LCC Second Derivatives (Hessian Blocks)

This page derives the second derivatives needed to extend the RobustHomotopyPowerFlow Hessian assembly (and the spectral-radius diagnostic) to systems with LCC HVDC lines. It is a companion to lcc_model.md, which lays out the residual and Jacobian rows.

The homotopy Hessian computes

\[H(x) = t_k \left( J(x)^\top J(x) + \sum_k F_k(x)\, \nabla^2 F_k(x) \right) + (1-t_k)\, \text{diag}(\mathbb{1}_{\text{PQ }V}).\]

The J^\top J term and the network-only ∑ F_k ∇² F_k block are already handled by _update_hessian_matrix_values! in homotopy_hessian.jl. The missing piece is the LCC contribution to ∑ F_k ∇² F_k. This page enumerates those contributions.

Notation and setup

For one LCC, write the rectifier-side and inverter-side AC variables as (V_r, t_r, α_r) and (V_i, t_i, α_i), and let

\[K := \frac{\sqrt{6}}{\pi}, \qquad \beta_s := \frac{x_s\, I_{dc}}{\sqrt{2}} \quad (s \in \{r, i\}),\]

with I_dc > 0 the (constant during the solve) DC-line current magnitude and x_s the side-s transformer reactance. Define the arccos arguments

\[u_r(V_r, t_r, \alpha_r) = \cos\alpha_r - \frac{\beta_r}{V_r\, t_r}, \qquad u_i(V_i, t_i, \alpha_i) = -\cos\alpha_i - \frac{\beta_i}{V_i\, t_i}.\]

(u_i differs from u_r only by the sign convention — the helper _calculate_ϕ_lcc is called with -I_dc on the inverter, which flips the overall sign of the bracket.) Then cos\phi_s = u_s and \sin\phi_s = \sqrt{1 - u_s^2} \ge 0 strictly in the interior.

All formulas below are the interior formulas (\sin\phi_s bounded away from 0). The clamp regime is handled in a separate section.

First derivatives of u_s

Useful intermediate quantities (rectifier shown; inverter differs only in sign of α terms):

\[\begin{aligned} \frac{\partial u_r}{\partial V_r} &= \frac{\beta_r}{V_r^2 t_r}, & \frac{\partial u_r}{\partial t_r} &= \frac{\beta_r}{V_r\, t_r^2}, & \frac{\partial u_r}{\partial \alpha_r} &= -\sin\alpha_r, \\ \frac{\partial u_i}{\partial V_i} &= \frac{\beta_i}{V_i^2 t_i}, & \frac{\partial u_i}{\partial t_i} &= \frac{\beta_i}{V_i\, t_i^2}, & \frac{\partial u_i}{\partial \alpha_i} &= +\sin\alpha_i. \end{aligned}\]

Second derivatives of u_s:

\[\begin{aligned} \frac{\partial^2 u_s}{\partial V_s^2} &= -\frac{2\beta_s}{V_s^3 t_s}, & \frac{\partial^2 u_s}{\partial t_s^2} &= -\frac{2\beta_s}{V_s\, t_s^3}, & \frac{\partial^2 u_s}{\partial V_s\,\partial t_s} &= -\frac{\beta_s}{V_s^2 t_s^2}, \\ \frac{\partial^2 u_r}{\partial \alpha_r^2} &= -\cos\alpha_r, & \frac{\partial^2 u_i}{\partial \alpha_i^2} &= +\cos\alpha_i, & \frac{\partial^2 u_s}{\partial V_s\,\partial \alpha_s} = \frac{\partial^2 u_s}{\partial t_s\,\partial \alpha_s} &= 0. \end{aligned}\]

(Sign of \partial^2 u/\partial V \partial t: \partial u/\partial V = \beta/(V^2 t), so \partial^2 u/\partial V \partial t = \beta/V^2 \cdot \partial(1/t)/\partial t = -\beta/(V^2 t^2). Easy to miss.)

Active power: closed form and Hessian

In the interior, the chain term in the dP/d· Jacobians cancels exactly against the leading factor, and P_s reduces to a particularly clean form:

\[P_r = V_r t_r\, K I_{dc} \cos\phi_r = K I_{dc}\,(V_r t_r \cos\alpha_r - \beta_r), \qquad P_i = K I_{dc}\,(-V_i t_i \cos\alpha_i - \beta_i).\]

That is, P_s is linear in the bilinear coordinate V_s t_s with an α_s-dependent coefficient, plus a constant. So its Hessian has only three non-zero entries (one mixed V·t term and the pure α·α, V·α, t·α terms):

Rectifier (P_r):

\[\begin{aligned} \frac{\partial^2 P_r}{\partial V_r^2} = \frac{\partial^2 P_r}{\partial t_r^2} &= 0, \\ \frac{\partial^2 P_r}{\partial V_r\,\partial t_r} &= K I_{dc} \cos\alpha_r, \\ \frac{\partial^2 P_r}{\partial V_r\,\partial \alpha_r} &= -K I_{dc}\, t_r \sin\alpha_r, \\ \frac{\partial^2 P_r}{\partial t_r\,\partial \alpha_r} &= -K I_{dc}\, V_r \sin\alpha_r, \\ \frac{\partial^2 P_r}{\partial \alpha_r^2} &= -K I_{dc}\, V_r t_r \cos\alpha_r. \end{aligned}\]

Inverter (P_i): same shape, opposite sign on every entry (since P_i = -K I_{dc} (V_i t_i \cos\alpha_i + \beta_i)):

\[\begin{aligned} \frac{\partial^2 P_i}{\partial V_i^2} = \frac{\partial^2 P_i}{\partial t_i^2} &= 0, \\ \frac{\partial^2 P_i}{\partial V_i\,\partial t_i} &= -K I_{dc} \cos\alpha_i, \\ \frac{\partial^2 P_i}{\partial V_i\,\partial \alpha_i} &= +K I_{dc}\, t_i \sin\alpha_i, \\ \frac{\partial^2 P_i}{\partial t_i\,\partial \alpha_i} &= +K I_{dc}\, V_i \sin\alpha_i, \\ \frac{\partial^2 P_i}{\partial \alpha_i^2} &= +K I_{dc}\, V_i t_i \cos\alpha_i. \end{aligned}\]

Reactive power: derivation and Hessian

Q_s = V_s t_s\, K I_{dc} \sin\phi_s. With C := \cos\phi_s = u_s, S := \sin\phi_s = \sqrt{1-u_s^2}, and using \partial S/\partial x = -(C/S)\,\partial u_s/\partial x, write Q_s = K I_{dc}\, A\, S where A := V_s t_s (so \partial A/\partial V_s = t_s, \partial A/\partial t_s = V_s, \partial A/\partial \alpha_s = 0, \partial^2 A/\partial V_s\partial t_s = 1, all other second partials of A are zero).

Differentiating once, then again, gives the product/chain expansion

\[\frac{\partial^2 Q_s}{\partial x\,\partial y} = K I_{dc}\Bigl[\,\frac{\partial^2 A}{\partial x \partial y}\, S + \frac{\partial A}{\partial x}\, \frac{\partial S}{\partial y} + \frac{\partial A}{\partial y}\, \frac{\partial S}{\partial x} + A\, \frac{\partial^2 S}{\partial x \partial y} \Bigr],\]

with

\[\frac{\partial^2 S}{\partial x\,\partial y} = -\frac{1}{S^3}\,\frac{\partial u_s}{\partial x}\,\frac{\partial u_s}{\partial y} - \frac{C}{S}\,\frac{\partial^2 u_s}{\partial x \partial y}.\]

(The 1/S^3 form comes from u^2 + S^2 = 1, which collapses the (1/S + u^2/S^3) algebra to 1/S^3.) Carrying out the substitution and collecting yields the closed-form entries below. Let \sigma_s = +1 for the rectifier, \sigma_s = -1 for the inverter — this encodes \partial u_s/\partial \alpha_s = -\sigma_s \sin\alpha_s and \partial^2 u_s/\partial \alpha_s^2 = -\sigma_s \cos\alpha_s. (The V/t derivatives of u_s are sign-invariant in \sigma_s, so all entries without an \alpha-derivative come out side-symmetric.)

Pure V_s, t_s block (no \alpha). Two of the three entries collapse to single-term expressions because the (\partial A/\partial x) \cdot (\partial S/\partial x) term cancels the leading piece of A \cdot (\partial^2 S/\partial x^2):

\[\begin{aligned} \frac{\partial^2 Q_s}{\partial V_s^2} &= -\frac{K I_{dc}\, \beta_s^2}{V_s^3 t_s\, S^3}, \\[4pt] \frac{\partial^2 Q_s}{\partial t_s^2} &= -\frac{K I_{dc}\, \beta_s^2}{V_s t_s^3\, S^3}, \\[4pt] \frac{\partial^2 Q_s}{\partial V_s\,\partial t_s} &= K I_{dc}\, S - \frac{K I_{dc}\, \cos\phi_s\, \beta_s}{V_s\, t_s\, S} - \frac{K I_{dc}\, \beta_s^2}{V_s^2 t_s^2\, S^3}. \end{aligned}\]

(The \partial^2 Q/\partial V^2 and \partial^2 Q/\partial t^2 single-term forms come from 2(\partial A/\partial V)(\partial S/\partial V) + A \cdot \partial^2 S/\partial V^2 = -2u\beta/(V^2 S) + (-\beta^2/(V^3 t S^3) + 2u\beta/(V^2 S)) = -\beta^2/(V^3 t S^3). The collapse is unconditional in \sigma_s.)

Mixed with \alpha_s:

\[\begin{aligned} \frac{\partial^2 Q_s}{\partial V_s\,\partial \alpha_s} &= \sigma_s\, K I_{dc}\, \sin\alpha_s \left[ \frac{t_s\, \cos\phi_s}{S} + \frac{\beta_s}{V_s\, S^3} \right], \\[4pt] \frac{\partial^2 Q_s}{\partial t_s\,\partial \alpha_s} &= \sigma_s\, K I_{dc}\, \sin\alpha_s \left[ \frac{V_s\, \cos\phi_s}{S} + \frac{\beta_s}{t_s\, S^3} \right]. \end{aligned}\]

Pure \alpha_s \alpha_s:

\[\frac{\partial^2 Q_s}{\partial \alpha_s^2} = -\frac{K I_{dc}\, V_s\, t_s\, \sin^2\alpha_s}{S^3} + \sigma_s\,\frac{K I_{dc}\, V_s\, t_s\, \cos\phi_s\, \cos\alpha_s}{S}.\]

(The first term, -V t K I \sin^2\alpha / S^3, is sign-invariant in \sigma: it picks up (\partial u/\partial \alpha)^2 = \sin^2\alpha either way. The second term carries \partial^2 u/\partial \alpha^2 = -\sigma_s \cos\alpha_s, which through the outer -\cos\phi_s/S factor becomes +\sigma_s \cos\phi_s \cos\alpha_s / S.)

Sanity check against the Jacobian

The existing Jacobian entry from _calculate_dQ_dV_lcc is

\[\frac{\partial Q_s}{\partial V_s} = t_s K I_{dc}\, S - \frac{K I_{dc}\, \beta_s\, \cos\phi_s}{V_s\, S}.\]

Differentiating once more w.r.t. V_s (treating u_s = \cos\phi_s and S = \sin\phi_s as functions of V_s) reproduces the \partial^2 Q_s/\partial V_s^2 = -K I_{dc} \beta_s^2 / (V_s^3 t_s S^3) formula above. Specifically:

\[\begin{aligned} \frac{\partial}{\partial V_s}\Bigl(\frac{\beta_s \cos\phi_s}{V_s S}\Bigr) &= \frac{\beta_s}{V_s} \cdot \frac{\partial(u_s/S)}{\partial V_s} - \frac{\beta_s u_s}{V_s^2 S} \\ &= \frac{\beta_s}{V_s} \cdot \frac{\beta_s}{V_s^2 t_s S^3} - \frac{\beta_s u_s}{V_s^2 S} \end{aligned}\]

(using \partial(u/S)/\partial V = \beta/(V^2 t S^3) from u^2 + S^2 =1), and \partial(t_s S)/\partial V_s = -t_s u_s \beta/(V_s^2 t_s S) =-u_s \beta/(V_s^2 S). Combining and simplifying gives the single-term result. The recommended numerical check is finite differences; this hand check is shown to demonstrate the cancellation that produces the clean -K I_{dc} \beta^2/(V^3 t S^3) form.

Hessian contributions per residual row

Per LCC, the residual rows that depend on LCC state and contribute to ∑_k F_k \nabla^2 F_k are:

So the additive contribution to the Hessian from LCC \ell is

\[\begin{aligned} H_\ell &= F_{P, f_b}\, \nabla^2 P_r \;+\; F_{Q, f_b}\, \nabla^2 Q_r \\ &\quad + F_{P, t_b}\, \nabla^2 P_i \;+\; F_{Q, t_b}\, \nabla^2 Q_i \\ &\quad + F_{t_r}\, \nabla^2 P_{r\text{ or }i}^{\;(\pm)} \;+\; F_{t_i}\,(\nabla^2 P_r + \nabla^2 P_i). \end{aligned}\]

Each \nabla^2 P_s and \nabla^2 Q_s has support only on its own side's (V_s, t_s, \alpha_s) columns/rows — the rectifier block and the inverter block of a single LCC do not share any second-derivative entries (they share no state variables, only constants like I_{dc} and R_{dc}, which are not differentiated). The two sides are therefore combined as a block-diagonal pair within each LCC's 6×6 LCC-state block.

Sparsity pattern of the LCC Hessian block

For LCC \ell with rectifier-side bus f_b and inverter-side bus t_b, the state coordinates touched are

\[(V_{f_b}, \theta_{f_b}, V_{t_b}, \theta_{t_b}, t_r, t_i, \alpha_r, \alpha_i).\]

The Hessian contributions above touch only V_{f_b}, t_r, \alpha_r and V_{t_b}, t_i, \alpha_i (the \theta coordinates do not appear in any LCC residual). The added sparse structure is therefore two 3×3 dense blocks (one per side), embedded into the global Hessian at the appropriate row/column indices:

• Rectifier block: rows/cols \{V_{f_b}, t_r, \alpha_r\} → \partial^2 (\text{linear comb of } P_r, Q_r).
• Inverter block: rows/cols \{V_{t_b}, t_i, \alpha_i\} → \partial^2 (\text{linear comb of } P_i, Q_i).

The new structural entries needed for the LCC Hessian are (V_{f_b}, t_r), (V_{f_b}, \alpha_r), (t_r, t_r), (t_r, \alpha_r), (\alpha_r, \alpha_r) (and symmetric), plus the analogous inverter block on (V_{t_b}, t_i, \alpha_i). None of these come from network-only rows of J (which have no t or \alpha columns at all). But the Jacobian has two other categories of rows that do:

1. LCC bus rows (P_{f_b}, Q_{f_b}, P_{t_b}, Q_{t_b}): these carry the LCC self-admittance contributions and thus have nonzero entries in all of V, t, and \alpha for the relevant side.
2. LCC tail rows (F_{t_r}, F_{t_i}, F_{\alpha_r}, F_{\alpha_i}): the F_t rows depend on V (via the chain rule through P_{lcc,from/to}) and on t, \alpha; the F_\alpha rows contain only the unit entry \partial F_{\alpha_s}/\partial \alpha_s = 1.

So when J^\top J is formed, the columns V_{f_b}, t_r, \alpha_r all share the LCC bus rows and the F_{t_r} / F_{t_i} tail rows as common nonzero positions, which gives J^\top J structural entries at every pairwise combination of those three columns — exactly the slots the LCC Hessian needs. Same on the inverter side. The sparsity pattern of J^\top J therefore already covers the LCC Hessian block without needing additional structural fill, provided A_plus_eq_BT_B!'s structural-zero preservation continues to hold (it asserts colptr and rowval equality on each call). The numeric updates from _update_hessian_lcc_contributions! go into existing slots.

Clamp regime

When \sin\phi_s < LCC_sinϕ_TOLERANCE, the residual treats \phi_s as locally pinned and the Jacobian helpers drop the chain term (see _dphi_dV_lcc and _dphi_dt_lcc). The corresponding Hessian behavior:

• P_s becomes linear in (V_s, t_s) with no α-dependence (since \cos\phi_s = \pm 1 constant): the only non-zero second partial is \partial^2 P_s / \partial V_s \partial t_s = \pm K I_{dc}. All α-mixed and α^2 partials vanish.
• Q_s = 0 at the clamp boundary (since \sin\phi_s = 0) and the derivative is pinned to 0 in the residual: drop all second partials of Q_s on the clamp branch.

Both clamp branches are degenerate one-sided regimes — the analytic formulas above have 1/S and 1/S^3 factors that blow up. Matching the Jacobian's clamp guard, the implementation should test \sin\phi_s and return zero (or the simple \pm K I_{dc} mixed-Vt entry for P_s) when below the tolerance.

Side comparison

A compact side-by-side reference for the most error-prone entries:

The V_s, t_s-only second partials of Q_s are identical on both sides because they depend only on (V_s, t_s, \cos\phi_s, \sin\phi_s, \beta_s) — quantities whose dependence on side enters only through the value of \cos\phi_s = u_s (whose sign already lives in \cos\phi_s itself and does not need an extra \sigma_s).