Key concepts

Per unit

Voltages are reported per unit of each bus’s own base voltage: 1.0 pu is nominal, whether the bus is an 11 kV substation bus or a 230 V house. Statutory limits are expressed the same way, so a single number compares across voltage levels.

Phases

Low-voltage customers are usually connected to one phase. A European feeder spreads its single-phase customers across all three phases; a North American one splits each house across the two legs of a centre-tapped transformer. popo keeps track of which phase each customer is on: results label every bus by the phase it is actually connected to, and rooftop arrays, chargers and batteries are installed on their customer’s own phase.

Statutory limits

A statutory limit is a band together with a rule for applying it. EN 50160 asks that 95 % of ten-minute means over a week sit within ±10 % of nominal, so a network can hold a low voltage for an hour and still comply; ANSI C84.1 Range A is ±5 % with no such allowance. VoltageLimits carries both the band and the rule, and a study selects a standard by name.

Busbar and customers

Most networks are measured at the secondary substation’s busbar. Voltage problems usually appear elsewhere: rooftop solar raises voltage most at the far end of the longest service. A feeder can be within limits at the busbar while its farthest customers are over the limit, so solve_time_series() can record every customer’s voltage as well as the busbar’s.

Convergence

A time-series solve whose inverter control loop fails to converge still reports a solved power flow, with meaningless reactive power. popo includes convergence in every result: a SolveReport counts the steps that did not converge.

Observability

A state estimator solves for a voltage magnitude and angle at every bus, less one reference angle: \(2N - 1\) unknowns for \(N\) buses. Each measurement supplies one equation, and each bus known to inject nothing supplies two. With fewer equations than unknowns no estimate exists, whichever solver is used. observability() counts both before an estimate is attempted.