OPF Formulations

This reference presents the mathematical formulations of Optimal Power Flow (OPF) variants implemented in GAT, from simplest to most accurate.

The OPF Problem

Optimal Power Flow minimizes generation cost while satisfying physical and operational constraints:

minimize    Total generation cost
subject to  Power flow equations (physics)
            Generator limits (equipment)
            Voltage limits (equipment protection)
            Thermal limits (line ratings)

The challenge: power flow equations are nonlinear and non-convex, making global optimization difficult.

Economic Dispatch (Copper Plate)

The simplest "OPF" ignores the network entirely.

Formulation

$$\min \sum_i C_i(P_i)$$

subject to:

$$\sum_i P_i = \sum_j D_j \quad \text{(power balance)}$$

$$P_i^{\min} \leq P_i \leq P_i^{\max} \quad \text{(generator limits)}$$

where:

• $C_i(P_i)$ is the cost function for generator i
• $D_j$ is demand at bus j

Solution

For quadratic costs $C_i(P) = a_i + b_i P + c_i P^2$:

At optimum, all generators have equal marginal cost (if unconstrained):

$$\frac{dC_i}{dP_i} = b_i + 2c_i P_i = \lambda \quad \text{(for all } i \text{ not at limits)}$$

where $\lambda$ is the system marginal price.

Limitations

• No network constraints → can dispatch infeasible solutions
• No losses → underestimates cost
• Single price → no locational pricing (LMP)

GAT Implementation

OpfSolver::new().with_method(OpfMethod::EconomicDispatch)

DC-OPF (Linear Program)

DC-OPF adds linearized network constraints to economic dispatch.

Assumptions

Same as DC power flow:

1. Voltage magnitudes = 1.0 p.u.
2. Small angle differences: $\sin(\theta) \approx \theta$
3. Lossless lines: $R = 0$

Formulation

Variables:

• $P_i$: generator output at bus i
• $\theta_i$: voltage angle at bus i

Objective:

$$\min \sum_i (a_i + b_i P_i + c_i P_i^2)$$

Note: Quadratic objectives make this a QP, not LP. For true LP, use piecewise-linear cost approximation.

Constraints:

Power balance at each bus:

$$P_i^{\text{gen}} - P_i^{\text{load}} = \sum_j B_{ij}(\theta_i - \theta_j) \quad \forall i$$

Generator limits:

$$P_i^{\min} \leq P_i \leq P_i^{\max} \quad \forall \text{ generators}$$

Branch flow limits:

$$|P_{ij}| = \left| \frac{\theta_i - \theta_j}{X_{ij}} \right| \leq P_{ij}^{\max} \quad \forall \text{ branches}$$

Reference angle:

$$\theta_{\text{ref}} = 0$$

Matrix Form

The power balance constraint in matrix form:

$$\mathbf{P}_{\text{gen}} - \mathbf{P}_{\text{load}} = \mathbf{B} \cdot \boldsymbol{\theta}$$

where $\mathbf{B}$ is the susceptance matrix (imaginary part of Y-bus).

Locational Marginal Prices (LMPs)

The dual variables of power balance constraints give LMPs:

$$\text{LMP}_i = \lambda_i = \frac{\partial(\text{Total Cost})}{\partial(\text{Demand at bus } i)}$$

LMP components:

• Energy: System marginal cost ($\lambda$)
• Congestion: Shadow price of binding flow limits
• Losses: Zero in DC-OPF (lossless assumption)

GAT Implementation

gat opf dc network.arrow --out results.parquet

Uses sparse LP/QP solver. Complexity: O(n) to O(n^1.5) depending on network structure.

SOCP Relaxation (Convex)

Second-Order Cone Programming relaxes the AC-OPF into a convex problem.

Branch Flow Model

Instead of bus injection formulation, SOCP uses branch flows as variables:

Variables:

• $w_i = |V_i|^2$: squared voltage magnitude
• $\ell_{ij} = |I_{ij}|^2$: squared current magnitude
• $P_{ij}, Q_{ij}$: branch power flows
• $P_i^{\text{gen}}, Q_i^{\text{gen}}$: generator outputs

Formulation

Objective:

$$\min \sum_i (a_i + b_i P_i + c_i P_i^2)$$

Branch flow equations (from bus i to bus j):

$$w_j = w_i - 2(r_{ij} P_{ij} + x_{ij} Q_{ij}) + (r_{ij}^2 + x_{ij}^2)\ell_{ij}$$

Power balance at each bus:

$$P_i^{\text{gen}} - P_i^{\text{load}} = \sum_j P_{ij} - \sum_k (P_{ki} - r_{ki}\ell_{ki})$$

$$Q_i^{\text{gen}} - Q_i^{\text{load}} = \sum_j Q_{ij} - \sum_k (Q_{ki} - x_{ki}\ell_{ki})$$

The SOC constraint (relaxed from equality):

$$P_{ij}^2 + Q_{ij}^2 \leq w_i \cdot \ell_{ij}$$

This is a second-order cone constraint:

$$\left| \left( P_{ij}, Q_{ij}, \frac{w_i - \ell_{ij}}{2} \right) \right|_2 \leq \frac{w_i + \ell_{ij}}{2}$$

Voltage limits:

$$(V_i^{\min})^2 \leq w_i \leq (V_i^{\max})^2$$

Thermal limits:

$$P_{ij}^2 + Q_{ij}^2 \leq (S_{ij}^{\max})^2$$

Generator limits:

$$P_i^{\min} \leq P_i^{\text{gen}} \leq P_i^{\max}$$

$$Q_i^{\min} \leq Q_i^{\text{gen}} \leq Q_i^{\max}$$

Why is SOCP a Relaxation?

The exact AC power flow requires:

$$P_{ij}^2 + Q_{ij}^2 = w_i \cdot \ell_{ij} \quad \text{(equality)}$$

SOCP relaxes this to:

$$P_{ij}^2 + Q_{ij}^2 \leq w_i \cdot \ell_{ij} \quad \text{(inequality)}$$

This makes the feasible region convex, enabling global optimization.

Exactness Conditions

The SOCP relaxation is exact (inequality is tight at optimum) when:

• Network is radial (tree topology)
• Voltage upper bounds are not binding
• Objective is strictly increasing in power injection

For meshed networks, the relaxation may be loose (gap between SOCP and true AC solution).

GAT Implementation

OpfSolver::new().with_method(OpfMethod::SocpRelaxation)

Uses Clarabel interior-point solver. Typical convergence: 15-30 iterations.

References

• Farivar & Low (2013), "Branch Flow Model: Relaxations and Convexification"
• Gan, Li, Topcu & Low (2015), "Exact Convex Relaxation of OPF for Radial Networks"

AC-OPF (Full Nonlinear)

The exact AC-OPF is a nonlinear, non-convex optimization problem.

Variables

• $V_i$: voltage magnitude at bus i (p.u.)
• $\theta_i$: voltage angle at bus i (radians)
• $P_i^{\text{gen}}, Q_i^{\text{gen}}$: generator real and reactive power

Formulation

Objective:

$$\min \sum_i (a_i + b_i P_i + c_i P_i^2)$$

Power balance (equality constraints):

$$P_i^{\text{gen}} - P_i^{\text{load}} = \sum_j V_i V_j (G_{ij} \cos\theta_{ij} + B_{ij} \sin\theta_{ij})$$

$$Q_i^{\text{gen}} - Q_i^{\text{load}} = \sum_j V_i V_j (G_{ij} \sin\theta_{ij} - B_{ij} \cos\theta_{ij})$$

where $\theta_{ij} = \theta_i - \theta_j$ and $G_{ij} + jB_{ij} = Y_{ij}$.

Voltage limits:

$$V_i^{\min} \leq V_i \leq V_i^{\max}$$

Generator limits:

$$P_i^{\min} \leq P_i^{\text{gen}} \leq P_i^{\max}$$

$$Q_i^{\min} \leq Q_i^{\text{gen}} \leq Q_i^{\max}$$

Thermal limits:

$$P_{ij}^2 + Q_{ij}^2 \leq (S_{ij}^{\max})^2$$

where branch flows are:

$$P_{ij} = V_i^2 G_{ij} - V_i V_j (G_{ij} \cos\theta_{ij} + B_{ij} \sin\theta_{ij})$$

$$Q_{ij} = -V_i^2 B_{ij} - V_i V_j (G_{ij} \sin\theta_{ij} - B_{ij} \cos\theta_{ij})$$

Reference angle:

$$\theta_{\text{ref}} = 0$$

Solution Methods

AC-OPF is NP-hard in general. No polynomial-time algorithm guarantees the global optimum. Practical approaches:

Penalty Method (GAT Default)

Convert constrained problem to unconstrained via penalty:

$$\min f(\mathbf{x}) + \mu |g(\mathbf{x})|^2 + \mu |\max(0, h(\mathbf{x}))|^2$$

where:

• $f(\mathbf{x})$ is the objective
• $g(\mathbf{x}) = 0$ are equality constraints (power balance)
• $h(\mathbf{x}) \leq 0$ are inequality constraints

Algorithm:

1. Start with small $\mu$
2. Minimize penalized objective using L-BFGS
3. If constraints violated, increase $\mu$ and repeat
4. Stop when constraints satisfied within tolerance

Pros: Simple, handles infeasible starts Cons: First-order convergence, ill-conditioning at large $\mu$

Interior Point Method (IPOPT)

Barrier method staying strictly inside inequality constraints:

$$\min f(\mathbf{x}) - \mu \sum_j \log(-h_j(\mathbf{x})) \quad \text{subject to } g(\mathbf{x}) = 0$$

Uses Newton steps with:

• Gradient of Lagrangian
• Hessian of Lagrangian (requires second derivatives)

Pros: Superlinear convergence, handles many constraints efficiently Cons: Requires Hessian computation, needs feasible start

GAT provides IPOPT backend via the solver-ipopt feature.

KKT Conditions

At a local optimum, the Karush-Kuhn-Tucker conditions hold:

$$\nabla f(\mathbf{x}^) + \sum_i \lambda_i \nabla g_i(\mathbf{x}^) + \sum_j \mu_j \nabla h_j(\mathbf{x}^*) = 0 \quad \text{(stationarity)}$$

$$g_i(\mathbf{x}^*) = 0 \quad \text{(primal feasibility)}$$

$$h_j(\mathbf{x}^*) \leq 0 \quad \text{(primal feasibility)}$$

$$\mu_j \geq 0 \quad \text{(dual feasibility)}$$

$$\mu_j h_j(\mathbf{x}^*) = 0 \quad \text{(complementarity)}$$

The dual variables $\lambda_i$ and $\mu_j$ provide sensitivity information and LMPs.

GAT Implementation

L-BFGS Penalty Method:

gat opf ac-nlp network.arrow --out results.json

IPOPT (if compiled with feature):

// Requires solver-ipopt feature
solve_with_ipopt(&problem, max_iter, tolerance)

Key source files:

• problem.rs: Problem construction
• solver.rs: L-BFGS penalty loop
• ipopt_solver.rs: IPOPT interface
• hessian.rs: Second derivatives for IPOPT

Comparison of Methods

Accuracy vs. Problem Size

For the PGLib-OPF benchmark suite:

Gap = (method cost - best known) / best known × 100%

Cost Function Details

Quadratic Cost Model

$$C(P) = c_0 + c_1 P + c_2 P^2$$

• $c_0$: No-load cost ($/hr) — fuel burned at minimum output
• $c_1$: Linear cost ($/MWh) — marginal fuel cost
• $c_2$: Quadratic cost ($/MW²h) — efficiency curve

Marginal cost: $\frac{dC}{dP} = c_1 + 2c_2 P$

Piecewise Linear Cost

$$C(P) = \sum_k m_k \cdot \max(0, P - P_k)$$

Linear segments approximate any convex cost curve. Requires additional variables in LP formulation.

Multi-Segment Representation in GAT

CostModel::Polynomial { coefficients: vec![c0, c1, c2] }
CostModel::PiecewiseLinear(vec![(P1, C1), (P2, C2), ...])

Constraint Handling

Active vs. Inactive Constraints

At optimum:

• Active (binding): Constraint is exactly satisfied (equality)
• Inactive: Constraint has slack (strict inequality)

Only active constraints affect the solution. Dual variables (shadow prices) are non-zero only for active constraints.

Soft Constraints via Penalty

Some constraints can be relaxed with penalty:

$$\min f(\mathbf{x}) + \rho \cdot (\text{violation})^2$$

Useful for:

• Detecting infeasibility
• Trading off constraint satisfaction vs. cost
• Load shedding in emergency scenarios

GAT Constraint Implementation

Warm Starting

Using a previous solution accelerates convergence:

solve_ac_opf_warm_start(&problem, &initial_solution, max_iter, tol)

Effective for:

• Contingency analysis (N-1 cases differ slightly)
• Time-series studies (sequential hours)
• Sensitivity analysis (parameter sweeps)

Typical speedup: 2-5x fewer iterations.

Further Reading

Foundational Papers

• Carpentier (1962): Original OPF formulation
• Dommel & Tinney (1968): Newton-based OPF solution

Convex Relaxations

• Jabr (2006): Radial network convexification
• Farivar & Low (2013): Branch flow SOCP relaxation
• Lavaei & Low (2012): SDP relaxation and exactness

Software

• MATPOWER: Reference OPF implementation
• PowerModels.jl: Julia OPF with multiple formulations
• IPOPT: Interior-point NLP solver

GAT Documentation

• OPF Guide — Practical usage
• Power Flow Theory — Underlying equations
• Glossary — Term definitions