Phase-domain conventions#
The phase-domain formulation represents only active electrical nodes and uses rectangular complex voltage as its Newton state. This representation preserves unbalanced phase voltage, missing phases, explicit neutrals, and floating conductors without introducing a polar-angle singularity.
Buses and electrical nodes#
A physical bus groups conductors at one location. An electrical node identifies one conductor at that bus by its OpenDSS node number. Node numbers remain general integers; the formulation does not assume that every bus contains nodes 1, 2, 3, and 4.
Ground uses OpenDSS node number 0, but it does not receive an electrical degree of freedom. A grounded conductor contributes its known zero voltage through the element connection. An explicit neutral uses an active nonzero node number and participates in the solved system.
Every active node stores a nominal line-to-neutral voltage. The nominal value is used for normalization and model limits; the solved voltage remains in volts.
Terminals and connection matrices#
Each element owns ordered terminals and ordered conductors. A ragged node map
connects the local conductor positions to global active nodes. The element
connection matrix maps node voltage v to terminal voltage u:
For a grounded wye load, each row of C selects one phase conductor. For a
delta load, each row subtracts the second conductor from the first conductor of
one phase-to-phase branch. The same mapping represents open-delta connections,
missing phases, transformer windings, and arbitrary conductor order.
The conjugate transpose maps terminal current to node current:
Linear, source, and voltage-dependent elements#
A linear element has a terminal admittance Y. Its node-current relation is
Voltage sources, current sources, and multi-terminal equivalents add their specified source terms to this relation.
A voltage-dependent element evaluates current from terminal voltage and any device state:
Loads, generators, photovoltaic systems, and storage use this form. Voltage-regulating generators and induction machines add bounded device state where required.
Power and current signs#
Terminal current is positive into an element. Complex terminal power follows
Therefore, a consuming load has positive active and reactive power under the
normal OpenDSS load convention. A generating device is represented by negative
power entering the element. element_losses sum the terminal powers of one
element using this orientation.
A mixed-system port uses positive current into its phase region. The mixed power-flow chapter defines the corresponding positive-sequence projection and three-phase power normalization.
Units#
The phase data plane uses SI units. Voltage uses volts, current uses amperes, power uses watts and vars, impedance uses ohms, admittance uses siemens, length uses metres, and frequency uses hertz. GPF converts OpenDSS engineering units to these units.
The phase formulation does not assign one global power base to every device. Each node carries its physical voltage base, and a mixed-system port performs the explicit conversion to the transmission system base.
Newton and control sequence#
GPF solves the nodal equations in rectangular coordinates with analytic device derivatives and damped Newton steps. GPF does not use a fixed-point fallback.
Static controls are evaluated only after an electrical solution. Each control measures the solved state and can change its controlled device. A selected action starts another electrical solve. Final results describe the last accepted state.