AC Optimal Power Flow

Formulation

The AC OPF minimizes generation cost subject to the full nonlinear AC power flow equations in polar coordinates:

\[\min_{\theta, |V|, p_g, q_g} \quad \sum_{i=1}^{k} \left(c_{q,i} \, p_{g,i}^2 + c_{l,i} \, p_{g,i} + c_{c,i}\right)\]

subject to:

Power Balance (Equality)

\[\begin{aligned} \sum_{(l,i,j) \in \mathcal{A}_i} p_{lij} + g_{s,i} |V_i|^2 &= \sum_{g \in \mathcal{G}_i} p_{g,g} - p_{d,i} & (\nu_{p,i}) \\ \sum_{(l,i,j) \in \mathcal{A}_i} q_{lij} - b_{s,i} |V_i|^2 &= \sum_{g \in \mathcal{G}_i} q_{g,g} - q_{d,i} & (\nu_{q,i}) \end{aligned}\]

where $\mathcal{A}_i$ are the arcs incident to bus $i$, $\mathcal{G}_i$ are the generators at bus $i$, and $g_{s,i}$, $b_{s,i}$ are shunt conductance/susceptance.

Reference Bus (Equality)

\[\theta_{\text{ref}} = 0 \qquad (\nu_{\text{ref}})\]

Branch Flow Equations

For each branch $l$ from bus $f$ to bus $t$ with switching state $\mathrm{sw}_l$:

\[\begin{aligned} p_{fr,l} &= \mathrm{sw}_l \left[\frac{g_l + g_{fr}^{sh}}{t_m^2} |V_f|^2 + \frac{-g_l t_r + b_l t_i}{t_m^2} |V_f||V_t| \cos(\theta_f - \theta_t) + \frac{-b_l t_r - g_l t_i}{t_m^2} |V_f||V_t| \sin(\theta_f - \theta_t)\right] \\ p_{to,l} &= \mathrm{sw}_l \left[(g_l + g_{to}^{sh}) |V_t|^2 + \frac{-g_l t_r - b_l t_i}{t_m^2} |V_t||V_f| \cos(\theta_t - \theta_f) + \frac{-b_l t_r + g_l t_i}{t_m^2} |V_t||V_f| \sin(\theta_t - \theta_f)\right] \end{aligned}\]

The reactive power flow equations follow the same structure with sine/cosine swapped:

\[\begin{aligned} q_{fr,l} &= \mathrm{sw}_l \left[-\frac{b_l + b_{fr}^{sh}}{t_m^2} |V_f|^2 - \frac{-b_l t_r - g_l t_i}{t_m^2} |V_f||V_t| \cos(\theta_f - \theta_t) + \frac{-g_l t_r + b_l t_i}{t_m^2} |V_f||V_t| \sin(\theta_f - \theta_t)\right] \\ q_{to,l} &= \mathrm{sw}_l \left[-(b_l + b_{to}^{sh}) |V_t|^2 - \frac{-b_l t_r + g_l t_i}{t_m^2} |V_t||V_f| \cos(\theta_t - \theta_f) + \frac{-g_l t_r - b_l t_i}{t_m^2} |V_t||V_f| \sin(\theta_t - \theta_f)\right] \end{aligned}\]

where $g_l + jb_l$ is the branch admittance, $t_r + jt_i$ is the complex tap ratio, $t_m^2 = \mathrm{tap}^2$, and $g_{fr}^{sh}$, $b_{fr}^{sh}$ are the from-side shunt elements of the pi-model.

Inequality Constraints

\[\begin{aligned} p_{fr,l}^2 + q_{fr,l}^2 &\leq r_l^2 & (\lambda_{\text{th},fr,l}) & \quad \text{Thermal limits (from)} \\ p_{to,l}^2 + q_{to,l}^2 &\leq r_l^2 & (\lambda_{\text{th},to,l}) & \quad \text{Thermal limits (to)} \\ \theta_f - \theta_t &\geq \alpha_{\min,l} & (\lambda_{\angle,lb,l}) & \quad \text{Angle difference bounds} \\ \theta_f - \theta_t &\leq \alpha_{\max,l} & (\lambda_{\angle,ub,l}) \\ V_{\min,i} \leq |V_i| &\leq V_{\max,i} & (\mu_{lb,i}, \mu_{ub,i}) & \quad \text{Voltage bounds} \\ p_{g,\min,i} \leq p_{g,i} &\leq p_{g,\max,i} & (\rho_{pg,lb,i}, \rho_{pg,ub,i}) & \quad \text{Generation bounds} \\ q_{g,\min,i} \leq q_{g,i} &\leq q_{g,\max,i} & (\rho_{qg,lb,i}, \rho_{qg,ub,i}) \\ -r_l \leq p_{fr,l} &\leq r_l & (\sigma_{p,fr,lb,l}, \sigma_{p,fr,ub,l}) & \quad \text{Flow variable bounds} \\ -r_l \leq q_{fr,l} &\leq r_l & (\sigma_{q,fr,lb,l}, \sigma_{q,fr,ub,l}) \end{aligned}\]

(with analogous bounds for the to-side flows $p_{to,l}$, $q_{to,l}$).

Reduced-Space Formulation

The implementation uses a reduced-space formulation where branch flows $p_{fr}$, $q_{fr}$, $p_{to}$, $q_{to}$ are treated as functions of the voltage state $(\theta, |V|)$ rather than separate primal variables. This means:

• The flow definition constraints are eliminated
• Stationarity conditions include all reduced-space chain-rule terms analytically
• Flow bound complementary slackness uses the computed flow expressions

KKT System

Variable Ordering

The KKT variable vector $z$ is structured as:

\[z = [\theta, |V|, p_g, q_g, \; \nu_p, \nu_q, \nu_{\text{ref}}, \; \lambda_{\text{th},fr}, \lambda_{\text{th},to}, \lambda_{\angle,lb}, \lambda_{\angle,ub}, \; \mu_{lb}, \mu_{ub}, \; \rho_{pg,lb}, \rho_{pg,ub}, \rho_{qg,lb}, \rho_{qg,ub}, \; \sigma_{p,fr,lb}, \ldots, \sigma_{q,to,ub}]\]

with total dimension $6n + 12m + 6k + n_{\text{ref}}$, where $n$ is the number of buses, $m$ is the number of branches, $k$ is the number of generators, and $n_{\text{ref}}$ is the number of reference buses.

KKT Conditions

1. Stationarity ($2n + 2k$ conditions): Assembled analytically from the reduced-space Lagrangian $\mathcal{L}(\theta, |V|, p_g, q_g)$ and branch-flow derivatives.

2. Primal feasibility ($2n + n_{\text{ref}}$ conditions):

   • Active power balance at each bus
   • Reactive power balance at each bus
   • Reference bus angle constraint

3. Complementary slackness ($4m + 2n + 4k + 8m$ conditions):

   • Thermal limits: $\lambda_{\text{th}} \cdot (p^2 + q^2 - r^2) = 0$
   • Angle differences: $\lambda_\angle \cdot (\theta_f - \theta_t - \alpha) = 0$
   • Voltage bounds: $\mu \cdot (|V| - V_{\text{bound}}) = 0$
   • Generation bounds: $\rho \cdot (x_g - x_{\text{bound}}) = 0$
   • Flow variable bounds: $\sigma \cdot (f \pm r) = 0$

Implicit Differentiation

Optimal AC OPF solutions are differentiated by applying the implicit function theorem to the KKT system, which is the constrained optimization counterpart of implicitly differentiating the power flow equations themselves. The same theory guarantees that the derivative exists under mild regularity (a nonsingular KKT Jacobian at a strictly complementary solution) for generic radial or meshed networks.

KKT Jacobian

The KKT Jacobian $\partial K / \partial z$ is assembled analytically as a sparse matrix. Branch-flow derivatives and Hessians are evaluated only where the reduced-space KKT terms require them.

Parameter Jacobians

For each parameter $p$, the parameter Jacobian $\partial K / \partial p$ is also assembled analytically. Single-column, JVP, and VJP paths avoid materializing the full parameter Jacobian when only one direction is needed.

The full derivative is:

\[\frac{dz}{dp} = -\left(\frac{\partial K}{\partial z}\right)^{-1} \frac{\partial K}{\partial p}\]

Supported Parameters

The AC OPF supports sensitivity w.r.t. 6 parameter types:

Each parameter type requires its own $\partial K / \partial p$ computation but shares the same KKT Jacobian factorization.

Caching Strategy

The ACSensitivityCache implements a two-level caching hierarchy:

1. KKT factorization (shared across all parameters): One LU factorization of $\partial K / \partial z$ is computed once and reused for all parameter types.

2. Parameter derivatives (shared across operands): For each parameter $p$, the full $dz/dp$ matrix is computed once. Different operand queries (:va, :vm, :pg, :qg, :lmp, :qlmp) simply extract different row blocks from the same cached matrix.

This means that querying all 6 operands for the same parameter costs essentially the same as querying 1 operand. And querying a second parameter type only requires the $\partial K / \partial p$ computation (the expensive KKT factorization is reused).

Operand Extraction

Given the full derivative $dz/dp$, individual operand sensitivities are extracted by selecting the appropriate row indices:

Supported Combinations

The AC OPF supports all 36 operand × parameter combinations (6 operands × 6 parameters):

LMP Computation

In the AC OPF, locational marginal prices are the active power balance duals $\nu_{p,i}$. These capture the marginal cost of serving an additional unit of active demand at bus $i$, accounting for network losses, congestion, and voltage constraints.

lmps = calc_sensitivity(ac_prob, :lmp, :d)   # dLMP/dd

Solver

The AC OPF uses Ipopt as the default nonlinear programming solver, accessed via JuMP.

References

The implicit differentiation framework underlying these sensitivities, which applies the implicit function theorem to power flow solutions and their optimization counterparts under admittance and topology changes, follows:

• S. Talkington, D. Turizo, S. A. Dorado-Rojas, R. K. Gupta & D. K. Molzahn, "Differentiating Through Power Flow Solutions for Admittance and Topology Control," 2025.