** **
25 Jul 2018

25 Jul 2018

# Empirical growth models for the renewable energy sector

Kristoffer Rypdal

**Kristoffer Rypdal**Kristoffer Rypdal

- Department of Mathematics and Statistics, UiT The Arctic University of Norway, 9037 Tromsø, Norway

- Department of Mathematics and Statistics, UiT The Arctic University of Norway, 9037 Tromsø, Norway

**Correspondence**: Kristoffer Rypdal (kristoffer.rypdal@uit.no)

**Correspondence**: Kristoffer Rypdal (kristoffer.rypdal@uit.no)

Received: 29 May 2018 – Revised: 19 Jul 2018 – Accepted: 20 Jul 2018 – Published: 25 Jul 2018

Three simple, empirical models for growth of power consumption in the
renewable energy sector are compared. These are the exponential, logistic,
and power-law models. The exponential model describes growth at a fixed
relative growth rate, the logistic model saturates at a fixed limit, while
the power-law model describes slowing, but unlimited, growth. The model
parameters are determined by regression to historical global data for solar
and wind power consumption, and model projections are compared to scenarios
based on macroeconomic modelling that meet the 2^{∘} target. It is
demonstrated that rational rejection of an exponential growth model in favour
of a logistic growth model cannot be made from existing data for the
historical evolution of global renewable power consumption *y*(*t*). It is also
shown that the logistic model yields saturation of growth at unrealistic low
levels. The power-law growth model is found to give very good fits to the
data through the last decade, and the projections align very well with the
scenarios. Power-law growth is equivalent to the simple law that the relative
growth rate ${y}^{\prime}/y$ decays inversely proportional to time. It is shown that
this is a natural model for growth that slows down due to various
constraints, yet not experiencing the effect of a strict upper limit defined
by physical boundaries. If the actual consumption follows the power-law curve
in the years to come the exponential-growth null hypothesis can be correctly
rejected around 2020.

It is widely recognised that economic growth in most sectors finally will
have to come to an end due to the constraints imposed by planetary boundaries
and that we need a new paradigm in Earth System science that integrates the
physical, biological, economic, social and cultural forces
(Donges et al., 2017). This idea has been developed for instance in the numerous
reports to the Club of Rome, starting with the seminal book “The Limits to
Growth” in 1972 (Meadows et al., 1972), which contains interesting reflections on the
Earth system limits to exponential growth. Later reports like the forty-year
follow up, “2052: A Global Forecast for the next Forty Years”
(Randers, 2012), draw rather pessimistic pictures of our energy future.
Somewhat more optimistic are the 450 scenario of the International Energy
Agency (IEA, 2016) and the REmap scenario of the International Renewable
Energy Agency (IRENA, 2018b). These scenarios represent emission pathways
that give a fair chance of limiting global warming to 2 ^{∘}C relative
to preindustrial global mean surface temperatures.

Energy production and distribution is the sector on which everything else
depends, and despite steady advances in energy efficiency, the growth of the
world economy relies on continuing growth of energy consumption. Without a
massive deployment of carbon capture and storage (CCS) and other negative
emission technologies, the target of global warming below 2 ^{∘}C from
preindustrial temperatures requires radical reduction of coal in electricity
production over next decades (Hansen et al., 2017a; IPCC, 2014). At present
there is doubt about the technical and economic feasibility of capturing and
storing 4 Gt CO_{2} annually by 2040. This is more than 10 % of
the emissions from fossil fuels and industry, and will require thousands of
large-scale CCS plants (Global CCS Institute, 2015; Le Quéré et al., 2016). The known reserves of
conventional oil and natural gas will set strict limitations to the growth in
the consumption of these fuels, although fracking technologies will extend
their time window somewhat (IPCC, 2014). In theory, large scale
implementation of fourth generation nuclear power with a total reformation of
the nuclear industry and the national and international regulatory systems,
could buy some time (Makabe, 2017), but the political feasibility of such a
project is highly questionable. Most integrated assessment models (IAMs) used
in IPCC (2014) include optimistic assumptions on implementation of negative
emission technology, but still the majority of such models conclude that a
future growth rate of 1–2 % of world gross domestic product constrained
by the 2^{∘} global temperature target will require that solar and wind
power production and consumption continues to grow at present rates without
fundamental constraints (Nordhaus, 2013). A fourth option,
geoengineering, has not been seriously implemented in the IAMs yet, since
these are the least matured set of technologies, and associated with profound
ethical issues. Thus, the rather depressing state of affairs is that the
prospect of meeting the IPCC temperature targets rests on the economic
feasibility of accelerating growth of world-wide, large-scale deployment of
at least one of four classes of technologies; CCS, 4th generation nuclear,
geoengineering, or renewables, and it is by no means obvious that any of them
can meet the world's demand for clean, safe, and affordable energy.

The present paper has focus on the possible constraints on the growth of the intermittent power sources; solar and wind. Consumption of hydropower and traditional bioenergy are considerably larger at present, but their growth potential is almost exhausted. For hydro this is true in the developed world, while some developing countries still have large unexploited resources. The installed capacity of hydro has doubled over the last thirty years, and the growth looks more linear than exponential. In contrast, solar and wind have been growing exponentially with a doubling time of 2 years for solar and three years for wind (IRENA, 2018b; WEC, 2016).

The continued exponential growth is compared to the IEA 450 scenario and the
IRENA REmap scenario in Fig. 1a. The figure shows projected
exponential growth of consumption of solar and wind power obtained by fitting
a linear model to the logarithm of the consumption time series for the period
1997–2016. The red curve is solar power, the blue is wind power. The
relative growth rate is ${y}^{\prime}/y=\mathrm{0.35}$ for solar and ${y}^{\prime}/y=\mathrm{0.23}$ and for wind,
which lets solar power overtake wind as the leading technology before 2030.
The red points to the right in the figure represent the total solar
photovoltaic (PV) power consumption in 2025 and 2040 according to IEA's 450
scenario and the REmap projection of solar PV in 2050. The blue points to the
right show the same data for wind power. The figure illustrates that with
continued exponential growth solar and wind together can deliver enough to
fulfil the 2^{∘} target demands for 2050 more than twenty years ahead of
time. This observation does not give us a great reason for optimism, though.
It rather suggests that the era of exponential growth will soon be over.

Forthcoming sections will discuss two alternative growth models, one is the
logistic model which describes initially exponential growth that saturates
and finally stabilises at a fixed limit. The other is a power-law model,
which does not saturate, but yields slower that exponential growth that
provides a description more in line with the 2^{∘} scenarios. Results of
power-law fit to the historical global consumption data are shown in
Fig. 1b. The fit to the wind consumption curve matches perfectly
the wind consumption projections in the 450 scenario for 2025 and 2040, and
the REmap projection for 2050. The solar consumption fit curve overshoots
those 2^{∘} target projections, but by less than a factor 2, and shows
the same tendency of decreasing relative growth rate. The details of the
models and the fitting procedures are presented in Sects. 2
and 3.

Solar and wind represent proven technologies of a certain maturity, but their intermittency is an obstacle that is held by some to be a fundamental constraint to further growth. These and other constraints have been discussed in many recent papers, e.g., Moriarty and Honnery (2011), Dale et al. (2011), Hall et al. (2014), and Davidsson et al. (2014). These outline a large number of restraining factors that that may slow, and possibly halt growth of renewable energies, whose low energy return on investment may negatively impact general economic growth. However, the majority of these papers do not present balanced treatments of impeding and accelerating factors, and do not make quantitative, integrated assessments of all these in a setting where energy markets develop in a world with effective implementation of climate change policies, including global pricing of carbon emissions.

In stark contrast to these papers are the most recent reports of the globally
levelised cost of energy (LCOE) for wind and solar photovoltaic power.
According to IRENA (2018a) the LCOE for these technologies are already in the
lower section of the range of 0.05–0.17 USD kWh^{−1} for fossil fuel
generation power in the G20 countries, and predicts that by 2020 all
renewable technologies now in commercial use will fall in the fossil fuel
fired cost range, with most in the lower end. Cost reduction drivers are
technology improvements, competitive procurement and a large base of
experienced, internationally active project developers. So far, scarcity of
natural resources, available land, or other planetary constraints do not seem
to play any significant limiting role in the development. Another interesting
observation made by IRENA (2018a) is that the total investment level in these
technologies has not increased significantly over the last decade, which
indicates that the expected redirection of investments from fossil to
renewables has not yet started. When this transformation gains momentum a new
impetus for increased growth will enter the stage, but it is difficult to
predict its impact on the growth rate.

This landscape of huge uncertainty in projections for the market of renewables, and the complexity of modeling them, have stimulated search for signs of stagnating growth in historical data for deployment of the fastest growing renewable energy technologies. Notably, Hansen et al. (2017b) attempt to make a model selection between exponential and logistic growth of wind and solar power based on standard curve fitting to historical data. The logistic growth curve has the form of a sigmoid, where the initial exponential growth converges to a maximum value due to a nonlinear saturation mechanism. They conclude that the logistic curve generally yields “better fit”, and that there is a statistically significant decline in the relative growth rate, signifying slower-than-exponential growth. They suggest that the fitted logistic curves indicate a stagnating optimum level of installed wind- and solar capacity not much higher than twice today's capacity, which effectively would remove solar and wind power from the list of potentially “life-saving” technologies. The harsh implications of these projections make it worthwhile to examine their substance in some detail, and to explore whether conclusions of this nature can be drawn from historical data via application of more rigorous methodologies.

The remainder of the paper is structured as follows. In Sect. 2 the inadequacy of usual least mean square fitting for models with multiplicative noise is explained and illustrated by an analysis of the growth curve for global consumption of wind power. The stochastic equations for exponential and logistic growth with multiplicative noise are then formulated, and an alternative least mean square fitting method, where the logarithms of these models are fitted to the logarithm of the data, is shown to be one that responds to the entire time series, not only to the greater values at its end. The section is concluded by formulating a test designed to reject the exponential-growth null hypothesis, and this test is applied to the wind data. According to this test, the exponential growth model is not rejected by these data. In Sect. 3 the results of fitting the exponential and the logistic models to the solar and wind power data are presented, and the slower-than-exponential power-law growth model is also explored and shown to yield good fits. Section 4 discusses the possible advantages of simple, empirical models over complex dynamical models in this particular context, and Sect. 5 summarises the main results.

Standard curve fitting is an example of regression where one estimates the
parameters (regression coefficients) ** α** of a statistical model of
the form;

where *t* is the *predictor variable* (in our case; time), *ϵ*
represents the “random” or “unexplained” part of the *response variable*
*y* (e.g., installed capacity), and *f*(*t*;** α**) is some specified
function. Suppose we have

*n*observations

*{*(

*t*

_{i},

*y*

_{i})

*}*, $i=\mathrm{1},\mathrm{\dots},n$ of the predictors and the response variable, then

*regression*means to find the regression coefficients

**such that the set of residuals $\mathit{\{}{r}_{i}\equiv {y}_{i}-f({t}_{i};\mathit{\alpha}\left)\mathit{\right\}}$, $i=\mathrm{1},\mathrm{\dots},n$ is minimised in a metric (norm) to be specified. A commonly used metric is the**

*α**least square*objective function

## 2.1 Multiplicative noise and fitting to log-data

Minimising the least-square deviation to yield the best estimate $\mathit{\alpha}=\widehat{\mathit{\alpha}}$ often leads to the best visual fit of the curve
(graph) of $f(t;\widehat{\mathit{\alpha}})$ to the data, but for data where the
fluctuation level $\mathrm{\Delta}{y}_{i}=|{y}_{i}-{y}_{i-\mathrm{1}}|$ is proportional to *y*_{i}
(multiplicative noise), this metric will not provide the best *model* for
the growth, since the estimated model parameters will be very sensitive to
the random fluctuations of the larger data points in the time series (this
sensitivity will be demonstrated when models are fitted to consumption data
in Figs. 3 and 4). A more relevant quantity to
minimise is the mean square of *z*−ln*f*, where *z*=ln*y*, since the
fluctuations $\mathrm{d}z=\mathrm{d}y/y$ will have magnitudes that no longer are proportional
to *y* (additive noise). The model to fit is then ln*f*(*t*,** α**);
for the exponential model this reduces to fitting a straight line to the
log-data, and for the logistic function fit, it corresponds to fitting a
function which has the slope of the initial relative growth rate for

*t*≪

*t*

_{s}and a zero slope for

*t*≫

*t*

_{s}, where

*t*

_{s}is the inflection point of the logistic growth curve. The logistic model and exact meaning of

*t*

_{s}is explained further in Sect. 2.3.

## 2.2 Exponential growth and the Black-Scholes stochastic equation

The rationale for operating on the logarithm ln*y* rather than on *y* can
be seen from the canonical Black-Scholes (BS) stochastic differential
equation (SDE) for asset prices, which is a general description of any
continuous-time variable stochastic process *y*(*t*) that grows at a rate *μ**y*(*t*) and is subject to random increments *y*(*t*) *σ* d*B*(*t*)
(McCauley, 2004). Since the growth rate is proportional to the asset price
*y*(*t*) this term contributes to exponential growth of *y*(*t*), while the
stochastic term gives rise to price fluctuations. Since the magnitude of the
fluctuations are proportional to *y*(*t*) this is an example of *multiplicative noise*. The multiplicative noise in *y* reduces to *additive noise* in *z*=ln*y*. The equation takes the form,

where *B*(*t*) is the Wiener process (also called Brownian motion), *μ*
represents the general economic growth rate and *σ* measures the
strength of the price fluctuations. The essential properties of the Wiener
process is that the increments d*B*(*t*) are identical and independently
distributed (i.i.d.), from which it also follows that the distribution is
Gaussian. With discrete time steps, e.g., a time series of annual data, the
Brownian motion reduces to a Gaussian random walk process. The equation for
the logarithm *z*=ln*y* is,

The non-intuitive term $-{\mathit{\sigma}}^{\mathrm{2}}/\mathrm{2}$ in the drift coefficient in
Eq. (4) arises because the equation is an SDE for the
stochastic process *y*(*t*). For a change of variable like $z=f\left(z\right)=\mathrm{ln}y$, we
have Itô's first lemma, which states that if *y*(*t*) satisfies
Eq. (3), and *f*(*y*) is a twice differentiable function,
then the stochastic process *z*=*f*(*y*) satisfies the SDE $\mathrm{d}z=\left[\mathit{\mu}{f}^{\prime}\right(y)+(\mathrm{1}/\mathrm{2}\left){f}^{\prime \prime}\right(y\left){\mathit{\sigma}}^{\mathrm{2}}{y}^{\mathrm{2}}\right]\phantom{\rule{0.125em}{0ex}}\mathrm{d}t+\mathit{\sigma}\phantom{\rule{0.125em}{0ex}}\mathrm{d}B\left(t\right)$. For *f*(*y*)=ln*y* this
equation reduces to Eq. (4). It implies that the stochastic
forcing gives rise to an additional drift. The solution to
Eq. (4) is a geometric Brownian motion (gBm);

The deterministic factor $\mathrm{exp}\left[\right(\mathit{\mu}-{\mathit{\sigma}}^{\mathrm{2}}/\mathrm{2}\left)t\right]$ grows exponentially and
the probability density function (PDF) of this stochastic process is skewed
and log-normal. The expected value of this distribution grows linearly as
𝔼[*y*]=*y*_{0}exp[*μ**t*] and the variance as
$\mathrm{Var}\left[y\right]={y}_{\mathrm{0}}^{\mathrm{2}}\mathrm{exp}\left[\mathrm{2}\mathit{\mu}t\right]\left(\mathrm{exp}\right[{\mathit{\sigma}}^{\mathrm{2}}t]-\mathrm{1})$. This variance
represents the statistical uncertainty associated with market fluctuations in
an exponentially expanding economy.

## 2.3 A stochastic equation for logistic growth

The debate over the growth of consumption of renewable energy is concerned
with whether the deterministic factor should be replaced by a function that
exhibits limited growth, such as a logistic function. In making this
assessment, however, one has to take into consideration the nature of the
sources of statistical uncertainty, which for exponential growth is
represented by the multiplicative noise factor exp[*σ**B*(*t*)]. While
standard curve fitting is based on the assumption that the statistical error
is additive random (white) noise, the actual market fluctuations is more
accurately represented as a multiplicative, autocorrelated process which is
the exponential of the Wiener process. Realisations of these processes are
shown in Fig. 2a. The blue curve is a discrete-time Gaussian
white noise, the yellow is its cumulative sum, which is a random walk, or a
discrete-time sampling of the Wiener process, and the green curve is the gBm
given by Eq. (5). Standard curve fitting assumes that the
deviation from the exponential growth is a Gaussian white noise.
Black-Scholes theory assumes that the exponential growth signal is multiplied
by the non-drifting geometric Brownian motion exp[*σ**B*(*t*)].

We can generalise the Black-Scholes equation to a stochastic logistic growth model (SLGM) (Capocelli and Ricciardi, 1974);

Without the stochastic forcing term, the solution to this equation is the logistic function

which has the shape of a sigmoid. Here, *μ* is the initial exponential
growth rate, ${y}_{m}={lim}_{t\to \mathrm{\infty}}y\left(t\right)$ is the asymptotic limit
to the growth, and *t*_{s} is is the time where
d*y*_{L}∕d*t* has its maximum value, which is also the
time at which $y\left({t}_{\mathrm{s}}\right)={y}_{m}/\mathrm{2}$. Hence, *t*_{s} is a
characteristic time for saturation of the logistic growth. Here
*t*_{s} is related to the initial value *y*(0) through the relation
$y\left(\mathrm{0}\right)={y}_{m}/[\mathrm{1}+\mathrm{exp}(\mathit{\mu}{t}_{\mathrm{s}}\left)\right]$.

## 2.4 The nature of the noise

From the analysis in the upcoming Sect. 3 (in particular
Figs. 3 and 4), we will observe that the residuals
obtained from subtracting *z*(*t*) from the log-data time series vary
relatively smoothly from one year to the next, but sampled on five years
intervals they may be consistent with a random walk. For instance, it will be
shown that the autocorrelation time is about four years for the residual wind
time series. The smooth appearance of the log-residuals on annual scales has
important implications for the statistical significance of the downward trend
of the relative growth rate claimed by Hansen et al. (2017b). The relative growth
rate is defined as the slope of the log-data curve, ${y}^{\prime}/y={z}^{\prime}$, and is
constant in time for exponential growth. Hansen et al. (2017b) make a linear
regression to the differences $\mathrm{\Delta}{z}_{i}={z}_{i}-{z}_{i-\mathrm{1}}$, $n=\mathrm{1},\mathrm{\dots},\mathrm{19}$ and
estimate a negative slope of this trend line which is claimed to be
significantly different from zero. Such significance estimates, however, are
only valid if the noise in Δ*z*_{i} on annual scale is a Gaussian i.i.d.
process. The statistical significance of the negative slope depends
critically on the number of independent data points, and if the fluctuations
in Δ*z*_{i} are independent only on time scales longer than four years,
there are effectively not more than five such points in the data record, and
this is clearly not enough to detect a significant trend in these data.

## 2.5 Testing the alternative logistic growth hypothesis

The problem we deal with in this paper is to search for evidence for saturation of exponential growth in time series data that to the first order are well described by an exponential function. More precisely, we try to find criteria by which we can reject the BS model (exponential growth) in favour of the SLGM (saturated growth). Hence, in this case it is natural to treat the BS model as the null hypothesis, and the SLGM as the alternative hypothesis. Here, we shall make the test for the wind-power data, since it will be shown in Sect. 3 that the SLGM can be rejected on other grounds for solar power data.

The first step is to perform a least-square fit of the logarithms of the
exponential and logistic models to the logarithm of the data.^{1} The exponential fit appears as a straight line in the
log-log plot, and the sigmoid logistic curve starts out as a straight line
with slope *μ*, gradually bending over to a straight line with zero slope
as the growth saturates. The exponential model contains two model parameters
*y*_{0} and *μ*, while the logistic model contains three; *y*_{m}, *μ*, and
*t*_{s}, and we note that the exponential model is a special case of
he logistic since the latter reduces to the exponential in the limit
$\mathit{\mu}({t}_{\mathrm{s}}-t)\gg \mathrm{1}$. Hence, the logistic model should provide a better
fit than the exponential to *any* data set in terms of the standard
deviation of the residual $\sqrt{{Q}_{\mathrm{2}}}$ given by Eq. (2).

The parameter estimation described above is associated with statistical
uncertainty, which will be provided by standard fitting routines as the
“standard error” of the estimation. This is presumably what is done to
obtain the confidence intervals on the logistic fit in Fig. 3 of
Hansen et al. (2017b). These error estimates are based on the assumption that the
observed data can be modelled by Eq. (1), where
*f*(*t*;** α**) is either the exponential or the logistic function,

**the corresponding model parameters, and**

*α**ϵ*(

*t*) a Gaussian white noise process. As explained in Sect. 2.2, it is generally recognised, however, that variables describing the volume of an expanding market is much more adequately described by models of the type Eqs. (3) or (6). This means that the estimation of the model errors (the uncertainty in the model coefficients) must be based on those stochastic models.

Since the question is whether or not we can reject the BS model (the null
hypothesis) from the data we explore the implications of that model. It is
important to keep in mind that when testing a null hypothesis one has to
assume that all assumptions of that hypothesis are true when we explore the
implications of the hypothesis, even if one does not believe that they are
true. For instance, according to Eq. (3) the residual
obtained by subtracting the fitted *z*(*t*) from the log-data should be
modelled as a Wiener process. To make sure that the downward curving
indicated by the three last data points does not contribute to the estimate
of the mean and variance of this Wiener process, these points are dropped
before estimating the process parameters. After estimating the mean and
variance of this process from the residual data,^{2} one can produce an ensemble of numerical
realisations of the processes described by Eq. (3). Such an
ensemble of realisations is shown for wind power as the cloud of thin wiggly
curves in Fig. 2b. The extent of this cloud indicates the
uncertainty of the realisations of the fitted model. We could have used this
cloud to compute 95 % confidence intervals, but this cloud of 100 realisations actually illustrates better that the observed time series (the
black dots) is a possible realisation of the BS model and hence consistent
with exponential growth. The blue curve is the fit of the logistic model, and
is a clearly better fit in terms of *Q*_{2}, but that is obvious due to the
additional freedom arising from an additional model parameter.

## 3.1 Exponential versus logistic growth

The blue and red curves in Fig. 3a show the exponential and
logistic least square fits to the time series of global solar power
consumption for the period 1997–2015, and the green dotted curves show the
effect of including the data point for 2016 (the green dot). It illustrates
that by fitting an exponential model to the exponentially growing consumption
data, the result becomes very sensitive to the last data points, since these
are so large compared to the early part of the time record. However, when we
attempt to fit the logarithm of the logistic model to the logarithm of these
data it turns out that the best fit (least *Q*_{2}) is obtained in the
exponential limit ${P}_{max}\to \mathrm{\infty}$, *t*_{p}→∞. In order to check this result, the fitting routine has been applied
with fixed ${P}_{max}=\mathrm{550}$, 10^{3}, 10^{4}, 10^{5}, 10^{6}. The results are
shown in Fig. 3b as the blue, gray, yellow, green, and red
curves, and the corresponding values of the least square deviation are
*Q*_{2}=0.1608, 0.1411, 0.1230, 0.1214, 0.1212. This means that, even though
the logistic model has one more degree of freedom, the exponential model
yields a better fit, and it is not possible to estimate a growth limit by
fitting the logistic model to the data. In Sect. 3.2 we shall
consider another growth model that makes more sense for these and other data.

A similar analysis has been done to wind power consumption data in
Fig. 4a and b. For these data, however, a well defined minimum
for *Q*_{2} is found for the logarithmic fit and is shown by the blue curve in
Fig. 4b. On top of this curve, but barely visible, is a green,
dashed curve representing the fit with the last green point of 2016 included.
It serves to demonstrate how the sensitivity to the last points in the time
series is reduced by fitting logarithms rather than the raw consumption data.
Fig. 4c shows *Q*_{2} versus *P*_{max}, when fitting is made with
fixed *P*_{max}. It shows a well defined minimum, which corresponds to the
blue curve in Fig. 4b. In Fig. 4d is plotted the
autocorrelation function (ACF) for the residual deviation between the data
and the exponential fit. It shows an autocorrelation time of around four
years, hence the residual for the entire data series contain no more than
five data points that can be considered independent.

## 3.2 Power-law growth – a middle ground

It is clear from Fig. 1a that continuing exponential growth for
solar and wind power beyond 2030 implies volumes that seem almost
unthinkable. Hence, it is not unreasonable to consider the possibility that
the declining relative growth rate ${z}^{\prime}={y}^{\prime}/y$ observed for wind power in
Fig. 2b is the manifestation of a declining trend in this growth
rate. What is demonstrated in that figure is not that this decline is not
real, but that it is not statistically significant for the time series data
at hand. But even if we assume that this decline is real, the correct model
does not have to be the logistic one. The logistic model requires complete
saturation of the growth for times well beyond *t*_{p}, while some regional
data, for instance wind consumption in Europe, seem to display non-saturating
growth slower than exponential. Actually a third-order polynomial can be very
accurately fitted to those data. By examining many regional data sets in
logarithmic plots there seems to be a curve which is linear (exponential
growth), or even curving upwards, up to a certain year, and then a
logarithmic-like curve after this year. This is in fact also what we observe
in the global solar consumption data.

Suppose we identify a year after which the logarithm of the consumption time record exhibits such a logarithmic growth, and let us drop all data prior to that year. A model that captures this logarithmic behaviour has the form

since it implies that $\mathrm{ln}y\left(t\right)=\mathrm{ln}p+q\mathrm{ln}(t+{t}_{\mathrm{0}})$. Here the origin of the
time axis is chosen at the first year of the new shorter time series and
*t*_{0} then is a positive number. The power-law model is a solution of the
differential equation

which is just the equation for exponential growth with a time-dependent relative growth rate $\mathit{\gamma}\left(t\right)=q/(t+{t}_{\mathrm{0}})$. By adding a multiplicative noise term on the right hand side, Eq. (9) can easily be cast into the form of a generalised BS equation.

When an exponential model was fitted to the data, the idea was to make use of the entire time series and give equal weight to the early and late stages of the growth. This is why it was appropriate to fit the linear model to the logarithm of the data. The power-law model, however, is expected to describe only the latest decade of the historical growth, and the advantage of fitting the logarithm of the power-law model to the logarithm of the data is not so obvious. The two methods give similar results, and those presented in Fig. 1b are for the power-law model fitted to the consumption data without taking logarithms.

For the global solar power consumption, the time series prior to 2007 does
not fit to the model and must be discarded. The fit for the period 2008–2016
is shown as the red curve in Fig. 1b. For global wind power
consumption, the entire time series can be used meaningfully, but the last
decade is expected to contain more relevant information about the
time-dependent growth rate. The latter yields a slightly lower *q* and the
fit is shown as the blue curve in Fig. 1b. The residual variances
*Q*_{2} are considerably lower than the corresponding for the fitted
exponential model (the fit is much better), but this is expected since the
number of model parameters is three for the power law model and two for the
exponential model.

The estimated growth exponents are *q*≈2.7 for solar power and
*q*≈2.0 for wind power. This implies that the solar power consumption
overtakes wind power around 2034, a few years later than predicted by the
exponential growth model. According to the REmap scenario this would happen
just after mid-century.

For the power-law model the underlying assumption is that it is not a good model up to certain date, where constraining factors start to kick in. That date can only be established by examination of the data. For wind power the result is quite insensitive to the choice of this date, but the fit is best if I choose it as late as 2007. For solar power the growth is faster than exponential before 2007, and the best fit is found if 2008 is chosen as a start date. That gives 10 data points to fit for wind and 9 data points for solar. Choosing a later start date has negligible effect on the fitted curves. Hence the results of the fitting procedure is quite robust as long as the start date is chosen after the saturation of the exponential growth has become visible in the data, i.e. after the slope of the logarithmic plot has started to decrease.

It should be emphasised that this is not cherry-picking, but construction of a model that has a limited range of validity and based on the available data. It is also important to keep in mind that there is also an upper time limit for the validity of the power-law model, since it exhibits unlimited growth as t goes to infinity. In this paper I have assumed that it holds at least up to 2050, but sooner or later the actual growth will have to stall due to planetary boundaries.

## 3.3 Why power-law growth?

In this subsection it is shown that power-law growth plays a particular role
in the class of growth models with relative growth rates *γ*(*t*) that
decay towards zero as *t*→∞. A general growth model for the
variable *y*(*t*) is one where we have a time-dependent relative growth rate
*γ*(*t*), i.e. we have the differential equation

with the solution

Note that the growth saturates to a finite value only if the integral
${\int}_{\mathrm{0}}^{\mathrm{\infty}}\mathit{\gamma}\left({t}^{\prime}\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}{t}^{\prime}$ is finite. Let us now consider a wider
class of relative growth rates than considered in Section 3.2,
namely those that decay algebraically towards zero as *t*→∞
with an arbitrary positive exponent *μ*;

The case *μ*=1 yields the power-law growth

which was treated in Sect. 3.2. For *μ*≠1 the general
solution is

Hence, for *μ*<1 we have unlimited growth which for *t*→∞
has the asymptotic form

which means faster-than-exponential growth for *μ*<0, exponential growth
for *μ*=0, and slower-than-exponential growth for $\mathrm{0}<\mathit{\mu}<\mathrm{1}$. For *μ*>1 it
is convenient to rewrite Eq. (14) in the form

The growth in this case is limited, and the solution increases monotonically towards the limit

The power-law growth for *μ*=1 is thus neatly poised between the
slower-than-exponential growth of Eq. (15) for $\mathit{\mu}\to {\mathrm{1}}^{-}$
and the limited growth of Eq. (16) for $\mathit{\mu}\to {\mathrm{1}}^{+}$. It is
thus a natural model for growth that slows down with time, but for which the
growth is not yet determined by a definite growth limit set by physical
boundaries.

The renewable energy sector is an integrated part of the global economy that grows in importance at the expense of fossil-fuel based energy production and consumption. Its growth is essentially governed by the same laws that govern other sectors of the economy. Sectors grow at different rates, depending on complex market mechanisms, economic policy strategies by governments and international bodies, and in some cases by boundaries set by the availability of natural resources and land. The effect of scarcity of resources is definitely felt in the fossil fuel sector. However, the decline, and eventually negative sign, of growth in this sector will not be determined by the physical limit of exploitable resources. The decline will be determined by the competition with the non-fossil energy sectors, by the pace of technological advances in those sectors, and by how well the international community will succeed on implementing carbon pricing reflecting the actual social cost of carbon.

The two IEA and IRENA scenarios used in this paper are the results of
macroeconomic modeling based on assumptions of rapid technological progress
in the renewable energy sector, of successful implementation of climate
mitigation policies, and absence of definite limits set by scarcity of
resources and land. Neither of these assumptions can be proven at present.
For instance, the physical availability of unexploited land up to 2050 is
unquestionable, but there is already strong public opposition against vast
wind farms and solar power installations in many countries. It is impossible
to predict or model the outcome of the political battles over such issues,
hence any attempt to justify a simple dynamical model by suggesting specific
constraining mechanisms will appear as arbitrary and ad hoc. This is why this
paper has a focus on the construction of simple *empirical* rather than
dynamical models.

Thus, in this paper, no attempts have been made to derive the three
statistical models under consideration from physical and/or economic laws.
That does not mean that they cannot be interpreted in the light of such laws.
The exponential growth law that underpins the BS-equation is based on the
assumption that the capital available to expand production at a given time is
proportional to the production volume at this time. This implies that the
relative growth rate $\mathit{\gamma}\left(t\right)={y}^{\prime}/y$ is independent of time. Since the
logistic growth law assumes that $\mathit{\gamma}\left(t\right)=\mathit{\mu}[\mathrm{1}-y(t)/{y}_{m}]$, it implies that
the relative growth rate goes to zero as the volume *y* reaches a certain
limit *y*_{m}. This typically models a situation where the growth rate depends
on a resource that is depleted as the volume grows. For solar and wind power
this could be the availability of suitable land areas, crucial raw materials,
or investment capital.

On the other hand, for renewable power there is no good reason why the growth
rate should go to zero at a particular volume of production, and the
macroeconomic modelling underlying the scenarios of IEA and IRENA does not
support that such a limit will be attained during the first half of this
century. The power-law model was found to give very good fits to the data
throughout the last decade. One cannot conclude that this model is preferable
based on the historical data only, but the predictions are close to the
2^{∘} scenarios of IEA and IRENA, while those of the exponential and the
logistic models are way off.

The paper has examined three empirical growth models whose parameters are determined by fitting to historical data for solar and wind power consumption. According to the principle of parsimony the simplest model consistent with the data is preferable, and in our study this is the exponential model, since it contains only two model parameters. The snag here, of course, is that the two more complex models yield better fit to the data, and hence it does not seem possible to select the best model from such principles.

It is clear from Fig. 2b that the exponential model is not rejected by the data, and it cannot be excluded that the near future will unfold as a realisation of a geometric brownian motion, i.e. as an exponential growth with multiplicative noise. In that case, the declining relative growth rate during the last five years should be interpreted as a market fluctuation. On the other hand, one cannot rule out that the decline of the growth rate is the start of a continuing trend. One model to describe such a trend is the logistic model, which describes a rapid convergence to a zero growth rate. For solar power data it is not possible to estimate the parameters of the logistic model, i.e. the optimal logistic model is reduced to the exponential. For wind power consumption the predicted limit is less than 50 % above the present level, which is only half of the wind consumption in the pessimistic IEA current policies scenario and one fourth of the prediction in the 450 scenario (IEA, 2016). Thus, the predicted limit of the logistic model seems unrealistically low, and hence casts doubt about the relevance of this growth model.

On the other hand, the $\mathit{\gamma}\left(t\right)\propto \mathrm{1}/t$ time dependence of the
power-law model is what yields good fit to the historical data as well as to
the 2^{∘} target scenario of IEA and IRENA. In fact, this match to
historical and scenario data suggests a remarkable simple empirical law:
*the relative growth rate of renewable energy consumption decays inversely proportional to time*. In Sec. 3.3 it was shown
that that this growth law appears as a special case of a wider class of
growth models for which the relative growth rate $\mathit{\gamma}\left(t\right)={y}^{\prime}/y$ decays
algebraically towards zero, i.e. as $\mathit{\gamma}\left(t\right)\sim (t+{t}_{\mathrm{0}}{)}^{-\mathit{\mu}}$. This
special case (*μ*=1) constitutes the borderline between slower than
potential unlimited growth (*μ*<1) and limited growth (*μ*>1).

The years to come will show if the data points will continue to fall on the power-law trajectories of Fig. 1b. If they do, they will begin falling outside the confidence cloud of Fig. 2b around 2020 and thus reject the exponential growth hypothesis. Thus, after this date we may be able to make a more educated selection among models for predicting the growth of these renewable energies through the first half of this century.

The author declares that there is no conflict of interest.

This article is part of the special issue “European Geosciences Union General Assembly 2018, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2018, Vienna, Austria, 8–13 April 2018.

The author is grateful for useful discussions with Martin Rypdal.

Edited by: Sonja Martens

Reviewed by: Johannes Schmidt and one anonymous referee

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