Question

In Economic Load Dispatch, the most efficient operation occurs when:

• A. All units operate at max capacity.
• B. Incremental fuel costs of all units are equal.
• C. Total fuel consumption is maximized.
• D. All units carry equal loads.

Hint:

If transmission losses are considered, the coordination equation becomes:
\[ L_i \frac{dF_i}{dP_i} = \lambda \] where \(L_i\) is the penalty factor of unit \(i\), defined as \(L_i = \frac{1}{1 - \frac{\partial P_L}{\partial P_i}}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical condition that minimizes the total operating fuel cost when sharing a given electrical load among multiple generating units (Economic Load Dispatch).

Step 2: Key Formula or Approach:
Let \(F_{\text{total}} = F_1(P_1) + F_2(P_2) + \dots + F_n(P_n)\) be the total fuel cost, and \(P_D\) be the total load demand such that:
\[ \sum_{i=1}^{n} P_i = P_D \] Using the Lagrange multiplier method to minimize \(F_{\text{total}}\) under this constraint, we obtain:
\[ \frac{\partial F_i}{\partial P_i} = \lambda \quad \text{for all } i = 1, 2, \dots, n \] where \(\frac{\partial F_i}{\partial P_i}\) is the incremental fuel cost of unit \(i\).

Step 3: Detailed Explanation:

• Economic Load Dispatch (ELD) determines the optimal output power of each generator to minimize the total operating cost.

• The incremental fuel cost represents the cost of producing one additional megawatt-hour (MWh) of electrical energy from a specific unit.

• If transmission line losses are neglected, the optimization condition dictates that all generating units must operate at the same incremental fuel cost:
\[ \frac{dF_1}{dP_1} = \frac{dF_2}{dP_2} = \dots = \frac{dF_n}{dP_n} = \lambda \]
• If one unit had a lower incremental cost than the others, power could be shifted to that unit to reduce the total cost.

• Therefore, minimum operating cost (most efficient operation) occurs when the incremental fuel costs of all participating units are equal.

Step 4: Final Answer:
The most efficient operation occurs when the incremental fuel costs of all units are equal.