Series Compensation (TCSC)

A TCSC (thyristor controlled series capacitor) is a variable series reactance inserted into 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! (issue #297). This page explains why a series reactance steers flow, how the physical device maps onto the model, and how the controller iterates; the outer-loop machinery itself is described in the Control Framework.

The branch as a four-terminal element

Every branch enters the power flow through its admittance matrix (see Branch Model for the derivation and sign conventions):

\[Y_{br} = \begin{bmatrix} \frac{1}{\tau^2}\left(y_{ser} + \frac{y_{shunt}}{2}\right) & -y_{ser}\,\frac{1}{\tau e^{-j\phi}} \\ -y_{ser}\,\frac{1}{\tau e^{j\phi}} & y_{ser} + \frac{y_{shunt}}{2} \end{bmatrix}, \qquad y_{ser} = \frac{1}{R + jX},\]

with the complex turns ratio $N = \tau e^{j\phi}$ and $N = 1$ for lines. The TCSC acts purely through $X$ inside $y_{ser}$: the controller assigns a new x_pu to its line branch, which changes exactly one branch stamp, and the outer loop re-stamps the Y-bus before the next solve. No solver internals are touched; this is the same actuator mechanism the phase-shifting transformer uses for its tap-dependent reactance $X(\alpha)$ in the tap controller.

Why a series reactance steers flow

For a lossless line between buses 1 and 2 the transferred active power is

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

Lowering the series reactance raises the power carried by that path at a given angle difference. In a meshed network the effect appears as a flow redistribution: two parallel corridors split the transfer in inverse proportion to their reactances, so compensating one corridor pulls flow onto it and off its neighbors. This loop-network mechanism is exactly what the example and the tests exercise: a corridor with twice the reactance of its parallel path carries one third of the transfer at baseline, and the TCSC moves that share onto the target.

Compensation degree

Series compensation is usually 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. Practical installations run at roughly $k = 0.2$ to $0.7$; beyond that, subsynchronous resonance concerns and protection coordination dominate. The model does not track $k$ explicitly: the controller works directly on the net reactance x_pu, clamped to [x_min_pu, x_max_pu]. A range for a line with $X_{line} = 0.20$ p.u. and up to 70 percent compensation is therefore simply 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 as a function of the firing angle passes through a resonance region between the capacitive and the inductive operating range; a real device must not dwell there, and vendor curves exclude it.

The model is deliberately simpler: one continuous, clamped reactance. Two consequences follow:

• The impedance-magnitude guard eps_z is the documented stand-in for the resonance exclusion. Registration rejects any range in which the series impedance magnitude $|R + jX|$ falls below the guard, checked at both range ends and, for a sign-crossing range, at the crossing itself (there the magnitude bottoms out at $|R|$). Splitting the admissible range into two disjoint intervals (capacitive and inductive, with the resonance band excluded between them) is a possible later refinement, mirroring the issue text.
• Negative net reactance (a net capacitive branch) is admissible by design: the guard already protects the singular neighborhood of $X = 0$, so the model does not need a positivity restriction. This is a documented design choice; most practical targets are reachable well inside the inductive range.

At a range end the controller stops moving and the branch behaves as a fixed compensated line; there is no special casing beyond the honest at_limit report.

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). Steering a remote branch through a TCSC elsewhere is a conceivable extension (an optional target_branch), but the first cut keeps measurement and actuator on the same element.

Numerical method

The controller runs inside the generic outer loop (Control Framework): measure after a converged solve, propose a step, apply it, re-solve.

Secant iteration. The scalar map from x_pu to the branch power is smooth in the operating range, so the controller uses the same secant update as the machine and shunt controllers: from the last two points $(x_k, P_k)$ and $(x_{k-1}, P_{k-1})$ it estimates the sensitivity and steps toward the target.

Bootstrap step. Before two points exist there is no measurable sensitivity. The first move is a bounded probe: a fixed fraction of the headroom toward the more distant range end. Unlike the shunt controller, 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, so the probe measures it and the secant uses the signed slope from then on.

Sensitivity guard. A slope magnitude below a minimum threshold would turn the secant step into noise amplification; the controller then falls back to the bounded probe instead, the same guard the machine controller uses.

Honest at limit. Every proposed step is clamped to [x_min_pu, x_max_pu]. at_limit is set only when the reactance is clamped at a range end and the target is still outside the deadband: the controller does not pretend convergence, and the solve itself remains valid with the branch as a fixed compensated line.

SSSC mode: injected-voltage limit (issue #297 Draft F)

The fixed window above models a TCSC, whose hardware defines an admissible reactance range independent of loading. A static synchronous series compensator (SSSC) is a voltage-source converter injecting a voltage in quadrature with the line current; in steady state that acts as a reactance deviation from the natural line reactance $x_{base}$, bounded by the injectable voltage magnitude:

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

addSeriesReactanceControl! with v_inj_max_pu (instead of x_min_pu/x_max_pu) switches the controller into this mode. The same secant machinery runs on a LIVE window:

• the bounds $x_{base} \pm V_{inj,max}/|I|$ are re-evaluated from the solved branch current (measured at the registered from side, $|I| = |S|/V$) before every outer step;
• the window SHRINKS with loading: at high transfer, exactly when a large flow correction would need a large reactance swing, the SSSC saturates, while a TCSC keeps its full window (the defining contrast between the two devices, see FACTS Devices);
• a floor on $|I|$ keeps the window finite on a currentless branch (physically an SSSC injects no voltage without current), and the eps_z resonance guard is applied as a clamp on the window instead of an error, so a transient operating point can never abort the run;
• an at-limit SSSC whose window still moves keeps adjusting and parks at_limit only once the window has settled; at the limit the effective injected voltage $|I| \cdot |x - x_{base}|$ sits at $V_{inj,max}$.

The element row and the summary printer show the live window, the natural reactance, the measured current, and the currently injected voltage.