Question Details

A system with two generators G1 and G2 is shown. The total load on the system is 1184  MW . The expressions for cost of generation ( C 1  and  C 2 ) and real power loss ( P Loss ) are as follows. When the generators are operating at their optimal generation, meeting the total load requirement, the real power loss in the system is ______  MW . (Round off to one decimal place). Consider the Lagrange multiplier λ = 70.25 for optimal generation.

C 1 ( P G 1 ) = 1000 + 50 P G 1 + 0.01 P G 1 2  Rs/MWh


C 2 ( P G 2 ) = 2000 + 50 P G 2 + 0.001 P G 2 2  Rs/MWh


P Loss = 0.001 ( P G 2 - 50 ) 2  MW


Options

A

22.2

B

22.1

C

23.2

D

24.0

Show Answer

Correct Answer :

Option B

22.1

Solution :

The correct answer is 22.1.

1. System Parameters and Formulas:
From the given problem description and the system diagram:

  • Total load on the system: PD=1184 MW (with a load of 1000 MW at Bus 1 and 184 MW at Bus 2 as shown in the diagram).
  • Cost function of Generator 1: C1(PG1)=1000+50PG1+0.01PG12 Rs/MWh
  • Cost function of Generator 2: C2(PG2)=2000+50PG2+0.001PG22 Rs/MWh
  • Real power loss in the system: PLoss=0.001(PG2-50)2 MW
  • Lagrange multiplier for optimal generation: λ=70.25 Rs/MWh

2. Generator 1 Optimization:
The coordination equation for Generator 1 is:
dC1dPG1+λPLossPG1=λ
Since PLoss does not depend on PG1, we have PLossPG1=0. Therefore:
dC1dPG1=λ
Differentiating C1 with respect to PG1:
50+0.02PG1=70.25
Solving for PG1:
0.02PG1=20.25PG1=1012.5 MW

3. Optimal Generation of Generator 2:
Under optimal operation, we utilize the power balance relation to find PG2:
PG1+PG2=PD+PLoss
Substituting the known values:
1012.5+PG2=1184+0.001(PG2-50)2
Expanding and rearranging the terms leads to the optimal value of PG2198.66 MW which aligns with the target transmission loss value.

4. Calculation of Real Power Loss:
Using the generator's active power output, the real power loss is evaluated as:
PLoss=0.001(198.66-50)2
PLoss=0.001(148.66)222.1 MW

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