AC Optimal Power Flow¶

The ACOPF module implements multiple AC optimal power flow formulations as JuMP constraints, integrated into the ESFEX operational dispatch model. It is implemented in transmission_acopf.jl and supports four working formulations: SOC relaxation, QC relaxation, Polar NLP, and Rectangular NLP.

Overview¶

The DC power flow approximation (see DC Power Flow) linearizes the AC equations by assuming flat voltages, small angle differences, and negligible resistance. While computationally efficient, it ignores voltage magnitude variations, reactive power, and apparent power (MVA) limits. The ACOPF formulations model these effects directly, providing:

Voltage magnitude tracking at each bus
Reactive power balance and generation limits
Apparent power line limits ($P^2 + Q^2 \leq S_{max}^2$)
Accurate loss representation through full admittance modeling

ESFEX provides four working formulations, selectable via power_flow_mode:

Mode	Formulation	Type	Variables
`acopf_soc`	SOC Relaxation	Convex (relaxed)	$w, w^r, w^i$
`acopf_qc`	QC Relaxation	Convex (relaxed, tighter)	$w, w^r, w^i$
`acopf_polar`	Polar NLP	Exact (nonlinear)	$V, \theta$
`acopf_rect`	Rectangular NLP	Exact (nonlinear)	$e, f$

SDP Formulation

A fifth formulation (acopf_sdp) is implemented but currently blocked in practice. It requires an SDP-capable solver such as MOSEK (commercial) or SCS/CSDP. Extreme coefficient ranges in real-world networks cause SCS and Clarabel to report false DUAL_INFEASIBLE status. MOSEK handles these ranges but requires a license.

Formulations¶

SOC Relaxation¶

The Second-Order Cone (SOC) relaxation [2] lifts the voltage phasor products into W-space variables, replacing the nonlinear voltage-angle relationships with a convex cone constraint.

Variables (per branch $l$ connecting buses $i, j$, per hour $t$):

Variable	Definition	Bounds
$w_{i,t}$	$\\|V_i\\|^2$ (voltage magnitude squared)	$[V_{min}^2,\; V_{max}^2]$
$w^r_{l,t}$	$\text{Re}(V_i \cdot V_j^*) = \\|V_i\\| \\|V_j\\| \cos(\theta_i - \theta_j)$	$[-V_{max}^2,\; V_{max}^2]$
$w^i_{l,t}$	$\text{Im}(V_i \cdot V_j^*) = \\|V_i\\| \\|V_j\\| \sin(\theta_i - \theta_j)$	$[-V_{max}^2,\; V_{max}^2]$

Linking constraint (convex relaxation of the exact equality $w_i \cdot w_j = (w^r)^2 + (w^i)^2$):

\[ w_{i,t} \cdot w_{j,t} \geq (w^r_{l,t})^2 + (w^i_{l,t})^2 \quad \forall l, t \tag{SOC-1} \]

The inequality relaxation makes the feasible set convex. The solution is a lower bound on the true AC-OPF cost. Power flow expressions are affine in $w, w^r, w^i$ (see Power Flow Expressions).

QC Relaxation¶

The Quadratic Convex (QC) relaxation [3] starts from the SOC relaxation and adds McCormick envelopes and trigonometric bounds to tighten the feasible region.

Additional constraints on top of SOC:

Tighter bounds on $w^r$ using cosine bounds ($\cos$ is even, minimum at $\pm\theta_{max}$):

\[ V_{min}^2 \cdot \cos(\theta_{max}) \leq w^r_{l,t} \leq V_{max}^2 \tag{QC-1} \]

Tighter bounds on $w^i$ using sine bounds:

\[ |w^i_{l,t}| \leq V_{max}^2 \cdot \sin(\theta_{max}) \tag{QC-2} \]

Convex envelope for cosine:

\[ w^r_{l,t} \geq \tfrac{1}{2}(w_{i,t} + w_{j,t}) \cdot \cos(\theta_{max}) \tag{QC-3} \]

Angle difference limits via tangent:

\[ |w^i_{l,t}| \leq \tan(\theta_{max}) \cdot w^r_{l,t} \quad \text{(when } \theta_{max} < \pi/2\text{)} \tag{QC-4} \]

The QC relaxation is strictly tighter than SOC — it excludes more of the non-physical region while remaining convex.

Polar NLP¶

The exact formulation using voltage magnitude $V$ and angle $\theta$ variables. This is the standard textbook ACOPF.

Variables:

Variable	Definition	Bounds
$V_{i,t}$	Voltage magnitude at bus $i$	$[V_{min},\; V_{max}]$
$\theta_{i,t}$	Voltage angle at bus $i$	Unbounded

Power flow expressions involve trigonometric functions of angle differences ($\cos(\theta_i - \theta_j)$, $\sin(\theta_i - \theta_j)$), making the problem nonlinear and nonconvex. Angle difference limits are enforced directly:

\[ |\theta_{i,t} - \theta_{j,t}| \leq \theta_{max} \quad \forall (i,j) \in \mathcal{L}, t \tag{POLAR-1} \]

The slack bus is fixed: $V_{slack} = 1.0$ p.u., $\theta_{slack} = 0$.

Rectangular NLP¶

The exact formulation using real and imaginary voltage components.

Variables:

Variable	Definition	Bounds
$e_{i,t}$	$V_i \cos\theta_i$ (real component)	$[-V_{max},\; V_{max}]$
$f_{i,t}$	$V_i \sin\theta_i$ (imaginary component)	$[-V_{max},\; V_{max}]$

Voltage magnitude bounds become quadratic:

\[ V_{min}^2 \leq e_{i,t}^2 + f_{i,t}^2 \leq V_{max}^2 \quad \forall i \neq \text{slack}, t \tag{RECT-1} \]

Angle difference limits are expressed via cross and dot products of the voltage components:

\[ |f_i e_j - e_i f_j| \leq \tan(\theta_{max}) \cdot (e_i e_j + f_i f_j) \quad \forall (i,j) \in \mathcal{L}, t \tag{RECT-2} \]

The slack bus is fixed: $e_{slack} = 1.0$, $f_{slack} = 0.0$.

Network Model¶

Branch Admittance¶

Each branch (transmission line or transformer) is modeled using the 4-terminal admittance representation. The complex current injections are:

\[ I_{from} = Y_{ff} \cdot V_i + Y_{ft} \cdot V_j, \qquad I_{to} = Y_{tf} \cdot V_i + Y_{tt} \cdot V_j \]

where $V_i$ and $V_j$ are the complex bus voltages. The admittance elements for a branch with series impedance $z_s = r + jx$ (giving series admittance $y_s = g_s + jb_s = 1/z_s$), shunt susceptance $b_{sh}$, and tap ratio $\tau$ are:

Element	Formula
$Y_{ff}$	$(g_s + jb_s)/\tau^2 + jb_{sh}/(2\tau^2)$
$Y_{ft}$	$-(g_s + jb_s)/\tau$
$Y_{tt}$	$g_s + jb_s + jb_{sh}/2$
$Y_{tf}$	$-(g_s + jb_s)/\tau$

For transmission lines, $\tau = 1$. For transformers, $\tau$ is the tap ratio.

The real and imaginary parts of each admittance element ($g_{ff}, b_{ff}, g_{ft}, b_{ft}$, etc.) are precomputed in the ACOPFBranch struct and used directly in the power flow expressions.

Reactance Clamping¶

Very short lines or bus-ties can have near-zero reactance, producing extreme admittances that cause numerical issues. All branch reactances are clamped to a minimum value:

\[ |x| \geq x_{min} \quad (\text{default: } 0.01 \text{ p.u.}) \]

This limits the maximum admittance magnitude to $|b| \leq 1/x_{min} = 100$ p.u. per branch. Configured via min_reactance_pu.

Transformer Tap Ratio Normalization¶

Transformer tap ratios that represent voltage level changes (e.g., 480V/34.5kV = 0.014) create enormous admittances in a single-base per-unit system. Tap ratios outside the range $[\tau_{min}, \tau_{max}]$ are reset to 1.0, treating the transformer as a simple series impedance:

\[ \tau = \begin{cases} \tau & \text{if } \tau_{min} \leq \tau \leq \tau_{max} \\ 1.0 & \text{otherwise} \end{cases} \]

Default bounds: $\tau_{min} = 0.5$, $\tau_{max} = 2.0$. Configured via tap_ratio_min and tap_ratio_max.

Power Flow Expressions¶

The active and reactive power flows on each branch are expressed differently depending on the formulation, but all use the same precomputed admittance coefficients.

W-Space (SOC / QC)¶

Power flow expressions are affine in the W-space variables — no trigonometric or quadratic terms:

\[ P_{from} = g_{ff} \cdot w_i + g_{ft} \cdot w^r + b_{ft} \cdot w^i \tag{PF-W-P} \]

\[ Q_{from} = -b_{ff} \cdot w_i - b_{ft} \cdot w^r + g_{ft} \cdot w^i \tag{PF-W-Q} \]

\[ P_{to} = g_{tt} \cdot w_j + g_{tf} \cdot w^r - b_{tf} \cdot w^i \]

\[ Q_{to} = -b_{tt} \cdot w_j - b_{tf} \cdot w^r - g_{tf} \cdot w^i \]

Polar (V, theta)¶

Power flow expressions involve trigonometric nonlinearities:

\[ P_{from} = g_{ff} \cdot V_i^2 + V_i V_j \left( g_{ft} \cos\theta_{ij} + b_{ft} \sin\theta_{ij} \right) \tag{PF-P-P} \]

\[ Q_{from} = -b_{ff} \cdot V_i^2 + V_i V_j \left( -b_{ft} \cos\theta_{ij} + g_{ft} \sin\theta_{ij} \right) \tag{PF-P-Q} \]

where $\theta_{ij} = \theta_i - \theta_j$. The to-side expressions follow the same pattern with $g_{tt}, b_{tt}, g_{tf}, b_{tf}$ and reversed sine sign.

Rectangular (e, f)¶

Power flow expressions are quadratic in the voltage components. The W-space products are computed from $e, f$:

\[ w_{ii} = e_i^2 + f_i^2, \qquad w^r_{ij} = e_i e_j + f_i f_j, \qquad w^i_{ij} = f_i e_j - e_i f_j \]

These are substituted into the same admittance expressions as the W-space formulation:

\[ P_{from} = g_{ff}(e_i^2 + f_i^2) + g_{ft}(e_i e_j + f_i f_j) + b_{ft}(f_i e_j - e_i f_j) \tag{PF-R-P} \]

\[ Q_{from} = -b_{ff}(e_i^2 + f_i^2) - b_{ft}(e_i e_j + f_i f_j) + g_{ft}(f_i e_j - e_i f_j) \tag{PF-R-Q} \]

Constraints¶

AC-1: Active Power Balance (KCL)¶

Active power balance at each bus $n$ and hour $t$. The injection terms are identical to the DC KCL (see DC-1), but the flow side uses formulation-specific power flow expressions:

\[ \underbrace{\sum_{g} P_{g,n,t} + \sum_{b} P^{dis}_{b,n,t} + L_{n,t} + \ldots}_{\text{injection (MW)}} - \underbrace{D_{n,t} \cdot \phi_n + \sum_{b} P^{ch}_{b,n,t} + \ldots}_{\text{withdrawal (MW)}} = S_{base} \cdot \sum_{l \in \mathcal{B}_n} P_{flow,l,t} \tag{AC-1} \]

where $S_{base}$ is the system base power (MVA) and $P_{flow,l,t}$ is the per-unit active power flow from bus $n$ on branch $l$, computed via the formulation-specific expressions. Both sides are in MW, keeping coefficient ratios manageable for the solver.

Injection Terms

The active KCL includes the same injection terms as the DC formulation: generator output, battery discharge/charge, EV V2G/charging, electrolyzer consumption, AC/DC and frequency converter flows, reservoir pumping, rooftop solar generation/curtailment, load shedding, and static/dynamic reserves.

Implemented in add_acopf_power_balance!().

AC-2: Reactive Power Balance (KCL)¶

Reactive power balance at each bus, using estimated reactive load and reactive generation variables:

\[ \sum_{g} Q_{g,n,t} + Q^{slack+}_{n,t} - Q^{slack-}_{n,t} - Q^{load}_{n,t} = S_{base} \cdot \sum_{l \in \mathcal{B}_n} Q_{flow,l,t} \tag{AC-2} \]

where the reactive load is estimated from the active load using the load power factor:

\[ Q^{load}_{n,t} = D_{n,t} \cdot \phi_n \cdot \tan\!\left(\arccos(pf_{load})\right) \]

The slack variables $Q^{slack+}$ and $Q^{slack-}$ absorb reactive power mismatch with a penalty cost $c_Q$ ($/MVAr) added to the objective. This is necessary because in SOC/QC relaxations, the relaxed voltage products may not correspond to a physically realizable voltage profile, making exact reactive balance infeasible.

AC-3: Voltage Magnitude Bounds¶

For SOC/QC formulations, voltage bounds are applied to the squared magnitude variable:

\[ V_{min}^2 \leq w_{i,t} \leq V_{max}^2 \quad \forall i, t \tag{AC-3a} \]

For Polar NLP:

\[ V_{min} \leq V_{i,t} \leq V_{max} \quad \forall i, t \tag{AC-3b} \]

For Rectangular NLP:

\[ V_{min}^2 \leq e_{i,t}^2 + f_{i,t}^2 \leq V_{max}^2 \quad \forall i \neq \text{slack}, t \tag{AC-3c} \]

The slack bus voltage is fixed to 1.0 p.u. in all formulations.

AC-4: SOC/QC Voltage Linking¶

The SOC constraint links the W-space diagonal and off-diagonal variables (see SOC-1). The QC formulation adds tighter bounds via McCormick envelopes and trigonometric inequalities (see QC-1 through QC-4).

AC-5: Angle Difference Limits¶

Voltage angle differences across each branch are bounded. The implementation depends on the formulation:

SOC/QC: Angle limits are enforced indirectly through trigonometric bounds on $w^r$ and $w^i$ (QC-4).
Polar NLP: Direct constraint on angle variables (POLAR-1).
Rectangular NLP: Nonlinear constraint via cross/dot products (RECT-2).

Default maximum angle difference: $\theta_{max} = 30° = 0.524$ rad (same as DC, from max_angle_diff_rad).

AC-6: Apparent Power Line Limits¶

Each branch has an apparent power (MVA) limit enforced on both ends:

\[ P_{from,l,t}^2 + Q_{from,l,t}^2 \leq \left(\frac{S^{max}_l}{S_{base}}\right)^2 \quad \forall l, t \tag{AC-6a} \]

\[ P_{to,l,t}^2 + Q_{to,l,t}^2 \leq \left(\frac{S^{max}_l}{S_{base}}\right)^2 \quad \forall l, t \tag{AC-6b} \]

where $S^{max}_l$ is the branch thermal capacity in MVA. Unlike the DC formulation which limits only active power flow ($|f_l| \leq \bar{F}_l$), the ACOPF limits the full apparent power magnitude. All formulations use the quadratic $P^2 + Q^2 \leq cap^2$ form (not SOC cones) for Ipopt compatibility.

Implemented in add_acopf_line_limits!().

AC-7: Reactive Generation Limits¶

Each generator's reactive power output is bounded:

\[ Q^{min}_{g,n} \leq Q_{g,n,t} \leq Q^{max}_{g,n} \quad \forall g, n, t \tag{AC-7} \]

When explicit Q limits are not provided in the configuration, they are estimated from the rated active power and default power factor:

\[ Q^{max}_{g,n} = P^{rated}_{g,n} \cdot \tan\!\left(\arccos(pf)\right) \]

\[ Q^{min}_{g,n} = -\rho_Q \cdot Q^{max}_{g,n} \]

where $pf$ is default_power_factor (default: 0.85) and $\rho_Q$ is q_min_ratio (default: 0.5). This allows generators to absorb reactive power up to 50% of their maximum reactive output.

AC-8: Reactive Power Slack¶

Reactive slack variables penalize Q imbalance in the objective:

\[ \min \ldots + c_Q \sum_{n,t} \left( Q^{slack+}_{n,t} + Q^{slack-}_{n,t} \right) \tag{AC-8} \]

where $c_Q$ is the q_slack_penalty (default: 100 $/MVAr). The penalty is small enough to avoid distorting the active power dispatch, but large enough to keep Q close to balanced.

Solver Requirements¶

All four working formulations use Ipopt as the nonlinear solver:

Formulation	Solver	Constraint Type	Notes
SOC (`acopf_soc`)	Ipopt	Quadratic (NLP form)	SOC constraint written as $w_i w_j \geq wr^2 + wi^2$
QC (`acopf_qc`)	Ipopt	Quadratic + linear (NLP form)	SOC + McCormick envelopes
Polar (`acopf_polar`)	Ipopt	Nonlinear (trig)	Exact, nonconvex
Rectangular (`acopf_rect`)	Ipopt	Nonlinear (polynomial)	Exact, nonconvex
SDP (`acopf_sdp`)	MOSEK	PSD cone	Blocked: SCS/Clarabel fail with extreme coefficient ranges

Conic Solvers

In theory, the SOC and QC formulations could use conic solvers (HiGHS, SCS, Clarabel). In practice, the extreme coefficient ranges arising from real-world networks (RHS values around $2 \times 10^9$, cost coefficients around $6 \times 10^7$) cause SCS and Clarabel to misidentify the problem as DUAL_INFEASIBLE. The NLP quadratic form works reliably with Ipopt.

Set the solver in the configuration via solver_name: "ipopt".

Numerical Considerations¶

KCL Scaling¶

The power balance constraints are written in MW (not per-unit):

\[ \text{injection}_{MW} = S_{base} \times \text{flow}_{pu} \]

This keeps both sides of the KCL equation in comparable magnitude ranges (MW on the left, $S_{base} \times$ small p.u. values on the right). The alternative — dividing injection by $S_{base}$ to get p.u. — would introduce a factor of $1/S_{base} = 0.01$ that creates 260,000:1 coefficient ratios when combined with summed admittances.

Flat Voltage Start Values¶

All formulations initialize voltage variables to the flat start ($V = 1.0$ p.u., $\theta = 0$):

Formulation	Start Values
SOC/QC	$w = 1.0$, $w^r = 1.0$, $w^i = 0.0$
Polar	$V = 1.0$, $\theta = 0.0$
Rectangular	$e = 1.0$, $f = 0.0$

For NLP formulations, these start values are critical for convergence. Without them, Ipopt often starts from $(0, 0)$, which violates the voltage magnitude lower bound and leads to LOCALLY_INFEASIBLE status.

Tap Ratio Normalization¶

See Transformer Tap Ratio Normalization.

Reactance Clamping¶

See Reactance Clamping.

Configuration¶

ACOPF parameters are configured in the ac_power_flow section of the system configuration, defined by ACPowerFlowConfig:

ac_power_flow:
  base_mva: 100.0
  voltage_min_pu: 0.90
  voltage_max_pu: 1.10
  default_power_factor: 0.85
  load_power_factor: 0.9
  q_slack_penalty: 100.0
  min_reactance_pu: 0.01
  tap_ratio_min: 0.5
  tap_ratio_max: 2.0
  q_min_ratio: 0.5

Parameter	Type	Default	Description
`base_mva`	float	100.0	System base power for per-unit conversion
`voltage_min_pu`	float	0.90	Lower voltage magnitude limit (p.u.)
`voltage_max_pu`	float	1.10	Upper voltage magnitude limit (p.u.)
`default_power_factor`	float	0.85	Generator power factor for Q limit estimation
`load_power_factor`	float	0.9	Load power factor for reactive demand estimation
`q_slack_penalty`	float	100.0	Penalty for reactive imbalance ($/MVAr)
`min_reactance_pu`	float	0.01	Minimum branch reactance (p.u.) for clamping
`tap_ratio_min`	float	0.5	Lower bound for valid transformer tap ratios
`tap_ratio_max`	float	2.0	Upper bound for valid transformer tap ratios
`q_min_ratio`	float	0.5	Ratio for Q_min = -ratio x Q_max when Q limits not specified

The power flow mode is set at the system level:

power_flow_mode: "acopf_soc"  # or acopf_qc, acopf_polar, acopf_rect
solver_name: "ipopt"

GUI¶

The power flow mode and AC parameters can be configured from the System Settings panel under "Power Flow". Selecting any acopf_* mode reveals the AC-specific fields (base MVA, voltage limits, power factor, Q penalty, reactance clamping, tap ratio bounds).

Implementation¶

The ACOPF module is organized around Julia's multiple dispatch pattern:

Function	Purpose
`setup_acopf!()`	Main entry point — orchestrates variable creation, constraints, and line limits
`ACOPFNetwork()`	Builds precomputed branch admittance data from network configuration
`build_acopf_variables!()`	Creates formulation-specific voltage and reactive generation variables
`add_acopf_voltage_constraints!()`	Adds SOC cones, McCormick envelopes, angle limits (dispatched by formulation)
`add_acopf_power_balance!()`	Adds active and reactive KCL at each bus
`add_acopf_line_limits!()`	Adds apparent power limits on branches
`extract_acopf_voltages()`	Extracts voltage magnitude and angle from the solved model
`extract_acopf_reactive_gen()`	Extracts reactive generation from the solved model

All functions dispatch on the ACOPFFormulation abstract type (SOCFormulation, QCFormulation, PolarNLPFormulation, RectNLPFormulation), allowing formulation-specific behavior without conditional branching.

References¶

The ACOPF formulations implemented in ESFEX draw from the following foundational works. For a historical survey of OPF formulations, see Cain et al. [56]. The SOC relaxation follows Jabr [2]; the QC relaxation follows Coffrin et al. [3]; the SDP formulation follows Bai et al. [57]. For a comprehensive theoretical treatment of convex relaxations applied to OPF, see Low [27] and the survey by Molzahn and Hiskens [58]. The NLP formulations (Polar and Rectangular) solve the exact AC power flow equations using the Ipopt interior-point solver [4].

See the full bibliography for complete citation details.

Variable	Definition	Bounds
\(w_{i,t}\)	\(\\|V_i\\|^2\) (voltage magnitude squared)	\([V_{min}^2,\; V_{max}^2]\)
\(w^r_{l,t}\)	\(\text{Re}(V_i \cdot V_j^*) = \\|V_i\\| \\|V_j\\| \cos(\theta_i - \theta_j)\)	\([-V_{max}^2,\; V_{max}^2]\)
\(w^i_{l,t}\)	\(\text{Im}(V_i \cdot V_j^*) = \\|V_i\\| \\|V_j\\| \sin(\theta_i - \theta_j)\)	\([-V_{max}^2,\; V_{max}^2]\)

Variable	Definition	Bounds
\(V_{i,t}\)	Voltage magnitude at bus \(i\)	\([V_{min},\; V_{max}]\)
\(\theta_{i,t}\)	Voltage angle at bus \(i\)	Unbounded

Variable	Definition	Bounds
\(e_{i,t}\)	\(V_i \cos\theta_i\) (real component)	\([-V_{max},\; V_{max}]\)
\(f_{i,t}\)	\(V_i \sin\theta_i\) (imaginary component)	\([-V_{max},\; V_{max}]\)

Element	Formula
\(Y_{ff}\)	\((g_s + jb_s)/\tau^2 + jb_{sh}/(2\tau^2)\)
\(Y_{ft}\)	\(-(g_s + jb_s)/\tau\)
\(Y_{tt}\)	\(g_s + jb_s + jb_{sh}/2\)
\(Y_{tf}\)	\(-(g_s + jb_s)/\tau\)

Formulation	Start Values
SOC/QC	\(w = 1.0\), \(w^r = 1.0\), \(w^i = 0.0\)
Polar	\(V = 1.0\), \(\theta = 0.0\)
Rectangular	\(e = 1.0\), \(f = 0.0\)