DC Optimal Power Flow

B-theta Formulation

The DC OPF minimizes generation cost subject to linearized power flow constraints using the susceptance-weighted Laplacian:

\[\min_{g, \theta, f, \text{psh}} \quad g^\top C_q g + c_l^\top g + c_{\text{shed}}^\top \text{psh} + \frac{\tau^2}{2} \|f\|^2\]

subject to:

\[\begin{aligned} G_{\text{inc}} g + \text{psh} - d &= B \theta & (\nu_{\text{bal}}) \\ f &= W A \theta & (\nu_{\text{flow}}) \\ -f_{\max} \leq f &\leq f_{\max} & (\lambda_{\text{lb}}, \lambda_{\text{ub}}) \\ g_{\min} \leq g &\leq g_{\max} & (\rho_{\text{lb}}, \rho_{\text{ub}}) \\ 0 \leq \text{psh} &\leq d & (\mu_{\text{lb}}, \mu_{\text{ub}}) \\ \alpha_{\min} \leq A\theta &\leq \alpha_{\max} & (\gamma_{\text{lb}}, \gamma_{\text{ub}}) \\ \theta_{\text{ref}} &= 0 & (\eta_{\text{ref}}) \end{aligned}\]

where:

$B = A^\top \operatorname{diag}(-b \circ \mathrm{sw}) \, A$ is the susceptance-weighted Laplacian
$W = \operatorname{diag}(-b \circ \mathrm{sw})$ is the branch weight matrix
$A$ is the $m \times n$ incidence matrix (branches × buses)
$G_{\text{inc}}$ is the $n \times k$ generator-bus incidence matrix
$C_q = \operatorname{diag}(c_q)$ contains quadratic cost coefficients
$c_l$ contains linear cost coefficients
$c_{\text{shed}}$ is the load-shedding cost vector
$\tau$ is a small regularization parameter for numerical conditioning

KKT System for Implicit Differentiation

OPF sensitivities are computed via the implicit function theorem applied to the KKT conditions. At an optimal solution $z^*$, the KKT residual satisfies $K(z^*, p) = 0$ where $z$ collects all primal and dual variables and $p$ is a parameter.

By the implicit function theorem:

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

KKT Variable Ordering

The KKT variable vector $z$ is structured as:

\[z = [\theta, g, f, \text{psh}, \lambda_{\text{lb}}, \lambda_{\text{ub}}, \gamma_{\text{lb}}, \gamma_{\text{ub}}, \rho_{\text{lb}}, \rho_{\text{ub}}, \mu_{\text{lb}}, \mu_{\text{ub}}, \nu_{\text{bal}}, \nu_{\text{flow}}, \eta_{\text{ref}}]\]

with total dimension $5n + 6m + 3k + 1$.

KKT Conditions

The KKT residual $K(z, p)$ consists of:

Stationarity w.r.t. $\theta$: $B^\top \nu_{\text{bal}} + (WA)^\top \nu_{\text{flow}} + e_{\text{ref}} \eta_{\text{ref}} + A^\top (\gamma_{\text{ub}} - \gamma_{\text{lb}}) = 0$
Stationarity w.r.t. $g$: $2 C_q g + c_l - G_{\text{inc}}^\top \nu_{\text{bal}} - \rho_{\text{lb}} + \rho_{\text{ub}} = 0$
Stationarity w.r.t. $f$: $\tau^2 f - \nu_{\text{flow}} - \lambda_{\text{lb}} + \lambda_{\text{ub}} = 0$
Stationarity w.r.t. psh: $c_{\text{shed}} - \nu_{\text{bal}} - \mu_{\text{lb}} + \mu_{\text{ub}} = 0$
Complementary slackness (flow bounds): $\lambda_{\text{lb}} \circ (f + f_{\max}) = 0$, $\lambda_{\text{ub}} \circ (f_{\max} - f) = 0$

5b. Complementary slackness (angle differences): $\gamma_{\text{lb}} \circ (A\theta - \alpha_{\min}) = 0$, $\gamma_{\text{ub}} \circ (\alpha_{\max} - A\theta) = 0$ 5c. Complementary slackness (generation/shedding bounds): $\rho \circ (\cdot) = 0$, $\mu \circ (\cdot) = 0$

Primal feasibility: $G_{\text{inc}} g + \text{psh} - d - B\theta = 0$
Flow definition: $f - WA\theta = 0$
Reference bus: $\theta_{\text{ref}} = 0$

Analytical Sparse KKT Jacobian

The KKT Jacobian $\partial K / \partial z$ is computed analytically as a sparse matrix, which is more efficient than automatic differentiation for large problems.

Parameter Derivatives

For each parameter type, we compute $\partial K / \partial p$ and then the full derivative via the IFT formula.

Demand ($d$)

Demand enters the power balance and the upper shedding bound:

\[\frac{\partial K_{\nu_{\text{bal}}}}{\partial d} = -I, \qquad \frac{\partial K_{\mu_{\text{ub}}}}{\partial d} = \operatorname{diag}(\mu_{\text{ub}})\]

Switching ($\mathrm{sw}$)

Switching affects the Laplacian $B$, weight matrix $W$, and flow definition through:

\[\frac{\partial B}{\partial \mathrm{sw}_e} = -b_e \, a_e a_e^\top, \qquad \frac{\partial W}{\partial \mathrm{sw}_e} = -b_e \, e_e e_e^\top\]

This propagates into the stationarity, power balance, and flow definition blocks of the KKT system.

Cost Coefficients ($c_q$, $c_l$)

Quadratic cost $c_q$ enters stationarity w.r.t. $g$:

\[\frac{\partial K_g}{\partial c_{q,i}} = 2 g_i e_i\]

Linear cost $c_l$ enters stationarity w.r.t. $g$:

\[\frac{\partial K_g}{\partial c_l} = I_k\]

Flow Limits ($f_{\max}$)

Flow limits enter the complementary slackness conditions:

\[\frac{\partial K_{\lambda_{\text{lb}}}}{\partial f_{\max}} = -\operatorname{diag}(\lambda_{\text{lb}}), \qquad \frac{\partial K_{\lambda_{\text{ub}}}}{\partial f_{\max}} = \operatorname{diag}(\lambda_{\text{ub}})\]

Susceptances ($b$)

Susceptances affect the same blocks as switching (through $B$ and $W$), but with different partial derivatives since $B = A^\top \operatorname{diag}(-b \circ \mathrm{sw}) A$.

LMP Decomposition

Locational marginal prices are the power balance duals $\nu_{\text{bal}}$, decomposed as:

\[\text{LMP} = \underbrace{\text{energy}}_{\text{uniform component}} + \underbrace{\text{congestion}}_{\text{flow-limit component}}\]

The congestion component is extracted by solving:

\[\text{congestion}[\text{non-ref}] = B_r^{-1} \left(A_r^\top W (\lambda_{\text{ub}}^{\text{std}} - \lambda_{\text{lb}}^{\text{std}}) + A_r^\top (\gamma_{\text{ub}}^{\text{std}} - \gamma_{\text{lb}}^{\text{std}})\right)\]

where $\lambda^{\text{std}}$, $\gamma^{\text{std}}$ use the standard sign convention (non-negative for binding constraints). The $\gamma$ terms capture congestion from binding phase angle difference limits. The energy component is uniform across all buses in a connected network and reflects the marginal cost of generation.