EBF 483
Introduction to Electricity Markets

8.5 LMP on a Three Node Network, Linear Marginal Cost

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Now we'll work through a LMP calculation problem on a three node network when generator marginal costs are linear. The principle here is basically the same - we'll find the economic dispatch, adjust that dispatch to avoid overloading transmission lines, and then calculate the LMPs based on that adjusted dispatch.

The figure below shows the three node network with marginal costs and 10 MWh of demand at Node C. The marginal cost of generator A is MC( G A )  = 6 + 2 G A and the marginal cost of generator B is MC( G B )  = 2 + G B . As an exercise for yourself, show that the economic dispatch (using the method for linear marginal costs from Lesson 4) is G A = 2 MWh ; G B = 8 MWh , and also show that the flows on the three transmission lines are F AB = -2 MW , F AC = 4 MW and F BC = 6 MW . The System Marginal Price in this case would be $10/MWh, which is the marginal cost of either generator at the economic dispatch point (use the marginal cost functions to verify this).

three node network with marginal costs and 10 MWh of demand at Node C
Figure 8.7: Three node network with marginal costs and 10 MWh of demand at Node C
Source: Seth Blumsack

Now let's congest the network. We'll set the following constraints on the transmission lines:

  • Flow on A-B  10 MW
  • Flow on A-C  10 MW
  • Flow on B-C  6 MW

Suppose that demand at Node C increases to 11 MWh. How should the RTO re-dispatch the generators? What are the LMPs?

What we need to do is to find a way to adjust the economic dispatch to satisfy two conditions:

  • ΔG A + ΔG B = 1 ( the system needs to serve one additional MWh )
  • ΔF BC = 0 ( our dispatch must keep the flow on line B,C constant so as not to overload that line ).

Using the distribution factors for the network and the fact that ΔG A  = 1 -  ΔG B , we get:

ΔF BC = 1 3 ΔG A + 2 3 ΔG B =0

                    = 1 3 ΔG A + 2 3 ( 1-ΔG A )=0

        - 1 3 ΔG A + 2 3 =0

                                     ΔG A =+2MWh,  ΔG B =-1MWh

The adjusted dispatch is thus GA = 4 MWh, GB = 7 MWh. You should check for yourself that the flows in the network become: FAB = -1 MW, FAC = 5 MW, FBC = 6 MW.

Now we'll move on to calculating LMPs. As with our previous example the LMPs at the generator nodes will be equal to the marginal cost of generation at the adjusted dispatch point.

LMP( A ) = MC( G A ) = 6 + 2×4 =$14/MWh

LMP( B ) = MC( G B ) = 2 + 7 =$9/MWh

To calculate the LMP at node C (the demand node), realize that the marginal demand at Node C is served partially by an increase at GA and a decrease in GB. By the same logic as in our first three-node example, the LMP at node C will be:

LMP( C )  = ΔG A × MC( G A )  + ΔG B × MC( G B )

When we evaluate this using the numbers in our problem, we get:

LMP(C) = 2 × MC(G A  = 4) + (-1) × MC(G B  = 7)

= 2 ×$14 - 1 ×$9

= $19/MWh.

Note that we have evaluated the marginal costs at the adjusted dispatch rather than the original economic dispatch that did not consider transmission constraints.

We can use the LMPs to calculate payments in the market and the congestion revenue collected by the RTO.

The RTO pays the following to generators:

  • To G A : 4 MWh × $14/MWh = $56
  • To G B : 7 MWh × $9/MWh = $63
  • Total payments: $119

The RTO collects 11 MWh × $19/MWh = $209 from the customer at node C.

Congestion revenue is thus $209 - $119 = $90.