If the total cost of electricity production from a power plant is linear in the amount of electricity produced:
where TC is total cost ($), Q is total output (MWh), and a and b are constants, then the marginal cost of electricity production is found by taking the derivative of the total cost function:
which is just a constant. In the previous lesson when we used a constant heat rate to derive a marginal cost, we had assumed this type of cost model.
The way that we construct a supply curve in the presence of constant marginal costs is to stack each of the power plants in increasing cost order. This will yield a supply curve that looks like a staircase.
Economic dispatch with constant marginal cost is best illustrated using an example. Suppose that you were an electric utility that had three generators that could be used to meet electricity demand, as shown in the table below. The fixed costs would represent land leases, fuel/transmission interconnections and any other costs that do not depend on the level of output. The marginal cost measures the amount of money that it costs each plant to produce one megawatt-hour of electric energy.
|Plant Name||Capacity (MW)||Fixed Costs||Marginal Cost ($/MWh)|
Suppose that demand during some time period was 150 MWh. The process of determining economic dispatch would follow three steps.
First, order the plants from lowest to highest marginal cost, which will tell you which plants would be utilized to produce electricity given some level of demand. This picture is called the "dispatch curve," shown in Figure 4.2. Each step in the dispatch curve represents one power plant. The height of the step represents the marginal cost of electricity production for each plant. The width of each step represents the capacity of each plant. Note that the x-axis represents total capacity for the entire utility, not the capacity for any individual power plant.
To determine which plants would be operating if demand was 150 MWh, you would draw a straight vertical line through the amount 150 MWh. Any capacity to the left of this line would be dispatched, while any capacity to the right of the line would not be dispatched. This is shown in Figure 4.3.
In this case, to meet demand of 150 MWh, the utility would dispatch Colchester at its maximum output (100 MWh) and would meet the remaining 50 MWh of demand using the Warren plant. The Burke plant (the most expensive of the lot) is not dispatched.
We can use the dispatch information to calculate several cost measures for our hypothetical electric utility.
Total Cost is the sum of all costs incurred to meet electricity demand. To obtain total cost, we would multiply quantity produced times marginal cost for each plant, and then add in the fixed costs for all plants. Note that we need to add fixed costs even for plants that do not produce anything.
Average Cost is Total Cost divided by the total amount of electricity produced.
System Marginal Cost is the marginal cost of the generating plant that meets the last MWh of electricity demanded. The system marginal cost is also referred to as the "system lambda."
We can now calculate these cost measures for our example. The economic dispatch of all power plants in the system is:
- Colchester produces 100 MWh
- Warren produces 50 MWh
- Burke produces 0 MWh.
Total cost = ($50 + $25 + $10) + $10/MWh × 100 MWh + $30/MWh × 50 MWh + $60/MWh × 0 MWh = $2,585.
Average Cost = $2,585 ÷ 150 MWh = $17.23/MWh.
The "marginal generator" (the plant used to meet the last MWh of demand) is the Warren plant. So, the system lambda would just be the marginal cost of the Warren plant, which is $30/MWh.
Note the different units for total cost ($) versus average cost and marginal cost ($/MWh).
Here is an exercise that you can try on your own. Take the same three-generator example and suppose that demand was 190 MWh. Verify that:
- Total cost = $4,235
- Average cost = $22.29/MWh
- System lambda = $60/MWh
Here is another example. Suppose that we had the following power plants for use in the economic dispatch, shown in the table below.
|Generator||Max Output||Marginal Cost||Fixed Cost|
Demand is 60 MW. Show that the total cost of generation at this dispatch is $525 and the system lambda is $20/MWh.