# 7Efficient Dispatch of Generation and Consumption Resources in the Presence of Network Congestion

In Chapter 6, we showed how we can express the physical limits of an electricity network as a set of linear constraint equations. In this chapter, we introduce these constraint equations into the optimal dispatch task to find the combination of production and consumption that maximises economic welfare subject to the physical constraints of the network.

## 7.1 Optimal Dispatch with Network Constraints

Previously, we defined the net injection of electric power at a node as the total production of producers at that node less the total consumption of consumers:

Using the benefit function introduced in Section 4.6 and the set of linear constraints derived from the DC Load Flow model introduced in Section 6.2, ignoring losses, the optimal dispatch problem can be written as follows:

subject to

- The energy balance equation and
- The generic network constraints

Recall that node *N* is chosen to be the reference node ( for all links *l*).

The KKT conditions for this constrained optimisation problem ...