Static series synchronous compenstion devices, e.g. smart wires
The series compensation devices inject a voltage in quadrature of the line voltage in order to control active and reactive power flows.
Parameters
Set of parameters used to model static series synchronous compenstion devices as defined in the input data
| name | symb. | unit | type | default | definition |
|---|---|---|---|---|---|
| index | $sc$ | - | Int | - | unique index of the sereies compensator |
| f_bus | $i$ | - | Int | - | unique index of the bus to which the series compensator is originating from |
| t_bus | $j$ | - | Int | - | unique index of the bus to which the series compensator is terminating at |
| sssc_r | $r_{sssc}$ | p.u. | Real | - | resistance of the series compensator |
| sssc_x | $x_{sssc}$ | p.u. | Real | - | inductance of the series compensator |
| rate_a | $\overline{S_{sssc}}$ | MVA | Real | - | long term rating of series compensator |
| rate_b | $\overline{S^{st}_{sssc}}$ | MVA | Real | - | short term term rating of series compensator |
| rate_c | $\overline{S^{em}_{sssc}}$ | MVA | Real | - | emergency rating of series compensator |
| sssc_status | $\delta_{sssc}$ | - | Int | - | status indicator of series compensator |
| vqmin | $\underline{v_{q}}$ | p.u. | Real | - | minimum quadrature voltage |
| vqmax | $\overline{v_{q}}$ | p.u. | Real | - | maximum quadrature voltage |
Variables
Optimisation variables representing SSSC behaviour
| name | symb. | unit | formulation | definition |
|---|---|---|---|---|
| vqsssc | $v_{q}$ | p.u. | ACP, ACR | Voltage injected in quadrature |
| alphaqsssc | $\alpha_{q}$ | rad | DCP | Equivalent phase angle shift induced by SSSC |
| psssc | $p_{sc,i,j}$ | p.u. | ACP, ACR, DCP | Active power flow through SSSC |
| qsssc | $q_{sc,i,j}$ | p.u. | ACP, ACR | Reactive power flow through SSSC |
Constraints
Flow, voltage, angle limits
Active, reactive, and apparent power limits:
\[\begin{align} - \overline{S_{sssc}} \leq p_{sc,i,j} \leq \overline{S_{sssc}} \\ - \overline{S_{sssc}} \leq q_{sc,i,j} \leq \overline{S_{sssc}} \\ p_{sc,i,j}^{2} + q_{sc,i,j}^{2} \leq \overline{S_{sssc}}^{2} \end{align}\]
Voltage range for the induced quadrature voltage:
\[\begin{align} \underline{v_{q}} \leq v_{q} \leq \overline{v_{q}} \end{align}\]
Range for equivalent phase angle shift:
\[\begin{align} -atan(\underline{v_{q}}) \leq \alpha_{q} \leq atan(\overline{v_{q}}) \end{align}\]
Constraints
SSSC admittance
\[\begin{align} g_{sssc} = \Re{\frac{1} {r_{sssc} + j \cdot x_{sssc}}} \\ b_{sssc} = \Im{\frac{1} {r_{sssc} + j \cdot x_{sssc}}} \\ \end{align}\]
Ohm's law
ACP model:
\[\begin{align} v_{i}^{*} &= \sqrt{v_{i}^{2} + 2 \cdot v_{i} \cdot v_{q} + v_{q}^{2}} \\ \theta_{i}^{*} &= atan(\frac{v_{i} \cdot sin(\theta_{i}) + v_{q}}{v_{i} \cdot cos(\theta_{i})}) \end{align}\]
\[\begin{align} p_{sc,i,j} &= g_{sssc} \cdot (v^{*}_{i})^{2} - g_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot cos(\theta_{i}^{*} - \theta_{j}) - b_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot sin(\theta_{i}^{*} - \theta_{j}) \\ q_{sc,i,j} &= -b_{sssc} \cdot (v^{*}_{i})^{2} + b_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot cos(\theta_{i}^{*} - \theta_{j}) - g_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot sin(\theta_{i}^{*} - \theta_{j}) \\ \end{align}\]
\[\begin{align} p_{j,i,sc} &= g_{sssc} \cdot (v_{j})^{2} - g_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot cos(\theta_{j} - \theta_{i}^{*}) - b_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot sin(\theta_{j} - \theta_{i}^{*}) \\ q_{j,i,sc} &= -b_{sssc} \cdot (v_{j})^{2} + b_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot cos(\theta_{j} - \theta_{i}^{*}) - g_{sssc} \cdot v^{*}_{i} \cdot v_{j} \cdot sin(\theta_{j} - \theta_{i}^{*}) \\ \end{align}\]
ACR model:
\[\begin{align} \underline{v_{i}^{*}} &= \underline{v_{i}} + j \cdot v_{q} \\ p_{sc,i,j} &= g_{sssc} \cdot (v_{i}^{*})^{2}- g_{sssc} \cdot (\Re{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Im{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) - b_{sssc} \cdot (\Im{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Re{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) \\ q_{sc,i,j} &= -b_{sssc} \cdot (v_{i}^{*})^{2} + b_{sssc} \cdot (\Re{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Im{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) - g_{sssc} \cdot (\Im{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Re{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) \\ \end{align}\]
\[\begin{align} p_{j,i,sc} &= g_{sssc} \cdot (v_{j})^{2}- g_{sssc} \cdot (\Re{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Im{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) - b_{sssc} \cdot (\Im{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Re{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) \\ q_{j,i,sc} &= -b_{sssc} \cdot (v_{j})^{2} + b_{sssc} \cdot (\Re{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Im{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) - g_{sssc} \cdot (\Im{\underline{v_{i}^{*}}} \cdot \Re{\underline{v_{j}}} + \Re{\underline{v_{i}^{*}}} \cdot \Im{\underline{v_{j}}}) \\ \end{align}\]
DCP model:
\[\begin{align} p_{sc,i,j} &= - b_{sssc} \cdot (\theta_{i} - \theta_{j} - \alpha_{q}) \\ p_{sc,i,j} + p_{j,i,sc} &= 0 \end{align}\]