Final answer:
The question deals with finding the marginal cost and optimal dispatch for different thermal plants to meet a specific load demand, with the first part requiring a solution for given cost functions and the second part for an updated cost function of one plant. Both scenarios involve economic dispatch and solving a system of equations considering the power generation boundaries for each thermal plant.
Step-by-step explanation:
The question revolves around the economic dispatch problem in power systems, where the goal is to determine the amount of power generation required from different plants to meet a load demand at minimum cost. The full-cost functions for the thermal plants include both fixed and variable costs as a function of power output (P1, P2, and P3 respectively). The marginal cost of each plant is derived from the derivative of the full-cost function in respect to the power output.
For the first scenario, the marginal costs (MC) and optimal dispatch are found by setting the derivatives of the cost functions equal to each other, ensuring that the sum of the power outputs meets the demand (P1 + P2 + P3 = 850 MW), and that each power output lies within its respective bounds. This ensures that the cost of producing the last MW of electricity is the same across all plants. In scenario B, we follow a similar method but with the new cost function for C1 which now has a different coefficient and a quadratic term with a negative coefficient implying economies of scale at higher outputs.
In both cases, it is essential to solve a system of equations to find the economically optimal power outputs (P1, P2, P3) that will minimize the total cost while satisfying the load demand.