The ﬁrst diﬀerence to note is that the optimization variable from the standard OPF formulation has been augmented with additional variables to form a new optimization variable , and likewise with the lower and upper bounds.
Second, there is an additional user-deﬁned cost term in the objective function. This cost consists of three pieces that will be described in more detail below.
Third, the nonlinear constraints and are augmented with user deﬁned additions and to give and .
Up through version 6.0 of
For the AC OPF, subsequent versions also include the general nonlinear
constraints and , and the quadratic and general nonlinear costs and
. The new quadratic cost terms can be handled by all of
Section 7 describes the mechanisms available to the user for taking advantage of the extensible formulation described here.
The creation of additional user-deﬁned variables can be done explicitly or implicitly based on the diﬀerence between the number of columns in and the dimension of the standard OPF optimization variable . The optional vectors and are available to impose lower and upper bounds on , respectively.
Each of these constraint sets is deﬁned by two M-ﬁle functions, similar to those required by MIPS, one that computes the constraint values and their gradients (Jacobian), and the other that computes Hessian values.
The user-deﬁned cost function consists of three terms for three diﬀerent types of costs: quadratic, general nonlinear, and legacy. Each term is a simple summation over all of the cost sets of that type.
where is deﬁned in several steps as follows. First, a new vector is created by applying a linear transformation and shift to the full set of optimization variables
then a scaled function with a “dead zone” is applied to each element of to produce the corresponding element of .
Here speciﬁes the size of the “dead zone”, is a simple scale factor and
is a pre-deﬁned scalar function selected by the value of . Currently,
This form for provides the ﬂexibility to handle a wide range of costs, from simple linear functions of the optimization variables to scaled quadratic penalties on quantities, such as voltages, lying outside a desired range, to functions of linear combinations of variables, inspired by the requirements of price coordination terms found in the decomposition of large loosely coupled problems encountered in our own research.
Some limitations are imposed on the parameters in the case of the DC OPF since