Please clear handwriting. There are three generating units in a three-bus power
ID: 2072853 • Letter: P
Question
Please clear handwriting. There are three generating units in a three-bus power system shown in Figure 1. Their cost functions are shown in Table 1. Line data are listed in Table 2. Bus loads are listed in Table 3. Assume that all three unites are committed. Choose Bus 2 as the reference bus. Consider two contingencies: i) the outage of Line 2. and ii) the outage of G3. Assume that no corrective action is allowed after the occurrence of the first contingency and the maximum allowable adjustment for all other generators is 50MW after the occurrence of the second contingency. Also assume that the two contingencies arc independent and they do not occur simultaneously. The ultimate goal is to determine the least cost generation dispatch that ensures the network security at both normal operating condition and contingency conditions. The above security-constrained economic dispatch problem can be solved using shift factor based method. With the solution from a pure economic dispatch problem (without network constraints), it is found that Line 2 is overloaded in the positive direction (i.e., l rightarrow 2) under the normal condition. Line 3 is overloaded in the positive direction (i.e.. l rightarrow 3) under contingency 1. Line 4 is overloaded in the negative direction (i.e., 2 rightarrow 3) under contingency 2. Requirement: Formulate the economic dispatch problem with the above three network constraints. The above security constrained economic dispatch problem can be solved using Benders decomposition based method. Requirements: Formulate the three subproblems and Benders cuts under normal operating condition, under contingency 1, and under contingency 2. Assume that you know the values of the objective functions (all none-zeros) and the dual variables of all constraints in the three subproblems.
Explanation / Answer
Shift Factor Based Method:
Concept of the shift factor, start with a sample FNM (buses and lines). Assume this network is a DC FNM. The arrow on line l represents the reference direction. it is also know as Power Transfer Distribution Factors (PTDFs), Power Distribution Coefficients (PDCs), Effectiveness Factors Impedance Factors.
Its has has 4 attributes:
The value of shift factor of line l with respect to bus i is defined to be:
flow of the line 1 to 2 is 0.4 and the bus is 1
0.4/4=0.4
flow of the line is 1 to 2 and the bus is 2
0.4/3=0.13
flow of the line is 1 to 3 andthe bus is 3
0.03/2=0.005
flow of the line is 2 to 2 and the bus is 3
0.03/2=0.005
B. Benders’ decomposition is a classical solution approach for combinatorial optimization problems based on partition and delayed constraint generatio. It is used for solving large scale combinatorial optimization problems and then several extensions were proposed.
One of the most important ones was presented by Geoffrion who proposed a “generalized Benders’ decomposition” approach. He used nonlinear duality theory and extended the Benders’ method to the case where the sub-problem was convex.