Question

The expressions of fuel cost of two thermal generating units as a function of the respective power generation $P_{G1$ and $P_{G2}$ are given as} \[ F_{1}(P_{G1}) = 0.1aP_{G1}^2 + 40P_{G1} + 120 \; \text{Rs/hour}, 0 \leq P_{G1} \leq 350 \, MW \] \[ F_{2}(P_{G2}) = 0.2P_{G2}^2 + 30P_{G2} + 100 \; \text{Rs/hour}, 0 \leq P_{G2} \leq 300 \, MW \] where $a$ is a constant. For a given value of $a$, optimal dispatch requires the total load of $290 \, MW$ to be shared as $P_{G1} = 175 \, MW$ and $P_{G2} = 115 \, MW$. With the load remaining unchanged, the value of $a$ is increased by $10%$ and optimal dispatch is carried out. The changes in $P_{G1}$ and in the total cost of generation, $F = F_{1} + F_{2}$, will be as follows:

• A. $P_{G1}$ will decrease and $F$ will increase
• B. Both $P_{G1}$ and $F$ will increase
• C. $P_{G1}$ will increase and $F$ will decrease
• D. Both $P_{G1}$ and $F$ will decrease

Hint:

In economic dispatch, if the cost coefficient of one unit increases, that unit is used less in optimal allocation, and the total cost of generation always increases.

Answer 1

The Correct Option is A

Step 1: Recall economic dispatch principle.
For economic load dispatch without transmission losses, the incremental cost (marginal cost) of both units must be equal: \[ \frac{dF_{1}}{dP_{G1}} = \frac{dF_{2}}{dP_{G2}} = \lambda \]

Step 2: Compute incremental costs.
\[ \frac{dF_{1}}{dP_{G1}} = 0.2aP_{G1} + 40 \] \[ \frac{dF_{2}}{dP_{G2}} = 0.4P_{G2} + 30 \] At initial $a$ and dispatch values $P_{G1} = 175, P_{G2} = 115$: \[ \lambda = 0.2a(175) + 40 = 35a + 40 \] \[ \lambda = 0.4(115) + 30 = 46 + 30 = 76 \] So, \[ 35a + 40 = 76 \Rightarrow 35a = 36 \Rightarrow a \approx 1.0286 \]

Step 3: Effect of increasing $a$ by 10%.
New $a' = 1.0286 \times 1.1 \approx 1.1314$ Now, the marginal cost of generator 1 increases faster with $P_{G1}$ because the coefficient $0.2a$ has increased. To restore equality of incremental costs: - $P_{G1}$ must decrease (so that $\frac{dF_{1}}{dP_{G1}}$ reduces). - Correspondingly, $P_{G2}$ must increase to keep the total load constant at $290 \, MW$.

Step 4: Effect on total cost $F$.
Since the unit 1 cost curve has become "steeper" (more expensive per MW), and more generation shifts to unit 2, the overall total cost $F$ will increase.

Step 5: Final outcome.
\[ P_{G1} \; \text{decreases}, F \; \text{increases}. \] % Final Answer \[ \boxed{\text{Option (A): $P_{G1}$ decreases and $F$ increases.}} \]