Series Compensation (TCSC)

A TCSC (thyristor controlled series capacitor) is a variable series reactance in a line branch. Steered onto a branch active-power target, it redistributes flow between the parallel paths of a meshed network. Sparlectra models it as the outer-loop controller SeriesReactanceControl, registered with addSeriesReactanceControl! on the Control Framework.

The branch as a four-terminal element

The TCSC acts through $X$ inside the series admittance $y_{ser} = 1/(R + jX)$ of the branch matrix (Branch Model) only: the controller assigns a new x_pu to its line branch, and the outer loop re-stamps the Y-bus before the next solve. The tap-dependent reactance $X(\alpha)$ of a phase-shifting transformer uses the same mechanism (tap controller).

Why a series reactance steers flow

For a lossless line between buses 1 and 2

\[P_{12} = \frac{V_1 V_2}{X}\,\sin(\delta_1 - \delta_2) .\]

A lower series reactance carries more power on that path at a given angle difference. Parallel corridors split the transfer in inverse proportion to their reactances, so compensating one corridor pulls flow onto it and off its neighbors.

Compensation degree

Series compensation is quoted as the compensation degree

\[k = \frac{X_C}{X_{line}}, \qquad X = X_{line} - X_C = (1 - k)\,X_{line},\]

with $X_C$ the inserted capacitive reactance and $X$ the net branch reactance the power flow sees. Installations run at about $k = 0.2$ to $0.7$; beyond that, subsynchronous resonance and protection coordination dominate. The controller works on the net reactance x_pu, clamped to [x_min_pu, x_max_pu]: a line with $X_{line} = 0.20$ p.u. and up to 70 percent compensation gets x_min_pu = 0.06, x_max_pu = 0.20.

Physical device versus model

A real TCSC is a firing-angle controlled parallel circuit of a capacitor bank and a thyristor-switched reactor; its apparent reactance passes through a resonance region, between the capacitive and the inductive range, where the device must not dwell. The model is one continuous, clamped reactance. The impedance-magnitude guard eps_z stands in for the resonance exclusion: registration rejects a range in which $|R + jX|$ falls below the guard, checked at both range ends and, for a sign-crossing range, at the crossing (where the magnitude bottoms out at $|R|$). A negative net reactance (a net capacitive branch) is admissible; the guard protects the singular neighborhood of $X = 0$. At a range end the controller stops and the branch is a fixed compensated line, reported as at_limit.

The measured quantity is the active power of the controlled branch itself, in the registered fromBus to toBus direction (the tap controller's achieved_p_mw convention).

Numerical method

The controller runs in the generic outer loop (measure after a converged solve, propose a step, apply it, re-solve) with the secant update of the machine and shunt controllers on the smooth map from x_pu to the branch power. Before two points exist, the first move is a fixed fraction of the headroom toward the more distant range end; the sign of $dP/dX$ is not hard-coded (in a meshed network it depends on where the branch sits relative to its parallel paths), the probe measures it, and a slope below a minimum threshold falls back to the probe. Every step is clamped to [x_min_pu, x_max_pu]; at_limit is set only when the reactance sits at a range end and the target is still outside the deadband.

SSSC mode: injected-voltage limit

The fixed window models a TCSC, whose admissible reactance range is independent of loading. A static synchronous series compensator (SSSC) injects a voltage in quadrature with the line current; in steady state that is a reactance deviation from the natural line reactance $x_{base}$, bounded by the injectable voltage:

\[|V_{inj}| = |I| \cdot |x - x_{base}| \le V_{inj,max} \quad\Longleftrightarrow\quad |x - x_{base}| \le \frac{V_{inj,max}}{|I|}\]

addSeriesReactanceControl! with v_inj_max_pu (instead of x_min_pu/x_max_pu) selects this mode. The window $x_{base} \pm V_{inj,max}/|I|$ is re-evaluated from the solved branch current (at the registered from side, $|I| = |S|/V$) before every outer step, so it shrinks with loading: at high transfer the SSSC saturates while a TCSC keeps its full window. A floor on $|I|$ keeps the window finite on a currentless branch, and the eps_z guard clamps the window instead of raising an error. At the limit $|I| \cdot |x - x_{base}|$ sits at $V_{inj,max}$. The live-bound mechanics (adjusting while the window still moves, at_limit once it has settled, the current window in the element row) are shared with the STATCOM: FACTS Devices.