The Lambda Iteration Method for Solving Optimal Dispatch

The lambda iteration method is one of the methods used in solving the system lambda and optimal power dispatch of generators. Other methods includes gradient method and newton method.
Lambda is the variable introduced in solving constraint optimization problem and called a Lagrange multiplier.
It is important to note that lambda can be solve at hand by solving systems of equations. Lambda iteration is introduced for the sake of computing lambda and other associated variables using a computer.
Unlike usual iteration methods like gauss seidel and newton-raphson, lambda iteration is somewhat different. In gauss seidel method, for example, the next value of the unknown variables can be solve using an equation, which is usually, a function of itself. In lambda interation method, the unknown variable, lambda, gets its next value based on intuition. That is, there is no equation that computes the next iteration of lambda. It is projected by interpolating the best possible value until a spicified mismatch has been reached.

The Lambda Iteration Method of Solving Optimal Power Dispatch
We are now going to discuss on how to set up and solve the optimal power dispatch and system lambda using lambda iteration method. To make the concept simple and explainable� through words without using illustrations and equations, we simply neglect the effect of power losses in the formulation.

Setting up the equations
1. Formulate the Lagrange function. Lagrange function is simply the objective function, in this case - minimize the fuel cost, plus the equality constraint - total power generated equals the total load, multiplied by lambda, plus the inequality constraint which will contain the multiplier mu (please find other literature on how mu is formulated, it’s hard to discuss here without illustrations).
2. Find the partial derivatives of the lagrange function with respect to each power generation and lambda.
3. After doing the calculus, you should come up with an equation of power generations as a function of lambda, and mu’s.

Do the iteration method
Before going to the iteration process, we will first assume that there is no violation of power generation limit. Using this assumption the power generation or dispatch will now only a function of lambda because mu’s are zero if there is no violation of power dispatch limit.
Step 1: Set any value of lambda.
Step 2: Solve for the value of power generation or dispatch using the equation of power that we have formulated earlier.
Step 3: Compute the mismatch or the absolute value of the difference between the actual load and the sum of the computed power generations.

If the sum of the power generation is too large compaired with the actual load, try so set another value of lambda that, depending on the equations for power, lower the sum of the power generations compaired with the load. Using this technique, you will have an idea on what the value of lambda that is that will make the total power generations equal to the system load and thereby solved the optimal power dispatch. This is how the iteration goes. You will have to project, through interpolation, the value of lambda.

If a violation of generation limit has occured. That limit will be automatically the dispacth power of that generator and what we only need to solve is the dispatch of other generators. In this case, the lambda of that generator that violate its limit is not equal to the system lambda.