We'll now turn to a more interesting (and complicated, and I dare say fun) example involving the three-node network. Assume that the marginal costs of the two generators are MC(g1) = 10 + 0.5g1 and MC(g2) = 35 + 0.1g2. Demand at node 3 is 520 MWh, and the resistances on all three branches are equal. We'll ignore any capacity limits on the generators in this problem.

The line flow limits we'll impose on this problem are:

- Line 1-3: 300 MW
- Line 1-2: 90 MW
- Line 2-3: 300 MW

First, go ahead and find the economic dispatch as we did earlier in the lesson. You should calculate g1* = 391.7 MWh and g2* = 128.3 MWh. Using the distribution factors from earlier in the lesson, we'll calculate flows on the three branches:

Line 1 - 3: (1/3) 391.7 + (2/3) (128.3) = 216.1 MW.

**Whew!** This line is operating below its rated limit.

Line 1 - 2: -(1/3) 391.7 + (1/3) (128.3) = -87.8 MW.

**Whew!** This line is operating below its rated limit.

Line 2 - 3: (2/3) 391.7 + (1/3) (128.3) = 303.9 MW.

**Uh oh!** This line is overloaded and the system is congested.

Just as in the previous simple example, the system operator must tell one of the generators to back off. This situation looks more complicated, though. Which generator should back off? How much should the generator back off? We can use distribution factors to help us solve this problem.

What we need is for flow on Line 2-3 to decrease by 3.9 MW. We'll write this as ΔF_{23} = −3.9 MW. From the flow equations and distribution factors, we know that F_{23} = (1/3) × g1 + (2/3) × g2. Because the power flows in our three-node network are linear functions of g1 and g2, we can also write:

$${\text{\Delta F}}_{\text{jk}}\text{=D}\left(\text{j,k;g1}\right)\text{}\times \text{}\text{\Delta}\text{g1+D}\left(\text{j,k;g2}\right)\text{}\times \text{}\text{\Delta}\text{g2}$$

for any line j-k. So, for Line 2-3 in particular we would have:

$${\text{\Delta F}}_{\text{23}}\text{=}\frac{\text{1}}{\text{3}}\text{\xd7\Delta g1+}\frac{\text{2}}{\text{3}}\text{\xd7\Delta g2}$$

Note that in the out-of-merit dispatch, we must increase and decrease the two generators by exactly the same magnitude. In other words, we must have Δg1 = −Δg2. So we can write the previous equation as:

$${\text{\Delta F}}_{\text{23}}\text{=}\frac{\text{1}}{\text{3}}\text{\xd7}\left(\text{\u2212\Delta g2}\right)\text{+}\frac{\text{2}}{\text{3}}\text{\xd7\Delta g2}$$

When we plug in ΔF_{23} = −3.9 MW into equation (10'), we get Δg1 = +11.7 MW and Δg2 = −11.7 MW.

Thus, the solution to the transmission congestion problem appears to be that we should increase g1 by 11.7 MW and decrease g2 by 11.7 MW, relative to the original economic dispatch solution. The adjusted economic dispatch would thus be g1 = 140 MWh and g2 = 380 MWh.

We can check to see if the adjusted economic dispatch solves the transmission congestion problem by plugging the adjusted g1 and g2 figures into the flow equations:

- Line 2 - 3: (2/3)*380+ (1/3)*(140) = 300 MW.
- Line 1 - 2: −(1/3)*380 + (1/3)*(140) = −80 MW.
- Line 1 - 3: (1/3)*380 + (2/3)*(140) = 220 MW.

None of the three lines is now operating above its limits.

Let's now calculate the cost of transmission congestion in this case.

- Without considering transmission limits, the total system cost is:

10*G1 + 0.25*G1^{2}+ 35*G2 + 0.05*G2^{2}

= 10*128.3 + 0.25*128.3^{2}+ 35*391.7 + 0.05*391.7^{2}

= $26,779 - After the out-of-merit dispatch, the total system cost is:

10*140 + 0.25*140^{2}+ 35*380 + 0.05*380^{2}

= $26,820. - The system cost of congestion is $26,820 − $26,779

= $41.

#### Here is an example that you can try yourself.

We'll use the same three-node network with equal resistances on all lines. Assume that demand is 600 MWh. Ignore any capacity limits on the generators.

Line flow limits are:

- Line 1-3: 400 MW
- Line 1-2: 90 MW
- Line 2-3: 340 MW

The total cost functions for the two generators are given by C1(g1) = 15g1 + 0.3g12 and C2(g2) = 30g2 + 0.1g2^{2}.

- Show that the unconstrained economic dispatch yields g1* = 168.75 MW and g2* = 431.25 MW, and that this dispatch would result in a violation of the flow limit on Line 2-3.
- Show that an adjusted dispatch of g1 = 180 MW and g2 = 420 MW would relieve the congestion on Line 2-3.
- Show that the system cost of transmission congestion is $50.62.