Economic Load Dispatch

Why it matters

Economic Load Dispatch (ELD) is crucial in power systems as it ensures that electricity is generated at the lowest possible cost while meeting the demand and adhering to operational constraints. This optimization helps in reducing fuel costs and improving the efficiency of power generation.

Key ideas

• Objective: Minimize the total generation cost while satisfying the load demand and operational constraints.
• Cost Function: Each generator has a cost function, typically quadratic, representing the cost of generation as a function of power output.
• Constraints: Include power balance (total generation equals total demand plus losses) and generator limits (minimum and maximum power output).
• Lambda Iteration Method: A common method used to solve ELD problems by iteratively adjusting the incremental cost (lambda) until the power balance is achieved.

Formulas

• C_i(P_i) = a_i + b_i * P_i + c_i * P_i^2
  • C_i(P_i): Cost of generation for the i-th generator (Rs/hr)
  • P_i: Power output of the i-th generator (MW)
  • a_i, b_i, c_i: Cost coefficients for the i-th generator
• ∑P_i = P_D + P_L
  • ∑P_i: Total power generated (MW)
  • P_D: Total power demand (MW)
  • P_L: Total transmission losses (MW)

Worked example

Given:

• Two generators with cost functions:
  • C_1(P_1) = 500 + 5.3 * P_1 + 0.004 * P_1^2
  • C_2(P_2) = 400 + 5.5 * P_2 + 0.006 * P_2^2
• Total demand P_D = 150 MW
• Neglect transmission losses and assume both units are online with limits that permit the resulting outputs.

1. Initial Guess: Assume P_1 = 75 MW and P_2 = 75 MW.
2. Calculate Incremental Costs:
   • dC_1/dP_1 = 5.3 + 0.008 * P_1
   • dC_2/dP_2 = 5.5 + 0.012 * P_2
3. Adjust Power Outputs:
   • Set dC_1/dP_1 = dC_2/dP_2 to find P_1 and P_2.
   • Solve 5.3 + 0.008 * P_1 = 5.5 + 0.012 * P_2.
   • With P_1 + P_2 = 150, solve the equations to find P_1 = 100 MW and P_2 = 50 MW.

Final Answer: P_1 = 100 MW, P_2 = 50 MW

At this solution both incremental costs equal 6.1 Rs/MWh. For a generator at a binding output limit, equal incremental cost need not hold. With transmission losses, the appropriate penalty factors enter the stationarity condition.

Common mistakes

• Ignoring transmission losses when the problem requires them.
• Incorrectly setting up the cost functions or constraints.
• Failing to iterate to convergence in methods like lambda iteration.

For GATE EE

Questions often involve calculating the optimal power output for generators given cost functions and demand. Practice setting up and solving the ELD problem using methods like lambda iteration and understanding the impact of constraints.

Quick check

1. What is the primary objective of Economic Load Dispatch?
2. How is the cost function of a generator typically expressed?
3. What method is commonly used to solve ELD problems?

Answers: 1. Minimize generation cost while meeting demand. 2. Quadratic function of power output. 3. Lambda Iteration Method.