The study employs 35-bus, and 53-bus distribution systems for dataset generation. Load flow analysis via the Forward Backward Load Flow Algorithm (FBLFA) in MATLAB simulates varying load conditions (10–150%) to compute voltage stability indices (FVSI). The FVSI values derived from Forward Backward Load Flow Algorithm (FBLFA) serve as ground truth.
The dataset, comprising bus voltages, power flows, and stability indices, is preprocessed and split into training/testing subsets. Four regression models LR, RF, GB, and SVM are implemented using scikit-learn in Python. Hyperparameters are tuned via grid search, and models are trained to predict FVSI.
Data preprocessing
The dataset, comprising bus voltages, active/reactive power flows, and line parameters, underwent preprocessing to ensure robustness. Input features were normalized using Min-Max scaling to a [0,1] range, mitigating bias from varying magnitudes. Feature selection was performed via Pearson correlation analysis, excluding variables with |r| > 0.85 to reduce redundancy (e.g., reactive power at adjacent buses). No missing data were present in the FBLFA-generated dataset, as simulations were fully controlled. To ensure robustness and reproducibility, the following preprocessing steps were applied:
Normalization
The input features like Vm, Vn, Pn, Qn were scaled to a [0,1] range using Min-Max normalization.
$$\:{X}_{norm}=\frac{X-{X}_{min}}{{X}_{max}-{X}_{min}}$$
(12)
Feature selection
Redundant features were avoided by Pearson correlation analysis. Variables with ∣r∣ > 0.85 such as reactive power at the load points or buses were excluded to reduce multicollinearity.
Missing data handling
The simulated dataset contained no missing values, as FBLFA ensured fully observable load-flow solutions. Here is the sample python code for preprocessing steps.
The feature ranking is tabulated in Table 1 for FVSI prediction of ML models to validate the importance of the variables in voltage stability of distribution system.
Hyperparameter tuning
Hyperparameters were tuned via grid search with 5-fold cross-validation. Parameter ranges included estimators (100–500) for RF/GB, max-depth (5–20) for RF, and C (0.1–100) for SVM and generally the summary of hyperparameter sensitivity analysis for ML models as shown in Table 2. The optimal values minimized RMSE on the validation set and the overall methodology flowchart is Proposed for FVSA assessment in Fig. 4.
Proposed FVSA assessment methodology flow chart.
The following are the performance evaluation metrics for this study includes, Mean Absolute Percentage Error (MAPE), Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE) and Coefficient of Determination (R²). These measures shed light on how accurate and consistent the model’s predictions are16.
$$\:RMSE=\sqrt{\frac{1}{N}{\sum\:}_{i=1}^{N}{\left(Ya-Yp\right)}^{2}}$$
(13)
$$\:MAPE=\frac{1}{N}{\sum\:}_{i=1}^{N}\left(\left|\frac{{Y}_{a}-{Y}_{p}}{{Y}_{a}}\right|\right)*100$$
(14)
$$\:MAE=\frac{1}{N}{\sum\:}_{i=1}^{N}\left({Y}_{a}-{Y}_{p}\right)$$
(15)
$$\:MSE=\frac{1}{N}{\sum\:}_{i=1}^{N}{\left({Y}_{a}-{\text{Y}}_{\text{p}}\right)}^{2}$$
(16)
$$\:{R}^{2}=1-\frac{{\sum\:}_{i=1}^{N}{\left({Y}_{a}-{Y}_{p}\right)}^{2}}{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\sum\:}_{i=1}^{N}{\left({Y}_{a}-\stackrel{-}{Ya}\right)}^{2}}$$
(17)
Where, \(\:{Y}_{a}\) = Actual value; \(\:{Y}_{p}\) = Model predicted value; and N =Number of the pattern.
Table 3 summarizes the computational efficiency of various machine learning models used for FVSI prediction.
Feature importance ranking with interpretability metrics
The feature importance ranking with interpretability importance of this study is tabulated in Table 4 as shown below.
SHAP-based model interpretability
To address the “black-box” nature of ensemble methods, we implement SHAP (SHapley Additive exPlanations) for enhanced model transparency34.
where \(\:{\varphi\:}_{i}\) represents the SHAP value for feature \(\:i\).
Feature importance of SHAP values at bus 36 FVSI prediction using GB model.
Figure 5 presents the SHAP-based feature importance for predicting the FVSI at bus 36 using the GB model. The voltage magnitude (\(\:{V}_{m}\) = 0.028) is identified as the most significant factor contributing to instability risk, followed by line reactance (\(\:{X}_{mn}\) = 0.015). Reactive power exhibits a negative influence, indicating its role in mitigating risk. This analysis confirms that bus 36 poses the highest FVSI-related risk in the 53-bus Papyrus feeder network, supported by a model accuracy of 99.98%.
Simulation results and analysis
Simulation results at nominal load (100%)
For the assessment and analysis of voltage stability FVSI method is used in this paper and predictions FVSI values are carried out in both case study systems. This paper work mainly considered two scenarios of at nominal load and load varying conditions. The selected ML algorithms predicted the values of FVSI in both case study feeder systems and the performance evaluation are conducted for all ML models used. The calculated and predicted results are tabulated in Tables 5 and 6 for both feeders of the case study respectively.
Comparison of ML models for FVSI prediction in 35 bus system.
The results confirm GB and RF as the most accurate predictors, closely aligning with FBLFA values across buses, while LR and SVM exhibit systematic deviations as indicated the result in Table 6 below. Buses 36, 32, and 21 display the highest instability (FVSI > 0.05), with SVM notably underestimating risks. These findings reinforce the reliability of ensemble methods for voltage stability assessment and highlight critical nodes requiring prioritized monitoring.
Comparison of ML models for FVSI prediction in 53 bus system.
The Figs. 6 and 7 the graphical representations of the stability indices FVSI of 35 bus and 53 bus system respectively. The figures show the comparison of the index with respect to the LR, RF, GB and SVM to the load flow analysis using FBLFA method. From this we can infer that the RF and GB have showed better performing results whereas LR and SVM have given poor results.
The comparison performance metrices of the ML algorithms results are presented in Table 7. The performance metrics indicate that the GB model excels in both the 35-bus and 53-bus systems, achieving the highest R² values of 0.999 for 35-bus and 0.9998 for 53-bus, suggesting excellent predictive accuracy which is tabulated as in Table 5. Conversely, SVM model underperformed, particularly in the 53-bus system, with a notably low R² of 0.635, indicating weaker predictive capabilities. The SVM’s lower accuracy (R² = 0.635 for 53-bus) stems from its sensitivity to nonlinear voltage-load dynamics. Trials with RBF kernels and regularization (C = 1–100) improved performance marginally but failed to match ensemble methods. Unlike GB/RF, which model nonlinearities via decision trees, SVM’s reliance on kernel transformations struggled with high-dimensional interactions and load variability. The LR has a medium performance in predicting the stability of the system.
Simulation result and analysis of both case study systems at load varying conditions
The following comparison results showed that the FVSI with ML models at load varying conditions with respect to nominal load.
Simulation result of FVSI values for bus-27 and bus − 16 at varying load conditions of 35 bus system.
Figure 8 demonstrates FVSI progression for critical buses 27 and 16 in the 35-bus system under load variation (10%-150%). Bus 27 exhibits higher vulnerability with FVSI values of 0.029 and 0.023 respectively at nominal load. The visualization compares machine learning model performance against FBLFA ground truth, revealing ensemble methods (GB, RF) provide superior accuracy while SVM shows significant deviations. Linear FVSI escalation with load validates the need for proactive stability monitoring and load management strategies.
Comprehensive Analysis: FVSI Prediction for critical buses in 35-bus system.
Figure 9 demonstrates comprehensive FVSI prediction analysis for critical buses in the 35-bus system, comparing machine learning models (FBLFA, GB, RF, SVM, LR) across four analytical perspectives. Subplot (a) shows Bus 27 as most critical with highest FVSI values, while subplot (b) confirms Bus 16’s secondary vulnerability. Prediction error analysis (c) reveals SVM’s significant deviations, while comparative analysis (d) validates ensemble method superiority. This multi-dimensional visualization enables operators to assess model reliability and identify critical stability monitoring priorities.
Simulation result of FVSI values of selected buses at varying load conditions for 53 bus system.
Figure 10 demonstrates FVSI simulation results for nine selected critical buses (36, 32, 21, 18, 15, 12, 9, 6, 3) in the 53-bus system under load variation (50%-200%). Bus 36 emerges as most vulnerable, with exponential FVSI growth approaching critical threshold (0.85) at 140% loading. The visualization reveals critical loading points where buses exceed warning thresholds (0.35), enabling proactive voltage stability monitoring. Yellow markers identify first critical points, validating the need for targeted reactive power compensation strategies.
Comprehensive FVSI Analysis Dashboard: Multi-Perspective Voltage Stability Assessment for 53-Bus Distribution System.
Figure 11 presents a comprehensive FVSI analysis for the 53-bus system through four analytical perspectives: (a) Top 3 critical buses showing exponential FVSI growth with Bus 36 as most vulnerable, (b) System maximum FVSI escalation reaching 0.5 at 200% loading, (c) Buses exceeding thresholds with critical count rising dramatically beyond 150% load, and (d) Safety margin analysis revealing diminishing stability buffers. This multi-dimensional visualization demonstrates voltage collapse proximity indicators and validates the need for proactive load management strategies in high-risk distribution networks.
The performance evaluation and comparison are carried out using various research papers on machine learning, as presented in Table 8.
Stability threshold analysis
The voltage stability threshold analysis for critical buses with different load levels is summarized in Table 9; Fig. 12 as shown below.
Stability threshold analysis with load variations.
FVSI Progression analysis for critical buses in Bahir Dar distribution systems under variable loading conditions.
Figure 13 presents a comprehensive stability threshold analysis dashboard for Ethiopian distribution systems, demonstrating multi-dimensional voltage stability assessment across nine analytical perspectives. The visualization encompasses FVSI progression analysis for critical buses in both 35-bus Bata and 53-bus Papyrus feeders under load variations (10%-150%). Panel (a) shows exponential FVSI growth with Bus 36 (Papyrus) and Bus 27 (Bata) emerging as most vulnerable nodes. Panel (b) reveals safety margin evolution, indicating diminishing stability buffers beyond 130% loading. Panel (c) displays Ethiopian grid resilience assessment, with both systems maintaining acceptable thresholds until 140% load. The analysis incorporates system-wide maximum FVSI comparison (d), grid stress factors (e), voltage profile monitoring (f), load shedding requirements (g), and overall resilience indices (h-i). This comprehensive framework enables Ethiopian Electric Power operators to implement proactive stability monitoring, optimize reactive power compensation strategies, and establish critical loading thresholds for enhanced grid reliability in challenging operating conditions.
Comparison of decision boundaries for stability region of SVM vs. RF.
The voltage stability region illustrated in Fig. 14 is the comparison between SVM and RF. The training convergence curve of ML models is depicted in Fig. 15 below.
Training convergence curve for the ML models.
Sensitivity analysis and feature impact assessment
RF analysis reveals voltage magnitude (Vm) as the most critical feature (importance: 0.42), followed by reactive power (Qn: 0.28), line reactance (Xmn: 0.18), and active power (Pn: 0.12). This ranking aligns with voltage stability theory, where voltage levels and reactive power balance dominate stability margins. Systematic variation of R/X ratios (± 20%), load compositions (± 30%), and voltage angles (± 15°) demonstrated model robustness with accuracy degradation 0.92 under all tested conditions while SVM performance degraded to R² 1000 predictions/second on standard hardware.
Comprehensive sensitivity analysis dashboard for FVSI-based voltage stability assessment in 53-bus distribution system.
Figure 16 presents a comprehensive six-panel sensitivity analysis dashboard for the 53-bus distribution system, validating machine learning model performance for FVSI prediction. The results demonstrate feature importance rankings matching paper specifications (Vm: 0.42, Qn: 0.28), with parameter variations showing acceptable degradation levels below 3%. Monte Carlo analysis confirms model robustness across 1000 iterations. Critical buses (36, 32, 21) exhibit increasing FVSI values under load variations (10%-150%), with ensemble methods (RF/GB) maintaining superior alignment with ground-truth FBLFA calculations compared to linear regression and SVM approaches.
Comprehensive sensitivity analysis dashboard for machine learning-based FVSI prediction in 35-bus distribution system.
Figure 17 presents a six-panel sensitivity analysis dashboard for the 35-bus distribution system, validating FVSI-based voltage stability assessment using machine learning algorithms. The dashboard demonstrates feature importance rankings consistent with paper specifications (Vm: 0.42, Qn: 0.28, Xmn: 0.18, Pn: 0.12), parameter variation impacts below 3% degradation threshold, and Monte Carlo distribution confirming model robustness. Critical buses 27 and 16 exhibit exponential FVSI growth under load variations, while feature sensitivity curves validate ensemble method superiority over linear approaches for real-time voltage stability monitoring applications28.
To mitigate risks near FVSI ≈ 1.0, we implement a multi-tier warning system:
$$\:Safety\:Margin=1.0-FVS{I}_{predicted}-{ϵ}_{uncertainty}$$
where \(\:{ϵ}_{uncertainty}\) accounts for model prediction uncertainty.
This approach allows system operators to dynamically assess the proximity to voltage instability and respond accordingly based on a clearly defined risk classification, as summarized in Table 10.
