Abstract
The proposed study introduces a long short-term memory architecture integrated with a novel comprehensive dynamic security index to enable multi-domain stability assessment beyond post-fault dependencies. The comprehensive dynamic security index unifies voltage, frequency, and transient stability metrics into a single interpretable scalar, quantifying real-time proximity to instability boundaries while classifying system states into five actionable categories. By prioritizing generator terminal dynamics, the framework operates with reduced PMU coverage through strategic feature engineering. Validated on IEEE 14 and 118-bus systems, the long short-term memory-deep neural network (LSTM-DNN) model outperforms state-of-the-art techniques in both prediction speed and operational granularity. By bridging static and dynamic data streams, a hybrid attention mechanism improves operator confidence by linking model decisions to physical grid components. Results demonstrate robustness to class imbalance.
Data availability
The datasets generated and analyzed during the current study are available in the Figshare repository at [https://doi.org/10.6084/m9.figshare.30590399](https:/doi.org/10.6084/m9.figshare.30590399) under the CC BY 4.0 license. Additional or extended data supporting this study are available from the corresponding author upon reasonable request.
Abbreviations
- DSA:
-
Dynamic security assessment
- OC:
-
Operating condition
- PMU:
-
Phasor measurement unit
- CDSI:
-
Comprehensive dynamic security index
- DSMV :
-
Dynamic security margin in term of STVS
- STVS:
-
Short-term voltage stability
- TSM:
-
Transient stability margin
- FSM:
-
Frequency stability margin
- RoCoF:
-
Rate of change of frequency
- RoCoFSM :
-
RoCoF stability margin
- \(\:{V}_{0}\) :
-
Pre-fault steady-state voltage magnitude
- \(\:{V}_{dev}\) :
-
Voltage deviation in postfault
- \(\:{V}_{pre\text{-}fault}\) :
-
Voltage magnitude in prefault
- \(\:{V}_{min}\) :
-
Minimum allowable voltage
- \(\:{V}_{max}\) :
-
Maximum allowable voltage
- \(\:{t}_{u}\) :
-
Duration of the threshold violation
- \(\:{N}_{record}\) :
-
Total number of study cases
- \(\:{N}^{FD}\) :
-
Number of selected fault durations
- \(\:{N}^{FT}\) :
-
Number of fault types
- \(\:{N}^{Line}\) :
-
Number of transmission lines
- \(\:{N}^{POL}\) :
-
Number of fault locations on each line
- \(\:{N}^{OC}\) :
-
Number of operating conditions
- \(\:{P}_{0},\text{\hspace{0.17em}}{Q}_{0}\) :
-
Pre-fault active/reactive power operating point
- \(\:{P}_{new},\text{\hspace{0.17em}}{Q}_{new}\) :
-
Active/reactive power after applied load event used for OC generation
- \(\:{t}_{end}\) :
-
Simulation stop time
- \(\:{I}_{line}\) :
-
Line current flow
- \(\:{I}_{thermal}\) :
-
Current flow limitation
- \(\:{\delta\:}_{i}\left(t\right)\) :
-
Rotor angle of generator i at time t
- \(\:{\delta\:}_{j}\left(t\right)\) :
-
Rotor angle of generator j at time t
- \(\:\varDelta\:{\delta\:}_{max}\) :
-
Maximum rotor angle separation between any pair of generators
- \(\eta\) :
-
Transient stability index
- CCT:
-
Critical clearing time
- \(\:CC{T}_{i}\) :
-
Estimated critical clearing time for ith case studied
- ACT:
-
Actual clearing time
- \(\:AC{T}_{i}\) :
-
Fault clearing time extracted from probability distribution
- \(\:\tau\:\) :
-
difference between current fault clearing time and previous one
- \(\:\stackrel{\prime}{\varepsilon}\) :
-
Convergence constant used in adaptive CCT search (e.g., \(\:0.0078115\)s)
- \(ACT^{\prime}\) :
-
Last clearing time that causes instability in previous steps
- \(ACT^{\prime\prime}\) :
-
Last stable clearing time in previous steps
- \(\:\varDelta\:t\) :
-
Time step used in CCT calculation
- \(\:{f}_{0}\) :
-
Nominal system frequency
- \(\:{f}_{t}\) :
-
System frequency measured at time t
- ft+100 :
-
System frequency measured 100 ms after time t
- \(\:F{N}_{lim}\) :
-
Frequency nadir stability limit
- \(\:RoCo{F}_{lim}\) :
-
Maximum allowable RoCoF limit
- \(\:{F}_{nadir}\) :
-
Frequency nadir
- \(\:F{N}_{ref}\) :
-
Maximum allowable deviation from the nominal frequency
- \(\:RoCo{F}_{ref}\) :
-
Normalization reference for RoCoF margin
- \(\:{W}_{FN}\) :
-
Weighting factor of frequency stability margin
- \(\:{W}_{R}\) :
-
Weighting factor of RoCoF stability margin
- \(\:{W}_{V}\) :
-
Weighting factor of voltage stability margin
- \(\:{W}_{T}\) :
-
Weighting factor of transient stability margin
- \(\:x\) :
-
Raw feature value
- \(\:{x}_{i}\) :
-
Input feature used in data normalization
- \(\:{x}_{min},\text{\hspace{0.17em}}{x}_{max}\) :
-
Minimum and maximum values of feature x
- \(\:{x}_{norm}\) :
-
Min-max normalization
- \(\:{\alpha\:}_{i}\) :
-
Attention weight of ith time step
- \(h_{{LSTM}}^{{(i)}}\) :
-
Hidden state of LSTM’s context integration layer at step i
- Xstatic :
-
Static system parameters
- \(\:{W}_{a}\) :
-
Trainable weight matrix in attention module
- \(\:{b}_{a}\) :
-
Bias
- \(\:\sigma\:(\cdot\:)\) :
-
Sigmoid activation function
- FL:
-
Federated learning
- LSTM-DNN:
-
Long short-term memory deep neural network
- GNN:
-
Graph neural network
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M. Shahriyari: Conceptualization, Software, Writing - original draft preparation; A. Safari: Supervision, Methodology, Writing - review and editing; A. Quteishat: Resources, formal analysis, Writing - review and editing; H. Afsharirad: Formal analysis and investigation. All authors read and approved the final manuscript.
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Shahriyari, M., Safari, A., Quteishat, A. et al. An LSTM architecture for real-time multi-domain stability boundary prediction beyond post-fault dependency in power systems. Sci Rep (2026). https://doi.org/10.1038/s41598-026-36571-6
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DOI: https://doi.org/10.1038/s41598-026-36571-6
