Empirical growth models for the renewable energy sector

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 μy(t) and is subject to random increments y(t) σ dB(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 dB(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 $z=f\left(z\right)=\ln y$, 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 $\mathrm{d}z=\left[\mathit{\mu }{f}^{\prime }\right(y)+(\mathrm{1}/\mathrm{2}\left)f^{\prime \prime }\right(y\left){\mathit{\sigma }}^{\mathrm{2}}{y}^{\mathrm{2}}\right]\mathrm{d}t+\mathit{\sigma }\mathrm{d}B\left(t\right)$. 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 $\exp \left[\right(\mathit{\mu }-{\mathit{\sigma }}^{\mathrm{2}}/\mathrm{2}\left)t\right]$ 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]=y₀exp⁡[μt] and the variance as $\mathrm{Var}\left[y\right]=y_{\mathrm{0}}^{\mathrm{2}}\exp \left[\mathrm{2}\mathit{\mu }t\right]\left(\exp \right[{\mathit{\sigma }}^{\mathrm{2}}t]-\mathrm{1})$. 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⁡[σB(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⁡[σB(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 dy_L∕dt 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 $y\left(\mathrm{0}\right)=y_m/[\mathrm{1}+\exp (\mathit{\mu }{t}_{\mathrm{s}}\left)\right]$.

Figure 2(a) The blue curve is a realisation of a Gaussian white noise process. The yellow curve is a realisation of the random walk given by the cumulative sum of the white noise. The green curve is a realisation of the corresponding geometric Brownian motion with drift coefficient $\mathit{\mu }-{\mathit{\sigma }}^{\mathrm{2}}/\mathrm{2}=\mathrm{0.002}$. (b) The black dots represent the global wind power consumption time series, and the red and blue thick curves represent the linear and log-logistic least-square fit to the logarithm of this time series, respectively. The black, dashed curve is the fit of the power-law model discussed in Sect. 3.2. The thin wiggly, red curves are 100 realisations of the BS process with model parameters that are estimated from the time series data.

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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 Δz_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 Δz_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.

Figure 3The black bullets represent global solar power consumption data for the period 1997–2015, and the green bullet for 2016; all according to BP Statistical Review of World Energy (2017). In panel (a), the red and blue curves are exponential and logistic model fits to the 1997–2015 data, respectively. The green, dashed curves are fits to the 1997–2016 data. In the logarithmic plot in panel (b), the blue curve, and the green, dashed curve on top of it (almost invisible) is a fit of the logarithm of the logistic model to the logarithm of the data with the parameter P_max⁡ fixed to 550 TWh. For the green curve, the green point is included, for the blue curve, it is not. The other curves are similar fits with P_max⁡ fixed to 10³, 10⁴, 10⁵, and 10⁶ TWh, respectively. The residual Q₂ decreases monotonically to the value corresponding to an exponential fit as $P_{max}\to \mathrm{\infty }$.

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Figure 4The black bullets represent global wind power consumption data for the period 1997–2015, and the green bullet for 2016; all according to BP Statistical Review of World Energy (2017). In panel (a), the red and blue curves are exponential and logistic model fits to the 1997–2015 data, respectively. The green, dashed curves are fits to the 1997–2016 data. The logarithmic plot in panel (b) shows the fits of the logarithms of these models to the logarithms of the data. Panel (c) shows Q₂ versus P_max⁡, when fitting is made with fixed P_max⁡. The blue curve in panel (b) is the fitted curve with the value of P_max⁡ fixed to the minimum of the graph in panel (c). Panel (d) shows the ACF for the residual deviation between the data and the exponential fit.

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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 y₀ 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, α the corresponding model parameters, and 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 Q₂, 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 Q₂) 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 $P_{max}=\mathrm{550}$, 10³, 10⁴, 10⁵, 10⁶. 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 Q₂=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 Q₂ 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 Q₂ 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 $\ln y\left(t\right)=\ln p+q\ln (t+t_{\mathrm{0}})$. Here the origin of the time axis is chosen at the first year of the new shorter time series and t₀ 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 Q₂ 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 ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathit{\gamma }\left(t^{\prime }\right)\mathrm{d}{t}^{\prime }$ 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.