Transient Stability-Constrained Optimal Power Flow

• TSC-OPF is an optimal power flow formulation that integrates transient stability constraints with steady-state conditions to ensure system safety post-disturbance.
• It employs methodologies like discretized DAEs, Lyapunov certificates, and sensitivity analyses to capture complex dynamic responses.
• Scalable strategies such as network reduction, reinforcement learning, and surrogate modeling mitigate computational burdens while maintaining transient security.

Transient Stability-Constrained Optimal Power Flow (TSC-OPF) is an optimal power flow formulation in which the dispatch is chosen not only to minimize operating cost and satisfy steady-state network constraints, but also to satisfy dynamic constraints associated with transient stability after specified disturbances. In the literature, this augmentation appears in several forms: direct embedding of discretized differential-algebraic equations over fault and post-fault horizons, algebraic stability certificates derived from Lyapunov functions, sensitivity-based linearized stability margins, data-driven correction constraints, and learned surrogate constraints for frequency or rotor-angle stability (Coelho et al., 7 Nov 2025, Li et al., 2017).

1. Canonical optimization structure

At its core, TSC-OPF extends AC-OPF by adding transient-stability constraints to the standard steady-state feasibility region. A representative formulation minimizes generation cost subject to AC power-flow equations, operational limits, and dynamic security conditions. One formulation writes the objective as

$$\min_{p^G} f=\sum_{i\in\mathcal N_G}\left(a_{i1}(p_i^G)^2+a_{i2}p_i^G\right),$$

with AC balance, line-flow, voltage, and generator limits, together with transient-stability constraints (Li et al., 2017). Another writes the objective as standard generation-cost minimization,

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$

subject to steady-state AC power balance at each bus, branch flow limits, voltage magnitude and angle limits, generator active/reactive limits, and transient stability constraints imposed over the fault and post-fault time horizon (Coelho et al., 7 Nov 2025).

This common structure makes TSC-OPF a preventive-control problem. The steady-state operating point determines both economic performance and the post-disturbance dynamic response. The problem is therefore not only to find a feasible dispatch, but to find a dispatch whose associated equilibrium and fault-cleared state remain inside an admissible transient-stability region (Li et al., 2017).

The formulation space is broader than generator redispatch alone. In the cited literature, decision variables may also include reactive power, reserve, power flow router settings, load shedding, renewable curtailment, and other preventive controls. This wider scope is explicit in formulations that coordinate generator-side and network-side control, or that couple transient-stability constraints with cut-set security, uncertainty handling, or microgrid planning (Chen et al., 2020, Sahoo et al., 2023, Su et al., 2024, Wang et al., 23 Jul 2025).

2. Dynamic models and stability criteria

The dynamic component of TSC-OPF is typically represented by generator swing dynamics or by a structure-preserving network model, embedded either directly or indirectly into the optimization. In a time-discretized formulation, the paper on load admittance approximation uses the classical generator model and the swing equations, discretized over time with the trapezoidal method. The rotor-angle and speed-deviation states are enforced at each time step, so each time step adds new variables and nonlinear constraints to the nonlinear program (Coelho et al., 7 Nov 2025).

A common rotor-angle stability constraint is expressed relative to the center of inertia (COI):

$$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$

with bounds

$$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$

These constraints ensure that generator rotor angles do not diverge excessively after a contingency (Coelho et al., 7 Nov 2025).

The literature also uses alternative transient-security criteria. In the Lur’e-system framework, the post-fault dynamics are rewritten as

$\dot x = Ax + B\phi(Cx) + B_u u(x),$

and transient stability is certified through a quadratic Lyapunov function

$W(x)=x^TPx,\qquad P\succ 0.$

The certified invariant set is

$$\Omega=\{x(t)\in\mathcal P \mid W(x(t_c))\le W^{\min}\},$$

so if the fault-cleared state lies in this sublevel set, the trajectory remains in the region and converges to the post-fault equilibrium (Li et al., 2017).

Other formulations shift the transient criterion from rotor-angle coherence to frequency security. One neural-surrogate TSC-OPF declares the contingency simulation stable if the frequency at every bus stays above a critical threshold, with nominal frequency $\omega_0=60$ Hz and minimum allowed frequency $\omega_{\min}=58.5$ Hz in the test case (Garcia et al., 3 Feb 2025). DynOPF-Net uses a rotor-angle threshold

$\delta^g(t)\le \delta^{\max}\qquad \forall t\ge 0$

for each synchronous generator, motivated by SIME-style transient-stability assessment (Vito et al., 2024). In wildfire-resilient operation, transient stability is quantified through a transient stability index,

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$0

with stability determined by the sign of the index (Sahoo et al., 2023).

These variations show that TSC-OPF is not tied to a single dynamic-security metric. The common requirement is that the OPF decision variables must imply acceptable post-disturbance trajectories under the adopted transient criterion.

3. Tractability and scalable reformulations

A central issue in TSC-OPF is computational burden. When differential-algebraic equations are discretized over a transient horizon, every time step adds state variables and nonlinear constraints; for realistic systems and multiple contingencies, convergence becomes harder and runtime rises sharply. One paper explicitly identifies this as the main computational bottleneck in practical TSC-OPF (Coelho et al., 7 Nov 2025).

Earlier TSCOPF approaches are summarized as typically using one of two strategies: discretize the DAE along the fault trajectory, turning the problem into a large NLP; or iterate OPF and transient stability assessment, checking stability separately at each candidate dispatch. Both are described as expensive and poorly scalable for large grids (Li et al., 2017).

Strategy	Mechanism	Representative paper
Time-discretized dynamic embedding	Discretized swing equations or DAEs over fault and post-fault horizons	(Coelho et al., 7 Nov 2025)
Lyapunov-certificate reformulation	Quadratic Lyapunov function, projected algebraic constraints, no full transient simulation inside the optimization loop	(Li et al., 2017)
Sensitivity/scenario reformulation	SIME-based linearized stability constraints with offline-online scenario handling	(Chen et al., 2020)
RL-based solver reformulation	One-step MDP with simulator-in-the-loop reward and improved DDPG training	(Xiao et al., 2023)

The Lyapunov-certificate approach is a prominent algebraic alternative. It constructs the stability region from a quadratic Lyapunov function, projects the stability region onto the feasible domain of the optimization problem, and then incorporates the resulting stability certificate through tractable algebraic constraints. In the IEEE 118-bus case, the original OPF solutions are not transiently stable after the bus-8 fault, while the TSCOPF solutions remain stable, with cost increases under 5% and moderate CPU-time overhead because only a limited number of algebraic constraints are added instead of time-discretized DAEs (Li et al., 2017).

Robust formulations use yet another tractability device. In RTSC-OPF-PFR, the transient stability condition is represented through a SIME-based sensitivity linearization of the stability margin, and the infinite robust constraint set over renewable realizations is approximated by a scenario approach. The computationally heavy tasks—Monte Carlo generation, scenario reduction, time-domain simulation, and derivation of linear transient-stability constraints—are performed offline; the online stage solves a low-dimensional deterministic problem with stored representative scenarios and stability coefficients (Chen et al., 2020).

Large-scale decision-making can also be cast as reinforcement learning. A DRL-based TSC-OPF reformulates the problem as a single-step MDP, transcribes transient constraints through a simulator mapping, reduces the observation space to load-level information, and replaces sparse transient-security rewards with a smoother instability-duration reward. On the IEEE 39-bus system and a practical 710-bus regional power grid, this formulation is reported to provide rapid solution results and strong scalability relative to PSO-based search (Xiao et al., 2023).

4. Network reduction, Kron reduction, and load modeling

A recurring tractability technique is network reduction. Many studies simplify the network by representing loads as constant admittances, which allows the use of Kron reduction. The enabling approximation is

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$1

Because the bus voltage magnitudes are unknown before solving the TSC-OPF, the recent WECC 9-bus study proposes computing the load admittance before optimization by assuming

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$2

so that constant-power loads are approximated as constant admittances using nominal voltage and the network can be Kron-reduced outside the optimization loop (Coelho et al., 7 Nov 2025).

The reduced admittance matrix is written as

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$3

and is then used to compute generator electrical power during the fault and post-fault periods (Coelho et al., 7 Nov 2025). The operational benefit is explicit: the reduced admittance matrix can be precomputed, and the optimization avoids repeatedly rebuilding and inverting it.

The advantage is not exactness but implementation practicality. The paper compares three cases: load admittances computed using the $$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$4 p.u. assumption, load admittances updated using the actual optimized bus voltages, and benchmark time-domain simulation using ANATEM. The approximation introduces errors in both rotor angle $$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$5 and speed deviation $$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$6; errors are larger without correction, smaller with correction and with smaller time step, and rotor-angle errors are generally more visible in the post-fault period. The paper notes that the approximation mainly affects the post-fault period, because during the fault the reduced admittance matrix is dominated by the large shunt admittance of the short circuit, making the voltage assumption less influential (Coelho et al., 7 Nov 2025).

The WECC 9-bus, 3-machine case study provides a concrete scale for these effects. The setup uses MATPOWER for steady-state data, Pavella et al. for dynamic parameters, all loads increased by a factor of 1.5, Julia-JuMP with IPOPT 3.14, a 5 s simulation horizon, time steps of 10 ms and 1 ms, and a stability limit of $$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$7 with respect to COI. For a mild three-phase short circuit at bus 4 cleared by opening line 4–5 at 150 ms, the stability constraint does not become active, dispatch remains unchanged, and the load-bus voltages are above 1.0 p.u. For a more severe three-phase short circuit at bus 7 cleared by opening line 7–5 at 300 ms, the stability constraints become active and force a different dispatch, with

$$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$8

In the mild case, the largest reported rotor-angle MAE is for G3 in the w/o correction case, with 4.8116 at 10 ms and 3.2151 at 1 ms, while the w correction case at 1 ms gives 0.1049. In the severe contingency, the reported G2 rotor-angle MAE reaches 16.3495 in the w/o correction case at 10 ms and 8.6702 in the w correction case at 1 ms (Coelho et al., 7 Nov 2025).

The methodological significance is two-sided. Offline Kron reduction reduces implementation complexity and mitigates convergence issues, but the $$\min \qquad f(P_g)=\sum_{g\in\mathcal G} C_g\cdot P_g,$$9 p.u. voltage assumption introduces modeling error, especially when voltages deviate noticeably from nominal or under more stressed contingencies (Coelho et al., 7 Nov 2025).

5. Surrogate, learning, and certification-based TSC-OPF

Recent TSC-OPF research increasingly replaces direct dynamic simulation inside the optimization loop with learned or approximate surrogates. One neural-surrogate framework embeds a binary neural-network classifier directly into AC-OPF through a constraint of the form

$$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$0

where larger $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$1 pushes the dispatch deeper into the stable region. The training data are not generated by uniform random sampling alone; instead, a model-driven active sampling algorithm iteratively solves an Active Search OPF that drives samples toward the current NN decision boundary. The paper emphasizes that validation should not rely only on classification accuracy over random points, but should solve TSC-OPF for many load instances and then run full dynamic simulation on the resulting optimal dispatch. In the Hawaii test case, AC-OPF produced 123 unstable dynamic simulations among 960 feasible solved instances, or 12.8% instability. For Input B, pushing instability to 0% at $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$2 increases cost by about 37% relative to AC-OPF and gives mean solve time 8.19 s, compared with 0.33 s for AC-OPF. The same paper identifies a trade-off between surrogate accuracy and pricing structure: input choices such as $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$3 are more accurate but generally yield discriminatory prices, whereas bus-level inputs such as $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$4 or $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$5 produce locational or uniform prices at some cost in surrogate accuracy (Garcia et al., 3 Feb 2025).

DynOPF-Net combines learning to optimize with Neural ODEs. A load-to-dispatch network predicts AC-OPF decisions, and generator-specific Neural ODE models predict rotor-angle and speed trajectories under the proposed dispatch. Stability is enforced during training through a Lagrangian-dual or penalty-based objective that includes AC feasibility and a differentiable penalty on violations of $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$6. On the WSCC 9-bus and IEEE 57-bus systems, DynOPF-Net reports stability violation 0.00, while the MSE, DC3, and LD baselines retain nonzero stability violations. The steady-state gap remains small, at $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$7 for WSCC 9-bus and $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$8 for IEEE 57-bus, with inference times of $$\delta_{\mathrm{COI}}^{t}=\frac{\sum_{g\in\mathcal G} H_g \delta_g^t}{\sum_{i\in\mathcal G} H_i},$$9 s and $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$0 s, respectively (Vito et al., 2024).

A distinct concern for learned TSC-OPF is robustness under unseen perturbations. Certification-informed preventive control addresses this by training a deep belief network to estimate transient stability and then verifying the surrogate with a sequential PGD, $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$1-CROWN, and $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$2-CROWN pipeline over a bounded perturbation set

$$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$3

In the modified western South Carolina 500-bus system, DBN-C achieves 99.15% classification accuracy, DBN-E achieves mean absolute error 1.86, and classification or estimation time is below 0.01 s. On 1,000 random samples with 5% perturbation range for IBRs, SGs, and loads, the verification outcomes are 63.4% unsafe by PGD, 29.7% safe by $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$4-CROWN, 6.5% safe by $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$5-CROWN, and 0.4% unknown, so the method certifies or identifies robustness in 99.6% of cases. In preventive control, the unverified TSC-OPF solution has total cost \$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$6\lambda=\mathrm{TSI}=63.28125$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$719,057.8, a cost increase of \$22.2 (Su et al., 2024).

A recurring point across these approaches is that steady-state feasibility, fast inference, or even high surrogate accuracy on random data is not by itself sufficient. OPF solutions that satisfy AC constraints can still be transiently unstable, and learned surrogates may require optimizer-focused validation or explicit certification under uncertainty (Garcia et al., 3 Feb 2025, Vito et al., 2024, Su et al., 2024).

6. Extensions to contingency management, uncertainty, and hybrid microgrids

TSC-OPF has also been specialized to multi-insecurity operating contexts. In wildfire-resilient operation, CSCOPF augments redispatch OPF with cut-set desaturation constraints derived from a Feasibility Test algorithm and with a transient-stability correction constraint based on a predicted correction factor $$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$8. The transient correction is enforced as

$$\delta^{\min}\le \delta_g^t-\delta_{\mathrm{COI}}^t\le \delta^{\max}.$$9

where the critical machines are identified from contingency analysis and the correction factor is predicted by a linear regression model trained offline. On the IEEE 118-bus wildfire case, RT-SCED remains transient unstable and cut-set insecure, TSCOPF fixes transient stability but not cut-set saturation, and CSCOPF fixes both transient stability and cut-set saturation with no load shedding required (Sahoo et al., 2023).

Renewable uncertainty motivates robust variants. RTSC-OPF-PFR coordinates generator redispatch, generator recourse through participation factors, and network-side PFR tuning. The transient stability condition is enforced through a linearized SIME-based stability-margin inequality, while robustness is handled through a scenario approach and an offline-online database. On the modified New England 39-bus system, the base OPF dispatch is transiently unstable under the tested contingency, and with 1,000 renewable scenarios the base case is unstable in 90.7% of cases. Under the proposed framework, the online selection of 54 representative scenarios achieves 100% robustness on the tested scenario set, with total generation cost increase of only 2.060% relative to base OPF and online computation time 49.81 s. In multi-contingency comparisons, RTSC-OPF-PFR achieves nearly 100% robustness under all tested contingencies and cost 3.0% lower than RTSC-OPF (Chen et al., 2020).

In AC/DC hybrid microgrids, TSC-OPF becomes a dynamic operational subproblem inside a larger defender-attacker-defender planning framework. The operational model couples hybrid AC/DC power flow, synchronous-generator rotor and excitation dynamics, PI governor response, IEEE Type I AVR, IBR current control and voltage droop, interlinking-converter frequency-voltage droop coupling, and transient constraints on system frequency and voltage. The paper states that the TSC-OPF can be solved on an hourly basis with a second-scale resolution through a Lyapunov optimisation approach that captures the time-coupling property of energy storages. The case study uses protection-style transient requirements: frequency nadir must stay above 49.2 Hz, RoCoF must remain below 1 Hz/s, and voltage must stay within 0.9–1.1 p.u. Relative to static planning, transient-stability-driven planning increases conventional DG capacity and reduces WT and PV capacities, reflecting the need for sufficient inertia, frequency reserve, and voltage support under severe contingencies (Wang et al., 23 Jul 2025).

These extensions show that TSC-OPF is not a single fixed model but a family of preventive-control formulations. Depending on the application, it may appear as a DAE-constrained nonlinear program, a certificate-constrained AC-OPF, a robust scenario program, a redispatch problem with cut-set and transient constraints, a learning-assisted real-time predictor, or an operational layer within resilience-oriented microgrid planning (Li et al., 2017, Sahoo et al., 2023, Wang et al., 23 Jul 2025).