# Bayesian Estimation of the Photovoltaic Balance-of-System Learning Curve.

JEL C11 * D00 * L10 * 040Photovoltaics (PV) is a renewable energy source technology that generates electricity. The time pattern of PV costs is crucial to understanding competitiveness in the future and to drafting more effective climate change policies (Nachtigall and Riibbelke, Resource and Energy Economics, 2016). The learning curve (LC), representing the inverse relationship between average cost and cumulative production, provides insight regarding future solar industry development and is often utilized to predict how the costs of a product or process may evolve based on historical trends (Schmidt et al., Nature energy, 2017). To this end, a Bayesian model was developed to estimate the LC for PV technology for more than 20 countries from 1983 to 2015 given the average cost and level of installed cumulative capacity (https://www.kapsarc.org/openkapsarc/kapsarc-solar-photovoltaic-toolkit). The Bayesian approach accounts for a joint stochastic distribution of parameters and concurrently allows for heterogeneity of the sample across countries. The novelty of this approach is the evaluation of the cost structure of PV systems using capital expenditures (CAPEX) which are the main component of the levelized cost (the present net unit-cost of electricity over the lifetime of the renewable generating technologies). The CAPEX were disaggregated into two main elements: the module, converting sunlight to electricity, and the balance of system (BOS), an all-encompassing term referring to all other components and services needed to make the PV system functional including mounts, cables, labor permitting, grid connection, and inverters. After almost four decades of technological advancements, the module cost has significantly shrunk and the BOS now accounts for more than half of the CAPEX (Basore, Progress in Photovoltaic, 2016). However, detailed studies of the global time evolution of the BOS are still scarce. The LC was estimated for the BOS in Germany at about 89% for 19902013 (Strupeit and Neij, Renewable and Sustainable Energy Reviews, 2017). A recent study (Elshurafa et al., Journal of Cleaner Production, 2018) derived the applicable country-specific LC applying for each cross-section ordinary least squares (OLS) and seemingly unrelated regression (SUR) method using an unbalanced panel database. Nevertheless, treating time series as separate cross-sections involved the assumption of complete heterogeneity of the LC among countries. Conversely, in this study the data were treated as an unbalanced panel data, allowing for the presence of a common temporal dynamic among cross-sections affecting each country's pattern.

Starting from the logarithmic form of LC: log([C.sub.Q]) - log ([C.sub.1]) + [beta]log(Q), where Q is the cumulative installed capacity of PV, [C.sub.1] and [C.sub.Q] are the associated BOS needed to produce the first and the 0th Megawatt of solar PV, respectively. Stationarity was checked using the Fisher test for unbalanced panel data. Tests rejected the unit root null hypothesis for all series.

In a preliminary step, panel regression was applied to derive coefficient estimates used as prior parameters of the joint stochastic distribution in the second step. The learning coefficient [beta] was estimated using the dynamic fixed effects (DFE) model: [[??].sub.FE]. Heteroskedasticity or within-panel serial correlation problems were overcome using the robust Hubert estimator for the variance. Results confirmed previous findings in the literature: [[??].sub.FE] is significant ([alpha] = 0.05) and around -0.19. This value involves a LC = [2.sup.-[beta]] (Trappey et al., Journal of Cleaner Production, 2016) equal to 0.88. When the cumulative PV production doubles, the amount by which costs fall is the progress ratio (1 - [2.sup.-[beta]] = 0.12). The variance [[??].sup.2.sub.v] of the fixed-effect parameter is 0.41, while [[??].sup.2.sub.[epsilon]] is 0.64. The DFE model allows the intercept (unobserved effect) to differ among cross-sections in order to account for country-specific characteristics, but it assumes complete homogeneity of the slope parameters.

To introduce some degree of heterogeneity in the slope parameters, in the second step a Bayes dynamic estimator was applied assuming that the slopes are random variables drawn from a probability distribution with prior mean and variance equal to [[??].sub.FE] and Var([[??].sub.FE]), respectively. In the panel data models with coefficient heterogeneity, Bayes estimators are superior to homogeneous and other heterogeneous estimators as far as predictive ability is concerned (Baltagi et al., Empirical Economics, 2003). Another advantage of the Bayesian framework is that, although coefficients can differ across countries, the number of parameters to be estimated is reduced, thanks to the random coefficient specification. Moreover, the Bayesian framework permits preservation of some degree of similarity because the slope parameters are drawn from the same distribution, though they can differ concurrently among countries. The empirical result was that the posterior distribution of [beta] converged to a normal distribution with a mean of-0.22 and standard deviation of 0.015. The Bayesian estimate [??] was higher than the DFE estimate in absolute value, implying a faster learning process (85%) and a progress ratio of 0.15.

Understanding the rate at which capital costs of PV decline will improve the effectiveness of renewable energy policy planning and budgeting. In this model, the prior mean and variance of [beta] are known. Further advancements on this study should include Bayesian hierarchical models with the prior parameters set as unknown. The posterior simulated distribution of [[??].sub.i] should be derived for each cross-section.

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Maria Chiara D'Errico (1) [iD]

Published online: 9 March 2019

[mail] Maria Chiara D'Errico

mariachiara.derrico@unipg.it

(1) Department of Economics, University of Perugia, Via Pascoli 20, 06123 Perugia, Italy

https://doi.org/10.1007/s11293-019-09608-7

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Title Annotation: | ANTHOLOGY |
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Author: | D'Errico, Maria Chiara |

Publication: | Atlantic Economic Journal |

Article Type: | Report |

Date: | Mar 1, 2019 |

Words: | 929 |

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