Developer Documentation

Function Naming Guidelines

Following the Julia style guidelines, functions that mutate arguments should end with !. Following the JuMP style guidelines functions beginning with an _ are intended for internal package use only (i.e. similar private scope functions). Underscores are used to separate multi-word function names and the words should typically be ordered from general to more specific, so that alphabetical sorting clusters similar functions together.

Variable and Parameter Naming Guidelines

Suffixes

_fr: from-side ('i'-node)
_to: to-side ('j'-node)

Power

Defining power $s = p + j \cdot q$ and $sm = |s|$

s: complex power (VA)
sm: apparent power (VA)
p: active power (W)
q: reactive power (var)

Voltage

Defining voltage $v = vm \angle va = vr + j \cdot vi$:

vm: magnitude of (complex) voltage (V)
va: angle of complex voltage (rad)
vr: real part of (complex) voltage (V)
vi: imaginary part of complex voltage (V)

Current

Defining current $c = cm \angle ca = cr + j \cdot ci$:

cm: magnitude of (complex) current (A)
ca: angle of complex current (rad)
cr: real part of (complex) current (A)
ci: imaginary part of complex current (A)

Voltage products

Defining voltage product $w = v_i \cdot v_j$ then $w = wm \angle wa = wr + j\cdot wi$:

wm (short for vvm): magnitude of (complex) voltage products (V$^2$)
wa (short for vva): angle of complex voltage products (rad)
wr (short for vvr): real part of (complex) voltage products (V$^2$)
wi (short for vvi): imaginary part of complex voltage products (V$^2$)

Current products

Defining current product $cc = c_i \cdot c_j$ then $cc = ccm \angle cca = ccr + j\cdot cci$:

ccm: magnitude of (complex) current products (A$^2$)
cca: angle of complex current products (rad)
ccr: real part of (complex) current products (A$^2$)
cci: imaginary part of complex current products (A$^2$)

Transformer ratio

Defining complex transformer ratio $t = tm \angle ta = tr + j\cdot ti$:

tm: magnitude of (complex) transformer ratio (-)
ta: angle of complex transformer ratio (rad)
tr: real part of (complex) transformer ratio (-)
ti: imaginary part of complex transformer ratio (-)

Impedance

Defining impedance $z = r + j\cdot x$:

r: resistance ($\Omega$)
x: reactance ($\Omega$)

Admittance

Defining admittance $y = g + j\cdot b$:

g: conductance ($S$)
b: susceptance ($S$)

Standard Value Names

network ids:network, nw, n
conductors ids: conductor, cnd, c
phase ids: phase, ph, h

DistFlow derivation

For an asymmetric pi section

Following notation of ^[1], but recognizing it derives the SOC BFM without shunts. In a pi-section, part of the total current $I_{lij}$ at the from side flows through the series impedance, $I^{s}_{lij}$, part of it flows through the from side shunt admittance $I^{sh}_{lij}$. Vice versa for the to-side. Indicated by superscripts 's' (series) and 'sh' (shunt).

\[\begin{align} & \mbox{Ohm's law: } U^{mag}_{j} \angle \theta_{j} = U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij} \nonumber \\ & \mbox{KCL at shunts: } I_{lij} = I^{s}_{lij} + I^{sh}_{lij}, I_{lji} = I^{s}_{lji} + I^{sh}_{lji} \nonumber \\ & \mbox{Observing: } Observing: I^{s}_{lij} = - I^{s}_{lji}, \vert I^{s}_{lij} \vert = \vert I^{s}_{lji} \vert \nonumber \\ & \mbox{Ohm's law times its own complex conjugate: } (U^{mag}_{j})^2 = (U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij})\cdot (U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij})^* \nonumber \\ & \mbox{Defining: } S^{s}_{lij} = P^{s}_{lij} + j\cdot Q^{s}_{lij} = (U^{mag}_{i}\angle \theta_{i}) \cdot (I^{s}_{lij})^* \nonumber \\ & \mbox{Working it out: } (U^{mag}_{j})^2 = (U^{mag}_{i})^2 - 2 \cdot(r^{s}_{lij} \cdot P^{s}_{lij} + x^{s}_{lij} \cdot Q^{s}_{lij}) + ((r^{s}_{lij})^2 + (x^{s}_{lij})^2)\vert I^{s}_{lij} \vert^2 \nonumber \\ \end{align}\]

Power flow balance w.r.t. branch total losses

\[\begin{align} & \mbox{Active power flow: } P_{lij} + P_{lji} = g^{sh}_{lij} \cdot (U^{mag}_{i})^2 + r^{s}_{l} \cdot \vert I^{s}_{lij} \vert^2 + g^{sh}_{lji} \cdot (U^{mag}_{j})^2 \nonumber \\ & \mbox{Reactive power flow: } Q_{lij} + Q_{lji} = -b^{sh}_{lij} \cdot (U^{mag}_{i})^2 + x^{s}_{l} \cdot \vert I^{s}_{lij} \vert^2 - b^{sh}_{lji} \cdot (U^{mag}_{j})^2 \nonumber \\ & \mbox{Current definition: } \vert S^{s}_{lij} \vert^2 = (U^{mag}_{i})^2 \cdot \vert I^{s}_{lij} \vert^2 \nonumber \\ \end{align}\]

Substitution:

\[\begin{align} & \mbox{Voltage from: } (U^{mag}_{i})^2 \rightarrow w_{i} \nonumber \\ & \mbox{Voltage to: } (U^{mag}_{j})^2 \rightarrow w_{j} \nonumber \\ & \mbox{Series current: } \vert I^{s}_{lij} \vert^2 \rightarrow l^{s}_{l} \nonumber \\ \end{align}\]

Note that $l^{s}_{l}$ represents squared magnitude of the series current, i.e. the current flow through the series impedance in the pi-model. Power flow balance w.r.t. branch total losses

\[\begin{align} & \mbox{Active power flow: } P_{lij} + P_{lji} = g^{sh}_{lij} \cdot w_{i} + r^{s}_{l} \cdot l^{s}_{l} + g^{sh}_{lji} \cdot w_{j} \nonumber \\ & \mbox{Reactive power flow: } Q_{lij} + Q_{lji} = -b^{sh}_{lij} \cdot w_{i} + x^{s}_{l} \cdot l^{s}_{l} - b^{sh}_{lji} \cdot w_{j} \nonumber \\ \end{align}\]

Power flow balance w.r.t. branch series losses:

\[\begin{align} & \mbox{Active power flow: } P^{s}_{lij} + P^{s}_{lji} = r^{s}_{l} \cdot l^{s}_{l} \nonumber \\ & \mbox{Reactive power flow: } Q^{s}_{lij} + Q^{s}_{lji} = x^{s}_{l} \cdot l^{s}_{l} \nonumber \\ \end{align}\]

Valid equality to link $w_{i}, l_{lij}, P^{s}_{lij}, Q^{s}_{lij}$:

\[\begin{align} & \mbox{Nonconvex current definition: } (P^{s}_{lij})^2 + (Q^{s}_{lij})^2 =w_{i} \cdot l_{lij} \nonumber \\ & \mbox{SOC current definition: } (P^{s}_{lij})^2 + (Q^{s}_{lij})^2 \leq w_{i} \cdot l_{lij} \nonumber \\ \end{align}\]

Adding an ideal transformer

Adding an ideal transformer at the from side implicitly creates an internal branch voltage, between the transformer and the pi-section.

\[\begin{align} & \mbox{New voltage: } w^{'}_{l} \nonumber \\ & \mbox{Ideal voltage magnitude transformer: } w^{'}_{l} = \frac{w_{i}}{(t^{mag})^2} \nonumber \\ \end{align}\]

W.r.t to the pi-section only formulation, we effectively perform the following substitution in all the equations above:

\[\begin{align} & w_{i} \rightarrow \frac{w_{i}}{(t^{mag})^2} \nonumber \\ \end{align}\]

The branch's power balance isn't otherwise impacted by adding the ideal transformer, as such transformer is lossless.

Adding total current limits

\[\begin{align} & \mbox{Total current from: } \vert I_{lij} \vert \leq I^{rated}_{l} \nonumber \\ & \mbox{Total current to: } \vert I_{lji} \vert \leq I^{rated}_{l} \nonumber \\ \end{align}\]

In squared voltage magnitude variables:

\[\begin{align*} & \mbox{Total current from: } (P_{lij})^2 + (Q_{lij})^2 \leq (I^{rated}_{l})^2 \cdot w_{i} \nonumber \\ & \mbox{Total current to: } (P_{lji})^2 + (Q_{lji})^2 \leq (I^{rated}_{l})^2 \cdot w_{j} \nonumber \\ \end{align*}\]

^[1] Gan, L., Li, N., Topcu, U., & Low, S. (2012). Branch flow model for radial networks: convex relaxation. 51st IEEE Conference on Decision and Control, 1–8. Retrieved from http://smart.caltech.edu/papers/ExactRelaxation.pdf