In addition to making this extensible OPF structure available to end users,

The standard OPF formulation in (6.1)–(6.4) does not directly handle the non-smooth piecewise linear cost functions that typically arise from discrete bids and oﬀers in electricity markets. When such cost functions are convex, however, they can be modeled using a constrained cost variable (CCV) method. The piecewise linear cost function is replaced by a helper variable and a set of linear constraints that form a convex “basin” requiring the cost variable to lie in the epigraph of the function .

Figure 6-3 illustrates a convex -segment piecewise linear cost function

| (6.52) |

deﬁned by a sequence of points , , where denotes the slope of the -th segment

| (6.53) |

and and .

The “basin” corresponding to this cost function is formed by the following constraints on the helper cost variable :

| (6.54) |

The cost term added to the objective function in place of is simply the variable .

Note that

A simple approach to dispatchable or price-sensitive loads is to model them as negative real power injections with associated negative costs. This is done by specifying a generator with a negative output, ranging from a minimum injection equal to the negative of the largest possible load to a maximum injection of zero.

Consider the example of a price-sensitive load whose marginal beneﬁt function is shown in Figure 6-4. The demand of this load will be zero for prices above , for prices between and , and for prices below .

This corresponds to a negative generator with the piecewise linear cost curve shown in
Figure 6-5. Note that this approach assumes that the demand blocks can be partially
dispatched or “split”. Requiring blocks to be accepted or rejected in their entirety would
pose a mixed-integer problem that is beyond the scope of the current

It should be noted that, with this deﬁnition of dispatchable loads as negative generators, if the negative cost corresponds to a beneﬁt for consumption, minimizing the cost of generation is equivalent to maximizing social welfare.

With an AC network model, there is also the question of reactive dispatch for such
loads. Typically the reactive injection for a generator is allowed to take on any value
within its deﬁned limits. Since this is not normal load behavior, the model used in

The power factor, which can be lagging or leading, is determined by the ratio of
reactive to active power for the load and is speciﬁed by the active and reactive limits
deﬁning the nominal load in the

Lagging Power Factor – The reactive injection is negative, meaning that reactive power is consumed by the load. Hence,QMIN is negative,QMAX is zero, andPG andQG must be set so thatQG is equal toPG * QMIN/PMIN .Leading Power Factor – The reactive injection is positive, that is, reactive power is produced by the load. Hence,QMAX is positive,QMIN is zero, andPG andQG must be set so thatQG is equal toPG * QMAX/PMIN .

The typical AC OPF formulation includes box constraints on a generator’s real and
reactive injections, speciﬁed as simple lower and upper bounds on ( and )
and ( and ). On the other hand, the true - capability curves of
physical generators usually involve some tradeoﬀ between real and reactive capability, so
that it is not possible to produce the maximum real output and the maximum (or
minimum) reactive output simultaneously. To approximate this tradeoﬀ,

The two sloped portions are constructed from the lines passing through the two pairs
of points deﬁned by the six parameters , , , , , and . If these
six parameters are speciﬁed for a given generator in columns

If one of the sloped portions of the capability constraints is binding for generator ,
the corresponding shadow price is decomposed into the corresponding and
or components and added to the respective column (

The diﬀerence between the bus voltage angle at the ^{39}