### 8.3 LMP on a Two Node Network

We'll start learning how to calculate LMPs using a simple two-node network shown in the figure below. Node 1 has 100 MWh of demand and a generator with a marginal cost of $20/MWh. Node 2 has 800 MWh of demand and a generator with a marginal cost of $40/MWh. The transmission line connecting the two has a capacity of 500 MW. Each of the two generators is very large, so we won't worry about either of the generators operating at their capacity limit.

First, let's look at what would happen if the transmission line could move as much power as we wanted. The RTO would solve the economic dispatch problem to find the lowest cost way to serve all 900 MWh of demand. Generator 1, which is cheaper, would be used to meet all 900 MWh of demand. Generator 1 would produce 100 MWh to be consumed at Node 1 and would export 800 MWh across the transmission line to Node 2. The LMP at both nodes would be $20/MWh.

Now let's look at the effect of transmission congestion. All of a sudden the maximum amount that Generator 1 can export to Node 2 is 500 MWh. So Generator 1 produces 600 MWh total. 100 MWh is consumed at Node 1, and 500 MWh is exported to Node 2. But total demand at Node 2 is 800 MWh, so the RTO needs to meet the extra 300 MWh of demand (800 MWh total minus the 500 MWh imported from Generator 1) by dispatching Generator 2.

Thus, the cost of serving an incremental unit of demand at Node 1 is different than at Node 2. At Node 1, the incremental cost of meeting electricity demand is $20/MWh (the marginal cost of Generator 1). At Node 2, the incremental cost of meeting electricity demand is $40/MWh ( the marginal cost of Generator 2). So the LMP at node 1 is $20/MWh and the LMP at Node 2 is $40/MWh.

Next, we'll look at how much the Generators are paid and how much the Customers pay. Customers at Node 1 are charged $20/MWh, while customers at Node 2 are charged $40/MWh *for all energy consumed*. Generator 1 is paid $20/MWh, while Generator 2 is paid $40/MWh *for all energy produced*. You may notice here that while customers at Node 2 pay $40/MWh for all 800 MWh that they consume, some of that energy was imported from Node 1 at a much lower cost. This is not a mistake - it's how the LMP pricing system works in the US.

The RTO collects revenue from the customers as follows:

$$\begin{array}{l}\text{FromNode1:100MWh\xd7\$20/MWh=\$2,000}\\ \text{FromNode2:800MWh\xd7\$40/MWh=\$32,000}\\ \text{TotalCollections:\$34,000}\end{array}$$

The RTO pays the generators as follows:

$$\begin{array}{l}\text{FromNode1:600MWh\xd7\$20/MWh=\$12,000}\\ \text{FromNode2:300MWh\xd7\$40/MWh=\$12,000}\\ \text{TotalCollections:\$24,000}\end{array}$$

The RTO collects excess revenue in the amount of $34,000 - $24,000 = $10,000. It collects this excess revenue because customers at Node 2 pay $40/MWh for all energy consumed, while some of those megawatt-hours only cost $20/MWh to produce.

This excess revenue is called "congestion revenue." In general, when there is congestion in the network and LMPs differ, then there will be some congestion revenue. We will discuss later in the course what the RTO does with this extra revenue. For now, just remember that whenever there is transmission congestion and LMPs at different nodes of the network aren't equal, the RTO will usually wind up with some congestion revenue.

As an exercise, calculate the LMPs and congestion revenue under this two-node example. Demand at Node 1 is 5 MWh, while demand at Node 2 is 10 MWh. The transmission line connecting Nodes 1 and 2 has a capacity of 5 MWh. The marginal cost of Generator 1 is $10/MWh while the marginal cost of Generator 2 is $15/MWh.

You should find that the LMP at Node 1 is $10/MWh; the LMP at Node 2 is $15/MWh; and that 5 MW of power is exported from Node 1 to Node 2 on the transmission line.

You should also find that the congestion revenue is equal to $25.