Linear AC Power Flow Modeling

• Linear AC power flow models are linear approximations of nonlinear AC systems that capture voltage magnitudes, angles, active and reactive power with improved fidelity over DC models.
• They enable fast simulation and optimization in applications like OPF, planning, and restoration by reducing computational complexity in iterative routines.
• Data-driven and learned linearizations enhance accuracy in predicting AC states, balancing approximation quality with the trade-offs between feasibility and operational performance.

Linear AC power flow denotes a family of linear or affine surrogates of the nonlinear AC power flow equations that retain more of the AC state than classical active-power-only DC power flow. In its narrower usage, it refers to first-order or state-independent models that include voltage magnitudes, angles, active power, and reactive power while preserving a linear algebraic structure suitable for fast simulation and optimization; in a broader usage, some literature treats the classical DC load-flow equations themselves as a linearization of AC power flow (Li et al., 2018, Hörsch et al., 2017). The subject spans analytical Taylor and rectangular-coordinate linearizations, LP- and QP-compatible formulations, linear relaxations, and data-driven or learned surrogates that fit affine or piecewise-linear maps directly from AC solutions (Dhople et al., 2015, Coffrin et al., 2012, Hu et al., 2019).

1. Nonlinear AC foundations and the rationale for linearization

For an $n$-bus AC system with bus voltages $V_i = |V_i| e^{j\theta_i}$ and bus admittance matrix $Y = G + jB$, the standard steady-state power balance equations are

$$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$

Branch flows have analogous expressions. These equations are nonlinear and nonconvex because they combine trigonometric terms with products of voltage magnitudes (Hu et al., 2019, Li et al., 2018).

The computational role of linear AC power flow follows directly from this structure. Full AC power flow is accurate, but its nonlinear and nonconvex character makes repeated solution expensive and can make embedded optimization, especially AC-OPF, difficult to solve robustly at realistic scale (Li et al., 2018). This is why linear approximations are widely used in market clearing, planning, dispatch, restoration, contingency analysis, and other settings where the same network equations appear inside larger optimization loops (Coffrin et al., 2012, Hörsch et al., 2017).

The most common baseline is DC power flow. It assumes $|V_i| \approx 1$ p.u., $G_{ij} \approx 0$, small angle differences so that $\sin\theta_{ij}\approx \theta_{ij}$ and $\cos\theta_{ij}\approx 1$, and it neglects reactive power and voltage magnitude variation. The result is the linear relation

$P = B' \theta,$

or, at branch level,

$f_\ell = \frac{\theta_i-\theta_j}{x_\ell}.$

This model is fast and robust, but its limitations are equally standard: it cannot represent reactive power, voltage magnitudes, or losses well, and its accuracy degrades for high $V_i = |V_i| e^{j\theta_i}$0, non-flat voltages, or stressed conditions (Hu et al., 2019, Li et al., 2022).

2. Analytical model families and coordinate choices

A central distinction in the literature is between DC-type models that keep only active power and angles, and linearized AC models that retain voltage magnitude and reactive power. A representative branch-level linearized AC model starts from the exact branch equations and, after assuming $V_i = |V_i| e^{j\theta_i}$1 and small $V_i = |V_i| e^{j\theta_i}$2, yields

$V_i = |V_i| e^{j\theta_i}$3

$V_i = |V_i| e^{j\theta_i}$4

These relations add conductance effects, voltage magnitude differences, reactive power, and shunt terms to the DC picture while remaining linear in the state variables (Li et al., 2018).

A different analytical route uses rectangular coordinates. Writing

$V_i = |V_i| e^{j\theta_i}$5

the exact bus injections become quadratic in $V_i = |V_i| e^{j\theta_i}$6, for example

$V_i = |V_i| e^{j\theta_i}$7

Linearization around a nominal voltage $V_i = |V_i| e^{j\theta_i}$8 with $V_i = |V_i| e^{j\theta_i}$9 leads to the affine complex system

$Y = G + jB$0

For lossless networks and flat nominal voltage, suppressing the real part of the perturbation yields a profile that satisfies active-power balance in the original nonlinear equations; under additional restrictions, the classical DC model appears as a special case of this rectangular-coordinate construction (Dhople et al., 2015).

A third analytical family keeps the problem in linear-program form. The LPAC models approximate $Y = G + jB$1 linearly and replace $Y = G + jB$2 with a convex piecewise-linear envelope, while using Taylor approximations for the remaining nonlinear voltage terms. Hot-start, warm-start, and cold-start variants were proposed to incorporate reactive power and voltage magnitudes in linear programs and to bridge the gap between DC and full AC power flow (Coffrin et al., 2012).

3. Linear relaxations, losses, and OPF-compatible formulations

Linear AC power flow is not only an approximation problem; it is also a modeling problem for optimization. One line of work formulates linear relaxations of AC transmission constraints. A network-flow relaxation introduces complex bus power balance, nonnegative line-loss inequalities, and phase-angle-difference constraints in linear form, while a copper-plate relaxation collapses the network to a system-wide balance. Theoretical results show that the network-flow model is a relaxation of the SOC model and the copper-plate model is a relaxation of the network-flow model, establishing a hierarchy between linear and nonlinear convex relaxations (Coffrin et al., 2015).

Another line of work keeps the full AC variable space $Y = G + jB$3 but replaces nonlinear losses by convex absolute-value approximations. Starting from

$Y = G + jB$4

and the analogous reactive relation, LP/QP formulations were built as LIN-OPF, LOLIN-OPF, LINLOLIN-OPF, and MIP-OPF. In a comprehensive case study, these methods produced reasonable errors on voltage magnitudes and angles while obtaining near-optimal results for typical scenarios and reducing computational complexity significantly relative to nonlinear AC-OPF (Fortenbacher et al., 2017).

Tight LP approximations can also be derived from stronger convex AC relaxations rather than from direct Taylor expansion. Two LP approximations based on tight polyhedral approximations of SOCP constraints were proposed for systems with up to 9241 buses, with computational efficiency comparable to, and in many instances better than, the SOCP models they approximate (Mhanna et al., 2016). This use of polyhedral outer approximations is distinct from DC linearization: it preserves reactive power, voltage magnitude, and thermal-limit structure much more explicitly.

For active-power-only formulations, the same physical linearization can be embedded with different optimization variables. Angle-based, PTDF-based, Kirchhoff-flow, and cycle-flow formulations are mathematically equivalent under the DC assumptions, but they differ materially in sparsity and computational behavior. In particular, cycle- and flow-based linear optimal power flow formulations exploit KCL and KVL directly and can be substantially faster than the angle formulation in large multi-period settings (Hörsch et al., 2017).

4. Data-driven and learned linearizations

A major recent development is the replacement of analytic linearization by regression on AC solutions. In the data-driven linearized AC model with regression analysis, the conventional branch-level LAC equations are retained as a physical template but their coefficients are learned from AC power flow data: $Y = G + jB$5

$Y = G + jB$6

On a 1,779-bus, 2,301-branch Tennessee Valley Authority system, trained on hour 1 and validated on hours 2–72, the fitted coefficients were approximately $Y = G + jB$7, $Y = G + jB$8, $Y = G + jB$9, $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$0, and $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$1, with $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$2 for active power and $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$3 for reactive power. Across 72 hours, the mean improvement in branch reactive power over regular LAC was $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$4, and the mean improvement in voltage error was $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$5, with average DLAC voltage error $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$6 (Li et al., 2018).

A more global data-driven construction learns affine maps directly in rectangular coordinates: $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$7 with analogous expressions for branch flows, where $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$8. Polynomial regression is used as a base learner, and gradient boosting or bagging combines multiple regressors. On IEEE 5-, 57-, and 118-bus systems, ensemble methods reduced RMSE substantially relative to a single polynomial regressor, and gradient boosting performed best. In OPF, the resulting DDCR-OPF model achieved objective gaps in $$\begin{aligned} P_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\cos(\theta_i-\theta_j) + B_{ij}\sin(\theta_i-\theta_j)\big),\ Q_i &= \sum_{j=1}^n |V_i||V_j|\big(G_{ij}\sin(\theta_i-\theta_j) - B_{ij}\cos(\theta_i-\theta_j)\big). \end{aligned}$$9 relative to ACOPF across the tested cases, while being faster than ACOPF and SDPOPF and comparable to DCOPF (Hu et al., 2019).

Least-squares distribution factors push the data-driven idea further in the active-power domain. Instead of deriving PTDFs analytically, the method fits the matrix $|V_i| \approx 1$0 in

$|V_i| \approx 1$1

from historical or simulated AC power flow solutions. Over the studied cases, the average error of LSDF was only about $|V_i| \approx 1$2 of the average error of PTDF, making it a cold-start linear model for branch active flows under large load variation (Shao et al., 2020).

Piecewise-linear learned models generalize affine regression to topology-aware surrogates. A ReLU network can be trained to approximate the common nonlinear terms

$|V_i| \approx 1$3

while fixed linear layers map these quantities to flows and injections. Because the full mapping is piecewise linear, it can be reformulated exactly as an MILP using big-$|V_i| \approx 1$4 ReLU constraints and McCormick line-status products. On IEEE 118-bus AC optimal transmission switching, the resulting PWL-OTS solutions were $|V_i| \approx 1$5 and $|V_i| \approx 1$6 of AC-OTS cost for switching budgets $|V_i| \approx 1$7 and $|V_i| \approx 1$8, with infeasible-solution rate $|V_i| \approx 1$9, compared with $G_{ij} \approx 0$0 and $G_{ij} \approx 0$1 cost and infeasible-solution rates $G_{ij} \approx 0$2 and $G_{ij} \approx 0$3 for DC-OTS (Cho et al., 2023).

5. Accuracy, feasibility, and computational trade-offs

The empirical comparison of linear power flow models depends strongly on which criterion is prioritized. A study of seven active-power approximations evaluated approximation accuracy, optimality, feasibility, and computation time on five systems. In that setting, the logarithmic-voltage model gave the most accurate line-flow approximation, whereas the classical DC model had the largest line-flow errors but remained the most robust overall in a $G_{ij} \approx 0$4-only OPF. Iterative DC models with quadratic or flow-proportional losses improved objective accuracy and feasibility relative to pure DC while increasing runtime by about a factor of four; on the 2000-bus case, the DC model solved in $G_{ij} \approx 0$5 s, the iterative loss-corrected variants in about $G_{ij} \approx 0$6 s, and the more detailed one-shot linear AC approximations required around $G_{ij} \approx 0$7 s (Li et al., 2022).

LPAC occupies a different point in this trade-off space. Its experimental evaluations on standard IEEE and MATPOWER benchmarks showed accurate values for active power, reactive power, phase angles, and voltage magnitudes, and the models were used in proof-of-concept studies in power restoration and capacitor placement (Coffrin et al., 2012). LP/QP linear AC models with absolute-value losses similarly target “near-optimal results for typical scenarios” while keeping the optimization problem in tractable linear or quadratic programming form (Fortenbacher et al., 2017).

The comparison between relaxation-based and approximation-based methods is also nontrivial. Tightened LP approximations of SOCP formulations retain stronger AC structure than DC power flow and scale to very large systems (Mhanna et al., 2016), whereas network-flow and copper-plate relaxations deliberately sacrifice fidelity to obtain simpler linear bounds (Coffrin et al., 2015). This suggests that “linear AC power flow” is not a single model class but a design space spanning local approximation, global regression, and polyhedral relaxation.

A recurring misconception is that lower equation-level approximation error automatically implies better OPF decisions. Recent work makes the opposite point explicitly: there is a linearization–application gap. A generalized linear PF model

$G_{ij} \approx 0$8

can be optimized directly for downstream AC dispatch performance, and on the IEEE 39-bus system such optimized linearizations traversed a cost-optimality versus operational-feasibility trade-off that classical DC coefficients do not target (Chen et al., 3 Apr 2025). This suggests that approximation quality and operational usefulness are related but not identical objectives.

6. Applications, limitations, and feasibility repair

Linear AC power flow is used where repeated full AC solves are impractical but active and reactive phenomena cannot be ignored. The surveyed applications include transmission expansion planning with voltage constraints, security-constrained economic dispatch and unit commitment, contingency screening, restoration, capacitor placement, market-clearing studies, probabilistic scenario analysis, topology optimization, and near-real-time dispatch (Li et al., 2018, Coffrin et al., 2012, Cho et al., 2023, Hu et al., 2019).

The principal limitations are also consistent across the literature. Analytical models inherit the operating-point and small-deviation assumptions used in their derivation; data-driven models depend on the range and diversity of their training set and typically require retraining when topology changes; global affine models can lose accuracy near voltage collapse or other strongly nonlinear regimes; and active-power-only formulations can perform well on objective value while still violating voltage or reactive limits under the true AC equations (Dhople et al., 2015, Li et al., 2018, Li et al., 2022). This suggests that linear AC power flow is most reliable when the operating region is well characterized and when the intended use matches the model’s retained variables.

A second limitation is that linear optimization outputs are not automatically AC-feasible. In practice, market and dispatch workflows often solve a DC-OPF or another linear approximation first and then run AC power flow as a feasibility check. Because standard AC power flow enforces equality constraints but not inequality bounds, the post-processed state can violate generator reactive limits, bus voltage magnitudes, or thermal limits. One proposed repair mechanism augments the AC feasibility step with two extra bus types, PQV and P, so that the voltage setpoint at a generator can be sacrificed in exchange for fixing the voltage at a load bus. On IEEE 14-, 57-, and 300-bus systems, this bus-type switching reduced overall network violations relative to baseline AC post-processing while keeping the market-cleared real-power setpoints unchanged (Boven et al., 10 Nov 2025).

Taken together, these developments define linear AC power flow as a layered modeling tradition rather than a single approximation. At one end are analytically derived linearizations in polar or rectangular coordinates; at another are LP-compatible relaxations and loss models; and at a third are data-driven affine or piecewise-linear surrogates. Their common aim is to recover enough of the AC structure—voltage magnitude, reactive power, conductance effects, losses, or topology dependence—to support tractable optimization without reverting to the full nonconvex AC equations.