Future of renewable energy consumption in France, Germany, Italy, Spain, Turkey and UK by 2030 using optimized fractional nonlinear grey Bernoulli model

The study addresses the increasing energy demand and depletion of conventional sources, focusing on forecasting renewable energy consumption and total gross final energy consumption (GFEC) in major European countries. The European Union increased the share of renewables in GFEC from 8.5% (2004) to 18.0% (2018), with targets of 20% by 2020 and 32% by 2030. Accurate forecasting of energy consumption and renewable shares is essential for energy management and policy evaluation. Grey prediction models, notably GM(1,1) and its improved forms, are well-suited to small datasets. Building on recent developments in fractional-order accumulation and nonlinear grey Bernoulli models, the research proposes optimizing background value λ (0–1), power index γ, and fractional order r to improve predictive accuracy. The aim is to forecast GFEC, RES consumption, and the share of RES in GFEC for France, Germany, Italy, Spain, Turkey, and the UK up to 2030, and to compare projections with national targets and prior literature.

The paper reviews grey prediction models, starting with GM(1,1) (Deng, 1982) and optimized forms that tune the background value λ from a fixed 0.5 to within [0,1], improving accuracy (OGM(1,1); Wen et al., 2000; Shang and Pei, 2009; Ma et al., 2013; Ene and Öztürk, 2017). Multivariable extensions (GM(1,N), GMC(1,N)) and metabolic rolling models (MGM(1,1), OMGM(1,1), NMGM(1,1), ONMGM(1,1)) further enhance performance. The nonlinear grey Bernoulli model NGBM(1,1) introduces a power index γ to capture nonlinearity; optimized variants (ONGBM/NNGBM) also show superior accuracy (Chen, 2008; Chen et al., 2008, 2010; Wang et al., 2011; Lu et al., 2016). Fractional-order accumulation was introduced to grey systems (Wu et al., 2013), leading to fractional models applied to multiple energy domains (electricity, nuclear, oil, wind, gas, coal, renewables). Wu et al. (2019a) proposed the fractional nonlinear grey Bernoulli model FANGBM(1,1), optimizing γ and r but assuming λ=0.5. The literature consistently shows λ-optimization improves forecasting accuracy (Table 1), motivating the present work to optimize λ alongside γ and r, yielding OFANGBM(1,1).

The optimized fractional nonlinear grey Bernoulli model OFANGBM(1,1) uses fractional-order accumulation (r-AGO) to transform an original non-negative sequence X^(0) into X^(r), where r>0. The model’s whitening equation for FANGBM(1,1) is dX^(r)/dt + a X^(r) = b (X^(r))^γ, with discrete form X^(r)(k) − X^(r)(k−1) + a z^(r)(k) = b (z^(r)(k))^γ. The background value z^(r)(k) is constructed as z^(r)(k) = λ X^(r)(k) + (1−λ) X^(r)(k−1), k≥2, with λ∈(0,1). Special cases include NGBM(1,1) when r=1, GM(1,1) when r=1 and γ=0, and their optimized forms when λ is allowed in (0,1). Given λ, γ, r, parameters a and b are estimated via least squares using matrix formulation with B comprising −z^(r)(k) and (z^(r)(k))^γ terms, and Y comprising first differences of X^(r). The predicted series is generated using the closed-form solution for X^(r), initialized with X^(0)(1). Optimization: A genetic algorithm (GA) is employed to find λ, γ, r that minimize mean absolute percentage error (MAPE) on the in-sample data (2004–2018), with GA iterations run in Excel until MAPE change ≤0.01%. Constraints: 0<λ<1, r>0, γ≠0. Performance metrics include APE (%) = |X(i) − X̂(i)| / X(i) × 100 and MAPE (%) as the mean of APE over observations, with lower MAPE indicating better performance; goodness of fit is defined as 100 − APE(%). A flowchart details steps: initialize parameters, perform r-AGO, compute z^(r), estimate a,b, generate predictions, evaluate MAPE, and iterate GA until convergence.

• OFANGBM(1,1) consistently achieved lower MAPE than FANGBM(1,1) for both GFEC and RES consumption across all six countries (Table 3), indicating improved predictive performance by optimizing λ alongside γ and r. Example MAPE reductions (GFEC/RES respectively): France 1.2943% vs 1.3019% and 3.5362% vs 3.6105%; Germany 1.7588% vs 1.7638% and 1.8707% vs 1.8769%; UK 1.5780% vs 1.6771% and 6.5416% vs 6.5525%. • Forecasted GFEC in 2030 (Mtoe): France 151.7; Germany 227.6; Italy 110.8; Spain 84.5; Turkey 173.4; UK 132.2. • Forecasted RES consumption in 2030 (Mtoe): France 28.5; Germany 53.8; Italy 22.2; Spain 23.2; Turkey 26.1; UK 39.3. • Forecasted share of RES in GFEC (%): France 18.8; Germany 23.6; Italy 20.0; Spain 27.5; Turkey 15.1 (reported also as ~15.07); UK 29.7. • Interim 2020 estimates for RES shares (%): France 17.2; Germany 17.6; Italy 18.7; Spain 19.0; Turkey 13.9; UK 13.4. • Trends and AAGR highlights: France GFEC declines (−0.2% AAGR 2018–2030) while RES rises (0.9%); Germany RES grows from 36.8 to 53.8 Mtoe (3.2% AAGR); Italy RES growth slows (0.2% AAGR) amid GFEC decline (−0.8%); Spain RES grows (3.4% AAGR) with GFEC decline; Turkey GFEC and RES both grow strongly (4.3% and 5.1% AAGR); UK RES rises rapidly (8.5% AAGR) as GFEC slightly declines. • Alignment with national 2020 targets: Germany’s 2020 estimate (17.6%) is close to the 18% target; France’s 17.2% and UK’s 13.4% are below their targets (France 23%, UK 15%). Spain’s 19.0% is near its 20% target; Italy’s 18.7% exceeds its 17% target; Turkey’s share remains below its 2023 target (20.5%), with 2023 estimated at 14.6%.

Optimizing λ in the fractional nonlinear grey Bernoulli framework enhances model flexibility and fit to real energy time series, addressing the research goal of producing accurate forecasts for GFEC, RES consumption, and their shares. The OFANGBM(1,1) consistently improved accuracy over FANGBM(1,1), demonstrating the importance of jointly optimizing λ, γ, and r. Forecasts indicate divergent trajectories: declining GFEC in several EU countries with rising renewable shares, suggesting progress toward EU 2030 ambitions, while Turkey exhibits robust growth in both GFEC and RES but a slower increase in RES share, highlighting a gap to national targets. Comparisons with prior studies (Cucchiella et al., 2018; Simionescu et al., 2020) show the 2020 share estimates are broadly consistent, validating the approach. The results provide actionable insights for policymakers to assess the feasibility of national and EU targets, plan renewable integration, and anticipate sectoral energy demand changes.

The study introduces OFANGBM(1,1), an improved fractional nonlinear grey Bernoulli model that optimizes background value λ (0–1), power index γ, and fractional order r via GA, outperforming the baseline FANGBM(1,1) on historical European energy data. Using OFANGBM(1,1), country-specific forecasts to 2030 were generated for GFEC, RES consumption, and the RES share, with clear trends toward higher renewable shares in most countries. The model provides a robust tool for energy planning where datasets are short and dynamics are nonlinear. Future research directions include: disaggregated forecasting by renewable type (hydro, solar, wind, geothermal, biomass); integrating rolling mechanisms, machine learning, or kernel methods; exploring alternative optimizers (PSO, GWO, ALO) for parameter tuning; testing forecasting performance with expanding window methods; and applying the model to forecast renewable investments and sectoral consumption, especially under post-COVID structural changes.

• Aggregation: RES is treated as a total category; disaggregated dynamics of individual technologies are not modeled and may differ. • Data horizon and small samples: Forecasts rely on Eurostat annual data from 2004–2018; grey models are designed for small samples, but limited history may constrain capturing structural breaks. • Parameter optimization: GA settings (stopping criterion at 0.01% MAPE change) and local optima risks may influence results; alternative algorithms could yield different parameter sets. • Model assumptions: The fractional accumulation and Bernoulli nonlinearity assume specific functional forms that may not capture all exogenous shocks or policy changes. • External shocks: COVID-19 and subsequent policy/investment shifts may alter energy consumption trajectories beyond historical patterns, potentially affecting forecast validity.