Question

Three generators supply a load centre with a demand Pd. The aim is to dispatch them so that the total cost of generation is minimised. The generators have the following cost functions: Ci(Pi) = 600 + 10P1 +0.016P[$/hour] C2 (P2) = 500 +8P2 +0.018P2 $ /hour) C3 (P3) = 700 + 12P3 + 0.018P} [$/hour] and the following technical limits: PL € (100 MW, 400 MW] P2 € [150 MW, 500 MW] P3 € (50 MW, 300 MW] Your task is to solve the economic dispatch problem for two scenarios: Plow = 392 MW and Phigh = 1192 MW using the following four approaches: (a) Formulate the problem as a quadratic programming problem and solve it in Matlab using quadprog; (b) Solve the problem directly from the optimality (KKT) conditions; (c) Solve the problem using dual decomposition; (d) Find the analytical expression of the Lagrange dual function g(v) and solve the Lagrange dual problem directly. In your report, you should discuss the following aspects: 1. What problem do you encounter when you use approaches (b) and (d) for the high-load scenario? 2. How does the choice of a affect the solution procedure in approach (c)? Hints: • @ Aspect 2: a lot! • For a minimisation problem, the Lagrange multipliers v associated with equality constraints are always non-positive (in this case strictly negative).