# Transmission Expansion Planning in Deregulated Power ... - tuprints Transmission Expansion Planning in Deregulated Power ... - tuprints

3 Probabilistic locational Marginal Prices 17

3.1.2 Mathematical Model for Computing Locational Marginal Prices

Consider a deregulated power system and suppose producers and consumers have submitted

their bids and other data to ISO for a specific period of times. ISO lists generators based on

their bids increasingly. Let’s call this list priority list. If there is no constraint in transmission

network, the generators are dispatched according to the priority list till sum of generation is

made equal to sum of loads and losses. The last dispatched generator, which is not dispatched

fully, is named marginal generator. If load of a bus is increased by 1 MW, regardless of the

bus location, this 1 MW load will be supplied by the marginal generator. Therefore, according

to the definition of LMP, LMPs of all buses are equal to the bid of marginal generator.

If there is constraint in transmission network, the generators are dispatched according to

priority list till reach the first constraint. The last dispatched generator can not be dispatched

more because of the constraint. Hence, the next generators of the priority list, which do not

increase the flow of the congested line, are dispatched respectively till reach the next

constraint or sum of generation is made equal to sum of loads and losses. If the generators

which have been dispatched after reaching the first constraints decrease the flow of the

congested lines, undispatch cheaper generations are dispatched. This process is continued till

sum of generation is made equal to sum of loads and losses. In the presence of constraints,

there are several marginal generators. If load of a bus is increased by 1 MW, depends on the

bus location, this 1 MW load is supplied by the marginal generators which do not violate lime

flow limits. Hence, LMP of each bus depends on its location in the network. Therefore, in the

presence of transmission constraints LMPs of buses are different. A simple example for LMP

is given in appendix C. In practice LMPs are computed using an optimization problem, which

is described in the following subsections.

3.1.2.1 Optimal Power Flow

Optimal power flow is modeled by an optimization problem. Objective function is the total

cost of operation including cost of running generators and load curtailment cost. Power flow

equations, line flow limits, generation limits, and load limits are the constraints of this

optimization problem. The objective function and constraints are modeled in (3.1)-(3.5).

Consider a power system with Nb buses, Ng generators, Nd loads, and Nl lines. Optimal power

flow is modeled as below:

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