Unit Commitment Using Lagrange Relaxation Method

Application of Lagrange Relaxation Method in Unit Commitment Problems Unit Commitment (UC) is a core optimization problem in power system dispatch, aiming to minimize total generation costs by合理安排 generator startup/shutdown decisions and power output allocations while satisfying load demand and operational constraints. The Lagrange Relaxation method handles complex coupling constraints (such as power balance and spinning reserve requirements) by relaxing them into penalty terms in the objective function, thereby decomposing the original problem into independently solvable subproblems, significantly reducing computational complexity. Implementation typically involves defining coupling constraints through Lagrange multipliers and solving separable subproblems using dynamic programming or mixed-integer linear programming techniques for individual units.

Typical Implementation for Three-Node Test Case In a small three-node system, the Lagrange Relaxation implementation generally follows these steps: Problem Formulation: Construct an objective function including unit startup/shutdown costs, fuel expenses, and ramp rate constraints, while treating system balance constraints as relaxed terms with Lagrange multipliers. Code implementation often initializes multipliers using heuristic values or historical data. Subproblem Decomposition: The original problem decomposes into unit-specific subproblems, each handling only local operational constraints (e.g., minimum uptime/downtime, power output limits). Each subproblem can be solved using dynamic programming with state transitions representing unit operational modes. Multiplier Update: Apply subgradient methods or heuristic rules to dynamically adjust Lagrange multipliers, iteratively approaching feasible solutions for coupling constraints. The update formula typically uses step-size parameters that decay over iterations to ensure convergence.

Technical Advantages and Challenges The method's primary advantage lies in parallelism—individual unit subproblems can be solved independently, making it suitable for large-scale systems. However, for small test cases like three-node systems, multiplier oscillation issues may arise. Common improvements include: adding quadratic penalty terms (enhancing convergence) or integrating dynamic programming for unit commitment logic. In practice, this method often serves as a preprocessing step for mixed-integer programming, rapidly generating high-quality initial solutions. Code implementation should incorporate adaptive step-size control and feasibility recovery mechanisms to handle convergence challenges.