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