Economic Dispatch With vs. Without Loss Coordination
How Transmission Losses Impact Power System Economics
Problem
A two-bus system shown in the below figure has the incremental fuel characteristics given by,
\[
\begin{aligned}
& IC_1=\dfrac{d F_1}{d t}=0.02 P_1+16 ~\mathrm{Rs} / \mathrm{MWh} \\
& IC_2=\dfrac{d F_2}{d t}=0.04 P_2+20 ~\mathrm{Rs} / \mathrm{MWh}
\end{aligned}
\]
When 100 MW of power is transmitted from plant 1 to the load, a transmission loss of 10 MW is incurred. If the incremental cost is Rs. 25/ MWh, then find the required generation for each power plant.
Problem Recap
System Data:
- Two generators:
\[
\begin{aligned}
IC_1 &= 0.02P_1 + 16 \\
IC_2 &= 0.04P_2 + 20
\end{aligned}
\]
- Loss coefficient: \( B_{11} = 10^{-3} \)
- \(\lambda = 25\, \text{Rs/MWh}\)
Calculating Loss Coefficient \(B_{11}\)
\[
P_L = B_{11} \times P_{G1}^2 \implies 10 = B_{11} \times 100^2 \implies B_{11} = \dfrac{10}{10,000} = 1 \times 10^{-3}\, \text{MW}^{-1}
\]
What is "Coordination"?
Economic Dispatch with Loss Coordination
Adjusts generation to account for transmission losses using penalty factors:
\[
L_i = \dfrac{1}{1 - \dfrac{\partial P_L}{\partial P_{Gi}}}
\]
Optimality Condition:
\[
\dfrac{dF_i}{dP_{Gi}} \cdot L_i = \lambda
\]
With Coordination:
- Minimizes total cost (generation + losses)
- Balanced generation
Without Coordination:
- Ignores losses (\(P_L = 0\))
- Leads to uneconomic dispatch
Like choosing between a traffic-aware GPS (with coordination) vs. shortest path (without).
Mathematical Comparison
With Coordination
\[
\begin{cases}
\dfrac{0.02P_1 + 16}{1 - 2 \times 10^{-3}P_1} = 25 \\
0.04P_2 + 20 = 25
\end{cases}
\]
Solution:
\[
P_1 = 128.57\, \text{MW}, \quad P_2 = 125\, \text{MW}
\]
Loss = 16.53 MW
Without Coordination
\[
\begin{cases}
0.02P_1 + 16 = 0.04P_2 + 20 \\
P_1 + P_2 = 237.04
\end{cases}
\]
Solution:
\[
P_1 = 275.17\, \text{MW}, \quad P_2 = 37.59\, \text{MW}
\]
Loss = 75.72 MW
Visualizing the Difference
- Without coordination: G1 is overloaded, causing huge losses.
- With coordination: Generators share load efficiently.
Cost Impact of Coordination
Cost Savings for G1:
\[
\int_{128.57}^{275.17} \!\!\!\! (0.02P_1 + 16)\, dP_1
\]
\[
= \text{Rs } 2937.69/\text{h}
\]
Cost Increase for G2:
\[
\int_{37.59}^{125} \!\!\!\! (0.04P_2 + 20)\, dP_2
\]
\[
= \text{Rs } 2032.43/\text{h}
\]
Net Savings
\[
2937.69 - 2032.43 = \text{Rs } 905.26/\text{h}
\]
Key Takeaways
- Loss coordination optimizes generation by accounting for transmission losses.
- Without coordination:
- Higher losses (75.72MW vs. 16.53MW)
- Uneconomic dispatch (overloads cheaper generators)
- With coordination:
- Saves Rs 905.26/hour
- Balances generation efficiently
