Question

Two units, rated at 100 MW and 150 MW, are enabled for economic load dispatch. When the overall incremental cost is 10,000 Rs./MWh, the units are dispatched to 50 MW and 80 MW respectively. At an overall incremental cost of 10,600 Rs./MWh, the power output of the units are 80 MW and 92 MW, respectively. The total plant MW-output (without overloading any unit) at an overall incremental cost of 11,800 Rs./MWh is ___________ (round off to the nearest integer).

Hint:

In economic load dispatch, the power output of each unit depends linearly on the incremental cost \(\lambda\). Use the given data points to find these linear relations and apply unit limits to get the final dispatch.

Answer 1

We are given dispatch data at two incremental cost points:

At \(\lambda_1 = 10,000\):
\(P_1 = 50\) MW, \(P_2 = 80\) MW
At \(\lambda_2 = 10,600\):
\(P_1 = 80\) MW, \(P_2 = 92\) MW

We can assume linear relationships for both units between \(P\) and \(\lambda\):
Let’s assume for Unit 1:
\[ P_1 = a_1 \lambda + b_1 \] Using the two points:

Subtracting (1) from (2):
\[ 30 = a_1 (600) \Rightarrow a_1 = \frac{30}{600} = 0.05 \] Substitute into (1):
\[ 50 = 0.05 \cdot 10,000 + b_1 \Rightarrow b_1 = 50 - 500 = -450 \] So,
\[ P_1 = 0.05 \lambda - 450 \] Similarly, for Unit 2:

\[ 12 = a_2 \cdot 600 \Rightarrow a_2 = \frac{12}{600} = 0.02 \] Substitute into (3):
\[ 80 = 0.02 \cdot 10,000 + b_2 \Rightarrow b_2 = 80 - 200 = -120 \] So,
\[ P_2 = 0.02 \lambda - 120 \] Now, at \(\lambda = 11,800\):
\[ P_1 = 0.05 \cdot 11,800 - 450 = 590 - 450 = 140~{MW} \] \[ P_2 = 0.02 \cdot 11,800 - 120 = 236 - 120 = 116~{MW} \] Total Plant Output:
\[ P_{{total}} = 140 + 76 = \boxed{216~{MW}} \] (Wait — typo! Earlier it says \(P_2 = 116\), not 76.)

Correct total:
\[ P_{{total}} = 140 + 116 = \boxed{256~{MW}} — exceeds ratings. \] But unit limits are:
\[ P_1^{\max} = 100~{MW}, \quad P_2^{\max} = 150~{MW} \Rightarrow {So we must cap } P_1 \leq 100 \] Hence:
\[ P_1 = 100~{MW (capped)}, \quad \lambda = \frac{P_1 + 450}{0.05} = \frac{550}{0.05} = 11,000 { (invalid)} \] Try finding \(\lambda\) where both stay within limits.
Try \(\lambda = 11,800\):
\[ P_1 = 0.05 \cdot 11,800 - 450 = 140~{MW}>100 \Rightarrow {Exceeds} \Rightarrow {Limit } P_1 = 100~{MW} \] Then compute corresponding \(\lambda\) for \(P_1 = 100\):
\[ 100 = 0.05 \lambda - 450 \Rightarrow \lambda = \frac{550}{0.05} = 11,000 \Rightarrow {Not valid since desired } \lambda = 11,800 \] So now reverse — for \(\lambda = 11,800\), set:
\[ P_1 = \min(0.05 \cdot 11,800 - 450, 100) = \min(140, 100) = 100~{MW} \] \[ P_2 = \min(0.02 \cdot 11,800 - 120, 150) = \min(236 - 120, 150) = \min(116, 150) = 116~{MW} \] Final Total Output:
\[ P_{{total}} = 100 + 116 = \boxed{216~{MW}} \]