Probabilistic forecasting of transition towards sustainable energy production

It is observed that cumulative production of electricity by renewable technologies increases exponentially with time, while the unit cost exponentially decreases. However, the exponential growth of annual production even by a sustainable technology will eventually flatten due to saturation of the global market. This should result in a linear growth of cumulative production with a passage of time. To capture this qualitative picture, we introduce the novel model of cumulative production by a sustainable technology with negative feedback in the different form than in the traditional logistic model. To account for irregularity of technological progress, we utilize the method of cumulant expansion that allows to make probabilistic forecasts for the unit cost, which do not depend on the specific model of underlying stochastic processes and a scenario of the future production growth. The derived results demonstrate that during the period of initial exponential growth of production, the effective rate of decrease of the unit cost is slightly lower than its average value. The respective confidence interval is growing linearly with the square root of the forecasting horizon. In the long run, the model of sustainable production forecasts the power law decline of the expected unit cost with time. In this regime, the forecasting error of the logarithm of the unit cost is slowly growing as the square root of the logarithm of the forecasting horizon. To illustrate the method, we make probabilistic forecasts for prices of solar photovoltaic modules up to 2060.