Question

The fuel cost functions of two power plants are

Plant \(P1:{C_1} = 0.05Pg_1^2 + AP{g_1} + B\)

Plant \({P_2}:{C_2} = 0.10Pg_2^2 + 3AP{g_2} + 2B\)

Where, P_g1 and P_g2 are the generated powers of two plants, and A and B are the constants. If the two plants optimally share 1000 MW load at incremental fuel cost of 100 Rs/MWh, the ratio of load shared by plants P₁ and P₂ is
1. 1 : 4
2. 2 : 3
3. 3 : 2
4. 4 : 1

Answer 1

Correct Answer - Option 4 : 4 : 1

Concept:

The condition for optimum operation and economic dispatch is

\(\frac{{d{F_1}}}{{d{P_1}}} = \frac{{d{F_2}}}{{d{P_2}}} = \frac{{d{F_3}}}{{d{P_3}}} \ldots \ldots \ldots \ldots = \frac{{d{F_n}}}{{d{P_n}}} = \lambda \)

Where,

\(\frac{{dF}}{{dP}}\) is incremental fuel costs in Rs/MWh for a power plant.

The above equation shows that the criterion for the most economical division of load between plants is that all the unit is must operate at the same incremental fuel cost.

Calculation:

Given fuel cost functions of two power plants are

Plant \(P1:{C_1} = 0.05Pg_1^2 + AP{g_1} + B\)

Plant \({P_2}:{C_2} = 0.10Pg_2^2 + 3AP{g_2} + 2B\)

The incremental fuel cost of the two plants will be

For plant 1

\(\frac{{d{C_1}}}{{dP{g_1}}} = 2 \times 0.05{P_{{g_1}}} + A = 100\)

\(\frac{{d{C_1}}}{{dP{g_1}}} = 0.1{P_{{g_1}}} + A = 100\)

For plant 2

\(\frac{{d{C_2}}}{{dP{g_2}}} = 2 \times 0.1{P_{{g_2}}} + 3A = 100\)

\(\frac{{d{C_2}}}{{dP{g_2}}} = 0.2{P_{{g_2}}} + 3A = 100\)

By solving both the incremental fuel cost equations

P_g1 = 800 MW

P_g2 = 200 MW

\(\Rightarrow \frac{{{P_{{g_1}}}}}{{{P_{{g_2}}}}} = \frac{4}{1}\)