but computational complexity increases cubically with the number of generators
Solution Approach 1:
The patent segments the quadratic programming problem into two parts: (1) an equality-constrained subproblem that can be solved efficiently using standard methods, and (2) an infeasibility detection component that checks whether a feasible solution exists. This segmentation allows the system to handle constraint satisfaction without the cubic computational complexity that would result from directly solving the full QP with all constraints.
Solution Approach 2:
The patent introduces a new variable dimension by adding a non-negative variable τ to the optimization problem. This transforms the original QP into a higher-dimensional problem where infeasibility can be detected by checking whether τ = 0 in the optimal solution. This dimensional extension provides a systematic way to detect infeasibility without increasing computational complexity for the main optimization.
2Reliability
If reformulation methods are used to address non-convexity and constraints, then feasibility can be improved, but computational complexity increases
Solution Approach 1:
The patent extracts the infeasibility detection function from the main optimization process by introducing a separate non-negative variable τ that specifically measures constraint violation. This extracted component can be evaluated independently after solving the main equality-constrained problem, avoiding the need to reformulate the entire QP and thus preventing exponential complexity increases.
3Reliability
If direct computation of reformulated QP is performed, then infeasibility can be detected, but computational complexity becomes prohibitively high
Solution Approach 1:
The patent performs preliminary computation by first solving the equality-constrained subproblem to obtain a candidate solution, and only then checking for infeasibility by evaluating whether this solution satisfies the inequality constraints. This preliminary action avoids the need to directly compute the full reformulated QP, thereby maintaining computational efficiency while still detecting infeasibility accurately.
Data Source
AI summary
The electric grid is controlled by formulating an original quadratic program (QP) for optimizing an objective function subject to equality constraints and inequality constraints, lifting the equality constraints and the inequality constraints into a lifted space by a lifting operation introducing an additional non-negative variable, and transforming the objective function of the original QP into a quadratic objective function. The quadratic objective function subject to the lifted equality and inequality constraints forms a homogeneous QP in the lifted space solved to produce a solution in the lifted space using a decomposition that replaces variables corresponding to individual generators with dual variables corresponding to the total demand of power, the additional nonnegative variable and its corresponding dual variable. That solution is transformed to a control command for controlling the electric grid.