• Dynamic programming

The basis for Dynamic Programming (DP) is the theory of optimality that can be used to explain crises in which many chronological conclusions are to be taken in defining the optimum operation of a system, which consists of distinct number of stages. The searching may be in forward or backward direction. Within a time period the combinations of units are known as the states. In Forward DP a excellent economic schedule is obtained by commencing at the preliminary stage amassing the total costs, then retracing from the combination of least accumulated cost starting at the last stage and finishing at the initial stage. The stages of the DP problem are the periods of the study horizon. Each stage usually corresponds to one hour of operation i.e., combinations of units steps forward one hour at a time, and arrangements of the units that are to be scheduled are stored for each hour. Finally, by backpedaling from the arrangement with smallest amount of total cost at the final hour throughout the finest path to the arrangement at the preliminary hour the most economical schedule is acquired. The estimation of each and every combination is not convenient evidently. Additionally, several of the combinations are prohibited due to insufficient existing capacity. The step by step procedure for dynamic programming approach is as follows:

1) Start randomly by considering any two units.
2) Assemble the collective output of the two units in the form of discrete load levels.
3) Determine the most economical combination of the two units for all the load levels. It is to be observed that at each load level, the economic operation may be to run either a unit or both units with a certain load sharing between the two units.
4) Obtain the more cost-effective cost curve for the two units in discrete form and it can be treated as cost curve of single equivalent unit.
5) Add the third unit and the cost curve for the combination of three units is obtained by repeating the procedure.
6) Unless all the existing units are considered the procedure is repeated.

The benefit of this method is that having the best way of running n units, it is simple to find out the best way for running n + 1 units. The DP approach is based on the subsequent recurring equation.

Fn (x) = min { fn (y) + Fn - 1 (x - y)}

Where,

Fn(x) = minimum cost in Rs./hr of generation of x MW by n generating units.

fn(y) = cost of generation of y MW by nth unit.

Fn - 1(x - y) = minimum cost of generation of (x - y) MW by the remaining (n -1) units.

In its elemental form, the dynamic programming algorithm for unit commitment problem inspects every possible state in every interval. The dimensionality of the problem is significantly declined which is the chief advantage of this technique.