Developer Documentation

Variable and parameter naming scheme

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 ($S$)

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).

• Ohm's law: $U^{mag}_{j} \angle \theta_{j} = U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij}$ $\forall lij$

• KCL at shunts: $ I_{lij} = I^{s}_{lij} + I^{sh}_{lij}$, $ I_{lji} = I^{s}_{lji} + I^{sh}_{lji} $

• Observing: $I^{s}_{lij} = - I^{s}_{lji}$, $ \vert I^{s}_{lij} \vert = \vert I^{s}_{lji} \vert $

• Ohm's law times its own complex conjugate: $(U^{mag}_{j})^2 = (U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij})\cdot (U^{mag}_{i}\angle \theta_{i} - z^{s}_{lij} \cdot I^{s}_{lij})^*$

• Defining $S^{s}_{lij} = P^{s}_{lij} + j\cdot Q^{s}_{lij} = (U^{mag}_{i}\angle \theta_{i}) \cdot (I^{s}_{lij})^*$

• 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}) $ + $((r^{s}_{lij})^2 + (x^{s}_{lij})^2)\vert I^{s}_{lij} \vert^2$

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

• 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 $

• 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 $

• Current definition: $ \vert S^{s}_{lij} \vert^2 $ $=(U^{mag}_{i})^2 \cdot \vert I^{s}_{lij} \vert^2 $

Substitution:

• Voltage from: $(U^{mag}_{i})^2 \rightarrow w_{i}$

• Voltage to: $(U^{mag}_{j})^2 \rightarrow w_{j}$

• Series current : $\vert I^{s}_{lij} \vert^2 \rightarrow l^{s}_{l}$

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

• 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} $

• 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} $

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

• Series active power flow : $P^{s}_{lij} + P^{s}_{lji}$ $ = r^{s}_{l} \cdot l^{s}_{l} $

• Series reactive power flow: $Q^{s}_{lij} + Q^{s}_{lji}$ $ = x^{s}_{l} \cdot l^{s}_{l} $

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

• Nonconvex current definition: $(P^{s}_{lij})^2$ + $(Q^{s}_{lij})^2$ $=w_{i} \cdot l_{lij} $

• SOC current definition: $(P^{s}_{lij})^2$ + $(Q^{s}_{lij})^2$ $\leq$ $ w_{i} \cdot l_{lij} $

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.

• new voltage: $w^{'}_{l}$

• ideal voltage magnitude transformer: $w^{'}_{l} = \frac{w_{i}}{(t^{mag})^2}$

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

• $ w_{i} \rightarrow \frac{w_{i}}{(t^{mag})^2}$

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

Adding total current limits

• Total current from: $ \vert I_{lij} \vert \leq I^{rated}_{l}$

• Total current to: $ \vert I_{lji} \vert \leq I^{rated}_{l}$

In squared voltage magnitude variables:

• Total current from: $ (P_{lij})^2$ + $(Q_{lij})^2 \leq (I^{rated}_{l})^2 \cdot w_{i}$

• Total current to: $ (P_{lji})^2$ + $(Q_{lji})^2 \leq (I^{rated}_{l})^2 \cdot w_{j}$

[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