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Adding SOC extended DistFlow (BFM) formulation (#229)
* Adding SOC BIM / extended DistFlow formulation * Updated current variable name + dev documentation for variable naming * defining opf_bf and pf_bf problem type * add arc_from to constraint_voltage_angle_difference * change signature of constraint_power_losses * updated docstrings * Derivation of distflow formulation with asymmetric shunts * moved/removed functions + derive bound for series current * Complete DF derivation with asymmetric shunts, including conductance * compatible with asymmetric pi-section
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| # Developer Documentation | ||
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| Nothing yet. | ||
| ## Variable and parameter naming scheme | ||
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| ### Suffixes | ||
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| - `_fr`: from-side ('i'-node) | ||
| - `_to`: to-side ('j'-node) | ||
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| ### Power | ||
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| 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) | ||
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| ### Voltage | ||
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| 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) | ||
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| ### Current | ||
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| 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) | ||
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| ### Voltage products | ||
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| 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$^2$) | ||
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| ### Current products | ||
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| 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$^2$) | ||
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| ### Transformer ratio | ||
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| 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 (-) | ||
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| ### Impedance | ||
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| Defining impedance | ||
| $z = r + j\cdot x$: | ||
| - `r`: resistance ($\Omega$) | ||
| - `x`: reactance ($\Omega$) | ||
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| ### Admittance | ||
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| Defining admittance | ||
| $y = g + j\cdot b$: | ||
| - `g`: conductance ($S$) | ||
| - `b`: susceptance ($S$) | ||
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| ## DistFlow derivation | ||
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| ### 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}) $ + $((r^{s}_{lij})^2 + (x^{s}_{lij})^2)\vert I^{s}_{lij} \vert^2$ | ||
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| 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 $ | ||
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| 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. | ||
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| 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} $ | ||
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| 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} $ | ||
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| 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} $ | ||
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| ### 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: $w^{'}_{l} = \frac{w_{i}}{(t^{mag})^2}$ | ||
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| 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}$ | ||
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| The branch's power balance isn't otherwise impacted by adding the ideal transformer, as such transformer is lossless. | ||
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| ### 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}$ | ||
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| 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$ + $(Q_{lji})^2 \leq (I^{rated}_{l})^2 \cdot w_{j}$ | ||
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| [^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 |
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