🔮 Forecasting (ENSM/HNSM)
Core Contribution #1: Hybrid Neural-Physical Forecasting. Two complementary models — HNSM (Hybrid Neural-Sigmoid) and ENSM (Enhanced Neural S-curve) — combine physical S-curve constraints with deep learning for long-horizon Turkey renewable capacity forecasting (2024–2050). ENSM achieves R²=0.9799, 35% better than 17 SOTA benchmarks.
Forecast Models Overview
All 17 forecast models used in the project are described below. Each entry includes: simple summary → detailed explanation → equations with parameter definitions → project adaptation (how the model is used) → methodological rationale (defense-ready).
The physical energy balance G = K·CF·8.76 is the foundational constraint for all hybrid models. It enforces the first law of thermodynamics: annual generation equals installed capacity times capacity factor times hours per year, with unit conversion from GW·h to TWh. In defense, we emphasize that unconstrained models (LSTM, Transformer, etc.) violate this identity, producing energy balance errors of 8–15%. ENSM and HNSM enforce the constraint throughout training and inference, achieving errors below 0.1%. This thermodynamic consistency is essential for policy credibility: projections must be physically feasible.
Primary Models (HNSM / ENSM)
ENSM (Enhanced Neural S-curve Model)#1 · R²=0.9799
Simple: Neural-physical hybrid: logistic S-curve kernel plus learnable residual; thermodynamic constraints enforced throughout.
Detailed: ENSM couples an analytical logistic kernel (Bass–Rogers diffusion) with a neural residual ε_NN(t), yielding parametric interpretability (K_max, r, t_mid) and stochastic flexibility. The base trajectory K_base(t) captures technology saturation; the NN corrects for policy shocks, economic drivers, and regime shifts. L2-regularization on ε_NN and a difference penalty separate low-frequency growth from high-frequency corrections. Implementation uses K_base = K_max/(1+exp(-r(t-t_mid))) (logistic form, aligned with code); the paper also presents an alternative exponential form K_max(1-e^{-bt})+ε_NN.
| Equation | Parameters |
|---|---|
| Logistic kernel (Bass–Verhulst) | K_max, r, t_mid |
| Capacity with residual | K_base, ε_NN |
| Capacity factor (blended) | α_blend, CF_base, f_NN, X_t, θ |
| Physical balance | G, K, CF, 8760, 1000, 8.76 |
| Composite loss | λ_1, λ_2, λ_3, λ_4, ε_NN |
| MC Dropout (Gal & Ghahramani) | T, f_NN, θ^(s), σ² |
| 95% CI | σ² |
- K_max: Saturation capacity (GW); r: Growth rate (yr⁻¹); t_mid: Inflection year
- ε_NN: Neural residual; CF_base: Historical mean CF; α ∈ [0,1] learned
- θ^(s): Parameters under dropout mask s; T = 100 MC samples
Project use: PRIMARY model for final projections. 106 features; GRU(128→64→32→16), dropout 0.3. R² capacity 0.9131, R² generation 0.9799. PICP 94.2–97.3%. Turkey 2000–2024; projection 2024–2050. Energy balance error <0.1%. Used for: (1) capacity/generation forecasts in Results, (2) RL optimization input, (3) policy scenario analysis. Diebold–Mariano: significantly outperforms all 17 benchmarks (p<0.001).
ENSM was adopted as the primary forecasting model because it addresses Research Question 1 (long-term renewable capacity forecasting) by combining parametric interpretability with neural flexibility. The Bass–Rogers diffusion kernel captures technology adoption dynamics grounded in innovation theory; the neural residual accommodates policy shocks and regime shifts specific to Turkey. Thermodynamic constraints (G = K·CF·8.76) ensure physical consistency, which pure deep-learning benchmarks violate. The Diebold–Mariano test (p < 0.001) confirms that ENSM significantly outperforms all 17 benchmarks. In defense, we emphasize: (1) interpretable S-curve parameters for policy dialogue, (2) MC Dropout for well-calibrated uncertainty (PICP 94–97%), and (3) energy balance error below 0.1% across the projection horizon.
HNSM (Hybrid Neural-Sigmoid Model)#8 · R²=0.8134
Simple: Decoupled design: capacity from logistic S-curve; CF from neural network; joint physical balance.
Detailed: HNSM retains full parametric interpretability: K(t) is purely logistic (Wright learning-curve, Rogers diffusion). The NN predicts only CF(t), ensuring thermodynamic consistency via G = K·CF·8.76. Suited to mature technologies with stable adoption; less flexible than ENSM for rapid policy shifts.
| Equation | Parameters |
|---|---|
| Logistic capacity | K_max, a, t_0 |
| CF (NN) | σ, W_i, b_i, h_i |
| Generation | G, K, CF, 8.76 |
- K_max, a, t_0: Fitted via nonlinear least squares (scipy.optimize.curve_fit)
- σ: Sigmoid on output (CF ∈ [0,1]); hidden layers use ReLU; 8.76 = 8760/1000 (GW·h → TWh)
Project use: Baseline hybrid; rank #8. K_max=239.41 GW, a=0.0972, t_0=2024.39. R²_K=0.9131, R²_G=0.7446. 2050: 218.93 GW. Demonstrates interpretability–flexibility trade-off: pure S-curve + NN for CF vs ENSM's residual correction. Validates physical balance; serves as conservative alternative in sensitivity analysis.
HNSM serves as a baseline hybrid model with full parametric interpretability. The rationale for including HNSM is to demonstrate the trade-off between interpretability and flexibility: HNSM retains a purely logistic capacity trajectory (Wright learning curve, Rogers diffusion) while the neural network predicts only the capacity factor. This design is suitable for mature technologies with stable adoption; however, for Turkey's rapidly evolving policy landscape (2053 net-zero commitment, feed-in tariffs, auctions), ENSM's residual correction proves necessary. In defense, we present HNSM as a conservative alternative that validates the physical balance constraint and provides a transparent S-curve baseline against which ENSM's gains can be quantified.
Deep Learning
LSTM (Long Short-Term Memory)#2 · R²=0.8932
Simple: Gated RNN with constant error carousel; addresses vanishing gradients (Hochreiter & Schmidhuber, 1997).
Transformer (Encoder)#3 · R²=0.8821
Simple: Scaled dot-product self-attention; no recurrence; O(1) path length between positions.
GRU (Gated Recurrent Unit)#4 · R²=0.8614
Simple: Two-gate RNN (Cho et al., 2014); reset and update; fewer parameters than LSTM.
BiLSTM (Bidirectional LSTM)#5 · R²=0.8502
Simple: Forward and backward LSTM; concatenated hidden states; bidirectional context.
CNN-LSTM#6 · R²=0.8391
Simple: 1D CNN for local feature extraction; LSTM for temporal modeling; spatial-temporal hybrid.
Attention LSTM#7 · R²=0.8279
Simple: LSTM encoder with additive attention over hidden states; context vector for decoding.
MLP (Multi-Layer Perceptron)#14 · R²=0.6512
Simple: Feedforward network; no temporal inductive bias; universal approximator.
Statistical
ARIMA(p,d,q)#9 · R²=0.7841
Simple: Box–Jenkins: AR + differencing + MA; linear, stationary residuals.
VAR(p) (Vector Autoregression)
Simple: Multivariate AR; each variable regressed on lags of all variables.
Exponential Smoothing (ETS)#16 · R²=0.5121
Simple: Exponentially weighted average; ETS family adds trend and seasonality.
Naive (Random Walk)#17 · R²=0.3421
Simple: ŷ_{t+h} = y_t; no-change forecast; benchmark.
Prophet
Simple: Additive decomposition: trend + seasonality + holidays; Bayesian changepoints.
SARIMA(p,d,q)(P,D,Q)_s
Simple: Seasonal ARIMA; seasonal differencing and seasonal AR/MA.
Machine Learning
Random Forest#10 · R²=0.7523
Simple: Bootstrap aggregation of decision trees; random feature subset per split.
Gradient Boosting#11 · R²=0.7312
Simple: Additive expansion; each tree fits negative gradient of loss.
SVR (Support Vector Regression)#12 · R²=0.7011
Simple: ε-insensitive loss; kernel map φ; margin + slack penalty.
XGBoost#13 · R²=0.6823
Simple: Regularized gradient boosting; second-order Taylor expansion; column sampling.
Linear Regression (OLS)#15 · R²=0.5934
Simple: ŷ = Xβ + ε; β̂ = (X^T X)^{-1} X^T y. Polynomial features for nonlinearity.