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 ($S$)
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).
- 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$ + $(Expr(:incomplete, "incomplete: premature end of input")){lij})^2 \leq (I^{rated}{l})^2 \cdot w{i}$
- Total current to: $ (P{lji})^2$ + $(Expr(:incomplete, "incomplete: premature end of input")){lji})^2 \leq (I^{rated}{l})^2 \cdot w{j}$
[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