Solving Unit Commitment Problems Using Lagrange Multiplier Method

The Lagrange multiplier method is a classic mathematical approach for handling constrained optimization problems, widely applied in power system unit commitment problems. The core of unit commitment involves optimizing generator startup/shutdown decisions and power output while satisfying power system operational constraints, aiming to minimize operational costs or maximize economic efficiency.

### Fundamental Principles The Lagrange multiplier method incorporates constraints into the objective function by introducing Lagrange multipliers, transforming the problem into an unconstrained optimization. In unit commitment, common constraints include power balance constraints, generator output limits, and ramp rate constraints. The method weights these constraints into the objective function to form the Lagrangian function, then iteratively adjusts multipliers to approach the optimal solution.

### Implementation Steps Construct Lagrangian Function: Combine the objective function (e.g., generation cost) with constraints (e.g., power balance, unit limits) to form a new optimization function. Code implementation typically involves defining the Lagrangian as L(x,λ) = f(x) + λ*g(x), where f(x) is the cost function and g(x) represents constraint violations. Solve Subproblems: For given Lagrange multipliers, independently optimize each unit's output or on/off status. This decomposition allows parallel computation using techniques like dynamic programming or mixed-integer linear programming for individual units. Update Multipliers: Adjust multipliers based on constraint violation levels, commonly using gradient ascent methods. In code, this translates to λ_{k+1} = λ_k + α * g(x_k), where α is the step size and g(x_k) measures constraint violations. Convergence Check: Terminate iterations when constraint deviations become sufficiently small or multiplier changes stabilize. Implementation typically uses tolerance thresholds like ||g(x)|| < ε or |λ_{k+1}-λ_k| < δ.

### Advantages and Limitations The method's key advantage lies in problem decomposition, making large-scale unit commitment problems computationally tractable through distributed optimization. However, it lacks guarantees for global optimality in non-convex problems, and multiplier adjustment strategies significantly impact convergence rates. In power system simulations, it's often combined with heuristic methods or mixed-integer programming to enhance solution quality.

This method provides crucial tools for power system economic dispatch, serving as a key technology for energy management and operational optimization.