AC Branches
AC Branches represent overhead lines, transformers and underground cables within AC grids.
Parameters
Set of parameters used to model DC branches as defined in the input data
| name | symb. | unit | type | default | definition |
|---|---|---|---|---|---|
| index | $b$ | - | Int | - | unique index of the AC branch |
| f_bus | $i$ | - | Int | - | unique index of the bus to which the AC branch is originating from |
| t_bus | $j$ | - | Int | - | unique index of the bus to which the AC branch is terminating at |
| r | $r_{ac}$ | p.u. | Real | - | resistance of the AC branch |
| l | $l_{ac}$ | p.u. | Real | - | inductance of the AC branch |
| c | $c_{ac}$ | p.u. | Real | - | capacitance of the AC branch |
| rateA | $\overline{S_{b}}$ | MW | Real | - | long term rating of the AC branch |
| rateB | $\overline{S^{st}_{b}}$ | MW | Real | - | short term term rating of the AC branch |
| rateC | $\overline{S^{em}_{b}}$ | MW | Real | - | emergency rating of the AC branch |
| status | $\delta_{b}$ | - | Int | - | status indicator of the AC branch |
| tap | $\tau$ | - | Real | - | tap ratio of potential transformer tap |
| shift | $\varphi$ | - | Real | - | phase shift induced by PST if modelled as a fixed element |
| angmin | $\underline{\theta}$ | - | Real | - | minimum allowable phase angle difference over AC branch |
| angmax | $\overline{\theta}$ | - | Real | - | maximum allowable phase angle difference over AC branch |
| construction_cost | $C_{ac}$ | - | Real | - | investment cost for AC branch used in TNEP problems |
Variables
Optimisation variables representing AC Branch behaviour
| name | symb. | unit | formulation | definition |
|---|---|---|---|---|
| p | $p_{b,i,j}$ | p.u. | ACP, ACR, LPAC, IVR, SOC, DCP, NF | Active power flow through AC branch b, connecting AC nodes i and j |
| q | $q_{b,i,j}$ | p.u. | ACP, ACR, LPAC, IVR, SOC, DCP, NF | Reactive power flow through AC branch b, connecting AC nodes i and j |
| cr | $\Re(i_{b,i,j})$ | p.u. | IVR | Real current flow through AC branch b, connecting AC nodes i and j |
| ci | $\Im(i_{b,i,j})$ | p.u. | IVR | Imaginary current flow through AC branch b, connecting AC nodes i and j |
| csr | $\Re(i^{s}_{b,i,j})$ | p.u. | IVR | Real current series flow through AC branch b, connecting AC nodes i and j |
| csi | $\Im(i^{s}_{b,i,j})$ | p.u. | IVR | Imaginary series current flow through AC branch b, connecting AC nodes i and j |
| cshr_fr | $\Re(i^{sh}_{b,i})$ | p.u. | IVR | Real shunt current of AC branch b, at node i |
| cshi_fr | $\Im(i^{sh}_{b,i})$ | p.u. | IVR | Imaginary shunt current of AC branch b, at node i |
| cshr_to | $\Re(i^{sh}_{b,j})$ | p.u. | IVR | Real shunt current of AC branch b, at node j |
| cshi_to | $\Im(i^{sh}_{b,j})$ | p.u. | IVR | Imaginary shunt current of AC branch b, at node j |
Constraints
Flow and voltage angle limits
\[\begin{align} - \overline{S_{b}} &\leq P_{d,i,j} \leq \overline{S_{b}} \\ - \overline{S_{b}} &\leq Q_{d,i,j} \leq \overline{S_{b}} \\ \sqrt{(P_{d,i,j})^2 +(Q_{d,i,j})^2 } &\leq \overline{S_{b}} \\ 0 &\leq I_{b,i,j} \leq \overline{I_{b}} \\ 0 &\leq W_{b,i,j} \leq max(V_{i}^{2}, V_{j}^{2}) \end{align}\]
Ohm's law
ACP model: consult power models for full implementation
\[\begin{align} p_{b,i,j} &= g_{b} \cdot (v_{i})^{2} - g_{b} \cdot v_{i} \cdot v_{j} \cdot cos(\theta_{i} - \theta_{j} ) - b_{b} \cdot v_{i} \cdot v_{j} \cdot sin(\theta_{i} - \theta_{j})\\ q_{b,i,j} &= -b_{b} \cdot (v_{i})^{2} + b_{b} \cdot v_{i} \cdot v_{j} \cdot cos(\theta_{i} - \theta_{j} ) - g_{b} \cdot v_{i} \cdot v_{j} \cdot sin(\theta_{i} - \theta_{j}) \\ p_{b,j,i} &= g_{b} \cdot (v_{j})^{2} - g_{b} \cdot v_{i} \cdot v_{j} \cdot cos(\theta_{j} - \theta_{i} ) - b_{b} \cdot v_{i} \cdot v_{j} \cdot sin(\theta_{j} - \theta_{i}) \\ q_{b,j,i} &= -b_{b} \cdot (v_{j})^{2} + b_{b} \cdot v_{i} \cdot v_{j} \cdot cos(\theta_{j} - \theta_{i} ) - g_{b} \cdot v_{i} \cdot v_{j} \cdot sin(\theta_{j} - \theta_{i}) \\ \end{align}\]
ACR model: consult power models for full implementation
\[\begin{align} p_{b,i,j} &= (g_{b}+g_{b,fr}) \cdot (\Re(v_{i})^2 + \Im(v_{i})^2) + (-g_{b} + b_{b}) \cdot (\Re(v_{i})\Re(v_{j}) + \Im(_{i})\Im(v_{j})) + (-g_{b} - b_{b}) \cdot (\Im(v_{i})\Re(v_{j}) - \Re(v_{i})\Im(v_{j})) \\ q_{b,i,j} &= -(b_{b}+b_{b,fr}) \cdot (\Re(v_{i})^2 + \Im(v_{i})^2) - (-g_{b} - b_{b}) \cdot (\Re(v_{i})\Re(v_{j}) + \Im(_{i})\Im(v_{j})) + (-g_{b} + b_{b}) \cdot (\Im(v_{i})\Re(v_{j}) - \Re(v_{i})\Im(v_{j})) \\ p_{b,j,i} &= (g_{b}+g_{b,to}) \cdot (\Re(v_{j})^2 + \Im(v_{j})^2) + (-g_{b} + b_{b}) \cdot (\Re(v_{i})\Re(v_{j}) + \Im(_{i})\Im(v_{j})) + (-g_{b} - b_{b}) \cdot (-(\Im(v_{i})\Re(v_{j}) - \Re(v_{i})\Im(v_{j}))) \\ q_{b,j,i} &= -(b_{b}+b_{b,to}) \cdot (\Re(v_{j})^2 + \Im(v_{j})^2) - (-g_{b} - b_{b}) \cdot (\Re(v_{i})\Re(v_{j}) + \Im(_{i})\Im(v_{j})) + (-g_{b} + b_{b}) \cdot (-(\Im(v_{i})\Re(v_{j}) - \Re(v_{i})\Im(v_{j}))) \\ \end{align}\]
IVR model: models the current flow explicitely, using a series current $c_{b}^{s}$ and a shunt current $c_{b}^{sh}$
Liniking powee flow to voltage and current:
\[\begin{align} p_{b,i,j} &= \Re(v_{i}) \cdot \Re(i_{b,i,j}) + \Im(v_{i}) \cdot \Im(i_{b,i,j}) \\ q_{b,i,j} &= \Im(v_{i}) \cdot \Re(i_{b,i,j}) - \Re(v_{i}) \cdot \Im(i_{b,i,j}) \\ p_{b,j,i} &= \Re(v_{j}) \cdot \Re(i_{b,j,i}) + \Im(v_{j}) \cdot \Im(i_{b,j,i}) \\ q_{b,j,i} &= \Im(v_{j}) \cdot \Re(i_{b,j,i}) - \Re(v_{j}) \cdot \Im(i_{b,j,i}) \\ \end{align}\]
Linking series and shunt currents:
\[\begin{align} \Re(i_{b,i,j}) &= \Re(i^{s}_{b,i,j}) + g_{b,fr} \cdot \Re(v_{i}) - b_{b,fr} \cdot \Im(v_{i}) \\ \Im(i_{b,i,j}) &= \Im(i^{s}_{b,i,j}) + g_{b,fr} \cdot \Im(v_{i}) - b_{b,fr} \cdot \Re(v_{i}) \\ \Re(i_{b,j,i}) &= \Re(i^{s}_{b,j,i}) + g_{b,to} \cdot \Re(v_{j}) - b_{b,to} \cdot \Im(v_{j}) \\ \Im(i_{b,j,i}) &= \Im(i^{s}_{b,j,i}) + g_{b,to} \cdot \Im(v_{j}) - b_{b,to} \cdot \Re(v_{j}) \\ \end{align}\]
Voltage drop:
\[\begin{align} \Re(v_{j}) &= \Re(v_{i}) - r_{b} \cdot \Re(i^{s}_{b,i,j}) + x_{b} \cdot \Im(i^{s}_{b,i,j}) \\ \Im(v_{j}) &= \Im(v_{i}) - r_{b} \cdot \Im(i^{s}_{b,i,j}) - x_{b} \cdot \Re(i^{s}_{b,i,j}) \\ \end{align}\]
SOC, QC bus injection model (BIM): #todo
SOC, QC branch flow model (BFM): #todo
DCP model:
\[\begin{align} p_{b,i,j} &= -b_{b} \cdot(\theta{i} - \theta{j}) \\ p_{b,j,i} &= -b_{b} \cdot(\theta{j} - \theta{i}) \\ \end{align}\]
NF model: In this model there are no losses, no impedances, as such only the active power limits are binding.