| (6.1) |

subject to

In both cases, the objective function consists of the polynomial cost of generator injections, the equality constraints are the power balance equations, the inequality constraints are the branch ﬂow limits, and the and bounds include reference bus angles, voltage magnitudes (for AC) and generator injections.

6.1 Standard AC OPF

6.1.1 Cartesian vs. Polar Coordinates for Voltage

6.1.2 Current vs. Power for Nodal Balance Constraints

6.2 Standard DC OPF

6.3 Extended OPF Formulation

6.3.1 User-deﬁned Variables

6.3.2 User-deﬁned Constraints

6.3.3 User-deﬁned Costs

6.4 Standard Extensions

6.4.1 Piecewise Linear Costs

6.4.2 Dispatchable Loads

6.4.3 Generator Capability Curves

6.4.4 Branch Angle Diﬀerence Limits

6.5 Solvers

6.6runopf

6.1.1 Cartesian vs. Polar Coordinates for Voltage

6.1.2 Current vs. Power for Nodal Balance Constraints

6.2 Standard DC OPF

6.3 Extended OPF Formulation

6.3.1 User-deﬁned Variables

6.3.2 User-deﬁned Constraints

6.3.3 User-deﬁned Costs

6.4 Standard Extensions

6.4.1 Piecewise Linear Costs

6.4.2 Dispatchable Loads

6.4.3 Generator Capability Curves

6.4.4 Branch Angle Diﬀerence Limits

6.5 Solvers

6.6