The optimization vector for the standard AC OPF problem consists of the vectors of voltage angles and magnitudes and the vectors of generator real and reactive power injections and .
The objective function in (6.1) is simply a summation of individual polynomial cost functions and of real and reactive power injections, respectively, for each generator:
The equality constraints in (6.2) are simply the full set of nonlinear real and reactive power balance equations from (4.2) and (4.3).
The inequality constraints (6.3) consist of two sets of branch ﬂow limits as
nonlinear functions of the bus voltage angles and magnitudes, one for the
The ﬂows are typically apparent power ﬂows expressed in MVA, but can be real power ﬂows (in MW) or currents,31 yielding the following three possible forms for the ﬂow constraints:
where is deﬁned in (3.9), in (3.15), and the vector of
ﬂow limits has the appropriate units for the type of constraint. It
is likewise for . The values used by
The variable limits (6.4) include an equality constraint on any reference bus angle and upper and lower limits on all bus voltage magnitudes and real and reactive generator injections:
Another variation of the standard AC OPF problem represents the bus voltages in cartesian, rather than polar, coordinates. That is, instead of and , the optimization vector includes the real and imaginary parts of the complex voltage, denoted respectively by and , where .
The objective function remains unchanged, but the nodal power balance constraints (6.7) and (6.8) and branch ﬂow constraints (6.9) and (6.10) are implemented as functions of and .
In this formulation, the voltage angle reference constraint (6.12) and voltage magnitude limits (6.13) cannot be simply applied as bounds on optimization variables. These constrained quantities also become functions of and .
Another variation of the standard AC OPF problem uses current balance constraints in place of the power balance constraints (6.7)–(6.8) or (6.17)–(6.18). If we let and represent the real and imaginary parts, respectively, of the current, we can express the current balance functions for the polar form as
and for the cartesian form as
where , and is a diagonal matrix whose -th diagonal entry is , that is or .
In this formulation, which can be selected by setting the