Short Answer:

The cost function of a power plant is determined by analyzing how much it costs to produce electricity at different output levels. It includes factors like fuel cost, operation and maintenance cost, and sometimes emission or penalty costs. The relationship between power output and cost is usually expressed as a mathematical equation.

For thermal power plants, the cost function is often modeled as a quadratic equation, where the total cost increases with the square of the output. This function helps in calculating the most economical generation schedule during power system operations like Economic Load Dispatch (ELD).

Detailed Explanation:

Cost function of a power plant

The cost function of a power plant represents the total cost of producing electricity as a function of the power generated. It is used in economic studies and operational decisions such as economic load dispatch, unit commitment, and pricing. The cost function is different for each power plant depending on its type, fuel, design, and efficiency.

To determine the cost function, various factors and components that contribute to the total operating cost of a power plant must be considered. These costs are then expressed as a function of the power output, usually in megawatts (MW).

Key Components of the Cost Function:

1. Fuel Cost:
   Fuel cost is the major part of the operating cost, especially in thermal power plants. It depends on the type of fuel (coal, gas, oil), fuel price, and plant efficiency.
   • Formula:
     Fuel Cost = (Fuel Rate × Fuel Price)
     where fuel rate depends on how much fuel is needed to generate a unit of electricity.
2. Operation and Maintenance Cost (O&M):
   These include labor, repairs, routine maintenance, spare parts, and servicing. They are usually fixed per unit of energy generated or based on time and usage.
3. Start-up and Shut-down Costs:
   Some plants, especially thermal units, incur additional costs during start-up and shut-down due to fuel burning and system heating.
4. Environmental and Emission Costs:
   Modern cost functions may also include carbon taxes, emission penalties, and pollution control costs, especially for coal-based plants.
5. Other Fixed and Variable Costs:
   Includes auxiliary power consumption, insurance, cooling water, etc.

Mathematical Form of Cost Function:

For thermal power plants, the cost function is usually written in the form of a quadratic equation:

C(P) = a + bP + cP²
Where:

• C(P) = total cost in ₹ or $
• P = power output in MW
• a, b, c = cost coefficients (determined through data and testing)
• ‘a’ represents the fixed cost
• ‘bP’ is the linear cost related to power output
• ‘cP²’ is the nonlinear cost due to fuel and operational inefficiency at high outputs

These coefficients are usually calculated by collecting data from the plant’s fuel usage and operating records over a range of output levels.

Steps to Determine Cost Function:

1. Record data on fuel consumed and power generated at different output levels.
2. Calculate cost per unit for each level.
3. Fit a curve (often quadratic) to the cost vs. output data using regression or estimation methods.
4. Derive the coefficients a, b, and c.
5. Use this function in optimization tools like ELD to minimize total generation cost.

Use in Power System Operations:

• Economic Load Dispatch (ELD): Uses cost functions to allocate generation among multiple units.
• Unit Commitment: Decides which units to operate based on their cost functions.
• Tariff Calculation: Helps determine the cost-based pricing of electricity.

Conclusion:

The cost function of a power plant is determined by analyzing all costs related to power generation, especially fuel and operational costs, as a function of output. It is usually expressed as a quadratic equation and is essential in economic operation of power systems. Accurate cost functions help reduce generation costs, improve efficiency, and support decision-making in energy planning and dispatch.